Heat Transfer

 

Mo Modu dule le 1: In Intr trod oduc ucti tion on to he heat at tra trans nsfe ferr 1. Introduction Heat transfer is a science, science, which deals deals with the flow of heat from from a higher temperature to lower temperature. temperature. Heat cannot be stored and it is defined defined as the energy in transit transit due to the difference in the temperatures temperatures of the hot and cold bodies. bodies. The study of heat transfer not only explains how how the heat energy transports transports but also predicts about about the rate of heat transfe tran sfer. r. When a cert certain ain amount amount of wat water er is evap evaporate orated d or cond condens ensed, ed, the amoun amountt of  heat transferred transferred in either of the processes processes is same. However, the the rate of heat transfer in both the cases may be different.  At this point, it is very important to understand about the basic information that the phases of a substance (solid, liquid, and gas) are associated with its energy content. In the solid phase, the molecules or atoms are very closely packed to give a rigid structure (fig.1.1a). In the liquid phase, sufficient thermal thermal energy is presen present, t, which keeps the molecules sufficiently apart and as a result the rigidity looses (fig.1.1b). In the gas phase, the presence of additional energy results in a complete separation and the molecules or atoms are free to move anywhere in the space (fig.1.1c). It must be noticed that whenever a change in phase occurs, a large amount of energy involves in the transition.

Fig. 1.1: Relative molecular distance of different phases of a substance at a fixed temperature temperatur e (a) gas/vapour, gas/vapour, (b) liquid, liquid, and (c) solid  As we are dealing with the heating and cooling of materials in almost our all the process processes, es, the heat transfer is an indispensable part of any of the industries. Therefore, heat transfer is a commo co mmon n su subj bjec ectt in ma many ny en engin ginee eeri ring ng di disc scip ipli line nes, s, es espe peci cial ally ly me mech chan anic ical al an and d

chem ch emic ical al

engineering. Study of heat transfer has a vital role in the chemical process industries. Chemical engineers must have a thorough knowledge knowledge of heat transfer principles and their applications applications.. There are three different modes in which heat may pass from a hot body to a cold one. These modes are conduction, convention, and radiation. It should be noted that the heat transfer takes

 

place in combination of two or three modes in any of the real engineering application. In this chapter, we will briefly discuss about the different modes of heat transfer along with the various basic information that will help us as a building block for further study. 1.1

Mode

of

heat

transfer

In this section, we will discuss about the three different modes of heat transfer. The discussion will help us to understand about the conduction, convection, and radiation. Moreover, we would be able to understand the basic difference between the three modes of heat transfer. 1.1.1 Conduction Conduction is the transfer of heat in a continuous substance without any observable motion of  the matter. Thus, heat conduction is essentially the transmission of energy by molecular motion. Consider a metallic rod being heated at the end and the other end of the rod gets heated automa aut omatic ticall ally. y. The hea heatt is tra trans nspo porte rted d fr from om one end to the ot othe herr end by the co cond nduc ucti tion on phenomenon. The molecules of the metallic rod get energy from the heating medium and collide with the neighbouring molecules. This process transfers the energy from the more energetic molecules to the low energetic molecules. Thus, heat transfer requires a temperature gradient, and the heat energy transfer by conduction occurs in the direction of decreasing temperature. Figure 1.2 shows an illustration for the conduction, where the densely packed atoms of the rod get energized on heating and vibration effect transfers transfers the heat as described in fig.1.2.

 

Fig.1.2: Diff Different erent stages during during conduction in a metallic rod 1.1.2 Convection When a macroscopic particle of a fluid moves from the region of hot to cold region, it carries with wi th it a de defi fini nite te am amou ount nt of en enth thal alpy py.. Su Such ch a fl flow ow of en enth thal alpy py is kn know own n as co conv nvec ecti tion on.. Convection may be natural or forced. In natural convection, the movement of the fluid particles is due to the buoyancy forces generated due to density difference of heated and colder region of  the fluid as shown in the fig.1.3a. Whereas, in forced convection the movement of fluid particles from the heated region to colder region is assisted by some mechanical means too (eg., stirrer) as shown in fig.1.3b.

 

Heat transfer through Fig.1.3: Heat through convection (a) natural, and (b) forced 1.1.3 Radiation We hav have e see seen n th that at a med medium ium is req requir uired ed fo forr th the e hea heatt tra trans nsfe ferr in ca case se of con conduc ductio tion n an and d convection. However, in case of radiation, electromagnetic electromagnetic waves pass through the empty space. Electromagnetic waves travel at the velocity of light in vacuum. These waves are absorbed, reflected, and/or transmitted by the matter, which comes in the path of the wave. We will limit our discussion (in this this NPTEL course) course) to the thermal radiation. Thermal radiation radiation is the term used to describe the electromagnetic electromagnetic radiation, radiation, which is observed observed to be emitted by the surface of the thermally excited body. The heat of the Sun is the the most obvious example of thermal radiation. There will be a continuous interchange of energy between two radiating bodies, with a net exchange of energy from the hotter to the colder body as shown in the fig.1.4. fig. 1.4.

Fig.1.4: Heat transfer through radiation

 

1.2 1. 2 Ma Mate teri rial al pr prop oper ertie ties s of im impo port rtan ance ce in he heat at tr tran ansf sfer er   Before understanding heat transfer laws, we have have to understa understand nd various properties properties of the material. This section section is devoted to a brief  brief  discussion of some of the important properties of the material. 1.2.1

Thermal

conductivity

 As discussed earlier, the heat conductio conduction n is the transmis transmission sion of energy by molecular action. Thermal conductivity is the property of a particular substance and shows the ease by which the process takes place. Higher the thermal conductivity more easily will be the heat conduction through the substance. It can be realized that the thermal conductivity of a substance would be dependent on the chemical composition, phase (gas, liquid, or solid), crystalline structure (if  solid),

temperature,

pressure,

and

its

The thermal conductivity of various substances substances is shown shown in table-1.1 and table 1.2. TableTab le-1.1 1.1:: The Therm rmal al con conduc ductiv tiviti ities es of var variou ious s sub substa stance nces s at 0oC

homogeneity.

 

Table-1.2: Thermal conductivity of mercury at three different different phases

 

The general results of the careful analysis of the table-1.1 and 1.2 are as follows, 

  Thermal conductivity depends on the chemical composition composition of the substance. substance.



  Thermal conductivity of the liquids is more than the gasses and the metals have the highest.



  Thermal conductivity of the gases and liquids increases with the increase in in temperature.



  Thermal conductivity of the metal decreases with the increase in temperature. temperature.



  Thermal conductivity conductivity is affected by the phase change. These The se dif diffe feren rences ces ca can n be exp expla laine ined d par partia tially lly by the fa fact ct th that at whi while le in ga gaseo seous us sta state te,, the molecules of a substance are spaced relatively far away and their motion is random. This means that energy transf transfer er by molecula molecularr impact is much slower than in the case of a liquid, in which the motion is still random but in liquids the molecules are more closely packed. The same is true concernin conce rning g the diff differen erence ce betw between een the therm thermal al cond conducti uctivity vity of the liquid and solid phases. phases. However, other factors are also important when the solid state is formed. Solid Sol id hav having ing a cry crysta stalli lline ne st stru ructu cture re ha hass hi high gh th therm ermal al con conduc ductiv tivity ity th than an a su subst bstanc ance e in an amorphous solid state. Metal, crystalline in structure, have greater thermal conductivity than non-metal (refer table-1.1). The irregular arrangement of the molecules in amorphous solids inhibits the effectiveness of the transfer of the energy by molecular impact. Therefore, the thermal conductivity of the non-metals is of the order of liquids. Moreover, in solids, there is an additional transfer of heat energy resulting from vibratory motion of the crystal lattice as a whole, in the direction of decreasing temperature. temperature. Many Man y fa facto ctors rs are kno known wn to inf influe luence nce th the e the therma rmall con conduc ductiv tivity ity of met metals als,, su such ch as che chemic mical al composition, atomic structure, phase changes, grain size, temperature, and pressure. Out of the above factors, the temperature, pressure, and chemical composition are the most important. However, if we are interested interested in a particular material then only the temperature temperature effects has to be accounted accoun ted for.  As per the previous discussion and the table it is now clear that the thermal conductivity of the metal is directly proportional to the absolute temperature and mean free path of the molecules. The mean free path decreases with the increase in temperature so that the thermal conductivity decreases with the temperature. It should be noted that it is true for the pure metal, and the presence of impurity in the metal may reverse the trend. It is usually possible to represent the

 

thermal conductivity of a metal by a linear relation k = k o o(  1 + bT),  where k o o is    is the thermal conductivity of the metal at 0oC, T is is the absolute temperature, temperature, and b  is  is a constant. In general the thermal conductivity of the liquids is insensitive to the pressure if the pressure is not very close to the critical temperature. Therefore, in liquids (as in solids) the temperature effects effe cts on the thermal conductivity conductivity are generally generally cons consider idered. ed. Liqu Liquids, ids, in gener general, al, exhi exhibit bit a decreasing thermal conductivity with temperature. However, water is a notable exception. Water has the highest thermal conductivity among the non-metallic liquids, with a maximum value occurring occurr ing at 450oC. The thermal conductivity of a gas is relatively independent of pressure if the pressure is near 1 atm. Vapours near the saturation point show strong pressure dependence. Steam and air are of  great engineering importance. Steam shows irregular behaving rather showing a rather strong pressure dependence for the thermal conductivity as well as temperature dependence. The above discussions concerning thermal conductivity were restricted to materials composed of  homogeneous or pure substan substances. ces. Many of the engineering engineering materials encountered in practice are not of this nature like building material, and insulating material. Some material may exhibit nonisotropic isotr opic cond conductiv uctivities ities.. The non-i non-isotro sotropic pic mate material rial show showss diff differen erentt cond conductivi uctivity ty in diffe different rent direction in the material. This directional directional preference is primarily the result of the fibrous fibrous nature of  the material like wood, asbestos etc. etc. 1.2.2

Specific

heat

capacity

Now we kno know w th that at the the therma rmall con condu ducti ctivit vity y fa facil cilita itate tess th the e hea heatt to pro propag pagate ate th throu rough gh the material due to the temperature gradient. Similarly, specific heat capacity or specific heat is the capacity of heat stored by a material due to variation in temperature. Thus the specific heat capacity capa city (unit (unit:: kJ/k kJ/kg· g·oC) is def define ined d as the am amou ount nt of the therma rmall ene energy rgy require required d to raise th the e temperature of a unit amount of material by 1 oC. Since heat is path dependent, so is specific heat. In general, the heat transfer processes used in the chemical process plant are at constant pressure; hence the specific heat capacity ( c o ) is generally used.

Frequently Asked Questions Questions (Module 1)

Q.1.

What Wh at

Q.2.

Define thermal What is the order of thermal conduc conductivity tivity of gas, What liquid, and metal in

Q.3. Q.4. Q.5.

is

the th e

basi ba sicc

should Discuss

the

be

diff di ffer eren ence ce

the effect

amon am ong g

approach of

to

cond co nduc ucti tion on,,

select

a

temperature   on on

conv co nvec ecti tion on,,

good

and an d ra radi diat atio ion? n? conductivity.

thermal

thermal

Q.6. What is the difference between thermal conductivity and and specific heat capacity?

general? insulator?

conductivity.

 

Module2: Modu le2: Conducti Conduction: on: one dimension dimensional al

The fundamentals of heat conduction were established over one and a half century and its contribution goes to a French mathematician and physicist, Jean Baptiste Joseph Fourier. You may be aware that any flow whether it is electricity flow, fluid flow, or heat flow needs a driving force. The flow is proportional to the driving force and for various kinds of flows the driving force is shown shown in the table 2.1. Table Tab le 2. 2.1. 1. Va Vari riou ous s fl flow ows s an and d the their ir dr drivi iving ng fo forc rces es

Thus the heat flow per unit area per unit time (heat flux,

) can be represented represented by the

followin follo wing g relati relation, on,

where, wher e, pro proport portiona ionality lity cons constant tant k   is th the e th ther erma mall co cond nduc ucti tivi vity ty of th the e mat mater eria ial, l, T   is th the e temperature and x  is  is the distance in the direction of heat flow. This is known as Fourier‟s Fourier‟s law  law of  conduction. The term steady-state conduction is defined as the condition which prevails in a heat conducting body when temperatures at fixed points do not change with time. The term one-dimensional is applied to a heat conduction problem when only one coordinate is required to describe the distribution of temperature within the body. Such a situation hardly exists in real engineering problems. However, by considering one-dimensional assumption the real problem is solved fairly upto the accuracy of practical engineering interest. interest. 2.1

Steady-state

conduction

through

constant

area

 A simple case of steady-state, one-dimensional one-dimensional heat conduction can be considered through a flat wall as shown in the fig.2.1.

 

Fig.2.1: Steady-stat Steady-state e conduction through a slab (constant area) The flat wall of thickness d x x is separated by two regions, the one region is at high temperature (T 1 1 ) and the other one is at temperature T 2 2 . The wall is very large in comparison of the   thickness so tha thatt the heat losse lossess fro from m the edges edges ar are e ne negl gligi igible ble.. Co Consi nsider der there there is no gen genera eratio tion n

or

accumulation of the heat in the wall and the external surfaces of the wall are at isothermal temperatures T 1 1 and T 2 2 . The area of the surface through which the heat transfer takes place   is A . Then the eq.2.2 can be written as,

The negative sign shows that the heat flux is from the higher temperature surface to the lower temperature surface and is the rate of heat transfer transfer through the wall. Now if we consider a plane wall made up of three different layers of materials having different thermal conductivities and thicknesses of the layers, the analysis of the conduction can be done as follows. Consider the area ( A   A ) of the heat conduction (fig.2.2) is constant and at steady state the rate of  heat transfer from layer-1 will be equal to the rate of heat transfer from layer-2. Similarly, the rate of heat transfer transfer through layer-2 will be equal to the rate of heat transfer through layer-3. If  we know the surface temperatures of the wall are maintained at T 1 1 and T 2 2 as shown in the   fig.2.2, the temperature of the interface of layer1 and layer 2 is assumed to be at T'  and the inte interfac rface e of  layer-2 and layer-3 asT" .

 

Fig.2.2: Heat different layers Heat conduction through three different The rate of heat transfer through through layer-1 to layer-2 will be,

and, The rate of heat transfer through through layer 2 to layer 3 will be,

and, The rate of heat transfer through layer 3 to the other side of the wall,

On adding the above three equations,

 

Where, R represents the thermal resistance of the layers. The above relation can be written analogous to the electrical circuit as,

Fig Fi g 2. 2.3: 3: Eq Equi uiva vale lent nt el elec ectr tric ical al cir circu cuit it of th the e fig fig.2 .2.2 .2 The wall is composed of 3-different layers in series and thus the total thermal resistance was represented by R (= R 1 1 + R 2 2 + R 3 3 ). The discussed concept can be understood by the illustrations shown show n below. The unit of the various parameters used above is summarized as follows,

Illustration

2.1

T he two two si side dess of a wall wall (2 ( 2 mm thi thick ck,, with wi th a cross-sect cross-sectii ona onall are ar ea of of 0.2 m2) are main mainta taii ned at 30oC and

90oC . The th the er mal cond conduct uctii vity of the wall mate materr i al i s 1.28 W/ W/(m· (m·oC) . Find F ind out the rat rate e of hea heat tran ransfe sfer  r  through the wall? Solution

2.1

Assumptions 1. 2.

  Thermal

3. 4. Given,

The

Steady-state cond co ndu ucti tiv vity heat

The

loss layers

is

conduction

one-dimensional con co nst stan antt

through are

for

the  

the

edge in

range

temperature sid si de  perfect

surrface su  

of

interest

is

insignificant

thermal

contact

 

 

Fig. 2.4: Illustration 2.1 2.1

Illustrati Illust ration on 2.2

Solution

2.2

Assumptions: 1. 2. 3.

