Bioreactor Simulation for Ethanol Production using GEKKO

Nice application! Here are some suggestions to improve the convergence.

1. Remove the lower and upper bounds when simulating. This was causing the "no solution found" error.

Vl     = m.Var(value=1000, name='Vl') # lb=-0.0, ub=0.75*V
Xt     = m.Var(value=0.1,  name='Xt') # lb=-0.0, ub=10
Xv     = m.Var(value=0.1,  name='Xv') # lb=-0.0, ub=10
S      = m.Var(value=400,  name='S')  # lb=+0.0, ub=10000
P      = m.Var(value=0, name='P')
Ol     = m.Var(value=0.0065, name= 'Ol')
Og     = m.Var(value=0.305, name='Og')
T      = m.Var(value=30, name='T')    # lb=20, ub=40
Tc     = m.Var(value=20, name='Tc')   # lb=0, ub=30

2. Re-arrange the equations to avoid divide-by-zero (when possible). For most of the equations, the volume term can be moved to the left-hand side of the equation to avoid a variable in the denominator.

m.Equation(Vl.dt()     == Qin - Qe)
m.Equation(Vl*Xt.dt()  == Qin*(Xtin-Xt) + mi*Vl*Xv)
m.Equation(Vl*Xv.dt()  == Qin*(Xvin-Xv) + Xv*Vl*(mi-Kd))
m.Equation(Vl*S.dt()   == Qin*(Sin-S) - qS*Vl*Xv)
m.Equation(Vl*P.dt()   == Qin*(Pin - P) + qP*Vl*Xv)
m.Equation(Vl*Ol.dt()  == Qin*(Ostar-Ol)   + Vl*kla*(Ostar-Ol) - qO*Vl*Xv)
m.Equation(Vg*Og.dt()  == Fair*(Ogasin-Og) - Vl*kla*(Ostar-Ol) + Og*(Qin-Qe))
m.Equation(Vl*T.dt()   == Qin*(Tin-T) - Tref*(Qin-Qe) 
                          + Vl*qO*Xv*deltaH/MO/rho/Chbr - KT*AT*(T-Tc)/rho/Chbr)
m.Equation(Vcj*Tc.dt() == Fc*(Tcin - Tc) + KT*AT*(T-Tc)/rhoc/Chc)
m.Equation(Sf_cum.dt() == Qin*Sin)

3. Use APOPT solver for improved speed and increase NODES=3 for improved accuracy. IMODE=7 is a sequential simulation to improve the solution speed when there are zero Degrees of Freedom (simulation, #equations=#variables).

m.options.SOLVER= 1
m.options.IMODE = 7
m.options.NODES = 3 

4. Here is an easier way to define the step inputs that is based on time.

# I want Qin to be a step function:
#   Qin = Qin0 + 15H(t-5) + 5H(t-10) - 6H(t-20) - 14H(t-35)
#   where H(t-t0) heaviside function
Qin_step = np.zeros(nt)
Qin_step[np.where(tm>=5)]  += 15
Qin_step[np.where(tm>=10)] += 5
Qin_step[np.where(tm>=20)] -= 6
Qin_step[np.where(tm>=35)] -= 14
Qin  = m.Param(value=Qin_step, name='Qin')

5. Avoid if3() (when possible) and replace with lower bounds in the variable definition:

#mi = m.if3(condition=mi, x1=0, x2=mi)
mi = m.Var(name='mi',lb=0)
m.Equation(mi == mimax * (S / (KSX+S)) * (Ol/(KOX + Ol)) 
                 * (1 - P/Pmax) * (1 / (1+m.exp(-(100-S)))))

or a slack variable slk that avoids a binary switching variable introduced with m.if3():

slk  = m.Var(0,lb=0)
mi_u = m.Var(name='mi_u')
mi   = m.Var(name='mi',lb=0)
m.Equation(mi = mi_u + slk)
m.Minimize(slk)

6. (Optional) Insert a few additional small steps at the start if there are problems converging. This is useful later when you start optimizing.

