Superheated steam at 1500 kPa, has a specific volume of 2.75 m3 /kmol and compressibility factor (𝒁) of 0.95. The temperature of steam is______ oC (round off to the nearest integer).

The correct option is 249.

Step-by-step Explanation:

To find the temperature of the superheated steam, we can use the compressibility factor equation. The compressibility factor, represented by $Z$, is a dimensionless quantity that describes the deviation of a real gas from ideal gas behavior. It is defined as:

$$Z=\frac{P\overline{v}}{\overline{R}T}$$

Where:
• $P$ is the pressure of the steam.
• $\overline{v}$ is the molar specific volume.
• $\overline{R}$ is the universal gas constant.
• $T$ is the absolute temperature in Kelvin (K).

1. Identify the given values from the problem statement:
• Pressure, $P=1500\text{ kPa}$
• Molar specific volume, $\overline{v}=2.75m^3\text{/kmol}$
• Compressibility factor, $Z=0.95$
• Universal gas constant, $\overline{R}\approx 8.314\text{ kJ/(kmol}·\text{K)}$

2. Rearrange the compressibility equation to solve for absolute temperature ($T$):

$$T=\frac{P\overline{v}}{Z\overline{R}}$$

3. Substitute the values into the equation:

$$T=\frac{1500\times 2.75}{0.95\times 8.314}$$

4. Perform the calculations:
• Numerator: $1500\times 2.75=4125$
• Denominator: $0.95\times 8.314\approx 7.8983$

Now, divide the numerator by the denominator to find the temperature in Kelvin:

$$T\approx \frac{4125}{7.8983}\approx 522.26\text{ K}$$

5. Convert the temperature from Kelvin to Celsius:
The relationship between Kelvin and Celsius is given by:

$$T\left({\text{°}}^{\text{C}}\right)=T\left(\text{K}\right)-273.15$$

Substituting the calculated value of $T$:

$$T\left({\text{°}}^{\text{C}}\right)\approx 522.26-273.15=249.11{\text{°}}^{\text{C}}$$

Rounding to the nearest integer gives 249 °C.