Radiator Maxumum Heat Transfer

V Mad

My first posting, so apologies if this is in any way inappropriate to the forum/section.

I am asking here for some scientific input to this question about a cooling system as, for example found in a classical motor vehicle; what is the likely relationship between coolant flow and heat transfer, given all other conditions being constant (heat input, air flow etc.).

As an electrinics engineers, I have little knowledge of thermodynamics, so please keep answers fairly basic if possible.

My suspicion is that there is a rate at which heat transfer peaks at some specific flow rate, ie faster flow rate does not necessarily result in greater heat transfer. In electronics, there is a maximum power transfer theorem that shows that maximum power is transferred from a generator when the load impedance matches the source impedance. I just wonder if something similar applies here?

Astronuc

Basically in heat transfer, the heat goes from high temperature to low temperature, and the maximum theoretical efficiency is described by the simple Carnot efficiency.
http://en.wikipedia.org/wiki/Carnot_cycle

There are three basic phenomena of heat transfer: conduction, convection (natural or forced) and radiation. The selection of which method to transfer heat will depend on the temperatures (hot and cold) involved, on the size, and on the environment in which the application is to serve. For example, in space surrounded by a vacuum, radiative heat transfer is the only possible means for power systems.

In general, conduction has the highest rate of heat transfer, convection less, and radiation the lowest.

Forced convection is quite common, and finds practical application in power generation systems where a working fluid is heating then passed through a mechcanical conversion system, e.g. Stirling, Brayton or Rankine cycles.

In general, the faster the flow rate, the better the heat transfer in forced convection. On the other hand, moving the fluid faster means doing more work on the fluid, either with a compressor or pump, and it is usually desirable to minimize the work done. The other consideration on the working fluid is the temperature and pressure of the fluid, which can go together (e.g. the higher the temperature, the high the pressure). A higher pressure means a stronger structure (fluid transfer circuit) is required.

The thermodynamic consideration is the temperature of the working fluid should be optimized to receive heat from the hot source, and reject heat to the cold sink, so perhaps that is a close analogy to the maximum power transfer theorem.

In conduction or radiation, on simply tries to maximize temperature differentials between hot and cold, but in practicality, there is a maximum allowable hot temperature and/or a maximum achievable cold temperature.

In the car, the radiator is not really a radiator, because it does not really radiate, but rather uses forced convection on two sides. In one circuit, the coolant water takes heat from the engine and delivers it to the radiator (really heat exchanger), where the heat is transferred to forced air. Both circuits used forced convection, but the heat is conducted through the tubes and fins of the radiator. There is a constraint on the max temperature of the engine, and also on the cooling water, which is related to the system pressure - one does not want the water to boil (or pressure to increase).

DaleSpam

I wouldn't say that it "peaks", but it definitely levels off. Heat is conducted from the engine into the coolant, the coolant is pumped to the radiator, and the heat is transfered by convection to the air.

In the low flow limit the coolant leaves the engine at thermal equilibrium and additional heat can only be removed by increased coolant flow (assuming constant engine temperature). Similarly with the radiator, in the low flow limit the coolant leaves the radiator at thermal equilibrium and additional heat can only be removed by increased coolant flow (assuming constant ambient temperature). In such a case the cooling can be described as flow-limited and increasing your flow rate will increase your cooling rate.

If you increase your coolant flow you will eventually get to the point where your coolant is no longer at thermal equilibrium with your engine. In other words, the coolant is leaving the engine at a lower temperature than above. At that point the system is no longer flow limited and further increases in coolant flow will serve to further reduce the temperature of the coolant leaving the engine.

If you still continue to increase your coolant flow you will eventually get to the point where your coolant is no longer at thermal equilibrium with your radiator. In other words, the coolant is leaving the radiator at a higher temperature than above. At that point the system is radiator limited and further increases in coolant flow will only serve to further increase the temperature of the coolant leaving the radiator. This would continue until the temperature of the coolant leaving the radiator was arbitrarily close to that of the coolant leaving the engine.

russ_watters

Carnot efficiency is for a heat engine. Heat transfer efficiency is a measure the effectiveness of a heat exchanger. For most heat exchanger applications (such as a car radiator), a faster flow rate will increase the heat transfer rate, but hyperbolically approaching a limit. A faster flow rate maximizes the delta-T (approach temperature) between the heat exchanger and the fluid. [edit: dale beat me to it]

Here's some info on heat exchanger operation:
http://en.wikipedia.org/wiki/Heat_exchanger

V Mad

Some interesting anwers here, but I need some time to reflect before commenting. I guess I was expecting some simplified (first order) mathematical models to back up the explanations, although I fear they may be beyond my comprehension.