Steady-state Thermal The

cond co nduc ucti tivi vity ty heat

loss

one-dimensional

is   con const staant through

4. The layers are in perfect thermal thermal contact.

the

for  fo r 

thee th

edge

conduction.

temp te mper erat atur uree   ra range side

surface

is

of  

interest.

insignificant.

 

On putting all the known values,

Fig. 2.5: Illustrati Illustration on 2.2 Thus,

The prev previous ious discussio discussion n sho showed wed the res resista istance ncess of diff differen erentt laye layers. rs. Now to unde understan rstand d the concept of equivalent resistance, we will consider the geometry of a composite as shown in fig.2.6a.

 

The wall is composed of seven different layers indicated by 1 to 7. The interface temperatures temperatures of  the composite are T 1 1 to T 5 5 as shown in the fig.2.6a. The equivalent electrical circuit of the above composite is shown in the fig 2.6b below,

Fig.2.6. (a) Composite Composite wall, wall, and (b) equivalent electrical electrical circuit The equivalent resistance resistance of the wall will be,

where,

 

Therefore, at steady state the rate of heat transfer through the composite can be represented by,

where, R is the equivalent resistance resistance..

Illustration

2.3

Consider a composite wall containing 5-different 5-different materials materials as shown in the fig. 2.7. Calculate the rate of heat flow through the composite from the following data?

Solution

2.3

Assumptions: 1. 2. 3. 4.

 

Steady-state St

Thermal The

conductivity heat

The

loss

is

constant

through

layers

conduction.

one-dimensional for

the

are

 

the

edge in

range

temperature sid si de

surrfac su acee

 perfect

 

of

interest.

is

insignificant.

thermal

contact.

2

5. Area in the direction of heat flow is 1 m .

The

height

of

th e

first

layer

is

4

m

(h 1 = h 2 + h 3 3 ).

The equivalent circuit diagram of the above composite is,

On calculating equivalent equivalent resistance resistance with the given given data (Note: thickness of layer 2 = thickness of layer 3 and thickness of layer 4 = thickness of layer 5, in the heat flow direction),

 

Fig. 2.7: Composite of illustration 2.3; (a) composite, (b) correspon corresponding ding electric elect rical al circuit Thus the heat flow rate through the composite,

 

2.2 Thermal contact resistance In the previous discussion, discussion, it was assumed that the different layers of the composite have perfect contact between any two layers. Therefore, the temperatures of the layers were taken same at the plane of contact. However, in reality it rarely happens, and the contacting surfaces are not in perfect contact or touch as shown in the fig.2.8(a). It is because as we know that due to the roughn roughness ess of the surface, the solid surfaces are not perfectly smooth. Thus when the solid surfaces are contacted the discrete points of the surfaces are in contact and the voids are generally genera lly fille filled d with the air. There Therefore, fore, the heat trans transfer fer acros acrosss the compo composit site e is due to the parallel effect of conduction at solid contact points and by convection or probably by radiation (for high temperature) through the entrapped air. Thus an apparent temperature drop may be assumed to occur between the two solid surfaces as shown in the fig.2.8b. If  T I I and T II  II are the theoretical tempe theoretical temperat rature ure of the plane inter interface face,, then the therm thermal al conta contact ct resi resista stance nce may be define def ined d as,

where R c crepresents r  epresents the thermal contact resistance. The utility of the thermal contact resistance (R c c ) is dependent upon the availability availability of the reliable data. The value of  R c c depends upon the solids involved, the roughness factor, contact   pressure, material occupying the void spaces, and temperature. The surface roughness of a   properly smooth metallic surface is in the order of micrometer. The values of  R c generally  obtained by the exper exp erime iment nts. s. How Howeve ever, r, th ther ere e are cer certai tain n th theo eorie riess wh which ich pre predic dictt the ef effe fect ct of th the e   various parameters on the R c c.  It can be seen in the fig.2.8, that the two main contributors to the heat transfer are (i) the conduction through entrapped gases in the void spaces and, (ii) the solid-solid conduction at the contact points. It may be noted that due to main contribution to the resistance will be through first factor because of low thermal conductivity conductivity of the gas.

 

Fig.2.8 (a) Contacting surfaces surfaces of two solids are not in perfect contact, (b) temperature temperatur e drop due due to imperfect contact If we denote the void area in the joint by A v and contact area at the joint by A c c,  then we may write heat flow across the joint as,

where, thickness of the void space and thermal conductivity of the fluid (or gas) is represented by l g gand  is the thickne  is thickness ss of solid-I and solid-II for evenly k f f,  respectively. It was assumed that l  /2    g  g  rough surfa surfaces. ces.

 

2.3 Steady-state heat conduction through a variable area It was observed observed in the previous discussion discussion that for the given plane wall the area for heat transfer transfer was constant along the heat flow direction. The plane solid wall was one of the geometries geometries but if  we take some other geometry (tapered plane, cylindrical body, spherical body etc.) in which the area changes in the direction of heat flow. Now we will consider geometrical configuration which will be mathematically simple and also of great engineering importance like hollow cylinder and hollow sphere hollow sphere.. In the these se cas cases es the hea heatt tra transf nsfer er are area a var varies ies in th the e ra radi dial al di direc rectio tion n of he heat at conduction. We will take up both the cases one by one in the following sections. 2.3.1 Cylinder Consider a hollow cylinder as shown in the fig.2.9a. fig.2.9a. The inner and outer radius is represented   by a r i iand   nd r  o  , whereas T  o  i  and T  i  o (T  o  i  > T  i  o  ) represent the uniform temperature of the inner and outer o  wall, respectively.

Fig. 2.9. (a) Hollow cylinder, (b) equivalent electrical electrical circuit Consider a very thin hollow cylinder of thicknessd r r  in   in the main geometry (fig.2.9a) at a radial distancer . If  d r r is small enough with respect to r , then the area of the inner and outer surface of  the thin cylinder may be considered to be of same area. In other words, for very small d r r with respect to r , the lines of heat flow may be considered parallel through the differential element in radial outward direction. We may ignore the heat flow through the ends if the cylinder is sufficiently large. We may thus eliminate any dependence of the temperature on the axial coordinate and for one dimensional steady state heat conduction, conduction, the rate of heat transfer for the thin cylinder,

 

 is the temperature difference between the inner and outer surface of the thin cylinder Where dT  is considered above and k  is  is the thermal conductivity of the cylinder. On rearrangin rearranging, g,

To get the heat flow through the thick wall cylinder, the above equation can be integrated between the limits,

On sol solvin ving, g,

Where , and the careful analysis of the above equation shows th that the expression is same as for heat flow through the plane wall of thickness ( r  –    ) except the o  r  o  i ) i  expression for the area. The A LM  known wn as log mea mean n are area a of the cyli cylind nder er,, who whose se len length gth LM  is kno is L and radius is r LM LM (=

). The fig.2.9b shows the equivalent electrical circuit of the

fig.2.9b. Now we have learnt that how to represent the analogous electrical circuit circuit for the cylindrical case. c ase. It will provide the building block for the composite cylinders similar to the plane composite we have learnt earlier. The following fig.2.10a shows a composite cylinder with 4-layers of solid material of different inner and outer diameter as well as thermal conductivity. The equivalent electrical circuit is shown below in fig.2.10b.

 

(a)

(b) Fig.2.10.(a) Fig.2.10. (a) Four layer composite hollow cylinder, (a) equivalent equivalent electrical electrical circuit The total heat transfer transfer at steady-state steady-state will be,

where R 1 , R 2 , R 3 , and R 4 are represented in the fig.2.10b. 2.3.2 Sphere The rate of heat transfer through a hollow sphere can be determined in a similar manner as for cylinder. The students are advised to derive the following expression shown shown below. The final expression for for the rate of heat flow is,

 

2.4 Heat conduction in bodies with heat sources The cases considered so far have been those in which the heat conducting solid is free of  internal heat generation. However, the situations where the internal heat is generated are very common cases in chemical industries for example, the exothermic reaction in the solid pallet of a catalyst. We have learnt that how the Fourier equation is used for the steady-state heat conduction through the composites of three different geometries that were not having any heat source in it. However, the heat generation term would come into the picture for these geometries. It would not be always easier to remember and develop heat conduction relations for different standard and non-standard geometries. Therefore, at this point we should learn how to develop a general relation for the heat conduction that should be applicable to the entire situation such as steadystate, unsteady state, state, heat source, different geometry, geometry, heat conduction in different direction, direction, etc.  Again here we will conside considerr that the solid is isotropic in nature, which means the thermal conductivity of the material is same in all the direction direction of heat flow. To get such a general equation the differential form of the heat conduction equation is most important import ant.. For simplicity, simplicity, we woul would d cons conside iderr an infinitesim infinitesimal al volum volume e elem element ent in a Cart Cartesia esian n coordinate system. The dimensions of the infinitesimal volume element are d x x , d y y , and d z z in the respective direction as shown in the fig.2.11.

Fig.2.11. Volume element for deriving general equation of heat conduction in in cartesian carte sian coor coordina dinate te The fig.2.11 shows that the heat is entering into the volume element from three different faces faces of the volume element and leaving from the opposite face of the control element. The heat source within the volume element generates the volumetric energy at the rate

of 

 

 According to Fourier‟s law of heat conduction, the heat flowing into the volume element from the left (in the x-direction) can be written as,

The heat flow out from the right surface (in the x-direction) of the volume element can be obtain obt ained ed by Tay Taylor lor ser series ies exp expan ansio sion n of the ab above ove equ equati ation on.. As th the e vol volume ume ele elemen mentt is of  infinitesimal volume, we may retain only first two element of the Taylor series expansion with a reasonable approximation approximation (truncating the higher order terms). Thus,

The left side of the above equation represent represent the net heat flow in the x-direction. If we put the value

of in the right side of the above equat equation, ion,

In a similar way we can get the net heat flow in the y and z -directions, -directions,

 As we know some heat is entering, some heat is leaving and some heat in generat generating ing in the volume element and as we have not considered any steady state assumption till now, thus because of all these phenomena some of the heat will be absorbed by the element. Thus the rate rat e of cha change nge of hea heatt ene energy rgy

within wit hin the vol volume ume ele elemen mentt can be wri writte tten n as,

 

where, c p is the specific heat capacity at constant pressure (J/(kg·K)), ρ is the density (kg/m3) of th e material, and t  is the time (s). We know all the energy term related to the above problem, and with the help of energy conservation,

On putting all the values in the above equation,

or,

 As we have considered considered that the thermal thermal conductivity conductivity of the the solid is isotropic in nature, nature, the above relation relati on reduces to,

or,

 

2 wh wher ere e is th the e th the erm rmal al di diff ffus usiv ivit ity y of th the e ma mate teri rial al an and d it itss un unit it m /s signifies the the rate at which heat diffuses in to the medium during change in temperature with time. Thus, the higher value of the thermal diffusivity diffusivity gives the idea of how fast the heat is conducting conducting into the medium, whereas the low value of the thermal diffusivity shown that that the heat is mostly absorbed

by the mater material ial and compar comparativ atively ely less amoun amountt is trans transfer ferred red for the condu conductio ction. n. The

called calle d

the Laplacian operator, and in Cartesian coordinate coordinate it is defined as Equation 2.19 is known as general heat conduction relation. relation. When there is no heat generation term the eq.2.19 will become,

and the equation is known as Fourie Fourierr Field Field Equati Equation  on .

General heat conduction c onduction relation in cylindrical coordinate coordinate system (fig. 2.12) is derived (briefly) below.

 

Cylindrical coordinate system (a) and Fig.2.12. Cylindrical and an element of the cylinder The energy conservation for the system is written as, Ӏ   +

ӀӀ   =

ӀӀӀ   +

ӀV

 

(2.21)

where, I II III IV

: :

Rate :

Rate of

heat

Rate :

Rate

and the above terms are defines as,

Thus,

of

heat

energy

generated

of

heat of

energy within energy

energy

conducted the

volume conducted

accumulated

in element out (ӀV)

 

On putting the values in equation 2.21,

Thus the Laplacian operator is,

 

Fi Fig. g.2. 2.13 13.. Sp Sphe heri rica call co coor ordi dina nate te sy syste stem m (a) an and d an el elem emen entt of th the e sp sphe here re In a similar way the general expression for the conduction heat transfer in spherical body with heat source can also be found out as per the previous discussion. The Laplacian operator for the spherical coordinate system (fig.2.13) is given below and the students are encouraged to derive the expression themselves. themselves.

Frequently Asked Questions and Problems Problems for Practice (Module 2) Q.1   Write the driving force force for electricity, electricity, fluid, and heat flow flow and discuss the similarity among them. Q.2   What is the ratio of heat flux through through area A1 and area A2 of an irregular pipeline pipeline shown in the figure below? The area A1 and A2 are same and the curved surface is well insulated for any kind of heat loss at steady state.

 

Q.3

What is the the unit of thermal thermal condu conductiv ctivity? ity?

Q.4

What is the the signi significa ficance nce of log mean area?

Q.5   A hollo hollow w cylinder has two different different layers of insulation insulation of same thickne thickness ss but different different thermal conductivity. conductivity. The outer diameter of the insulated insulated cylinder cylinder is double that of the inner diameter diamet er of the cylinder. cylinder. What will be the change in heat flow flow if the insulati insulation on layers layers are interchanged considering considering the same temperature temperature driving force? force? The thermal conductivity of  the inner layer is considered to be the the four times that of the other layer for for previous case.

Q.6   A th thic ick k wa wall ll of 30 cm thick thick and 20 W/ W/(m (m··oC) of thermal conductivity has one surface (maint (ma intain ained ed at 250oC) and th the e op oppos posite ite su surf rface ace is com comple plete tely ly ins insula ulate ted. d. The he heat at is generated in the wall at a uniform volumetric rate of 180 kW/m 3. Determ Determine ine the a. b.

the the

temperature maximum

distribution wall

in

the

temperature

wall and

at its

followin follo wing, g,

steady location,

state, and

c. the average wall temperature. temperature. Q.7   A hollow aluminium sphere sphere having inner diameter diameter of 5 cm and outer diameter of 10 cm is maintained at 100   oC and 50   oC at inside and outside of the sphere. Calculate the heat flux at the outer surface. Q.8   A hot steam pipe (k = 50 W/m·oC) having an inner diameter of 8 cm is at 250 oC. The thickness of the wall is 5.5 5.5 mm. The pipe is covered with a 90 mm layer of insulation insulation

(k =

0.2 W/m·oC) followed by a 40 mm layer of insulation (k = 0.3 W/m·oC) . The outside temperature of the insulation is 20 oC. Calculate the heat loss per unit of the pipe length. Q.9   Consider a plane wall having uniformly uniformly distributed distributed heat sources sources and one face maintained maintained at a temperature T1 while the other face is maintained at a temperature T2. The thickness of  the wall may be taken as 2t. Derive an expression for the temperature distribution in the plane plan e wal wall. l. Q.10   Derive an expression for the temperature temperature distribution distribution in a sphere of radius R with uniform heat generation and constant surface temperature. temperature.

 

Module Modu le 3: Convecti Convective ve heat transf transfer: er: One dimensiona dimensionall

The rate of heat transfer transfer in a solid body or medium can be calculated by Fourier‟s Fourier‟s law.  law. Moreover, the Fourier law is applicable to the stagnant fluid also. However, there are hardly a few physical situations in which the heat transfer in the fluid occurs and the fluid remains stagnant. The heat transfer in a fluid causes convection (transport of fluid elements) and thus the heat transfer in a fluid mainly occurs by convection. 3.1 Prin inc cip iple le of he heat at flow in flui uid ds an and d co conc nce ept of he heat at tr tra ans nsffer coeffic iciien entt   It is learnt by day-to-day experience experience that a hot plate of metal will cool faster faster when it is placed placed in front of a fan than than exposed to air, which is stagnant. stagnant. In the process, the heat is convected convected away, and we call the process convecti convective ve heat transfer transfer.. The term convective convective refers refers to tran transpo sport rt of heat (or mass) mas s) in a flu fluid id med medium ium due to th the e mot motion ion of the fluid. fluid. Con Convec vectiv tive e hea heatt tra trans nsfer fer,, thu thus, s, associated with the motion of the fluid. The term convection provides an intuitive concept of the heat transfer process. process. However, this intuitive concept must be elaborated to enable one to arrive at anything like an adequate analytical analytical treatment of the problem. It is well known that the velocity at which the air blows over the hot plate influences the heat transfer rate. A lot of questions come into the way to understand the process thoroughly. Like, does the air velocity influence the cooling in a linear way, i.e., if the velocity is doubled, will the heat transfer rate double. We should also suspect that the heat-transfer rate might be differen differentt if  we cool the plate with some other fluid (say water) water) instead of air, but again how much difference would there be? These question questiones es may be answered with the help of some basic basic analysis in the later part of this module. The physical mechanism of convective heat transfer for the problem is shown in fig.3.1.

