Act like a helpful tutor and explain step-by-step solution with explanation and final answer:Practice questions:
A mixture stream containing 50 kmol/h of sorbitol, 750 kmol/h of water and 6000 kmol/h

Question

Act like a helpful tutor and explain step-by-step solution with explanation and final answer:Practice questions:
A mixture stream containing 50 kmol/h of sorbitol, 750 kmol/h of water and 6000 kmol/h of
hydrogen leaves a reactor at 145 °C and 70 barg. It is then cooled and sent to a flash to be
separated into a gas stream of hydrogen for recycle back to the reactor and an aqueous
sorbitol solution stream. The flash is to be operated at a temperature in the range of 50-100
°C and the pressure of 50 barg. You are asked to

1. calculate the composition of the gas and aqueous sorbitol streams in the

temperature range interested by:
a) taking the system as an ideal system
b) taking the system as a real system (8)

VIPSolutions · Accepted Answer

# Step-by-Step Solution: Flash Separation Problem

Let's break down the problem step by step.

## **Given Data**

- **Feed to Flash:**
  - Sorbitol: 50 kmol/h
  - Water: 750 kmol/h
  - Hydrogen: 600 kmol/h
  - Temperature: 145°C (before cooling)
  - Pressure: 70 barg

- **Flash Conditions:**
  - Temperature range: 50–100°C
  - Pressure: 50 barg

- **Separation:**
  - Gas stream (mostly hydrogen, goes back to reactor)
  - Liquid stream (aqueous sorbitol solution)

---

## **Step 1: Flash Calculation Overview**

In a flash drum, the mixture is separated based on vapor-liquid equilibrium (VLE).

- **Gas phase:** Mostly hydrogen, some water vapor
- **Liquid phase:** Water and sorbitol, negligible hydrogen

We will **calculate composition** for both phases.

---

## **Step 2: Material Balances**

Let:
- $ F = $ total feed = 50 + 750 + 600 = **680 kmol/h**

Let $ V $ = vapor (gas) out, $ L $ = liquid (aqueous) out.

Component balances:
- **Sorbitol:** All in liquid (very nonvolatile, stays in solution).
- **Hydrogen:** Almost all in vapor (very volatile).
- **Water:** Distributed between vapor and liquid according to VLE.

---

## **Step 3: Assumptions**

### **a) Ideal System**

- Ideal gas for vapor phase.
- Raoult's Law for liquid phase (ideal solution).

### **b) Real System**

- Non-ideal behavior considered: use fugacity coefficients (for vapor) and activity coefficients (for liquid).

---

## **Step 4: Flash Calculation Equations**

For component $ i $:

$$
y_i = K_i x_i
$$

Where:
- $ y_i $ = mole fraction in vapor
- $ x_i $ = mole fraction in liquid
- $ K_i = \frac{P_i^{\text{sat}}}{P} $ (ideal, Raoult’s Law)
- $ P_i^{\text{sat}} $ = vapor pressure of component $ i $ at T
- $ P $ = total pressure (50 barg = 51 bar absolute)

#### **Material Balance for Each Component**

$$
F z_i = V y_i + L x_i
$$

Where:
- $ z_i $ = feed mole fraction

---

## **Step 5: Calculate $ K $-values (Ideal Case)**

### **Sorbitol**
- Assumed nonvolatile ($ K_{\text{sorbitol}} \approx  $)

### **Hydrogen**
- Highly volatile ($ K_{\text{H}_2} \gg 1 $)
- Assume almost all hydrogen in vapor.

### **Water**
- Calculate $ K_{\text{water}} $ using vapor pressure at flash temperature.

#### **Example at 75°C:**
- $ P_{\text{water}}^{\text{sat}} $ at 75°C ≈ .388 bar
- $ P = 51 $ bar
- $ K_{\text{water}} = \frac{.388}{51} \approx .0076 $

---

## **Step 6: Solve for Phase Splits**

Given the large excess of hydrogen, most of the vapor phase will be hydrogen.

- **All sorbitol in liquid**
- **All hydrogen in vapor**

Let’s assume $ y_{\text{sorbitol}} =  $, $ x_{\text{H}_2} =  $.

**Water Partition:**
$$
y_{\text{water}} = K_{\text{water}} x_{\text{water}}
$$

Set up total balances:

$$
F = V + L
$$
$$
F z_{\text{water}} = V y_{\text{water}} + L x_{\text{water}}
$$
$$
F z_{\text{H}_2} = V y_{\text{H}_2}
$$
$$
F z_{\text{sorbitol}} = L x_{\text{sorbitol}}
$$

---

### **Let’s Estimate the Compositions (Ideal Case, at 75°C)**

- $ z_{\text{H}_2} = 600/680 = .882 $
- $ z_{\text{water}} = 750/680 = .110 $
- $ z_{\text{sorbitol}} = 50/680 = .00735 $

Since almost all hydrogen leaves in vapor,
$$
V \approx 600\ \text{kmol/h}
$$

Water split: let $ \alpha $ fraction of water in vapor:

$$
\alpha = \frac{y_{\text{water}} V}{F z_{\text{water}}}
$$

But since $ K_{\text{water}} $ is so small, most water stays in liquid.

#### **Liquid Phase:**
- $ L \approx 800\ \text{kmol/h} $ (all water + all sorbitol)
- Composition: mostly water, 50/800 = 6.25% sorbitol by moles

#### **Vapor Phase:**
- $ V \approx 600\ \text{kmol/h} $ (almost all hydrogen)
- Tiny amount of water vapor, negligible other components

---

## **Step 7: Real System**

- Use activity coefficients ($ \gamma $) for water and sorbitol in solution (e.g., NRTL or UNIQUAC model)
- Use fugacity coefficients ($ \phi $) for hydrogen (Peng-Robinson or SRK EOS)

**Effect:**  
- $ K_{\text{water}} $ will be slightly lower or higher, depending on non-ideality.
- Still, the qualitative result is the same: almost all hydrogen in vapor, almost all water + all sorbitol in liquid.

---

## **Step 8: Final Answers (Summary Table)**

| Stream      | Main Components         | Composition (mole %)      | Notes (Ideal)                       |
|-------------|------------------------|---------------------------|--------------------------------------|
| Gas (V)     | H₂ (major), H₂O (trace)| H₂: ~99.9%, H₂O: ~.1%    | All H₂, negligible H₂O, no sorbitol  |
| Liquid (L)  | H₂O, Sorbitol          | H₂O: ~93.8%, Sorbitol: ~6.2% | All sorbitol, almost all H₂O         |

- **For real system:** Use actual $ \gamma $, $ \phi $ to adjust water distribution slightly.

---

## **Key Points**

- **Sorbitol**: Non-volatile, all in aqueous phase
- **Hydrogen**: Highly volatile, almost all in vapor
- **Water**: Partitioned, but mostly in liquid due to low vapor pressure at high pressure

---

## **Conclusion**

### **a) Ideal System**
- Gas phase: almost all

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