Modeling of Thermal Systems • Importance of Modeling in Design • The process of simplifying a given problem so that it may be represented in terms of a system of equations, for analysis, or a physical arrangement, for experimentation, is termed modeling. • Modeling is needed for understanding and predicting the behavior and characteristics of thermal systems. • Once a model is obtained, it is subjected to a variety of operating conditions and design variations. • If the model is a good representation of the actual system under consideration, the outputs obtained from the model characterize the behavior of the given system. • This information is used in the design process as well as in the evaluation of a particular design to determine if it satisfies the given requirements and constraints.

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Modeling of Thermal Systems • Basic Features of Modeling • The model may be descriptive or predictive. We are all very familiar with models that are used to describe and explain various physical phenomena. • A working model of an engineering system, such as a robot, an internal combustion engine, a heat exchanger, or a water pump, is often used to explain how the device works. • Frequently, the model may be made of clear plastic or may have a cutaway section to show the internal mechanisms. Such models are known as descriptive and are frequently used in classrooms to explain basic mechanisms and underlying principles. • Predictive models are of particular interest to our present topic of engineering design because these can be used to predict the performance of a given system.

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Modeling of Thermal Systems • The equation governing the cooling of a hot metal sphere immersed in an extensive cold-water environment represents a predictive model because it allows us to obtain the temperature variation with time and determine the dependence of the cooling curve on physical variables such as initial temperature of the sphere, water temperature, and material properties. • Models such as the control mass and control volume formulations in thermodynamics, representation of a projectile as a point to study its trajectory, and enclosure models for radiation heat transfer are quite common in engineering analysis for understanding the basic principles and for deriving the governing equations. • A few such models are sketched in Figure 3.1.

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• FIGURE 3.1 A few models used commonly in engineering: (a) Control volume, (b) control mass, (c) graphical representation, and (d) enclosure configuration for thermal radiation analysis. Shiv Sharma

Modeling of Thermal Systems • In many practical systems, it is not possible to simplify the problem enough to obtain a sufficiently accurate analytical or numerical solution. • In such cases, experimental data are obtained, with help from dimensional analysis to determine the important dimensionless parameters. • Experiments are also crucial to the validation of the mathematical or numerical model and for establishing the accuracy of the results obtained. • Material properties are usually available as discrete data at various values of the independent variable, e.g., density and thermal conductivity of a material measured at different temperatures. • For all such cases, curve fitting is frequently employed to obtain appropriate correlating equations to characterize the data. • These equations can then serve as inputs to the model of the system, as well as to the design process. • Curve fitting can also be used to represent numerical results in a compact and convenient form, thus facilitating their use. • Figure 3.2 shows a few examples of curve fitting as applicable to thermal processes, indicating best and exact its to the given data. Shiv Sharma

• FIGURE 3.2 Examples of curve fitting in thermal processes.

Types of Models There are four main types of predictive models that are of interest in the design and optimization of thermal systems. These are: 1. Analog models 2. Mathematical models 3. Physical models 4. Numerical models

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Modeling of Thermal Systems • Analog models are based on the analogy or similarity between different physical phenomena and allow one to use the solution and results from a familiar problem to obtain the corresponding results for a different unsolved problem. • The use of analog models is quite common in heat transfer and fluid mechanics. • An example of an analog model is provided by conduction heat transfer through a multilayered wall, which may be analyzed in terms of an analogous electric circuit with the thermal resistance represented by the electrical resistance and the heat flux represented by the electric current, as shown in Figure 3.3(a). • The temperature across the region is the potential represented by the electric voltage. • Then, Ohm’s law and Kirchhoff’s laws for electrical circuits may be employed to compute the total thermal resistance and the heat flux for a given temperature difference.

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Modeling of Thermal Systems •

FIGURE 3.3 Analog models. (a) Conduction heat transfer in a composite wall; (b) analog model of plume flow in a room fire; and (c) flow diagram for material flow in an industry.