 

heat transfer from a heated wall to a fluid Fig. 3.1: Convective heat Consider a heated wall shown shown in fig.3.1. fig.3.1. The temperature of the wall and bulk fluid is denoted by respectively. The velocity of the fluid layer at the wall will be zero. Thus the heat will be transferred through the stagnant film of the fluid by conduction only. Thus we can compute the heat transfer using Fourier‟s Fourier‟s law  law if the thermal conductivity of the fluid and the fluid temperature gradient at the wall is known. Why, then, if the heat flows by conduction in this layer, do we speak of convective heat transfer and need to consider the velocity of the fluid? The answer is that the temperature gradient is dependent on the rate at which the fluid carries the heat away; a high velocity produces a large temperature gradient, and so on. However, it must be remembered that the physical mechanism of heat transfer at the wall is a conduction process. It is apparent from the above discussion that the prediction of the rates at which heat is convected away from the solid surface by an ambient fluid involves thorough understanding of  the principles of heat conduction, fluid dynamics, and boundary layer theory. All the complexities involved in such an analytical approach may be lumped together in terms of a single parameter by introduction of Newton‟s of Newton‟s law  law of cooling,

where, h  is  is known as the heat transfer coefficient or film coefficient. It is a complex function of  the fluid composition and properties, the geometry of the solid surface, and the hydrodynamics of the fluid motion. If  k  is   is the thermal conductivity of the fluid, the rate of heat transfer can be written directly by following the Fourier‟s Fourier‟s law.  law. Therefore, we have,

where,

is the temperature gradient in the thin film film where the temperature gradient is

linear. On comparing eq.3.1 and 3.2, we have,

 

transfer coefficient It is clear from the above expression that the heat transfer coefficient can be calculated if k  and  and δ  are known. Though the k   values are easily available but the δ   is is not easy easy to

dete de term rmin ine. e.

Theref The refor ore, e, th the e abo above ve eq equat uation ion looks looks sim simple ple but not really really eas easy y fo forr th the e cal calcul culat ation ion of

reall rea

problems due to non-linearity of k  and  and difficulty in determining δ . The heat heat transfer transfer coefficient is is important to visualize the convection convection heat transfer phenomenon phenomenon as discussed discussed before. In fact, δ  is  is the thicknes thicknesss of a heat tra transf nsfer er resistanc resistance e as that really really exists in the fluid under the given given hydrodynamic conditions. Thus, we have to assume a film of  δ  thickness   thickness on the surface and the heatt tr hea trans ansfe ferr coe coeffi fficie cient nt is determi determine ned d by the propert properties ies of th the e flu fluid id fil film m suc such h as den densit sity, y, viscosity, viscosit y, specific heat, heat, thermal conductivity conductivity etc. The effects of all these parameters parameters are lumped or clubbed together to define the film thickness. Henceforth, Henceforth, the heat transfer coefficient c oefficient (h ) can be found out with a large number of correlations developed over the time by the researchers. These correlations will be discussed in due course of time as we will proceed throu th rough gh th the e mo modul dules es.. Tab Table le 3. 3.1 1 sh show owss th the e ty typi pical cal va value luess of the con conve vecti ctive ve hea heatt tr tran ansf sfer er coefficient under different situations. situations. TableTab le-3.1 3.1:: Typ Typica icall valu values es of  h  differen rentt situat situations ions  h under diffe

3.2 Individual and overall heat transfer coefficient If two fluids are separated by a thermally conductive wall, the heat transfer from one fluid to another fluid is of great importance in chemical engineering process plant. For such a case the rate of heat transfer is done by considering an overall heat transfer coefficient. However, the overall heat transfer coefficient depends upon so many variables that it is necessary to divide it into individual heat transfer coefficients. The reason for this becomes apparent if the above situation can be elaborated as discussed in the following sub-sections.

 

3.2.1 Heat transfer between fluids separated by a flat solid wall  As shown in fig.3.2 fig.3.2,, a hot fluid is separated by solid wall from a cold fluid. The thickness of the solid wall is l , the temperature of the bulk of the fluids on hot and cold sides are T h h  and T c c,  respectively. The average temperature of the bulk fluid is T 1 1 and    and T 4 4,  for hot and cold fluid, respectively. The thicknesses of the fictitious thin films on the hot and cold sides of the flat solid are shown by δ1 and δ2. It may be assumed that the Reynolds numbers of both the fluids are sufficiently large to ensure turbulent flow flow and the surfaces of the solid wall are clean.

Fig.3.2. Real temperature temperature profile It ca can n be see seen n th that at th the e te tempe mperat ratur ure e gr gradi adient ent is la large rge nea nearr th the e wal walll (th (throu rough gh the vis viscou couss sublayer), small in the turbulent core, and changes rapidly in the buffer zone (area near the interface of sublayer and bulk fluid). The reason was discussed earlier that the heat must flow through the viscous sublayer by conduction, thus a steep temperature gradient exists because of  the low temperature gradient of most of the fluids. The average temperatures of the warm bulk fluid and cold bulk fluids are slightly less than the maximum temperature T h h  (bulk   (bulk tempe temperat rature ure of hot fluid fluid)) and sligh slightly tly more than the minimu minimum m

 

temperature T c (bulk temperature of cold fluid), respectively. The average temperatures are shown

by T 1 and T 4 4, 

for fo

the

hot

and

cold

fluid

streams,

respectively.

Figure 3.3 shows the simplified diagram of the above case, c ase, where T2 and T3 are the temperatures of the fluid wall interface.

Fig.3.3. Simplified Simplified temperature temperature profile for fig.3.2 If the thermal conductivity of the wall is k , and the area of the heat transfer is  A , the electrical analogy of the fig.3.3 can be represented by fig.3.4, where h 1 1   and h 2 2 are    are the individual heat transfer coefficient coefficient of the hot and and cold side of the fluid.

Fig.3.4. Equivalent electrical electrical circuit for fig. 3.3

 

Considering that the heat transfer transfer is taking place at the steady-state steady-state through through a constant area and the heat loss from other other faces faces are negligible, negligible, then the rate of heat transfer transfer on two sides of the wall will be represente represented d by eq. 3.4-3.6. Rate of heat transfer from the hot fluid to the wall,

Rate of heat transfer through the wall,

Rate of heat transfer from the wall to cold fluid,

 At steady state, the rate of heat transfers by

. Th Ther eref efor ore, e,

On adding equations (3.7 to 3.9)

are same same and can be represented

 

where,

Thus,

The Th e qu quan anti tity ty

is ca call lled ed

the th e ov over eral alll he heat at tr tran ansf sfer er co coef effi fici cien entt (c (can an be ca calc lcul ulat ated ed if 

the the are kn kno own) n).. Th Thu us fr fro om the sy sysste tem m de desscr criibe bed d is es esttab abli lish shed ed tha hatt the ove vera rall ll heat transfer coefficient is the function of individual heat transfer coefficient of the fluids on the two sides of the wall, as well as the thermal conductivity of the flat wall. The overall heat transfer trans fer coefficien coefficientt can be used to intr introduce oduce the cont controllin rolling g term concept. concept. The cont controll rolling ing resistance is a term which possesses possesses much larger thermal resistance resistance compared to the sum of the other resistances. resistances. At this point it may be noted that in general the resistance resistance offered by the solid wall is much lower. Similarly, if a liquid and a gas are separated by a solid wall the resistance offered by the gas film may generally be high. Illustrat Illust ration ion 3.1.

The steady state temperature distribution in a wall is , where x (in meter) is the   positi  pos ition on in th the e wa wall ll an and d T is the the te tempe mpera ratu ture re (in o C). Th The e th thic ickn knes ess s of th the e wa wall ll is 0. 0.2 2 m  and the therm thermal al conduct conductivity ivity of the the wall is 1.2 (W/m· o C). The The wal walll dissipate dissipates s the the hea heat  t  to the ambient at 30   o C. Calculate the heat heat transfer coefficient coefficient at at the surface surface of the  wall wa ll at 0.2 0.2 m. Solution

3.1

The rate of heat transfer through the wall by conduction will be equal to the rate of heat transfer from the surface to the ambient by convention at steadystate, steadystate,

Rate of heat transfer by conduction at at x=0.2 is given by,

where T a is the ambient temperatu temperature. re.

 

On putting the values and solving,

3.2.2 Heat transfer between fluids separated by a cylindrical wall In the above section we have seen that how the rate of heat transfer is calculated when the two fluids flu ids are sep separa arated ted by a fl flat at wal wall. l. Ano Anothe therr com common monly ly enc encou ount nter ered ed sh shape ape in th the e che chemic mical al engineering plant is the heat transfer between fluids separated by a cylindrical wall. Therefore, we will see them to understand the overall heat transfer coefficient in such a system. Consider a double pipe heat exchanger which consists of of two concentric pipes arrange arrange as per the fig. 3.4.

Fig. 3.5: Schematic of a co-current double pipe pipe heat exchanger

 

The purpose of a heat exchanger is to increase the temperature temperature of a cold fluid and decrease that of the hot fluid which is in thermal contact, in order to achieve heat transfer. The fig. 3.5 shows that the hot fluid passes through the inner tube and the cold fluid passes through the outer tube of the double pipe heat exchanger. The inner and the outer radii of the inne in nerr pipe pipe are

, re resp spec ecti tive vely ly,, wh wher erea eass th the e in inne nerr ra radi dius us of th the e ou oute terr tu tube be is

heat trans transfer fer coeff coefficien icientt of the fluid in the inner pipe is the fluid over the inner

pipe pi pe is

. Th The e

and the heat trans transfer fer coeff coefficien icientt of 

are the inner and outer wall tem temperatures of 

the inner pipe. The bul ulk k fluid temp mpe eratures of the hot and cold flu luiids are

,

respectively, at steady state condition and assumed to be fairly constant over the length of the pipe (say L ). ). The construction in fig. 3.6 provides pr ovides a better understanding understanding..

Fig. 3.6: Cross-section Cross-section of the double pipe heat exchanger exchanger shown in fig. 3.5 The rate of heat transfer from the hot fluid to the inner surface which is at temperature (3.12)

The rate of heat transfer through through the pipe wallis, wall is, (3.13)

(Refer to the section, heat conduction through varying area.) The rate of heat transfer from the outer surface of the inner pipe to the cold fluid is,

 

transfers will be same, thus The rate of heat transfers

Thus on rearranging rearranging above equations,

where,

If we compare the overall heat transfer coefficient shown above with the overall heat transfer coefficient discussed in eq.3.11 (for flat plate). It can be seen that due to the different inside and outside radii of the pipe, the overall heat transfer coefficient will be different. Therefore, the overall heat transfer coefficient coefficient can be defined either by U i i (overall heat transfer transfer coefficient coefficient based on inside surface area) or U o o (overall heat transfer coefficient based on outside surface   area). area). But it shoul should d be noted tha thatt the rate of heat tran transfe sferr and the driving force remain the same. Therefore Therefore,, we have (3.19)

 

where,

or,

Similarly,

In terms of thermal resistance, we can use eq. 3.19

Illustrat Illust ration ion 3.2.

Warm methanol flowing in the inner pipe of a double pipe heat exchanger is being  cooled by the flowing water in the outer tube of the heat exchanger. The thermal  conductivity of the exchanger, inner and outer diameter of the inner pipe are 45  W/(m· o C), 26 mm, and 33 mm, respectively. The individual heat transfer coefficients  are: 

Methan Met hanol, ol, hi

 

Coefficient (W/(m2·oC)) 1000

 

Wat Water, er, ho

 

1750

Calculate the overall heat transfer coefficient based on the outside area of the inner tube. Solution Solut ion 3.2 Using following equation,

It is apparent that all the values are known. Thus, on putting the values theU o is 519 W/(m2·oC).

3.3

Enhanced

heat

transfer:

concept

of

fins

In the previous discussion, we have seen that the heat transfer from one fluid to another fluid needs nee ds a so solid lid bounda boundary. ry. The rat rate e of he heat at tr trans ansfer fer dep depen ends ds on man many y fa fact ctor orss inc includ ludin ing g th the e individual heat transfer coefficients of the fluids. The higher the heat transfer coefficients the higher will be the rate of heat transfer. There are many situations where the fluid does not have a high heat transfer coefficient. For example, the heat lost by conduction through a furnace wall must be dissipated to the surrounding by convection through air. The air (or the gas phase in general) has very low heat transfer coefficient, since the thermal conductivities of gases are very low, as compared to the liquid phase. Thus if we make heat transfer device for gas and a liquid (of course separated by a heat conducting wall), the gas side film will offer most of the thermal resist res istanc ance e as com compar pared ed to the liqu liquid id si side de fi film. lm. The Theref refor ore, e, to mak make e the hea heatt tr tran ansf sfer er mos mostt effective we need to expose higher area of the conductive wall to the gas side. This can be done by making or attaching fins to the wall of the surface. A fin (in general) is a rectangular metal strip or annular rings to the surface surface of heat transfer. transfer. Thus, a fin is a surface that extends extends from an obje ob ject ct to in incr crea ease se th the e ra rate te of he heat at tr tran ansf sfer er to or fr from om th the e en envi viro ronm nmen entt by in incr crea easi sing ng convections. Fins are sometimes known as extended surface. Figure 3.7 shows photographs of  an electric motor with the fins on the motor body and a computer processor with the fins to dissipate the generate generated d heat into the environment. environment. Figure 3.8 shows the different types of finned surfaces.

 

Fig. 3.7. Cooling fins of (a) electric motor, (b) computer processor

 

Figu Figure re 3.9 sho shows ws a sim simple ple str straig aight ht rec rectan tangu gular lar fin on pla plane ne wal wall. l. The fi fin n is pro protr trude uded d a distance l  from  from the wall. The temperature of the plane wall (in fact the base of the fin) is T w  w and that of the ambient is T ∞  ; thicknes thicknesss t ; and the breadth b . ∞.   The distances of the fin are: length l  The heat is conducted through the body by conduction and dissipates to the surrounding by convection. The heat dissipation to the surrounding occurs from both top, bottom, and side surfaces of the fin. Here, it is assumed that the thickness of the fin is small and thus the temperature does not vary in the Y-direction. However, the fin temperature varies in the Xdirection directi on only.

Fig. 3.9. 1-D heat conduction conduction and convection through through a rectangular fin Consider a thin element of thickness d x of the fin at a distance x from the fin base. The energy balance on the fin element at steady state is discussed below.

 

where, P  is  is the perimeter [2(b +t)] +t)] of the element, T  is  is the local temperature of the fin, h  is  is the film heat transfer coefficient, and bt  is   is the fin area ( A   A ) perpendicular to the direction of heat transfer.

Thus,

at

steady

state,

Rate of heat input – input – Rate  Rate of heat output – output – Rate  Rate of heat heat loss loss = 0

However, the other boundary conditions depend on the physical situation of the problem. A few o f se I: thelong and thus the temperature typical casfin es is same as that of the are, Case Ca I: The  The fin is very at the end of the

 

ambient Case II: The fin is of finite length and looses looses heat from its end by convection.

fluid.

