Design and Optimization of Thermal Systems Episode 2 Part 2 ppt

In summary, the numerical model for the complete thermal system may tain programs that have been developed by the user, those in the public domain, standard programs available on the com

become uniform at Q  1 Figure 4.18 shows the variation of the temperature at several locations in the plate with time T Again, the approach to steady state at large time is clearly seen The Crank-Nicolson method is a very popular choice for such one-dimensional problems because of the second-order accuracy in time and space Tridiagonal sets of equations are generated for one-dimensional problems, and these may be solved conveniently and accurately by the Thomas algorithm to yield the desired solution.

4.3 NUMERICAL MODEL FOR A SYSTEM

We now come to the numerical model for the overall system, which may comprise several parts, constituents, or subsystems The model may be relatively simple, as is the case for systems with a small number of components such as a refrigerator, or may be very involved, as is the case for a major undertaking such as a power plant The numerical model may be developed by the users themselves or it may be based

on a commercially available general-purpose code such as Fidap, Ansys, Phoenics, Simpler, or Fluent Specialized programs for specific applications are also available Since the development of computer codes for large thermal systems is a very elabo-rate and time-consuming process, it is often more convenient and efficient to use

a commercially available program Consequently, such codes are employed sively in industry and form the basis for the numerical simulation and design of a variety of thermal systems ranging from electronic packages to air-conditioning and energy systems However, it is important to be conversant with the algorithm used

exten-in the software and to be aware of its applicability, accuracy, limitations, and ease with which inputs may be given to simulate different circumstances

Even if the numerical model is being developed indigenously, software able on the computer or in the public domain may be employed effectively This

avail-is particularly the case for graphics programs and standard programs, such as matrix methods for solving sets of linear algebraic equations and the Runge-Kutta method for the solution of ODEs Again, we must be familiar with the numerical approach used in the software and must have information on its accuracy and pos-sible limitations It is rarely necessary to develop the numerical code for graphics, because a wide variety of programs, such as Tecplot, are conveniently available and easy to use for different needs, ranging from line graphs to contour plotting Similarly, programs for curve fitting are widely used for the analysis of experi-mental or numerical data and for the derivation of appropriate correlations

In summary, the numerical model for the complete thermal system may tain programs that have been developed by the user, those in the public domain, standard programs available on the computer, and even commercially available general-purpose programs, with all of these linked to each other to simulate dif-ferent aspects or components of the system In addition to these programs, the numerical model may be linked with available information on material proper-ties, characteristics of some of the devices or components in the system, heat transfer correlations, and other relevant information The range of applicability

con-of the complete numerical model and the expected accuracy con-of the results are determined through validation studies

4.3.1 M ODELING OF I NDIVIDUAL C OMPONENTS

Isolating System Parts

The first step in the mathematical and numerical modeling of a thermal system

is to focus on the various parts or components that make up the system In many cases, the choice of individual components is obvious For instance, in a vapor compression refrigeration system, the compressor, the condenser, the evapora-tor, and the throttling value may be taken as the components of the system (see Figure 4.19) Each component here may be considered as a separate entity, in terms of the thermodynamic process undergone by the refrigerant and geom-etry, design, and location of the component Similar subdivisions are employed in many thermodynamic systems such as those in energy generation, heating, cool-ing, and transportation The components are chosen so that these are relatively self-contained and independent in order to facilitate the modeling However, all such components will ultimately be linked to each other through energy, material, and momentum transport For instance, in a refrigeration system, the refrigerant flows from one component to the other, conveying the energy stored in the fluid,

as shown in Figure 1.8 In each component, energy exchanges occur, leading to the resulting thermodynamic state of the fluid at the exit of the component

In many cases, the choice of the individual components is not so obvious However, differences in geometry, material, function, thermodynamic state, loca-tion, and other such characteristics may be used to separate the components For instance, the walls and ceiling of a room may be treated as separate components because of the different transport mechanisms they are exposed to The walls and the outside insulation in a furnace may be treated as different components because of the difference in material The main thing to remember is that the component must be substantially separate or different from the others and must

be amenable to modeling as an individual item

The given system may also be broken down into subsystems, each with its own components Then each subsystem is treated as a system for model development,

with all the individual models being brought together at the end For instance, the cooling system in an automobile, the boiler in a power plant, and the cooling arrangement in an electronic system may be considered as subsystems for model-ing and design Frequently, the subsystems are designed separately and the results obtained are employed in the design of the overall system, treating the subsystem simply as a component whose characteristics are known.

