6.2 Modeling of Systems and Components

Transcription

1 Chapter 6 Modelling of Equipment, Processes, and Systems 61 Introduction Modeling is one of the most important elements of thermal system design Most systems are analyzed by considering equations which represent a physical process or behaviour We may also resort to other means such as experimentation with a physical or scale model Nowadays, most engineers resort to some form of numerical modelling with a computer This may be a simple process or systems analysis software or may include a full numerical solution using Finite Element Analysis (FEA) or Computational Fluid Dynamics (CFD) However, these tools are both costly, monetarily and time wise In this chapter we will examine several approaches to modelling thermal systems using a number of different tools 62 Modeling of Systems and Components 621 Physical, Mathematical, Numerical, and Analog Models In engineering design we often encounter four types of models These include: Mathematical Models, Physical Models, Numerical Models, and Analog Models Each is discussed below Mathematical Models: represent the performance and behaviour of a system by means of equations which describe the physical phenomena These types of models are the most important since they offer the greatest flexibility in the design process Once constructed, they allow the system to be analyzed under many types of input conditions This is an important issue for undertaking a system simulation or optimization 91

2 92 Mechanical Equipment and Systems Physical Models: are ones that represent the system closely and are used to obtain experimental data to model the performance of a real system These models may be constructed at full scale (prototype) or reduced scale Often in the early design stages a scale model is constructed to obtain some preliminary design data, while at the end of a design cycle a full scale (and usually operational) model or prototype is constructed This is not always possible In many cases where a one of kind design design is being constructed the final system is also the prototype In other cases, it is not possible to build a complete scaled model In some cases a model which only satisfies certain elements of the system is built and tested under one set of conditions, while another model which represents other aspects of the system is built and tested under another set of conditions These types of models are often referred to as distorted models since the system cannot be fully scaled under all conditions Numerical Models: are based on a mathematical model, but are used when the equations describing the system cannot be solved easily using traditional methods More often than not, the engineer must resort to numerical models for undertaking analysis of complex systems Numerical models are conventionally based on a discretised form of the system equations These most frequently occur in FEA and CFD simulations, where a system or component is broken down in to tiny elements and the equations of elasticity or fluid dynamics are solved for the collective system of elements These approaches offer the engineer very detailed information, but can be time consuming to setup and run Analog Models: are based on an analogy or similarity between physical phenomena In fluid mechanics and heat transfer these types of models are frequently used, ie thermal and hydraulic circuits Example 61 Consider heat transfer through a simple composite wall The system consists of four materials each having its own unique thermal conductivity The system is arranged such that heat flows through a series-parallel-series element as shown Consider each of the four types of models and discuss how each could be used to model the system Solution We begin by first considering analog models Although the system is two dimensional, we choose to model the system as a thermal circuit containing a number of thermal resistors We denote these resistances as: R A, R B, R C, and R D which connect the two temperatures T 1 and T 2 with the heat flow Q There are two possible thermal circuits for the given system: and 1 R T,series = R A R D (61) R B R C [ ] R T,parallel = + (62) R A,l + R B + R D,l R A,u + R C + R D,u

3 System Modelling 93 where R i = L i k i A i (63) By using the simple concept of thermal resistance and the electrical analog rules for combining resistances, we may obtain two equivalent circuits using series and parallel flow concepts In fact the results for the actual two dimensional system must lie between these two idealized results: R T,parallel < R T < R T,series (64) This leads us to another useful concept in systems analysis: system bounds Bounding is a powerful way of bracketing system performance when developing models from first principles The use of bounding allows the engineer to apply simple principles in a manner that two extreme values of system performance may be obtained The actual system performance should then lie between these two limits We will delve into this concept further a little later on An exact mathematical model may be formulated by writing the equation of conduction: 2 T i x T i y 2 = 0 (65) for each material and developing the appropriate boundary conditions and interface conditions The resulting system of four partial differential equations and sixteen boundary and interface conditions is too difficult to solve The mathematical statement only serves as the first logical step to a discretised numerical solution using a finite difference approach The resulting finite difference equation for interior nodes is easily developed for a uniform grid of (m by n) nodes for each material: T i m,n+1 + T i m,n 1 + T i m+1,n + T i m 1,n 4T i m,n = 0 (66) In addition to writing the above equation for each interior node, additional equations are required for nodal points at each interface and boundary Ultimately a linear system of equations is obtained which may easily be solved using direct or numerical methods Once the temperature is known at each node, heat transfer rates can be calculated and ultimately a thermal resistance determined Finally, a simple scaled model using actual materials could be constructed and simple measurements made to obtain the overall thermal resistance A typical experiment may entail supplying heat at a known rate with an electric heater and measuring the mean temperature at each of the two principal surfaces The actual measured resistance becomes: R T = T 1 T 2 Q (67)

