Temperature Variations in a fin 2

[theonlydave]

Hi,

I have a problem regarding the temperature distribution in a fin. The problem is different from the usual fin problem, instead of heat flowing from a body to the fin attached on it, there is heat generation at the fin tip and the heat transfer is actually to the environment through convection and also conduction to the body. The body is in fact a rotating cylinder and fins are attached to the cylinder surface. Convective heat coefficient is found to be 1705 W/m^2K. Fin tip temperature within the heat generation is 1200°C. Do I ned to derive the temperature profile from scratch? What are the boundary conditions I can use? I tried to setup the conditions similar to the derivations of the traditional fin problem but made great mistakes. The fin base temperature was found to be 5000°C , more than the heat generation region. Can anyone please help?

 

[IRstuff]

Well, 5000º is pretty close to absurd, although I'm a bit dubious about 1200º as well.

What materials are these?? Ceramic? Refractory metals?
Fin aspect ratio?
Does the fin get equal air flow from tip to base?
Does the cylinder get access to something cooler or is it thermally floating?

You should still be able to do a lumped model analysis of the fin, segmenting it into some number of segments and then attaching the other end to the cylinder body.

You'll basically have to iterate the process to get the correct values of temperatures along the way and HTCs along the way.



TTFN

 

[sailoday28]

theonlydave (Mechanical)asks" What are the boundary conditions?"
There is heat generation at the tip.
Fin tip temperature within the heat generation is 1200°C.
Convective heat coefficient is found to be 1705 W/m^2K.

It seems to me that you are specifing some of the boundary conditions?

What does your fin look like? What is the boundary condition at its base?
Is it thin enough so that you can neglect the temp gradient normal to the direction that you are specifing the heat transfer coef?
What is the ambient temp surrounding the fin.
Heat generation at the tip? Or do you mean the heat flux is specified at the tip?



Get the above cleared up, then the governing equations for the model can be made.

 

[prex]

1705 W/m²°K ? Surprising, unless this is the overall equivalent coefficient of the finned surface of the tube.
Concerning your boundary conditions, you probably already know that only for a rectangular fin of constant section you can develop simple solutions by formula: variable section fins, including annular ones, may be also solved, but require the use of Bessel functions, so it is easier to use a numerical approach.
In the case of the rectangular fin the general solution for the temperature along the fin is
[θ]=C₁e^nx+C₂e^-nx
and this is valid irespective of the direction of the heat flux.
The constants may be easily determined if you can impose two end conditions on temperature and/or flux: if you know the tip temperature, then this is a condition. Also if you know for example that the conduction to the cylinder is negligible (because all the heat is evacuated by the fin), then you have zero flux at the base, and so on.

prex

http://www.xcalcs.com

Online tools for structural design

 

[theonlydave]

Hi, in case I did not specify the problem properly, I written the problem including diagrams to the post. The link is here:

http://home.pacific.net.sg/~emailcs/Finproblem.pdf

I tried the general temperature equation of the fin but got erroneous results. I specified the tip temperature at 1200°C and extend the tip end by a negative dimension equal to the (cross section area)/(Perimeter), to take into account of the tip end convection. So, now I have insulated tip end but I forgot about the heat generation region in between the condction and convection end. In this case, I have assumed that the convective heat coefficient is constant throughout the fin surface as well as the tip end. I cannot specify the temperature or boundary condition at the fin base conected to the cylinder because that is the concern region, I need to find out the temperature of the fin base subjected to the conditions. Then I realised that the equation cannot be applied over the heat generation region. On the left end of the heat generation zone is conduction and convection over the fin surface and over to the right of the heat generation (fin tip), I have convective heat transfer at the fin tip. How can I setup equations to solve the problem?

 

[theonlydave]

Hi Prex,

I tried solving the fin equation but got erronous results. This is what I did.

T§ means the temperature of the air far away which is taken to be 30°C.

The equation of the fin is T-T§ = C1 sinh(mx) + C2 cosh(mx)
where C1 and C2 are constants to be found and m^2 = (hP)/(kA) where h is the average convective heat transfer, P is the perimeter of the fin, k is the thermal conductivity of fin, A is the cross sectional area of fin.

The cartesian framework has x positive from right to left (from tip to base with x=0 at the tip end, and x=6.3mm at the base connected to the cylinder) and boundary conditions I set up are at x=0, T=1200°C, at x=0, heat flux = 1000W/mm^2, so I solved C1=-1.58 X 10^6 and C2 = 1170. When x=6.3mm at the base, T=--9.6 X 10^6 °C. This is absurd. Can anyone enlighten me?

BTW, this is a design problem and the heat flux and heat generation is assumed to be 1000W/mm^2 and 1000W/mm^3 with heat penetration to 1mm. Source of heat generation is likely to be a YAG laser. Please advise.

