Heat Transfer

HEAT TRANSFER There are three modes of heat transfer: conduction, convection, and radiation.

Conduction Through a Cylindrical Wall

Conduction Fourier’s Law of Conduction Qo =- kA dT , where dx oQ = rate of heat transfer (W) k = the thermal conductivity [W/(m•K)] A = the surface area perpendicular to direction of heat transfer (m2)

T2

r2 Cylinder (Length = L)

2rkL ^T1 - T2h Qo = r ln d r2 n 1

Qo = hA _Tw - T3i, where

h = the convection heat transfer coefficient of the fluid [W/(m2•K)] A = the convection surface area (m2) Tw = the wall surface temperature (K) T∞ = the bulk fluid temperature (K) Radiation The radiation emitted by a body is given by Qo = fvAT 4, where ε = the emissivity of the body σ = the Stefan-Boltzmann constant = 5.67 × 10-8 W/(m2•K4) A = the body surface area (m2) T = the absolute temperature (K)

Critical Insulation Radius k rcr = insulation h3

Thermal Resistance (R)

k

k insulation

Qo = DT Rtotal

Composite Plane Wall

T2

Fluid 1 T∞1 T1 h1

Q

- kA ^T2 - T1h Qo = , where L

kA

Fluid 2 T∞2 h2

kB T2

Q

A = wall surface area normal to heat flow (m2) L = wall thickness (m) T1 = temperature of one surface of the wall (K) T2 = temperature of the other surface of the wall (K)

1 h1 A Q

HEAT TRANSFER

r insulation

Plane Wall Conduction Resistance (K/W): R = L , where kA L = wall thickness r ln d r2 n 1 Cylindrical Wall Conduction Resistance (K/W): R = , 2rkL where L = cylinder length Convection Resistance (K/W) : R = 1 hA

Conduction Through a Plane Wall

84

h∞

Resistances in series are added: Rtotal = RR, where

CONDUCTION

L

r1

k

Convection Newton’s Law of Cooling

T1

Q

T1

BASIC HEAT TRANSFER RATE EQUATIONS

T∞1

T1

LA

LB

LA kA A

LB kB A T2

T3 1 h2 A T3

T∞2

To evaluate Surface or Intermediate Temperatures: T2 - T3 T - T2 Qo = 1 RA = RB

Transient Conduction Using the Lumped Capacitance Method The lumped capacitance method is valid if

Steady Conduction with Internal Energy Generation The equation for one-dimensional steady conduction is o d 2T + Qgen = 0, where 2 k dx Qo gen = the heat generation rate per unit volume (W/m3) For a Plane Wall

Biot number, Bi = hV % 1, where kAs h = the convection heat transfer coefficient of the fluid [W/(m2•K)] V = the volume of the body (m3) k = thermal conductivity of the body [W/(m•K)] As = the surface area of the body (m2) Fluid h, T∞ Body

T(x) Ts1

Ts2 k

Q gen Q"1

Q"2 x −L

T ^ xh =

As

0

L

2 Qo gen L2 T T T T d1 - x 2 n + c s2 - s1 m b x l + c s1 - s2 m L 2 2 2k L

Qo 1" + Qo 2" = 2Qo gen L, where Qo " = the rate of heat transfer per area (heat flux) (W/m2) Qo 1" = k b dT l and Qo 2" = k b dT l dx - L dx L For a Long Circular Cylinder

Constant Fluid Temperature If the temperature may be considered uniform within the body at any time, the heat transfer rate at the body surface is given by Qo = hAs ^T - T3h =- tV ^cP h b dT l, where dt T T∞ ρ cP t

= the body temperature (K) = the fluid temperature (K) = the density of the body (kg/m3) = the heat capacity of the body [J/(kg•K)] = time (s)

The temperature variation of the body with time is T - T3 = _Ti - T3i e-bt, where

hAs b= tVcP

Ts

ρ, V, c P, T

where b = 1x and x = time constant ^ sh

The total heat transferred (Qtotal) up to time t is Qtotal = tVcP _Ti - T i, where