Case III: The end of the fin is insulated insulated so that at

3.3.1

Analytical

solution

of

the

Case

above

cases I:

The boundary b oundary conditions conditions will be

Using boundary conditions, the solution of the equation 3.23 becomes,

 All of the heat lost lost by the the fin must have conducted from the base at at x= 0. 0. Thus, we can compute the heat loss by the fin using the equation for temperature distribution, distribution,

Similarly, for Case Case – II,, the boundary conditions are:  – II

 

The second boundary condition is a convective boundary condition which implies that the rate at which heat is conducted from inside the solid to the boundary is equal to the rate at which it is transported to the ambient fluid by convection. convection. The temperature temperature profile profile is,

or we can write write,,

Therefore, the boundary conditions led to the following solution to the eq.3.23.

 

distribution can be easily found Thus, the heat loss by the fin, using the equation for temperature distribution out by the following equation, equation,

In

a

similar

fashion

we

can

solve

the case

 –

III also.

The boundary boundary conditions are,

Thus, on solving eq.3.23,

Thus the heat loss by the fins, using the equation for for temperature distribution,

It is to be not noted ed tha thatt the gen genera erall ex expre press ssion ion fo forr the tem temper peratu ature re gra gradie dient nt (e (eq.3 q.3.2 .23) 3) wa wass developed develo ped by assumin assuming g the tempera temperature ture gradi gradient ent in the x-direct x-direction. ion. It is really applic applicable able

with wit h

very less error, if the fin is sufficiently thin. However, However, for the practical practical fins the error introduced introduced by this assumption assumption is less than 1% only. Moreover, Moreover, the practical fin calculation calculation accuracy is limited by the uncertainties in the value of  h . It is because the h  value   value of the surrou surroundin nding g fluid is hard hardly ly uniform over the entire surface of the the fin.

 

3.3.2

Fin

efficiency

It was seen that the temperature temperature of the fin decreases with distance distance x from the base of the body. Therefore, the driving force (temperature difference) also decreases with the length and hence the th e he heat at fl flux ux fr from om th the e fi fin n al also so de decr crea ease ses. s. It ma may y al also so be vis visua uali lize zed d th that at if th the e th ther erma mall conductivity of the fin material is extremely high. Its thermal resistance will be negligibly small and the temperature temperature will rema remain in almos almostt con constan stantt (Tw) thro througho ughout ut fin. In this condition condition the maximum heat transfer can be achieved and of-course it is an ideal condition. It is therefore, interesting and and useful to calculate the efficiency of the fins. The fin efficiency may be define as,

Thuss dep Thu depend ending ing up upon on the con condit ditio ion, n, th the e act actual ual hea heatt tr tran ansfe sferr can be ca calcu lculat lated ed as sho shown wn previously. As an example, for case – case  – III  III (end of the fin is insulated), the rate of heat transfer was

The maximum heat would be transferred from the fin in an ideal condition in which the entire fin area was at T w  b e, w.   In this ideal condition the heat transferred to the surrounding will be,

Therefore, under such conditions, the efficiency of the fin will be;

 

If the fin is quite deep as compared to the thickness, the term 2b will be very large as compared to 2t ,and , and

The equation shows that the efficiency (from eq.3.30) of a fin which is insulated at the end can be easily calculated, which is the case-III discussed earlier. The efficiency for the other cases may also be evaluated in a similar fashion. The above derivation is approximately same as of practical purposes, where the amount of heat loss from the exposed end is negligible. It can be noted that the fin efficiency efficiency is maximum for the zero length of the fin ( l  =  = 0) or if there is no fin. Therefore, we should not expect to be able to maximize fin efficiency with respect to the fin length. However, the efficiency maximization should be done with respect to the quantity of the fin material keeping economic consideration consideration in mind. Sometimes the performance of the fin is compared on the basis of the rate of heat transfer with the fin and without without the fin as shown,

Illustrat Illust ration ion 3.3.

 A ste steel el pipe having having inner inner diamet diameter er as 78 mm and outer outer diamet diameter er as 89 mm has 10  external longitudinal rectangular fins of 1.5 mm thickness. Each of the fins extends  30 mm from the pipe. The thermal conductivity of the fin material is 50 W/m   o C. The  temperature of the pipe wall and the ambient are 160   o C, and 30   o C, respectively,  whereas the surface heat transfer coefficient is 75 W/m 2 o C. What is the percentage  increas incr ease e in the rat rate e of hea heatt tran transfe sferr after after the fin arr arrang angeme ement nt on on the pla plane ne tub tube?  e?  Solution Solut ion 3.3  As the fins are rectangular, rectangular, the perimeter of the fin, P = 2(b + t).  The thickness (t ) of the fin is quite small as compared to the width (b ) of the fin. Thus, P = 2b as   well as we may assume that there is no heat transfer transfer from the tip of the fin. Under such condition we can treat it as case-III,

 

whe here re the here re was no he heat at tra rans nsffer to

the at atmo mossph pher ere e

due du e to in insu sula latted fin

tip ip..

Using eq. 3.30,

 As the pipe length is not given, we will work-out considering the length of the pipe as 1 m and henceforth the breadth of the fins should also be considered as 1 m. We have to consider the area of the fins in order to consider the heat dissipation from the fins. However, we may neglect the fin area at the y-z plane and x-y plane (refer fig. fig . 3.8) as compared to the area of x-z plane. The area of all the fins = (number of fins) (2 faces) (1) (0.03) = 0.6 m2 The maximum rate of heat transfer from the fins

Actual rate of heat transfer transfer = The total rate of heat transfer from the finned tube will be the sum of actual rate of heat transfer from the fins and the rate of heat transfer from the bare pipe, the pipe portion which is not covered by the fins. Therefore, There fore, the remaining area will be calculated as follows,

The remaining area = Total pipe area - base area covered by the the 10 fins

Pipe are =  Attached The

area

remaining

of area

10 fins = comes out to

(10) be

(1) (0.28

(0.0015) =  –   0. 0.015) =

0.015 0.265

Th e corresponding heat transfer = ( 75 ) (0.265) (160-30) = 2583.75 The total heat transfer transfer from the finned tube = 3802.5 + 2583.75 = 6386.25 W Rate of heat transfer from the tube if it does not have any fins = (75) (0.28) (160-30) = 2730 W The percentage p ercentage increase increase in the heat transfer =

m2 m2 W

 

3.4

Thermal

insulation

We have seen how heat transfer is important in various situations. Previous discussion indicates that we are all the time interested in the flow of the heat from one point to another point. However, there are many systems; in fact it is a part of the system, in which we are interested to minimize the losses through heat transfer. For example, in a furnace we want to have high heat transfer inside inside the furnace; however we do not want any heat loss through the furnace wall. Thus to prevent the heat transfer transfer from from the furnace to the atmosphere atmosphere a bad heat conductor conductor or a very good heat insulator insulator is required. In case of furnace the wall is prepared prepared by multiple layers of  refractory refracto ry materials to to minimize the heat losses. Therefore, Therefore, wall insulation is required in various process proce ss equipment equipment,, reactors, reactors, pipelines pipelines etc. to minimize minimize the heat loss loss from the system to the envir env ironm onment ent or he heat at gai gain n fro from m th the e en envir vironm onment ent to the sy syste stem m (li (like ke cr cryog yogen enic ic sy syst stems ems). ). However, there are situations in which we want to maximize the losses for example, insulation to electric wires. The petr petroleu oleum m con conser servati vation on rese research arch associatio association n (PC (PCRA) RA) prov provides ides a good data database base on the properties

and

applications

of

industrial

thermal

insulations

(http:  ).   The The ta table ble 3. 3.2 2 sh show owss so some me co comm mmon on  //w  //www ww.pc .pcra. ra.org org/En /Engli glish/ sh/edu educat cation ion/li /liter teratu ature. re.ht htm  m ). insulations used in chemical process industries for for various process equipment and pipelines. Tab able le--3.2 .2:: The herrmal prope pert rtie ies s of a few of th the e in insu sula lati tio ons bein ing g use sed d in the chemic ica al process proc ess indust industries ries Material  Asbestos Glass Gla ss woo wooll

Temperature (°C)

 Approximate thermal  Approximate thermal conductivity(W/(m° conductivity(W/(m °C))

Density (kg/m3)

-200 to 0 −7 to 38

0.074 0.031

469 64

38 to 93 Fibre insulating board 21 Hard rubber 0

0.041 0.049 0.151

64 237 2000

Polyurethan Polyur ethane e foa foam m

0.018

32

−170 to  to 110

 An interesting application of the heat heat loss from a surface of some practical significance is found in the case of insulation of cylindrical cylindrical surfaces surfaces like small pipes or electrical wires. wires. In many a cases we desire to examine the variation in heat loss from the pipe with the change in insulation thickness, assuming that the length of the pipe is fixed. As insulation is added to the pipe, the outer exposed surface surface temperature temperature will decrease, but at the same time the surface area available to the convective heat heat dissipation dissipation will increase. increase. Therefore, Therefore, it would would be interesting interesting to study these opposing opposi ng effec effects. ts.

 

dissipation from an insulated pipe Fig. 3.10: Heat dissipation Let us consider a thick insulation insulation layer which is installed around a cylindrical pipe as shown in fig. 3.10 (equivalent electrical circuit is shownin figure 3.11). Let the pipe radius be R   and the insulation radius is r . This (r -R ) will represent the thickness of the insulation. If the fluid carried by the pipe is at a temperature T  and  and the ambient temperature is T a a.  The insulation of the pipe will alter pipe surface temperature T  in  in the radial direction. That is the temperature of the inner surfac sur face e of the pi pipe pe an and d th the e out outer er su surf rface ace (b (bel elow ow ins insula ulati tion on)) of the pi pipe pe wil willl be dif diffe feren rent. t. However, if the thermal resistance offered by the pipe is negligible, it can be considered that the temperature (T ) is same across the pipe wall thickness and it is a common insulation case (please (pleas e refer prev previous ious discu discussio ssion). n). It can also be assum assumed ed that the heat trans transfer fer

coeffici coef ficient ent

inside the pipe is very high as compared to the heat transfer coefficient at the outside of the insulated pipe. Therefore, only two major resistances in series will be available (insulation layer and gas film of the the ambient ambient). ).

Fig.3.11: Resistance offered by the insulation and ambient gas film Therefore,

 

where,

k

is

the

thermal

conductivity

of

the

material.

On differentiating above equation with respect to r will show that the heat dissipation reaches a maximum,

So it is maxima, where the insulation radius is equal to

where, r c denotes the critical radius of the insulation. insulation. The heat dissipation dissipation is maximum at r c which is the result of the previously mentioned mentioned opposing effects.

Fig. 3.12: The critical insulation thickness of the pipe insulator Therefore, the heat dissipation from from a pipe increases by the addition of the insulation. However, However, above r c the heat dissipation reduces. The same is shown in fig. 3.12.

The careful analysis of the r c reveals that it is a fixed quantity determined determined by the thermal properties of the insulator. If R  <  T f f  , the flui fluid d temp temperat erature ure appro approache achess asym asymptot ptoticall ically y and the tem temper peratu ature re prof profile ile at a Ts s   > distance x is is shown in fig.4.3. However, a thermal boundary may be defined (similar to velocity boundary) as the distance from the surface to the point where the temperature is within 1% of  the free stream fluid temperature (T f f ). Outside the thermal boundary layer the fluid is assumed to be a heat sink at a uniform temperature of T  T f f . The thermal boundary layer is generally not coincident with the velocity boundary layer, although it is certainly dependant on it. That is, the velocity, velocit y, bou bounda ndary ry laye layerr thic thicknes kness, s, the var variati iation on of velo velocity city,, whe whethe therr the flo flow w is lamin laminar ar or turbulen turb ulentt etc are all the factors factors whi which ch dete determin rmine e the tempe temperatu rature re variation variation in the therm thermal al boundary layer. The thermal boundary layer and velocity boundary layer are related by the Prandtl number, is called the momentum diffusivity and is called the thermal diffusivity; is less than unity, the momentum boundary layer (or velocity boundary layer) remains within the thermal boundary layer. If  P r r >1, the boundary layers will be reversed rever sed as shown in the fig.4.4. fig.4.4. The thermal boundary boundary layer and velo velocity city boundary boundary laye layerr coincides at P r r =1.    =1.

Fig.4.4: The relation relation of two boundary layers at different Pr numbers The above boundary layer theory will be helpful to understand the heat transfer in the process. Through the boundary layers heat transfer is covered in a separate chapter, but the detailed derivati deri vation on and deve developm lopment ent of all the relationsh relationships ips havin having g engin engineeri eering ng impor importanc tance e for the prediction of forced convection heat transfer coefficient is beyond the scope of the course. The reader may consult any standard fluid mechanics and and heat transfer books for for detailed knowledge knowledge.. The purpose of this chapter is to present a collection of the most useful of the existing relations for the most frequently encountered cases of forced convection. Some of these relations will be having theoretical bases, and some will be empirical dimensionless correlations of experimental data. In some situations, situations, more than one relation relation will be given.

The discussion on heat transfer correlations consists of many dimensionles dimensionlesss groups. Therefore, before we we discuss the importance of heat transfer transfer coefficients, coefficients, it is important to understand the

 

phy physic sical al si signi gnific ficanc ance e of the these se dim dimen ensio sionle nless ss gro group ups, s, whi which ch are fr frequ equent ently ly us used ed in fo force rced d convection heat transfer. The table 4.1 shows some of the dimensionless numbers used in the forced convection heat transfer. Table-4.1: Some important dimensionless numbers used in forced heat transfer convection

 

4.3 3 Fl Flo ow th thro roug ugh h a pip ipe e ortu ortube be 4. 4.3.1

Turbulent

flow

 A classical expres expression sion for calculatin calculating g heat transf transfer er in fully develope developed d turbulent flow in smooth tubes/pipes of diameter (d) and length (L) is given by Dittus and Boelter (4.3)

where, n = 0.4, for heating heating of the fluid n = 0.3, for cooling of the fluid The properties in this equation are evaluated at the average fluid bulk temperature. Therefore, the temperature difference between between bulk fluid and the wall should not be significantly high.

 

 Application of eq. 4.3 lies in the following following limits

Gnielinski suggested that better results for turbulent flow in smooth pipe may be obtained from the following relations relations

When the temperature difference between bulk fluid and wall is very high, the viscosity of the fluid and thus the fluid properties changes substantially. Therefore, the viscosity correction must be accounted using Sieder – Sieder – Tate  Tate equation given below

However, the fluid properties have to be evaluated at the mean bulk temperature of the fluid exceptμ w  temperature. re.   The earlier relations were were applicable w which should be evaluated at the wall temperatu for fully developed flow when when entrance length was was negligibl negligible. e. Nusselt recommended recommended the following relation for the entrance entrance region when the the flow is not fully developed. (4.7)

 

where, L is the tube length and d is the tube diameter. The fluid properties in eq. 4.7 should be evaluated at mean bulk temperature of the fluid.

 Applicability conditions, .  As different different temperature temperature terms will appear in the course course therefore therefore to understand these terms see the following details. details. Bulk temperature/mixing cup temperature: temperature: Average temperature in a cross-sectio cross-section. n.  Average bulk bulk temperature temperature:: Arithmetic average temperature of inlet and outlet bulk temperatures. Wall temperature: temperature: Temperature of the wall. Film temperatur temperature e: Arithmetic average temperature of the wall and free stream temperature.

