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172 Design and Optimization of Thermal Systems local temperature. Here, 2 t 2 /tx 2 t 2 /ty 2 . For the solid region, the energy equa- tion is ()R T C T kT ss 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 eld, at the inner surface of the enclosure, continuity of the temperature and the heat ux gives TT T n k T n ss s t t ¤ ¦ ¥ ³ µ ´ t t ¤ ¦ ¥ ³ µ ´ and where n is the coordinate normal to the surface. Also, at the left source, an energy balance gives QL k T y k T y ss s t 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 t t k T n hT T si () At the inlet, the temperature is uniform at T i and at the outlet developed tempera- ture conditions, tT/ty 0, may be used. Therefore, the governing equations and boundary conditions are written for this coupled conduction-convection problem. The main characteristic quantities in the problem are the conditions at the inlet and the energy input at the sources. The energy input governs the heat transfer processes and the inlet conditions deter- mine the forced airow 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(v i /H i ). The nondimensional temperature Q is dened as Q TT T T Q k is $ $,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 s . Modeling of Thermal Systems 173 The governing equations and the boundary conditions may now be nondimen- sionalized to obtain the important dimensionless parameters in the problem. The dimensionless equations for the convective ow are obtained as t t ¤ ¦ ¥ ³ µ ´ V V VV Gr Re 0 2 T Q() p e 11 1 2 2 Re V V RePr () .( ) ( ) t t Q T QQ The dimensionless energy equation for the solid is t t ¤ ¦ ¥ ³ µ ´ ¤ ¦ ¥ ³ µ ´ Q T A A Q s 1 2 RePr () where the asterisk denotes dimensionless quantities. The dimensionless pressure ppv i * /R 2 and A is the thermal diffusivity. Therefore, the dimensionless parame- ters that arise are the Reynolds number Re, the Grashof number Gr, and the Prandtl number Pr, where these are dened as Re Pr vH g H T ii i N B N N A Gr = 3 2 $ In addition, the ratio of the thermal diffusivities A s /A arises as a parameter. Here, NM/R is the kinematic viscosity of the uid. The Reynolds number determines the characteristics of the ow, 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 xed for a given uid 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 ´ tQQ n k kn s ** fluid solid QQ Q 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, uid, and heat inputs at the sources are xed, the main governing parameters are Re, Gr, Bi, and the material property ratios A s /A and k s /k. These 174 Design and Optimization of Thermal Systems 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 nite-volume-based numerical scheme for solving the dimensionless equations, are shown in Figure 3.21 and IsothermsStreamlines (a) (b) (c) FIGURE 3.21 Calculated streamlines and isotherms for the steady solutions obtained in the LR conguration 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.) Modeling of Thermal Systems 175 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 ow eld, in terms of streamlines, and the temperature eld, in terms of isotherms, are shown for one, LR, congu- ration. Such results are used to indicate if there are any stagnation regions or hot spots in the system. The conguration may be changed to improve the ow and temperature distributions to obtain greater uniformity and/or lower temperatures. Figure 3.22 shows the maximum temperatures of the electronic components for different congurations as functions of the parameter Gr/Re 2 . We can use these FIGURE 3.22 Calculated maximum temperature for different source locations in the congurations 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.) 10 2 10 1 10 0 (a) Gr/Re 2 10 –1 10 –2 LB BR LR LB 0.0 0.3 0.6 0.9 10 2 10 1 10 0 (b) Gr/Re 2 10 –1 10 –2 0.0 0.3 0.6 0.9 10 2 10 1 10 0 (c) Gr/Re 2 10 –1 10 –2 LR BR 0.0 0.3 0.6 0.9 176 Design and Optimization of Thermal Systems results to determine if the allowable temperatures are exceeded in a particular case and also to vary the conguration and ow 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 spa- 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 of the system. In this problem, we may be interested in nding the optimal loca- tion of the heat sources, appropriate dimensions, airow 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. 