Steady State Heat Transfer - Contribution to Reaching Equilibrium

[KleZMeR]

Hi,

I'm interested in the theory of an Aluminum sphere that is initially at some temperature that is hotter than its surroundings. The sphere is surrounded by air at some small radius, say one meter, and then the air and sphere are both enclosed by an infinite shell of Aluminum at some ambient temperature about 100K less than the inner sphere. I am also assuming there is a temperature gradient already established due to radiation, conduction, and forced convection (with vent).

If I close the vents and turn off the forced convection mechanism, I'm then interested in the mechanisms which contribute to the model reaching equilibrium temperature, i.e. ambient.

From what I gather I will have free convection, radiation, and conduction as my heat transfer mechanisms.

My question is, where do start? In time, I will have a decreasing temperature gradient across my air boundary. I also do not know if free convection will really play a role, even though my current model has free convection dominating the energy transfer from the hot body when I close up all vents and turn off fan. Another question is, if free convection IS a mechanism in reaching the equilibrium state what do I use for my convective heat transfer coefficient? I am finding many values for this and am wondering if it too will change during the cool-down.

I know I have a lot of questions here, please let me know if I have misstated or under-supplied info.

Any help would be appreciated, thanks in advance.

 

[Naty1]

hot air rises...

 

[KleZMeR]

Any non-condescending comments are also welcome...

 

[Chestermiller]

Hi,

I'm interested in the theory of an Aluminum sphere that is initially at some temperature that is hotter than its surroundings. The sphere is surrounded by air at some small radius, say one meter, and then the air and sphere are both enclosed by an infinite shell of Aluminum at some ambient temperature about 100K less than the inner sphere. I am also assuming there is a temperature gradient already established due to radiation, conduction, and forced convection (with vent).

If I close the vents and turn off the forced convection mechanism, I'm then interested in the mechanisms which contribute to the model reaching equilibrium temperature, i.e. ambient.

From what I gather I will have free convection, radiation, and conduction as my heat transfer mechanisms.

My question is, where do start? In time, I will have a decreasing temperature gradient across my air boundary. I also do not know if free convection will really play a role, even though my current model has free convection dominating the energy transfer from the hot body when I close up all vents and turn off fan. Another question is, if free convection IS a mechanism in reaching the equilibrium state what do I use for my convective heat transfer coefficient? I am finding many values for this and am wondering if it too will change during the cool-down.

I know I have a lot of questions here, please let me know if I have misstated or under-supplied info.

Any help would be appreciated, thanks in advance.

Let me say it back so that I can see if I understand what you are asking. You have two concentric spheres, with air in the space between them. I think you are talking about a transient heat transfer problem, but I am not sure. It could be a steady state heat transfer problem.

Are you holding the temperature of the inner sphere constant (say by electric heating), or is it being allowed to drop?

If it is being allowed to drop, then you have to take that into account as part of the transient heat transfer response. In this case, the final steady state temperature will be that of the outer sphere.

If its temperature of the inner sphere is being held constant, then you will eventually reach a steady state in which there is a spatial temperature variation within the air space, but not a time variation. However, you might also be interested in the transient response leading up to the final steady state. I don't know from your description.

Whatever the conditions you are imposing, this problem can definitely be modeled. The system is going to be axisymmetric, so the gas velocity and the temperature distribution will be two dimensional. For such a small temperature difference, it is unlikely that the radiative heat transfer will be significant. But the conduction and the convection will be.

You can bound the answer by neglecting the natural convection, and only including the conduction. This will provide one limiting case, and would involve only radial variations. If the spheres were in outer space with no gravity, this would give you the solution.

If you only need to get the steady state solution to the case where the inner sphere temperature is maintained constant, you can set up the pde's for the heat transfer and gas flow. These equations can be solved numerically using finite differences. '

You can also solve these problems in their full complexity using CFD.

Please let me know the answers to the questions about the system I have asked, and I may have some better advice for you.

Chet

 

[KleZMeR]

Hi Chet,

Thanks for the response. Yes I have 2 concentric spheres.

