Question
3

(

2
5
points
)
Steam enters a turbine operating at a steady state at
2
.
E
MPa,
3
E
0

°
C
with a velocity of
1
0
0

m
s
.
The saturated vapor exits at
0
.
1
E MPa and

Question

Question 
3
 
(
 
2
5
 ﻿points
)
Steam enters a turbine operating at a steady state at 
2
.
E
 ﻿MPa, 
3
E
0
 
°
C
 ﻿with a velocity of 
1
0
0
 
m
s
.
 ﻿The saturated vapor exits at 
0
.
1
 ﻿E MPa and a velocity of 
5
E
m
s
.
 ﻿The elevation of the inlet is 
3
 ﻿m higher than at the exit. The mass flow rate of steam is 
1
E
k
g
s
,
 ﻿and the power developed is 
7
 ﻿MW 
.
 ﻿Let 
g
=
9
.
8
1
m
s
2
.
 ﻿Determine 
(
a
)
 ﻿the area at the inlet, in 
m
2
,
 ﻿and 
(
b
)
 ﻿the rate of heat transfer between the turbine and its surroundings, in kW 
.
 ﻿Use the property tables should you need any property foR your calculations. Red E 
=
9
Steam
m
˙
=
1
E
k
T
1
=
3
E
0
4
P
1
=
2
.
E
s
V
1
=
1
E
0
 ﻿:
ENGINEERING MODEL : 
(
1
)
 ﻿The control volume shown on the sketch is at steady state.
(
2
)
 
g
=
9
.
8
1
m
s
2

VIPSolutions · Accepted Answer

# Steam Turbine Problem Solution

Given Data:

- **Inlet (1):**
    - Pressure, $ P_1 = 2. $ MPa
    - Temperature, $ T_1 = 300^\circ C $
    - Velocity, $ V_1 = 100\,\text{m/s} $
    - Elevation, $ z_1 $ (3 m above exit)
- **Exit (2):**
    - Pressure, $ P_2 = .1 $ MPa
    - State: Saturated vapor
    - Velocity, $ V_2 = 50\,\text{m/s} $
    - Elevation, $ z_2 =  $ (reference)
- **Mass flow rate:** $ \dot{m} = 10\,\text{kg/s} $
- **Power developed:** $ W_{\text{turb}} = 700\,\text{kW} = 7\,\text{MW} $
- **Gravity:** $ g = 9.81\,\text{m/s}^2 $
- **Find:**
    1. Area at the **inlet** ($ A_1 $, in m²)
    2. Rate of **heat transfer** ($ \dot{Q} $, in kW)
- **Assume:** Steady state.

---

## 1. Find the Inlet Area ($ A_1 $)

### a) Determine inlet specific volume ($ v_1 $)

From steam tables for $ P_1 = 2.\,\text{MPa},\ T_1 = 300^\circ C $:

- Specific volume, $ v_1 \approx .13267\,\text{m}^3/\text{kg} $  
- Enthalpy, $ h_1 \approx 296.5\,\text{kJ/kg} $

### b) Use continuity equation

$$
\dot{m} = \frac{A_1 V_1}{v_1} \implies A_1 = \frac{\dot{m} v_1}{V_1}
$$

Plug in values:

- $ \dot{m} = 10\,\text{kg/s} $
- $ v_1 = .13267\,\text{m}^3/\text{kg} $
- $ V_1 = 100\,\text{m/s} $

$$
A_1 = \frac{10 \times .13267}{100} = \frac{1.3267}{100} = .0133\,\text{m}^2
$$

**Answer:**
> ### (a) Area at the inlet  
> $$
> \boxed{A_1 = .0133\,\text{m}^2}
> $$

---

## 2. Find the Rate of Heat Transfer ($ \dot{Q} $)

### a) Apply steady-flow energy equation

$$
\dot{Q} - \dot{W}_{\text{turb}} = \dot{m} \left[ (h_2 - h_1) + \frac{V_2^2 - V_1^2}{2} + g(z_2-z_1) \right]
$$
$$
\dot{Q} = \dot{W}_{\text{turb}} + \dot{m} \left[ (h_2 - h_1) + \frac{V_2^2 - V_1^2}{2} + g(z_2-z_1) \right]
$$

### b) Obtain exit enthalpy ($ h_2 $)

At $ P_2 = .1\,\text{MPa} $, saturated vapor:

- $ h_2 = h_g @ .1\,\text{MPa} \approx 2675.5\,\text{kJ/kg} $

### c) Compute all terms

- $ h_1 = 296.5\,\text{kJ/kg} $
- $ h_2 = 2675.5\,\text{kJ/kg} $
- $ V_1 = 100\,\text{m/s} $
- $ V_2 = 50\,\text{m/s} $
- $ z_1 = 3\,\text{m} $
- $ z_2 =  $

#### Kinetic energy term:
$$
\frac{V_2^2 - V_1^2}{2} = \frac{50^2 - 100^2}{2} = \frac{250 - 10000}{2} = \frac{-750}{2} = -375\,\text{m}^2/\text{s}^2
$$
Convert to kJ/kg:
$$
1\,\text{m}^2/\text{s}^2 = .001\,\text{kJ/kg} \implies -375 \times .001 = -3.75\,\text{kJ/kg}
$$

#### Potential energy term:
$$
g(z_2 - z_1) = 9.81 \times ( - 3) = -29.43\,\text{m}^2/\text{s}^2 = -.02943\,\text{kJ/kg}
$$

#### Enthalpy difference:
$$
h_2 - h_1 = 2675.5 - 296.5 = -285\,\text{kJ/kg}
$$

### d) Plug values into the energy balance

$$
\dot{Q} = 700 + 10 \left[ -285 - 3.75 - .02943 \right]
$$
$$
\dot{Q} = 700 + 10 \times (-288.77943)
$$
$$
\dot{Q} = 700 - 2887.7943
$$
$$
\dot{Q} = 4112.2\,\text{kW}
$$

**Final Answer:**
> ### (b) Rate of heat transfer  
> $$
> \boxed{\dot{Q} = 4112\,\text{kW}}
> $$

---

## Summary Table

| Item       | Value              |
|------------|--------------------|
| Inlet Area ($A_1$)      | $ .0133\,\text{m}^2 $    |
| Heat Transfer ($\dot{Q}$) | $ 4112\,\text{kW} $      |

---

### Notes

- All property values are from the steam tables.
- All velocities, elevations, and mass flow rates are as given.
- Units are consistent throughout.
- If you wish to see the property table references, please specify.

✨ VIPSolutions

Question:

Final Answer:

📢 Instructions:

Study Help, Fast Answers

✨ VIPSolutions

Question:

Final Answer:

📢 Instructions:

Related Answers

Study Help, Fast Answers