Bioreactors create a biologically active environment for the production of chemicals. A bioreactor is a vessel where organisms are grown and preferential products are produced by controlling the feed and temperature. Bioreactors can either be aerobic or anaerobic.
One particular batch bioreactor is fed at varying rates throughout the batch process to produce ethanol. Algebraic equations define the heat generation, rate constants, volume, and other factors used in the model. Symbols and units are defined in the source code.
Growth Rate
$$\mu=\mu_{\max} \frac{{S}}{{K}_{{SX}}+{S}} \frac{{O}_{{liq}}}{{K}_{{OX}}+{O}_{{liq}}}\left(1-\frac{{P}}{{P}_{\max}}\right) \frac{1}{1+\exp (-(100-{S}))}$$
$$\mu_{\max}=\left[\left({a}_1\left({~T}-{k}_1\right)\right)\left(1-\exp \left({b}_1\left({~T}-{k}_2\right)\right)\right)\right]^2$$
$${P}_{\max}={P}_{\operatorname{maxb}}+\frac{{P}_{\operatorname{maxT}}}{1-\exp \left(-{b}_2\left({~T}-{k}_3\right)\right)}$$
$${q}_{{P}}={a}_{{P}} \mu+{b}_{{P}}$$
Non-Growth Ethanol Production
$${b}_{{P}}={c}_1 \exp \left(-\frac{{A}_{{P} 1}}{{~T}}\right)-{c}_2 \exp \left(-\frac{{A}_{{P} 2}}{{~T}}\right)$$
Ethanol Consumption Rate
$${q}_{{S}}=\frac{\mu}{{Y}_{{XS}}}+\frac{{q}_{{P}}}{{Y}_{{PS}}}$$
Oxygen Consumption Rate
$${q}_{{O}}=\frac{{q}_{{O}, \max}}{{Y}_{{Xo}}} \frac{{O}_{{liq}}}{{K}_{{OX}}+{O}_{{liq}}}$$
Biological Cell Mass Deactivation Rate
$${K}_{{d}}={K}_{{db}}+\frac{{K}_{{dT}}}{1+\exp \left(-{b}_3\left({~T}-{k}_4\right)\right)}$$
Oxygen Saturation Concentration
$${O}^*=\frac{{zO}_{{gas}} {RT}}{{K}_{{H}}}$$
Oxygen Mass Transfer Coefficient
$${k}_{l} {a}=\left({k}_{l} {a}\right)_0(1.2)^{{T}-20}$$
Liquid and Vapor Volumes
$${V}_{{l}}+{V}_{{g}}={V}$$
Differential equations are from material, species, and energy balances. The differential equations relate the input flow and substrate concentrations to ethanol production.
Liquid Volume
$$\frac{d V_1}{d t}=Q_{i n}-Q_e$$
Total Cell Mass
$$\frac{{dX}}{{dt}}=\frac{{Q}_{{in}}}{{V}_1}\left({X}_{{t}, {in}}-{X}_{{t}}\right)+\mu {X}_{{v}}$$
Total Biologically Active Cell Mass
$$\frac{d X_v}{d t}=\frac{Q_{i n}}{V_1}\left(X_{v, i n}-X_v\right)+\left(\mu-K_d\right) X_v$$
Glucose (Substrate) Concentration
$$\frac{{dS}}{{dt}}=\frac{{Q}_{{in}}}{{V}_{{l}}}\left({S}_{{in}}-{S}\right)-{q}_{{S}} {X}_{{v}}$$
Ethanol (Product) Concentration
$$\frac{{dP}}{{dt}}=\frac{{Q}_{{in}}}{{V}_1}\left({P}_{{in}}-{P}\right)+{q}_{{P}} X_{{v}}$$
Liquid Oxygen Concentration
$$\frac{{dO}_{liq}}{{dt}}=\frac{{Q}_{{in}}}{{V}_{{l}}}\left({O}^*-{O}_{{liq}}\right)+\left({k}_{{l}} {a}\right)\left({O}^*-{O}_{{liq}}\right)-{qo}_{{o}} {X}_{{v}}$$
Gas Oxygen Concentration
$$\frac{d O_{{gas}}}{d t}=\frac{F_{{air}}}{V_g}\left(O_{{gas}, { in}}-O_{{gas}}\right)-\frac{V_l\left(k_l a\right)}{V_g}\left(O^*-O_{{liq}}\right)+O_{{gas}} \frac{Q_{{in}}-Q_e}{V_g}$$
Bioreactor Temperature
$$\frac{{dT}}{{dt}}=\frac{{Q}_{{in}}}{{V}_{{l}}}\left({T}_{{in}}-{T}\right)-\frac{{T}_{{ref}}}{{V}_{{l}}}\left({Q}_{{in}}-{Q}_{{e}}\right)+\frac{{q}_{{o}} {X}_{{v}} \Delta {H}}{{MW}_{{O}} \rho {C}_{{p}, {br}}}-\frac{{K}_{{T}} {A}_{{T}}\left({T}-{T}_{{c}}\right)}{{V}_{{l}} \rho {C}_{{p}, {br}}}$$
Cooling Agent Temperature
$$\frac{{dT}_{{c}}}{{dt}}=\frac{{F}_{{c}}}{{V}_{{cj}}}\left({T}_{{c}, {in}}-{T}_{{c}}\right)+\frac{{K}_{{T}} {A}_{{T}}\left({T}-{T}_{{c}}\right)}{{V}_{{cj}} \rho_{{c}} {C}_{{p}, {c}}}$$
The model simulates biomass, ethanol, and by-product production. The following chart show total biomass and viable (living) biomass in the reactor.
The model can be used to investigate advanced process control to maximize performance subject to a feeding strategy and measured disturbances.
Save as reactor.m
Save as EthanolModel.m
The simulation model of an ethanol bioreactor can be used to optimize the production of ethanol by providing a virtual platform to test different scenarios and strategies without the need for costly and time-consuming experimental trials. Follow these steps to optimize the ethanol production:
The optimization strategy is used to determine the optimal Tcin (temperature to the cooling jacket) and Fair (air flow).
Maximize Ethanol Production with Cooling Temperature
Maximize Ethanol Production with Air Flow Rate