# Create time vector: t=[0, 0.1, 0.2,...,36.9,37], [hours]
tm = np.linspace(0,37,371)
# Insert smaller time steps at the beginning
tm = np.insert(tm,1,[0.001,0.005,0.01,0.05])

7. (Later) When you need to optimize, it is often helpful to converge to a simulation as an initial guess for the optimization problem.

m.options.IMODE=7
m.solve()
m.options.IMODE=6
m.solve()

Here is the complete script.

from gekko import GEKKO
import numpy as np
import matplotlib.pyplot as plt

m = GEKKO(remote=False)
# Create time vector: t=[0, 0.1, 0.2,...,36.9,37], [hours]
tm = np.linspace(0,37,371)
# Insert smaller time steps at the beginning
tm = np.insert(tm,1,[0.001,0.005,0.01,0.05])
m.time = tm
nt = len(tm)

# Define constants and parameters
#################################
# Kinetic Parameters
a1     = m.Const(value=0.05, name='a1')     # Ratkowsky parameter [oC-1 h-0.5]
aP     = m.Const(value=4.50, name='aP')     # Growth-associated parameter for EtOh production [-]
AP1    = m.Const(value=6.0, name='AP1')     # Activation energy parameter for EtOh production [oC]
AP2    = m.Const(value=20.3, name='AP2')    # Activation energy parameter for EtOh production [oC]
b1     = m.Const(value=0.035, name='b1')    # Parameter in the exponential expression of the maximum specific growth rate expression [oC-1]
b2     = m.Const(value=0.15, name='b2')     # Parameter in the exponential expression of the maximum specific growth rate expression [oC-1]
b3     = m.Const(value=0.40, name='b3')     # Parameter in the exponential expression of the specific death rate expression [oC-1]
c1     = m.Const(value=0.38, name='c1')     # Constant decoupling factor for EtOh [gP gX-1 h-1]
c2     = m.Const(value=0.29, name='c2')     # Constant decoupling factor for EtOh [gP gX-1 h-1]
k1     = m.Const(value=3, name='k1')        # Parameter in the maximum specific growth rate expression [oC]
k2     = m.Const(value=55, name='k2')       # Parameter in the maximum specific growth rate expression [oC]
k3     = m.Const(value=60, name='k3')       # Parameter in the growth-inhibitory EtOH concentration expression [oC]
k4     = m.Const(value=50, name='k4')       # Temperature at the inflection point of the specific death rate sigmoid curve [oC]
Pmaxb  = m.Const(value=90, name='Pmaxb')    # Temperature-independent product inhibition constant [g L-1]
PmaxT  = m.Const(value=90, name='PmaxT')    # Maximum value of product inhibition constant due to temperature [g L-1]
Kdb    = m.Const(value=0.025, name='Kdb')   # Basal specific cellular biomass death rate [h-1]
KdT    = m.Const(value=30, name='KdT')      # Maximum value of specific cellular biomass death rate due to temperature [h-1]
KSX    = m.Const(value=5, name='KSX')       # Glucose saturation constant for the specific growth rate [g L-1]
KOX    = m.Const(value=0.0005, name='KOX')  # Oxygen saturation constant for the specific growth rate [g L-1]
qOmax  = m.Const(value=0.05, name='qOmax')  # Maximum specific oxygen consumption rate [h-1]

# Metabolic Parameters
YPS    = m.Const(value=0.51, name='YPS')    # Theoretical yield of EtOH on glucose [gP gS-1]
YXO    = m.Const(value=0.97, name='YXO')    # Theoretical yield of biomass on oxygen [gX gO-1]
YXS    = m.Const(value=0.53, name='YXS')    # Theoretical yield of biomass on glucose [gX gS-1]