Free stream temperature: temperature: Temperature Temperature free from the the effect of wall. Log mean temperature difference: difference: It will be discussed in due course of time

Illustration

4.1

Pr essuri zed ai ai r i s to be he hea ated by by flowi flowi ng i nt nto o a pi pi pe of 2.54 cm di ameter. The ai r at 200oC and 2 atm  pres  p ressu sure re ent nte ers in the thep pip ipe e at 10 m/s. The temperat rature ure of the thee ent ntir ir e pipe ipeis is maint inta aine ined d at 22 220 0oC . E va valua luatte the he hea at trans transfer fer co coeff ffii cien cientt for a unit le length ngth of a tu tub be co conside nsiderr i ng the the co const nsta ant he hea at flux co cond ndii tio tions ns ar e mai nt nta ai ne ned d at the pi pe wall. What What wi ll be the bulk te temper atur ure e of the the ai r at the end of 3 m le length ngth of the the tube?  The foll followin owing g data data for the ent enterin ering g air (at (at 200  200 o C) ha has s be been en given given,  , 

 

Solut Solution ion 4.1 Reynolds number can be calculated from the above data,

The value of Reynolds number shows that that the flow is in turbulent zone. Thus the Dittus-Boelter equation (eq.4.3) should be used,

Thus h can be calculated for the known values of k , and d , which comes out to be

Energy balance is required to evaluate the increase in bulk temperature in a 3 m length of the tube,

 

Therefore the temperature of the air leaving the pipe will be at 210.81oC. 4.3.2 Laminar

flow

Hausen presents the following empirical relations for fully developed laminar flow in tubes at constant wall temperature temperature..

The heat transfer coefficient coefficient calculated from eq. 4.8 is the average value over the entire

(including

entrance

length)

of

length

tube

.

Sieder and Tate suggested a simple simpl e relation for laminar heat transfer in tubes.

The condition c ondition for applicability applicability of eq. 4.9:

where, μ   is the viscosit viscosity y of th the e flu fluid id at th the e bul bulk k tem tempe perat rature ure and μ w  that at at th the e wa wall ll w  is th temperatureT w  w . The other fluid properties are at mean bulk temperature of the fluid. Here also the heat transfer coefficient calculated from eq. 4.9 is the average value over the entire length

(including entrance length) of o f tube

.

The empirical relations shown in eq. 4.2-4.9 are for smooth pipe. However, it case of rough pipes, it is sometimes appropriate that the Reynolds analogy between fluid friction and heat transfer be used to effect a solution under these conditions and can be expressed in terms of  Stanton Stant on number number.. In order to account the variation of the thermal properties of different fluids the following equati equ ations ons may be use used d (i. (i.e. e. Sta Stanto nton n num number ber mul multip tiplie lied d by

),

 

 

where,

is the mean free veloci velocity. ty. The frict friction ion facto factorr can can be be evalua evaluated ted from Moody‟s Moody‟s chart.  chart.

4.3.3

Flow

through

non-circular

ducts

The same co-relations as discussed in section 4.4.1 can be used for the non-circular ducts. However, the diameter of the tube has to be replaced by the hydraulic diameter or equivalent diameter for the non-circular ducts. The hydraulic diameter diameter is defined as

Where r h h is    is hydraulic radius. radius. 4.3.4.

Flow

over

a

flat

plate

Heat transfer in flow over a plate occurs through the boundary layer formed on the plane. Therefore at any location the heat transfer coefficient will depend on the local Reynolds and Prandtl number. For local heat transfer coefficient in laminar boundary layer flow, the following correlation can be used to find the local Nusselt number. It depends upon the distance from the leading edge (x ) of the plate. (4.13)

where,

and an d

are ar e

the lo loca call

Nus usse sellt

and an d

Reyn Re ynol old d

respectively.  An average value value of the heat transfer transfer coefficient over a distance distance l may be obtained by,

num umbe berrs,

 

4.3.5 Flow across cylinders and spheres

4.3.5.1

Flow

across

a

cylinder  

The heat transfer coefficient coefficient can be found out by the correlations correlations given by many researcher researcherss

 

 Applicability of eq. 4.19: 102 < Re < 107, and Re Pr >0.2 . However, the following equation equation (eq. 4.20) 4.20) is more accurate for the condition where where 20,000 < Re < 4,00,000 and Re Pr > 0.2.

 

4.3.5.2. Flow acro across ss a sphere 

The above correlation is applicable to both gases and liquids.

4.3.5.3

Flow

over

a

bank

of

tubes  

Flow over bank of tubes is one of the very important phenomena in chemical process industries. Heat exchanger, air conditioning for cooling and heating etc. involve a bank or bundle of tube over which a fluid flows. The two most common geometric arrangements of a tube bank are shown show n in fig. 4.5.

 

Fig.4.5: Tube banks: (a ) aligned; (b ) staggered In any of the arrangements, arrangements, D is the diameter of tube, S L is the longitudinal spacing, and ST is   the transver trans verse se tube spacin spacing. g. The flow over a tube is quite different than the flow over bank of tubes. In case of bank of tube, the flow is influenced by the effects such as the “shading the  “shading”  ”  of  of one tube by another etc. Moreover, the heat transfer for any particular tube thus not only determined by the incident fluid   conditions,

v ∞  ∞ andT  ∞, ∞    but also by D, S  L , and ST and the tube positions in the bank. It is now clear   that the L  heat transfer coefficient coefficient for the first first row of tubes is much like that for a single cylinder in cross flow.. Howe flow However, ver, the heat transfer transfer coefficient coefficient for the tubes in the inner rows is generally generally larg larger er because of the wake generation generation by the previous previous tubes. For the heat transfer correlations, correlations, in tube banks, the Reynolds number is defined by

where v m is the maximum fluid velocity occurring at the minimum vacant area of the tube bank. For the aligned ali gned tube arrangement,

 

In case of bank of tubes, generally we are interested interested for a single tube but interested interested to know the average avera ge heat transfer transfer coefficie coefficient nt for the enti entire re bank of tubes. tubes. Zuka Zukauska uskass has summarized summarized his extensive for for the heat transfer transfer coefficients coefficients for fluid past past a bank of tubes, (4.27)

The applicability ap plicability of eq. 4.27:

, and number of tubes are

atleast

20. The constants C   and and m   of of co-relation 5.26 can be found out from any standard book on heat transfer. It may be noted that the above relation is for the inner rows of bank, or for banks of  many man y row rows. s.

4.4 Mo Mome ment ntum um an and d he heat at tr tran ansf sfer er an anal alog ogie ies s Consider a fluid flows in a circular pipe in a laminar low (fig.6.6). The wall of the pipe is maintained at Tw temper temperature, ature, which is higher than the flowing fluid temperature. The fluid being in relatively lower temperature than the wall temperature will get heated as it flows through the pipe. Moreover, the radial transport of the momentum in the pipe occurs as per the Newton‟s   law of viscosity. For a circular pipe momentum transport and heat transport may be written in a similar way as shown in the eq. 4.28, Momentum flux = momentum diffusivit di ffusivity y × gradient of concentratio concentration n of momentum

 

4.28(a)

It may be noted that the fluid velocit velocity y

is a funct function ion of radius of the pipe.

Heat flux= thermal diffusivity × gradient of concentration of heat energy

Now, the question comes, why are we discussion about the similarities? The answer is straight forward that it is comparatively easy to experimentally/theoretically evaluate the momentum transport under various conditions. However, the heat transport is not so easy to find out. Therefore, we will learn different analogies to to find the heat transport relations. Equation 4.28 is for the laminar flow but if the flow is turbulent, eddies are generated. Eddy is a lump/chunk of fluid elements that move together. Thus it may be assumed that the eddies are the molecules of the fluid and are responsible for the the transport of momentum and heat energy in the turbulent flow. Therefore, in turbulent situation the momentum and heat transport is not only by b y the molecular diffusion but but also by the eddy diffusivities. Thus, turbulent transport of momentum and turbulent transport of heat may be represented by eq. 4.29a and 4.29b, respectively. respectively.

The Th e te term rmss

repr re pres esen entt th the e ed eddy dy di diff ffu usi sivi viti ties es fo forr mo mome ment ntum um an and d he heat at,, re resp spec ecti tive vely ly..

 At the wall of the pipe, the momentum equation equation (eq. 4.29a) becomes,

 

Where f is the fanning friction factor (ratio of shear force to inertial force) and average averag e fluid veloci velocity. ty.

is the

Equation eq.4.30 can be rearranged as,

The eq.4.32 is the dimensionless velocity gradient at the wall using momentum transport. We may get the similar relation using using heat transport as shown shown below. below. Wall heat flux can be written as,

Where Tav is the wall temperature and the Tav is the average temperature of the fluid. Thus, the dimensionless temperature temperature gradient at the wall using heat transfer will be,

Where the heat transfer coefficient is represented by h and dimension dimensionless less temperature represented by

.

is

 

Based on the above discussion many researchers have given their analogies. These analogies are represented represente d in the subsequent section.

4.4.1

Reynolds

analogy

Reynolds has taken the following assumptions to find the analogy between heat and momentum transport. 1.

Gradients

of

the

dimensionless

parameters

at

the

wall

are

equal.

2. The diffusivity terms are equal. equal. That is Thus if we use the above assumptions along with the eq.4.32 and 4.33,

Thus if we use the above assumptions along with the eq.4.32 and 4.33,

Equation

4.34

is

known

as

Reynolds ‟s

analogy.

The above relation may also be written in terms of the Darcy‟s friction factor (fD) instead fanning

friction

Where Stanton number (St) is defined as,

factor

(fD  =

of  4f )

 

The Th e ad adva vant ntag age e of th the e an anal alo ogy li lies es in th that at th the e h   may may not be av avai aila labl ble e fo forr ce cert rtai ain n geometries/situations however, for which f  value   value may be available as it is easier to perform momentum transport experiments and then to calculate the f . Thus by using using the eq.4. eq.4.34 34 the h  may be found out without involving involving into the exhaustive and difficult heat transfer transfer experiments. 4.4.2

The

Chilton-Colburn

analogy

The Reyn Reynolds olds analogy analogy doe doess not alw always ays give sati satisfa sfactor ctory y res results ults.. Thu Thus, s, Chil Chilton ton and Colb Colburn urn experimentally modified the Reynolds‟  analogy. The empirically modified Reynolds‟ Reynolds‟   analogy is known as Chilton-Colburn analogy and is given by eq.4.35,

It can be noted that for unit Prandtl number the Chilton-Colburn analogy becomes Reynolds analogy. 4.4.3

The

Pradntl

analogy

In the turbulent core the transport is mainly by eddies and near the wall, that is laminar sublayer lay er,, th the e tr tran ansp spor ortt is by mol molecu ecular lar dif diffu fusio sion. n. The Theref refor ore, e, Pra Prand ndtl tl mod modifi ified ed the abo above ve tw two o analogies using universal velocity profile while driving the analogy (eq. 4.36).

4.4.4 4.4 .4 Th The e Va Van n Ka Karm rman an an anal alog ogy y Though Prandtl considered the laminar and turbulent laminar sublayers but did not consider the buffer zone. Thus, Van Karman included the buffer zone into the Prandtl analogy to further improve the analogy.

 

Frequently Asked Questions and Problems Problems for Practice (Module 4) Q.1

What is the local Reyn Reynolds olds number number??

dimensionlesss groups groups in heat transfer? transfer? Explain their their physical Q.2   What are the important dimensionles significance. Q.3

What is the the difference between Reynol Reynolds ds and Prandt Prandtll anologies?

Q.4   Explain why there is more heat transfer in forced convection as compared to natural convection. Q.5

What is Dittus-Boelter Dittus-Boelter equation and when is it applied?

Q.6

Water is to be heate heated d from 50oC to 100oC in a smooth hot pipe. The pipe is maintained at a constant temperature above 30oC that of bulk water temperature under the condition of  consta con stant nt he heat at flu flux. x. Cal Calcul culate ate the len lengt gth h of the pi pipe pe re requi quired red fo forr hea heatin ting, g, if th the e tu tube be diameter is 0.6 m and the Reynolds number of the water inside the the pipe is 95000?

array of 100 tubes arrange arranged d in an in-line position is at 100oC. Q.7   A tube bank having a square array The diameter and length of the tubes are 15 mm and 100 cm, centre to centre tube spacing is 20 mm. Atmospheric air enters in the tube bank at 25 oC and at the free stream velocity of 5.5 m/s. Determine the total heat loss by the tubes. o

Q.8 Q. 8

Wa Wate terr at 15 C past a are sphere at the stream velocity of 4 m/s.the The diameter temperature offlow the sphere 30 mm andfree 70oC, respective respectively. ly. Calculate heat loss by and the sphere.

Q.9

Atmosphe Atmos pheric ric air flows flows at 10 m/ m/ss of free stream stream velocit velocity y in a re recta ctang ngula ularr duc ductt hav havin ing g  o dimensions of 25 cm by 50 cm. The air and wall temperature of the duct are 25 C and 50oC, respectively. Calculate the mean exit temperature of the air per unit length of the duct.

Q.10  Air at 25   oC flows in a 10 mm diameter tube at a Reynolds number of 50,000. If the length of the tube is 100 cm, estimate the average heat transfer coefficient for a constant heat flux at the wall.

 

Module Modu le 5: Heat transfe transferr by natural natural convectio convection n

5.1 Intro Introduct duction ion In the previous chapter, we have discussed about the forced convective heat transfer when the fluid motion relative to the solid surface was caused by an external input of work by means of  pump, fan, blower, stirrer, etc. However, in this chapter we will discuss about the natural or free convec con vectio tion. n. In nat natura urall con convect vection ion,, th the e flu fluid id vel veloci ocity ty fa farr from from the sol solid id bod body y wil willl be ze zero. ro. However, near the solid body there will be some fluid motion if the body is at a temperature different from that of the free fluid. In this situation there will be a density difference between the fluid near the solid surface and that far away from the system. There will be a positive or negative buoyancy force due to this density difference. Hot surface will create positive buoyancy force whereas the cold surface will create the negative buoyancy force. Therefore, buoyancy force will be the driving force which produce and maintain the free convective process. Figure 5.1 shows the natural convective process for for a hot and cold vertical surface.

Fig.5.1: Free convection boundary layer for vertical (a) hot surface and (b) cold surface Consider a vertical flat plate with contact of a fluid (say liquid) on one side of the plate. Now assume that we raise the temperature of the plate to Ts, a natural convective boundary layer forms as shown in fig. 5.2. The velocity profile in this boundary layer is slightly different as compared to forced convection boundary layer. At the wall the velocity is zero because of no slip condition. The velocity increases to maximum and then reduces to zero at the end of the boundary layer because the fluid is at rest in the bulk. Initially the laminar flow is achieved in the boundary layer, but at some distance from the leading edge, depending on the fluid properties and the temperature difference between plate and bulk fluid, turbulent eddies are found thus laminar to transition region comes. On further away from the leading edge the boundary layer

 

may become Instability of the b ecome turbulent and the boundary layer instability comes in to picture. Instability boundary layer is quite complex and does not fall into the scope of this study material.

Fig. 5.2: Boundary layer on a hot vertical flat plate (Ts: surface temperat t emperature; ure; Tb: bulk fluid temp tempera erature ture)) It has been found over the years that the average Nusselt Nusselt number (or the average heat transfer transfer coeffi coe fficien cient) t) fo forr con conve vecti ctive ve hea heatt tr tran ansf sfer er can be re repre presen sente ted d by th the e fo follo llowi wing ng fu funct nction ional al dependence (say viscous flow past a hard body). bod y).

