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Modeling of Thermal Systems 197 PROBLEMS Note: In all problems dealing with model development, list the assumptions, approximations, and idealizations employed; give the resulting governing equa- tions; and, whenever possible, give the analytical solution. Symbols may be used for the appropriate physical quantities. 3.1. An energy storage system consists of concentric cylinders, the inner being of radius R 1 , the outer of radius R 2 and both being of length L, as shown in Figure P3.1. The inner cylinder is heated electrically and supplies a constant heat ux q to the material in the outer cylinder, as shown. The annulus is packed with high conductivity metal pieces. Assuming that the system is well insulated from the environment and that the annular region containing the metal pieces may be taken as isothermal, (a) Obtain a mathematical model for the system. (b) If the maximum temperature is given as T max , obtain the time for which heating may be allowed to occur, employing the usual symbols for properties 3.2. Solid plastic cylinders of diameter 1 cm and length 30 cm are heat treated by moving them at constant speed U through an electric oven of length L, as shown in Figure P3.2. The temperature at the oven walls is T s and the air in the oven is at temperature T a . The convective heat transfer coefcient at the plastic surface is given as h and the surface Metal pieces R 2 R 1 Insulation q FIGURE P3.1 FIGURE P3.2 L U T s h, T a 198 Design and Optimization of Thermal Systems emissivity as E. The cylinders are placed perpendicular to the direc- tion of motion and are rotated as they move across the oven. Develop a simple mathematical model for obtaining the temperature in the plastic cylinders as a function of the temperatures T s and T a , h, L, and U, for design of the system. Clearly indicate the assumptions and approxima- tions made. 3.3. A chemical industry needs hot water at temperature T c o$T c for a chemical process. For this purpose, a storage tank of volume V and surface area A is employed. Whenever hot water is withdrawn from the tank, cold water at temperature T a , where T a is the ambient tem- perature, ows into the tank. A heater supplying energy at the rate of Q turns on whenever the temperature reaches T c – $T c and turns off when it is reaches T c $T c . The heater is submerged in the water con- tained in the tank. Assuming uniform temperature in the tank and a convective loss to the environment at the surface, with a heat transfer coefcient h, obtain a mathematical model for this system. Sketch the expected temperature T of water in the tank as a function of time for a given ow rate m of hot water and also for the case when there is no outow, m 0. 3.4. Consider a cylindrical rod of diameter D undergoing thermal process- ing and moving at a speed U as shown in Figure P3.4. The rod may be assumed to be innite in the direction of motion. Energy transfer occurs at the outer surface, with a constant heat ux input q and convec- tive loss to the ambient at temperature T a and heat transfer coefcient h. Assuming one-dimensional, steady transport, obtain the governing equation and the relevant boundary conditions. By nondimensional- ization, determine the governing dimensionless parameters. Finally, obtain T(x) for (a) h 0 and (b) q 0. 3.5. Give the governing equations and boundary conditions for the steady- state, two-dimensional case for the preceding physical problem. Derive the governing dimensionless parameters using the nondimensionaliza- tion of the equations and the boundary conditions. 3.6. During the heat treatment of steel bolts, the bolts are placed on a con- veyor belt that passes through a long furnace at speed U as shown in Figure P3.6. In the rst section, the bolts are heated at a constant heat ux q. In the second and third sections, they lose energy by convection D x q h, T a U T FIGURE P3.4 Modeling of Thermal Systems 199 to the air at temperature T a at convective heat transfer coefcients h 1 and h 2 in the two sections, respectively. (a) Assuming lumped mass analysis is valid, obtain the governing equations for the three sections and outline the mathematical model thus obtained. (b) Sketch the temperature variation qualitatively as a function of dis- tance x from the entrance. 3.7. In a manufacturing process, a metal block of surface area A and vol- ume V is melted in a furnace. The initial temperature of the block is T i , the melting point is T m , and the nal temperature is T f , where T f T m T i . The block is exposed to a constant heat ux input q due to radiation and also loses energy by convection to the surrounding air at T a with a convective heat transfer coefcient h. Employing the usual symbols for the properties and assuming no temperature variation in the block: (a) Obtain a suitable mathematical model for the process. (b) Qualitatively sketch the temperature variation with time. (c) If the temperatures T i , T m , and T f are given, what are the design variables? 