# Physicochemical and thermodynamic parameters
Chbr   = m.Const(value=4.18, name='Chbr')      # Heat capacity of the mass of reaction [J g-1 oC-1]
Chc    = m.Const(value=4.18, name='Chc')       # Heat capacity of cooling agent [J g-1 oC-1]
deltaH = m.Const(value=518000, name='deltaH')  # Heat of reaction of fermentation [J mol-1 O2]
Tref   = m.Const(value=20, name='Tref')        # Reference temperature [oC]
KH     = m.Const(value=200, name='KH')         # Henry's constant for oxygen in the fermentation broth [atm L mol-1]
z      = m.Const(value=0.792, name='z')        # Oxygen compressibility factor [-]
R      = m.Const(value=0.082, name='R')        # Ideal gas constant [L atm mol-1 oC-1]
kla0   = m.Const(value=100, name='kla0')       # Temperature-independent volumetric oxygen transfer coefficient [-h]
KT     = m.Const(value=360000, name='KT')      # Heat transfer coefficient [J h-1 m-2 oC-1]
rho    = m.Const(value=1080, name='rho')       # Density of the fermentation broth [g L-1]
rhoc   = m.Const(value=1000, name='rhoc')      # Density of the cooling agent [g L-1]
MO     = m.Const(value=32.0, name='MO')        # Molecular weight of oxygen [g mol-1]

# Bioreactor design data
AT     = m.Const(value=1, name='AT')          # Bioreactor heat transfer area [m2]
V      = m.Const(value=1800, name='V')        # Bioreactor working volume [L]
Vcj    = m.Const(value=50, name='Vcj')       # Cooling jacket volume [L]
Ogasin = m.Const(value=0.305, name='Ogasin')  # Oxygen concentration in airflow inlet [g L-1]

# Define variables
##################
mi = m.Var(name='mi',lb=0)
# I want Qin to be a step function:
#   Qin = Qin0 + 15H(t-5) + 5H(t-10) - 6H(t-20) - 14H(t-35)
#   where H(t-t0) heaviside function
Qin_step = np.zeros(nt)
Qin_step[np.where(tm>=5)]  += 15
Qin_step[np.where(tm>=10)] += 5
Qin_step[np.where(tm>=20)] -= 6
Qin_step[np.where(tm>=35)] -= 14
Qin  = m.Param(value=Qin_step, name='Qin')
# Fixed variables, they are constant throughout the time horizon
Xtin = m.FV(value=0, name='Xtin')
Xvin = m.FV(value=0, name='Xvin')
Qe   = m.FV(value=0, name='Qe')
Sin  = m.FV(value=400, lb=0, ub=1500)
Pin  = m.FV(value=0, name='Pin')
Fc   = m.FV(value=40, name='Fc')
Fair = m.FV(value=60000, name='Fair')
Tin  = m.FV(value=30, name='Tin')
Tcin = m.FV(value=15, name='Tcin')

Vl     = m.Var(value=1000, name='Vl') # lb=-0.0, ub=0.75*V
Xt     = m.Var(value=0.1,  name='Xt') # lb=-0.0, ub=10
Xv     = m.Var(value=0.1,  name='Xv') # lb=-0.0, ub=10
S      = m.Var(value=50,  name='S')   # lb=+0.0, ub=10000
P      = m.Var(value=0, name='P')
Ol     = m.Var(value=0.0065, name= 'Ol')
Og     = m.Var(value=0.305, name='Og')
T      = m.Var(value=30, name='T')    # lb=20, ub=40
Tc     = m.Var(value=20, name='Tc')   # lb=0, ub=30
Sf_cum = m.Var(value=0, name='Sf_cum')
#t      = m.Var(value=0, name='Time')