heat transfer coefficient h at the wall surface This decouples the solutions for the

two heat transfer regions since the conditions at the boundary

T n

(4.34a)

which requires a solution for the flow and heat transfer in air, are replaced by

tt

n wall h T( wall Tair) (4.34b)

where n is in the direction normal to the surface and Tair is a specified ture Similarly, the air is modeled separately for heat transfer with a specified wall temperature Then, the two regions are modeled as separate entities without

Air flow

Door

Walls

Wall Solar

flux

Solar flux

FIGURE 4.20 Decoupling a wall and enclosed air for modeling thermal transport in a

room.

linking the two Similarly, the condenser in a home air-conditioning system may

be modeled using given, fixed inflow conditions of the refrigerant to decouple it from the compressor that provides the input to the condenser in the actual system Then each component can be subjected to mathematical modeling procedures and the resulting mathematical equations derived

Different simplifications and idealizations may apply for different ponents, resulting in different types of governing equations For example, one component may be modeled as a lumped mass, giving rise to an ODE for the temperature as a function of time, while another component may be modeled

com-as a one-dimensional transient problem, governed by Equation (4.27) A single nonlinear algebraic equation may arise from an energy balance for determining the temperature at the surface of a body The continuity, momentum, and energy equations may be needed for modeling the flow

The different mathematical models obtained for the various system parts are based on simplifications, approximations, and idealizations that are, in turn, based on the material, geometry, transport processes, and boundary conditions Estimates of the contributions of the various mechanisms play an important part

in the modeling process However, the mathematical model derived is not unique and further improvements may be needed, depending on the numerical results from simulation and on comparisons with experimental data Therefore, it is important to maintain the link between mathematical and numerical models and

to be prepared to improve both as the need arises It is generally best to start with the simplest possible mathematical and numerical models and to improve these gradually by including effects that may have been neglected earlier

Numerical Modeling

The governing mathematical equations for each component must be solved to study its behavior Numerical algorithms applicable to the different types of equa-tions that arise are employed to solve these equations Thus, a numerical model, which is decoupled from the others, is obtained for each component The results from this model indicate the basic characteristics of the component under the idealized or approximated boundary conditions used The behavior of the compo-nent, as some of these conditions or related parameters are varied, may be studied

in order to ensure that the individual model is physically realistic For instance, the flow rate of the colder fluid in a heat exchanger may be increased, keeping the other variables fixed It is expected that the temperature rise of this fluid in the heat exchanger will decrease because a larger amount of fluid is to be heated The results from the numerical model must show this trend if the model is physi-cally valid Grid refinement is also done to ensure the accuracy of the results Some simple analytical results, if available, may also be employed to check the accuracy of the numerical results obtained Particularly simple cases may be con-sidered to obtain analytical solutions and thus provide a method of validating the individual numerical models

4.3.2 M ERGING OF D IFFERENT M ODELS

Once all the individual numerical models for the various components or parts

of the given thermal system have been obtained and tested on the basis of cal reasoning and analytical results, these must be merged to obtain the model for the overall system Such a merging of the models requires bringing back the coupling between the different parts that had been neglected in the development

physi-of individual models For instance, if two parts A and B physi-of the system exchange energy by radiation, their temperatures T A and T B are coupled through a boundary condition of the form

Q A T A A T B

AA AB

temperature environment in radiation exchange with the surface Thus, the

tem-perature of part B is eliminated from the model for part A, which may then

be modeled separately Similarly, such approximations with constant or known

parameters in the boundary conditions may be used for modeling part B In the

process of merging the two models, these approximations must be replaced by the actual boundary condition, Equation (4.35), which couples the temperatures

of the two parts

Similarly, the approximation made in Equation (4.34b) is removed by ing the convective condition by the correct boundary conditions given by Equation (4.34a) This couples the transport in the wall with that in the air For the air-conditioning system considered earlier, the temperature and pressure at the inlet

replac-of the condenser are set equal to the corresponding values at the exit replac-of the pressor to link these two parts of the system Proceeding in this way, other parts are also coupled through the boundary and inflow/outflow conditions

com-This approach of modeling individual parts and then coupling them may appear to be an unnecessarily complicated way of deriving the numerical model for the system Indeed, for relatively simple systems consisting of a small number

of parts, it is often more convenient and efficient to develop the numerical model for the system without considering individual parts separately However, if the system has a large number of parts, it is preferable to develop individual numerical models and to test and validate them separately before merging them to obtain the model for the system This allows a complicated problem to be broken down into simpler ones that may be individually treated and tested before final assembly This approach is used extensively in industry to model complex systems A direct modeling of the entire system has little chance of success because many coupled

equations are involved Figure 4.21 shows a schematic of the process described here for a general thermal system.

