# Solution ## Problem Restatement **Sketch and discuss the operation of a steady-state reactor system with a recycle stream. Identify the key operating variable and its limits at which the system can achieve mixing behavior of an ideal plug flow reactor (PFR) or an ideal mixed flow reactor (CSTR).** --- ## 1. **Sketch of Reactor System with Recycle Stream** ```plaintext +-----------------------------+ | | | Feed (F) | | | | | v | | +-------+ Product (P) | | | |------------------>| | | | | | | | | | |REACTOR| | | | | | | +-------+ | | | | | v | | Recycle (R) | | | | +----|------------------------+ | +------------------------+ ``` *Alt text: Diagram showing a reactor with a recycle loop from the outlet back to the inlet, combining with the fresh feed.* --- ## 2. **Operation Discussion** - **Feed (F₀):** Fresh reactants enter the system. - **Recycle (R):** A portion of the reactor outlet is sent back and mixed with the fresh feed. - **Combined Feed (F):** The total input to the reactor is \( F = F_ + R \). - **Reactor:** Can be of any type (PFR or CSTR); the recycle stream influences the mixing behavior. --- ## 3. **Key Operating Variable** - **Recycle Ratio (R/F₀):** The ratio of recycled stream to fresh feed. --- ## 4. **Mixing Behavior and Limits** ### **A. Plug Flow Reactor (PFR)** - **No recycle (R = ):** - All fluid passes through the reactor once. - System behaves as an ideal PFR (no back-mixing). - **Mathematically:** \( R/F_ = \) ### **B. Mixed Flow Reactor (CSTR)** - **Infinite recycle (R → ∞):** - Recycle dominates, causing complete mixing. - System approaches CSTR behavior. - **Mathematically:** \( R/F_ \to \infty \) - **In practice:** Very high recycle ratio makes the reactor content nearly uniform, approximating a CSTR. --- ## 5. **Summary Table** | Recycle Ratio (R/F₀) | Reactor Behavior | |----------------------|--------------------| | | Ideal PFR | | ∞ (very large) | Ideal CSTR | | Intermediate | Intermediate (between PFR and CSTR) | --- ## 6. **Final Answer** - **Key Operating Variable:** The recycle ratio (\( R/F_ \)). - **Limits:** - At \( R/F_ = \): Reactor acts as an **ideal plug flow reactor (PFR)**. - At \( R/F_ \to \infty \): Reactor acts as an **ideal mixed flow reactor (CSTR)**. --- ### **Conclusion** By adjusting the recycle ratio, the system can be tuned to behave anywhere between a PFR and a CSTR. This is a powerful method for process control and optimization in chemical reactor design.