# Define algebraic equations
############################
# Specific growth rate of cell mass
mimax = m.Intermediate(((a1*(T-k1))*(1-m.exp(b1*(T-k2))))** 2)
Pmax = m.Intermediate(Pmaxb + PmaxT/(1-m.exp(-b2*(T-k3))))
m.Equation(mi == mimax * (S / (KSX+S)) * (Ol/(KOX + Ol)) 
                 * (1 - P/Pmax) * (1 / (1+m.exp(-(100-S)))))
#mi = m.if3(condition=mi, x1=0, x2=mi)
# Specific production rate of EtOH
#bP = m.if3(condition=S, x1=0, x2=c1*m.exp(-AP1/T) - c2*m.exp(-AP2/T))
bP = m.Intermediate(c1*m.exp(-AP1/T) - c2*m.exp(-AP2/T))
qP = m.Intermediate(aP*mi + bP)
# Specific consumption rate of glucose
qS = m.Intermediate(mi/YXS + qP/YPS)
# Specific consumption rate of oxygen
qO = m.Intermediate(qOmax*Ol/YXO/(KOX+Ol))
# Specific biological deactivation rate of cell mass
Kd = m.Intermediate(Kdb + KdT/(1+m.exp(-b3*(T-k4))))
# Saturation concentration of oxygen in culture media
Ostar = m.Intermediate(z*Og*R*T/KH)
# Oxygen mass transfer coefficient
kla = m.Intermediate(kla0*1.2**(T-20))
# Bioreactor phases equation
Vg = m.Intermediate(V - Vl)

# Define differential equations
###############################
m.Equation(Vl.dt()     == Qin - Qe)
m.Equation(Vl*Xt.dt()  == Qin*(Xtin-Xt) + mi*Vl*Xv)
m.Equation(Vl*Xv.dt()  == Qin*(Xvin-Xv) + Xv*Vl*(mi-Kd))
m.Equation(Vl*S.dt()   == Qin*(Sin-S) - qS*Vl*Xv)
m.Equation(Vl*P.dt()   == Qin*(Pin - P) + qP*Vl*Xv)
m.Equation(Vl*Ol.dt()  == Qin*(Ostar-Ol)   + Vl*kla*(Ostar-Ol) - qO*Vl*Xv)
m.Equation(Vg*Og.dt()  == Fair*(Ogasin-Og) - Vl*kla*(Ostar-Ol) + Og*(Qin-Qe))
m.Equation(Vl*T.dt()   == Qin*(Tin-T) - Tref*(Qin-Qe) 
                          + Vl*qO*Xv*deltaH/MO/rho/Chbr - KT*AT*(T-Tc)/rho/Chbr)
m.Equation(Vcj*Tc.dt() == Fc*(Tcin - Tc) + KT*AT*(T-Tc)/rhoc/Chc)
m.Equation(Sf_cum.dt() == Qin*Sin)
#m.Equation(t.dt()      == 1)

# solve ODE
m.options.SOLVER= 1
m.options.IMODE = 7
m.options.NODES = 3
# m.open_folder()
m.solve(disp=False)

# Plot results
plt.figure(1)
plt.title('Total & Viable Cellular Biomass')
plt.plot(m.time, Xv.value, label='Xv')
plt.plot(m.time, Xt.value, label='Xt')
plt.legend()
plt.ylabel('Biomass concentration [g/L]')
plt.xlabel('Time [h]')
plt.grid()
plt.minorticks_on()
plt.ylim(0)
plt.xlim(m.time[0],m.time[-1])
plt.tight_layout()

plt.figure(2)
plt.title('Substrate (S) & Product (P) concentration')
plt.subplot(2,1,1)
plt.plot(m.time, S.value, label='S')
plt.legend(); plt.grid()
plt.ylabel('Conc [g/L]')
plt.subplot(2,1,2)
plt.plot(m.time, P.value, label='P')
plt.legend(); plt.grid()
plt.ylabel('Conc [g/L]')
plt.xlabel('Time [h]')
plt.minorticks_on()
plt.ylim(0)
plt.xlim(m.time[0],m.time[-1])
plt.tight_layout()