4.3.3 A CCURACY AND V ALIDATION

We discussed the validation of the mathematical model and the numerical scheme earlier Numerical models for individual parts of the system are similarly tested and validated before merging them to yield the overall numerical model of the thermal system under consideration The validation of this complete model, therefore, is based largely on the testing and validation employed at various steps along the process The main considerations that form the basis for validation of the numerical model of the entire system are, as before,

1 Results should be independent of arbitrary numerical parameters

2 Physical behavior

3 Comparison with analytical and experimental results

4 Comparisons with prototype results

The arbitrary numerical parameters refer to the grid, time step, and other quantities chosen to obtain a numerical solution It is important to ensure that the results from the model are essentially independent of these parameters, as was done earlier for the numerical solution of individual equations and components The physical behavior now refers to the thermal system, so that the results from the model are considered in terms of the expected physical trends to ascertain that the model does indeed yield physically realistic characteristics The numeri-cal model is subjected to a range of operating conditions and the results obtained examined for physical consistency

Numerical model of system

Physical system

Numerical models

Mathematical models Components

FIGURE 4.21 Schematic of the general approach of developing an overall model for a

thermal system.

Analytical and experimental results are rarely available for validation ever, as discussed earlier, analytical results may be obtained for a few highly idealized situations Similarly, experimental data may be obtained or may be available for a few simple geometries and conditions Such analytical results and experimental data are used for validating mathematical and numerical models for individual parts of the system For the overall system, experimental data may

How-be available from existing systems For instance, existing cooling and heating systems may be numerically modeled in order to compare the results against data available on these systems Before going into production, a prototype may be developed to test the model and the design This provides the best information for the quantitative validation of the model and a check on the accuracy of the results obtained from the model

4.4 SYSTEM SIMULATION

System simulation refers to the process of obtaining quantitative information on the behavior and characteristics of the real system by analyzing, studying, or examin-ing a model of the system The model may be a physical, scaled-down version of the given system, derived on the basis of the similarity principles outlined in Chapter 3 Such a model may be subjected to a variety of operating and environmental condi-tions and the performance of the system determined in terms of variables such as pressure, flow rate, temperature, energy input/output, and mass transfer rate that are

of particular interest in thermal systems The results from such a simulation may

be expressed in terms of correlating equations derived by curve-fitting techniques Physical modeling and testing of full-size components such as compressors, pumps, and heat exchangers are often used to derive the performance characteristics of these components This approach is rarely used for the entire system because of the cost and effort involved in fabrication and experimentation

Sometimes the given physical system may be simulated by investigating another system that is governed by the same equations and that may be easier

to fabricate or assemble Such a model, called an analog model in the preceding chapter, also has a limited range of applicability, and, therefore, this simulation is not often used in the design of thermal systems Electrical circuits used to simulate fluid-flow systems, consisting of pipes, fittings, valves, and pumps, and conduction heat transfer through a multi-layered wall are examples of analog simulation

In the remaining portion of this chapter, we will consider only system lation based on mathematical and numerical modeling Therefore, the govern-ing equations obtained from the mathematical model are solved by analytical

simu-or numerical methods to yield the system behavisimu-or under a variety of ing conditions as well as for different design variables, in order to provide the quantitative inputs needed for design and optimization Mathematical solutions are obtained in only a few, often highly idealized, circumstances, and numerical modeling is generally needed to obtain the desired results for practical problems Performance characteristics of components, as obtained from separate physical modeling and tests, as well as material properties, form part of the overall model

operat-and are assumed to be known These may be available in the form of data or

as correlating equations The governing equations may be algebraic equations, ordinary or partial differential equations, integral equations, or a combination