The density differences that arise in room fires due to temperature differences are often simulated experimentally by the use of pure and saline water, the latter being more dense and thus representative of a colder region. •The flows generated in a fire can then be studied in an analogous saltwater/pure-water arrangement, which is often easier to fabricate, maintain, and control.

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Modeling of Thermal Systems • Figure 3.3(b) shows the analog modeling of a fire plume in an enclosure. The low is closely approximated. However, the jet is inverted as compared to an actual fire plume, which is buoyant and rises; salt water is heavier than pure water and drops downward. • A graph is itself an analog model because the coordinate distances represent the physical quantities plotted along the axes. Flow charts used to represent computer codes and process flow diagrams for industrial plants are all analog models of the physical processes they represent; see Figure 3.3(c). • Mathematical Models • A mathematical model is one that represents the performance and characteristics of a given system in terms of mathematical equations. • These models are the most important ones in the design of thermal systems because they provide considerable versatility in obtaining quantitative results that are needed as inputs for design. • Mathematical models form the basis for numerical modeling and simulation, so that the system may be investigated without actually fabricating a prototype.

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Modeling of Thermal Systems • Physical Models • A physical model is one that resembles the actual system and is generally used to obtain experimental results on the behavior of the system. • An example of this is a scaled down model of a car or a heated body, which is positioned in a wind tunnel to study the drag force acting on the body or the heat transfer from it, as shown in Figure 3.4. FIGURE 3.4 Physical modeling of (a) fluid flow over a car and (b) heat transfer from a heated body.

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Modeling of Thermal Systems • Numerical Models • Numerical models are based on mathematical models and allow one to obtain, using a computer, quantitative results on the system behavior for different operating conditions and design parameters. • Numerical modeling refers to the restructuring and discretization of the governing equations in order to solve them on a computer. The relevant equations may be algebraic equations, ordinary or partial differential equations, integral equations, or combinations of these, depending upon the nature of the process or system under consideration. • Numerical modeling involves selecting the appropriate method for the solution, for instance, the finite difference or the finite element method; discretizing the mathematical equations to put them in a form suitable for digital computation; choosing appropriate numerical parameters, such as grid size, time step, etc.; and developing the numerical code and obtaining the numerical solution.

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Modeling of Thermal Systems

• FIGURE 3.5 Numerical modeling. (a) A computer flowchart for a hot-water storage system and (b) various inputs and components that constitute a typical numerical model for a thermal system.

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Modeling of Thermal Systems • Other Classifications • There are several other classifications of modeling frequently used to characterize the nature and type of the model. • A steady-state model is one whose properties and operating variables do not change with time. If time-dependent aspects are included, the model is dynamic. • Thus, the initial, or start-up, phase of a furnace would require a dynamic model, but this would often be replaced by a steady-state model after the furnace has been operating for a long time and the transient effects have died down. • Deterministic models predict the behavior of the system with certainty, whereas probabilistic models involve uncertainties in the system that may be considered as random or as represented by probability distributions. • Models for supply and demand are often probabilistic, while typical thermal systems are analyzed with deterministic models. • Lumped models use average values over a given volume, whereas distributed models provide information on spatial variation. Shiv Sharma

Modeling of Thermal Systems • Discrete models focus on individual items, whereas continuous models are concerned with the flow of material in a continuum. • In a heat treatment system, for instance, a discrete model may be developed to study the transport and temperature variation associated with a given body, say a gear, undergoing heat treatment. • The flow of hot gases and thermal energy, on the other hand, is studied as a continuum, using a continuous model. • Mathematical Modeling • Transient/steady state • One of the most important considerations in modeling is whether the system can be assumed to be at steady state, involving no variations with time, or if the time-dependent changes must be taken into account. Since time brings in an additional independent variable, which increases the complexity of the problem, it is important to determine whether these effects can be neglected. • Most thermal processes are time dependent, but for several practical circumstances, they may be approximated as steady. Shiv Sharma