plt.figure(3)
plt.title('Bioreactor & Cooling jacket temperature')
plt.plot(m.time, T.value, label='T')
plt.plot(m.time, Tc.value, label='Tc')
plt.legend()
plt.ylabel('Temperature [oC]')
plt.xlabel('Time [h]')
plt.grid()
plt.minorticks_on()
plt.ylim(0)
plt.xlim(m.time[0],m.time[-1])
plt.tight_layout()

fig4, ax = plt.subplots()
ax.title.set_text('Dissolved & Gaseous Oxygen concentration')
lns1 = ax.plot(m.time, Ol.value, label='[Oliq]', color='c')
ax.set_xlabel('Time [h]')
ax.set_ylabel('Oliq [g/L]', color='c')
ax.minorticks_on()
ax2 = ax.twinx()
lns2 = ax2.plot(m.time, Og.value, label='[Ogas]', color='y')
ax2.set_ylabel('Ogas [g/L]', color='y')
ax2.minorticks_on()
lns = lns1 + lns2
labs = [l.get_label() for l in lns]
ax.legend(lns, labs, loc='best')
ax.grid()
fig4.tight_layout()
plt.figure(4)

plt.figure(5)
plt.title('Feeding Policy')
plt.plot(m.time, Qin.value, label='Qin')
plt.legend()
plt.ylabel('Qin [L/h]')
plt.xlabel('Time [h]')
plt.grid()
plt.minorticks_on()
plt.ylim(0)
plt.xlim(m.time[0],m.time[-1])
plt.tight_layout()

plt.figure(6)
plt.title('Check >=0 Constraints')
plt.subplot(2,1,1)
plt.plot(tm,bP.value,label='bP')
plt.legend(); plt.grid()
plt.subplot(2,1,2)
plt.plot(tm,mi.value,label='mi')
plt.legend(); plt.grid()

plt.show()

The plots replicate the Matlab plots with the updated parameters (thanks for the suggestion). Let us know if we can help with further questions.

Python Version

Here is an equivalent Python version. I created this to help with the equivalency testing, if it is still needed.

import numpy as np
from scipy.integrate import odeint
from scipy.interpolate import interp1d
import matplotlib.pyplot as plt

#Simulate fed-batch operation
# Specify the simulation time (hrs)
tspan = np.linspace(0,37,371); t=tspan
# Specify values of control variables [Qin0 Xtin Xvin Qe Sin Fc Fair Tin Tcin]
u0 = [0.0,0.0,0.0,0.0,0.0,400,40,60000,30,15]
# Specify initial conditions [Xt Xv S P Oliq Ogas T Tc Vl Gloss MP]
x0 = [0.1,0.1,50,0,0.0065,0.305,30,20,1000,0.0,0.0]
ux0 = tuple(u0 + x0)

#Qin = 15*heaviside(t-5) + 5*heaviside(t-10)
#      - 6*heaviside(t-20) - 14*heaviside(t-35)
Qin = np.zeros_like(tspan)
Qin[np.where(tspan>=5)]  += 15
Qin[np.where(tspan>=10)] += 5
Qin[np.where(tspan>=20)] -= 6
Qin[np.where(tspan>=35)] -= 14
QinInterp = interp1d(tspan,Qin,bounds_error=False)

