Download presentation

Published byEdward Hoover Modified over 4 years ago

1
**Selected Differential System Examples from Lectures**

2
**Liquid Storage Tank Standing assumptions Other possible assumptions**

V = Ah wi wo Standing assumptions Constant liquid density r Constant cross-sectional area A Other possible assumptions Steady-state operation Outlet flow rate w0 known function of liquid level h

3
**Mass Balance Mass balance on tank Steady-state operation:**

Valve characteristics Linear ODE model Nonlinear ODE model

4
**Stirred Tank Chemical Reactor**

Overall mass balance Component balance Assumptions Pure reactant A in feed stream Perfect mixing Constant liquid volume Constant physical properties (r, k) Isothermal operation

5
**Plug-Flow Chemical Reactor**

z qi, CAi qo, CAo CA(z) Dz Assumptions Pure reactant A in feed stream Perfect plug flow Steady-state operation Isothermal operation Constant physical properties (r, k)

6
**Plug-Flow Chemical Reactor cont.**

z qi, CAi qo, CAo CA(z) Dz Component balance Overall mass balance

7
**Continuous Biochemical Reactor**

Fresh Media Feed (substrates) Exit Gas Flow Agitator Exit Liquid Flow (cells & products)

8
**Cell Growth Modeling Specific growth rate Yield coefficients**

Biomass/substrate: YX/S = -DX/DS Product/substrate: YP/S = -DP/DS Product/biomass: YP/X = DP/DX Assumed to be constant Substrate limited growth S = concentration of rate limiting substrate Ks = saturation constant mm = maximum specific growth rate (achieved when S >> Ks)

9
**Continuous Bioreactor Model**

Assumptions Sterile feed Constant volume Perfect mixing Constant temperature and pH Single rate limiting nutrient Constant yields Negligible cell death Product formation rates Empirically related to specific growth rate Growth associated products: q = YP/Xm Nongrowth associated products: q = b Mixed growth associated products: q = YP/Xm+b

10
**Mass Balance Equations**

Cell mass VR = reactor volume F = volumetric flow rate D = F/VR = dilution rate Product Substrate S0 = feed concentration of rate limiting substrate

11
Exothermic CSTR Scalar representation Vector representation

12
**Isothermal Batch Reactor**

CSTR model: A B C Eigenvalue analysis: k1 = 1, k2 = 2 Linear ODE solution:

13
**Isothermal Batch Reactor cont.**

Linear ODE solution: Apply initial conditions: Formulate matrix problem: Solution:

14
**Isothermal CSTR Nonlinear ODE model**

Find steady-state point (q = 2, V = 2, Caf = 2, k = 0.5)

15
**Isothermal CSTR cont. Linearize about steady-state point:**

This linear ODE is an approximation to the original nonlinear ODE

16
**Continuous Bioreactor**

Cell mass balance Product mass balance Substrate mass balance

17
**Steady-State Solutions**

Simplified model equations Steady-state equations Two steady-state points

18
**Model Linearization Biomass concentration equation**

Substrate concentration equation Linear model structure:

19
**Non-Trivial Steady State**

Parameter values KS = 1.2 g/L, mm = 0.48 h-1, YX/S = 0.4 g/g D = 0.15 h-1, S0 = 20 g/L Steady-state concentrations Linear model coefficients (units h-1)

20
**Stability Analysis Matrix representation Eigenvalues (units h-1)**

Conclusion Non-trivial steady state is asymptotically stable Result holds locally near the steady state

21
**Washout Steady State Steady state:**

Linear model coefficients (units h-1) Eigenvalues (units h) Conclusion Washout steady state is unstable Suggests that non-trivial steady state is globally stable

22
**Gaussian Quadrature Example**

Analytical solution Variable transformation Approximate solution Approximation error = 4x10-3%

23
**Plug-Flow Reactor Example**

z qi, CAi qo, CAo CA(z) Dz L

24
**Plug-Flow Reactor Example cont.**

Analytical solution Numerical solution Convergence formula Convergence of numerical solution

25
**Matlab Example Isothermal CSTR model**

Model parameters: q = 2, V = 2, Caf = 2, k = 0.5 Initial condition: CA(0) = 2 Backward Euler formula Algorithm parameters: h = 0.01, N = 200

26
**Matlab Implementation: iso_cstr_euler.m**

h = 0.01; N = 200; Cao = 2; q = 2; V = 2; Caf = 2; k = 0.5; t(1) = 0; Ca(1) = Cao; for i=1:N t(i+1) = t(i)+h; f = q/V*(Caf-Ca(i))-2*k*Ca(i)^2; Ca(i+1)= Ca(i)+h*f; end plot(t,Ca) ylabel('Ca (g/L)') xlabel('Time (min)') axis([0,2,0.75,2.25])

27
Euler Solution >> iso_cstr_euler

28
**Solution with Matlab Function**

function f = iso_cstr(x) Cao = 2; q = 2; V = 2; Caf = 2; k = 0.5; Ca = x(1); f(1) = q/V*(Caf-Ca)-2*k*Ca^2; >> xss = xss = >> df iso_cstr(x); >> [t,x] = ode23(df,[0,2],2); >> plot(t,x) >> ylabel('Ca (g/L)') >> xlabel('Time (min)') >> axis([0,2,0.75,2.25])

29
**Matlab Function Solution**

30
CSTR Example Van de Vusse reaction CSTR model Forward Euler

31
**Stiff System Example CSTR model: A B C Homogeneous system:**

Eigenvalue analysis: q/V = 1, k1 = 1, k2 = 200

32
**Explicit Solution Forward Euler First iterative equation**

Second iterative equation

33
**Implicit Solution Backward Euler First iterative equation**

Second iterative equation

34
**Matlab Solution function f = stiff_cstr(x) Cai = 2; qV = 1; k1 = 1;**

Ca = x(1); Cb = x(2); f(1) = qV*(Cai-Ca)-k1*Ca; f(2) = -qV*Cb+k1*Ca-k2*Cb; f = f'; >> xo = 1]) xo = >> df stiff_cstr(x); >> [t,x] = ode23(df,[0,2],[2 0]); >> [ts,xs] = ode23s(df,[0,2],[2 0]); >> size(t) ans = >> size(ts) ans = 30 1

35
Matlab Solution cont. >> subplot(2,1,1) >> plot(t,x(:,1)) >> hold Current plot held >> plot(ts,xs(:,1),'r') >> ylabel('Ca (g/L)') >> xlabel('Time (min)') >> legend('ode23','ode23s') >> subplot(2,1,2) >> plot(t,x(:,2)) >> plot(ts,xs(:,2),'r') >> ylabel('Cb (g/L)')

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google