Design and Optimization of Thermal Systems Episode 1 Part 9 pdf

The governing equations and the boundary conditions may now be sionalized to obtain the important dimensionless parameters in the problem.. This example is given mainly to illus- trate s

local temperature Here,  2  t 2 /tx 2 t 2 /ty 2 For the solid region, the energy tion is

equa-( R ) T

C s T k s T

t

2

where the subscript s denotes solid material properties The Boussinesq

approxi-mations have been used for the buoyancy term The pressure work and viscous dissipation terms have been neglected The boundary conditions on velocity are the no-slip conditions, i.e., zero velocity at the solid boundaries At the inlet and outlet, the given velocities apply.

For the temperature field, at the inner surface of the enclosure, continuity of the temperature and the heat flux gives

T n

t t

¤

¦¥

³ µ´

t 

t t

¤

¦¥

³ µ´

where Q s is the energy dissipated by the source per unit width Similar equations may be written for other sources At the outer surface of the enclosure walls, the convective heat loss condition gives

mine the forced airflow in the enclosure Therefore, v i , H i , T i , and Q s are taken as the characteristic physical quantities The various dimensions in the problem are

nondimensionalized by H i and the velocity V by v i Time T is nondimensionalized

by H i /v i to give dimensionless time T`  T(vi /H i) The nondimensional temperature

Q is defined as

Q k

$ , where$Here $T is taken as the temperature scale based on the energy input by a given

source The energy input by other sources may be nondimensionalized by Q.

The governing equations and the boundary conditions may now be sionalized to obtain the important dimensionless parameters in the problem The dimensionless equations for the convective flow are obtained as

V V

¤

¦¥

³ µ´ 

Q T

A

RePr ( )

where the asterisk denotes dimensionless quantities The dimensionless pressure

p*  p v/ R i2 and A is the thermal diffusivity Therefore, the dimensionless ters that arise are the Reynolds number Re, the Grashof number Gr, and the Prandtl number Pr, where these are defined as

parame-Re  v H i i g H i T Pr  N

B N

N A

Gr =

3 2

$

In addition, the ratio of the thermal diffusivities As/A arises as a parameter Here,

N  M/R is the kinematic viscosity of the fluid The Reynolds number determines the characteristics of the flow, particularly whether it is laminar or turbulent, the Grashof number determines the importance of buoyancy effects, and the Prandtl number gives the effect of momentum diffusion as compared to thermal diffusion and is fixed for a given fluid at a particular temperature.

Additional parameters arise from the boundary conditions The conditions at the inner and outer surfaces of the walls yield, respectively,

t t

¤

¦¥

³ µ´ 

¤

¦¥

³ µ´

t t

¤

¦¥

³ µ´

n

hH k i s

*

Therefore, the ratio of the thermal conductivities k s /k and the Biot number Bi 

hH i /k s arise as parameters A perfectly insulated condition at the outer surface is achieved for Bi  0 In addition to these, several geometry parameters arise from

the dimensions of the enclosure (see Figure 3.20), such as d i /H i , H o /H i , L s /H i, etc

Heat inputs at different sources lead to parameters such as (Q s)2/Q s , (Q s)3/Q s, etc.,

where (Q s) 2 and (Q s) 3 are the heat inputs by different electronic components.

The above considerations yield the dimensionless equations and boundary conditions, along with all the dimensionless parameters that govern the thermal transport process Clearly, a large number of parameters are obtained However,

if the geometry, fluid, and heat inputs at the sources are fixed, the main governing parameters are Re, Gr, Bi, and the material property ratios A /A and k/k These

may be varied in the simulation of the given system to determine the effect of materials used and the operating conditions Similarly, geometry parameters may

be varied to determine the effect of these on the performance of the cooling system, particularly on the temperature of the electronic components, whose performance

is very temperature sensitive.

Some typical results, obtained by the use of a finite-volume-based numerical scheme for solving the dimensionless equations, are shown in Figure 3.21 and

Isotherms Streamlines

(a)

(b)

(c)

FIGURE 3.21 Calculated streamlines and isotherms for the steady solutions obtained in

the LR configuration for the problem considered in Example 3.7 at Re  100 and Gr/Re 2 values of (a) 0.1, (b) 1.0, and (c) 10.0 (Adapted from Papanicolaou and Jaluria, 1994.)

