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.

Question

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.

Answer 1

a) The species mass conservation equations for alcohol, bacteria, sugar, and water can be written as follows:

1. Alcohol Mass Conservation Equation:
Input Alcohol = Output Alcohol
0.3 * Input Process Stream = 0.005 * Output Cell-Rich Stream + ??? * Output Cell-Free Stream
(Note: ??? represents the unknown fraction of alcohol in the cell-free stream)

2. Bacteria Mass Conservation Equation:
Input Bacteria = Output Bacteria
0.1 * Input Process Stream = 0.9 * Output Cell-Rich Stream

3. Sugar Mass Conservation Equation:
Input Sugar = Output Sugar
0.05 * Input Process Stream = 0.025 * Output Cell-Rich Stream + ??? * Output Cell-Free Stream
(Note: ??? represents the unknown fraction of sugar in the cell-free stream)

4. Water Mass Conservation Equation:
Input Water = Output Water
(1 - (0.3 + 0.1 + 0.05)) * Input Process Stream = 0.07 * Output Cell-Rich Stream + ??? * Output Cell-Free Stream
(Note: (1 - (0.3 + 0.1 + 0.05)) represents the fraction of water in the process stream)

b) The number of linearly independent mass balance equations can be determined by examining the number of unknowns and the degree of freedom of the system. In this case, we have 4 unknown fractions (alcohol in the cell-free stream, sugar in the cell-free stream, water in the cell-free stream, and the composition of the cell-free stream itself) and 3 independent mass balance equations (alcohol, sugar, and water). Therefore, the system is over-determined and not all equations are linearly independent.

c) To determine the composition of the cell-free stream, we need additional information or assumptions. Without knowing the specific composition or properties of the cell-free stream, we cannot determine its exact composition solely based on the given information.