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.

 

[rezeew]

I can understand your confusion in analyzing this system. Let's break it down step by step.

Firstly, the system consists of a vertically-oriented tube submerged in water with heating elements around the middle. This means that the heat energy is being added to the water in the tube, causing it to have a higher temperature compared to the surrounding water. This temperature difference creates a buoyant force, which will drive the flow of water in the tube.

Initially, there will be minimal flow as the temperature difference is not high enough to overcome the resistance of the water in the tube. However, as the temperature difference increases, the steady flow energy equation will be satisfied. This means that the energy of the water flowing out of the tube will equal the energy input from the heating elements. This will result in a steady flow rate.

To determine the flow rate, we need to consider the equations governing this system. The first equation to consider is the buoyancy force equation, which is given by:

Fb = ρgV

Where Fb is the buoyancy force, ρ is the density of the water, g is the acceleration due to gravity, and V is the volume of the water displaced.

Next, we need to consider the energy equation, which is given by:

Q = mCpΔT

Where Q is the heat energy input, m is the mass of the water, Cp is the specific heat capacity of water, and ΔT is the temperature difference between the water in the tube and the surrounding water.

Combining these equations, we can determine the flow rate by setting the buoyancy force equal to the energy input:

ρgV = mCpΔT

Solving for the volume V, we get:

V = (mCpΔT)/ρg

This equation gives us the volume of water displaced per unit time, which is the flow rate. However, we need to consider the control volume of the system. This means that we need to take into account the volume of water that is already in the tube and the volume of water that is entering and leaving the tube.

If we assume that the tube is long enough that the volume of water entering and leaving can be neglected, then the flow rate through the system will be equal to the volume of water displaced per unit time.

In summary, the flow rate through the system will depend on the buoyancy force, which is driven by the temperature difference between