Design and Optimization of Thermal Systems Episode 1 Part 10 pdf

Assuming uniform temperature in the tank and a convective loss to the environment at the surface, with a heat transfer con-coefficient h, obtain a mathematical model for this system.. En

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 R1, the outer of radius R2 and both being of length L,

as shown in Figure P3.1 The inner cylinder is heated electrically and

supplies a constant heat flux 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 Tmax, 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 coefficient at the plastic surface is given as h and the surface

Metal pieces

emissivity as E The cylinders are placed perpendicular to the tion of motion and are rotated as they move across the oven Develop a simple mathematical model for obtaining the temperature in the plastic

direc-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 tions made

approxima-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chemical process For this purpose, a storage tank of volume V and surface area A is employed Whenever hot water is withdrawn from the tank, cold water at temperature T a , where T a is the ambient tem-perature, flows into the tank A heater supplying energy at the rate of

Q turns on whenever the temperature reaches T c – $Tc and turns off

when it is reaches T c $T c The heater is submerged in the water tained in the tank Assuming uniform temperature in the tank and a convective loss to the environment at the surface, with a heat transfer

con-coefficient 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 flow rate m of hot water and also for the case when there is no

outflow, m 0

3.4 Consider a cylindrical rod of diameter D undergoing thermal ing and moving at a speed U as shown in Figure P3.4 The rod may

process-be assumed to process-be infinite in the direction of motion Energy transfer

occurs at the outer surface, with a constant heat flux input q and tive loss to the ambient at temperature T a and heat transfer coefficient

convec-h Assuming one-dimensional, steady transport, obtain the governing

equation and the relevant boundary conditions By ization, determine the governing dimensionless parameters Finally,

nondimensional-obtain T(x) for (a) h  0 and (b) q  0.

3.5 Give the governing equations and boundary conditions for the 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

steady-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 first section, the bolts are heated at a constant heat

flux q In the second and third sections, they lose energy by convection

Modeling of Thermal Systems 199

to the air at temperature T a at convective heat transfer coefficients h1and h2 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 ume V is melted in a furnace The initial temperature of the block is

vol-T i , the melting point is T m , and the final temperature is T f , where T f

T m  T i The block is exposed to a constant heat flux input q due to

radiation and also loses energy by convection to the surrounding air

at T a with a convective heat transfer coefficient 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 flow rate m Assume a cubical tank of

cold water surrounded by insulation of uniform thickness Water at the ambient temperature flows into the tank to make up the cold-water out-flow The refrigeration unit turns on if the water temperature reaches a

value Tmax and turns off when it is drops to Tmin, 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

coefficient 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 III

II I

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 coefficient is 20 W/m2K, 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 flue gases at temperature T f (x) exchange energy with an overall heat transfer coefficient h f On the other side, inert gases at temperature

T g (x) have a heat transfer coefficient 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 briefly 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 fluids

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 fixed, 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 counterflow heat exchanger, the heat loss to the environment is to

be included in the mathematical model Considering the case of the hot fluid on the outside and the colder fluid 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 flat plate at room temperature is immersed in the chamber at time zero Develop a simple mathematical

Energy loss

Hot Cold

FIGURE P3.16

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 simplifications and assumptions, indicating these in your answer.

3.21 A flat steel (R  10000 kg/m3, C  500 J/kg•K, k  100 W/m•K) sheet

emerges from a furnace at 10 cm/s and 800nC At distances of 10 meach, 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 coefficient of 120 W/m2•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 fit to these data is to be obtained for the design of air-conditioning systems Will an exact or a best fit be more appropriate? Suggest a suitable form

of the function for curve fitting

3.23 The average daily air temperature at a location is available for each day

of 2005 We wish to obtain a best fit 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 fit the data and outline the reasons for your choice Outline the mathematical procedure to determine the constants of the equation chosen for the curve fit

3.24 A steel sphere at initial temperature T o is immersed in a cold fluid at

temperature T a and allowed to cool rapidly for hardening At 20 time intervals Ti , 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 fit 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/m3, and specific heat 300 J/kg•K is heated by convection from a hot

fluid 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 tion of time T Obtain the general form of the solution If the measured temperature values at different time intervals are given as

func-Modeling of Thermal Systems 203

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 coefficient h.

