Solution to Homework 1 qi h1 h2 q1 qo a1 a2 v1 v2 Chapter 2 Examples of Dynamic Mathematical Models Solution to Homework 1 qi A1 : Cross-sectional area of the first tank [m2] A2 : Cross-sectional area of the second tank [m2] h1 : Height of liquid in the first tank [m] h2 : Height of liquid in the second tank [m] h1 h2 q1 qo a1 a2 v1 v2 The process variable are now the heights of liquid in both tanks, h1 and h2. The mass balance equation for this process yields:

Chapter 2 Examples of Dynamic Mathematical Models Solution to Homework 1 Assuming ρ, A1, and A2 to be constant, we obtain: After substitution and rearrangement,

Chapter 2 Examples of Dynamic Mathematical Models Homework 2 Build a Matlab-Simulink model for the interacting tank-in-series system and perform a simulation for 200 seconds. Submit the mdl-file (softcopy) and the screenshots of the Matlab-Simulink file and the scope of h1 and h2 as the homework result (hardcopy). Use the following values for the simulation. a1 = 210–3 m2 a2 = 210–3 m2 A1 = 0.25 m2 A2 = 0.10 m2 g = 9.8 m/s2 qi = 510–3 m3/s tsim = 200 s

Chapter 2 Examples of Dynamic Mathematical Models Homework 2A Build a Matlab-Simulink model for the triangular-prism-shaped tank and perform a simulation for 200 seconds. Submit the mdl-file (softcopy) and the screenshots of the Matlab-Simulink file and the scope of h as the homework result (hardcopy). Use the following values for the simulation. a = 210–3 m2 Amax= 0.5 m2 hmax = 0.7 m hinitial= 0.05 m (inside the integrator!) g = 9.8 m/s2 qi1 = 510–3 m3/s qi2 = 110–3 m3/s tsim = 200 s Deadline: Tuesday, 31 January 2017

Chapter 2 Examples 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.

Chapter 2 Examples of Dynamic Mathematical Models Heat Exchanger Consider a heat exchanger for the heating of liquids as shown below. Tl q Tl : Temperature of liquid at inlet [K] Tj : Temperature of jacket [K] T : Temperature of liquid inside and at outlet [K] q : Liquid volume flow rate [m3/s] V : Volume of liquid inside the tank [m3] ρ : Liquid specific density [kg/m3] cp : Liquid specific heat capacity [J/(kgK)] V ρ T cp T q Tj

Heat Exchanger The heat balance equation becomes: Rearranging: Chapter 2 Examples of Dynamic Mathematical Models Heat Exchanger The heat balance equation becomes: A : Heat transfer area of the wall [m2] a : Heat transfer coefficient [W/(m2K)] Rearranging: The heat exchanger will be in steady-state if dT/dt = 0, so the steady-state temperature at outlet is:

Series of Heat Exchangers Chapter 2 Examples of Dynamic Mathematical Models Series of Heat Exchangers Consider a series of heat exchangers where a liquid is heated. T0 T1 T2 Tn–1 Tn V1 T1 V2 T2 Vn Tn w1 w2 wn Assumptions: Heat flows from heat sources into liquid are independent from liquid temperature. Ideal liquid mixing and zero heat losses. Accumulation ability of exchangers walls is neglected Flow rates and liquid specific heat capacity are constant

Series of Heat Exchangers Chapter 2 Examples of Dynamic Mathematical Models Series of Heat Exchangers Under these circumstances, the following heat balances result: t : Time variable [s] T0 : Liquid temperature in the first tank inlet [K] Ti : Liquid temperature inside the i-th heat exchanger [K] Vi : Liquid temperature inside the i-th heat exchanger [m3] q : Volume flow rate [m3/s] ρ : Liquid density [kg/m3] wi : Heat inputs [W] The process input variables are heat inputs wi and the first tank inlet temperature T0. The process state variables are temperatures T1, ... Tn. Initial conditions, i.e., initial temperatures in heat exchangers, are arbitrary. T1(0) = T10, ..., Tn(0) = Tn0. The output variables can be chosen up to the interest of the user.