3.8. A water cooler is to be designed to supply cold drinking water with a given time-dependent mass ow rate m. Assume a cubical tank of cold water surrounded by insulation of uniform thickness. Water at the ambient temperature ows into the tank to make up the cold-water out- ow. The refrigeration unit turns on if the water temperature reaches a value T max and turns off when it is drops to T min , thus maintaining the temperature between these two values. Develop a simple mathematical model for this system. 3.9. It is necessary to model and simulate a hot-water distribution system consisting of a tank, pump, and pipes. The heat input Q to the tank is given and the ambient temperature is T a , with h as the heat transfer coefcient for heat loss. Develop a simple mathematical model for this system. 3.10. We wish to model a vapor compression cooling system, such as the one shown in Figure 1.8(a). For a simple model based on the thermodynamic Conveyor IIIIII x q h 1 h 2 L 2 L 1 L 3 U FIGURE P3.6 200 Design and Optimization of Thermal Systems cycle, list the main approximations and idealizations you would employ to obtain the model. Justify these in a few sentences. 3.11. A mathematical model is to be developed to simulate a power plant, such as the one shown in Figure 2.17. For a simple model based on the thermodynamic cycle, list the approximations and idealizations you would employ to obtain the model. Justify these in a few sentences. 3.12. In the hot-water storage system considered in Example 3.5, if the ambi- ent temperature is 20nC and the heat transfer coefcient is 20 W/m 2 K, sketch the temperature distribution in the steady-state case. What are the governing parameters in this problem? How does the solution vary with these parameters? 3.13. In a heat treatment furnace, a thin metallic sheet of thickness d, height L, and width W is employed as a shield. On one side of the sheet, hot ue gases at temperature T f (x) exchange energy with an overall heat transfer coefcient h f . On the other side, inert gases at temperature T g (x) have a heat transfer coefcient h g , as shown in Figure P3.13. The sheet also loses energy by radiation. If L d and W d, obtain a mathematical model for calculating the temperature T in the sheet. Assume that T f and T g are known functions of height x. Also, take h f and h g as known constants. Give the resulting governing equation and its solution, if easily obtainable analytically. 3.14. For the following systems, consider and briey discuss the various approximations and idealizations that can be made to simplify the mathematical model. When are these approximations valid and how would you relax them? Outline the nature and type of governing equa- tions that you expect to obtain for the different systems. (a) Food-freezing plant to chill vegetables to –10nC by circulating chilled air past the vegetables. (b) A shell and tube heat exchanger, with hot and cold water as the two uids. x L d h g h f W T f (x) T g (x) FIGURE P3.13 Modeling of Thermal Systems 201 (c) A system consisting of pumps and pipe network to transport water from ground level to a tank 100 m high. (d) A vapor compression system for cooling a cold storage room. (e) Flow equipment such as compressors, fans, pumps, and turbines. 3.15. In the electronic system considered in Example 3.7, if the geometry and heat inputs are xed, what are the design variables in terms of dimen- sionless parameters? If the maximum temperature in the electronic components is to be restricted for an acceptable design, what physical quantities may be adjusted to reach an acceptable design? 3.16. In a counterow heat exchanger, the heat loss to the environment is to be included in the mathematical model. Considering the case of the hot uid on the outside and the colder uid on the inside, as shown in Figure P3.16, sketch qualitatively the change that the inclusion of this consideration will have on the temperature distribution in the heat exchanger. Also, give the energy equation taking this loss into account. 3.17. Scale-up from a laboratory system to a full-size version is a very impor- tant consideration in industry. For the problems considered in Example 3.5 and Example 3.6, determine the important parameters that may be used for scale-up and whether it is possible to achieve the desired similarity. 3.18. A scaled-down version of a shell and tube heat exchanger is to be used to simulate the actual physical system to be used in a chemical plant. Determine the dimensionless parameters that must be kept the same in order to ensure similarity between the full-size and scaled-down systems. 3.19. Obtain the dimensionless parameters that govern the scale-down and scale-up of a vapor compression refrigeration system. 3.20. Consider the condensation soldering system discussed in Chapter 2 (Figure 2.4 and Figure 2.6), with boiling liquid at the bottom of a chamber and water-cooled condensing coils at the top, generating a condensing vapor region in the tank. A large electronic circuit board that may be approximated as a thick at plate at room temperature is immersed in the chamber at time zero. Develop a simple mathematical Energy loss Hot Cold FIGURE P3.16 202 Design and Optimization of Thermal Systems model to compute the temperature distribution in the plate, giving the governing equation(s) and boundary and initial conditions. Also, write down the global energy balance equation to determine the energy input into the liquid needed before and after immersion of the board. Make suitable simplications and assumptions, indicating these in your answer. 3.21. A at steel (R 10000 kg/m 3 , C 500 J/kgK, k 100 W/mK) sheet emerges from a furnace at 10 cm/s and 800nC. At distances of 10 m each, there are three rolling dies; see Figure 1.10(d). The initial thick- ness of the sheet is 2 cm and at each die, a reduction of 20% in thickness occurs. In addition, a temperature rise of 50nC occurs due to friction at each of the rolling die. The sheet loses energy to the environment, at 20nC, at an overall heat transfer coefcient of 120 W/m 2 K. It is neces- sary to maintain the temperature of the material higher than 700nC. Using a simple mathematical model of the process, determine the level of heating, or cooling, needed between the rolling stations. 3.22. The average daily temperature in New Brunswick, NJ, is obtained by taking data over several years. The results are given as 365 data points, with each point corresponding to a day during the year. A curve t to these data is to be obtained for the design of air-conditioning systems. Will an exact or a best t be more appropriate? Suggest a suitable form of the function for curve tting. 3.23. The average daily air temperature at a location is available for each day of 2005. We wish to obtain a best t to these data and use the equation obtained in a computer model for an environmental thermal system. Choose an appropriate form of the equation that may be employed to curve t the data and outline the reasons for your choice. Outline the mathematical procedure to determine the constants of the equation chosen for the curve t. 3.24. A steel sphere at initial temperature T o is immersed in a cold uid at temperature T a and allowed to cool rapidly for hardening. At 20 time intervals T i , the corresponding temperature T i in the sphere is mea- sured, where i 1, 2, 3, . . . , 20. The temperature variation across the sphere may be taken as negligible. We wish to obtain the best t to the data collected. What function f(T), where T f(T), will you employ for the purpose? Justify your answer. 3.25. In a heat treatment process, a metal cube of side 2 cm, density 6000 kg/m 3 , and specic heat 300 J/kgK is heated by convection from a hot uid at temperature T f 220nC. The initial temperature of the cube is T i 20nC. If the temperature T within the cube may be taken as uni- form, write down the equation that governs the temperature as a func- tion of time T. Obtain the general form of the solution. If the measured temperature values at different time intervals are given as Modeling of Thermal Systems 203 T (min) 0 0.5 1.0 2.0 3.0 6.0 TT TT f if 1.0 0.85 0.72 0.5 0.4 0.14 obtain a best t to these data using information from the analytical solu- tion for T(T). Sketch the resulting curve and plot the original data to indicate how good a representation of the data is obtained by this curve. From the results obtained, compute the heat transfer coefcient h. 3.26. Obtain a linear best t to the data given below from a chemical reactor by using the method of least squares: Concentration (g/m 3 ) 0.1 0.2 0.5 1.0 1.2 Reaction Rate (g/s) 1.75 1.91 2.07 2.32 2.4 Is a linear t satisfactory in this case? 3.27. The temperature variation with height in the large oil res in Kuwait was an important consideration. Measurements of the temperature T versus the height H were taken and presented in dimensionless terms as H : 1.0 2.0 3.0 4.0 5.0 T : 10.0 7.9 6.9 6.3 5.9 It is given that T varies as T A(H) a . Using linear regression methods, as applied to such equations, obtain the values of A and a from these data. How accurate is your correlation? 3.28. Experimental runs are performed on a compressor to determine the relationship between the volume ow rate Q and the pressure differ- ence P. It is expected that Q will be proportional to P b , where b is a constant. The measurements yield the mass ow rate Q for different pressure differences P as P (atm) 5.0 10.0 15.0 20.0 25.0 30.0 Q (m 3 /h) 7.4 13.3 16.5 19.0 20.6 24.3 It is known that there is some error in the data. Will you use a best or an exact t? Use the appropriate t to these data and determine the coefcients. Is your equation a good t? 204 Design and Optimization of Thermal Systems 3.29. Tests are performed on a nuclear power system to ensure safe shutdown in case of an accident. The measurements yield the power output P versus time T in hours as T (hours) 13591012 P (MW) 13.0 7.0 5.4 4.7 4.5 4.2 From theoretical considerations, the power is expected to vary as a b/T, where a and b are constants. It is also known that there is experimental error in the data. Will you use a best or an exact t? Use an appropriate t to these data points and determine the relevant constants. Is it a good curve t? Briey explain your answer. 