4 94 Mechanical Equipment and Systems It is clear, that the simplest approach for this not so simple system, is to use the analog model and define two limiting values which bound the actual thermal resistance Once these bounds are known, a number of simple averages of these bounds may be taken to represent the actual system 622 Analysis of Thermal-Fluid Systems Most thermal system analyses begin with an idealization of the actual systemusually the engineer determines the types of issues to be considered in the analysis For example possible issues include but are not limited to: Steady State versus Dynamic System Analysis One Dimensional versus Multi-Dimensional Linear versus Non-Linear Behaviour Laminar versus Turbulent Flow Single Phase versus Two Phase Flow Once the level of complexity of the system analysis is established, equations which describe the system are developed In most cases the system model comprises of many algebraic or differential equations, ie, f 1 (x 1,x 2,x 3,x n ) = 0 f 2 (x 1,x 2,x 3,x n ) = 0 f 3 (x 1,x 2,x 3,x n ) = 0 f n (x 1,x 2,x 3,x n ) = 0 or, if a simple dynamic lumped analysis is undertaken, (68) f 1 (x 1,x 2,x 3,x n,t) = dx 1 dt f 2 (x 1,x 2,x 3,x n,t) = dx 2 dt f 3 (x 1,x 2,x 3,x n,t) = dx 3 dt f n (x 1,x 2,x 3,x n,t) = dx n dt (69) where x i represents a variable of interest in the system In most cases these systems are combinations of linear and non-linear equations

5 System Modelling Rapid Design Tools Rapid design tools are finding wide spread use in industry They are intended to provide a means of establishing performance of systems or components using simple robust models A robust model characterizes fundamental phenomena in a simple manner over a wide range of a particular variable For example, the now popular Churchill equation for laminar-transition-turbulent friction in circular pipes, Eq (327), would be considered to be a robust model since it is valid for all Reynolds numbers and all pipe roughness These tools are usually computer based such as a C program, a comprehensive Excel TM spreadsheet analysis, or a Matlab TM program with a simple interface for data exchange These tools are often developed by engineers for application specific tasks to assist in the early design stages They are often not intended to provide final design analysis, but merely are used in the early stages of analysis to get some preliminary data using minimal effort Once a design approaches the final stages, the engineer would likely turn to a more refined analysis using an FEA or CFD type system to provide more accurate answers Many such systems which are available commercially are based on simple one dimensional flow principles An example is the package MacroFlow TM which uses simple flow networks to simulate system performance for thermal-fluid system design 63 Fitting of Equations to Data Engineers often rely on tabulated or graphical data for systems design However, data in this form is often cumbersome to use if a system simulation is to be undertaken Therefore it is beneficial to have any useful design data in the form of a mathematical equation This is accomplished through the art of equation fitting The main reasons for fitting data to an equation are: facilitate the process of system simulation develop a mathematical statement for system optimization assist in interpolation or extrapolation of a data set Often data from manufacturer s catalogs are required in the form of an equation, or if the model for a particular process is complex or cumbersome, it may be easier to fit the results to a simpler expression This is especially important if an optimization is to be undertaken In these cases we often require the derivative of an expression with respect to a particular variable This is more easily achieved through a fit of the model rather than the model itself, especially if the model is implicit in the variable of interest Data can be fit using a number of methods These include: exact fits, best fits, and fits based on physical laws Ultimately, no matter what form we choose, we will be required to solve a system of linear or non-linear equations Solving linear

