Problem
4

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A supply line that carries steam at
5
MPa and
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5
0
0
^
{
\
circ
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\
mathrm
{
C
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)
is connected to a pistoncylinder device in which steam undergoes a constant

Question

Problem 
4
 
(
2
0
'
)
 ﻿A supply line that carries steam at 
5
 ﻿MPa and 
\
(
 
5
0
0
^
{
\
circ
}
 
\
mathrm
{
C
}
 
\
)
 ﻿is connected to a pistoncylinder device in which steam undergoes a constant pressure expansion process of 
\
(
 
\
mathbf
{
8
 
0
 
0
 ﻿k P a
}
 
\
)
.
 ﻿The piston
-
cylinder device initially contains saturated vapor with a volume of 
\
(
 
0
.
1
 
\
mathrm
{
~m
}
^
{
3
}
 
\
)
.
 ﻿Now the valve is opened and steam is allowed to flow slowly into the cylinder until the volume of the cylinder doubles and the temperature reaches 
\
(
 
2
5
0
^
{
\
circ
}
 
\
mathrm
{
C
}
 
\
)
,
 ﻿at which the valve is closed. Determine the amount of mass that enters and the amount of heat transfer during this process.

VIPSolutions · Accepted Answer

# Solution

Let's break down the problem and solve step by step.

---

## **Given Data**

- **Supply line steam:**  
  Pressure $P_s = 5\,\text{MPa}$  
  Temperature $T_s = 500^\circ\text{C}$

- **Piston-cylinder device:**  
  Constant pressure process, $P = 800\,\text{kPa} = .8\,\text{MPa}$  
  **Initial state:**  
  - Saturated vapor at $P = 800\,\text{kPa}$  
  - Volume $V_1 = .1\,\text{m}^3$
  
  **Final state:**  
  - Volume doubles: $V_2 = 2V_1 = .2\,\text{m}^3$
  - Temperature $T_2 = 250^\circ\text{C}$

---

## **1. Find Initial and Final Properties**

### **Initial State (1):**
- $P_1 = .8\,\text{MPa}$
- Saturated vapor ($x_1 = 1$)
- $V_1 = .1\,\text{m}^3$

From steam tables:

At $P = .8\,\text{MPa}$:

- $T_{\text{sat}} = 170.41^\circ\text{C}$
- $v_g = .2403\,\text{m}^3/\text{kg}$ (specific volume of vapor)
- $h_g = 2778.1\,\text{kJ}/\text{kg}$ (enthalpy of vapor)
- $u_g = 2583.9\,\text{kJ}/\text{kg}$ (internal energy of vapor)

Find initial mass:

$$
m_1 = \frac{V_1}{v_1} = \frac{.1}{.2403} = .416\,\text{kg}
$$

---

### **Final State (2):**

- $P_2 = .8\,\text{MPa}$
- $V_2 = .2\,\text{m}^3$
- $T_2 = 250^\circ\text{C}$

From steam tables, at $P = .8\,\text{MPa}$ and $T_2 = 250^\circ\text{C}$:

- $v_2 = .30624\,\text{m}^3/\text{kg}$
- $h_2 = 2943.\,\text{kJ}/\text{kg}$
- $u_2 = 271.9\,\text{kJ}/\text{kg}$

Final mass:

$$
m_2 = \frac{V_2}{v_2} = \frac{.2}{.30624} = .653\,\text{kg}
$$

---

## **2. Mass of Steam Entered**

$$
\Delta m = m_2 - m_1 = .653 - .416 = .237\,\text{kg}
$$

---

## **3. Energy Balance (First Law, Unsteady Flow)**

For a control mass (piston-cylinder), with mass flow in but not out:

$$
Q - W_b = (m_2 u_2 - m_1 u_1) - m_{\text{in}} h_s
$$

But more generally for a filling process:
$$
Q - W_b = (m_2 u_2 - m_1 u_1) - m_{\text{in}} (h_s)
$$

Where:
- $Q$: heat added to system
- $W_b$: boundary work (expansion at constant pressure)
- $u$: internal energy
- $h_s$: enthalpy of supply steam ($5\,\text{MPa}, 500^\circ\text{C}$)
- $m_{\text{in}}$: mass that entered

**Find $h_s$ from steam tables at $5\,\text{MPa}, 500^\circ\text{C}$:**

$$
h_s = 341.1\,\text{kJ}/\text{kg}
$$

---

### **Boundary Work ($W_b$):**

$$
W_b = P(V_2 - V_1) = 800\,\text{kPa} \times (.2 - .1)\,\text{m}^3
$$
$$
W_b = 800 \times .1 = 80\,\text{kJ}
$$

---

### **Plugging In Values:**

- $m_1 = .416\,\text{kg}$, $u_1 = 2583.9\,\text{kJ}/\text{kg}$
- $m_2 = .653\,\text{kg}$, $u_2 = 271.9\,\text{kJ}/\text{kg}$
- $m_{\text{in}} = .237\,\text{kg}$, $h_s = 341.1\,\text{kJ}/\text{kg}$

$$
Q = (m_2 u_2 - m_1 u_1) + W_b - m_{\text{in}} h_s
$$

Calculate terms:

1. $m_2 u_2 = .653 \times 271.9 = 1771.2\,\text{kJ}$
2. $m_1 u_1 = .416 \times 2583.9 = 1074.\,\text{kJ}$
3. $m_{\text{in}} h_s = .237 \times 341.1 = 808.2\,\text{kJ}$

$$
Q = (1771.2 - 1074.) + 80 - 808.2
$$
$$
Q = (697.2) + 80 - 808.2
$$
$$
Q = 777.2 - 808.2 = -31.\,\text{kJ}
$$

---

## **4. Answers**

### **a) Mass of steam that enters:**
$$
\boxed{.237\,\text{kg}}
$$

### **b) Heat transfer during process:**
$$
\boxed{-31.\,\text{kJ}}
$$
(The negative sign means heat is lost from the system during the process.)

---

## **Summary Table**

| Quantity            | Value         |
|---------------------|--------------|
| Mass entered ($\Delta m$) | $.237\,\text{kg}$ |
| Heat transfer ($Q$)       | $-31.\,\text{kJ}$ |

---

## **Diagram**

![Schematic of piston-cylinder device filling with steam](https://i.imgur.com/3fKp7xW.png)
*Alt text: Schematic of a piston-cylinder device with inlet valve, showing initial and final volumes and relevant states for the steam.*

---

**References:**  
- Thermodynamics steam tables (IAPWS or ASME)  
- First Law for control mass with mass flow-in

If you need step-by-step steam table lookup, let me know!

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