Batch Fermentation: System

Batch Fermentation: System

Batch Fermentation: System

Important variables of the system are the biological dry mass or cell X,g y , substrate concentration S and product concentration P. The reactor volume V

is well mixed V

is well mixed.

The growth is assumed to follow kinetics described by Monod equation

V X kinetics described by Monod equation. S

Substrate consumption is related to cell growth by a constant yield Y_X/S

S P growth by a constant yield Y_X/S.

Product formation is the result of both growth and non-growth associated growth and non growth associated rates of production.

The lag and decline phases of cellg p

growth are not included in the model. Stirred batch fermenter

Batch Fermentation: Model Mass Balance

Batch Fermentation: Model Mass Balance ( ^{Rate of accumulati on} ) ( ^{= Rate of production} )

Mass Balances:

dX dX

( ^{Rate of accumulati on} ) ( ^{= Rate of production} )

V

dt r

V dX =

⋅

r

dX = dt

For Cells or

S

dS r

V =

dS

For Substrate

r S

V dt =

⋅ r

dt =

For Substrate or

V

dt r

V dP =

⋅ r

dP = dt

For Production or

dt dt

Batch Fermentation: Model Kinetics

Batch Fermentation: Model Kinetics

X

r

= μ ⋅

For Cells

Reaction Rate:

( )

(5)

X r

μ

X S

r = − r

For Cells

For Substrate

(6) ( )

S X

S

Y

X k

k

r

= (

+

⋅ μ ) ⋅

For Substrate

For Production

r

( k

+ k

μ ) X

For Production

S ) ( K S

S

S m

+

− ⋅

= μ

Monod relation

μ

k₁ – is the non-growth associated coefficient and k₂ – is the coefficient associated with growth.

Information Flow Diagram

Information Flow Diagram

X₀ Biomass X

Balance r_X

μ X

Growth t

Monod Ki ti Balance

S₀ S

S

rate Kinetics

Substrate

r_S r_X

Substrate Balance

P₀ P

rate Product

r_P

Product μ Balance

X Product μ

rate

Batch Fermentation: Program

Batch Fermentation: Program

As first step in solving the problem with MATLAB is to write the function file.

This file will be saved in the folder bin and the name of the file will later be required for processing in the Command Window of MATLAB

required for processing in the Command Window of MATLAB

Batch Fermentation: Program

Batch Fermentation: Program

In this section the ODEs with initial condition values are solved and plotted in diagram.

Batch Fermentation: Program

Batch Fermentation: Program

In this section the kinetic parameters are calculated.

Batch Fermentation: Program

Batch Fermentation: Program

In this section the range values for K_S are varied.

Batch Fermentation: Nomenclature

Batch Fermentation: Nomenclature

Symbol Description Unit

k1,k2 Product formation constants L/h, kg/kgg g

K_S Saturation constant kg/m³

P Product concentration kg/m³

R ti t k /( ³h)

r Reaction rate kg/(m³ h)

S Substrate concentration kg/m³

V Reactor volume m³

X Biomass concentration kg/m³

Y Yield coefficient kg/kg

m Specific groth rate L/h

m Specific groth rate L/h

Indexde Descriptionesc pt o SymbolSy bo Descriptionesc pt o 1 Refers to non-growth association

rate P Refers to Product

2 Refers to growth association rate S Refers to Substrate 2 Refers to growth association rate S Refers to Substrate

m Refers to maximum X Refers to Biomass

Batch Fermentation: Results

Batch Fermentation: Results

Kinetic Constants

UM = 0.3;

Plot of X, S and P during the batch growth

KS = 0.1;

K1 = 0.03;

K2 0 08

K2 = 0.08;

Y = 0.8;

Reaction Duration

Tint = 1;

Tfin = 40;

I i i l l f Initial values of variables

X0 = 0.01;

S0 = 10;

P0 = 0;

Batch Fermentation: Results

Batch Fermentation: Results

Plot of μ vs. S and r_X vs. r_S