of these Therefore, a numerical model is developed to solve the resulting taneous equations, many of which are typically nonlinear for thermal systems Simulation of the system is carried out by means of this model

simul-4.4.1 I MPORTANCE OF S IMULATION

System simulation is one of the most important elements in the design and mization of thermal systems Since experimentation on a prototype of the actual thermal system is generally very expensive and time consuming, we have to depend on simulation based on a model of the given system to obtain the desired information on the system behavior under different conditions A one-to-one cor-respondence is established between the model and the physical system by valida-tion of the model, as discussed earlier Then the results obtained from a simulation

opti-of the model are indicative opti-of the behavior opti-of the actual system

There are several reasons for simulating the system through its mathematical and numerical model Simulation can be used to

1 Evaluate different designs for selection of an acceptable design

2 Study system behavior under off-design conditions

3 Determine safety limits for the system

4 Determine effects of different design variables for optimization

5 Improve or modify existing systems

6 Investigate sensitivity of the design to different variables

yield-a heyield-at exchyield-anger yield-applicyield-ation involving given fluids yield-and given requirements on the temperatures or the heat transfer rates Instead of fabricating each of these heat exchanger designs, mathematical and numerical modeling may be employed to obtain a satisfactory and accurate model This model is then used for simulating the actual system in order to obtain the desired outputs in terms of heat transfer rates and temperatures Operating conditions for which the system is designed are considered first to determine if the design meets the given requirements and constraints These conditions are often termed design conditions because they form the basis for the design Even if only one design has been developed for a given application, it must be evaluated to ensure that it is acceptable

Off-Design Performance and Safety Limits

Predicting the behavior of the system under off-design conditions, i.e., values beyond those used for the design, is another important use of system simulation Such a study provides valuable information on the operation of the system and how it would perform if the conditions under which it operates were to be altered,

as under overload or fractional-load circumstances Systems seldom operate at the design conditions and it is important to determine the range of operating condi-tions over which they would deliver acceptable results The deviation from design conditions may occur due to many reasons, such as variations in energy input, differences in raw materials fed into the system, changes in the characteristics

of the components with time, changes in environmental conditions, and shifts in energy load on the system The results obtained from simulation under off-design conditions would indicate the versatility and robustness of the system It is obvi-ously desirable to have a wide range of off-design conditions for which the system performance is satisfactory A narrow range of acceptability is generally not suit-able for consumer products because large variations in the operating conditions are often expected to arise For instance, a residential hot-water system designed for a particular demand and given inlet temperature must be able to perform satis-factorily if either of these were to vary substantially In manufacturing processes,

it is common to encounter variations in the shape, dimensions, and material erties of the items undergoing thermal processing

prop-These outputs also indicate the safety limits of the system It is important to determine the maximum thermal load an air conditioner can take, the maximum power input to a furnace that can be given, and so on, without damage to the system or the user Safety features can then be built into the system, such as man-datory shutdown of the system if the safety levels are exceeded or warning lights

to indicate possible damage to the system We are all familiar with such features

in cars, lawn mowers, and other systems in daily use Some of these aspects were also considered earlier in Section 2.3.6

Optimization

System simulation plays an important role in optimization of the system As will

be seen in later chapters, the outputs from the system must be obtained for a range

of design variables in order to select the optimum design The optimization of the system may involve minimization of parameters such as cost per item, weight, and energy consumption per unit output or maximization of quantities such as out-put, return on investment, and rate of energy removal Whatever the criterion for optimization, it is essential to change the variables over the design domain, deter-mined by physical limitations and constraints, and to study the system behavior Then, using the various techniques for optimization presented later, the optimal design is determined The results obtained from simulation may sometimes be curve fitted to yield algebraic equations, which greatly facilitate the optimiza-tion process For instance, if the cooling system for an electronic equipment has been designed using a fan, different locations, flow rates, and dimensions of the

fan may be considered to derive algebraic equations to represent the dependence

of the heat removal rate on these variables Then, an optimal configuration that delivers the most effective cooling per unit cost may be obtained easily

