# Water Thermodynamics Problem

I was wondering if anyone could help me calculate something based on the following information. My Thermodynamics training in college is eluding me at the moment.

The point of this question is to determine whether or not a vessel needs a relief device.

Consider a vessel filled 100 % with water, with all outlets blocked in. The vessel is sealed. The Temperature of the water is 90 degrees F. The pressure is 84.7 psi absolute (70 psi gauge).

so P1 = 84.7 psia, T1 = 90 deg F (550 deg R)

Next, we raise the temperature to 225 deg. F. Assuming the vessel has enough time to reach steady state, what is the resulting pressure?

so T2 = 225 deg F (685 deg R), P2 = ?

I don’t think the volume of the vessel should matter, considering the vessel is 100 % full and you know the properties of the water.

I tried using the steam tables, but I had trouble because I think these conditions may be off the normal scales used.

I am NOT looking for the ideal gas answer. At the resulting temperature, the water will still be a liquid. I am looking for the resulting pressure due to thermal expansion of the water due to the temperature increase.

Anyone care to take a stab at this?

On the other hand, I don’t think you can extrapolate from the thermal expansion of liquid water, either, given that it’s above the normal boiling point. I think I’d work from a phase diagram, and see what pressure is needed to keep the water a liquid at that temperature, and just completely ignore the thermal expansion.

Unless I’m misunderstanding the situation, what you’re looking for is the Saturated pressure at T2 = 225F. According to the Saturated water tables in Çengel and Boles Thermodynamics, that pressure is 18.998 psia.

Disagree. I think it’s easier to work the other way around.

Consider: The volumetric coefficient of thermal expansion of water is about 207 E-6/degC. The bulk modulus of water is about 2.2×E+9 Pa.

An increase of 135degF is 75degC. At 207 E-6/degC, the temperature increase would give a volumetric increase of 0.0155, ir about 1-1/2%. With a bulk modulus of 2.2×E+9 Pa, that correspondes to a pressure increase of 34 million Pascals, or 340 atm, which is 5000psi end pressure, give or take.

That’s pretty safely in the range of liquid water on a phase diagram, I think.

The analysis disregards expansion of the container due to temperature and stress, which is probably non-negligible, but I think it’s a pretty good bet that a pressure relief device is required.

Clearly, I really did misunderstand! Sorry!

Thanks for the help so far guys. I had a feeling the pressure might be very high like that. I will double check the API’s pressure vessel codes to see if they have any applicable examples.

Where will the vessel be? If the approach of asking anonymous web forum participants doesn’t pan out for you, I want to avoid the area.

Going back to the OP, the vessel NEEDS a relief device per ASME Code Section VIII. However, the next question is what are the credible failure scenarios for which the relief device should be sized. Are you certain that thermal expansion of liquid is worst-case? Usually, this is overshadowed by various other credible failure scenarios.

…Are you sure this isn’t a homework problem??? :dubious:

One caveot: The ASME Code Sect VIII applies to vessels over a minimum threshold volume AND operating at a threshold pressure above what is deemed an “atmospheric tank”.

For all interested, the vessel in question is a sample cooler. It is a shell-and-tube type heat exchanger. The process side is the tube side and the cooling water side is the shell side. The existing relief device protects the shell side of the vessel.

`````` ASME Section VIII does apply to pressure vessels like this, but all pressure vessels do NOT need relief devices. They only need relief devices if there is a credible relief scenario. At one time it was true that a pressure vessel always needed a relief device, but this is not true anymore. I know this because I attended a class on pressure relieving systems taught by the American Petroleum Institute based on the API 520/521 standards. They call unnecessary releif valves on older vessels "ASME courtesy valves."

Only two relief scenarios apply to this vessel, hydraulic expansion and exterior fire. The exterior fire case is an obvious one, but we are investigating whether or not the hydraulic expansion case is credible. Under most circumstances, exterior fire requires the highest relief rate, and thus the relief valve is usually sized for this case. However, this is one of the rare cases that this is not true, and calculations show that thermal expansion would require a higher relief rate.

The trouble is that the calculations are based on the ASME and API codes, which do allow you to calculate the relief rate needed and valve size needed, but not whether the relief scenario is credible. There are 18 cases we consider. I can calculate relief rates for all of them if I want, but usually only one or two apply.``````