Modelling of Equipment, Processes, and Systems

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1 1 Modelling of Equipment, Processes, and Systems

2 2 Modelling Tools Simple Programs Spreadsheet tools like Excel Mathema7cal Tools MatLab, Mathcad, Maple, and Mathema7ca Special Purpose Codes Macroflow, HySys, PipeSym, Finite Element Codes Abacus, Ansys, Algor, Comsol, CFD Codes Fluent, Flowtherm, Flow3d, CFX,

3 3 Approaches to Analysis Steady State versus Dynamic System Analysis One Dimensional versus Mul7dimensional 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, equa7ons which describe the system are developed. In most cases the system model comprises of many algebraic or differen7al equa7ons.

4 4 System Modelling Equa<ons Steady State Dynamic

5 5 Four Types of Models Mathema<cal Models: represent the performance and behaviour of a system by means of equa7ons which describe the physical phenomena. These types of models are the most important since they offer the greatest flexibility in the design process. 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. Numerical Models: are based on a mathema7cal model, but are used when the equa7ons describing the system cannot be solved easily using tradi7onal methods. More o_en than not, the engineer must resort to numerical models for undertaking analysis of complex systems. 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, i.e. thermal and hydraulic circuits.

6 6 Example 4 (Problem 6.1) Consider a system composed similar to that sketched earlier, but with the following dimensions: L A = 5 cm, L B = 5 cm, L C = 5 cm, L D = 5 cm, H A =10 cm, H B = 3 cm, H C = 7 cm, H D = 10 cm. The system is also 10 cm deep into the page. k A = 5 W/mK, k B = 50 W/mK, k C = 10 W/ mk, and k D = 1 W/mK. Find the upper and lower bounds on the thermal resistance.

7 7 FiHng Equa<ons to Data Engineers rely on graphical or tabulated data for design O_en we require this data in the form of an equa7on This is accomplished through curve fihng: assists in interpola<on or extrapola<on of data facilitates the process of system simula7on and/or op7miza7on we may use exact fits, best fits, and fits based on physical laws

8 8 Polynomial Approxima<on Polynomial approxima7on is an exact fihng process which is used for interpola7on of data. With proper insight, we can also use it for extrapola7on purposes. We may fit one and two dimensional data sets. Polynomial approxima7on gives systems of linear equa7ons which must be solved for the unknown coefficients.

9 9 Polynomial Approxima<on: 1D Posi7ve Exponents: Nega7ve Exponents:

10 10 Example 5 (Problem 6.2) A fan curve for an axial flow fan has the following data: Fit the data to a 6th order polynomial of the form: Once you have found the solu7on, examine the fit for interpola7on and extrapola7on purposes.

11 11 Example 5 (cont d) Polynomial Fit of Fan Curve Data Points are captured exactly Interpola7on can be poor in certain regions Extrapola7on is definitely not possible

12 12 Polynomial Approxima<on: 2D In pump and heat exchanger applica7ons, performance is o_en a func7on of two variables:

13 13 Example 6 (Problem 6.3) Using the pump curves supplied in the handout and the chosen points, develop an exact fit for interpola7on of the pump performance for opera7onal speeds between 1760 RPM and 3600 RPM over the flow range 200 gpm to 600 gpm. Use nine points and a quadra7c polynomial form.

14 14 Example 6 (cont d)

15 15 Least Squares Fits Least squares fihng is used to fit data to models when there are more data points than undetermined coefficients. Simplest form is linear regression. Assuming we have m points, we start with:

16 16 Least Squares Fits This leads to a system of equa7ons: or

17 17 Least Squares Fits This matrix system can be generalized for polynomial forms of the form:

18 18 Generalized Least Squares A simple matrix form for least squares can be developed in the form of: Where x i can be any func7onal form, e.g. x, sin (x), exp(x), for example:

19 19 Generalized Least Squares We define the following matrices: Such that the solu7on matrix for the coefficients is:

20 20 Other Forms Simple Power Law Forms

21 21 Example 7 (Problem 6.10) The following data for fric7on factor (f) as a func7on of Reynolds number (Re) were obtained from a compact heat exchanger design text: Using these data points, and your knowledge of laminar and turbulent boundary layer fric7on, you propose to fit the data to a simple curve using the least squares method and the following equa7on: where a and b are unknown constants to be determined. Obtain a best fit and use your fit to compare to the exact values at Re = 600 and Re = 1000.

