1. Protein Digestion: A Dynamic Model Student Name 1 & Student Name 2 Process Control: Design Challenge I

2. Our System F 3, X a, X b, X c F 1, X ai F 2, X bi The Stomach V Assumptions: Well mixed Constant: V, X bi, density, flow rates Reaction: ODEs

3. Steady State Solution Original Conditions INPUT: Clear [xa,xb,xc]; eqns = {F1*Xai-V*k*xa*xb-F3*xa == 0, F2*Xbi-V*k*xa*xb-F3*xb == 0, V*k*xb*xa-F3*xc == 0, F1 == 15, F2 == 7, F3 == F1+F2, k == 5, Xai == 150, Xbi == 1000, V == 1000}; NSolve[eqns, {xa,xb,xc}] OUTPUT: {{xc®102.271,xa®0.00208415,xb®215.911},{xc®318.188,xa®-215.916,xb®- 0.00648415}} Exit concentrations:Protein: 0.0021 mg/ml Enzyme: 216.0 mg/ml Tryptophan: 102.3 mg/ml Decrease Initial Enzyme Concentration INPUT: Clear [xa,xb,xc]; eqns = {F1*Xai-V*k*xa*xb-F3*xa == 0, F2*Xbi-V*k*xa*xb-F3*xb == 0, V*k*xb*xa-F3*xc == 0,F1 == 15, F2 == 7, F3 == F1+F2, k == 5, Xai == 150, Xbi == 500, V == 1000}; NSolve[eqns, {xa,xb,xc}] OUTPUTl: {{xc®102.265,xa®0.00791828,xb®56.8261},{xc®159.103,xa®-56.8305,xb®- 0.0123183}} Exit concentrations:Protein: 0.008 mg/ml Enzyme: 56.8 mg/ml Tryptophan: 102.3 mg/ml K5 ml·min/mg F115 ml/min F27 ml/min V1000 ml Xai150 mg/ml Xbi1000 mg/ml Given Values

4. Steady State Solution Increase Inlet Flow Rate INPUT Clear [xa,xb,xc]; eqns = {F1*Xai-V*k*xa*xb-F3*xa == 0, F2*Xbi-V*k*xa*xb-F3*xb == 0, V*k*xb*xa-F3*xc == 0, F1 == 30, F2 == 7, F3 == F1+F2, k == 5, Xai == 150, Xbi == 1000, V == 1000}; NSolve[eqns, {xa,xb,xc}] OUTPUT {{xc®121.608,xa®0.0133159,xb®67.5809},{xc®189.21,xa®-67.5883,xb®- 0.0207159}} Exit concentrations:Protein: 0.013 mg/ml Enzyme: 67.6 mg/ml Tryptophan: 121.6 mg/ml INPUT: Clear [xa,xb,xc]; eqns = {F1*Xai-V*k*xa*xb-F3*xa == 0, F2*Xbi-V*k*xa*xb-F3*xb == 0, V*k*xb*xa-F3*xc == 0,F1 == 15, F2 == 7, F3 == F1+F2, k == 5, Xai == 150, Xbi == 1000, V == 1000}; NSolve[eqns, {xa,xb,xc}] OUTPUT: {{xc®102.271,xa®0.00208415,xb®215.911},{xc®318.188,xa®-215.916,xb®- 0.00648415}} Exit concentrations:Protein: 0.0021 mg/ml Enzyme: 216.0 mg/ml Tryptophan: 102.3 mg/ml Original Conditions

5. Mathematical Solution ODEs Laplace Procedure 1.Linearize ODE 2.Subtract original steady state 3.Find τ’s and k’s 4.Laplace 5.Rearrange to find transfer functions 6.Substitute to get one equation 7.Inverse Laplace with Maple®

6. Mathematical Solution Laplace Transform: Where

7. Mathematical Solution Cont. Inverse Laplace via Maple® Step Response Impulse Response Fuzzy Math

8. Step Response Matlab® Input function hdot=stomachstep(t,h) hdot=zeros(3,1); F1=15; F2=7; F3=22; V=1000; k=5; Xbi=1000; if t<100 Xai=100; else Xai=150; end; hdot(1)=(F1*Xai-V*k*h(1)*h(2)-F3*h(1))/V; hdot(2)=(F2*Xbi-V*k*h(1)*h(2)-F3*h(2))/V; hdot(3)=(k*V*h(1)*h(2)-F3*h(3))/V; t0=0; tf=400; h0=[0.021 216 102.3]; [t,h]=ode23(@stomachstep,[t0 tf],h0); plot(t,h(:,2),t,h(:,3))

9. Step Response Matlab® Output

10. Step Response Simulink® Diagram

11. Step Response Simulink® Output for Enzyme

12. Step Response Simulink® Output for Tryptophan

13. Impulse Response function hdot=stomachimpulse(t,h) hdot=zeros(3,1); F1=15; F2=7; F3=22; V=1000; k=5; Xbi=1000; if t 105 Xai=200; else Xai=500; end; hdot(1)=(F1*Xai-V*k*h(1)*h(2)-F3*h(1))/V; hdot(2)=(F2*Xbi-V*k*h(1)*h(2)-F3*h(2))/V; hdot(3)=(k*V*h(1)*h(2)-F3*h(3))/V; t0=0; tf=400; h0=[0.0021 216 102.3]; [t,h]=ode23(@stomachimpulse,[t0 tf],h0); plot(t,h(:,2),t,h(:,3)) Matlab® Input

14. Impulse Response Matlab® Output

15. Impulse Response Simulink® Diagram

16. Impulse Response Simulink® Output for Enzyme

17. Impulse Response Simulink® Output for Tryptophan

18. Final Thoughts Linear approximation is valid Step response is as expected Impulse response as expected Matlab® & Simulink® agree Model will provide basis for more complex systems Conclusions Vary k values Explore larger/smaller step and impulse values Make model more realistic Model multiple food components Recommendations

19. References Biglione, N., Yousef, M., & Rodgers, V.G.J., (2004). Process Dynamics & Control in Design 52:185: Lab # 1: Programming and Simulation using Matlab® and Simulink®. Seborg, D.E., Edgar, T.F., & Millichamp, D.A. (2004). Process Dynamics and Control (2nd ed.). Hoboken, NJ: John Wiley & Sons. Vander, A., Sherman, J., Luciano, D. (2001). Human Physiology: The Mechanisms of Body Function (8th ed.). Boston: McGraw Hill.