Modeling of Thermal Systems • Thus, even though the hot rolling process, sketched in Figure 1.10(d), starts out as a transient problem, it generally approaches a steady state condition as time elapses. • Similarly, the solar heat flux incident on the wall of a house clearly varies with time. Nevertheless, over certain short periods, it may be approximated as steady. • Two main characteristic time scales need to be considered. • The first, Tr, refers to the response time of the material or body under consideration, and the second, Tc, refers to the characteristic time of variation of the ambient or operating conditions. • Therefore, Tc indicates the time over which the conditions change. • For instance, it would be zero for a step change and the time period Tp for a periodic process, where Tp = 1/f, with f being the frequency. As mentioned in Chapter 2 and discussed later in this chapter, the response time Tr for a uniform-temperature (lumped) body subjected to a step change in ambient temperature for convective cooling or heating is given by the expression Shiv Sharma

Modeling of Thermal Systems

• where R is the density, C is the specific heat, V is the volume of the body, A is its surface area, and h is the convective heat transfer coefficient. Several important cases can be obtained in terms of these two time scales, as follows: 1. Tc is very large, i.e., Tc→∞: In this case, the conditions may be assumed to remain unchanged with time and the system may be treated as steady state. • At the start of the process, the variables change sharply over a short time and transient effects are important. • However, as time increases, steady-state conditions are attained. • Examples of this circumstance are the extrusion, wire drawing, and rolling processes, as sketched in Figure 3.6(a). • Clearly, as the leading edge of the material moves away from the die or furnace, steady-state conditions are attained in most of the region away from the edge. Shiv Sharma

Modeling of Thermal Systems •Thus, except for the starting transient conditions and in a region close to the edge, the system may be approximated as steady. • FIGURE 3.6 Attainment of steady-state conditions at large time. (a) Modeling of heated moving material, and (b) temperature variation of an electronic chip heated electrically. A similar situation arises in many practical systems where the initial transient is replaced by steady conditions at large time; for instance, in the case of an initially unheated electronic chip that is heated by an electric current and finally attains steady state due to the balance between heat loss to the environment and the heat input [see Figure 3.6(b)].

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Modeling of Thermal Systems • The transient terms, which are of the form δφ/δT, where φ is a dependent variable, are dropped and the steady-state characteristics of the system are determined. • 2. Tc << Tr: In this case, the operating conditions change very rapidly, as compared to the response of the material. • Then the material is unable to follow the variations in the operating variables. • An example of this is a deep lake whose response time is very large compared to the fluctuations in the ambient medium. • Even though the surface temperature may reflect the effect of such fluctuations, the bulk fluid would essentially show no effect of temperature fluctuations. • Then the system may be approximated as steady with the operating conditions taken at their mean values.

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Modeling of Thermal Systems • Such a situation arises in many practical systems due to rapid variations in the heat input or the flow rate. • If the mean value itself varies with time, then the characteristic time of this variation is considered in the modeling. • In addition, if the operating conditions change rapidly from one set of values to another, the system goes from one steady-state situation to another through a transient phase. • Again, away from this rapid variation, the problem may be treated as steady. • 3. Tr << Tc: This refers to the case where the material or body responds very fast but the operating or boundary conditions change very slowly. An example of this is the slow variation of the solar flux with time on a sunny day and the rapid response of the collector. • Similarly, an electronic component responds very rapidly to the turning on of the system, but the walls of the equipment and the board on which it is located respond much more slowly.

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Modeling of Thermal Systems • Another example is a room that is being heated or cooled. • The walls respond very slowly as compared to the items in the room and the air. • It is then possible to take the surroundings as unchanged over a portion of the corresponding response time. • Therefore, in such cases, the part may be modeled as quasi-steady, with the steady problem being solved at different times. • This implies that the part or system goes through a sequence of steady states, each being characterized by constant operating or environmental conditions. • Figure 3.7 shows a sketch of such quasi-steady modeling. • This is one of the most frequently employed approximations in timedependent problems, since many practical systems involve such slowly varying operating, boundary, or forcing conditions.

