Act like a helpful tutor and exlain me :
Give step-by-step solution with explanation and final answer:solve all parts completely , Steam enters a Turbine operating at steady state at 700 degree F and

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Act like a helpful tutor and exlain me :
Give step-by-step solution with explanation and final answer:solve all parts completely , Steam enters a Turbine operating at steady state at 700 degree F and 450 ( lbf )/( in^2 ) and leaves as a saturated vapor at 0.8 ( lbf )/( in^2 )  The turbine develops 12000 hp , and heat transfer from the turbine to the surrounding occurs at a rate of 2×10^6 ( Btu )/h Neglec kinetic and potential energy changes from inlet to exit
Determine the exit temperature in degree F and the volumetric flow rate of the steam in ( ft^3 )/s

VIPSolutions · Accepted Answer

# Turbine Problem: Step-Step Solution

**Given Data- **In Pressure ($_1\**: 450bf/in²psia)
 **Inlet (\(T1$): 700°F- **Exit ($P2$): 8 lbf² (psia- **Exit State**: Saturated vapor- **Turbine Power ($\dot{W}_{\text{turbine}}$)**: 12,000 hp
- **Heat Loss ($\dot{Q}$)**: 2 × 10⁶ Btu/hr (from steam to surroundings)
- **Neglect**: Kinetic and potential energy changes

## **Unknowns to Find**

1. **Exit Temperature ($T_2$)** in °F
2. **Volumetric Flow Rate at Exit ($\dot{V}_2$)** in ft³/s

---

## **Step 1: Find Exit Temperature ($T_2$)**

The steam **leaves as a saturated vapor at .8 psia**.

- For saturated vapor, $T_2 = T_{sat}$ at $P_2 = .8$ psia.

Consulting the **steam tables**:

- At $P = .8$ psia:
  - **Saturated Temperature ($T_{sat}$)**: ≈ **104.08°F**
  - (Check: At .8 psia, $T_{sat}$ is slightly above 100°F; actual value from tables is ~104°F.)

**Final Answer for (1):**
> **Exit Temperature, $T_2 = 104.08^\circ$F**

---

## **Step 2: Find Volumetric Flow Rate at Exit ($\dot{V}_2$)**

### **A. Find Enthalpy at Inlet and Exit**

#### **Inlet ($P_1 = 450$ psia, $T_1 = 700^\circ$F):**

From superheated steam tables:

- At **450 psia, 700°F**:
  - **Enthalpy ($h_1$)**: ≈ 1325.6 Btu/lbm
  - **Specific Volume ($v_1$)**: ≈ .5256 ft³/lbm

#### **Exit ($P_2 = .8$ psia, Sat. Vapor):**

From saturated steam tables:

- At **.8 psia, Sat. Vapor ($x=1$)**:
  - **Enthalpy ($h_2$)**: ≈ 1049.7 Btu/lbm
  - **Specific Volume ($v_2$)**: ≈ 235.6 ft³/lbm

### **B. Apply Energy Balance (First Law for Steady-Flow Devices)**

Neglecting kinetic and potential energy changes:

$$
\dot{W}_{\text{turbine}} = \dot{m}(h_1 - h_2) - \dot{Q}
$$

Where:
- $\dot{W}_{\text{turbine}}$: Power output (Btu/hr)
- $\dot{Q}$: Heat loss (Btu/hr)
- $\dot{m}$: Mass flow rate (lbm/hr)
- $h_1, h_2$: Enthalpy at inlet, exit (Btu/lbm)

#### **Convert Power to Btu/hr**

- 1 hp = 2545 Btu/hr

$$
\dot{W}_{\text{turbine}} = 12,000 \text{ hp} \times 2545 \frac{\text{Btu}}{\text{hr·hp}} = 30,540,000 \text{ Btu/hr}
$$

#### **Plug in Values**

$$
30,540,000 = \dot{m}(1325.6 - 1049.7) - 2,000,000
$$

$$
30,540,000 + 2,000,000 = \dot{m}(275.9)
$$

$$
32,540,000 = \dot{m} \times 275.9
$$

$$
\dot{m} = \frac{32,540,000}{275.9} \approx 117,900~\text{lbm/hr}
$$

### **C. Find Volumetric Flow Rate at Exit ($\dot{V}_2$)**

$$
\dot{V}_2 = \dot{m} \cdot v_2
$$

Convert mass flow rate to lbm/s:

$$
\dot{m} = \frac{117,900~\text{lbm/hr}}{360~\text{s/hr}} \approx 32.75~\text{lbm/s}
$$

Now, plug in $v_2 = 235.6~\text{ft}^3/\text{lbm}$:

$$
\dot{V}_2 = 32.75~\text{lbm/s} \times 235.6~\text{ft}^3/\text{lbm}
$$
$$
\dot{V}_2 \approx 7,714~\text{ft}^3/\text{s}
$$

---

## **Final Answers**

### **1. Exit Temperature**
> **$T_2 = 104.08^\circ$F**

### **2. Volumetric Flow Rate at Exit**
> **$\dot{V}_2 \approx 7,714~\text{ft}^3/\text{s}$**

---

### **Summary Table**

| Quantity                   | Value           | Units           |
|----------------------------|-----------------|-----------------|
| Exit Temperature ($T_2$) | 104.08          | °F              |
| Volumetric Flow Rate ($\dot{V}_2$) | 7,714           | ft³/s           |

---

### **Key Steam Table Values Used**

- At 450 psia, 700°F: $h_1 = 1325.6~\text{Btu/lbm}$
- At .8 psia, sat. vapor: $h_2 = 1049.7~\text{Btu/lbm}$, $v_2 = 235.6~\text{ft}^3/\text{lbm}$
- 1 hp = 2545 Btu/hr

---

#### **If you need to see the process for a different exit state (e.g., not saturated vapor), or for more detail on the steam table lookups, just ask!**

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