President UniversityErwin SitompulSMI 2/1 Dr.-Ing. Erwin Sitompul President University Lecture 2 System Modeling and Identification

President UniversityErwin SitompulSMI 2/2 Solution of Homework 1: Interacting Tank-in-Series System v1v1 qiqi h1h1 Chapter 2Examples of Dynamic Mathematical Models h2h2 v2v2 q1 q1 a1 a1 a2 a2 A 1 :Cross-sectional area of the first tank [m 2 ] A 2 :Cross-sectional area of the second tank [m 2 ] h 1 : Height of liquid in the first tank [m] h 2 : Height of liquid in the second tank [m] qo qo The process variable are now the heights of liquid in both tanks, h 1 and h 2. The mass balance equation for this process yields:

President UniversityErwin SitompulSMI 2/3 Solution of Homework 1: Interacting Tank-in-Series System Chapter 2Examples of Dynamic Mathematical Models Assuming ρ, A 1, and A 2 to be constant, we obtain: After substitution and rearrangement,

President UniversityErwin SitompulSMI 2/4 Homework 2: Interacting Tank-in-Series System Chapter 2Examples of Dynamic Mathematical Models Build a Matlab-Simulink model for the interacting tank-in- series system and perform a simulation for 200 seconds. Submit the mdl-file and the screenshots of the Matlab- Simulink file and the scope of h 1 and h 2 as the homework result. Use the following values for the simulation. a 1 = 210 –3 m 2 a 2 = 210 –3 m 2 A 1 = 0.25 m 2 A 2 = 0.10 m 2 g = 9.8 m/s 2 q i = 510 –3 m 3 /s t sim = 200 s

President UniversityErwin SitompulSMI 2/5 Homework 2: Triangular-Prism-Shaped Tank Chapter 2Examples of Dynamic Mathematical Models Build a Matlab-Simulink model for the triangular-prism- shaped tank and perform a simulation for 200 seconds. Submit the mdl-file and the screenshots of the Matlab- Simulink file and the scope of h as the homework result. Use the following values for the simulation. NEW a = 210 –3 m 2 A max = 0.5 m 2 h max = 0.7 m h 0 = 0.05 m (!) g = 9.8 m/s 2 q i1 = 510 –3 m 3 /s q i2 = 110 –3 m 3 /s t sim = 200 s

President UniversityErwin SitompulSMI 2/6 Chapter 2Examples of Dynamic Mathematical Models Heat Exchanger Consider a heat exchanger for the heating of liquids as shown below. Assumptions: Heat capacity of the tank is small compare to the heat capacity of the liquid. Spatially constant temperature inside the tank as it is ideally mixed. Constant incoming liquid flow, constant specific density, and constant specific heat capacity.

President UniversityErwin SitompulSMI 2/7 Chapter 2Examples of Dynamic Mathematical Models Heat Exchanger Consider a heat exchanger for the heating of liquids as shown below. TjTj q TlTl q T V ρ T cpcp T l :Temperature of liquid at inlet [K] T j :Temperature of jacket [K] T:Temperature of liquid inside and at outlet [K] q:Liquid volume flow rate [m 3 /s] V:Volume of liquid inside the tank [m 3 ] ρ: Liquid specific density [kg/m 3 ] c p : Liquid specific heat capacity [J/(kgK)]

President UniversityErwin SitompulSMI 2/8 Chapter 2Examples of Dynamic Mathematical Models Heat Exchanger The heat balance equation becomes: A:Heat transfer area of the wall [m 2 ]  :Heat transfer coefficient [W/(m 2 K)] Rearranging: The heat exchanger will be in steady-state if dT/dt = 0, so the steady-state temperature at outlet is:

President UniversityErwin SitompulSMI 2/9 τ:Space variable [m] T i : Liquid temperature in the inner tube [K]  T i (τ,t) T o : Liquid temperature in the outer tube [K]  T o (t) q:Liquid volume flow rate in the inner tube[m 3 /s] ρ: Liquid specific density in the inner tube [kg/m 3 ] c p : Liquid specific heat capacity [J/(kgK)] A:Heat transfer area per unit length [m] A i :Cross-sectional area of the inner tube [m 2 ] Chapter 2Examples of Dynamic Mathematical Models Double-Pipe Heat Exchanger L T o,ss τ dτ A single-pass, double-pipe steam heat exchanger is shown below. The liquid in the inner tube is heated by condensing steam. T i,ss τ q