3.26 Obtain a linear best fit 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 fit satisfactory in this case?

3.27 The temperature variation with height in the large oil fires 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 flow rate Q and the pressure ence P It is expected that Q will be proportional to P b, where b is a

differ-constant The measurements yield the mass flow rate Q for different pressure differences P as

P (atm) 5.0 10.0 15.0 20.0 25.0 30.0

Q (m3 /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 fit? Use the appropriate fit to these data and determine the coefficients Is your equation a good fit?

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

3.30 Experiments are carried out on a plastic extrusion die to determine the

relationship between the mass flow 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 flow rate m for ferent pressure differences P as

dif-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 fit to these data and determine the coefficients A and

n Is this a good best fit, or should we consider other functional

3.32 The flow rate F is given at various values of the pressure P as

F 1.41 2.54 4.2 5.9 6.9 7.6 7.8

Use the last five points to get an exact fit Use extrapolation with this fit

to obtain values at 0.025 and 0.05 Compare with given data Comment

on the results

3.33 Obtain the first-, second-, and third-order best fits 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/m3 ) 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 fifth order, obtain a best fit 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(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 fit to the given data and determine the two constants A and a.

3.36 The displacement x of a particle in a flow 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 fit to these data From this fit calculate the values

atT  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 fit?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

(a) Obtain the highest-order polynomial E(V) that exactly fits the

given data

(b) Obtain the best linear fit, 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 fluid 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 fits the data

and give the result as T  F(V) Also, obtain a linear best fit 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 flux q is measured at four values

of the flow velocity, which is related to the fluid flow rate The

veloc-ity V was measured as 0, 1, 2, 3, and 4 m/s and the corresponding heat

flux as 1, 2, 9, 29, and 65 W/m2 It is desired to fit 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 fit to the given data

3.40 The volume flow rate Q in m3/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:

be varied in an experimental or analytical study to characterize the behavior of the given thermal system Physical modeling, which involves experimentation on

a scale model of the system, is used as a means to obtain results that are not easily extracted from mathematical modeling Curve fitting is often used to derive alge-braic equations and expressions to represent experimental or numerical results,

as well as data on material properties, environmental conditions, financial trends, and equipment characteristics

As a consequence of mathematical and physical modeling, along with curve fitting, mathematical equations that govern the behavior of the thermal system are obtained These equations are generally linked to each other through mate-rial properties, 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 character-istics 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, including both ordinary and partial differential equa-tions, arise in typical thermal systems, analytical solutions are rarely possible and

we have to resort to numerical techniques to obtain the desired results

A numerical model of the thermal system refers to a computational or

numeri-cal representation of the system on a computer, which may be used to approximate the behavior and characteristics of the system It consists of a numerical scheme or procedure that would yield a solution to the governing mathematical equations, with numerically imposed boundary and initial conditions, relevant property data, com-ponent characteristics, and other inputs needed for representing the entire system The numerical algorithm, as well as its implementation on a computer, constitutes the numerical model Once the model is confirmed to be a valid and accurate rep-resentation 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

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 in the development of a numerical model are first presented in this chapter, followed by a discussion of numerical simulation of a thermal system

The numerical solution of mathematical equations that are commonly tered in engineering applications is covered in a variety of courses dealing with numerical analysis A large number of books available in this area discuss various methods for different types of equations, along with important aspects such as accuracy, convergence, and stability of these methods (Smith, 1965; Hornbeck, 1975; Atkinson, 1978; Gerald and Wheatley, 1994; Ferziger, 1998) A few oth-ers are concerned with problems of engineering interest and discuss the imple-mentation of the algorithm on the computer (Carnahan et al., 1969; James et al., 1985; Jaluria, 1996; Chapra and Canale, 2002) There has also been a substantial increase in interest in the solution of the relevant mathematical problems by the use of MATLAB, as presented by Recktenwald (2000), Mathews and Fink (2004), Chapra (2005), and others This chapter presents a brief discussion of numerical modeling in order to indicate the main concerns with respect to thermal systems and the most commonly used techniques For further details, the aforementioned books and others available on this subject may be consulted

encoun-4.1.1 G ENERAL F EATURES

The main purpose of numerical modeling is to develop a computational code, implemented on a digital computer, which provides a physically valid and accu-rate representation of the real system and allows the behavior of the system to

be determined under different conditions Thus, a one-to-one correspondence is established between the physical thermal system and the numerical model so that the desired information on system characteristics and behavior can be obtained

by subjecting the numerical model to different conditions As shown cally in Figure 4.1, the inputs into the physical system, arising due to changes in

schemati-Outputs

System behavior

Physical system

Numerical model