Selected Differential System Examples from Lectures

Liquid Storage Tank l Standing assumptions »Constant liquid density  »Constant cross-sectional area A l Other possible assumptions »Steady-state operation »Outlet flow rate w 0 known function of liquid level h V = Ah wiwi wowo

Mass Balance l Mass balance on tank l Steady-state operation: l Valve characteristics l Linear ODE model l Nonlinear ODE model

Stirred Tank Chemical Reactor l Assumptions »Pure reactant A in feed stream »Perfect mixing »Constant liquid volume »Constant physical properties ( , k) »Isothermal operation l Overall mass balance l Component balance

Plug-Flow Chemical Reactor l Assumptions »Pure reactant A in feed stream »Perfect plug flow »Steady-state operation »Isothermal operation »Constant physical properties ( , k) z q i, C Ai q o, C Ao CA(z)CA(z) zz

Plug-Flow Chemical Reactor cont. l Overall mass balance z q i, C Ai q o, C Ao CA(z)CA(z) zz l Component balance

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

Cell Growth Modeling l Specific growth rate l Yield coefficients »Biomass/substrate: Y X/S = -  X/  S »Product/substrate: Y P/S = -  P/  S »Product/biomass: Y P/X =  P/  X »Assumed to be constant l Substrate limited growth »S = concentration of rate limiting substrate »K s = saturation constant »  m = maximum specific growth rate (achieved when S >> K s )

Continuous Bioreactor Model Assumptions l Sterile feed l Constant volume l Perfect mixing l Constant temperature and pH l Single rate limiting nutrient l Constant yields l Negligible cell death l Product formation rates »Empirically related to specific growth rate »Growth associated products: q = Y P/X  »Nongrowth associated products: q =  »Mixed growth associated products: q = Y P/X 

Mass Balance Equations l Cell mass »V R = reactor volume »F = volumetric flow rate »D = F/V R = dilution rate l Product l Substrate »S 0 = feed concentration of rate limiting substrate

Exothermic CSTR l Scalar representation l Vector representation

Isothermal Batch Reactor l CSTR model: A  B  C l Eigenvalue analysis: k 1 = 1, k 2 = 2 l Linear ODE solution:

Isothermal Batch Reactor cont. l Linear ODE solution: l Apply initial conditions: l Formulate matrix problem: l Solution:

Isothermal CSTR l Nonlinear ODE model l Find steady-state point (q = 2, V = 2, C af = 2, k = 0.5)

Isothermal CSTR cont. l Linearize about steady-state point: l This linear ODE is an approximation to the original nonlinear ODE

Continuous Bioreactor l Cell mass balance l Product mass balance l Substrate mass balance

Steady-State Solutions l Simplified model equations l Steady-state equations l Two steady-state points

Model Linearization l Biomass concentration equation l Substrate concentration equation l Linear model structure:

Non-Trivial Steady State l Parameter values »K S = 1.2 g/L,  m = 0.48 h -1, Y X/S = 0.4 g/g »D = 0.15 h -1, S 0 = 20 g/L l Steady-state concentrations l Linear model coefficients (units h -1 )

Stability Analysis l Matrix representation l Eigenvalues (units h -1 ) l Conclusion »Non-trivial steady state is asymptotically stable »Result holds locally near the steady state

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

Gaussian Quadrature Example l Analytical solution l Variable transformation l Approximate solution l Approximation error = 4x10 -3 %

Plug-Flow Reactor Example z q i, C Ai q o, C Ao CA(z)CA(z) zz 0 L

Plug-Flow Reactor Example cont. l Analytical solution l Numerical solution l Convergence formula l Convergence of numerical solution

Matlab Example l Isothermal CSTR model l Model parameters: q = 2, V = 2, C af = 2, k = 0.5 l Initial condition: C A (0) = 2 l Backward Euler formula l Algorithm parameters: h = 0.01, N = 200

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])

Euler Solution >> iso_cstr_euler

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])

Matlab Function Solution

CSTR Example l Van de Vusse reaction l CSTR model l Forward Euler

Stiff System Example l CSTR model: A  B  C l Homogeneous system: l Eigenvalue analysis: q/V = 1, k 1 = 1, k 2 = 200

Explicit Solution l Forward Euler l First iterative equation l Second iterative equation

Implicit Solution l Backward Euler l First iterative equation l Second iterative equation

Matlab Solution function f = stiff_cstr(x) Cai = 2; qV = 1; k1 = 1; k2 = 200; 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

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)) >> hold Current plot held >> plot(ts,xs(:,2),'r') >> ylabel('Cb (g/L)') >> xlabel('Time (min)') >> legend('ode23','ode23s')

Binary Flash Unit l Schematic diagram l Vapor-liquid equilibrium Feed F, x F Liquid V, y L, x Vapor Q Heat H l Assumptions » Saturated liquid feed » Perfect mixing » Negligible heat losses » Negligible vapor holdup » Negligible energy accumulation » Constant heat of vaporization » Constant relative volatility

Model Formulation l Mass balance l Component balance

Model Formulation cont. l Steady-state energy balance l Index 0 DAE model

Binary Flash Unit Revisited l DAE model Parameter values: H = 5, F = 10, x F = 0.5, V = 2,  = 10 SolverProblemsMethod ode15sStiff DAEs up to index 1Numerical differentiation ode23tModerately stiff DAEs up to index 1 Trapezoidal MATLAB DAE Solution Codes

binary_flash.m function f = binary_flash(x) H = 5; F = 10; xf = 0.5; V = 2; alpha = 10; xv = x(1); yv = x(2); f(1) = F/H*(xf-xv)-V/H*(yv-xv); f(2) = yv-alpha*xv/(1+(alpha-1)*xv); f = f';

Matlab Commands Results for V = 2 >> f binary_flash(x); >> xss = fsolve(f,[1 1],[]) xss = >> df binary_flash(x); >> M = [1 0; 0 0]; >> options=odeset('Mass',M); >> [t1,y1]=ode15s(df,[0,10],xss,options); Change V = 1 >> [t2,y2]=ode15s(df,[0,10],xss,options);

Results for V = 2 and V = 1