Developing Empirical Nusselt Number Correlations For Single Phase Flow Through A Plate Heat Exchanger

 

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CHAPTER   1 1

 INTRODUCTION

Heatt exchang Hea exchangers ers are impor importan tant, t, and used used frequen frequently tly in the proces process, s, hea heatt and power, power, air air-conditioning and refrigeration, heat recovery, transportation and manufacturing industries. Such equipment is also used in electronics cooling and for environmental issues like thermal pollution, waste disposal disposal and sustainabl sustainablee development development.. Various Various types of heat exchangers exchangers exist. The study concerns plate heat exchangers (PHEs), which are one of the most common types in practice. Plate heat exchangers are widely used in dairy, pharmaceutical and paper/pulp industry as well as in HVAC applications. Flow of the substances to be heated and cooled takes place between alternating metal sheets allowing heat transfer between the fluids. Gaskets are placed between the plates to avoid mixing of the fluids. In the majority of the industrial applications, the plate heat exchanger is the design of choice because of its many advantages. Among these are: •

Superior thermal performance; plate heat exchangers have heat transfer coefficients as high as three to four times that of tubular types because of smaller hydraulic diameter. The turbulent conditions are achieved at much lower Reynolds number hence higher heat transfer coefficients.

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Compact design; the superior thermal performance of the plate heat exchanger and the space spa ce effici efficient ent design design of the plate arrangeme arrangement nt result resultss in a very very compact compact piece of equipment. Space requirements for the plate heat exchanger generally run 10% to 50% that of a shell and tube unit for the same amount of heat transfer. In addition, tube cleaning and replacing clearances are eliminated.

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Expa Ex panda ndabi bili lity ty an and d mult multip iple lex x ca capa pabi bili lity ty;; the the natur naturee of th thee pl plat atee heat heat ex exch chang anger er construction permits expansion of the unit should heat transfer requirements increase

 

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after installation. In addition, two or more heat exchangers can be housed in a single frame, thus reducing space requirements and capital costs.

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Ea Ease se of main mainte tenan nance; ce; the the co cons nstr truct uctio ion n of th thee he heat at ex excha chang nger er is su such ch th that at,, up upon on disassembly, all heat transfer areas are available for inspection and cleaning. Disassembly consists only of loosening a small number of tie bolts.

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Availability of a wide variety of corrosion resistant alloys; since the heat transfer area is constructed of thin plates, stainless steel or other high alloy construction is significantly less costly than for a shell and tube exchanger of similar material.

1.1 Need Statement

A number of analytical and experimental studies have been conducted to study the heat transfer and fluid flow characteristics of tubular heat exchangers. However a very limited work is found in open literature regarding the heat transfer through plate heat exchangers. Computerized design software has been developed by the manufacturers of plate heat exchangers but not a lot of data are available for research purposes about the design of these heat exchangers. Therefore, there is an urgent need for comprehensive and systematic research in this field.

1.2 Project Scope

In this project, to investigate the heat transfer characteristics and thermal performance of plate heat exchangers, a series of experimentation were performed. The test rig was used to conduct single phase experiments in order to develop Nusselt number correlation for the plate heat exchang exc hanger er with with a mixed mixed plate plate configu configurat ration ion.. The operati operating ng range range of flow flow rat rates es and fl fluid uid temperatures correspond to a maximum Reynolds number (Re) of 4500 and Prandtl number (Pr) in the range of 5.6 to 8. The project was divided into five different phases: 1. Literature review of Nusselt number correlations

 

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2. Writing the code for Modified Wilson Plot 3. Setting up the apparatus 4. Experimentation 5. Analysis of the result and conclusion 1.3 Some Important Definitions 1.3.1 Nusselt Number

 Nusselt number is equal to the dimensionless temperature gradient at the surface, and it essentially provides a measure of convective heat transfer.  

(1)

The Nusselt number is to the thermal boundary layer what the friction friction coefficient coefficient is to velocity  boundary layer. Equation (1) implies that for a given geometry, the Nusselt number must be some universal function of x of  x*, *, Reynolds number and Prandtl number. (2) where: = characteristic length (m) = thermal conductivity of the fluid (W/m.K) = convective heat transfer coefficient (W/m2.K) Selection of the characteristic length should be in the direction of growth (or thickness) of the  boundary layer. Some examples of characteristic length are: the outer diameter of a cylinder in (ext (e xter ernal nal)) cros crosss flow flow (per (perpen pendi dicul cular ar to the the cy cyli linde nderr ax axis is), ), th thee le lengt ngth h of a verti vertica call pl plat atee undergoing natural convection, or the diameter of a sphere. For complex shapes, the length may  be defined as the volume of o f the fluid body bod y divided by the surface area. The thermal conductivity of the fluid is typically (but not always) evaluated at the film temperature, which for engineering  purposes may be calculated as the mean-average of the bulk fluid temperature and wall surface

 

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temperature. For relations defined as a local Nusselt number, one should take the characteristic length to be the distance from the surface boundary to the local point of interest. However, to obtain an average Nusselt number, one must integrate said relation over the entire characteristic length.