I already solved the steady state problem; Inner sphere was at constant temperature transporting heat energy via radiation, conduction, and forced convection across air barrier and out into environment. I've been able to estimate what temperature gradient I can reach between my environment and inner sphere that will ensure constant transport of the Joules that are being fed into the center sphere, creating a steady state.

My question was, when I close all the vents, and rely on only natural convection, conduction, and radiation, and I then allow the center sphere to cool to environmental temperature, how long will this take.
I guess I am interested in the transient response from steady state to equilibrium.

From my current model I see instantaneously when shutting off energy input into the center sphere and closing all the vents that natural convection and radiation both dominate conduction, and my radiation is about 1/3 of the convection, so I am wondering why I would ignore the convection and/or radiation?
Will the dominance of convection and radiation change as temperature drops?

Hope this answers your questions.

Thanks again,
Sam

 

[Chestermiller]

You are saying that you allow the system to come to steady state before you shut off the heat input to the center sphere. After that, you have a transient situation in which the air and the center sphere are cooling, while the outer sphere is being held at constant temperature.

The center sphere has thermal inertial (heat capacity) and you have to take that into account during the transient period.

The inner sphere is 1 meter in diameter, but the outer sphere is "infinite". Precisely how big is the outer sphere? What is the inner sphere like, considering that you are heating it initially? Is the inner sphere solid? What is the geometry of the heater on the inner sphere? You may have to include transient conductive heat transfer within the inner sphere.

If natural convection is important, then please know that, in natural convection, heat transfer between natural convection streamlines takes place by conduction. Therefore, conduction has to be significant.

Look up in the transport phenomena literature natural convection from a sphere. This may give you a decent idea of the heat transfer coefficient on the gas side immediately adjacent to the inner sphere.

Have you tried writing down the transient pde's for this system?

Chet

 

[Chestermiller]

I looked up natural convection heat transfer to a sphere in Transport Phenomena (Bird, Stewart, and Lightfoot). The equation they gave for the Nussult Number in air was

Nu = 0.452 (Gr Pr)^1/4

where Gr is the Grashoff number and Pr is the Prantdl number (0.71 for air). The length scale for the Grashoff number is the diameter of the inner sphere. The equation assumes an unbounded region surrounding the sphere, with most of the temperature variation occurring in the boundary layer (in your case, adjacent to the surface of the inner sphere). How does this compare with the equation you used for the heat transfer coefficient?

If you neglect the thermal inertia of the air in the region between the spheres, the above equation should apply equally well to the initial state before the heat to the sphere is turned off, and to the transient state following switching off the heat to the sphere. In the transient situation, of course, you have to include transient heat conduction within the sphere. The experimental results prior to turning off the heater should provide a check on the heat transfer coefficient.

Is this a real physical system that actually exists, or just a hypothetical example? If it is a real physical system, then you can do some experiments on it to measure the heat transfer coefficient under steady state conditions before shutting off the heater to the inner sphere.

 

[KleZMeR]

Yes your description is right, the outer sphere is an absorber and is essentially the environment.

I have taken into account the capacity of the aluminum to retain energy, at shut down it has about 500KJ.

I do understand that the conduction of the heat from the inner sphere will contribute to the convection, that is why this model will be in a loop that should account for the temperature changes of the gradient across the boundary.

I have seen that the heat transfer coefficient in the convective term is very important as far as rate of convection, I think this is what you are edging me towards as far as the Nussult Number?

This is a real physical system, an engine in an aluminum insulating box that will be buried in the arctic.

I'm not looking for exact seconds of cooldown right now, just a rough model, could be off by minutes.

What I am using to model right now is:

Q/dt = h*A*dT + σ*ε*A*[(Th^4)-(Tc^4)] + [4*pi*R1*R2*k*(Th-Tc)]/(R2-R1)

I am guessing that my convective term might be what I am oversimplifying? This is ok for now because to be honest, the exact geometry of this engine and all the components inside the box will definitely result in dynamic turbulence etc.

 

[Chestermiller]

Yes your description is right, the outer sphere is an absorber and is essentially the environment.