def ethanol(x,t,Qin0,Qin,Xtin,Xvin,Qe,Sin,Fc,Fair,
            Tin,Tcin,Xt0,Xv0,S0,P0,Oliq0,Ogas0,T0,
            Tc0,Vl0,Sf_cum0,Time0):
    ##    #Initial Conditions
    ##    Xt0 = u[10]     # Initial total cellular biomass, [g L-1]
    ##    Xv0 = u[11]     # Initial viable cellular biomass, [g L-1]
    ##    S0 = u[12]      # Initial substrate/Glucose concentration, [g L-1]
    ##    P0 = u[13]      # Initial product/Ethanol concentration, [g L-1]
    ##    Oliq0 = u[14]   # Initial Dissolved oxygen concentration, [g L-1]
    ##    Ogas0 = u[15]   # Initial Gas phase oxygen (bubbles) in the fermentation broth, [g L-1]
    ##    T0 = u[16]      # Initial Temperature in the bioreactor, [oC]
    ##    Tc0 = u[17]     # Initial Temperature of the cooling agent in the jacket, [oC]
    ##    Vl0 = u[18]     # Initial Culture volume in the bioreactor, [L]
    ##    Sf_cum0 = u[19] # Initial Cumulative substrate/glucose fed to the bioreactor, [g]
    ##    Time0 = u[20]   # Initial batch time, [h]
    ##    
    ##    #Control variables
    ##    Qin0 = u[0]        # Volumetric inflow rate, [l/h-1]
    ##    Qin = u[1]         # Volumetric inflow rate, [l/h-1]
    ##    Xtin = u[2]        # Total biomass concentration in the bioreactor feed, [g L-1]
    ##    Xvin = u[3]        # Viable biomass concentration in the bioreactor feed, [g L-1]
    ##    Qe = u[4]          # Volumetric outflow rate, [l/h-1]
    ##    Sin = u[5]         # Substrate/Glucose concentration in bioreactor feed, [g L-1]
    ##    Fc = u[6]          # Cooling agent inlet volumetric flowrate, [L h-1]
    ##    Fair = u[7]        # Airflow inlet volumetric flowrate, [L h-1]
    ##    Tin = u[8]         # Temperature of bioreactor feed, [oC]
    ##    Tcin = u[9]        # Temperature of cooling agent inlet, [oC]

    # 1D Interpolation for Qin
    Qin = QinInterp(t)
    
    #Definition of model parameters
    #Kinetic parameters
    a1 = 0.05    # Ratkowsky parameter [oC-1 h-0.5]
    aP = 4.50    # Growth-associated parameter for ethanol production, [-]
    AP1 = 6.0    # Activation energy parameter for ethanol production, [oC]
    AP2 = 20.3   # Activation energy parameter for ethanol production, [oC]
    b1 = 0.035   # Parameter in the exponential expression of the maximum specific growth rate np.expression, [oC-1]
    b2 = 0.15    # Parameter in the exponential expression of the growth inhibitory ethanol concentration np.expression, [oC-1]
    b3 = 0.40    # Parameter in the exponential np.expression of the specific death rate expression,[oC-1]
    c1 = 0.38    # Constant decoupling factor for ethanol production, [gP gX-1 h-1]
    c2 = 0.29    # Constant decoupling factor for ethanol production, [gP gX-1 h-1]
    k1 = 3.00    # Parameter in the maximum specific growth rate expression, [oC]
    k2 = 55.0    # Parameter in the maximum specific growth rate expression, [oC]
    k3 = 60.0    # Parameter in the growth-inhibitory ethanol concentration expression, [oC]
    k4 = 50.0    # Temperature at the inflection point of the specific death rate sigmoid curve, [oC]
    Pmaxb = 90   # Temperature-independent product inhibition constant, [g L-1]
    PmaxT = 90   # Maximum value of product inhibition constant due to temperature, [g L-1]
    Kdb = 0.025  # Basal specific cellular biomass death rate, [h-1]
    KdT = 30.00  # Maximum value of specific cellular biomass death rate due to temperature, [h-1]
    KSX = 5      # Glucose saturation constant for the specific growth rate, [g L-1]
    KOX = 0.0005 # Oxygen saturation constant for the specific growth rate, [g L-1]
    qOmax = 0.05 # Maximum specific oxygen consumption rate, [h-1]
    
    #Metabolic parameters
    YPS = 0.51 # Theoretical yield of ethanol on glucose, [gP gS-1]
    YXO = 0.97 # Theoretical yield of biomass on oxygen, [gX gO-1]
    YXS = 0.53 # Theoretical yield of biomass on glucose, [gX gS-1]
    
