Question

A real gas undergoes throttling from 50 bar to 10 bar from an initial temperature of 300 K. If \(\mu_{JT} = -0.05 \text{ K/bar}\) for the initial condition, what is the approximate value of the exit temperature?

• A. \(298 \text{ K} \)
• B. \(302 \text{ K} \)
• C. \(300 \text{ K} \)
• D. \(304 \text{ K} \)

Hint:

When $\mu_{JT}$ is negative, the gas warms up during expansion ($\Delta P$ is always negative in a flow restriction). A quick qualitative check reveals that since $\mu_{JT} < 0$, $T_2$ must be greater than $T_1$. This instantly eliminates options (A) and (C).

Answer 1

The Correct Option is B

Concept: Throttling is a steady-flow expansion process across a restriction (such as a valve or porous plug) characterized by negligible heat transfer, no external shaft work, and negligible changes in kinetic and potential energies. Consequently, it is a constant enthalpy (isenthalpic) process ($h_1 = h_2$). The behavior of temperature relative to pressure changes during this expansion is defined by the Joule-Thomson coefficient ($\mu_{JT}$): \[ \mu_{JT} = \left(\frac{\partial T}{\partial P}\right)_h \approx \frac{\Delta T}{\Delta P} = \frac{T_2 - T_1}{P_2 - P_1} \] Where:
• $\mu_{JT} > 0$ implies cooling occurs during expansion ($\Delta P < 0 \rightarrow \Delta T < 0$).
• $\mu_{JT} < 0$ implies heating occurs during expansion ($\Delta P < 0 \rightarrow \Delta T > 0$).

Step 1: Gather and list the given state properties.
The parameters specified in the problem statement are:
• Initial Pressure, \(P_1 = 50 \text{ bar}\)
• Final Pressure, \(P_2 = 10 \text{ bar}\)
• Initial Temperature, \(T_1 = 300 \text{ K}\)
• Joule-Thomson coefficient, \(\mu_{JT} = -0.05 \text{ K/bar}\)

Step 2: Compute the change in pressure (\(\Delta P\)).
The pressure difference experienced by the real gas during this throttling expansion is: \[ \Delta P = P_2 - P_1 = 10 \text{ bar} - 50 \text{ bar} = -40 \text{ bar} \]

Step 3: Relate the parameters to find the final exit temperature (\(T_2\)).
Using the finite difference approximation for the Joule-Thomson relation: \[ T_2 - T_1 = \mu_{JT} \times \Delta P \] Substituting the known values into the algebraic equation: \[ T_2 - 300 = (-0.05 \text{ K/bar}) \times (-40 \text{ bar}) \] Multiplying the two negative numbers together produces a positive value: \[ T_2 - 300 = 2 \text{ K} \] Isolating $T_2$ by moving 300 to the right-hand side: \[ T_2 = 300 + 2 = 302 \text{ K} \] Because the Joule-Thomson coefficient is negative, the gas undergoes heating during expansion, raising its temperature to approximately 302 K, which corresponds to Option (B).