I have taken into account the capacity of the aluminum to retain energy, at shut down it has about 500KJ.

I do understand that the conduction of the heat from the inner sphere will contribute to the convection, that is why this model will be in a loop that should account for the temperature changes of the gradient across the boundary.

I have seen that the heat transfer coefficient in the convective term is very important as far as rate of convection, I think this is what you are edging me towards as far as the Nussult Number?

This is a real physical system, an engine in an aluminum insulating box that will be buried in the arctic.

I'm not looking for exact seconds of cooldown right now, just a rough model, could be off by minutes.

What I am using to model right now is:

Q/dt = h*A*dT + σ*ε*A*[(Th^4)-(Tc^4)] + [4*pi*R1*R2*k*(Th-Tc)]/(R2-R1)

I am guessing that my convective term might be what I am oversimplifying? This is ok for now because to be honest, the exact geometry of this engine and all the components inside the box will definitely result in dynamic turbulence etc.

The Nussult Number is the dimensionless heat transfer coefficient, and is equal to hD/k.

In your equation above, I think you meant the first couple of terms to be:

dQ/dt = h*A*(Th - Tc) + ...

I don't think that the motor is going to be levitated in the box. What is the actual geometry?

You might be able to simplify your equation conceptually by writing:

dQ/dt = h_overall*A*(Th - Tc)

where

h_overall = h + σ*ε*[(Th^2)+(Tc^2)](Th + Tc) + (R2/R1)*k/(R2-R1)

In this equation, h = k/(2*R1)*0.452*(Gr Pr)^1/4

From past experience, I'm confident that h_overall is going to be greater than 1 Btu/hr-ft²-F, and will probably be lower than 100 Btu/hr-ft²-F. A good starting value would be ~10 But/hr-ft²-F. Check and see what the above equation for h_overall predicts at the anticipated temperatures. Also note that there are probably going to be temperature variations from inside to outside of the motor as it cools, and this could have a significant effect. You should consider doing some scouting calculations to assess the effects of the transient conduction problem within the motor.

 

[KleZMeR]

I've done the calculation with a radius of .15 meters. My result is 0.00263, far from my original value of 25. I used 25 because it's a general value for air from what I can see. Please let me know if you think my calculation is off, 0.0026 seems very low and changes my time constant considerably.

 

[Chestermiller]

I've done the calculation with a radius of .15 meters. My result is 0.00263, far from my original value of 25. I used 25 because it's a general value for air from what I can see. Please let me know if you think my calculation is off, 0.0026 seems very low and changes my time constant considerably.

This value seems very low, even for the case of pure conduction with a sphere diameter of 1 meter. For pure conduction, I calculate a contribution that's about 5X as high. Have you calculated a Grashoff number yet? What value do you get?

 

[KleZMeR]

I recalculated it, I was doing something wrong. I'm still getting a value of about .379, which gives an unrealistic answer of about 11 days for the cool down (using the model I previously mentioned). My generally used air constant of 25 gives off a much more realistic time of about an hour. Thoughts?

 

[Chestermiller]

I did a quick calculation for the natural convection contribution to the overall h, and estimated a value of about 0.67 BTU/hr-ft²-F. This is on the same order as your estimate. I haven't looked at the radiative contribution yet. I don't feel like dragging out my books to find the value of σ and reasonable values for ε. If you want to give me the values you used, I will double check your calculation of the radiative contribution.

What makes you so sure that a cooling time of 1 hour is more realistic? What is the mass of the motor and the heat capacity of the aluminum? What is the actual surface area of the motor? Was the surface to volume ratio of the motor what motivated you to choose a 1m diameter for the sphere?

P.S., Are you in a klezmer band?

 

[KleZMeR]

I used 5.6704E-8 for sigma and and .9 and .5 for inner and outer ε, respectfully.
Sphere radius is .15 m for inner and .45 m for outer.
I don't know the surface area of the motor, I am estimating a radius of 0.15 meters, I do see that this radius affects the calculation drastically.
Mass of motor = 11.3398 kg, specific heat of aluminum is 910 J/kg*K.
Initial temp of Alum is 331K, final and also ambient is 273.15K

I chose a one-meter diameter because our casing that will hold the motor will be about that size.