3.4.2 MODELING AND SIMILITUDE 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 ow 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 specic 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. Modeling of Thermal Systems 177 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 L 1 (3.25) where the subscripts p and m refer to the prototype and the model, respectively, and L 1 is the scaling factor. Similarly, other shapes and geometries may be consid- ered, with the scale model representing a geometrically similar representation of the full-size system. This is the rst 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 a fraction of the oven length. Models of solar ponds often scale 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 cor- 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 ows are related by a constant factor, and, therefore, they must also be geometri- cally similar. The ow regime, for instance, whether the ow is laminar or tur- 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, kine- matic similarity requires that u u v v w w p m p m p m L 2 (3.26) where L 2 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 prototype must both have the same length-scale ratio and the same time-scale ratio. Conse- quently, derived quantities such as acceleration and volume ow 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. 178 Design and Optimization of Thermal Systems 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/(RV 2 D 2 ), 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 uids, 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 f 2 in Equation (3.21) and then this equation can be used for predicting the drag for a wide range of diameters, velocities, and uid properties. Figure 3.23 shows the sketches of a few examples of physical modeling of the ow to obtain similitude. Thermal Similarity This is of particular relevance to thermal systems. Thermal similarity requires that the temperature proles 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 cooling of electronic component Heat rejection Heat loss Channel Electronic components T a T 0 Flow FIGURE 3.23 Experiments for physical modeling of thermal processes and systems. Modeling of Thermal Systems 179 predictions on the temperatures in the prototype. The Nusselt number Nu char- acterizes the heat transfer in a convective process. Thus, in forced convection, if two ows are geometrically and kinematically similar and the ow regime, as determined by the Reynolds number Re, is the same, the Nusselt number is the same if the uid Prandtl number Pr is the same. The Grashof number Gr arises as an additional parameter if buoyancy effects are signicant. This relationship can be expressed as Nu f 3 (Re, Gr, Pr) (3.27) where f 3 is obtained by analytical, numerical, or experimental methods. For con- 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). 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 f 3 in Equation (3.27). Radiative transport is often difcult to model because of the T 4 dependence of heat transfer rate on temperature. Similarly, temperature-dependent material properties and thermal volumetric sources are difcult 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 proles 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 f 4 (Re, Gr c ,Sc) (3.28) where Sh is the Sherwood number, Sc is the Schmidt number (Table 3.1), and Gr c 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. 180 Design and Optimization of Thermal Systems 3.4.3 OVERALL PHYSICAL MODEL 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 uids 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 ow is disturbed to induce an earlier onset of turbu- lence in order to approximate the turbulent ow at larger Re. Similarly, thermal and mass transfer similarities may lead to conditions that are difcult to match. An attempt is generally made to match the temperature and concentration lev- 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 dened uniquely and approximate representations are generally used to provide the inputs needed for design. 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 tting. Results are obtained at a nite 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 tting, 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 tting 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 tting is particularly useful in representing calibration results and material Modeling of Thermal Systems 181 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 tting. The rst one is known as an exact t 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 t, 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 t. Sketches of curve tting 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 tting considerably facilitates the design and optimization process. 3.5.1 EXACT FIT This approach for curve tting 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 t. 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 t is both desir- able and appropriate. Many methods are available in the literature for obtaining an exact t to a given set of data points (Jaluria, 1996). Some of the important ones are: 1. General form of a polynomial 2. Lagrange interpolation 3. Newton’s divided-difference polynomial 4. Splines A polynomial of degree n can be employed to exactly t (n 1) data points. The general form of the polynomial may be taken as yfx a axax ax ax n n () 01 2 2 3 3 ! (3.29) where y is the dependent variable, x is the independent variable, and the a’s are constants to be determined by curve tting of the data. If (x i , y i ), where i 0, 1, 2,z, n, represent the (n 1) data points, y i being the value of the dependent [...]