1.3.2 Nusselt Number Correlation

Generally, for single phase heat transfer, Nu is represented by an empirical expression of the form:  

(3)

where C, m, and n are independent of the nature of fluid used. The last term in the expression accounts for the variable viscosity effect.

1.3.3 Prandtl Number

The Prandt Prandtll number number Pr is a dimens dimension ionles lesss number number approxim approximati ating ng the rat ratio io of moment momentum um diffusivity (kinematic viscosity) and thermal diffusivity.  

(4)

where: : dynamic viscosity, (Pa s) : thermal conductivity of the fluid, (W/m.K) : specific heat, (J/kg.K)  Note that whereas the Reynolds number is subscripted with a length scale variable, Prandtl number contains no such length scale in its definition and is dependent only on the fluid and the fluid state. As such, Prandtl number is often found in property tables alongside other properties such as viscosity and thermal conductivity. This number essentially delineates a ratio which is

 

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the thickness of the momentum boundary layer to the thermal boundary layer. When Pr is small, it means that the heat diffuses very quickly compared to the velocity (momentum). This means thatt for liquid tha liquid metals metals the thickn thickness ess of the thermal thermal boundar boundary y layer layer is much bigger bigger than the velocity boundary layer.

1.3.3 Reynolds Number

Reynolds number (Re) is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions.  

(5)

where: is the mean fluid velocity, (m/s) is a characteristic linear dimension, (m)  is the dynamic viscosity of the fluid, (Pa·s or N·s/m² or kg/m·s) is the density of the fluid, (kg/m³) Reynolds number can be defined for a number of different situations where a fluid is in relative motion to a surface (the definition of the Reynolds number is not to be confused with the Reynol Rey nolds ds Equati Equation on or lubri lubricati cation on equati equation) on).. These These defini definiti tions ons general generally ly includ includee the fluid fluid  properties of density and viscosity, plus a velocity and a characteristic length or characteristic dimension. For flow in a pipe or a sphere moving in a fluid the internal diameter is generally used today. Other shapes (such as rectangular pipes or non-spherical objects) have an equivalent diameter defined.

 

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CHAPTER   2 2 MODIFIED WILSON PLOT

Modified Wilson Plot Technique is used to determine the value of multiplier and exponent of Reynolds  Number in the Nusselt Number Number Correlation. The heat transfer coefficients for the cold and hot sides of plate heat exchanger are obtained by the following equations respectively

 

(1)

 

(2)

The following basic relation (3) is algebraically manipulated in two different ways to obtain two equations, which are subsequently used for obtaining the Modified Wilson Plot:  

2.1 Linear Modification Substituting equation (1) and (2) into the basic relation (3):

(3)

 

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(4)

The above equation (4) is a linear modification of the basic relation and it is of the form:   Where,

Slope: 

Intercept:  An initial guess ‘p’ is used to obtain a plot between X 1 and Y1 .This plot yields C c and Ch

2.2 Logarithmic Modification Given below is the logarithmic modification of (3):

 

 

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= (5) Eq. (5) is also of linear form

Slope:  Intercept:  A plot between X2 and Y2 provides the iterated value of p and Cc

2.3 Iterative Procedure

Steps for the iterative procedure are as follows: •

The value of C h obtained from the linear plot (X 1, Y1) coupled with the initial guess value of ‘ p’  ‘ p’   is is used to obtain a logarithmic plot (X2, Y2) as delineated by the logarithmically modified version (5)

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The gradient of the logarithmic equation is the new value of ‘p’

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Reinsert ‘p’ obtained from the second plot (X 2, Y 2) into (4) to acquire yet another value of ‘p’ in the second plot. The new value approaches closer and closer to the root (i.e. converges)

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Repeat this this procedure procedure until the difference difference between consecuti consecutive ve values of “p” tends tends to a value smaller than the prescribed error

 

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CHAPTER   3 3 EXPERIMENTAL PARAMETERS AND APPARATUS

The literature review has established that the general form of the Nusselt number correlation for  plate type heat exchangers is in the form of power law and given by:

Where, C and m are constants which were evaluated by experimentation. The purpose of the experimental setup is to provide a means to control certain dimensionless parameters which are explained below.