    #Physicochemical and thermodynamic parameters
    Chbr = 4.18     # Heat capacity of the mass of reaction, [J g-1 oC-1]
    Chc = 4.18      # Heat capacity of the cooling agent, [J g-1 oC-1]
    DeltaH = 518000 # Heat of reaction of fermentation, [J mol-1 O2]
    Tref = 20       # Reference temperature, [oC]
    KH = 200        # Henry's constant for oxygen in the fermentation broth, [atm L mol-1]
    z = 0.792       # Oxygen compressibility factor, [-]
    R = 0.082       # Ideas gas constant, [L atm mol-1 oC-1]
    kla0 = 100      # Temperature-independent volumetric oxygen transfer coefficient, [h-1]
    KT = 360000     # Heat transfer coefficient, [J h-1 m-2 ??C-1]
    rho = 1080      # Density of the fermentation broth, [g L-1]
    rhoc = 1000     # Density of the cooling agent, [g L-1]
    MO = 32.0       # Molecular weight of oxygen (O2), [g mol-1]
    
    #Bioreactor design data
    AT = 1.0       # Bioreactor heat transfer area, [m2]
    V = 1800       # Bioreactor working volume, [L]
    Vcj = 50       # Cooling jacket volume, [L]
    Ogasin = 0.305 # Oxygen concentration in airflow inlet, [g L-1]
    
    #Definition of model variables
    #State variables
    Xt = x[0]      # Total cellular biomass, [g L-1]
    Xv = x[1]      # Viable cellular biomass, [g L-1]
    S = x[2]       # Substrate/Glucose concentration, [g L-1]
    P = x[3]       # Product/Ethanol concentration, [g L-1]
    Oliq = x[4]    # Dissolved oxygen concentration, [g L-1]
    Ogas = x[5]    # Gas phase oxygen (bubbles) in the fermentation broth, [g L-1]
    T = x[6]       # Temperature in the bioreactor, [oC]
    Tc = x[7]      # Temperature of the cooling agent in the jacket, [oC]
    Vl = x[8]      # Culture volume in the bioreactor, [L]
    Sf_cum = x[9]  # Cumulative amount of substrate/glucose fed to the bioreactor, [g]
    Time = x[10]   # Batch time, [h]
        
    # Definition of model equations
    # Kinetic rates
    # -----------------------------
    # Specific growth rate, [h-1] 
    mmax = ((a1*(T-k1))*(1-np.exp(b1*(T-k2))))**2
    Pmax = Pmaxb + PmaxT/(1-np.exp(-b2*(T-k3)))
    m1 = mmax * S/(KSX + S) * Oliq/(KOX + Oliq) * (1 - P/Pmax) * 1/(1+np.exp(-(100-S)/1)) # Specific growth rate, [h-1] 
    if m1 >= 0:
        m = m1
    else:
        m=0.0
    # Non-growth-associated ethanol specific production rate, [h-1]
    if S > 0:
        bP = c1 * np.exp(-AP1/T) - c2 * np.exp(-AP2/T)  # Non-growth-associated ethanol specific production rate, [h-1]
    else:
        bP = 0.0
    qP = aP*m + bP
    # Ethanol consumption specific rate
    qS = m/YXS + qP/YPS
    # Oxygen consumption specific rate
    qO = qOmax*Oliq/YXO/(KOX + Oliq)
    # Specific biological deactivation rate of cell mass
    Kd = Kdb + KdT/(1+np.exp(-b3*(T-k4)))
    # Saturation concentration of oxygen in culture media
    Osat = z*Ogas*R*T/KH
    # Oxygen mass transfer coefficient
    kla = kla0*1.2**(T-20)
    # Volume of the gas phase in the bioreactor
    Vg = V - Vl
        