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Modeling of Thermal Systems •

FIGURE 3.7 Replacement of the ambient temperature variation with time by a finite number of steps, with the temperature held constant over each step.

4. Periodic processes: • In many cases, the behavior of the thermal system may be represented as a periodic process, with the characteristics repeating over a given time period Tp. • Environmental processes are examples of this modeling because periodic behavior over a day or over a year is of interest in many of these systems. • The modeling of solar energy collection systems, for instance, involves both the cyclic nature of the process over a day and night sequence, as well as over a year.

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• Long term energy storage, for instance, in salt-gradient solar ponds, is considered as cyclic over a year. • Similarly, many thermal systems undergo a periodic process because they are turned on and off over fixed periods. • The main requirement of a periodic variation is that the temperature and other variables repeat themselves over the period of the cycle, as shown in Figure 3.8(a) for the temperature of a natural water body such as a lake. • In addition, the net heat transfer over the cycle must be zero because, if it is not, there is a net gain or loss of energy. • This would result in a consequent increase or decrease of temperature with time and a cyclic behavior would not be obtained.

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Modeling of Thermal Systems • These conditions may be represented as

where Q(T) is the total heat transfer rate from a body as a function of time T. • For a deep lake with a large surface area, Q(T) is essentially the surface heat transfer rate because very little energy is lost at the bottom or at the sides. • Either one of the above conditions may be used in the modeling of a periodic process. • The main advantage of modeling a system as periodic is that results need to be obtained only over the time of the cycle. • The conditions given by Equation (3.2) and Equation (3.3) can be used for validation as well as for the development of the numerical code.

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Modeling of Thermal Systems • FIGURE 3.8 Periodic temperature variation in (a) a natural lake over the year, and (b) a body subjected suddenly to a periodic variation in the heat input. • Frequently, the system undergoes a starting transient and finally attains a periodic behavior. • This is typical of many industrial systems that are operated over fixed periods following a start-up. • Figure 3.8(b) shows the typical temperature variation in such a process. • The time-dependent terms are retained in the governing equations and the problem is solved until the cyclic behavior of the system is obtained. • Because of the periodic nature of the process, analytical solutions can often be obtained, particularly if the periodic process can be approximated by a sinusoidal variation. Shiv Sharma

• 5. Transient: • If none of the above approximations is applicable, the system has to be modeled as a general time-dependent problem with the transient terms included in the model. • Since this is the most complicated circumstance with respect to time dependence, efforts should be made, as outlined above, to simplify the problem before resorting to the full transient, or dynamic, modeling. • However, there are many practical systems, particularly in materials processing, that require such a dynamic model because transient effects are crucial in determining the quality of the product and in the control and operation of the system. • Heat treatment and metal casting systems are examples in which a transient model is essential to study the characteristics of the system for design.

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• Spatial Dimensions • This consideration refers to the determination of the number of spatial dimensions needed to model a given system. • Though all practical systems are three dimensional, they can often be approximated as two- or one-dimensional to considerably simplify the modeling. • Thus, this is an important simplification and is based largely on the geometry of the system under consideration and on the boundary conditions. • As an example, let us consider the steady-state conduction in a solid bar of length L, height H, and width W, as shown in Figure 3.9. • Let us also assume that the thermal boundary conditions are uniform, though different, on each of the six surfaces of the solid.

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• FIGURE 3.9 Three-dimensional conduction in a solid block.

• Now the temperature distribution within the solid T(x, y, z), where x, y, z are the three coordinate distances, is governed by the following partial differential equation, if the thermal conductivity is constant and no heat source exists in the material:

• This equation may be generalized by using the dimensionless variables

to yield the dimensionless equation

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where Tref is a reference temperature and may simply be the ambient temperature or the temperature at one of the surfaces. • Other definitions of the non-dimensional variables, particularly for θ, are used in the literature. • Lumped Mass Approximation • In this model, which is extensively used and is thus an important circumstance, the temperature, species concentration, or any other transport variable is assumed to be uniform within the domain of interest. • Thus, the variable is lumped and no spatial variation within the region is considered. • For steady-state conditions, algebraic equations are obtained instead of differential equations. • Most thermodynamic systems, such as air conditioning and refrigeration equipment, internal combustion engines, power plants, etc., are analyzed assuming the conditions in the different components as uniform and, thus, as lumped.