Nu = f(R f(Re,G e,Gr,Ec r,Ec,Pr)  ,Pr) 

 

(5.1)

The Reynolds number (Re ) is the ratio of inertia forces in the fluid to the viscous forces. The Grashof number (Gr ) is the ratio of buoyant forces to the viscous forces. The Eckert number (Ec ) is a measure of the thermal equivalent of kinetic energy of the flow to the imposed temperature differences. The Eckert number arises due to the inclusion of viscous dissipation. Thus Ec   is absent where dissipation is neglected. The Prandtl number, Pr , is the ratio of the momentum diffusivity (kinematic viscosity) to the thermal diffusivity. In other words, Prandtl number is a measure of the relative magnitude of the diffusion of momentum, through viscosity, and the diffusion of heat through conduction, in the fluid. In case of perfect natural-convection natural-convection and in absence of heat dissipation, the the eq. 5.1 reduces to,

 

Nu = f( f(Gr Gr,P ,Pr)  r) 

 

(5.2)

It is to be b e noted that in case of perfect natural convection, the the main fluid stream is absent, thus Reynolds number is no longer longer significant. The Th e

dime di men nsio ionl nle ess

temp te mper erat atur ure, e,

num umbe berrs

inv nvo olved

in

eq.. eq

5.2 5. 2

eval ev alua uatted

at

the

aver av era age

fil ilm m

It ca can n be ea easi sily ly fo foun und d th that at in ca case se of th the e fo forc rced ed co conv nvec ecti tion on an and d in

absence of heat dissipati di ssipation on the function for average heat transfer will be,

Nu = f(R f(Re,P e,Pr)  r) 

 

(5.3)

On comparing eq. 5.2 and 5.3, one can c an see that the Grashof number will perform p erform for free free convection in a same way as the Reynolds number for forced convection.  Another parameter, parameter, the the Rayleigh number is also used for perfect natural-convection is defined defined as,

Ra = Gr Gr . Pr 

 

(5.4)

 

(5.5)

Thus the functional relation is eq. 5.2 can be written as,

Nu = f(R f(Ra,P a,Pr)  r) 

 As discussed earlier that all free convection flow flowss are not limited to laminar flow. If instab instability ility occurs, the problem becomes complex. A general rule one may expect that transition will occur for critical Rayleigh number of  (5.6)

The Grashof number is defined as

where, g = accele accelerati ration on due due to to gravity gravity β= coefficient of volume expansion expansion = Ts = surface temper temperature ature b

T  = bulk fluid temperature L= charac cha racter terist istic ic len lenght ght v = Momentum diffusivity (kinematic viscosity)

 

5.2 Empirical relations relations for natural-convective natural-convective heat transfer 5.2.1 Natural convection around a flat vertical surface  Churchill and Chu provided the correlation for average heat transfer coefficient for natural convection for different ranges of Rayleigh number. Case Ca se I:

 

If RaL < 109 (5.7)

Case Ca se II II::

 

If 10-1 < RaL < 1012 (5.8)

It should be noted that the eq. 5.7 and 5.8 are also applicable for an inclined surface upto less than inclination from the vertical plane. The above relations can be used for the vertical cylinder if the boundary layer thickness is quite small as compared to the diameter of the cylinder. The criteria to use the above relation for vertical cylinder is, (5.9)

where, is the diameter and is the height of the the cylinder. cylinder. 5.2.2 Natural convection around a horizontal cylinder  Churchill and Chu has provided p rovided the following expression expression for natural-convective heat transfer. (5.10)

Condition of applicability of the eq.5.10: 5.2.3 Natural convection around a horizontal flat surface 

 

In the previous case of vertical flat surface, the principal body dimension was in-line with the gravity (i.e. vertical). Therefore, the flow produced by the free convection was parallel to the surface surfa ce regardless regardless of whe whethe therr the surface surface was hott hotter er or cooler compared compared to the bulk fluid around. However, in case of horizontal flat plate the flow pattern will be different and shown in fig. 5.3 5.3..

Fig. Fi g. 5. 5.3: 3: A re repr pres esen enta tati tive ve fl flow ow pa patt tter ern n (n (nat atur ural al co conv nvec ecti tion on)) fo forr (a (a)) ho hott su surf rfac ace e do down wn,, (b)) ho (b hott su surf rfac ace e up up,, (c (c)) col old d su surf rfac ace e down wn,, an and d (d (d)) col old d su surf rfac ace e up Thus from fig. 5.3 it is understood that there are in fact two cases (i) when the heated plate facing up or cooled plate facing down, and (ii) heated plate facing down or cooled plate facing up. Case Ca se I:

Heated plate facing facing up, cooled plate facing dow down  n  (5.11) (5.12)

Case Ca se II II::

Heated plate facing facing down, down, cooled plate facing up  (5.13)

where, Lc is characteristic characteristic length defined as below.

 

5.2.4 Natural convection around sphere Churchill proposed proposed,, (5.14)

Condition for applicability: Pr ≥ 0.7; Ra ≤ 1011 5.2.5 5.2 .5 Natu Natura rall con convec vectio tion n in enc enclos losure ure It is an anot other her cla class ss of pro proble blems ms fo forr wh which ich the there re ar are e man many y ca case sess an and d th thei eirr cor corre respo spondi nding ng correlations are also available in the literature. Here two cases will be discussed, (i) in which a fluid is contained between two vertical plates separated by a distance d, (ii) the other where the fluid is in an annulus formed by two two concentric horizontal cylinders. In the case first, the plates are at different temperature, T 1 and T2. Heat transfer will be from higher temperature (T1) to lower temperature temperature (T2) through through the fluid. The corresponding corresponding Grashof Grashof number will will be

McGregor and Emery proposed the following correction for free convection heat transfer in a vertical rectangular enclosure, where the vertical walls are heated or cooled and the horizontal surfaces many be assumed adiabatic, (5.15)

 Applicability conditions for the above equation are,

or,

 

(5.16)

 Applicability conditions are,

Here L/d is known as the aspect ratio.  At steady state condition, condition, the heat flux (qx) is equal thus,

q x = h(T 1 - T  )  2  2  or, or, or,

or,

where, k c c(= conductivity.. (  = Nu x xk) k  ) is known as the apparent thermal conductivity In the second case the heat transfer is involved in the enclosure formed by two concentric cylinders in horizontal position, the correlation given by Raithby and Holland, (5.17)

is the modified Rayleigh number given by,

where, d i and d 0 are the outer and inner diameter of the inner and outer cylinders, respectively. The enclosure characteristic characteristic length l is defined as (d 0 - d i i). )  .

The applicability of the eq. 5.17 is 102 <

> 107.

 

It should be noted that the rate of heat flow by natural convection per unit length is same as that through the annular cylindrical region having effective thermal conductivity k e for the case,

where, T1 and T2 are the temperatures of the inner and outer cylindrical walls, respectively. respectively.

Illustrati Illust ration on 5.1

 A hot hot oven oven is main mainta tain ined ed at at 18 180  0   o C having vertical door 50 cm high is exposed to the  atmospheric atmosph eric air at 20 o C. Calculat Calculate e the avera average ge heat transfer coefficient at the surfac surface  e  of th the e do door or.. The various air properties at the average temperature [(180+20)/2 = 100 o C] are, k = 0.03 0.032 2 W/ W/m  m   o C;

Pr = 0. 0.7; 7;

Kine Ki nema mati tic c vi visc scos osit ity y = 24 x 10 -6 m 2 /s 

 At T b = 20oC, Solution Solut ion 5.1 First we have to find the Grashof number,

With the help of Gr and Pr , we can estimate the R a number, Ra = GrPr = 1.16 X 108 X 0.7 = 8.12 X 107  As Ra < 109, the eq.5.7 can be used,

 

5.3 Co Comb mbin ined ed na natu tura rall an and d fo forc rced ed co conv nvec ecti tion on  As we know that there is hardly any situati situation on in which only natural or forced convecti convective ve heat transfer occurs. Generally, in all the processes natural and forced convection heat transfer occur but depending upon the contribution made, the process may be approximated as either natural or forced convection problem. However, certain situation needs to be addressed as combined natural and forced convection problem. The following is a thumb rule to determine the individual situations.

1. Forced convection region i.e. negligible negligible natural natural convection convection contribution

2. Natural convection convection region i.e. negligible negligible forced convection contribution 3. Mixed convection i.e., significant contributio contribution n by both natural and forced convections In this situation, the following equation may be used,

where, Nu is the Nusselt number due to mixed convection, Nu n is the Nusselt number due to natural convection, and Nuf is the Nusselt number due to forced convection. The value of m is generally taken as 3, whereas positive and negative signs can be used for the convection in the same and opposite directions, respectively. respectively. Illustrati Illust ration on 5.2

In the oven door described in illustration 5.1 is subjected to an upward flow of air  (that is forced convection). What would be the minimum free stream velocity for  which whic h nat natura urall convect convection ion may be neglected?  neglected? 

 

Solution Solut ion 5.2. Section 5.3 above shows that for the following condition the effect of natural convection may be neglected,

The value of Gr number calculated in the previous illustration was was 1.16 X 108 Thus,

U >> 0.24 0.24 m/ m/s  s  Therefore, the bulk velocity of the air should be far greater that 0.24 m/s.

Frequently Asked Questions and Problems for for Practice (Module 5)

Q.2

Define Grashof Grashof number and its physical significance. What is the analogous in forced convection? Whatt are the crit Wha criteri eria a to kno know w nat natura urall and for forced ced con convec vectio tion? n?

Q.3

Define Rayle Rayleigh igh number and its physi physical cal signi significa ficance. nce.

Q.1

 

Show the flow pattern of natural convection for a (a) hot surface down, (b) hot surface up, (c) cold surface down, and (d) cold surface up. Q.5   Warm air at 65oC, 3 m/s enters into a square (25 cm) duct made up of steel meta me tal. l. Th The e un un-i -ins nsu ula late ted d du duct ct is 10 m lo long ng an and d is in co cont ntac actt wit ith h th the e o atmos atm osphe pheric ric air at 25 C. If the heat lo loss sses es to th the e at atmos mosph pher ere e by na natu tural ral convection then what will be the exit warm air temperature from the duct. It may be assumed that the natural convective convective flow at one surface does not affect the others as the two surfaces are horizontal, one at the top and at the bottom, and the two surfaces surfaces are vertical. vertical. Calculate the heat transfer transfer coefficient for the obviously forced convection inside of the duct. horizontal tube having diameter of 10 mm is heate heated d to a surface temperature temperature Q.6.   A horizontal o o of 240 C. The tube is exposed to air at 25 C. Calculate the natural convective heat trans transfer. fer. sphere having having diameter diameter of 25 25 cm is heated to a surface temperature of 240 Q.7.   A sphere o C. The sphere is exposed to air at 25oC. Calculate Calculate the the natural convectiv convective e heat transfer. Q.4

Q.8.. Q.8

 

A squ square are (1 m) fl flat at pla plate te in incli clined ned at 45   oC with the horizontal is exposed to air at 25oC and 1 atm. The plate receives 750 W/m2 from the Sun and dissipated the heat heat to the atmospher atmosphere e by natu natural ral convec convectio tion. n. What will be the avera average ge

 

temperature of of the plate at steady-state?

Module Modu le 6: Heat transfer transfer in boiling and condens condensati ation on

Heat tra Heat trans nsfe ferr in bo boil ilin ing g an and d co cond nden ensa sati tion on In the previous chapter we have discussed about the convective heat transfer in which the homogeneous single phase system was considered. The heat transfer processes associated with the change of fluid phase have great importance in chemical process industries. In this chapter, we will focus our attention towards the phase change change from liquid to vapour and vice-versa. 6.1 He Heat at tra trans nsfe ferr du duri ring ng bo boil ilin ing g The conversion of a liquid into a vapour is one of the important and obvious phenomena. It has been found that if water (say) is totally distilled and degassed so that it does not have any impur imp urit ity y or di diss ssol olve ved d ga gase ses, s, it wi will ll un unde derg rgo o li liqu quid id to va vapo pour ur ph phas ase e ch chan ange ge wi with thou outt th the e appearance of bubbles, when it is heated in a clean and smooth container. However, in normal situat sit uation ion,, as can be un unde derst rstood ood,, th the e pre presen sence ce of impu impurit rities ies,, dis disso solve lved d gas gases, es, an and d su surf rface ace irregularities causes the appearance of vapour bubble on the heating surface, when the rate of  heat input is significantly high. The boiling may be in general of two types. The one in which the heating surface is submerged in a quiescent part of liquid, and the heat transfer occur by free convection and bubble agitation. The process is known as pool boiling. The pool boiling may further be divided into sub-cooled or local boiling and saturated or bulk boiling. If the temperature of the liquid is below the saturation temperature, the process is known as sub-cooled, or local, boiling. If the liquid is maintained at saturation temperature, the process is known as saturated or bulk boiling. The other form of the boiling is known as forced convective boiling in which the boiling occurs simultan simu ltaneou eously sly wit with h fluid moti motion on ind induced uced by ext extern ernally ally impo imposed sed pres pressur sure e diff differen erence. ce. In thi thiss chapter, we will mostly consider the pool boiling.  As generally the bubbles are formed during boilin boiling, g, we will first refresh the following basic information. Consider a spherical spherical bubble of of radius in a liquid as shown in fig. 6.1

 

Fig. 6.1: Force balance on a submerge spherical bubble in a liquid The pressure of vaporisation inside the bubble, Pvap, must excee exceed d that in the surrounding liquid, Pliq, because of the surface tension (σ) acting on the liquid-vapour interface. interface. The force balance on the equatorial plane

πr 2 (P vap  ) = 2πrσ  vap - P  liq  liq  (6.1)

The eq. 6.1 shows that to create a bubble of small radius, it would be necessary to develop very large pressure in the vapour. In other word, a high degree super heat is necessary for the generation of a tiny bubble (or nucleus) in the bulk liquid. This is the reason, the bubble are usually formed at bits existing in the surface irregularities, where a bubble of finite initial radius may form, or where gasses gasses dissolved in the the system of the liquid come out of the solution. 6.2 6. 2 Bo Boil ilin ing g of sa satu tura rate ted d li liqu quid id In this section, we will discuss about the boiling curve which is a log-log plot between heat flux (q/A ) or heat transfer coefficient ( h ) and excess temperature (ΔT) (ΔT).. Excess temperature (ΔT (ΔT =  =   Tw Tsat) is the temperature difference between heating surface (T w) and saturated saturated temperature temperature of the liqui liq uid d (Tsat). Figure 6.2 shows a typical representative pool boiling curves for water contained in a container where the water is heated by an immersed horizontal wire. Consider we are measuring measuring the heat

 

flux (thus,h ) and the temperature difference (ΔT) (ΔT) between  between the boiling water (Tsat) and the wall temperature of the heater wire (Tw). The temper temperature ature of the heater wire is gradually raised while measuring the heat flux between heated surface and boiling water until a large value of   ΔT reaches. The corresponding plot is prepared at the log-log scale. The plot shows six different sections in the boiling curve shown in the fig.6.2.

Fig. 6.2: Saturated water boiling curve The different regimes of the boiling plot (fig. 6.2) have different mechanism. We will see those mechanisms in-brief in the following section.

Sect Se ctio ion n PQ PQ::  In section PQ , initially when the wire temperature is slightly above the saturation temperature of the liquid, the liquid in contact with the heating surface get slightly superheated. The free convection of this heated fluid element is responsible for motion of the fluid, and it subse su bseque quentl ntly y eva evapo porat rated ed wh when en it ri rise sess to the su surfa rface ce.. Thi Thiss reg regime ime is cal called led th the e int interf erfaci acial al evaporation regime. Sect Se ctio ion n QS QS::  The section QS  is  is composed of section QR  and  and section RS . In QR  section,  section, bubbles begin to form on the surface of the wire and are dissipated in the liquid after detaching from the heating surface. If the excess temperature ( further increases, bubbles form rapidly on the

 

surface of the heating wire, and released from it, rise to the surface of the liquid, and are discharged into the top of the water surface (fig 6.3). This particular phenomenon is shown in section RS . Near the pointS , the vapour bubbles rise as columns and bigger bubbles are formed. The vapour bubbles break and coalesce thus thus an intense motion of the liquid occurs which in-turn increases the heat transfer coefficient or heat transfer flux to the liquid from the heating wire. The section QS  is  is known as nucleate boiling.