22 22 Example 8 (Problem 6.13) The following data for fluid viscosity as a func7on of absolute temperature were obtained from an experiment: Using these data points and the observed behaviour, you propose to fit the data to a simple inverse curve using the least squares method and the following equa7on: Where a and b are unknown constants to be determined. What is the root mean square (rms) error of your fit?

23 23 Example 9 (Problem 6.16) A performance curve for a compact heat exchanger surface was determined experimentally and found to have a linear behaviour on a log- log plot. The equa7on for fihng data is chosen to have the form: where a, b, and c, are constants. Given the data below, obtain a fit of the data using the generalized least squares method: a) Obtain the solu7on for the constants a,b, and c, for the best fit. b) Test the solu7on at Re = 300, 500, and 800.

24 24 Example 10 (Problem 6.17) The following sample of data (8 points) were obtained from an experiment for a par7cular variable which is a func7on of x and y: Using these data points, you propose to fit the data to a simple power law model using the least squares method and the following equa7on: where A, m and n are unknown constants to be determined. Set up the necessary system of equa7ons in matrix form, i.e. find the three required equa7ons and solve them.

25 25 FiHng Thermal Property Data Thermal- Fluid property data are best using the following forms: Viscosity needs careful amen7on: There are other forms as well.

26 26 Model Development Frequently in the analysis of a system, you will have to model por7ons of the problem your self using physical principles, e.g. pressure drop in a flow system: iden7fy the system or component use physical laws to describe the behaviour combine necessary rela7onships to obtain as simple an expression which describes the behaviour in terms of explicit variables

27 27 Example - 11 Develop models for the thermal resistance and pressure drop of a liquid cooled heat sink containing four discrete heat sources, each dissipa7ng 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 6.75 mm and each pass is approximately 175 mm in length. Three 180 degree return bends are used, each with a radius of curvature of 12.5 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 solu7on.

28 28 Example 11 (cont d) m [kg/s] Re f Δp [Pa] h [W/m 2 K] R [K/W] T s [C] To [C] dp (Pa) and T (C) Pressure Drop Surface Temperature m (kg/s)

29 29 Solving Non- Linear Equa<ons Newton- Raphson in one variable: Func7on must be differen7able Can be divergent if a poor guess is made Is rapidly convergent if a good guess is made Can be extended to mul7ple variables Easy to implement or carry out for one variable For the (i+1) th itera7on it takes the form:

30 30 Example 12 (Example 6.5) A simple piping system requires fluid to be pumped through a finite length of pipe and through a fixed eleva7on. Assuming turbulent pipe fric7on, the resul7ng equa7on is non- linear in Q, the volumetric flow rate. Solve the simple pump/piping system equa7ons given by: Use Newton- Raphson itera7on. Note: the maximum flow for the pump (when H p =0) is 20 m 3 /s and the pump head is given in m.

31 31 Solving Non- Linear Equa<ons For mul7variable systems we can extend the Newton- Raphson method as follows: Given: Solve: Upda7ng the solu7on with: *x o is the guess for all values

32 32 Example 13 (Example 6.6) Solve the system below using mul7- variable Newton- Raphson itera7on: Use an ini7al guess of x o =[w,x,y,z]=[5,3,1.5,750]. As a further exercise consider other guesses and examine the stability of the solu7on process.

33 33 Example 13 (cont d) w x y z df i /dx i ([x o ]) f([x o ]) Initial Calc Inverse Matrix Δx x o New

34 34 Example 13 (cont d) w x y z Revised #1 df i /dx i ([x o ]) f([x o ]) Inverse Matrix Δx x o New

35 35 Example 13 (cont d) w x y z Revised # df i /dx i ([x o ]) f([x o ]) Inverse Matrix Δx x o New

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