In an industrial process to make alcohol, bacteria, sugar, and water are fed into a bioreactor. The bacteria make alcohol, and water as well as leftover sugar. We desire to remove all the cells from the process stream so we may purify our alcohol product. The process stream enters a separator where the cellular components are separated from the rest of the stream. The entering process stream contains 30 wt% alcohol, 5 wt% sugar, 10 wt% cells, and the rest water. Two product streams, a cell-rich stream and a cell-free stream, leave the separator. The cell-rich stream is 90 wt% cells, 2.5 wt% sugar, 0.5 wt% alcohol, and 7wt% water.
a) Write our species mass conservation equations for alcohol, bacteria, sugar, and water. Write a total mass conservation equation.
b) How many of the mass balance equations are linearly independent ?
c) Determine the composition of the cell-free stream.
a) In order to write the species mass conservation equations, we need to consider the mass balance for each component: alcohol, bacteria, sugar, and water. Let's denote the mass fractions of alcohol, bacteria, sugar, and water in the entering process stream as A_in, B_in, S_in, and W_in, respectively. The mass fractions in the cell-rich stream are denoted as A_rich, B_rich, S_rich, and W_rich, and in the cell-free stream as A_free, B_free, S_free, and W_free, respectively.
The mass conservation equations for each component can be written as follows:
1. Alcohol mass balance:
A_in + A_rich = A_free
2. Bacteria mass balance:
B_in + B_rich = B_free
3. Sugar mass balance:
S_in + S_rich = S_free
4. Water mass balance:
W_in + W_rich = W_free
b) To determine how many of the mass balance equations are linearly independent, we need to check the rank of the coefficient matrix. In this case, we have 4 equations and 4 unknowns (A_free, B_free, S_free, W_free). If the rank of the coefficient matrix is 4, then all 4 equations are linearly independent.
c) To determine the composition of the cell-free stream, we need to apply the mass conservation equations and solve for the unknowns (A_free, B_free, S_free, W_free). We can use the given information to substitute the known values into the equations:
From the given information:
A_in = 0.30
B_in = 0.10
S_in = 0.05
W_in = 1 - (A_in + B_in + S_in) = 0.55
Next, we can substitute these values into the mass conservation equations:
1. Alcohol mass balance:
0.30 + A_rich = A_free
2. Bacteria mass balance:
0.10 + B_rich = B_free
3. Sugar mass balance:
0.05 + S_rich = S_free
4. Water mass balance:
0.55 + W_rich = W_free
We also know the composition of the cell-rich stream:
A_rich = 0.005
B_rich = 0.90
S_rich = 0.025
W_rich = 0.07
Substituting these values into the equations, we can solve for the unknowns:
1. Alcohol mass balance:
0.30 + 0.005 = A_free
A_free = 0.305
2. Bacteria mass balance:
0.10 + 0.90 = B_free
B_free = 1.00
3. Sugar mass balance:
0.05 + 0.025 = S_free
S_free = 0.075
4. Water mass balance:
0.55 + 0.07 = W_free
W_free = 0.62
Therefore, the composition of the cell-free stream is:
Alcohol (A_free) = 0.305
Bacteria (B_free) = 1.00
Sugar (S_free) = 0.075
Water (W_free) = 0.62