Q gen

Ti = initial body temperature (K)

r0 Q'

o 1 d b r dT l + Qgen = 0 r dr dr k T ]r g =

2 Qo genr02 1 - r 2 p + Ts f 4k r0

Qo l = rr02Qo gen, where Qo l = the heat transfer rate from the cylinder per unit length of the cylinder (W/m) HEAT TRANSFER

85

Variable Fluid Temperature If the ambient fluid temperature varies periodically according to the equation T3 = T3, mean + 1 _T3, max - T3, min i cos ^~t h 2 The temperature of the body, after initial transients have died away, is b ; 1 _T3, max - T3, min iE T= 2 cos =~t - tan- 1 c ~ mG + T3, mean b ~2 + b2 Fins For a straight fin with uniform cross section (assuming negligible heat transfer from tip), Qo = hPkAc _Tb - T3i tanh _ mLci, where

h = the convection heat transfer coefficient of the fluid [W/(m2•K)] P = perimeter of exposed fin cross section (m) k = fin thermal conductivity [W/(m•K)] Ac = fin cross-sectional area (m2) Tb = temperature at base of fin (K) T∞ = fluid temperature (K) hP kAc

m=

Terms D = diameter (m) h = average convection heat transfer coefficient of the fluid [W/(m2•K)] L = length (m) Nu = average Nusselt number c n Pr = Prandtl number = P k um = mean velocity of fluid (m/s) u∞ = free stream velocity of fluid (m/s) µ = dynamic viscosity of fluid [kg/(s•m)] ρ = density of fluid (kg/m3) External Flow In all cases, evaluate fluid properties at average temperature between that of the body and that of the flowing fluid. Flat Plate of Length L in Parallel Flow ReL =

tu3 L n

NuL = hL = 0.6640 Re1L 2 Pr1 k

3

NuL = hL = 0.0366 Re0L.8 Pr1 k

3

_ReL < 105i

_ReL > 105i

Cylinder of Diameter D in Cross Flow

A Lc = L + c , corrected length of fin (m) P

ReD =

Rectangular Fin

tu3 D n

NuD = hD = C Re nD Pr1 3, where k

T∞ , h

P = 2w + 2t Ac = w t t

Tb

CONVECTION

w

L

C 0.989 0.911 0.683 0.193 0.0266

ReD 1–4 4 – 40 40 – 4,000 4,000 – 40,000 40,000 – 250,000

n 0.330 0.385 0.466 0.618 0.805

Flow Over a Sphere of Diameter, D Pin Fin T∞ , h

P= π D D

Tb

86

HEAT TRANSFER

L

Ac =

πD 2 4

NuD = hD = 2.0 + 0.60 Re1D 2 Pr1 3, k ^1 < ReD< 70, 000; 0.6 < Pr < 400h

Internal Flow

ReD =

tumD n

Laminar Flow in Circular Tubes For laminar flow (ReD < 2300), fully developed conditions NuD = 4.36

(uniform heat flux)

NuD = 3.66

(constant surface temperature)

For laminar flow (ReD < 2300), combined entry length with constant surface temperature NuD = 1.86 f

1 3

ReDPr L p D

d

nb n ns

0.25

tl2 gh fg D3 H NuD = hD = 0.729 > k nlkl _Tsat - Tsi

0.14

, where

L = length of tube (m) D = tube diameter (m) µb = dynamic viscosity of fluid [kg/(s•m)] at bulk temperature of fluid, Tb µs = dynamic viscosity of fluid [kg/(s•m)] at inside surface temperature of the tube, Ts Turbulent Flow in Circular Tubes For turbulent flow (ReD > 104, Pr > 0.7) for either uniform surface temperature or uniform heat flux condition, SiederTate equation offers good approximation:

Outside Horizontal Tubes

n NuD = 0.027 Re0D.8 Pr1 3 d nb n s

0.14

Non-Circular Ducts In place of the diameter, D, use the equivalent (hydraulic) diameter (DH) defined as DH = 4 # cross -sectional area wetted perimeter