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• For transient problems, the variables change only with time, resulting in ordinary differential equations instead of partial differential equations. • Consider, for instance, a heated body at an initial temperature of To cooling in an ambient medium at temperature Ta by convection, with h as the convective heat transfer coefficient. • Then, if the temperature T is assumed to be uniform in the body, the energy equation is where the symbols were defined for Equation (3.1). • If the temperature difference (T – Ta) is substituted by θ the governing equation and its solution are obtained as

where θo = To – Ta. Shiv Sharma

• The quantity ρCV/hA represents a characteristic time and is the time needed for the temperature difference from the ambient, T – Ta, to drop to 1/e of its initial value, where e is the base of the natural logarithm. • This e-folding time is also known as the response time of the body, as given earlier in Equation (3.1). This model and the corresponding temperature variation are shown in Figure 3.10. FIGURE 3.10 Lumped mass approximation of a heated body undergoing convective cooling.

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• Simplification of Boundary Conditions • Most practical systems and processes involve complicated, non-uniform, and time-varying boundary conditions. • However, considerable simplification can be obtained, without significant loss of accuracy or generality, by approximating the boundaries as smooth, with simpler geometry and uniform conditions, as sketched in Figure 3.11.

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• Thus, roughness of the surface is neglected unless interest lies in scales of that size or the effect of roughness is being investigated. • The geometry may be approximated in terms of simpler configurations such as flat plate, cylinder, or sphere. • The human body is, for example, often approximated as a vertical cylinder for calculating the heat transfer from it. • A large cylinder is itself approximated as a flat surface for convective transport if the thickness of the boundary layer δ adjacent to the surface is much smaller than the diameter D of the cylinder, i.e., δ /D << 1. • Conditions that vary over the boundaries or with time are often approximated as uniform or constant to considerably simplify the model. • Negligible Effects • Major simplifications in the mathematical modeling of thermal systems are obtained by neglecting effects that are relatively small. • Estimates of the relevant quantities are used to eliminate considerations that are of minor consequence. Shiv Sharma

• For instance, estimates of convective and radiative loss from a heated surface may be used to determine if radiation effects are important and need to be included in the model. • If Qc and Qr are the convective and radiative heat transfer rates, respectively, these may be estimated for a surface of area A as where approximated or expected values of the surface temperature may be employed to estimate the relative magnitudes of these transport rates. • Clearly, at relatively low temperatures, the radiative heat transfer may be neglected and at high temperatures it may be the dominant mechanism. Similarly, the change in the volume of a material as it changes phase from, say, liquid to solid, may be neglected in several cases if this change is small. • Changes in dimensions due to temperature variation are usually neglected, unless these changes are significant or lead to an important consideration in the problem. • Potential energy effects are usually neglected, compared to the kinetic energy changes, in a gas turbine. Shiv Sharma

• Idealizations • Practical systems and processes are certainly not ideal. • There are undesirable energy losses, friction forces, fluid leakages, and so on, that affect the system behavior. • However, idealizations are usually made to simplify the problem and to obtain a solution that represents the best performance.

• FIGURE 3.12 Idealizations used in mathematical modeling. (a) Ideal turbine behavior; • (b) step change in heat flux; and (c) perfectly insulated outer surface of a heat exchanger.

