# Dimensionless analysis in engine cooling

1. Jan 7, 2013

### Zirkus

Hi, I have a problem with measuring cooling performance of a car on a chassis dynamometer. But maybe I should start with the theoretical part of my problem, namely with the dimensional analysis.

If I have a physical system and I am interested in finding one variable as a function of all other variables on which this one depends, I can use the Buckingham pi theorem and reduce the number of parameters involved. I think I understand how to do that. But what if most of these parameters are kept constant all the time and I am not interested in how their changes influence my output variable, can I still use this theory?

In my particular case I measure temperatures in several places of an engine and of the cooling circuit as a function of mass flow of the coolant, blowing air velocity and temperature, braking power and rpm of the dynamometer and gas pedal position (not sure about this one though), some of which are themselves functions of time. The temperatures will be definitely dependant on changes of other parameters as well (thermal capacities, viscosities, densities etc.) but I am not changing these. Can I still use the dimensional analysis on my (probably) seven variables? And how many dimensionless parameters would that give me?

Thank you for any reactions.

2. Jan 7, 2013

### Staff: Mentor

You can't really leave out the physical properties, because they will be critical to arriving at the dimensionless groups. You will not be able to form the proper dimensionless groups without including the physical properties, since their dimensions will cancel in the formulation of the key dimensionless groups. For example, in your system, some of the groups involved will the the Prantdl number, the Reynolds number, and the Nussult number. All of these involve the physical properties.

3. Jan 8, 2013

### Zirkus

I see, that is what I suspected, thank you.

If I tried to list all the relevant quantities in this problem I would get to something like:
$$T_i=T_i(u_{air}, T_{air},\rho_{air},c_{p,{air}},\nu_{air},\dot{m}_{coolant},\rho_{coolant},c_{p,{coolant}},\nu_{coolant},P_b(t),N(t),\beta(t)),$$
where the last three symbols denote braking power, rpm and gas pedal position. Did I forget about something? How is it with dimensions (lengths) actually...I don't measure on a model but on the real thing. Should one length appear anyway? And how about pressure drops, water pump power...I'm really confused. Thanks for any help.

4. Jan 8, 2013

### Staff: Mentor

You are trying to do too much to begin with. One of the basic principles that I recommend in doing good modeling is to break the problem down into smaller bite sized chunks. Why? if you can't do the smaller simpler problem(s), you will never be able to solve the fully integrated problem. Plus, once you get some results for the simpler problem, you will already have your overall problem partially solved, and it may give you insight into the larger problem.

With that said, I recommend you start out by first focusing attention exclusively on the radiator. You have coolant going in at one temperature, and coming out at a cooler temperature. The radiator is a finned heat exchanger. The tube diameters, lengths, and layout are important. The fin geometry and spacing is important. Coolant properties and air properties are important. If you know that air velocity from the fan, that is a parameter to include. Have you learned yet how to set up the differential equation for the coolant heat balance in this type of heat exchanger? If you have, then, rather than using Buckingham, you can reduce the differential equations to dimensionless form and elucidate the key dimensionless groups very easily. Have you learned about heat transfer coefficients? on the air side? and on the coolant side of the tube wall? The heat transfer coefficient on the coolant side of the boundary is a function of the coolant flow rate (velocity), the coolant density, the coolant viscosity, the tube ID, the coolant thermal conductivity, the coolant heat capacity. The dimensionless heat transfer coefficient is called the Nussult Number.

Until I know your background regarding fluid mechanics and convective heat transfer, I will have trouble advising you further. But, all I can say is that you need to be able to do the radiator first.

5. Jan 23, 2013

### Zirkus

That is completely true, as a matter of fact I won't be doing the measurment at all in the end, so your nice answer goes in vain...sorry about that. At least I lerned a bit about dimensional analysis, mostly from these lecture notes from MIT, which describe the abovementioned situation of changing only some of the relevant quantities (starting at page 50):

http://web.mit.edu/2.25/www/pdf/DA_unified.pdf

Thank you once again!