Fig. 6.3: (a) Formation Formation of tiny bubbles, and (b) Grown up bubbles Section  ST :  At the beginning of the section ST  or   or at the end of the section , the maximum number of bubbles are generated from the heating surface. The bubbles almost occupy the full surface of the heating wire. Therefore, Therefore, the agitation becomes highest as they discharge from the surface. Thus, maximum heat transfer coefficient is obtained at point S . However, once the population of the bubbles reaches to maximum, the nearby bubbles coalesce and eventually a film fi lm of va vapo pour ur fo form rmss on th the e he heat atin ing g su surf rfac ace. e. Th This is la laye yerr is hi high ghly ly un unst stab able le an and d it fo form rmss momentarily and breaks. This is known as transition boiling (nucleate to film). In this situation the vapour film (unstable) imparts a thermal resistance and thus the heat transfer coefficient reducess rapidl reduce rapidly. y. Section   TU :   If the excess temperature is further increased, the coalesced bubbles form so rapidly that they blanket the heating surface (stable vapour film) and prevent the inflow of fresh liquid from taking their place. The heat conducts only by the conduction through this stable vapour film. As a result the flux of heat transfer decreases continuously and reaches a minimum at point U . All the resistance to the heat transfer is imposed by this layer stable layer of vapour film. Section   UV :  At very high excess temperature the heat transfer is facilitated by the radiation through the the vapour film and thus the heat transfer transfer coefficient start increasing. increasing. Infact the excess

 

temperature in this regime is so high that the heating wire may get melted. This situation is known as boiling crises. The combine regime of ST , TU , and UV  is  is known as film boiling regime.  At this stage it would be interest interesting ing to know the Leidenfrost phenomenon, which was observed by Leidenfrost in 1756. When water droplets fall on a very hot surface they dance and jump on the hot surface and reduces in size and eventually the droplets disappear. The mechanism is related to the film boiling of the water droplets. When water droplet drops on to the very hot surface, a film of vapour forms immediately immediately between between the droplet and the hot surface. surface. The vapour film generated generated provide and up-thrust to the the droplet. Therefore, Therefore, the droplet droplet moves up and when again the droplet comes comes in the contact of the hot surface, the vapour vapour generated out of the water droplet and the phenomenon continues till it disappears disappears.. The effectiveness of nucleate boiling depends primarily on the ease with which bubbles form and free themselves from the heating surface. The important factor in controlling the rate of bubble detachment is the interfacial interfacial tension between between the liquid and the heating surface. surface. If this interfacial tension tens ion is large the bubbles bubbles tends to spread spread along the surface surface and blocked blocked the heat tran transfer sfer area, are a, ra rath ther er tha than n lea leavin ving g th the e su surfa rface, ce, to mak make e roo room m fo forr ot other her bu bubbl bbles es.. The he heat at tr trans ansfe ferr coefficient obtained during the nucleation boiling is sensitive to the nature of the liquid, the type and con condit dition ion of the hea heatin ting g su surfa rface, ce, the com compos positi ition on an and d pur purity ity of th the e liq liquid uid,, agi agitat tation ion,, temperature and pressure.

Fact :  Film boiling is not normally desired in commercial equipment because the heat transfer rate is low for for such a large temperature temperature drop. 6.2.1 Nucle Nucleatio ation n boili boiling ng Rohsenow correlation correlation may be used for calculating pool boiling heat transfer (6.2)

where, q is the heat heat flux (W/m2) μ l is the liquid viscosity (Pa.s) (J/kg) λ is the enthalpy of liquid vaporisation (J/kg) density, respectively, respectively, (kg/m3)  ρ l and ρ v are the liquid and vapour density, cp1 is the specific heat of liquid (J/kg/° (J/kg/°C) σ is the surface tension (N/m) T e is the excess temperature of the boiling surface, T w - T sat  sat , (K) Pr1 is the liquid Prandtl number

 

Csf and n are the constants and depend on the liquid l iquid and heating surface combination for for boiling operation, for example,

 All the properties properties are to be evaluated evaluated at film temperature.

6.2.2 Maximum heat flux

Maximum heat flux corresponding corresponding to the point S in the fig.6.2 can be found by Leinhard correlation, (6.3)

The notations are same as for eq.6.2. 6.2.3 Film boili boiling ng

(6.4)

where, k v is the thermal conductivity of the vapour, µ v is the viscosity of the vapour, d   is the characteristic length (tube diameter or height of the vertical plate), other notations are same as for eq. eq. 6.2 6.2.. If the surface temperature is high enough to consider the contribution of radiative heat transfer, the total heat transfer coefficient coefficient may be calculation by, (6.5)

where, hr is the radiative heat transfer coefficient and and is given in eq.6.4.

 

Upto this section, we have discussed about the boiling phenomenon where the liquid phase changes to vapour phase. In the subsequent sections, we will study the opposite phenomena of  boiling that is condensatio condensation, n, where the vapour phase changes to the liquid phase. 6.3

Heat

transfer

during

condensation

Condensation of vapours on the surfaces cooler than the condensing temperature of the vapour is an important phenomenon in chemical process industries like boiling phenomenon. It is quite clear that in condensation the phase changes from vapour to liquid. Consider a vertical flat plate which is exposed to a condensable vapour. vapour. If the temperature of the plate is below the saturat saturation ion temperature temperatu re of the vapour, condensate will form on the surface and flows down down the plate due to gravity. It is to be noted noted that a liquid at its boiling point is a saturated saturated liquid and the vapour in equilibrium equilibriu m with the saturated saturated liquid is is saturated saturated vapour. vapour. A liquid or vapour vapour above the satura saturation tion temperature is called superheated. superheated. If the non-condensable non-condensable gases will present present in the vapour the rate of condensatio condensation n of the vapour will reduce significantly. Condensation may be of two types, film condensation and dropwise condensation. If the liquid (condensate) wets the surface, a smooth film is formed and the process is called film type condensation. In this process, the surface is blocked by the film, which grows in thickness as it moves down the plate. A temperature gradient exists in the film and the film represents thermal resistance in the heat transfer. The latent heat is transfe transferred rred through the wall to the cooling fluid on the other side of the wall. However, if the liquid does not wet the system, drops are formed on the surface in some random fashion. This process is called dropwise condensation. Some of  the surface will always be free from the condensate drops (for a reasonable time period). period). Now, with the help of the above discussion one can easily understand that the condensate film offers significant heat transfer resistance as compared to dropwise condensation. In dropwise condensation the surface is not fully covered by the liquid and exposed to the vapour for the condensation. Therefore, the heat transfer coefficient will be higher for dropwise condensation. Thus the dropwise condensation is preferred over the film condensation. However, the dropwise condensation is not practically easy to achieve. We have to put some coating on the surface or we have to add some additive to the vapour to have dropwise condensation. Practically, these techniques for dropwise condensation are not easy for the sustained dropwise condensation. Because of these reasons, in many instances we assume film condensation because the film condensation condensatio n sustained on the surface and it is comparatively easy to quantify and analyse.

6.4 6. 4 Fi Film lm co cond nden ensa sati tion on on a ve vert rtic ical al fl flat at pl plat ate e Figure 6.4 shows a vertical wall very long in z-direction. The wall is exposed to a condensable vapour. The condensate film is assumed to be fully developed laminar flow with zero interfacial shear and constant c onstant liquid properties. properties. It is also assumed assumed that the vapour is saturated and the heat

 

transfer through the condensate film occurs by condensation only only and the temperature profile is assumed to be linear.

Fig. 6.4: Condensation Condensation of film in laminar flow The wall tempera temperature ture is maintained at temperatur temperature e T w  vapour temperature temperature at the edge of  w and the vapour the film is the saturation temperature T v v.  The condensate film thickness is represented by δ x x,  a function of  x . A fluid element of thickness dx  was   was assumed with a unit width in the z-direction. The force balance on the element provides, F1 = F2 - F3

where, shear force

is the viscosity of the condensate (liquid).

In the

subsequent sections sections of this module, the subscripts l  and  and v  will  will represent represent liquid and vapour phase. Gravity force, F 2 = ρ l lg ; and   (δ  x - y)dx  Buoyancy force, F 3 = ρ v vg   (δ  x - y)dx 

Thus,

 

On integrating for for the following boundary boundary condition, condition, u = 0 at y = 0; no slip condition. (6.6)

Equation 6.6 shows the velocity profile in the condensate falling film. The correspo c orresponding nding mass flow rate of the condensate for dy thickness and and unit width of the film,

(6.7)

where dy  is   is the length of the volume element at y distance. The rate of condensation for dx .1 .1 (over element surface) area exposed to the vapour can be obtained from the rate of heat transfer trans fer through this area.

The rate of heat transfer The thermal conductivity c onductivity of of the liquid liq uid is represented by k l l.  The above rate of heat transfer is due to the latent l atent heat of condensation of the vapour. Thus, (6.8)

The specific spe cific latent heat of condensation is represented by λ. On solving eqs.6.7 and 6.8, for boundary layer conditions (x = 0; δx = 0) (6.9)

The eq. 6.9 gives the local condensate film thickness at any location x. If h is the film heat transfer coefficient coefficient for the condensate condensate film, film, heat flux through through the film at any location is,

 

(6.10a)

The local Nusselt Nusselt number will be,

We can also calculate the average heat transfer coefficient along the length of the surface, (6.10b)

In eq eq.. 6. 6.10 10,, th the e li liqu quid id pr prop oper erti ties es ca can n be ta take ken n at th the e me mean an fi film lm te temp mper erat atur ure e

equation 6.10 is applicable for for Pr > 0.5 and

The Th e

≤   1.0

It can also be understood that at any location on the plate the liquid film temperature changes fromT v v to T w  w.   It indicates that apart from latent heat some amount of sensible heat will also be removed. remove d. Thus Thus,, to take thi thiss into accou account nt and to furth further er improve the accur accuracy acy of  equation (eq. 6.10), a modified latent

heat term

Nusselt‟s

can be used in place

of λ of  λ. The term Ja is called the Jacob number as is defined by eq. 6.11. All the properties properties are to   be evaluated at film temperature. temperature. (6.11)

In the previous discussion we have not discussed about the ripples or turbulent condition of the condensate film as it grows while coming down from the vertical wall. The previous discussion was applicable only when the flow in the condensate film was 1-D and the velocity profile was half parabolic all along the length of the wall. However, if the rate of condensation is high or the height of the condensing wall is more, the thicknes thicknesss of the condensat condensate e film neither remains small nor the flow flow remains laminar. The nature of the flow is determined by the film Reynolds number (Re f ). The local average liquid velocity in the film can be obtained obtained by eq. 6.6.

 

Now, the Ref can be calculated by, (6.12)

where D is th where the e hy hydra draul ulic ic dia diamet meter er of th the e con conde dens nsate ate fi film. lm. The hyd hydrau raulic lic dia diamet meter er can be calculated by the flow area (δx.1) and wetted perimeter perimeter (unit breadth, thus 1). It has been found that, if  Case 1: 1: Re  Ref ≤  30; the film remains laminar and the free surface of the film remains wave free. Case 2:  30 < Ref  < 1600; the film remains laminar but the waves and ripples appear on the surface. Case Ca se 3:   Ref  ≥ 1600; the film becomes turbulent and surface becomes wavy wavy.. The cor corre resp spond ondin ing g av aver erage age hea heatt tr tran ansfe sferr co coeff effici icient ent can be cal calcula culated ted by th the e fo follo llowi wing ng correlation, : for for Case Case 1

(It is same as eq. 6.10 if Ref is taken at the bottom of the wall.) : for for Case Case 2

: for for Case Case 3

The Nusselt number in case-1 is defined as Modified Nusselt number or condensation number (Co ). ). The above relations may also be used for condensation inside or outside of a vertical tube if the tube diameter is very large in comparison to condensate film thickness. Moreover, the relations are valid for the tilted surfaces also. If the surface make an angle  “θ ”  ”  from from the vertical plane the  “g”  will  will be replaced by “ by “g.cosθ”  in  in the above equations

Illustrati Illust ration on 6.1 Saturated steam at 70.14 kPa is condensing on a vertical tube 0.5 m long having an outer diameter of 2.5 cm and a surface temperature of 80oC. Calculate the average heat-transfer coefficient.

 

Solution Solut ion 6.1 It

is

a

problem

of

condensation

on

a

vertical

plate,

thus

eq.6.10b

can

be

used, where, different liquid and steam properties are evaluate at average film temperature,

Using steam table, the temperature of the steam corresponding to 70.14kPa pressure is 90oC. The average film temperature will then be the average of 80   oC and 90   oC and it comes out to be 85  oC, Using given data the different properties can be found using steam table and other relevant tables given in the standard literature. literature. The data is tabulates below at 85oC,

On putting the above values in the above equation, hav = 1205.2 W/m2 °C 6.5 Condensation Condensation for horizontal tube

 

6.5. 6. 5.1 1 Con onde dens nsat atio ion n ou outs tsid ide e ho hori rizo zont ntal al tu tube be or ba bank nk of tu tube be   This type of condensation condensation is very common especially for for the shell and tube heat exchanger. In case of condensation condensation outside the vertical array of horizontal horizontal tubes the condensate condensate flows flows as a film along the cylindrical surface surface or it may drop down. In case of another another tube below, below, the condensate condensate film flows down down from the bottom edge of the upper tube to the upper edge of the bottom bottom tube. As it goes on to the lower tubes, the thickness of the condensate film increases. Some of the correlations are given below.

6.5.1.1 6.5. 1.1 Co Cond nden ensa sati tion on on a sin single gle ho horiz rizon onta tall tube  tube  (6.13)

6.5.1.2 Condensation on a vertical tube of N horizontal tubes  (6.14)

6.5.1.3

Condensation

inside

a

horizontal

tube  

Figure 6.5 shows the physical picture of the condensation inside a horizontal tube (like an open channel flow).

Fig. 6.5: Film condensation condensation inside a horizontal horizontal tube Case 1: The length is small or the rate of condensation condensation is low. This situation will have small thickness of the flowing condensate layer at the bottom of the tube and the following coefficient can be used,

 

(6.15)

where,

 

and the vapour Reynolds number (Rev) should be less than 35,000.

The Rev is calculated based on inlet condition of vapour and inside diameter of tube. Case 2: The length is high or the rate of condensatio condensation n is high. In this the following relation can be used. (6.16)

where,

(Condition: Rel >5000, Rev > 20,000) where, Gl and Gv are the liquid and vapour mass velocities calculated on the basis of the crosssection 6.6

of the tu t ube. Correlations

for

6.6.1

packed

and

Packed

fluidized

bed bed

The heat transfer correlation correlation for gas flow through through a packed bed is given as, (6.17)

Conditions to use eq.6.12 are,

where,

is the Stanton number number..

 

is the particle Reynolds number effective diameter of of a particle d p = Diameter or the effective velocity. It is the velocity based on the cross-section of the bed). v 0 = Superficial fluid velocity. ∈

: Bed porosity or void fraction

∈

: 0.3

 0.5

→

Theoretically:

∈ =

0.69 for uniform shape = 0.71 bed of cubes

= 0.79 bed of cylinder

Colburn Colbur n facto factorr 6.6.2 Fluidized Fluidized bed The heat transfer coefficient to or from particles in a fluidized bed can be estimated with the help of following correlation, correlation, (6.18)

where, v 0 is the superficial velocity.

Frequently Asked Questions and Problems Problems for Practice (Module 6) Q.1 Q.2

Whatt is Leid Wha Leidenf enfro rost st phe phenom nomen enon? on?  

How the presence of non-conden non-condensable sable gases affect the condensation rate of

Q.3

vapour? What is the diffe difference rence betwe between en drop drop-wis -wise e and film conden condensat sation? ion?

Q.4

Discusss the pheno Discus phenomena mena of conde condensa nsation tion on a verti vertical cal and a horizo horizontal ntal plate.

Q.5

 

Saturated vapour of methanol methanol condenses on a vertical plate at 1 atm. The vertical plate is maintained at 55 oC by cooling water at at the other other

side.

 

Calculate

the

following,

(a)Length of the plate over which the condensate film remains laminar. (b) What is the thickness of the film at the end of the laminar region? (c)Determine the average heat transfer transfer coefficient and the rate of condensatio condensation n in

the

laminar

r egion. re

(c) What is the average heat transfer coefficient coefficient for for the entire plate? Q.6.   Saturated steam flows flows in a horizontal horizontal tube having having 5 cm diameter and 15 cm of length. Calculate the condensation for a tube wall temperature of 96   oC. Q.7.   An uninsu uninsulated lated water pipe carrying water at 2   oC passes through hot and humid area where the temperature is 32   oC and the relative humidity is 75%. Estimate the condensate if the pipe is 50 mm in diameter and the exposed length is 75 cm. It may be assumed that the pipe is exposed to saturated vapour at the partial pressure of the water vapour in the air. Q.8.   Compare the heat transfer transfer coefficients for boiling water and condensing condensing steam on a horizontal tube for normal atmospheric conditions. conditions.

