1 / 21

Solution of Homework 1: Interacting Tank-in-Series System

Chapter 2. Examples of Dynamic Mathematical Models. Solution of Homework 1: Interacting Tank-in-Series System. q i. 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]

feleti
Download Presentation

Solution of Homework 1: Interacting Tank-in-Series System

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 2 Examples of Dynamic Mathematical Models Solution of Homework 1: Interacting Tank-in-Series System 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 v2 v1 • The process variable are now the heights of liquid in both tanks, h1 and h2. • The mass balance equation for this process yields:

  2. Chapter 2 Examples of Dynamic Mathematical Models Solution of Homework 1: Interacting Tank-in-Series System • Assuming ρ, A1, and A2 to be constant, we obtain: • After substitution and rearrangement,

  3. Chapter 2 Examples of Dynamic Mathematical Models Homework 2: Interacting Tank-in-Series System 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 h1 and h2 as the homework result. 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10–3m3/s tsim = 200 s

  4. Chapter 2 Examples of Dynamic Mathematical Models Homework 2: Triangular-Prism-Shaped Tank 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 m2 Amax= 0.5 m2 hmax = 0.7 m h0 = 0.05 m (!) g = 9.8 m/s2 qi1 = 510–3 m3/s qi2 = 110–3 m3/s tsim = 200 s

  5. 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.

  6. Chapter 2 Examples of Dynamic Mathematical Models Heat Exchanger • Consider a heat exchanger for the heating of liquids as shown below. Tl 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)] q V ρ T cp T q Tj

  7. 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:

  8. 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

  9. 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:

  10. 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).

  11. 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

  12. 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]

  13. 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:

  14. 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]

  15. 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-stateTs(x) can be dervied when∂T(x,t)/∂t = 0.

  16. 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:

  17. 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

  18. 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

  19. 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 functionsf are defined as: Then the set of state equatios can be written compactly as:

  20. 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 equatios can be written compactly as:

More Related