Let's break down the problem step-by-step and provide thorough solution. --- ## **Step 1: Understand the Problem** You need to **compress acetylene** (C₂H₂) from 1 atm (300 K) to the **highest safe pressure** using one of two configurations: - **Option A:** One very efficient compressor (η ≈ 100%) - **Option B:** Two compressors (η = .8 each) with a heat exchanger (chilled water at 280 K) in between ### **Safety Limits** - **Decomposition risk:** At P > 200 kPa (2 atm), intense heat or vibrations can cause explosive decomposition. - **Decomposition temperature:** Spontaneous decomposition above 600 K. - **You must keep both P and T within safe limits after each compression stage.** --- ## **Step 2: Thermodynamics of Compression** For an **ideal gas**, the temperature after adiabatic (isentropic) compression is: \[ \frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{(\gamma-1)/\gamma} \] where \( \gamma = C_p/C_v \). For an ideal diatomic gas, \( C_p \approx 3.5R \), so: \[ C_v = C_p - R = 2.5R \implies \gamma = \frac{3.5}{2.5} = 1.4 \] --- ## **Step 3: Option A Analysis (Single Compressor, η ≈ 100%)** **Isentropic Compression:** - \( P_1 = 1 \) atm, \( T_1 = 300 \) K - Let \( P_2 \) = final pressure (to be determined) - Want \( T_2 < 600 \) K (decomposition risk) - Also, \( P_2 \) should not exceed much above 2 atm (preferably stay below for safety) \[ \frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{.286} \] \[ T_2 = 300 \left(\frac{P_2}{1}\right)^{.286} \] **Set \( T_2 = 600 \) K:** \[ 600 = 300 \left(P_2 \right)^{.286} \implies 2 = P_2^{.286} \implies \ln(2) = .286 \ln(P_2) \implies \ln(P_2) = \frac{\ln(2)}{.286} \approx 2.42 \implies P_2 = e^{2.42} \approx 11.3 \text{ atm} \] ### **BUT - Safety Note!** The notes say: **At P > 2 atm, intense heat or vibrations may cause explosive decomposition**. Even though temperature is below 600K, you should **not exceed 2 atm for safety**. --- ## **Step 4: Option B Analysis (Two Compressors + Intercooling, η = .8)** ### **Compression in Two Stages with Intercooling** #### **First Compressor (COM-2):** - \( P_1 = 1 \) atm, \( T_1 = 300 \) K - Compress to \( P_x \), then cool to 280 K (HX-1), then compress to \( P_2 \) #### **Assume equal pressure ratios:** \( P_x^2 = P_2 \implies P_x = \sqrt{P_2} \) #### **Stage 1:** \[ T_{x1} = 300 \left(\frac{P_x}{1}\right)^{.286} \] After heat exchanger, back to \( T = 280 \) K. #### **Stage 2:** \[ T_2 = 280 \left(\frac{P_2}{P_x}\right)^{.286} = 280 \left(\sqrt{P_2}\right)^{.286} = 280 (P_2^{.143}) \] But **efficiency is .8** for each compressor. For a non-ideal compressor, the actual temperature rise is: \[ T_{2, \text{actual}} = T_1 + \frac{T_{2, \text{ideal}} - T_1}{\eta} \] #### **First Compressor:** \[ T_{x1,\text{ideal}} = 300 \left(P_x\right)^{.286} \] \[ T_{x1,\text{actual}} = 300 + \frac{300 \left(P_x\right)^{.286} - 300}{.8} \] #### **Intercooler:** Cools gas back to 280 K. #### **Second Compressor:** \[ T_{2,\text{ideal}} = 280 \left(\frac{P_2}{P_x}\right)^{.286} \] \[ T_{2,\text{actual}} = 280 + \frac{280 \left(\frac{P_2}{P_x}\right)^{.286} - 280}{.8} \] Set final \( T_{2,\text{actual}} = 600 \) K (safety limit): \[ 600 = 280 + \frac{280(P_2/P_x)^{.286} - 280}{.8} \] \[ 600 - 280 = \frac{280[(P_2/P_x)^{.286} - 1]}{.8} \] \[ 320 \times .8 = 280 [(P_2/P_x)^{.286} - 1] \] \[ 256 = 280 [(P_2/P_x)^{.286} - 1] \] \[ \frac{256}{280} + 1 = (P_2/P_x)^{.286} \] \[ 1.914 = (P_2/P_x)^{.286} \] \[ \ln(1.914) = .286 \ln(P_2/P_x) \implies \ln(P_2/P_x) = \frac{\ln(1.914)}{.286} \approx 2.33 \implies P_2/P_x = e^{2.33} \approx 10.3 \] Recall, \( P_x = \sqrt{P_2} \implies P_2/P_x = \sqrt{P_2} \) So: \[ \sqrt{P_2} = 10.3 \implies P_2 = 106 \] But this far exceeds the **2 atm safety limit**. --- ## **Step 5: Check Pressure Safety** Both options, thermodynamically, could reach much higher pressures before hitting 600 K — **BUT** process safety limits final pressure to ~2 atm due to risk of spontaneous/explosive decomposition. Thus, **the maximum feasible pressure in both options is 2 atm**. --- ## **Step 6: Which Option is Better?** ### **(a) Which configuration would you choose and why?** - **Option B** is generally preferred in industrial practice because: - Intercooling between compression stages reduces the work required and keeps temperatures lower after each stage. - Even though each compressor is less efficient, the overall system is safer and consumes less energy due to heat removal. - **Option A** achieves the same maximum pressure but is less safe, as the gas temperature after compression is higher, increasing decomposition risk if a malfunction occurs. - **Conclusion:** **Choose Option B for safety and lower energy cost.** ### **(b) If you had unlimited budget and only wanted to minimize energy cost?** - **Optimal design:** Use **as many compression stages as possible** with **intercooling to the lowest possible temperature** (as close to ambient as practical). - This approach minimizes the total compression work and energy cost, as each stage operates closer to an isothermal process. - Use **high-efficiency compressors** if possible. --- ## **Final Answers** ### **(a) Which configuration and why?** > **Option B** (two-stage compression with intercooling) is preferred. This configuration allows heat to be removed between compression stages, reducing the exit temperature and overall energy required, while keeping the acetylene within safe temperature and pressure limits. Although each compressor is less efficient, the benefit of intercooling outweighs the lower efficiency. Option A results in higher gas temperatures, increasing the risk of decomposition and explosion. ### **(b) Unlimited budget, objective: minimize energy cost?** > Use **multiple compression stages with intercooling** after each stage, cooling to the lowest possible temperature (e.g., chilled water at 280 K or lower if feasible). Use the most efficient compressors available. This setup approaches isothermal compression, which minimizes energy use. --- ### **Summary Table** | Option | Max Safe Pressure | Final T (at max P) | Energy Cost | Safety | |----------------|------------------|--------------------|-------------|-----------------------| | A (Single) | 2 atm | Lower than B | Higher | Higher risk | | B (Two-stage) | 2 atm | Lower than A | Lower | Safer (preferred) | | Unlimited $ | 2 atm (safety) | Lowest possible | Lowest | Safest | --- If you need detailed calculation for the work/energy or more explanation about the thermodynamics, let me know!