by Jeffrey Kantor (jeff at nd.edu). The latest version of this notebook is available at https://github.com/jckantor/CBE30338.

This example provides an introduction to the use of python for the simulation of a simple process modeled by a pair of ordinary differential equations. See SEMD textbook example 2.1 for more details on the process.

Unlike Matlab, in Python it is always necessary to import the functions and libraries that you intend to use. In this case we import the complete `pylab`

library, and the function `odeint`

for integrating systems of differential equations from the `scipy`

library. The command `%matplotlib inline`

causes graphic commands to produce results directly within the notebook output cells.

In [13]:

```
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint
```

In [14]:

```
rho = 900.0 # density, kg/m**3
w1 = 500.0 # stream 1, kg/min
w2 = 200.0 # stream 2, kg/min
w = 650.0 # set outflow equal to sum of inflows
x1 = 0.4 # composition stream 1, mass fraction
x2 = 0.75 # composition stream 2, mass fraction
```

In [15]:

```
def func(y,t):
V,x = y
dVdt = (w1 + w2 - w)/rho
dxdt = (w1*(x1-x)+w2*(x2-x))/(rho*V)
return [dVdt, dxdt]
```

In [16]:

```
V = 2.0 # initial volume, cubic meters
x = 0.0 # initial composition, mass fraction
t = np.linspace(0,10.0)
y = odeint(func,[V,x],t)
```

In [17]:

```
plt.plot(t,y)
plt.xlabel('Time [min]')
plt.ylabel('Volume, Composition')
plt.legend(['Volume','Composition'])
plt.ylim(0,3)
plt.grid()
#plt.savefig('BlendingTankStartUp.png')
```

The blending tank is a system with two state variables (volume and composition). Suppose a mechanism is put in place to force the inflow to equal the outflow, that is

$$w = w_1 + w_2$$The mechanism could involve the installation of an overflow weir, level controller, or some other device to force a balance between the outflow and total inflows. In this case,

$$\frac{dV}{dt} = 0$$which means volume is at *steady state*.

In that case there is just one remaining differential equation

$$\frac{dx}{dt} = \frac{1}{\rho V}( w_1(x_1 - x) + w_1(x_2 - x)) = 0$$Solving for the steady value of $x$,

$$\bar{x} = \frac{w_1x_1 + w_2x_2}{w_1 + w_2}$$In [19]:

```
w1 = 500.0 # stream 1, kg/min
w2 = 200.0 # stream 2, kg/min
x1 = 0.4 # composition stream 1, mass fraction
x2 = 0.75 # composition stream 2, mass fraction
x = (w1*x1 + w2*x2)/(w1 + w2)
print('Steady State Composition =', x)
```

Steady State Composition = 0.5

In [ ]:

```
```