3.1 Experimental Parameters 3.1.1 Reynolds Number: 

In the case of the plate heat exchangers, the hydraulic diameter is very small, of the order of mm, so the turbulent conditions are achieved quite early i.e., at a very low value of Reynolds number. Simpson reported that the turbulent condition can be achieved at Reynolds numbers as low as 150 [8]. Reynold Reynoldss number number is a functi function on of flu fluid id flow flow rat rate. e. In the experimen experimental tal setup, setup, the Reynolds number is varied by changing the fluid flow rate. Variable frequency drive (VFD) along with bypass loops and valves were used to vary the flow rate of water. Fluid flow through the heat exchanger was varied by the mutual adjustment of these valves and the settings on the VFD, allowing the desired amount of fluid to flow through the heat exchanger.

3.1.2 Prandtl number

Prandtl number is defined as the ratio of the momentum boundary layer to the thermal boundary layer in heat transfer problems. Mathematically, it is important to note that the Prandtl number is

 

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only a function of fluid properties. These fluid properties depend upon the nature of fluid and the fluid temperature. In our experimental setup, the Prandtl number was varied by changing the fluid temperature.

3.2 Experimental Apparatus

Fig.3.1: Photograph of the Experimental Setup

Fig. 3.2: Schematic of the Apparatus [1]

 

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  A schematic of the experimental apparatus is shown in Fig.3.2. The central piece of equipment is the plate heat exchanger. The setup consists of a hot fluid loop and a cold fluid loop. The hot fluid flui d loop consists of a hot fluid tank and a pump that is pumping hot fluid from the tank to the heat exchanger. The hot fluid tank has the capacity of 150 US gallons and is equipped with 8 electr ele ctric ic immer immersio sion n heater heaters. s. One of the heater heaterss is att attach ached ed to a tempera temperatur turee contro controlle llerr and magnetic contactor. When the desired temperature is reached in the hot fluid tank, the controller sends a signal to the magnetic contactor which acts as a relay and disconnects the supply from the heater. The cold fluid loop consists of a cold fluid tank and a pump that is pumping cold fluid from the cold fluid tank to the heat exchanger. The cold fluid tank has the capacity of 35 US gallons. Temperatures are measured at the inlet and exit of the plate heat exchangers for both the hot and cold cold stream streams. s. Temper Temperatu atures res are measur measured ed at variou variouss locati locations ons using using the Resist Resistance ance Temperature Device (RTDs). The fluid is cooled by a 2 TR Packaged Air Cooled Water Chiller. R22 is used as the refrigerant for chilling the fluid in the chiller. The desired temperatures of the cold and hot fluids were achieved and the fluids of both the loops were pumped into the Plate heat exchanger where they exchanged heat. The hot fluid flowed from top to bottom within a channel of the plate heat exchanger while the cold fluid flowed flowe d from bottom to the top of the channel achieving achieving counter-flow. counter-flow. Flow rate measurements measurements were taken by the conventional bucket and stop watch method. Reynolds number was varied by changi cha nging ng the fluid fluid flow flow rate rate through through variab variable le freque frequency ncy drives drives and bypass bypass valves valves.. Prandt Prandtll nu numb mber er was was varie varied d by ch chang angin ing g the the te temp mper erat atur uree of th thee fl flui uid d us usin ing g ch chil ille ler, r, he heat ater erss an and d temperature controllers. Experiments were conducted at various temperatures and flow rates of hot and cold fluids using plates of a fixed chevron angle, β = 45o. Such plate configuration was achieved using a combination of 30 o and 60o plates. The system was allowed to reach the steady state before any reading was made.

 

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3.2.1 Equipment Details

The following table lists the equipment used in our experiment and their specifications:

Table 3.1: Details of the Equipment [7]

 

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3.3 Plate Heat Exchanger

In order to completely understand the plate heat exchanger, it is very important to understand the  plate geometry. Discussions of various parameters that define the plate geometry are provided  below.

Fig 3.3: Plate Geometry [4] Different geometric parameters of plate heat exchangers are defined below

Chevron Angle

Usually termed β and varies between 22◦–65◦. This angle also defines the thermal hydraulic softness (low thermal efficiency and pressure drop) and hardness (high thermal efficiency and  pressure drop).

Enlargement Factor

This factor φ is the ratio of the developed de veloped length to the protracted length.

 

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Mean Flow Channel Gap

This is defined as the actual gap available for the flow.  b = p – t

Channel Flow Area

This is the actual flow area defined as: Ax = b*w

Channel Equivalent Diameter

Defined as: Dh = 4Ax/P where P = 2(b + φw) = 2φw. Since b