Figure 3.22 from a detailed numerical simulation carried out by Papanicolaou and Jaluria (1994) Two electronic components are taken, placing these on the left wall (L), the right wall (R), or the bottom (B) The flow field, in terms of streamlines, and the temperature field, in terms of isotherms, are shown for one, LR, configu- ration Such results are used to indicate if there are any stagnation regions or hot spots in the system The configuration may be changed to improve the flow and temperature distributions to obtain greater uniformity and/or lower temperatures Figure 3.22 shows the maximum temperatures of the electronic components for different configurations as functions of the parameter Gr/Re 2 We can use these

FIGURE 3.22 Calculated maximum temperature for different source locations in the

configurations considered, at various values of Gr/Re 2 for (a) left wall location, (b) bottom wall location, and (c) right wall location (Adapted from Papanicolaou and Jaluria, 1994.)

102

101

100(a) Gr/Re2

10–1

10–2

LB BR

LR LB

10–1

10–20.0 0.3 0.6 0.9

10 –1

10 –2

LR BR

0.0 0.3 0.6 0.9



results to determine if the allowable temperatures are exceeded in a particular case and also to vary the configuration and flow rate to obtain an acceptable design Thus, the simulation results may be used to change the design variables over given ranges in order to obtain an acceptable or optimal design of the system.

This is clearly a very complicated problem because transient effects and tial variations are included Many practical systems involve complicated govern- ing equations and complex geometry Finite-element methods are particularly well suited for generating the numerical results needed for the design and optimization

spa-of the system In this problem, we may be interested in finding the optimal tion of the heat sources, appropriate dimensions, airflow rate, wall thickness, and materials for the given electronic circuitry This example is given mainly to illus- trate some of the complexities of practical thermal systems and the derivation of governing dimensionless parameters The results indicate typical outputs obtained and their relevance to system design Cooling of electronic systems has been an important area for research and design over the past two to three decades In many cases, commercially available software, such as Fluent, is used to simulate the sys- tem and obtain the results needed for design and optimization.

loca-3.4.2 M ODELING AND S IMILITUDE

In order for a scale model to predict the behavior of the full-scale thermal system, there must be similarity between the model and the prototype Scaling factors must be established between the two so that the results from the model can be applied to the system These scaling laws and the conditions for similitude are obtained from dimensional analysis As mentioned earlier, if the dimensionless parameters are the same for the model as well as for the prototype, the flow and transport regimes are the same and the dimensionless results are also the same This can be seen easily in terms of the dimensionless governing equations, such

as Equation (3.6) and Equation (3.20) The governing equations are the same for the model and the full-size system If the nondimensional parameters for the two cases are the same, the results obtained, in dimensionless terms, will also be the same for the model and the system

Several different mechanisms usually arise in typical thermal systems, and it may not be possible to satisfy all the parameters for complete similarity However, each problem has its own specific requirements These are used to determine the dominant parameters in the problem and thus establish similitude Several common types of similarities may be mentioned here These include geometric, kinematic, dynamic, thermal, and chemical similarity It is important to select the appropriate parameters for a particular type of similarity (Schuring, 1977; Szucs, 1977)

Geometric Similarity

The model and the prototype are generally required to be geometrically similar This requires identity of shape and a constant scale factor relating linear dimensions

Thus, if a model of a bar is used for heat transfer studies, the ratio of the model lengths to the corresponding prototype lengths must be the same, i.e.,

L L

H H

W W p

m p m p m

where the subscripts p and m refer to the prototype and the model, respectively,

and L1 is the scaling factor Similarly, other shapes and geometries may be ered, with the scale model representing a geometrically similar representation of the full-size system This is the first type of similitude in physical modeling and is commonly required of the model However, sometimes the model may represent only a portion of the full system For example, a long drying oven may be studied with a short model that is properly scaled in terms of the cross-section but is only

consid-a frconsid-action of the oven length Models of solconsid-ar ponds often scconsid-ale the height but not the large surface area of typical ponds In these cases, the model is chosen to focus on the dominant considerations