... constants C1, C2, and C3 are obtained as 16 C1 6.243C2 0 .11 4C3 9. 555 6.243C1 8 .17 3C2 0.044C3 9. 490 0 .11 4C1 0.044C2 3.481C3 3.762 These equations are solved to yield the three constants as C1 C2 1. 50 39 2.30 39 C3 1. 1005 Therefore, B exp(C1) 4. 49 91 , a C2 2.30 39, and b C3 1. 1005 Rounding these off to the second place of decimal, the best fit to the given data is given by the equation Q 4.5D 2.3 ( p )1. 1 It can... 0.5 1. 0 1. 4 0.5 0 .13 0.43 2 .1 4.55 0 .9 0.25 0. 81 4.0 8. 69 1. 2 0.34 1. 12 5.5 11 .92 1. 8 0.54 1. 74 8. 59 18 .63 p (atm) Obtain a best fit to these data, assuming a power-law dependence of Q on the two independent variables D and p 19 4 Design and Optimization of Thermal Systems Solution The variation of Q with D and p may be written for a power-law variation as BDa( p)b Q Taking the natural logarithm of. .. 3.8 The temperature T of a small copper sphere cooling in air is measured as a function of time to yield the following data: (s) T ( C) 0.2 0.6 1. 0 1. 8 2.0 3.0 5.0 6.0 8.0 14 6.0 12 9. 5 11 4.8 90 .3 85 .1 63.0 34.6 25.6 14 .1 An exponential decrease in temperature is expected from lumped mass modeling Obtain a best fit to represent these data 19 2 Design and Optimization of Thermal Systems Solution The given... computations A numerical scheme provides flexibility and versatility so that different data sets can easily be considered for best fit The resulting values of C1 and C2 are C1 5.04 31 and C2 Therefore, A exp(5.04 31) 15 4 .94 8 and a the best fit to the given data as T 0. 299 8 0. 299 8 This gives the equation for 15 4 .94 8 exp( 0. 299 8 ) This may be approximated as T 15 4 .95 exp(–0.3 ) The given data may be compared with... or B is given by exp(C) and a or b by D Therefore, from this linear fit, the constants A, a, B, and b can be calculated Similarly, other nonpolynomial forms such as f (x) ax b x f (x) a b x f (x) a b x (3.48) 19 0 Design and Optimization of Thermal Systems can be written as 1 f (x) b 1 a x 1 a f (x) a b 1 x 1 f (x) b a 1 x a and linearized as b X a 1 a Y Y a bX Y b a 1 X a (3. 49) by substituting Y for... Fortran90 and C However, MATLAB is particularly well suited for such problems because the command Polyfit yields the best fit to a chosen order of the polynomial for curve fitting (see Appendix A) For instance, the following program may be used: %Input Data tau [0.2 0.6 1. 0 1. 8 2.0 3.0 5.0 6.0 8.0]; t0 [14 6.0 12 9. 5 11 4.8 90 .3 85 .1 63.0 34.6 25.6 14 .1] ; t log(t0); % Cutve Fit t1 polyfit(tau,t ,1) ; a t1 (1) ... equation The nine values of T from this Modeling of Thermal Systems 19 3 equation are calculated as 14 5 .93 , 12 9. 44, 11 4. 81, 90 .33, 85.07, 63.03, 34. 61, 25.64, and 14 .08 Therefore, the given data are closely represented by this equation As mentioned previously, a computer program may be developed to calculate the summations needed for generating the two algebraic equations for C1 and C2, using programming... 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 frequently used for an exact fit in engineering problems (Carnahan, et al., 19 69; Hornbeck, 19 75; Gerald and Wheatley, 19 94 ; Jaluria, 19 96 ) Splines approach the problem as a piece-wise fit and, therefore,... first and then the intercept These are given by t1 (1) and t1(2) in the program Then a is simply t1 (1) and A is the exponential of t1(2) The program yields the same results for a and A as given above Further details on such algorithms in MATLAB may be obtained from Recktenwald (2000) and Mathews and Fink (2004) Example 3 .9 The flow rate Q in circular pipes is measured as a function of the diameter D and. .. McGraw-Hill, New York Doebelin, E.O ( 19 80) System Modeling and Response, Wiley, New York Eckert, E.R.G and Drake, R.M ( 19 72) Analysis of Heat and Mass Transfer, McGrawHill, New York Fox, R.W and McDonald, A.T (2003) Introduction to Fluid Mechanics, Wiley, 6th ed., New York Gebhart, B ( 19 71) Heat Transfer, 2nd ed., McGraw-Hill, New York Gerald, C.F and Wheatley, P.O ( 19 94 ) Applied Numerical Analysis, 5th . location, and (c) right wall location. (Adapted from Papanicolaou and Jaluria, 19 94.) 10 2 10 1 10 0 (a) Gr/Re 2 10 1 10 –2 LB BR LR LB 0.0 0.3 0.6 0 .9 10 2 10 1 10 0 (b) Gr/Re 2 10 1 10 –2 0.0 0.3 0.6 0 .9 10 2 10 1 10 0 (c) Gr/Re 2 10 1 10 –2 LR BR 0.0 0.3 0.6 0 .9 17 6. 19 94.) 10 2 10 1 10 0 (a) Gr/Re 2 10 1 10 –2 LB BR LR LB 0.0 0.3 0.6 0 .9 10 2 10 1 10 0 (b) Gr/Re 2 10 1 10 –2 0.0 0.3 0.6 0 .9 10 2 10 1 10 0 (c) Gr/Re 2 10 1 10 –2 LR BR 0.0 0.3 0.6 0 .9 17 6 Design and Optimization of Thermal Systems results to determine if the. considered in Example 3.7 at Re 10 0 and Gr/Re 2 values of (a) 0 .1, (b) 1. 0, and (c) 10 .0. (Adapted from Papanicolaou and Jaluria, 19 94.) Modeling of Thermal Systems 17 5 Figure 3.22 from a detailed
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