Series of Heat Exchangers Chapter 2 Examples of Dynamic Mathematical Models Series of Heat Exchangers The series of heat exchangers will be in a steady-state if: Let the steady-state values of the process inputs wi, T0 be given, the steady-state temperatures inside the exchangers are:

Double-Pipe Heat Exchanger Chapter 2 Examples of Dynamic Mathematical Models Double-Pipe Heat Exchanger A single-pass, double-pipe steam heat exchanger is shown below. The liquid in the inner tube is heated by condensing steam. τ : Space variable [m] Ti : Liquid temperature in the inner tube [K]  Ti(τ,t) To : Liquid temperature in the outer tube [K]  To(t) q : Liquid volume flow rate in the inner tube[m3/s] ρ : Liquid specific density in the inner tube [kg/m3] cp : Liquid specific heat capacity [J/(kgK)] A : Heat transfer area per unit length [m] Ai : Cross-sectional area of the inner tube [m2] q τ To,ss Ti,ss τ dτ L Heat transfer modes: convection (through the moving fluid) and conduction (across the metal of the inner tube)

Double-Pipe Heat Exchanger Chapter 2 Examples of Dynamic Mathematical Models Double-Pipe Heat Exchanger The profile of temperature Ti of an element of heat exchanger with length dτ for time dt is given by: To(t) Ti(τ,t) (taken as approximation) dτ The heat balance equation of the element can be derived as: Find how to increase output temperature Ti(L,t)

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

Heat Conduction in a Solid Body Chapter 2 Examples of Dynamic Mathematical Models Heat Conduction in a Solid Body 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. q(0) q(L) q(x) q(x+dx) x dx L

Heat Conduction in a Solid Body Chapter 2 Examples of Dynamic Mathematical Models Heat Conduction in a Solid Body q(0) q(L) q(x) q(x+dx) x dx L t : Time variable [s] x : Space variable [m] T : Rod temperature [K]  T(x,t) ρ : Rod specific density [kg/m3] A : Cross-sectional area of the rod [m2] cp : Rod specific heat capacity [J/(kgK)] q(x) : Heat flow density at length x [W/m2] q(x+dx) : Heat flow density at length x+dx [W/m2]

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

Heat Conduction in a Solid Body Chapter 2 Examples of Dynamic Mathematical Models 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 [m2/s]

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

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

General Process Models Chapter 2 General Process Models General Process Models A general process model can be described by a set of ordinary differential and algebraic equations, or in matrix-vector form. The set of ordinary differential equations that constructs a model is called a state space model, consisting of state equations and output equations. For control purposes, linearized mathematical models are used, to maintain the simplicity of the control design. Later in this section, the conversion from partial differential equations that describes processes into models with ordinary differential equations will be shown.

Chapter 2 General Process Models State Equations A suitable model for a large class of continuous theoretical processes is a set of ordinary differential equations of the form: t : Time variable x1,...,xn : State variables u1,...,um : Manipulated variables r1,...,rs : Disturbance, nonmanipulable variables f1,...,fn : Functions

Chapter 2 General Process Models Output Equations A model of process measurement can be written as a set of algebraic equations: t : Time variable x1,...,xn : State variables u1,...,um : Manipulated variables rm1,...,rmt : Disturbance, nonmanipulable variables at output y1,...,yr : Measurable output variables g1,...,gr : Functions

State Equations in Vector Form Chapter 2 General Process Models State Equations in Vector Form 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 equations can be written compactly as:

Output Equations in Vector Form Chapter 2 General Process Models Output Equations in Vector Form If the vectors of output variables y, disturbance variables rm, and vectors of functions g are defined as: Then the set of algebraic output equations can be written compactly as: