MATLAB MODEL FERMENTATION

Brewing beer involves putting sugar (S) and yeast (Y) together in a vessel so that alcohol (A) is produced as a by-product of the metabolism of sugar by yeast. Two important aspects of fermentation are the diminishing supply of sugar and the formation of excessive alcohol that will kill yeast cells. The model has some analogy to enzyme kinetics (sugar is a substrate for yeast) and predator prey interaction (alcohol preys yeast).

The set of differential equations, initial conditions and constants are:

S=1000 (rel. conc.)

Y =100 (rel. conc.)

A = 0 (rel. conc.)

dS/dt = -(a*b*S*Y) - (a*f*S*Y)

dY/dt = a*c*S*Y - (d*Y*A)

dA/dt = a*b*S*Y

Constants:

a=0.1

b=0.01

c=0.05

d=0.01

f=0.01

Use Eulers method with a step size of h = 0.1 to find approximate values of the state variables S, Y and A from t=0 up to t=6 hours. Produce a graph showing the change of state variables. Do not use Matlab ODE solvers, but develop your own Euler solver.

Develop a Runga-Kutta solver (RK4) with a step size of h = 0.1 . Graph the functions for all three state variables.

Determine the absolute and relative errors for all three predicted state variables at t=6 between Euler and RK4.

Summarize your results with 2-3 sentences. What do your predictions tell us about the process of fermentation? To what degree is RK4 better than Euler and does it depend on the behavior of the state variable?