6 96 Mechanical Equipment and Systems systems can be tedious when the number of equations exceeds three For the most part, we can often get away with fitting data to an equation with only two or three coefficients without sacrificing accuracy When systems larger than three equations are required, the use of mathematical software or spreadsheets should be used to facilitate computations 631 Polynomial Approximation: Exact Fits Polynomial approximation is a form of an exact fit An exact fit requires that the number of data points equal the number of undetermined coefficients in the trial equation We will consider a number of useful types of exact fits such as single and multi-variable fits and exponential fits It is important at this point to understand that in all of the fits being considered, it is not important whether the trial equation is linear in the desired variable(s), but rather that it is linear in the undetermined coefficients Functions of a Single Variable We begin by first considering functions of a single variable For example, if we consider the family of simple polynomial functions: y = a 0 + a 1 x y = a 0 + a 1 x + a 2 x 2 y = a 0 + a 1 x + a 2 x 2 + a 3 x 3 y = a 0 + a 1 x + a 2 x 2 + a 3 x a n x n (610) These equations require a data point for each of the undetermined coefficients a i That is if the order of the polynomial is n, then (n + 1) distinct data points must be used to develop the system of (n + 1) equations For large data sets, using all data points would lead to a very large system of equations In these cases we use best fit methods, to be discussed later Polynomial approximation is often used when only a few points are required to capture the trend of a data set This is usually sufficient for interpolation or extrapolation of desired data Further, we are not limited to positive exponents In fact, there are many cases where negative exponents are more appropriate: y = a 0 + a 1 x 1 y = a 0 + a 1 x 1 + a 2 x 2 y = a 0 + a 1 x 1 + a 2 x 2 + a 3 x 3 y = a 0 + a 1 x 1 + a 2 x 2 + a 3 x a n x n (611) These forms are most appropriate when when data sets show a decaying characteristic, that is the dependent variable decreases as a function of the independent

7 System Modelling 97 variable Choosing the correct form is especially important if extrapolation is desired In general both forms (positive or negative exponents) will allow for a fit to be obtained for interpolation purposes But one should match the trend of the equation to the trend of the data if reliable extrapolation is to be undertaken Example 62 We wish to fit the data points: (x 1,y 1 ), (x 2,y 2 ), and (x 3,y 3 ) to a simple quadratic or second order polynomial: y = a 0 + a 1 x + a 2 x 2 (612) Once you have the solution, develop an expression to predict the following data points: (10,259), (16,27), and (22,284) Solution Application of Eq (612) leads to the following system of equations: y 1 = a 0 + a 1 x 1 + a 2 x 2 1 y 2 = a 0 + a 1 x 2 + a 2 x 2 2 y 3 = a 0 + a 1 x 3 + a 2 x 2 3 or in matrix form we may write: 1 x 1 x x 2 x x 3 x 2 3 a 0 a 1 a 2 = y 1 y 2 y 3 (613) (614) The solution may be found using Cramer s rule, or by finding the inverse of the coefficient matrix such that: a 0 a 1 a 2 = 1 x 1 x x 2 x x 3 x y 1 y 2 y 3 (615) Finding the inverse of the coefficient matrix is easily accomplished in all mathematics packages and all spreadsheet programs such as Excel TM Thus we shall consider most problems solved once the coefficient matrix is obtained Finally, substituting the numeric values for the data points yields: a 0 a 1 a 2 = Finding the inverse (which is left as an exercise) subsequently gives: (616) a 0 = a 1 = a 2 = (617)

8 98 Mechanical Equipment and Systems It is left as an exercise to verify that the predicted points are the same as actual data points within the numerical round off error Functions of Two Variables In many applications, engineering data is found to be a function of more than one variable For example, consider the pump performance curves shown in class handout The pump head H varies with pump speed RPM and the discharge Q For each of the three selected pump curves (at a given speed) we may write: H 1 = a 1 + b 1 Q 1 + c 1 Q 2 1 H 2 = a 2 + b 2 Q 2 + c 2 Q 2 2 H 3 = a 3 + b 3 Q 3 + c 3 Q 2 3 (618) We may also write using the fact that each set of coefficients varies with pump speed That is: a = A 0 + A 1 ω 1 + A 2 ω 2 1 b = B 0 + B 1 ω 2 + B 2 ω 2 2 c = C 0 + C 2 ω 3 + C 3 ω 2 3 (619) This gives the following general equation which must be solved using nine data points since there are exactly nine undetermined coefficients: H = (A 0 + A 1 ω 1 + A 2 ω 2 1) + (B 0 + B 1 ω 2 + B 2 ω 2 2)Q + (C 0 + C 2 ω 3 + C 3 ω 2 3)Q 2 (620) Later, we shall see that we may combine the above approach with a best fit method In this hybrid approach, each unique curve may be fit using a best fit method if more than three points are required, then use an exact fit to predict the coefficients of each model curve This is frequently the case with most engineering data such as pump curves Example 63 Using the pump curves supplied in the handout and the chosen points, develop an exact fit for interpolation of the pump performance for operational speeds between 1760 RPM and 3600 RPM over the flow range 200 gpm to 600 gpm Use nine points and a quadratic polynomial form The solution for the chosen points will be completed in class Exponential Forms: Linearization Finally, a number of usefeul non-polynomial forms arise in many applications These are the exponential forms They often take the form: y = bx ±m (621)