    #Material balances
    #-----------------
    # Volume of liquid culture
    dVl = Qin - Qe
    # Total cell mass
    dXt = m*Xv + Qin/Vl*(Xtin-Xt)
    # Total mass of biologically active cells
    dXv = (m-Kd)*Xv + Qin/Vl*(Xvin-Xv)
    # Glucose concentration
    dS = Qin/Vl*(Sin-S) - qS*Xv
    # Ethanol concentration
    dP = Qin/Vl*(-P) + qP*Xv
    # Disolved oxygen
    dOliq = Qin/Vl*(Osat - Oliq) + kla*(Osat-Oliq) - qO*Xv
    # Oxygen gas phase
    dOgas = Fair/Vg*(Ogasin-Ogas) - Vl*kla/Vg*(Osat - Oliq) + Ogas*(Qin-Qe)/Vg
    
    # Energy balances
    #---------------
    # Bioreactor temprature
    dT = Qin/Vl*(Tin-T) - Tref/Vl*(Qin-Qe) + qO*Xv*DeltaH/MO/rho/Chbr - KT*AT*(T-Tc)/Vl/rho/Chbr
    # Cooling agent temperature
    dTc = Fc/Vcj*(Tcin-Tc) + KT*AT*(T-Tc)/Vcj/rhoc/Chc
    
    # Yields & Productivity
    #---------------------
    # Cumulative amount of glucose fed to the bioreactor
    dSf_cum = Sin*Qin
    dTime = 1
    
    # Definition of state derivatives vector
    # State derivatives
    dxdt = [dXt,dXv,dS,dP,dOliq,dOgas,dT,dTc,dVl,dSf_cum,dTime]
    
    # [dxdt,mmax,Pmax,bP,m,Kd,Qin]
    return dxdt

# test function
print(ethanol(x0,0.0,*ux0))

# Simulate the bioreactor operation until the selected time tf
x = odeint(ethanol,x0,tspan,args=ux0)

#plots Results
#Total and Viable Cellular Biomass
plt.figure()
plt.plot(tspan,x[:,0])
plt.plot(tspan,x[:,1])
plt.title('Total & Viable Cellular Biomass')
plt.ylabel('Biomass concentration [g/L]') 
plt.xlabel('t [h]')
plt.legend(['Xt','Xv'])

plt.figure()
plt.title('Substrate (S) & Product (P) concentration')
plt.plot(tspan,x[:,2], label='S')
plt.plot(tspan,x[:,3], label='P')
plt.legend(); plt.grid()
plt.ylabel('Conc [g/L]')
plt.xlabel('Time [h]')
plt.minorticks_on()
plt.ylim(0)
plt.xlim(t[0],t[-1])
plt.tight_layout()

plt.figure()
plt.title('Bioreactor & Cooling jacket temperature')
plt.plot(tspan,x[:,6], label='T')
plt.plot(tspan,x[:,7], label='Tc')
plt.legend()
plt.ylabel('Temperature [oC]')
plt.xlabel('Time [h]')
plt.grid()
plt.minorticks_on()
plt.ylim(0)
plt.xlim(t[0],t[-1])
plt.tight_layout()

fig4, ax = plt.subplots()
ax.title.set_text('Dissolved & Gaseous Oxygen concentration')
lns1 = ax.plot(t,x[:,4], label='[Oliq]', color='c')
ax.set_xlabel('Time [h]')
ax.set_ylabel('Oliq [g/L]', color='c')
ax.minorticks_on()
ax2 = ax.twinx()
lns2 = ax2.plot(t,x[:,5], label='[Ogas]', color='y')
ax2.set_ylabel('Ogas [g/L]', color='y')
ax2.minorticks_on()
lns = lns1 + lns2
labs = [l.get_label() for l in lns]
ax.legend(lns, labs, loc='best')
ax.grid()
fig4.tight_layout()

plt.figure()
plt.title('Feeding Policy')
plt.plot(tspan, Qin, label='Qin')
plt.legend()
plt.ylabel('Qin [L/h]')
plt.xlabel('Time [h]')
plt.grid()
plt.minorticks_on()
plt.ylim(0)
plt.xlim(tspan[0],tspan[-1])
plt.tight_layout()

plt.show()