# # Blending Tank Simulation
# ## Summary
#
# 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.
# ## Basic Simulation of the Blending Tank
#
# \begin{align*}
# \frac{dV}{dt} & = \frac{1}{\rho}(w_1 + w_2 - w)\\
# \frac{dx}{dt} & = \frac{1}{\rho V}(w_1 (x_1 - x) + w_2 (x_2 - x))
# \end{align*}
# ### Step 1. Initialize Python Workspace
#
# 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]:
get_ipython().run_line_magic('matplotlib', 'inline')
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint
# ### Step 2. Establish Parameter Values
# 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
# ### Step 3. Write a function to compute the RHS's of the Differential Equations
# 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]
# ### Step 4. Set the Initial Conditions, Time Grid, and Integrate
# 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)
# ### Step 5. Visualize the Solution
# 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')
# ## Steady State Analysis
#
# 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)
# In[ ]:
#
# < [Gravity Drained Tank](http://nbviewer.jupyter.org/github/jckantor/CBE30338/blob/master/notebooks/02.02-Gravity-Drained-Tank.ipynb) | [Contents](toc.ipynb) | [Continuous Product Blending](http://nbviewer.jupyter.org/github/jckantor/CBE30338/blob/master/notebooks/02.04-Continuous-Product-Blending.ipynb) >