Modifications in Existing Systems

The use of simulation for correcting a problem in an existing system or for fying the system for improving its performance is also an important application Rather than changing a particular component in order to correct the problem

modi-or improve the system, simulation is first used to determine the effect of such a change Since the simulation closely represents the actual physical system, the usefulness of the proposed change can be determined without actually carrying out the change For instance, if a flow system is unable to deliver the expected flow rates, the problem may lie with various sections of the piping, pipe fittings, valves, or pumps Instead of proceeding to change a given valve or pump, simula-tion may be used to determine if indeed the problem is caused by a particular item and if an improved version of the item will be worthwhile It may be shown by the simulation that the lack of flow at a given point is due to some other cause, such as blockage in a particular section of the piping Clearly, considerable savings may

be obtained by using simulation in this manner

Sensitivity

A question that arises frequently in design is the effect of a given variable or component on the system performance For instance, if the dimensions of the channels or of the collectors in a solar collection system were varied, what would

be the overall effect on the system? Similarly, if the capacity of the fan or blower

in a cooling system were varied, how would it affect the heat removal rate? Such questions relate to the sensitivity of the system performance to the design vari-ables and are important from a practical viewpoint A substantial reduction in the cost of the system may be obtained by slight changes in the design in order to use standard items available in the market Pipes and tubings are usually available at fixed dimensions and if these could be employed in the system, rather than the exact custom-made dimensions, substantial savings may result Similarly, fluid flow components such as blowers, pumps, and fans are often cheaply and eas-ily available for given specifications At different values, these may have to be fabricated individually, raising the price substantially System simulation is used

to determine the sensitivity of the system performance to such variables and to decide if slight alterations can be made in the interest of reducing the cost without significant sacrifice in system characteristics

4.4.2 D IFFERENT C LASSES

Several types of simulation are used for thermal systems We have already tioned analog and physical simulations, which are based on the corresponding form of modeling, as discussed in Chapter 3 In this chapter, we have focused on

men-numerical modeling and men-numerical simulation, which are based on the ical modeling of the thermal system Three main classes of this form of system simulation are discussed here.

mathemat-Dynamic or Steady State

The simulation of a system may be classified as dynamic or steady state The

former refers to the circumstances where changes in the operating conditions and relevant system variables occur with respect to time Many thermal systems are time-dependent in nature and a dynamic simulation is essential This is particu-larly true for the start-up and shutdown of the system Also, in most manufactur-ing processes the temperature and other attributes of the material undergoing thermal processing vary with time, as shown in Figure 2.1 The system itself may vary with time over the duration of interest due to energy input as in welding, gas cutting, heat treatment, and metal forming In processes such as crystal growing, ingot casting, and annealing, the system varies with time along with the tempera-ture of the material being processed Dynamic simulation is also needed to study the response of the system to changes in the operating conditions such as a sharp increase in the heat load on a food freezing plant The results obtained from a dynamic simulation are also useful in the design and study of the control scheme for a satisfactory operation of the system

Steady-state simulation refers to situations where changes with respect to time are negligible or do not occur Since the dependence of the variables on time

is eliminated, a steady-state simulation is much simpler than the corresponding dynamic simulation In addition, the steady-state approximation can be made in a large number of practical cases, making steady-state simulation of greater interest and importance in thermal systems Except for times close to start-up and shut-down, many systems behave as if they are under steady-state conditions Thus,

a blast furnace may be treated as essentially steady over much of its operation

A typical system that is transient at the beginning and end of its operation and steady over the rest is shown in Figure 4.22 In addition, the system itself may be approximated as steady even though the temperature of the material undergoing

Time

Shutdown Steady

Start-up

FIGURE 4.22 Temperature variation in a typical thermal system that is steady over most

of the duration of operation and is time-dependent only near start-up and shutdown.

thermal processing varies with time An example of this is a circuit board being baked The baking oven may be approximated as being unchanged and operating under steady-state conditions while the board undergoes a relatively large tem-perature change as it moves through the oven.

in a continuum Particles, if present in the flow, are not treated separately but as part of the average properties of the fluid Most thermodynamic systems can be simulated as continuous because energy and fluid flow are generally continuous [see Figure 4.23(a)]

On the other hand, if discrete pieces, such as ball bearings, fasteners, and gears, undergoing thermal processing, are considered, the simulation focuses on

a finite number of such items In the manufacture of television sets, individual glass screens are heat treated as they pass through a furnace on a conveyor belt,

as sketched in Figure 4.23(b) In such cases, the mass, momentum, and energy

(b) (a)

q

Reflector Discrete items

q

FIGURE 4.23 (a) Continuous and (b) discrete simulation.