# What limits the mass transfer from a boiler to a condenser in a sealed vessel?

If we set up an evacuated vessel where we keep one end at a higher temperature than the other, and then introduce a liquid at the warm end, it will evaporate and condense at the cooler end. If we trap the liquid there, all the liquid will move to the cooler end.

What limits the speed of this mass transfer?

To have example values, let's use water for the liquid, 25 °C for the warm and 4 °C for the cold end. The most simple geometry of the vessel I can think of that disallows the liquid to flow back to the warm end would be a pipe bent into an upturned U shape.

The pressure inside the vessel will eventually settle at the vapour pressure of the liquid at the cooler temperature. How would it look during the mass transfer?

• I edited the question entirely after some more research because I had not understood precisely what piece of information I was missing before. – Hanno Fietz Sep 20 '13 at 6:25
• I imagine in most cases the answer will be the rate of evaporation in the hot end or the rate of condensation in the cold end, or a combination of the two. In most cases, I would guess, the actual transport of gas from one end to the other would be much faster than these processes. Then again, it probably depends quite a lot on the geometry of the vessel, properties of the fluid involved, and the temperatures used. My knowledge is somewhat limited, so I'll leave it to someone else to give a proper answer. – Nathaniel Sep 23 '13 at 10:38
• As currently defined, there is a limitless source of thermal energy to keep the warm end at 25 C, and a limitless heat sink to keep the cool end at 4 C. In this case, wouldn't the limiting factor just be the diameter of the vessel? – Mark Sep 24 '13 at 0:15
• @Mark, no, I believe the vapour will propagate very quickly, so that won't limit the whole process. – Hanno Fietz Sep 29 '13 at 17:46
• @Hanno, I believe in any practical test, the limiting factor will be either the rate at which heat is supplied to the hot end, or the rate at which heat is taken from the cold end - whichever is less. However, if you fix the temperatures as in the example, then that limitation is removed. In that case, I believe the pressure drop as the vapor moves through the vessel becomes limiting. As flow increases, the frictional pressure drop will also increase. The maximum vapor flow rate will be reached when the frictional pressure drop reaches the pressure difference between the two ends. – Mark Sep 30 '13 at 10:04

I tried to piece the answer together from some more research, here's what I found. If you spot mistakes or have additions to make, please edit.

When introducing the liquid into the chamber, it undergoes what is called flash evaporation: because the liquid is now superheated due to the pressure loss, part of it vaporizes. If there is not enough surface area to let this happen quickly enough, gas bubbles will form, just like in a pot of water on the stove. So there's the surface area as a limiting factor.

The vapour then expands to fill the vessel at the speed of sound.

The process cools both vapour and liquid down to the saturation temperature corresponding to the low pressure, so the next limiting factor would be how quickly heat can be transferred to the liquid to bring it back to the "boil". And then it's the surface area again.

• Stir the pot. Have nucleation centers in the liquid (boiling chips). Avoid the Leidenfrost effect. Heat the liquid with an IR or microwave source to avoid container wall heat conductance limits. – Uncle Al Mar 2 '14 at 15:59

There are basically three limitting factors in the system you set up. Both evaporation as well as condensation require in practice som sort of liquid films, which provide thermal resistance between the walls and the vapour phase and therefore temperature gradients are required to allow for heat transfer. This means that besides geometry and fluid dynamic processes in the assumed liquid films, the temperature difference between hot and cold end influence the transfer rate. Third factor, as you say, would be the duct in which the vapour flows, beeing limitted to the speed of sound and producing pressure loss and correspondingly a further temperature gradient. Apart from the above, there is also a theoretical limit for mass transfer, having to do with number of particle impacts per time and area, which is proportional to the vapour density if I remeber correctly.

If the Reynolds number of laminar flow exceeds about 4000, it will go turbulent. Turbulent flow will choke mass transfer, and worse if it a supersonic shock. Rate of heat transfer through source and sink walls is another limit. And if you really juice it (accelerator)...the beta factor in special relativity. That last can be real world - ion engines in spacecraft.