Understanding temperature drop within throttling processes

Question

I am studying a particular throttling process (https://en.m.wikipedia.org/wiki/Isenthalpic_process, https://en.m.wikipedia.org/wiki/Joule%E2%80%93Thomson_effect): cryogenic flow going through an orifice (please see below)

Experimental measurements of both static upstream and downstream pressures $p_1$ and $p_2$ yield a pressure drop $\Delta P_{12} = p_1 - p_2 >0$. Same occurs with upstream and downstream temperatures $T_1$ and $T_2$ (so $\Delta T_{12} = T_1 - T_2 >0$).

I want to explain the temperature drop $\Delta T$ based on just classical thermodynamics, using the concept of enthalpy.

Taking a control volume way upstream and another downstream we have $H_u = U_u + P_uV_u$ and $H_d = U_d + P_dV_d$ respectively. Besides, I made the assumption that the volume upstream and downstream is the same given that the expansion caused by the formation of bubbles is negligible way downstream

Enthalpy is conserved so it follows that

\begin{equation*} H_1 = U_1 + P_1 V = U_2 + P_2 V = H_2 \end{equation*}

\begin{equation*} \Rightarrow \Delta U_{21} = \Delta P_{12} V \neq 0 \end{equation*}

At this stage I made another assumption: the internal energy is a function only of temperature. Hence

\begin{equation*} \Delta U_{21} = \Delta P_{12} V \neq 0 \Rightarrow \Delta T_{12} \neq 0 \end{equation*}

But here's the problem: $\Delta P_{12} V > 0$ so $\Delta U_{21} > 0$. But this cannot be, as it implies $T_{12} >0$ rather than $T_{12} < 0$.

What am I missing?

I am not taking into account the kinetic energy due to the flow, this might need to be incorporated.

Answer 1

To analyze this change, you should work directly with enthalpy. For the exit liquid, $H_L=C_P(T_2-T_1)$. For the exit vapor, $H_V=C_P(T_2-T_1)+\lambda (P_2)$ where $\lambda$ is the heat of vaporization at P2. These are the exit enthalpies per unit mass.

So if X represents the final mass fraction vapor, $$(1-X)H_L+XH_V=0$$,together with $$T_2=T_2(P_2)$$, where $P_2$ is the equilibrium vapor pressure at temperature $T_2$.