# How do I go about calculating final temperature of tank draining at saturation?

I am working on calculating pressure in a tank where the fluid is sitting at its vapor pressure. For example, imagine a 12 in diameter 100 L tank of nitrous oxide at room temperature (745 psi vapor pressure). This tank would be drained at 1 L/s from the liquid side, and some of the nitrous oxide would boil off in order to maintain vapor pressure, but would come at the cost of reducing the temperature and vapor pressure in the fluid.

How do I find what the final fluid temperature at drain would be (assuming no heat being input into the system)?

I was thinking there would be some combination of Antoine's, Clausius-Clapeyron, ideal gas law, and some derivation of an energy balance equation, but I am not too sure.

The experimental situation would literally be a tank (assume perfect insulation) that is filled with nitrous oxide, say a 100 L tank because it is arbitrary. This tank would then be drained through a choked orifice into the atmosphere. The flowrate at the start would be 1 L/s, but as pressure drops, would decrease in flowrate. I want to find out the ending pressure and temperature when the liquid would run out. Experimental data points to the tank draining from liquid before running out, similar to how a propane barbeque runs out of liquid and becomes cold with lower pressures.

• If you can truly assume an isolated system, then the final temperature would be about -88.5 Deg C (nitrous oxide boiling point), varying with atmospheric pressure. Assume it would be high T and high p scenario for water. Final temp would be 100 Deg C. Similarly for LPG with complication of being a mixture. But in all cases it would be asymptotical final state. Apr 15 at 6:24
• Unless you mean the lowest temperature still able to supply 1 L/s. Final T would be then the T with pressure able to provide the given flow with the fully open valve. There is not enough data for that. Apr 15 at 7:00
• I guess my question is what would be the temperature where the fluid runs out? The drain rate would be out of a choked orifice, so upstream pressure would affect the downstream flowrate. We have empirical data that shows ending pressure is around 75% of the starting pressure but want to be able to actually calculate/model this. Apr 15 at 7:32
• How well insulated the real system is? As with total insulation it would not run out, being in mechanical equilibrium with atmosphere. // I advice to properly describe scenario in the question to prevent task misinterpretation by readers. Apr 15 at 7:38
• The situation would literally be a tank (assume perfect insulation) that is filled with nitrous oxide, say a 100L tank because it is arbitrary. This tank would then be drained through a choked orifice into the atmosphere. The flowrate at the start would be 1 L/sec, but as pressure drops, would decrease in flowrate. I want to find out the ending pressure and temperature when the liquid would run out. Experimental data points to the tank draining from liquid before running out, similar to how a propane barbeque runs out of liquid and becomes cold with lower pressures. Apr 15 at 7:45

Let m be the mass of N2O in the tank at any time, x be the mass fraction liquid N2O,V be the tank volume, v be saturated volume per unit mass, u be saturated internal energy per unit mass, and h be saturated enthalpy per unit mass. At any time, the internal energy of the tank contents will be: $$U=m[xu_L(T)+(1-x)u_V(T)]$$If we apply the open system version of the 1st law of thermodynamics to the contents of this adiabatically insulated tank, we obtain: $$dU=h_L(T)dm$$or$$[xu_L(T)+(1-x)u_V(T)]dm-m(u_V-u_L)dx+m\left[x\frac{du_L}{dT}+(1-x)\frac{du_V}{dT}\right]dT=(u_L+Pv_L)dm$$or $$-(u_V-u_L)dx+\left[x\frac{du_L}{dT}+(1-x)\frac{du_V}{dT}\right]dT=-(1-x)(u_V-u_L)d\ln{m}+Pv_Ld\ln{m}$$This equation is subject to the constraint that the total volume is constant: $$m[xv_L+(1-x)v_V]=V$$or$$x=\frac{v_V-\frac{V}{m}}{v_V-v_L}$$These equations provide the basis for determining x(m) and T(m).