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• Material Properties • For a satisfactory mathematical modeling of any thermal system or process, it is important to employ accurate material property data. • The properties are usually dependent on physical variables such as temperature, pressure, and species concentration. • In polymeric materials such as plastics, the viscosity of the fluid also depends on the shear rate and thus on the flow field. • Even though the properties vary with temperature and other variables, they can be taken as constant if the change in the property, say thermal conductivity k, is small compared to the average value kavg, i.e., Δk/kavg << 1. • However, in many practical systems the constant property approximation cannot be made because of large changes in these variables. I • n such cases, curve fitting is often used to represent the variation of the relevant properties.

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• For instance, the variation of the thermal conductivity with temperature may be represented by a function k(T), where • Here, ko is the thermal conductivity at a reference temperature To and a and b are constants obtained from the curve fitting of the data on this property at different temperatures. • Higher order polynomials and other algebraic functions may also be used to represent the property data. • Similarly, curve fitting may be used for other properties such as density, specific heat, viscosity, etc. • Such equations are very valuable in the mathematical modeling of thermal systems. • Conservation Laws • The conservation laws for mass, momentum, and energy form the basis for deriving the governing equations for thermal systems and processes. • The resulting equations may be algebraic, differential, or integral. Shiv Sharma

• Algebraic equations arise mainly from curve fitting, such as Equation (3.13), and also apply for steady-state, lumped systems. • As mentioned earlier, thermodynamic systems are often approximated as steady and lumped, resulting in algebraic governing equations. • In some cases, overall or global balances could also lead to algebraic equations. • For instance, the energy balance at a furnace wall, under steady-state conditions, yields the equation • where Th is the temperature of the heater radiating to the inner surface at temperature T, Ta is the temperature of air adjacent to the inner surface, and Ts is the outer surface temperature of the wall. • The temperature T at the wall may then be obtained by solving Equation (3.14), which is a nonlinear equation and will generally require iterative methods. For systems of algebraic equations as well as for a single nonlinear equation, numerical methods are generally needed to obtain the solution. Shiv Sharma

• Differential approaches are the most frequently employed conservation formulation because they apply locally, allowing the determination of variations in time and space. • Ordinary differential equations arise in a few idealized situations for which only one independent variable is considered. Therefore, if the lumped mass assumption can be applied and transient effects are important, Equation (3.7), would be the relevant energy equation.

• Partial differential equations are obtained for distributed models. • Thus, Equation (3.4) is the applicable energy equation for threedimensional, steady conduction in a material with constant properties.

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• Similarly, one-dimensional transient conduction in a wall, which is large in the other two dimensions, is governed by the equation

• if the material properties are taken as variable. • Curve Fitting • An important and valuable technique that is used extensively to represent the characteristics and behavior of thermal systems is that of curve fitting. • Results are obtained at a finite number of discrete points by numerical computation and experimentation. • If these data are represented by means of a smooth curve, which passes through or as close as possible to the points, the equation of the curve can be used to obtain values at intermediate points where data are not available and also to model the characteristics of the system.

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• Physical reasoning may be used in the choice of the type of curve employed for curve fitting, but the effort is largely a data-processing operation, unlike mathematical modeling discussed earlier, which was based on physical insight and experience. • The equation obtained as a result of curve fitting then represents the performance of a given equipment or system and may be used in system simulation and optimization. • This equation may also be employed in the selection of equipment such as blowers, compressors, and pumps. • Curve fitting is particularly useful in representing calibration results and material property data, such as the thermodynamic properties of a substance, in terms of equations that form part of the mathematical model of the system. • There are two main approaches to curve fitting. • The first one is known as an exact it and determines a curve that passes through every given data point.

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• This approach is particularly appropriate for data that are very accurate, such as computational results, calibration results, and material property data, and if only a small number of data points are available. • If a large amount of data is to be represented, and if the accuracy of the data is not very high, as is usually the case for experimental results, the second approach, known as the best fit, which obtains a curve that does not pass through each data point but closely approximates the data, is more appropriate. • The difference between the values given by the approximating curve and the given data is minimized to obtain the best fit. • Sketches of curve fitting using these two methods were seen earlier in Figure 3.2. • Both of these approaches are used extensively to represent results from numerical simulation and experimental studies. • The availability of correlating equations from curve fitting considerably facilitates the design and optimization process.