9 System Modelling 99 or y = a + bx ±m (622) In the above equation, the constant a represents the initial value if the data set does not begin at the origin The value of a must be known for the data set These equations are not linear in the undetermined coefficients b, m, but can be linearized in the following form: or ln y = lnb ± m ln x (623) ln(y a) = lnb ± m ln x (624) The above equations are now linear in the undetermined coefficients, since lnb is now a new constant We also frequently find the similar forms which contain the natural log e: or y = be ±mt (625) y = a + be ±mt (626) Once again, the value of a must be known from the initial conditions of the data set These equations may also be linearized in the following form: or ln y = lnb ± mt (627) ln(y a) = lnb ± mt (628) It is important to note that if the data are offset by the constant a, the initial value at x = 0 or t = 0, that this constant be subtracted before the natural logarithms are applied An example of the above equations finds applications in heat transfer experiments, where through a series of experiments it is desired to obtain the mean heat transfer coefficient h A time-temperature history is obtained for a given set of conditions Then by means of the appropriately applied lumped capacitance model we find: θ = T T ( = exp ha ) t (629) θ i T i T ρv C p Here we see that b = θ i = T i T, m = ha/ρv C p, and a = T 632 Least Squares Approximation: Best Fits The method of least squares uses all points in a data set to obtain a best fit from the point of view of minimizing the square of the errors Classic linear regression is based on the method of least squares Therefore we start the discussion with this in mind

10 100 Mechanical Equipment and Systems Linear Regression Recall in basic statistics, that a simple linear fit of the form: y = a + bx (630) is desired for a finite number of pairs of (x i,y i ) for i = 1 = m, where m is the number of data points in the set We may define the following objective function which is the sum of the errors squared: S = (a + bx i y i ) 2 minimize (631) In order to minimize the the function S, we take partial derivatives of S with respect to the free variables a and b and equate them to zero: m S a = 2(a + bx i y i ) = 0 m S b = 2(a + bx i y i )x i = 0 This leads to the following set of equations which must be solved for a and b: ma + b a x i = x i + b x 2 i = y i (632) (633) x i y i This system can now be solved for a and b once the data set is specified These values of a and b will give the equation which provides the best fit overall from the point of view of providing the least error on an aggregate basis Non-Linear Regression In general we can apply least squares methods to any polynomial expression with constant coefficients or to any other equation provided that it is linear in the unknown coefficients For example, if we take the following polynomial: y = a o + a 1 x + a 2 x 2 + a 3 x 3 a n x n (634) and apply the method outlined above, it is not difficult to show that one obtains the following expanded matrix form:

11 System Modelling 101 m x i x 2 i x n i x i m x 2 i x 3 i x n+1 i m x 2 i x 3 i x 4 i x n+2 i x n i x n+1 i x n+2 i x 2n i a o a 1 a 2 a n = y i x i y i x 2 iy i x n i y i (635) Example 64 Ten data points were obtained from a pump test The following set of data represents efficiency versus flowrate, (Q[m 3 /s],η) : [(20, 073), (20, 078), (30, 085), (40, 090), (40, 091), (50, 087), (50, 086), (50, 091), (60, 075), (70, 065)] Fit a quadratic model using a least squares fit to the above data and determine the value of Q in which the maximum efficiency occurs Fit Indicators In order to determine the quality of the obtained fit, we must consider various fit indicators Two popular approaches are the r 2 coefficient and the root mean square (rms) error The r 2 correlation coefficient varies between 0 and 1 If there is a strong correlation between the independent and dependent variables then r 2 1, while if there is no relationship then r 2 0 This parameter finds widespread usage in statistics, but really says nothing about the accuracy of the fit The root mean square error is defined as: where N e rms = N e 2 i (636) e i = (y i y p ) y i 100% (637) The (rms) error is useful for comparing the accuracy of various types of fits For example, if we considered a polynomial fit, the (rms) error could be computed for each fit to determine the best order of polynomial for modelling the data set

