Step 1: Assemble (Gather Information)

Based on the problem statement and the diagram provided:

1. Stream 1 (W, D):

   • Contains 0.5 wt% penicillin (D) and the rest is water (W).

2. Stream 4 (BA):

   • Contains 100% butyl acetate (BA), and its mass flow rate is 30% of Stream 1.

3. Stream 3 (D, BA):

   • Contains 3.0 wt% penicillin and the rest is butyl acetate.

4. Stream 6 (W, D, BA):

   • Contains water, penicillin, and butyl acetate, where the mass fraction of penicillin (D) is 1/4000 of the water (W) mass fraction.

5. Penicillin Separation Efficiency in Unit I:

   • 98% of the penicillin entering Unit I is retained in Stream 2.

6. Stream 2 (W, D, BA):

   • Contains 1.7 wt% penicillin and 2.0 wt% water.

Step 2: Analyze

Define Variables:

• Let $\dot{m}_1$ be the mass flow rate of Stream 1.
• Let $\dot{m}_3$ be the mass flow rate of Stream 3.
• Let $\dot{m}_4$ be the mass flow rate of Stream 4.
• Let $\dot{m}_6$ be the mass flow rate of Stream 6.

Key Relationships:

1. Stream 1 (Penicillin and Water):

   • Penicillin mass flow rate in Stream 1: $0.005 \cdot \dot{m}_1$.
   • Water mass flow rate in Stream 1: $0.995 \cdot \dot{m}_1$.

2. Stream 4 (Butyl Acetate):

   • $\dot{m}_4 = 0.3 \cdot \dot{m}_1$.

3. Stream 3 (Penicillin and Butyl Acetate):

   • Penicillin mass flow rate in Stream 3: $0.03 \cdot \dot{m}_3$.

4. Stream 6 (Water, Penicillin, Butyl Acetate):

   • The mass fraction of penicillin in Stream 6 is $\frac{1}{4000}$ of the water mass fraction.
   • Total mass of Stream 6 is $\dot{m}_6$.

5. Mass Balance for Penicillin (D):

   • In Unit I, 98% of penicillin from Stream 1 is retained in Stream 2, and the remaining 2% goes to Stream 6.
   • Overall penicillin mass balance: $\dot{m}_{D,1} = \dot{m}_{D,3} + \dot{m}_{D,6}$.

6. Mass Balance for Butyl Acetate (BA):

   • The total butyl acetate in Stream 6 and Stream 3 must sum up to the butyl acetate from Stream 4.

Step 3: Calculate

Let's start solving for the mass flow rates and compositions. We will perform the necessary mass balances for water, penicillin, and butyl acetate across both Unit I and Unit II.

I'll calculate this now.From the calculations:

1. The mass flow rate of Stream 1 ($\dot{m}_1$) is $3.33 \times \dot{m}_4$, meaning it is about 3.33 times the mass flow rate of Stream 4.
2. The mass flow rate of Stream 3 ($\dot{m}_3$) is related to $\dot{m}_4$ and $\dot{m}_6$ by $\dot{m}_3 = 0.555 \times \dot{m}_4 - 0.00833 \times \dot{m}_6$.

These equations establish the relationships between the flow rates of the streams. Next, we need to find the exact values of the mass flow rates for each stream by using additional conditions, such as the total mass balance for water and butyl acetate across the system.

Would you like to continue solving for the exact flow rates, or do you have questions about the relationships so far?

Related Questions:

1. How do we apply mass balance to determine the mass flow rate of Stream 6?
2. How does the efficiency of Unit I affect the mass flow rates of Streams 2 and 6?
3. What is the role of butyl acetate in the overall extraction process?
4. How can we calculate the composition of each stream using the flow rate relationships?
5. How would changes in the separation efficiency affect the overall system performance?

Tip: Always check units carefully when solving mass balance problems to ensure consistency across equations.