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• Exact Fit • This approach for curve fitting is somewhat limited in scope because the number of parameters in the approximating curve must be equal to the number of data points for an exact fit. • If extensive data are available, the determination of the large number of parameters that arise becomes very involved. • Then, the curve obtained is not very convenient to use and may be ill conditioned. • In addition, unless the data are very accurate, there is no reason to ensure that the curve passes through each data point. • However, there are several practical circumstances where a small number of very accurate data are available and an exact it is both desirable and appropriate.

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• Many methods are available in the literature for obtaining an exact fit to a given set of data points. • 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 fit (n + 1) data points. • The general form of the polynomial may be taken as • where y is the dependent variable, x is the independent variable, and the a’s are constants to be determined by curve fitting of the data. • If (xi, yi), where i = 0, 1, 2, , n, represent the (n + 1) data points, yi being the value of the dependent variable at x = xi , these values may be substituted in Equation (3.29) to obtain (n + 1) equations for the a’s. Shiv Sharma

• Thus, • Since xi and yi are known for the given data points, (n + 1) equations are obtained from Equation (3.30), and these can be solved for the unknown constants in Equation (3.29). • Thus, two data points yield a straight line, y = a0 + a1x, three points a second-order polynomial, y = a0 + a1x + a2x2, four points a third-order polynomial, and so on. • The method is appropriate for small sets of very accurate data, with the number of data points typically less than ten. • For larger data sets, higher order polynomials are needed, which are often difficult to determine, inconvenient to use, and inaccurate because of the many small coefficients that arise for higher-order terms. • Different forms of interpolating polynomials are used in other methods. • In Lagrange interpolation, the polynomial used is known as the Lagrange polynomial and the nth-order polynomial is written as Shiv Sharma

• The coefficients ai, where i varies from 0 to n, can be determined easily by substitution of the (n + 1) data points into Equation (3.31). • Then the resulting interpolating polynomial is

• where the product sign Π denotes multiplication of the n factors obtained by varying j from 0 to n, excluding j = i, for the quantity within the parentheses. • It is easy to see that this polynomial may be written in the general form of a polynomial, Equation (3.29), if needed. • Lagrange interpolation is applicable to an arbitrary distribution of data points, and the determination of the coefficients of the polynomial does not require the solution of a system of equations, as was the case for the general polynomial. Shiv Sharma

• Because of the ease with which the method may be applied, Lagrange interpolation is extensively used for engineering applications. • In Newton’s divided-difference method, the nth-order interpolating polynomial is taken as

• A recursive formula is written to determine the coefficients. • The higher-order coefficients are determined from the lower-order ones. • Therefore, we evaluate the coefficients by starting with a0 and successively calculating a1, a2, a3, and so on, up to an. • Once these coefficients are determined, the interpolating polynomial is obtained from Equation (3.33). Several simplified formulas can be derived if the data are given at equally spaced values of the independent variable x. These include the Newton-Gregory forward and backward interpolating polynomials. • This method is particularly well suited for numerical computation and is frequently used for an exact fit in engineering problems. Shiv Sharma

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

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• FIGURE 3.24 Interpolation with single polynomials over the entire range and with piecewise cubic splines for a step change in the dependent variable.

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• Functions of more than one independent variable also arise in many problems of practical interest. • An example of this circumstance is provided by thermodynamic properties like density, internal energy, enthalpy, etc., which vary with two independent variables, such as temperature and pressure. • Similarly, the pressure generated by a pump depends on both the speed and the flow rate. • Again, a best it is usually more useful because of the inaccuracies involved in obtaining the data. • However, an exact it may also be obtained. • Curve fitting with the chosen order of polynomials is applied twice, first at different fixed values of one variable to obtain the curve fit for the other variable. • Then the coefficients obtained are curve fitted to reflect the dependence on the first variable. • As shown in Figure 3.25, 9 data points are needed for second-order polynomials. Shiv Sharma

• For third-order polynomials, 16 points are needed, and for fourthorder polynomials, 25 points are needed. • The resulting general equation for the curve fit shown in Figure 3.25 is

• FIGURE 3.25 A function f(x1, x2) of two independent variables x1 and x2, showing the nine data points needed for an exact fit with second-order polynomials.