 

[prex]

You are using an inconsistent value for the flux.
1000 W/mm² means 1E9 W/m² and, taking for brevity k=10 W/m°K you would have, in the absence of convection, a gradient of 1E8 °K/m or 10000 °K/mm along the fin.
I think you should revise your assumptions.
As you are trying to estimate the base temperature from the tip temperature and flux, you'll need a very good estimate for these values in order to have a good estimate of end conditions.
Moreover the actual conditions will depend on cylinder conditions, as you have some conduction also at the fin base: this means that you cannot independently state the flux at the tip (unless the fin is sufficiently long to get to zero flux at the base).


prex

http://www.xcalcs.com

Online tools for structural design

 

[IRstuff]

And if the flux to the base is zero, then the temperature of the base should equal the temperature of the cylinder and therefore the temperature of the ambient.

TTFN

 

[theonlydave]

Hi Prex and IRstuff,

I guess I made a mistake in the flux. I guess that 250W of energy is generated in the tip of the fin. The volume of heat generation would be cross sectional area multiplied by the penetration depth which is about 1mm. However, the heat flux 250W/(1.5X15X10^-6) still cannot solve the equation. My question is can I still state 2 boundary condition at x=0? Is it still valid to solve the equation or does the 2 boundary condition clash? or do I need to state another boundary condition at x=6.3mm (base tip)?

Unfortunately, the cylinder is 350mm which is much larger than the fin (6.3mm), I cannot ignore any flux flowing into the cylinder. Also interesting, rotating the cylinder alone without heat generation at the fin will cause the cylinder and fins to rise to about 100 degrees celsius while the ambient condition is 30 celsius. Frictional effects with the air caused a temperature rise.

 

[sailoday28]

theonlydave (Mechanical)
ASKS "Is it still valid to solve the equation or does the 2 boundary condition clash? or do I need to state another boundary condition at x=6.3mm (base tip)? "

See the solution of PREX. To obtain the constants C1 and C2 you need the boundary conditions at x=0 and x=L

You can specify a flux at one or both the boundaries boundaries by taking the derivative of Temp with respect to X.

I am curious as to how you know the flux and temp at one boundary.

 

[sailoday28]

Correction to my post
I am curious as to how you know the heat generation and temperature at the one boundary?

 

[theonlydave]

The heat generation is applied so we know the volumetric heat genertion, and here I assumed 250W is generated in the fin. Btw, i know that the flux q at x=0 and the temperature at x=0, can i still solve the equation. I cannot assume anything at x=L because the fin is short, there is, I presume heat conduction into the cylinder, so flux cannot be zero and the interest is to determine the temperature at the base of the fin. Can you advise?

 

[sailoday28]

Another approach to the boundary conditions.
Consider the heat generation portion as a lumped mass.
The heat generated will be convected/radiated at of all sides except that directly connected to fin at x=0

1-At x=0, the heat from the lumped mass is conducted into the fin. Heat into fin -kAdt/dx=heat generated - heat loss by convection/radiation from lumped mass.

2- For max temp at cylinder, and everywhere else, let dt/dx at cylinder boundary =0
3- From 1 and 2 above solve fin equation given by PREX.

 

[prex]

As I stated above you can not state any two boundary conditions independently of all the others: otherwise you get physically unrealistic results, as you did.
This means that, if in your conditions there is a flux through the fin base (what seems quite likely), then you need to estimate this from local conditions (at base), you can't get it from something that happens elsewhere (the tip).
If for instance the cylinder is effectively cooled by some means, then you would have a constant known temperature at base, and this would be your boundary condition. If on the contrary the cylinder is made with a bad conducting material, then you could approach the base condition with zero flux. If the actual condition is an intermediate one, then you need to make some assumptions on the temperature distribution in the cylinder.
A first guess you could do is with an infinitely long fin: the heat removal by the cylinder will hardly be much better than this, unless you have a very effective cooling system.
On the other side, if you know the power generated at fin tip (250 W), and also you know that all this power is evacuated through the fin (what is much less sure), then the gradient close to the tip in the fin would be of the order of 1000 °K/mm, and of course your estimate of 1200 °C at fin tip would be unrealistic.
Assuming you have some heat evacuated by radiation directly from the tip to the surrounding, an estimate of this (for 1200 °C and 0.5 emissivity) is 3 W: this means that your assumption of 250 W generated is also unrealistic.
I repeat: before getting into mathematical tricks, you need to physically set up and understand your problem.

prex

http://www.xcalcs.com

Online tools for structural design

 

[theonlydave]

Hi Prex and Sailoday28,

I organised my thoughts and came to the following conclusions. The only confirmed boundary condition is that the tip of the fin must be kept at 1200°C (at x=0 which is fin tip), heat input would cause the fin tip to rise to 1200°C and we have then a steady state condition. I am going to assume that the base of the fin is kept at 100°C and I solve the fin equation to arrive at the result that the heat flux at the fin tip is 7.5W/mm^2, ie 170W will start to flow at the fin tip and will be lost through convection. Hence the temperature gradient is 4722°C/mm at the fin tip. Lets say again that the base fin is at 800°C, then i have 6.7W/mm^2 of heat flux at the fin tip. ie a power lost of 150W. So my conclusion would be the lower the power input to the fins, the higher the temperature at the base of the fin at steady state. if i am going to maintain the fin base temperature at a lower temperature, then i need to increase the fin power input. Sounds interesting but according to the math, the system would behave like this. What do u all think?