Module Modu le 7: Radia Radiation tion heat transfer transfer

In the previous chapters it has been observed that the heat transfer studies were based on the fact that the temperature of a body, a portion of a body, which is hotter than its surroundings, tends to decrease with time. The decrease in temperature indicates a flow of energy from the body. In all the previous chapters, limitation was that a physical medium was necessary for the transport of the energy from the high temperature source to the low temperature sink. The heat trans tra nspor portt was re relat lated ed to co condu nduct ctio ion n and con conve vect ction ion and the rat rate e of he heat at tr tran ansp spor ortt wa wass proportional to the temperature difference between the source and thesink. the sink. Now, if we observe the heat transfer from the Sun to the earth atmosphere, we can understand that there is no medium exists between the source (the Sun) and the sink (earth atmosphere). Howeve How ever, r, sti stillll the hea heatt tr tran ansf sfer er tak takes es pla place, ce, wh which ich is ent entire irely ly a dif diffe feren rentt ene energy rgy tra trans nsfe ferr mechanism takes place and it is called thermal radiation. Thermal radiation is referred when a body is heated or exhibits the loss of energy by radiation. However, Howev er, more gene general ral form  “radiat  “radiation ion energy”   is used used to cover cover all the ot othe herr for forms ms.. Th The e emission of other form of radiant energy may be caused when a body is excited by oscillating electrica elect ricall curr current ent,, elec electro tronic nic bomb bombard ardment ment,, chem chemical ical reac reaction tion etc. More Moreover over,, when rad radiati iation on energy strikes a body and is absorbed, it may manifest itself in the form of thermal internal energy, a chemical reaction, an electromotive force, etc. depending on the nature of the incident radiation and the substance of which which the body is composed. In thi thiss chap chapter, ter, we will concentrat concentrate e on ther thermal mal radia radiation tion (emission (emission or absorptio absorption) n) that on radiation produced by or while produces thermal excitation excitation of a body.

 

There The re ar are e man many y th theo eorie riess ava availa ilable ble in lit litera eratu ture re wh which ich ex expla plain inss th the e tr trans anspor portt of en ener ergy gy by radiation. However, a dual theory is generally accepted which enables to explain the radiant ener en ergy gy in th the e ch char arac acte teri risa sati tion on of a wa wave ve mo moti tion on (e (ele lect ctro roma magn gnet etic ic wa wave ve mo moti tion on)) an and d discontinuouss emission (discrete packets or quanta of energy). discontinuou  An electroma electromagnetic gnetic wave propagat propagates es at the speed of light (3 (3× ×108 m/s). It is characterised by its wavelength λ or its frequency ν related by

c = λv 

 

(7.1)

Emission of radiation is not continuous, but but occurs only in the form of discrete quanta. Each quantum quant um has energ energy y

E = hv 

 

(7.2)

where, = 6.6246× 6.6246×10-34 J.s, is known as Planck‟s Planck‟s constant.  constant. Table 7.1 shows the electromagnetic radiation covering the entire spectrum of wavelength Table 7.1: Electromagnetic Electromagnetic radiation radiation for entire spectrum spectrum of wavelength Type

Band of wavelength (µm)

Cosmic Cos mic ray rayss

 

upto 4×10⁻7

Gamma Gamm a ray rayss

 

4×10⁻7 to 1.4× 1.4×10⁻4

 

X-rays

1×10⁻5 to 2× 2×10⁻2  

Ultraviolet rays

 Visible light Infrared Infrar ed rays Thermal radiation

5×10⁻3 to 3.9× 3.9×10⁻1 ⁻1

 

⁻1

3.9× 3.9 ×10 to 7.8× 7.8×10  

7.8× 7.8 ×10⁻1 to 1×103  

Microwave, radar, radio waves

1×10⁻1 to 1× 1×102 1×103 to 5× 5×1010

It is to be noted that the above band is in approximate values and do not have any sharp boundary. 7.1

Basic

definition

pertaining

to

radiation

Before we further study about the radiation it would be better to get familiarised with the basic terminology and properties of the radiant energy and how to characterise it.

 

 As observed in the table 7.1 that the thermal radiation is defined between wavelength of about 1×10-1 and 1× 1×102 μm μm of  of the electromagnetic radiation. If the thermal radiation is emitted by a surface, which is divided into its spectrum over the wavelength band, it would be found that the radiation is not equally distributed over all wavelength. Similarly, radiation incident on a system, refl re flec ecte ted d by a sy syst stem em,, ab abso sorb rbed ed by a sy syst stem em,, et etc. c. may be wa wave vele leng ngth th de depe pend nden entt. Th The e dependence on the wavelength is generally different from case to case, system to system, etc. The wavelength dependency of any radiative quantity or surface property will be referred to as a spectral spect ral depen dependenc dency. y. The rad radiati iation on quan quantit tity y may be monoc monochrom hromatic atic (ap (applica plicable ble at a sing single le wavelength) or total (applicable at entire thermal radiation spectrum). It is to be noted that radiation quantity may be dependent on the direction and wavelength both but we will not consider any directional dependency. This chapter will not consider directional effect and the emissiv emis sive e pow power er wil willl al alwa ways ys us used ed to be (h (hemis emisphe pheric rical) al) sum summed med ove overal ralll dir direct ection ion in th the e hemisphere above the surface. 7.1.1

Emissive

power

It is the emitted thermal radiation leaving a system per unit time, per unit area of surface. The total emissive power of a surface is all the emitted energy, summed over all the direction and all wavelengths, and is usually denoted as E . The total emissive power is found to be dependent upon the temperature of the emitting surface, the subsystem which this system is composed, and the nature of the surface structure or texture. texture. The monochromatic emissive power E λ, is defined as the rate, per unit area, at which the surface emitss the emit therma rmall ra radia diatio tion n at a par parti ticul cular ar wa wavel veleng ength th λ. Thu Thuss th the e to total tal an and d mon monoch ochrom romati aticc hemispherical emissive emissive power are related by (7.3)

and the functional dependency dep endency of E λ on λ must be known to evaluate E . 7.1.2

Radiosity

It is the term used to indicate all the radiation leaving leaving a surface, per unit time and unit area. (7.4)

where, J  and  and J λ are

th e

total

and

monochromatic

radiosity.

The rad radiosi iosity ty inclu includes des reflected reflected ener energy gy as well as orig original inal emission emission wher whereas eas emissi emissive ve pow power er consists of only original emission leaving the system. The emissive power does not include any energy leaving a system that is the result of the reflection of any incident radiation.

 

7.1.3 Irra Irradia diation tion It is the term used to denote the rate, per unit area, at which thermal radiation is incident upon a sur surfa face ce (f (from rom all the dir direct ection ions) s).. The irr irradi adiati ative ve inc incide ident nt up upon on a su surfa rface ce is th the e res result ult of  emission and reflection from other surfaces and and may thus be spectrally dependent. (7.5)

where, G  and  and G λ are

the

total

and

monochromatic

irradiation.

Reflection from a surface may be of two types specular or diffusive as shown in fig.7.1.

Fig. 7.1: (a) Specular, and (b) diffusive radiation Thus,

 ρG  J = E +  ρG 

 

(7.6)

7.1.4 Absorptivity, reflectivity, and transmitting The emissive power, radiosity, and irradiation of a surface are inter-related by the reflective, absorptive,

and

transmissive

properties

of

the

system.

When thermal radiation is incident on a surface, a part of the radiation may be reflected by the surface, a part may be absorbed by the surface and a part may be transmitted through the surface as shown in fig.7.2. These fractions of reflected, absorbed, and transmitted energy are interpreted as system properties called reflectivity, absorptivity, and transmissivity, respectively respectively..

 

Fig. 7.2: Reflection, absorption absorption and transmitted transmitted energy Thus using energy conservation, conservation, (7.7)

(7.7)

wher wh ere, e,

are ar e to tota tall re refl flec ecti tivi vity ty,, to tota tall ab abso sorp rpti tivi vity ty,, an and d to tota tall tr tran ansm smis issi sivi vity ty..

The Th e

subscript λ indicates the monochromatic property. In gen genera erall th the e mon monoch ochrom romati aticc and to tota tall su surf rface ace pro proper pertie tiess are de depen penden dentt on the sys system tem composition, its roughness, roughness, and on its temperature temperature.. Monochromatic properties are dependent on the wavelength of the incident radiation, and the total properties are dependent on the spectral distributi d istribution on of the incident energy.

Most gases have high transmissivity, i.e. (like air at atmospheric pressure). However, some other gases (water vapour, CO 2  etc.) may be highly absorptive to thermal radiation, at least l east at certain wavelength. Most solids encountered in engineering practice are opaque to thermal radiation

Thus

for thermally opaque solid surfaces,

 ρ + α = 1

 

(7.6)

 Another important property property of the surface of a substance is its ability to emit radiation radiation.. Emission and radi radiatio ation n have different different conce concept. pt. Refl Reflecti ection on may occur only when the surf surface ace rece receives ives radiation whereas emission always always occurs if the temperature of the surface is above the absolute zero. Emissivity of the surface is a measure of how good it is an emitter.

 

7.2 Blac Blackbo kbody dy rad radiat iatio ion n In order to evaluate the radiation characteristics and properties of a real surface it is useful to define an ideal surface such as the perfect blackbody. The perfect blackbody is defined as one which wh ich abs absor orbs bs all inc incide ident nt ra radia diatio tion n re regar gardle dless ss of the sp spect ectral ral dis distr tribu ibutio tion n or dir direct ection ional al characteristic of the incident radiation. radiation.

 A blackbody is black because it does not reflect any radiation radiation.. The only radiation leaving a blackbod blac kbody y surf surface ace is ori original ginal emission emission sin since ce a blac blackbod kbody y abso absorbs rbs all inci incident dent radiation. radiation. The emissive power of a blackbody is represented represented by , and depends on the surface temperature only.

Fig. 7.3: Example of of a near perfect blackbody It is possible to produce p roduce a near perfect blackbody as shown in fig.7.3. Figure 7.2 shows a cavity with a small opening. The body is at isothermal state, where a ray of  incident radiation enters through the opening will undergo a number of internal reflections. A  portion of the radiation absorbed at each internal reflection and a very little of the incident beam ever find the way out through the small hole. Thus, the radiation found to be evacuating from the hole will appear to that coming from a nearly perfect blackbody. blackbody. 7.2.1

Planck’s  

law

 A surface emits radiatio radiation n of different wavelengt wavelengths hs at a given temperature (theoretically (theoretically zero to infi in fini nite te wa wave vele leng ngth ths) s).. At a fi fixe xed d wa wave vele leng ngth th,, th the e su surf rfac ace e ra radi diat ates es mo more re en ener ergy gy as the temperature increases. Monochromatic Monochromatic emissive power of a blackbody is given by eq.7.10.

 

(7.7)

where,

Planck‟s constant  constant h = 6.6256 X 10-34 JS; Planck‟s c = 3 X 108 m/s; speed of light T = absolute temperature of the blackbody λ = wavelenght of of the monochromatic radiation radiation emitted k = Boltzmann constant constant

Equation 7.10 is known as Planck‟s law. Figure 7.4 shows the represent representative ative plot for Planck‟s distribution.

Fig. 7.4: Representati Representative ve plot for Planck’s distribution

7.2.2

Wien’ s

law

Figure 7.4 shows that as the temperature increases the peaks of the curve also increases and it shif sh iftt to towa ward rdss th the e sh shor orte terr wa wave vele leng ngth th.. It ca can n be ea easi sily ly fo foun und d ou outt th that at th the e wa wave vele leng ngth th corresponding to the peak of the plot ( λmax ) is inversely proportional to the temperature of the blackbody (Wein‟s (Wein‟s law)  law) as shown in eq. 7.11.

λmax T = 2898

(7.11)

 

Now with the Wien‟s Wien‟s law  law or Wien‟s Wien‟s displacement  displacement law, it can be understood if we heat a body, initially the emitted radiation does not have any colour. As the temperature rises the λ  of the radiation reach the visible spectrum and we can able to see the red colour being height λ (fo (forr red colour). Further Further increase in temperature temperature shows the white colour indicating indicating all the colours in the light. 7.2.3 7.2. 3 The Stef Stefan an-Bo -Boltz ltzma mann nn law for blac blackb kbod ody y Josef Stefan based on experimental facts suggested that the total emissive power of a blackbody is pro propor portio tional nal to th the e fo fourt urth h pow power er of the abs absolu olute te tem temper peratu ature. re. Lat Later er,, Lud Ludwig wig Bol Boltz tzman mann n derive der ived d the sa same me us using ing cla class ssica icall th therm ermod odyna ynami mics cs.. Thu Thuss th the e eq. 7.1 7.12 2 is kn know own n as Ste Stefan fan-Boltzmann law,

E b = σT 4 

(7.12)

where, E b is the emissive power of a blackbody, T  is  is absolute temperature, and σ  (=   (= 5.67 X 108 2 4 W/m /K  ) is the Stefan-Boltzm Stefan-Boltzmann ann constant. The Stefan-Boltzmann law for the emissive power gives the total energy emitted by a blackbody defined by eq.7.3. 7.2.4 7.2. 4 Spe Specia ciall cha charac racter terist istic ic of bla blackb ckbod ody y ra radi diati ation on It has been shown that the irradiation field in an isothermal cavity is equal to E b b.  Moreover, the irradiation was same for all planes of any orientation within the cavity. It may then be shown that the intensity of the blackbody radiation, I b b,  is uniform. Thus, blackbody radiation is defined as,

πI b b  E b =  = πI 

whe herre,

is the total intensity of the

 

(7.13)

radiation and is calle rad led d the spe pecctral

radiation intensity of the blackbody. 7.2.5

Kirchhoff’s  

law

Consider an enclosure as shown in fig.7.2 and a body is placed inside the enclosure. The radiant heat flux (q) is incident onto the body and allowed to come into temperature equilibrium. The rate of energy absorbed at equilibrium by the body must be equal to the energy emitted.

 

where, E is the emissive power of the body,

is absorptiv absorptivity ity of the of the body at equilibrium

temperature, and A is the area of the body. Now consider the body is replaced by a blackbody i.e. E

→

E b and

= 1, the equation7.14

becomes

E b b= q   

 

(7.15)

Dividing eq. 7.14 by eq.7.15, (7.16)

 At this point we may define define emissivity, which which is a measure of how good the body is an emitter as compared to blackbody. Thus the emissivity can be written as the ratio of the emissive power to that of a blackbody, blackbody, (7.17)

On comparing eq.7.16 and eq.7.17, we get g et (7.18)

Equation 7.18 is the Kirchhoff‟s   law, which states that the emissivity of a body which is in thermal equilibrium with its surrounding is equal to its absorptivity of the body. It should be noted not ed tha thatt the so sour urce ce te tempe mperat rature ure is equ equal al to the te tempe mperat rature ure of the irr irradi adiate ated d su surfa rface ce.. However, in practical purposes it is assumed that emissivity and absorptivity of a system are equal equ al eve even n if it is not in the therma rmall equ equili ilibri brium um wi with th th the e su surro rroun undin ding. g. Th The e rea reaso son n be bein ing g the absorptivity of most real surfaces is relatively insensitive to temperature and wavelength. This particular assumption leads to the concept of grey body. The emissivity is considered to be independent of the wavelength of radiation for grey body. b ody. 7.3

Grey

body

If grey body is defined d efined as a substance whose monochromatic emissivity and absorptivity are independent

of

wavelength.

blackbody is shown in the table 7.2.

A

comparative

study

of

grey

body

and

 

Table-7.2: Comparison Comparison of grey and blackbody Blackbody

 

Grey Grey body

Ideal body

Ideal body

Emissivity

independent

of

wavelength

Emissivity

is

independent

of

wavelength

Absor ptivity

s

independent

of

Absorptivity  (α) is

wavelength

wavelength

ε =1

ε