Circular Annulus (Do > Di) In place of the diameter, D, use the equivalent (hydraulic) diameter (DH) defined as DH = Do - Di Liquid Metals (0.003 < Pr < 0.05) NuD = 6.3 + 0.0167 Re0D.85 Pr0.93 (uniform heat flux) NuD = 7.0 + 0.025 Re0D.8 Pr0.8 (constant wall temperature) Condensation of a Pure Vapor On a Vertical Surface 0.25

tl2 gh fg L3 H NuL = hL = 0.943 > k nlkl _Tsat - Tsi

, where

ρl = density of liquid phase of fluid (kg/m3) g = gravitational acceleration (9.81 m/s2) hfg = latent heat of vaporization [J/kg] L = length of surface [m] µl = dynamic viscosity of liquid phase of fluid [kg/(s•m)] kl = thermal conductivity of liquid phase of fluid [W/(m•K)] Tsat = saturation temperature of fluid [K] Ts = temperature of vertical surface [K] Note: Evaluate all liquid properties at the average temperature between the saturated temperature, Tsat, and the surface temperature, Ts.

, where

D = tube outside diameter (m) Note: Evaluate all liquid properties at the average temperature between the saturated temperature, Tsat, and the surface temperature, Ts. Natural (Free) Convection Vertical Flat Plate in Large Body of Stationary Fluid Equation also can apply to vertical cylinder of sufficiently large diameter in large body of stationary fluid. hr = C b k l RaLn, where L L = the length of the plate (cylinder) in the vertical direction gb _Ts - T3i L3 RaL = Rayleigh Number = Pr v2 Ts = surface temperature (K) T∞ = fluid temperature (K) β = coefficient of thermal expansion (1/K) 2 (For an ideal gas: b = with T in absolute temperature) Ts + T3 υ

= kinematic viscosity (m2/s) Range of RaL 104 – 109 109 – 1013

C 0.59 0.10

n 1/4 1/3

Long Horizontal Cylinder in Large Body of Stationary Fluid h = C b k l Ra nD, where D RaD =

gb _Ts - T3i D3 Pr v2

RaD 10 – 102 102 – 104 104 – 107 107 – 1012

C 1.02 0.850 0.480 0.125

–3

n 0.148 0.188 0.250 0.333

Heat Exchangers The rate of heat transfer in a heat exchanger is Qo = UAFDTlm, where A = any convenient reference area (m2) F = heat exchanger configuration correction factor (F = 1 if temperature change of one fluid is negligible) U = overall heat transfer coefficient based on area A and the log mean temperature difference [W/(m2•K)] ∆Tlm = log mean temperature difference (K) HEAT TRANSFER

87

Heat Exchangers (cont.) Overall Heat Transfer Coefficient for Concentric Tube and Shell-and-Tube Heat Exchangers D ln d o n R R fo Di fi 1 1 1 , where UA = hiAi + Ai + 2rkL + Ao + hoAo

Effectiveness-NTU Relations C Cr = min = heat capacity ratio Cmax For parallel flow concentric tube heat exchanger f=

2

1 - exp 8- NTU ^1 + Cr hB 1 + Cr

Ai = inside area of tubes (m ) Ao = outside area of tubes (m2) NTU =- ln 81 - f ^1 + Cr hB 1 + Cr Di = inside diameter of tubes (m) Do = outside diameter of tubes (m) For counterflow concentric tube heat exchanger hi = convection heat transfer coefficient for inside of tubes [W/(m2•K)] 1 - exp 8- NTU ^1 - Cr hB f= ho = convection heat transfer coefficient for outside of tubes 1 - Crexp 8- NTU ^1 - Cr hB [W/(m2•K)] f = NTU k = thermal conductivity of tube material [W/(m•K)] 1 + NTU Rfi = fouling factor for inside of tube [(m2•K)/W] NTU = 1 ln c f - 1 m Cr - 1 fCr - 1 Rfo = fouling factor for outside of tube [(m2•K)/W] NTU = f Log Mean Temperature Difference (LMTD) 1-f For counterflow in tubular heat exchangers