Kinematic Similarity

The model and the system are kinematically similar when the velocities at responding points are related by a constant scale factor This implies that the velocities are in the same direction at corresponding points and the ratio of their magnitudes is a constant The streamline patterns of two kinematically similar flows are related by a constant factor, and, therefore, they must also be geometri-cally similar The flow regime, for instance, whether the flow is laminar or tur-

cor-bulent, must be the same for the model and the prototype Thus, if u, v, and w

represent the three components of velocity in a model of a thermal system, matic similarity requires that

kine-u u

v v

w w p

m p m p m

where L2 is the scale factor and the subscripts p and m again indicate the

proto-type and the model For kinematic similarity, the model and the protoproto-type must both have the same length-scale ratio and the same time-scale ratio Conse-quently, derived quantities such as acceleration and volume flow rate also have

a constant scale factor For a given value of the magnitude of the gravitational

acceleration g, the Froude number Fr represents the scaling for velocity and

length Therefore, this kinematic parameter is used for scaling wave motion in water bodies

Dynamic Similarity

This requires that the forces acting on the model and on the prototype are in the same direction at corresponding locations and the magnitudes are related by a constant scale factor This is a more restrictive condition than the previous two and, in fact, requires that these similarity conditions also be met All the impor-tant forces must be considered, such as viscous, surface tension, gravitational, and buoyancy forces If dynamic similarity is obtained between the model and the prototype, the results from the model may be applied quantitatively to deter-mine the prototype behavior The various dimensionless parameters that arise in the momentum equation or that are obtained through the Buckingham Pi theo-rem may be used to establish dynamic similarity For instance, in the case of the drag on a sphere, Equation (3.21), if the Reynolds numbers for the model and the

prototype are equal, the dimensionless drag forces, given by F/(RV2D2), are also equal Then the results obtained from the model can be used to predict the drag force on the full-size component Clearly, the tests could be carried out with dif-ferent fluids, such as air and water, and over a convenient velocity range, as long

as the Reynolds numbers are matched In fact, the model can be used in a wind

or water tunnel to determine the functional dependence given by f2 in Equation (3.21) and then this equation can be used for predicting the drag for a wide range

of diameters, velocities, and fluid properties Figure 3.23 shows the sketches of a few examples of physical modeling of the flow to obtain similitude

Thermal Similarity

This is of particular relevance to thermal systems Thermal similarity requires that the temperature profiles in the model and the prototype be geometrically similar at corresponding times If convective motion arises, kinematic similar-ity is also a requirement Thus, the temperatures are related by a constant scale factor and the results from a model study may be applied to obtain quantitative

Cooling of moving plate

in hot rolling Natural convection

predictions on the temperatures in the prototype The Nusselt number Nu acterizes the heat transfer in a convective process Thus, in forced convection, if two flows are geometrically and kinematically similar and the flow regime, as determined by the Reynolds number Re, is the same, the Nusselt number is the same if the fluid Prandtl number Pr is the same The Grashof number Gr arises

char-as an additional parameter if buoyancy effects are significant This relationship can be expressed as

Nu  f3 (Re, Gr, Pr) (3.27)

where f3 is obtained by analytical, numerical, or experimental methods For duction in a heated body with convective loss at the surface, the Biot number Bi arises as an additional dimensionless parameter from the boundary condition, as seen in Equation (3.23)

con-Thus, thermal similarity is obtained if these parameters are the same between the model and the system As mentioned earlier, the dimensionless governing equations and corresponding boundary conditions indicate the dimensionless parameters that must be kept the same between the model and the system in order to apply the model-study results to the system Experiments may be carried

out to obtain the functional dependence, such as f3 in Equation (3.27) Radiative

transport is often difficult to model because of the T4 dependence of heat transfer rate on temperature Similarly, temperature-dependent material properties and thermal volumetric sources are difficult to model because of the often arbitrary, nonlinear variations with temperature that arise Consequently, physical model-ing of thermal systems is often complicated and involves approximations simi-lar to those discussed with respect to mathematical modeling Relatively small effects are neglected to obtain similarity

Mass Transfer Similarity

This similarity requires that the species concentration profiles for the model and the system be geometrically similar at corresponding times At small concentra-tion levels, the analogy between heat and mass transfer may be used, resulting

in expressions such as Equation (3.27), which may be written for mass transfer systems as

