10 m3 of gas at initial pressure of 100 kPa (gauge) and at a temperature of 400 C is

Question

compressed in a cylinder to 6 m3, the pressure of the gas being raised to 155kPa (gauge).
Calculate the rise in temperature of the gas after compression.

Answer 1

To solve the problem, we can use the ideal gas law and assuming that the process is an ideal process (we may assume it behaves like an ideal gas). The parameters are:

• Initial Volume, \( V_1 = 10 , \text{m}^3 \)
• Initial Gauge Pressure, \( P_1 = 100 , \text{kPa} + 101.3 , \text{kPa} = 201.3 , \text{kPa} \) (absolute pressure)
• Initial Temperature, \( T_1 = 400 , °C = 400 + 273.15 = 673.15 , K \)
• Final Volume, \( V_2 = 6 , \text{m}^3 \)
• Final Gauge Pressure, \( P_2 = 155 , \text{kPa} + 101.3 , \text{kPa} = 256.3 , \text{kPa} \) (absolute pressure)

Using the ideal gas law relation \( PV = nRT \), we know that for both states:

\[ n = \frac{P_1 V_1}{RT_1} = \frac{P_2 V_2}{RT_2} \]

Since \( n \) and \( R \) (gas constant) are the same for both states, we can disregard \( R \) and rearrange this relationship:

\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]

Rearranging for \( T_2 \):

\[ T_2 = T_1 \cdot \frac{P_2 V_2}{P_1 V_1} \]

Now substitute in the values:

\[ T_2 = 673.15 , K \cdot \frac{256.3 , \text{kPa} \cdot 6 , \text{m}^3}{201.3 , \text{kPa} \cdot 10 , \text{m}^3} \]

Calculate \( T_2 \):

1. Calculate the right-hand side:

\[ \frac{256.3 \times 6}{201.3 \times 10} = \frac{1537.8}{2013} \approx 0.764 \]

2. Now multiply by \( T_1 \):

\[ T_2 \approx 673.15 , K \cdot 0.764 \approx 513.35 , K \]

Finally, convert \( T_2 \) back to Celsius:

\[ T_2 = 513.35 - 273.15 \approx 240.20 , °C \]

To find the rise in temperature:

\[ \text{Rise in Temperature} = T_2 - T_1 = 240.20 , °C - 400 , °C = -159.80 , °C \]

However, this indicates a decrease in temperature rather than an increase, and since this conflicts with expectations of compression typically resulting in an increase in temperature, it's possible that additional properties of the gas (real gas behavior) or heat transfer considerations may have influenced the system.

Thus, before drawing conclusions about the temperature change during compression, it's essential to confirm operating conditions and gas properties regarding ideality and heat transfer in the specific scenario. If you wish to compute an alternative approach including thermodynamic processes, please clarify the scenario, and we can assess accordingly.