DTlm =

_THo - TCi i - _THi - TCoi

ln d

THo - TCi n THi - TCo

For parallel flow in tubular heat exchangers DTlm = ∆Tlm THi THo TCi TCo

_THo - TCoi - _THi - TCi i

T - TCo n ln d Ho THi - TCi

, where

= log mean temperature difference (K) = inlet temperature of the hot fluid (K) = outlet temperature of the hot fluid (K) = inlet temperature of the cold fluid (K) = outlet temperature of the cold fluid (K)

Heat Exchanger Effectiveness, ε Qo actual heat transfer rate f= o = maximum possible heat transfer rate Qmax f=

C _T - TCi i CH _THi - THoi or f = C Co Cmin _THi - TCi i Cmin _THi - TCi i

where C = mc o P = heat capacity rate (W/K) Cmin = smaller of CC or CH Number of Transfer Units (NTU) NTU = UA Cmin

88

HEAT TRANSFER

^Cr< 1h ^Cr = 1h ^Cr< 1h

^Cr = 1h

RADIATION Types of Bodies Any Body For any body, α + ρ + τ = 1 , where α = absorptivity (ratio of energy absorbed to incident energy) ρ = reflectivity (ratio of energy reflected to incident energy) τ = transmissivity (ratio of energy transmitted to incident energy) Opaque Body For an opaque body: α + ρ = 1 Gray Body A gray body is one for which α = ε, (0 < α < 1; 0 < ε < 1), where ε = the emissivity of the body

For a gray body: ε + ρ = 1



Real bodies are frequently approximated as gray bodies.

Black body A black body is defined as one which absorbs all energy incident upon it. It also emits radiation at the maximum rate for a body of a particular size at a particular temperature. For such a body

α=ε=1

Shape Factor (View Factor, Configuration Factor) Relations Reciprocity Relations AiFij = AjFji, where

One-Dimensional Geometry with Thin Low-Emissivity Shield Inserted between Two Parallel Plates Radiation Shield Q12

Ai = surface area (m2) of surface i Fij = shape factor (view factor, configuration factor); fraction of the radiation leaving surface i that is intercepted by surface j; 0 ≤ Fij ≤ 1

ε3, 1

Summation Rule for N Surfaces

A1 , T1, ε1

N

! Fij = 1

j=1

Qo 12 =

Qo 12 = the net heat transfer rate from the body (W) ε = the emissivity of the body σ = the Stefan-Boltzmann constant [σ = 5.67 × 10-8 W/(m2•K4)] A = the body surface area (m2) T1 = the absolute temperature [K] of the body surface T2 = the absolute temperature [K] of the surroundings

A1 , T1 , ε1

Qo 12 = A1F12 v `T14 - T24j

Generalized Cases

A1 , T1 , ε1

v `T14 - T24j 1 - f3, 1 1 - f3, 2 1 - f1 1 - f2 1 1 f1A1 + A1F13 + f3, 1A3 + f3, 2A3 + A3F32 + f2A2

Reradiating Surface Reradiating Surfaces are considered to be insulated or adiabatic _Qo R = 0i .

Net Energy Exchange by Radiation between Two Black Bodies The net energy exchange by radiation between two black bodies that see each other is given by

Net Energy Exchange by Radiation between Two DiffuseGray Surfaces that Form an Enclosure

A2 , T2 , ε2 A3 , T3

Net Energy Exchange by Radiation between Two Bodies Body Small Compared to its Surroundings Qo 12 = fvA `T14 - T24j, where

ε3, 2

Q12

AR , TR , εR

A2 , T2 , ε2

Qo 12 =

v `T14 - T24j 1 - f1 1 - f2 1 -1 + f A f1A1 + 2 2 A1F12 + =c 1 m + c 1 mG A1F1R A2F2R

A2 , T2 , ε2 Q12

Q12

A1 , T1 , ε1 A2 , T2 , ε2

Qo 12 =

v `T14 - T24j 1 - f1 1 - f2 1 f1A1 + A1F12 + f2A2

HEAT TRANSFER

89