A simple pump driven by buoyancy

[joe87]

I'm a bit lost getting started analyzing the following system, a pump of sorts:

http://i.imgur.com/P82vtZ1.png

It consists of a vertically-orientated tube submerged in water with heating elements around the middle. Heat energy goes in, causing a temperature difference which drives a buoyant flow.

I reckon at first, there won't be much flow and the water in the tube will heat up. But once the temperature difference is high enough, the steady flow energy equation will be satisfied (the temperature and flow rate will be high enough that the enthalpy of the water leaving equals the heat energy and enthalpy coming in),

What I'm failing at completely is figuring out the flow rate through the system. I would be happy finding the body force on a block of warm fluid, but don't know how it will work with a control volume.

Could someone please help me write down the equations governing this system? I am happy to make every simplifying instruction, and am aiming to find what flow rate the system will settle at for a given heat input (assuming the tank is sufficiently large that ALL the water heating up is not a factor).

 

[Simon Bridge]

Welcome to PF;
Off the diagram - the heating element raises the temperature of the water close to it, that water expands (against the surrounding pressure) and rises ... more water flows in from around the element - some of that water will flow into the tube.

Presumably you want to have very good insulation around the outside so the element won't directly heat the surrounding water?

The rate of flow depends on the temperature of the water inside the pipe, which will depend on how fast you can deliver heat to the water compared with how fast the water takes it away.

The surrounding water will eventually get warmer, making the pump less efficient.
(You don't need all the water to heat up, just the surrounding water.)

Aside: this pump is not "driven" by buoyancy... it is driven by the energy supplied to the heating element.

 

[joe87]

Thanks Simon. I agree with each of your points.

In the first instance I am interested in 'setting up' the problem in a very simplified way - neglecting the surrounding water heating up, writing a momentum and energy equation for a control volume in the middle of the pipe. I can calculate the density of heated water but I don't know how to work it in.

(By the way, in case the diagram isn't clear, the heating elements are meant to be heating the fluid inside the tube only, and are assumed to be well insulated on the outside).

 

[Simon Bridge]

I'd work the problem from two ends ... how much power I have to supply to heat cold water to temp T when it is moving through a pipe area A at speed v ... the other end is how fast (speed v) does water at temp T rise through buoyancy.

That should get you into the ballpark ... there is also the energy to expand the water against the surrounding pressure (heated water has a bigger volume, so it must be displacing some cold water). You'll also still lose heat via conduction etc as prev mentioned... so all this will be minimum figures.

All that will end up telling you the speed vs energy input.