12 102 Mechanical Equipment and Systems Regression Analysis Using Generalized Matrix Forms The method of least squares can be generalized in such a manner that it can be applied to any functional form provided it is linear in the undetermined coefficients Further, by considering matrix algebra, we can simplify the analysis dramatically once the necessary data is input in a spreadsheet We begin, by defining a general functional relationship: y = n b i x i (638) i=0 where when i = 0, x 0 = 1 is assumed such that an intercept is provided in the equation This may be expanded such that: y = b 0 + b 1 x 1 + b 2 x 2 + b n x n (639) Now given any functional relationship such as a polynomial of the form: y = b 0 + b 1 x 1 + b 2 x 2 + b n x n (640) we find that x 1 = x 1, x 2 = x 2, and so on The method of least squares may now be applied, and a matrix system generated similar to the procedure outlined earlier We may then define the following compact matrix form: where b = (X T X) 1 (X T Y) (641) 1 x 11 x 12 x 1n 1 x 21 x 22 x 2n X = 1 x p1 x p2 x pn and X T is the transpose of the matrix X The remaining matrices are: y 1 y 2 Y = y p (642) (643) and b = b 0 b 1 b n (644)

13 System Modelling 103 The matrices X and Y are generated from the data pairs In the case of the X matrix, each column represents the x datapoint evaluated as the x i function For example if Eq (640) is used, the first column is composed of 1 s, the second column is composed of 1/x for each value of x, the third column is composed of 1/x 2 for each value of x, and so on The method is best learned through an example Example 65 Repeat Example 64 using the generalized matrix approach The matrix approach is significantly more efficient, especially when spreadsheets are used The data points are simply input in two columns Then the X matrix is generated using the functional form of the fit Next, the transpose is generated Finally, the coefficient matrix, (X T X) 1 (X T Y), is generated in a number of simple steps Care must be taken to understand the correct matrix algebra in each step, ie (X T X) 1 and X T Y must be computed first, before their product can be found 64 Mathematical Modelling of Thermophysical Data Thermophysical data are required in some form for all thermal-fluid systems analyses It is useful to have the properties k - thermal conductivity [W/mK], ρ - density [kg/m 3 ], µ - viscosity [Pa s], and C p - specific heat [J/kgK], as functions of temperature Other properties such as kinematic viscosity ν, thermal diffusivity α, and Prandtl number P r, may be computed from the fundamental properties above In thermodynamic analyses we also require u - internal energy, h - enthalpy, and s - entropy, as functions of temperature and pressure Some useful forms for fitting thermal properties are: or ρ = a 0 + a 1 T + a 2 T 2 (645) k = a 0 + a 1 T 1/2 + a 2 T (646) C p = a 0 + a 1 T 1/2 + a 2 T + a 3 T 3/2 + a 4 T 2 + a 5 T 5/2 (647) In the case of viscosity, the following forms are useful log 10 (µ) = a 0 + a 1 T 1/2 + a 2 T + a 3 T 3/2 (648) µ 1/3 = a 0 + a 1 T 1/2 + a 2 T + a 3 T 3/2 + a 4 T 2 (649) as viscosity tends to vary dramatically with temperature, or have very small values, ie large negative exponents Several useful fluids have been fit to these expressions and are provided in the Excel spreadsheet made available in class Finally, it is advantageous to work in the Kelvin K temperature scale as all values of temperature are positive

14 104 Mechanical Equipment and Systems 65 Model Development using Physical Principles An important issue in the analysis of a component or system is the development of models using first principles, which describe the behaviour or performance of the system or component Usually we tend to work with mathematical models The following steps are taken to build a model: identify the system or component use physical laws to describe the system or component behaviour combine all necessary relationships to obtain a single multi-variable function which can be solved in an explicit form for the parameter(s) of interest If this is not possible, then data can be generated and fit using methods described earlier We will illustrate the approach using the example of the chip cooler outlined earlier Example 66 Develop models for the thermal resistance and pressure drop of a liquid cooled heat sink containing four discrete heat sources, each dissipating 150 W The heat sink has dimensions of 127 mm by 127 mm by 13 mm and contains four equally spaced tube passes in series The diameter of the tube is 675 mm and each pass is approximately 175 mm in length Three 180 degree return bends are used, each with a radius of curvature of 125 mm The heat sink is composed of aluminum while the tubing is copper In general we may consider three levels of analysis, lumped, discrete-lumped, and discrete These will be discussed in the context of the solution

15 System Modelling References Boehm, R, Design Analysis of Thermal Systems, Wiley, 1987 Davis, PJ, Interpolation and Approximation, Dover, 1963 Draper, NR and Smith, H, Applied Regression Analysis, Wiley, 1966 Edgar, TF and Himmelblau, DM, Optimization of Chemical Processes, McGraw- Hill, 1988 Stoecker, WF, Design of Thermal Systems, McGraw-Hill, 1989

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