(A quick mention of DaleSpams reply which seems to allude to a situation where maximum transfer occurs when there is a match between the thermal resistance of the engine/coolant interface, and the coolant/radiator interface. But maybe thats just my pet theory looking for some credibility?)

I suspect that the reality is that simplified models dont really represent a typical system, but nevertheless, it is a good start for further thought and discussion.

One second order effect I can suggest might occur is that, at high coolant flow rates, turbulence/cavitation will no doubt occur, and in that case equations possibly even become non-linear.

Astronuc

A system would be designed to preclude cavitation. Some turbulence is inherent. In fact, some system promote turbulence, particularly at the hot (high temp) souce in order to get better heat transfer coefficients.

Pumps in a forced convection system are usually put after the cold sink, because that is the least likely place to induce a phase change in a liquid system, e.g. pressurized water or organic fluid.

In gas systems, turbulence is generally assured.

DaleSpam

The basic first order equation is:
dQ/dt = k A (Te-Tc)
where Q is heat entering the coolant, k is the heat transfer coefficient, A is the area over which heat transfer occurs, Te is the temperature of the environment (engine or radiator) and Tc is the temperature of the coolant. With the heat capacity of the coolant you can relate Q and Tc.

You can do more fancy things like including the heat enetering and leaving a differential element of pipe through coolant flow and then integrating along the length of pipe, etc.

V Mad

That formula seems adequate as a basic formula. Presumably this would also apply to the coolant/radiator interface, ie
dQ/dt = kA (Tc/Ta) whre Ta is the ambinet air temp.
If that is so, I can then build this into an overall system formula.

But, to solve my problem about the effect of coolant flow, I need the eqations to include coolant flow as a variable. Again, a first order equation for a start.

Would this just add Fc (coolant flow) as a multiple in the equation, ie
dQ/dt = kA Fc (Tc/Ta) etc. ?

PS: is Q in units of energy (joules) ? So then dQ/dT is power (watts) ?

V Mad

Interesting point about turbulence. I suppose it locally increases flow rate and therefore improves heat transfer.

On cavitation, I never understood this phenomenon. If water cavitates, what is in the cavity and where does it come from?

DaleSpam

Sorry I wasn't clear, but it appears that you understood anyway. The formula above is the heat transfer by conduction across the pipe wall, not through mass transfer within the pipe. For mass transfer within the pipe look at the formula for specific heat:
dQ = c m dTc
where c is the specific heat of the coolant, m is the mass of the coolant, and again Tc is the temperature of the coolant. The coolant flow is dm/dt.

Consider a differential element of the pipe. Heat enters the element through conduction across the wall and through coolant flow in. Heat leaves the element through coolant flow out. By conservation of energy the heat entering is equal to the heat leaving. You can build your system description from there, but I am sure if you do an internet search you will be able to find someone who has already worked this out.

One hint if you work it yourself, once you get the system of equations, set any dT/dt terms to zero. You are not interested in how the temperatures change as the engine is warming up, you want to see its steady-state behavior. You are interested in dT/dL where dL is a differential length of pipe, spatial derivatives, not time derivatives.

V Mad

I dont think thats correct. The coolant mass is a constant, therefore dm/dt is zero. I think coolant flow might be dv/dt where v=coolant volume within the relevant heat exchanger (cylinder block/heads, or radiator). Do you agree?

DaleSpam

No, it is correct. Remember we are talking about conservation of energy in a (differential) *segment* of pipe. Although the total mass of coolant in the system is constant there is mass flux in and out of any given segment of pipe. This is dm/dt, and this mass flux carries heat energy as well.

Of course, with a known density of coolant you can easily change between mass and volume if you prefer to work with dv/dt.