 

[sailoday28]

The key to solving your problem is a heat balance.
This includes:
A heat balance of the heat generation volume.
A heat balance of the fin.
This has been discussed in previous responses.

As Prex has stated, knowledge of a cooling fluid temp will give a reasonable handle for the remaining boundary condition.
I have prev suggested dt/dx at the cylinder for max temp conditions.
With the above input, the temp of the heat generating volume may be then determined.

 

[hacksaw]

sailoday28 makes a good point, you must use a consistent set of well defined boundary conditions otherwise the results will be nonsense...

 

[prex]

Once again: you can't mathematically state any condition, without analysing its physical meaning.
If you impose 800°C at fin base, then you need to supply heat through the base too to maintain this temperature and this goes into the heat balance, so that of course you get more heat input for higher temperatures.
Your fin BTW is almost of infinite length: it loses almost 1/3 of the temperature level and heat input in the first 2.5 mm of length and drops down below 100°C at the base. This is due to a heat transfer coefficient of 1705 W/m²°K that appears very high: can you justify this value?

prex

http://www.xcalcs.com

Online tools for structural design

 

[theonlydave]

Hi Prex,

This value is found from empirical results. The cylinder is rotating at 20000rpm. I assumed cosistent convective heat coeffient throughout the entire fin from tip to base because it is really very turbulent.

I dont understand the point about the fin being infinite. How did you arrive at the conclusion that the temperature drops below 1/3 of the temperature in the first 2.5 mm of length. From the heat balance equation, the amount of heat flux and hence the heat generation would determine the base temperature. I prove that by "reverse engineering". I have the fin equation:
T(x) - T§ = C1 sinh (mx) + C2 cosh (mx) as stated in the above post. The heat generature depth would be assumed negligible, so I take the entire length and ignore the heat generation depth. At x=0, T=1200°C, at x= 6.3mm (fin base) I assumed T=100°C, so I get C1=-1174.5 and C2 = 1170. With C1 and C2 found for the above boundary condition,then I differentiate the fin equation and set it at x=0, which I can then equate to the heat flux as q=-k(dT/dx) so
dT/dx)=-q/k= the diffentiation of fin equation at x=0. There I have it, the required heat flux to maintain the fin base at 100°C, which is 7.5W/mm^2 or 170W of heat generation. Say if I need 800°C at the fin base, I redo the above step and arrive at q=4.3W/mm^2. so Q=150W. I guess the fin can expel more heat if heat flux from tip to base is higher. I dont have heat generation in the cylinder. Instead of the conventinal fin where heat is flowing from base to tip, I have heat transfer from tip to base, because the tip has heat generation and it will be at 1200°C. The only heat generation of the cylinder is only through surface friction with the air as it is rotating but at equilibrium without any other heat input, we have the cylinder at 100°C at 20000rpm.

I dont think I need to supply heat to the fin base because the fin equation is generic for any fin with only one dimensional heat transfer, here i have it from right to left, and I use it to obtain the amount of heat flux I need at the tip so that I get a certain temperature at the base. so now, I am able to know the amount of heat generation I need at the tip to control the temperature at the base. I do not understand Prex's statements. Would anyone like to explain a bit and comment on my work?

 

[prex]

Please use the simpler (equivalent) equation
[θ]=C₁e^nx+C₂e^-nx
An infinite fin is a fin that is sufficiently long to have all the heat input evacuated by the fin surface, so that a zero flux is present at the other end. For such a fin C₁=0 (or close to zero).
Now let say that a negligible heat flux is 1% of the input flux: so as n=395 m^-1 with your conditions, the condition above means e^-nx=0.01 or nx=4.6 or x=11.6 mm: this proves that your fin is only half length of an 'infinite' fin. You can also see that under the above conditions the flux at 6.3 mm is 8% of the input flux and the temperature is 130 °C: hence here you are close to the condition you took of 100 °C at the base (and in fact you calculate C₁=0.255 °C and C₂=1170 °C for that case).
This means that you can't impose a temperature of 800 °C at fin base, unless you want to heat up the cylinder to that temperature, but using a different heat source of course.
So you have above the solution to your problem: the gradient at fin tip is 460 W/mm, the heat flux is 7.4 W/mm² and the heat input 170 W as you said (that is not very far from your preceding assumption of 250 W that I erroneously defined as unrealistic).

prex

http://www.xcalcs.com

Online tools for structural design