3.30. Experiments are carried out on a plastic extrusion die to determine the relationship between the mass ow rate m and the pressure difference P. We expect the relationship to be of the form m AP n , where A and n are constants. The measurements yield the mass ow rate m for dif- ferent pressure differences P as m (kg/h) 12.8 15.5 17.5 19.8 22.0 P (atm) 10.0 15.0 20.0 25.0 30.0 Obtain a best t to these data and determine the coefcients A and n. Is this a good best t, or should we consider other functional relationships? 3.31. Use Polyt in MATLAB to get the best t to the following data, using rst-, second-, and third-order polynomials. Then plot the data as well as the three best-t curves obtained. Which is the best t? x: 0 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.2 y: 0 0.87 1.82 2.86 4.0 5.26 6.65 9.88 13.8 18.52 3.32. The ow rate F is given at various values of the pressure P as P 0.025 0.05 0.1 0.2 0.3 0.4 0.5 F 1.41 2.54 4.2 5.9 6.9 7.6 7.8 Use the last ve points to get an exact t. Use extrapolation with this t to obtain values at 0.025 and 0.05. Compare with given data. Comment on the results. 3.33. Obtain the rst-, second-, and third-order best ts to the above data. Plot the three curves and the data to determine the best curve to use. Modeling of Thermal Systems 205 3.34. In a chemical reaction, the effect of the concentration C of a catalyst on the reaction rate is investigated and the experimental results are tabulated as C (g/m 3 ) 0.1 0.2 0.5 1.0 1.2 1.8 2.0 2.6 3.5 4.0 R(g/s) 1.75 1.91 2.07 2.32 2.40 2.54 2.56 2.53 2.03 1.24 Using the method of least squares and considering polynomials up to the fth order, obtain a best t to these data. Which curve provides the best approximation to the given data? Also, compare the results with those obtained in Problem 3.26. 3.35. A small heated metal block cools in air. Its temperature T is measured as a function of time T and the results are given as T (s) 1 2 5 10 15 20 25 30 T(nC) 109.58 99.25 73.78 45.15 26.78 17.24 9.85 6.97 From the physical considerations of this problem, the temperature is expected to decay exponentially, as sketched in Figure P3.35. Obtain a best t to the given data and determine the two constants A and a. 3.36. The displacement x of a particle in a ow is measured as a function of time T. The data obtained are T(sec) : 0.0 1.0 2.0 3.0 4.0 5.0 x(m) : 0.0 2.0 8.0 20.0 40.0 62.0 Obtain a linear best t to these data. From this t calculate the values at T 2.0 and 4.0. Compare these with the given data and comment on the difference. How would you improve the accuracy of the curve t? 3.37. In an experiment, the signal from a sensor is measured over the veloc- ity range of 0 – 3 m/s. If the signal E is measured as 2, 9, 24, and 47 volts at the velocity V of 0, 1, 2, and 3 m/s, respectively: FIGURE P3.35 206 Design and Optimization of Thermal Systems (a) Obtain the highest-order polynomial E(V) that exactly ts the given data. (b) Obtain the best linear t, employing the method of least squares. (c) Determine the value of E at V 5 as calculated from the two curves obtained above and comment on the comparison between the two. 3.38. A thermocouple is being calibrated for temperature measurements by measuring its voltage output V in millivolts and the corresponding uid temperature T in nC, using a calibration device. For voltage values of 0, 0.1, 0.2, and 0.3 millivolts, the temperature is measured as 15, 18.5, 24, and 31.5nC. Determine the highest-order polynomial that ts the data and give the result as T F(V). Also, obtain a linear best t to these data using the method of least squares. Compare the two expressions obtained and comment on the difference. 3.39. In a heat transfer experiment, the heat ux q is measured at four values of the ow velocity, which is related to the uid ow rate. The veloc- ity V was measured as 0, 1, 2, 3, and 4 m/s and the corresponding heat ux as 1, 2, 9, 29, and 65 W/m 2 . It is desired to t a polynomial to these points so that q may be expressed as q f(V). What is the highest-order polynomial that may be obtained from these data? Also determine a linear best t to the given data. 3.40. The volume ow rate Q in m 3 /s of water in an open channel with a slight downward slope S and a hydraulic radius R is measured to yield the following data: R (m) 0.5 1.0 1.5 2.0 S 1.5 r 10 3 1.91 3.10 4.11 5.03 5 r 10 3 3.48 6.66 7.51 9.19 9 r 10 3 4.67 7.59 10.08 12.33 It is expected from theoretical considerations that Q varies with R and S as AR b S c , where A, b, and c are constants. Obtain a best t to the given data and determine these constants. [...]