Let's break down and solve the problem step by step. --- ## **Problem Statement** > There are two first-order, irreversible, parallel reactions of reactants A and B given by: > > ``` > A ──> D > \ > └─> C > ``` > > Follows the kinetics: > - \( r_D = 50C_A \) mol/(m³·hr) > - \( r_C = 10C_A \) mol/(m³·hr) > > If **D** is the desired product and **C** is the undesired by-product, recommend: > 1. Feeding pattern for a tubular reactor. > 2. The order in which two stirred tank reactors of volume ratio 2 should be arranged and fed to maximize the productivity of D. --- ## **Step 1: Understand the Reactions and Kinetics** Both reactions are first-order and occur in parallel: - \( A \to D \) with rate constant \( k_D = 50 \) hr\(^{-1}\) - \( A \to C \) with rate constant \( k_C = 10 \) hr\(^{-1}\) ## **Step 2: Tubular Reactor (Plug Flow Reactor, PFR)** ### **Selectivity & Productivity** - **Selectivity** is defined as the ratio of the rate of formation of D to C: \[ S = \frac{r_D}{r_C} = \frac{k_D}{k_C} = \frac{50}{10} = 5 \] (This is constant since both are first-order and parallel.) #### **Feeding Pattern** - For parallel reactions where the desired product has a higher rate constant, **high concentration of A favors the desired product D**. - For a PFR, the concentration of A decreases along the length of the reactor. - **Thus, using a high initial concentration of A (i.e., feed all A at once) will maximize D formation.** - **No benefit to feeding A gradually or in stages.** **Recommendation:** **Feed all reactant A at the inlet (single-point feed) to maximize productivity of D.** --- ## **Step 3: Two CSTRs (Continuous Stirred Tank Reactors) in Series** Given: - Two reactors of volume ratio 2 (let's call them \( V_1 \) and \( V_2 \), with \( V_1 = 2V_2 \) or vice versa). ### **Which Arrangement Maximizes D?** For parallel reactions, **high concentration of A favors D** (since \( k_D > k_C \)). In a CSTR, the exit concentration is lower than in a PFR for the same conversion. - **To maximize D:** Put the **smaller reactor first**. This gives higher A concentration in the first reactor, maximizing the rate of desired reaction. Let: - \( V_1 = 2V \), \( V_2 = V \) (total volume = 3V) - Two possible arrangements: - **Case 1:** Feed to small reactor first (\( V \)), then to big reactor (\( 2V \)) - **Case 2:** Feed to big reactor first (\( 2V \)), then to small reactor (\( V \)) #### **General Rule** For parallel reactions, **put the smaller reactor first**. --- ## **Step 4: Final Answers** ### **1. Feeding Pattern for Tubular Reactor** - **Feed all reactant at once (single-point feed at the inlet).** - **Do not use distributed or staged feed.** ### **2. Order of Two CSTRs** - **Arrange the smaller reactor first, followed by the larger one, to maximize the productivity of D.** - **So, feed should enter the reactor of volume V first, then go to the reactor of volume 2V.** --- ## **Summary Table** | Reactor Type | Feeding/Arrangement Recommendation | |---------------------|-------------------------------------------------------------| | Tubular (PFR) | Single-point feed at inlet | | Two CSTRs (V:2V) | Smaller reactor (V) first, then larger reactor (2V) | --- ### **Reasoning** - For parallel first-order reactions with desired product having higher k, keep A's concentration high as long as possible. - Smaller CSTR first keeps A higher in early stage, maximizing D. --- **If you need a detailed calculation for the CSTRs, let me know!**