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• Best Fit • The data obtained in many engineering applications have a significant amount of associated error. • Experimental data, for instance, would generally have some scatter due to error whose magnitude depends on the instrumentation and the arrangement employed for the measurements. • In such cases, requiring the interpolating curve to pass through each data point is not appropriate. • In addition, large data sets are often available and a single curve for an exact fit leads to high-order polynomials that are again not satisfactory. • A better approach is to derive a curve that provides a best it to the given data by somehow minimizing the difference between the given values of the dependent variable and those obtained from the approximating curve. • Figure 3.26 shows a few circumstances where a best fit is much more satisfactory than an exact fit.

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• FIGURE 3.26 Data distributions for which a best fit is more appropriate than an exact fit.

Shiv Sharma

• The curve from a best fit represents the general trend of the data, without necessarily passing through every given point. • It is useful in characterizing the data and in deriving correlating equations to quantitatively describe the thermal system or process under consideration. • For instance, correlating equations derived from experimental data on heat and mass transfer from bodies of different shapes are frequently used in the design of the relevant thermal process. • Similarly, correlating equations representing the behavior of an internal combustion engine under various fuel-air mixtures are useful in the analysis and design of engines. • Several criteria can be used to derive the curve that best its the data. • If the approximating curve is denoted by f(x) and the given data by (xi, yi), as before, the error ei is given by ei = yi – f(xi). • Then, one method for obtaining a best fit to the data is to minimize the sum of these individual errors; that is, minimize Σei. Shiv Sharma

• Since errors tend to cancel out in this case, being positive or negative, the sum of absolute values of the error, Σ|ei|, may be minimized instead. • However, this is not an easy condition to apply and may not yield a unique curve. • The most commonly used approach for a best fit is the method of least squares, in which the sum S of the squares of the errors is minimized. • The expression for S, considering n data points, is

• This approach generally yields a unique curve that provides a good representation of the given data, if the approximating curve is properly chosen. • The physical characteristics of the given problem may be used to choose the form of the approximating function. • For instance, a sinusoidal function may be used for periodic processes such as the variation of the average daily ambient temperature at a given location over the year. Shiv Sharma

• Linear Regression • The procedure of obtaining a best fit to a given data set is often known as regression. • Let us first consider fitting a straight line to a data set. • This curve fitting is known as linear regression and is important in a wide variety of engineering applications because linear approximations are often satisfactory and also because many nonlinear variations such as exponential and power-law forms can be reduced to a linear best fit, as seen later. • Let us take the equation of the straight line for curve fitting as • where a and b are the coefficients to be determined from the given data. For a best it, the sum S is to be minimized, where

• The minimum occurs when the partial derivatives of S with respect to a and b are both zero. Shiv Sharma

• This gives

• These equations may be simplified and expressed as • which may be written for the unknowns a and b as

• where the summations are over the n data points, from i = 1 to i = n. • These two simultaneous linear equations may be solved to obtain the coefficients a and b. • The resulting equation f(x) = a + bx then provides a best it to the given data by a straight line, as sketched in Figure 3.26(a). Shiv Sharma

• The spread of the data before regression is applied is given by the sum Sm where • yavg being the average, or mean, of the given data. • Then the extent of improvement due to curve fitting by a straight line is indicated by the reduction in the spread of the data, given by the expression • where r is known as the correlation coefficient. • A good correlation for linear regression is indicated by a high value of r, the maximum of which is 1.0. • The given data may also be plotted along with the regression curve in order to demonstrate how good a representation of the data is provided by the best it, as seen in Figure 3.26.

Shiv Sharma