Sh  f4 (Re, Grc ,Sc) (3.28)where Sh is the Sherwood number, Sc is the Schmidt number (Table 3.1), and

Grc is based on the concentration difference $C, instead of the temperature

dif-ference $T in Gr Thus, the conditions for mass transfer similarity are close to

those for thermal similarity in this case If chemical reactions occur, the reaction rates at corresponding locations must have a constant scale factor for similitude between the model and the prototype Since reaction rates are strongly dependent

on temperature and concentration, the models are usually studied under the same temperature and concentration conditions as the full-size system

3.4.3 O VERALL P HYSICAL M ODEL

Based on dimensional analysis, which indicates the main dimensionless groups that characterize a given system, and the appropriate similarity conditions, a physical model may be developed to represent a component, subsystem, or system However, even though a substantial amount of work has been done on these con-siderations, particularly with respect to wind and water tunnel testing for aerody-namic and hydrodynamic applications, physical modeling of practical processes and systems is an involved process This is mainly because different aspects may demand different conditions for similarity For instance, if both the Reynolds and the Froude numbers are to be kept the same between the model and the prototype for the modeling of viscous and wave drag on a ship, the conditions of similarity cannot be achieved with practical fluids and dimensions Then complete similar-ity is not possible and model testing is done with, say, only the Froude number matched The data obtained are then combined with results from other studies on viscous drag Sometimes, the flow is disturbed to induce an earlier onset of turbu-lence in order to approximate the turbulent flow at larger Re Similarly, thermal and mass transfer similarities may lead to conditions that are difficult to match

An attempt is generally made to match the temperature and concentration els in order to satisfactorily model material property variations, reaction rates, thermal source, radiative transport, etc However, this is frequently not possible because of experimental limitations Then, the matching of the dimensionless groups, such as Pr, Re, and Gr, may be used to obtain similarity and hence the desired information Again, the dominant effects are isolated and physical mod-eling involves matching these between the system and the model Because of the complexity of typical thermal systems, the physical model is rarely defined uniquely and approximate representations are generally used to provide the inputs needed for design

lev-3.5 CURVE FITTING

An important and valuable technique that is used extensively to represent the characteristics and behavior of thermal systems is that of curve fitting Results are obtained at a finite number of discrete points by numerical computation and experimentation If these data are represented by means of a smooth curve, which passes through or as close as possible to the points, the equation of the curve can

be used to obtain values at intermediate points where data are not available and also to model the characteristics of the system Physical reasoning may be used in the choice of the type of curve employed for curve fitting, but the effort is largely a data-processing operation, unlike mathematical modeling discussed earlier, which was based on physical insight and experience The equation obtained as a result of curve fitting then represents the performance of a given equipment or system and may be used in system simulation and optimization This equation may also be employed in the selection of equipment such as blowers, compressors, and pumps Curve fitting is particularly useful in representing calibration results and material

property data, such as the thermodynamic properties of a substance, in terms of equations that form part of the mathematical model of the system.

There are two main approaches to curve fitting The first one is known as

an exact fit and determines a curve that passes through every given data point This approach is particularly appropriate for data that are very accurate, such

as computational results, calibration results, and material property data, and if only a small number of data points are available If a large amount of data is to

be represented, and if the accuracy of the data is not very high, as is usually the case for experimental results, the second approach, known as the best fit, which obtains a curve that does not pass through each data point but closely approxi-mates the data, is more appropriate The difference between the values given by the approximating curve and the given data is minimized to obtain the best fit Sketches of curve fitting using these two methods were seen earlier in Figure 3.2 Both of these approaches are used extensively to represent results from numeri-cal simulation and experimental studies The availability of correlating equations from curve fitting considerably facilitates the design and optimization process

3.5.1 E XACT F IT

This approach for curve fitting is somewhat limited in scope because the number

of parameters in the approximating curve must be equal to the number of data points for an exact fit If extensive data are available, the determination of the large number of parameters that arise becomes very involved Then, the curve obtained is not very convenient to use and may be ill conditioned In addition, unless the data are very accurate, there is no reason to ensure that the curve passes through each data point However, there are several practical circumstances where

a small number of very accurate data are available and an exact fit is both able and appropriate

desir-Many methods are available in the literature for obtaining an exact fit to a given set of data points (Jaluria, 1996) Some of the important ones are:

1 General form of a polynomial

variable at x  x i, these values may be substituted in Equation (3.29) to obtain

(n 1) equations for the a’s Thus,

y ia0a x1 i a x2 i2a x3 i3!a x n i n fori0, 1,2, n, (3.30)

Since x i and y i are known for the given data points, (n 1) equations are obtained from Equation (3.30), and these can be solved for the unknown constants in Equa-

tion (3.29) Thus, two data points yield a straight line, y  a0 a1x, three points

a second-order polynomial, y  a0 a1x a2x2, four points a third-order mial, and so on The method is appropriate for small sets of very accurate data, with the number of data points typically less than ten For larger data sets, higher-order polynomials are needed, which are often difficult to determine, inconve-nient to use, and inaccurate because of the many small coefficients that arise for higher-order terms

polyno-Different forms of interpolating polynomials are used in other methods In

Lagrange interpolation, the polynomial used is known as the Lagrange mial and the nth-order polynomial is written as

i n

where the product sign 0 denotes multiplication of the n factors obtained by

vary-ing j from 0 to n, excludvary-ing j  i, for the quantity within the parentheses It is easy

to see that this polynomial may be written in the general form of a polynomial, Equation (3.29), if needed Lagrange interpolation is applicable to an arbitrary distribution of data points, and the determination of the coefficients of the poly-nomial does not require the solution of a system of equations, as was the case for the general polynomial Because of the ease with which the method may be applied, Lagrange interpolation is extensively used for engineering applications

In Newton’s divided-difference method, the nth-order interpolating

(

!

A recursive formula is written to determine the coefficients The higher-order coefficients are determined from the lower-order ones Therefore, we evaluate

the coefficients by starting with a0 and successively calculating a1, a2, a3, and so

on, up to a n Once these coefficients are determined, the interpolating polynomial

is obtained from Equation (3.33) Several simplified formulas can be derived if

the data are given at equally spaced values of the independent variable x These

include the Newton-Gregory forward and backward interpolating polynomials This method is particularly well suited for numerical computation and is fre-quently used for an exact fit in engineering problems (Carnahan, et al., 1969; Hornbeck, 1975; Gerald and Wheatley, 1994; Jaluria, 1996)

Splines approach the problem as a piece-wise fit and, therefore, can be used

for large amounts of accurate data, such as those obtained for the calibration of equipment and material properties Spline functions consider small subsets of the data and fit them with lower-order polynomials, as sketched in Figure 3.24 The cubic spline is the most commonly used function in this exact fit, though poly-nomials of other orders may also be used Spline interpolation is an important technique used in a wide range of applications of engineering interest Measure-ments of material properties such as density, thermal conductivity, mass diffu-sivity, reflectivity, and specific heat, as well as the results from calibrations of equipment and sensors such as thermocouples, often give rise to large sets of very accurate data

Functions of more than one independent variable also arise in many problems

of practical interest An example of this circumstance is provided by namic properties like density, internal energy, enthalpy, etc., which vary with two independent variables, such as temperature and pressure Similarly, the pressure generated by a pump depends on both the speed and the flow rate Again, a best fit

thermody-is usually more useful because of the inaccuracies involved in obtaining the data However, an exact fit may also be obtained Curve fitting with the chosen order

of polynomials is applied twice, first at different fixed values of one variable to obtain the curve fit for the other variable Then the coefficients obtained are curve fitted to reflect the dependence on the first variable As shown in Figure 3.25,

9 data points are needed for second-order polynomials For third-order als, 16 points are needed, and for fourth-order polynomials, 25 points are needed The resulting general equation for the curve fit shown in Figure 3.25 is

polynomi-ya0a x1 2a x2 2 b0b x1 2b x x3 2 1c0c x1 2c x x2 22

1 (3.34)

3.5.2 B EST F IT

The data obtained in many engineering applications have a significant amount

of associated error Experimental data, for instance, would generally have some scatter due to error whose magnitude depends on the instrumentation and the arrangement employed for the measurements In such cases, requiring the inter-polating curve to pass through each data point is not appropriate In addition,