Ok, so my 11 day model was also wrong, I had the convection being turned on as the only transport.

With free convection, radiation, and conduction and using my h value of .379 my model says about 2.4 hours until cool down. A bit more realistic. Why is there some "general knowledge" on the internet that says to choose h=25 for air? Does this assumption totally ignore geometry of my hot body? Seems strange, nevertheless one to two hours still tells us that we will need some insulation.

Another question is, with convection being dominant right now, if I add insulation at a radius say between my inner radius and outer, how does this affect convection? I only see it affecting conduction.

ps
hahha no I'm not in a klezmer band, I play piano solo.
pps
Do you know who Chet is from Weird Science?

 

[KleZMeR]

What I have been thinking of is how I am doing my calculation in my model. The convection transfers energy through the fluid (air) and I'm not accounting for this temperature change across the barrier. I am assuming that each portion of energy from my three terms is transported out of my system every second. I think this is how it should work but I am really questioning the convective term, dQ/dt = h*A*dT, and how this will be affected by added insulation, because right now it's not, but I know adding insulation at a lesser radius than the ambient will definitely affect the energy transfer. Currently my insulation only affects the conductive term which has little to no affect on the system.

 

[KleZMeR]

Maybe it's not a radius issue, I think I'm going about this all wrong. I think what I am to do is use a convective heat transfer coefficient for my insulating material and simplify the model by assuming this material is 'wrapped' around my engine.

 

[Chestermiller]

I used 5.6704E-8 for sigma and and .9 and .5 for inner and outer ε, respectfully.
Sphere radius is .15 m for inner and .45 m for outer.
I don't know the surface area of the motor, I am estimating a radius of 0.15 meters, I do see that this radius affects the calculation drastically.
Mass of motor = 11.3398 kg, specific heat of aluminum is 910 J/kg*K.
Initial temp of Alum is 331K, final and also ambient is 273.15K

I chose a one-meter diameter because our casing that will hold the motor will be about that size.

Ok, so my 11 day model was also wrong, I had the convection being turned on as the only transport.

With free convection, radiation, and conduction and using my h value of .379 my model says about 2.4 hours until cool down. A bit more realistic. Why is there some "general knowledge" on the internet that says to choose h=25 for air? Does this assumption totally ignore geometry of my hot body? Seems strange, nevertheless one to two hours still tells us that we will need some insulation.

Another question is, with convection being dominant right now, if I add insulation at a radius say between my inner radius and outer, how does this affect convection? I only see it affecting conduction.

ps
hahha no I'm not in a klezmer band, I play piano solo.
pps
Do you know who Chet is from Weird Science?

Let me understand. This motor only weights 25 lb, and is about 1 ft across in linear dimension. It sounds tiny for any practical application.

Now, you are going to run it in an enclosure, but, every so often, it is shut off, and you are worried about it cooling down because of the cold walls of the enclosure. Are you worried that you won't be able to get it started again, if the walls of the enclosure are at around 0 C and it cools down to that temperature. Is it a gasoline motor? Why would a gasoline motor not be able to start at 0 C?

The motor sits in a casing that is on the order of 1 m in linear dimension. Is the casing closed from the remainder of the enclosure, and is there air in the casing, or is the casing an open framework?

Where do you want to put the insulation? How thick. Do you want it on the motor or on the walls of the enclosure? The presence of insulation is going to substantially reduce the heat transfer coefficient, because its resistance is in series with the other elements of resistance in the system. 1/U = 1/h_overall+ H/k, where H is the thickness of the insulation and k is the thermal conductivity of the insulation. (This assumes that the insulation is on the motor).

I'm having trouble getting my head around the precise details of this system, which seem to be important. If you don't want to share these details with the world, perhaps you would feel more comfortable sending me a private message. I just don't have a good idea of the real geometry.

Chet

P.S., I also play piano, but not always solo. My genres are blues, jazz, rock 'n' roll. I'm in a ballroom dance band.