President UniversityErwin SitompulSMI 2/10 Double-Pipe Heat Exchanger To(t)To(t) Ti(τ,t)Ti(τ,t) dτ The profile of temperature T i of an element of heat exchanger with length dτ for time dt is given by: (taken as approximation) The heat balance equation of the element can be derived as: Chapter 2Examples of Dynamic Mathematical Models

President UniversityErwin SitompulSMI 2/11 Chapter 2Examples of Dynamic Mathematical Models Double-Pipe Heat Exchanger The equation can be rearrange to give: The boundary condition is T i (0,t) and T i (L,t). The initial condition is T i (τ,0).

President UniversityErwin SitompulSMI 2/12 Heat Conduction in a Solid Body L q(0)q(L)q(L) q(x)q(x)q(x+dx)q(x+dx) x dxdx Consider a metal rod of length L with ideal insulation. Heat is brought in on the left side and withdrawn on the right side. Changes of heat flows q(0) and q(L) influence the rod temperature T(x,t). The heat conduction coefficient, density, and specific heat capacity of the rod are assumed to be constant. Chapter 2Examples of Dynamic Mathematical Models

President UniversityErwin SitompulSMI 2/13 Heat Conduction in a Solid Body t:Time variable [s] x:Space variable [m] T:Rod temperature [K]  T(x,t) ρ:Rod specific density [kg/m 3 ] A:Cross-sectional area of the rod [m 2 ] c p : Rod specific heat capacity [J/(kgK)] q(x) : Heat flow density at length x [W/m 2 ] q(x+dx): Heat flow density at length x+dx [W/m 2 ] L q(0)q(L)q(L) q(x)q(x)q(x+dx)q(x+dx) x dxdx Chapter 2Examples of Dynamic Mathematical Models

President UniversityErwin SitompulSMI 2/14 Heat Conduction in a Solid Body The heat balance equation of at a distance x for a length dx and a time dt can be derived as: L q(0)q(L)q(L) q(x)q(x)q(x+dx)q(x+dx) x dxdx Chapter 2Examples of Dynamic Mathematical Models

President UniversityErwin SitompulSMI 2/15 Heat Conduction in a Solid Body According to Fourier equation: λ:Coefficient of thermal conductivity [W/(mK)] Substituting the Fourier equation into the heat balance equation: :Heat conductifity factor [m 2 /s] Chapter 2Examples of Dynamic Mathematical Models

President UniversityErwin SitompulSMI 2/16 The boundary conditions should be given for points at the ends of the rod: Heat Conduction in a Solid Body The temperature profile of the rod in steady-state T s (x) can be dervied when ∂T(x,t)/∂t = 0. Chapter 2Examples of Dynamic Mathematical Models The initial conditions for any position of the rod is:

President UniversityErwin SitompulSMI 2/17 Thus, the steady-state temperature at a given position x along the rod is given by: Chapter 2 Heat Conduction in a Solid Body Examples of Dynamic Mathematical Models

President UniversityErwin SitompulSMI 2/18 A suitable model for a large class of continuous theoretical processes is a set of ordinary differential equations of the form: State Equations Chapter 2General Process Models t:Time variable x 1,...,x n :State variables u 1,...,u m :Manipulated variables r 1,...,r s :Disturbance, nonmanipulable variables f 1,...,f n :Functions

President UniversityErwin SitompulSMI 2/19 A model of process measurement can be written as a set of algebraic equations: Output Equations Chapter 2General Process Models t:Time variable x 1,...,x n :State variables u 1,...,u m :Manipulated variables r m1,...,r mt :Disturbance, nonmanipulable variables at output y 1,...,y r :Measurable output variables g 1,...,g r :Functions

President UniversityErwin SitompulSMI 2/20 State Equations in Vector Form Chapter 2General Process Models If the vectors of state variables x, manipulated variables u, disturbance variables r, and the functions f are defined as: Then the set of state equatios can be written compactly as:

President UniversityErwin SitompulSMI 2/21 Output Equations in Vector Form Chapter 2General Process Models If the vectors of output variables y, disturbance variables r m, and vectors of functions g are defined as: Then the set of algebraic output equatios can be written compactly as: