## Bioreactor Simulation

This simulation is meant to demonstrate aspects of microbial growth in a bioreactor. A summary of this system and its governing equations may be found below the simulation. Set up your initial conditions at the bottom of the applet and press play to run the reaction simulation in "real-time". Press fast forward to run as quickly as your computer is capable. Mouse-over each input to see a description of that option or variable.

Batch or Continuous Bioreactor Simulation

by Anthony Butterfield

0 s
0

 x ln(x) y ln(y)
 Plot What: [X] [S] [P] [T] [G] μ
Simulation Constants and Variables:
Reactor

 t shrmindayweekmonthyrAEmsμspsdscsfsaszs V m3LmLbbl (UK)bbl (US, liq.)cm3ft3in3yd3cupgalfl ozpintqt F m3/sgpmL/hin3/sin3/mingphm3/minL/minL / s Δt shrmindayweekmonthyrAEmsμspsdscsfsaszs
Cells

 [X] g/Lkg/m3mg/L μmax Hz1/h1/s μ Hz1/h1/s tlag shrmindayweekmonthyrAEmsμspsdscsfsaszs
Substrate

 [S] g/Lkg/m3mg/L Ks g/Lkg/m3mg/L Ys/x
Gas

 [G] g/Lkg/m3mg/L Kg g/Lkg/m3mg/L Yg/x Hz1/h1/s [G]* g/Lkg/m3mg/L kla Hz1/h1/s
Product

 [P] Yp/x Hz1/h1/s
Toxin

 [T] g/Lkg/m3mg/L kd Hz1/h1/s Yt/x Hz1/h1/s [T]max g/Lkg/m3mg/L

Governing Equations:

Cellular growth is modeled by:

${ \left( \frac { \partial [X] }{ \partial t } \right) }_{ growth }=\mu [X]$

where [X] is the concentration of cells, t is time, and μ is the specific growth rate.

$\mu ={ \mu }_{ max }\left( \frac { [S] }{ { K }_{ S }+[S] } \right) \left( \frac { [G] }{ { K }_{ G }+[G] } \right) \left( \frac { 1+erf\left( t-{ t }_{ lag } \right) }{ 2 } \right)$

The specific growth rate is dependent upon monod-like relationships to substrate concentration, [S], and dissolved gas concentration, [G], with a maximum specific growth rate of μmax and half-velocity constants of KS and KG. The last term in the above equation accounts for the lag phase; when time, t, is less than the lag time, tlag, this term is zero and it quickly approaches 1 if time is greater than the lag time.

Cellular death is modeled by the following:

${ \left( \frac { \partial [X] }{ \partial t } \right) }_{ death }=\left( \frac { erf\left( [T]-{ [T] }_{ max } \right) +1 }{ 2 } \right) { k }_{ d }$

The first term in this equation, in parentheses, is zero until the concentration of the toxin, [T], nears its maximum, [T]max, at which point it becomes one and the death rate is a constant, kd.

The volume of the reactor, V, is held constant. As such, the flow rate into the reactor is always the same as the flow rate out, F. No cells are in the feed to the reactor. If this flow rate is non-zero then cells may also be lost out the drain:

${ \left( \frac { \partial [X] }{ \partial t } \right) }_{ drain }=\frac { [X]F }{ V }$

All together, the rate of change in the cell concentration is:.

${ \frac { \partial [X] }{ \partial t } }={ \left( \frac { \partial [X] }{ \partial t } \right) }_{ growth }-{ \left( \frac { \partial [X] }{ \partial t } \right) }_{ death }-{ \left( \frac { \partial [X] }{ \partial t } \right) }_{ drain }$

Substrate is consumed by cell division; for each gram of cells created the yield coefficient, YSX, of substrate is consumed. It may also be added at its initial concentration, [S]0, through the inlet media and lost in the reactor exit flow:

${ \frac { \partial [S] }{ \partial t } }=-{ Y }_{ SX }\frac { \partial [X] }{ \partial t } +\frac { \left( { [S] }_{ 0 }-[S] \right) F }{ V }$

A similar equation governs the concentration of dissolved gas, except all live cells consume gas at a rate of YGX grams per gram cells per hour, and gas is added by sparging and is governed by the mass transfer constant, kla, and the difference between the gas concentration and its saturation concentration, [G]*.

${ \frac { \partial [G] }{ \partial t } }=-{ Y }_{ GX }[X]+\frac { \left( { [G] }_{ 0 }-[G] \right) F }{ V } +{ k }_{ l }a\left( { [G] }^{ * }-[G] \right)$

The toxin and desired product are produced by the cells and are lost through the reactor outlet. They both have zero concentration in the inlet:

${ \frac { \partial [P] }{ \partial t } }={ Y }_{ PX }[X]-\frac { [P]F }{ V } ,\quad \quad { \frac { \partial [T] }{ \partial t } }={ Y }_{ TX }[X]-\frac { [T]F }{ V }$