finned plate - material mass optimization

[moio92]

Hello everyone, I have a problem that I can't solve and I need urgently because I have to deliver it as a relationship to support an exam.

" a square plate (lxl) is cooled with cylindrical fins (copper material). evaluate number of fins, diameter and length so that the amount of material used is minimal by switching a certain q power (please) "

then analyzing the problem I immediately said: the unknowns are three: No, d and l
I must minimize the mass of the fins, that is: m= (n ρ π l d^2)/4 where I will have to minimize only (n l d^2) (°°°°°) being the other constant quantities.
and I have only one equation, that of power: q=an h (ts-t∞) + n qaletta (°)
where an is the surface of the untapped plate and the qaletta is the power exchanged by an aletta, that is:
√(d^3) tanh(l/(√(k/(4) √d))

Now I thought of acting putting a certain value of n, through the equation (°) get an addiction between l and keeping all the other greatnesses as constant.
insert this dependence into (°°°°) , through the derivative find the minimum value of l (or d) and then repeat the path for various n until you find the minimum value.

the problem is: how do I get an addiction between l and d from (°) being a nonlinear equation in two unknowns? ?
Is that right for you as a procedure?

Thank you so much for your help,

 

[paulpaul]

Hi.

some questions:

- the convection coefficient h presumably is known, correct?
- if I'm not mistaken, in expressing the heat exchanged by the aletta, you used the condition of adiabacity at the end of the cylindrical fin (no heat exchange ends/environment), right?

That said, I think you should express the heat exchanged according to the overall performance of the alettata surface (which takes into account the variables such as areas, number of fins, etc.) and maximise the latter: if you have not already consulted it, on the riot (fundamentals of heat and mass transfer) the argument is very well treated.
the heat exchanged (and the efficiencies, yields, etc.) of a fin depends on hyperbolic functions, difficult to maximise except for the case - precisely - of adiabatic ends in which, if well remember, there is only tanh in the expression of heat therefore easily maximizeable: growing monotonous, limited to 1 per x - > inf, which already around 2 is worth almost 1. Given the school nature of the question, it seems to me that you can realistically consider only this case.

In practice, experimental data is often used, or formulas are thrown on xls and is optimized for "tentatives" by changing the various parameters.

 

[moio92]

Hi.

some questions:

- the convection coefficient h presumably is known, correct?
- if I'm not mistaken, in expressing the heat exchanged by the aletta, you used the condition of adiabacity at the end of the cylindrical fin (no heat exchange ends/environment), right?

That said, I think you should express the heat exchanged according to the overall performance of the alettata surface (which takes into account the variables such as areas, number of fins, etc.) and maximise the latter: if you have not already consulted it, on the riot (fundamentals of heat and mass transfer) the argument is very well treated.
the heat exchanged (and the efficiencies, yields, etc.) of a fin depends on hyperbolic functions, difficult to maximise except for the case - precisely - of adiabatic ends in which, if well remember, there is only tanh in the expression of heat therefore easily maximizeable: growing monotonous, limited to 1 per x - > inf, which already around 2 is worth almost 1. Given the school nature of the question, it seems to me that you can realistically consider only this case.

In practice, experimental data is often used, or formulas are thrown on xls and is optimized for "tentatives" by changing the various parameters.

Thank you very much for the answer
regarding your questions:
- the coefficent h is not known, but since I realized that his knowledge is necessary (even for ts-tinf), I asked the professor (opening in response in the coming days) if these values I have to choose them myself or can give them to me him.
- Exactly, I had chosen the case of adiabatic ends.
- so you say you take this equation:and put the end of the tanh so that it is already to the maximum (poning for example ml=2)? and then maximise ηf? say that maximizing ηf, at the same time minimizing the mass of the material?

Thanks again

 

[moio92]

unexpectedly the professor answered me immediately by email, the h and deltat I can choose them but placing sensible values.
I thought about a deltat of about 50° (between ambient air and a plate at about 75°c)
as regards h I thought to use 1000 w/km^2
I have taken inspiration from the examples of incropera, but there is not even an example of minimization of the material used for fins.

 

[paulpaul]

Hi.

I would say deltat and h can choose them as you want from realistic values, they don't move the problem. I don't know if there is any problem about this. I quoted that text only because I think the part on the wings is done pretty well.
I would use the formula you posted, taking into account that the efficiency of the fin is the relationship between the heat exchanged and the maximum heat exchanged if the fin had the same temperature as the base to which it is connected: maximizing it, surely maximize the heat exchanged.
If, however, maximise tanh(ml) by placing for example ml=2 (beyond you don't have large efficiency increments) you get l=2/m, and you already have a limit to the length (and then also to the volume) of the fin.
I would at least proceed like this...

 

[moio92]

Hi.

I would say deltat and h can choose them as you want from realistic values, they don't move the problem. I don't know if there is any problem about this. I quoted that text only because I think the part on the wings is done pretty well.
I would use the formula you posted, taking into account that the efficiency of the fin is the relationship between the heat exchanged and the maximum heat exchanged if the fin had the same temperature as the base to which it is connected: maximizing it, surely maximize the heat exchanged.
If, however, maximise tanh(ml) by placing for example ml=2 (beyond you don't have large efficiency increments) you get l=2/m, and you already have a limit to the length (and then also to the volume) of the fin.
I would at least proceed like this...

Thank you very much for the answers! I will put myself to work through matlab to evaluate a bit the best solution, but now I have a clear vision of the problem. thank you! if you have any problems in the solution, I will write here again.
a greeting