... thermal systems usually lead to nonlinear equations that have to be solved by iteration 218 Design and Optimization of Thermal Systems Number of iterations 200 10 0 opt 0 1. 0 1. 5 2.0 Relaxation factor FIGURE 4.3 Typical variation of the number of iterations needed for convergence of a linear system with the relaxation factor Example 4 .1 An industrial organization produces four items x1, x2, x3, and x4... differential equations often end up requiring the solution of sets of linear algebraic equations In addition, many applications, such as those concerned with fluid flow circuits, chemical reactions, conduction heat transfer, and data analysis, are often governed by linear systems A system of n linear equations may be written in the general form a11x1 a12 x 2 a1n x n b1 a21x1 a22 x 2 a2 n x n b2 an1x1 an 2 x 2... types of equations, along with important aspects such as accuracy, convergence, and stability of these methods (Smith, 19 65; Hornbeck, 19 75; Atkinson, 19 78; Gerald and Wheatley, 19 94; Ferziger, 19 98) A few others are concerned with problems of engineering interest and discuss the implementation of the algorithm on the computer (Carnahan et al., 19 69; James et al., 19 85; Jaluria, 19 96; Chapra and Canale,... taken as known, even though it is a function of the flow, which, in turn, depends on the geometry, dimensions, fluids, and flow rates (Incropera and Dewitt, 20 01) But, 210 Design and Optimization of Thermal Systems experimental results and simplified analysis are often used to avoid solving the full convective heat transfer equations because of the complexity of the resulting problem In addition, there... eigenvalue problem, which is of particular interest in stability of systems and flows Most of the direct methods are based on matrix inversion or on elimination, so that the given set of equations is reduced to a form that is amenable to 214 Design and Optimization of Thermal Systems a solution by algebraic analysis The important direct methods available in the literature are: 1 2 3 4 Gaussian elimination... a valid and accurate representation of the system, the model is subjected to changes in the design variables and operating conditions This process of studying the behavior of the system by means of a model, rather than by fabricating a prototype, is known as simulation The results obtained allow us to consider many different design possibilities as 207 208 Design and Optimization of Thermal Systems. .. boundary conditions, flow of material and energy, and interaction between the various components of the system We are interested in obtaining solutions to this coupled set of equations to determine the behavior and characteristics of the system for wide ranges of design variables and operating conditions Because of the coupled nature of these equations and because nonlinear algebraic and differential equations,... convergence characteristics of the Gauss-Seidel method can often be significantly improved by the use of point relaxation, given by xi(l 1) xi(l 1) GS (1 ) xi(l ) (4.9) where is a constant in the range 0 2 and [ xi(l 1) ]GS is the value of xi obtained for the (l 1) th iteration by using the Gauss-Seidel iteration If 0 1, the scheme is known as successive under-relaxation (SUR), and if 1 2, it is known as successive... the use of such available software is quite prevalent because interest often lies in obtaining the desired results as quickly as possible If a particular computer program has been successfully employed in the past, it is a good idea to use it for future applications However, it is necessary to understand the algorithm adopted by the available software so 212 Design and Optimization of Thermal Systems. .. Thermal Systems well as a variety of operating conditions Different designs may thus be evaluated to choose an acceptable design and safe levels may be established for the operating conditions These results are also used for optimization of the system Therefore, the success of the design and optimization process is strongly dependent on the numerical modeling and simulation of the system The basic considerations . m 3 /s of water in an open channel with a slight downward slope S and a hydraulic radius R is measured to yield the following data: R (m) 0.5 1. 0 1. 5 2.0 S 1. 5 r 10 3 1. 91 3 .10 4 .11 5.03 5 r 10 3 3.48. furnace at 10 cm/s and 800nC. At distances of 10 m each, there are three rolling dies; see Figure 1. 10( d). The initial thick- ness of the sheet is 2 cm and at each die, a reduction of 20% in thickness. is a function of the ow, which, in turn, depends on the geometry, dimensions, uids, and ow rates (Incropera and Dewitt, 20 01) . But, 210 Design and Optimization of Thermal Systems experimental
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