Thermal Diffusivity

[homerphish]

How can I figure out how long it will take to thaw out an 8'x8' plot of mud 17' deep. I have 100 degree F air above it and the mud is probably 0 degrees with thermal conductivity of 0.62. Is there any way to calculate how long the heat transfer would take to bring the mud temperature to 32 degrees?

 

[sailoday28]

Thermal diffusivity is the right lead in to this problem. What are the units of k? Also what is the sp. heat and density of the mud.

The above allows computation of the thermal diffusivity.
What is below and to the sides of the plot of mud. These boundary conditions are needed. In fact, if the boundaries are in contact with other mediums, such as the air above, those conditions must be specified.

 

[JStephen]

If it wasn't for the "thawing" part, it would be a fairly straightforward conduction problem, although not an easy one, necessarily.

I'll suggest a shortcut that should give you a lower bound on the time required. Assume the surface of the ground is at 0 degrees. You should be able to find convection heat transfer coefficients for that case in a standard "Heat Transfer" textbook (one of which is available online, although I haven't perused it). And you should be able to calculate the energy required to thaw that chunk of ice. Dividing that amount of energy by the maximum heat transfer rate will give you the minimum possible time required.

I assume that by the time you thawed out that area that deep, you would have thawed out a much larger area to a shallower depth (or else lost a lot of heat through going up through the surrounding ground). So whatever answer you get from the approach above could problably be increased by 4 times or better. And of course, as the surface heats up, the heat transfer slows down at the surface.

Based on my experiences thawing meat in the kitchen, it will take ages to thaw mud to 17' deep. It might help to drill some holes down through it and pipe in steam or hot water.

 

[ko99]

17' frost depth is unusual. How do you know it's frozen that deep?


ko (

http://www.ecooling.biz)

 

[sailoday28]

Does the 32F requirement include thawing out?

 

[corus]

The general solution to this problem for a temperature T at a distance, x, at time t is given by
T =sum(Ai.cos(Pi.x+Bi.sinPi.x).exp(-a.Pi^2.t)) where a is the diffusivity. Solve for the suffixes Ai, Bi, and Pi given your boundary conditions at time t=0, and x=0, and that there is no heat flow at infinity, and there you have it. Use the resulting equation to give you the time at which the temperature is 32 at a distance of 17.

A finite element program would probably be best or extrapolate from some data from tests might be another option.

corus

 

[JStephen]

Corus, does that take into effect the energy required to melt the ice? or is that just the conduction problem?

 

[amorrison]

Some discussion points.

Since the top surface is only 8x8ft. while the depth is 17ft the heat flow will spread out a lot before it reaches the 17 ft depth.
Suggest your top surface be 17+8+17=42ft square if possible to allow for the spreadout which means a lot more volume will have to be thawed out than you actually need - but that will happen anyway if your 8x8 patch is used.
The surface heat transfer coefficient will be of little importance in the big picture as "soil" has a thermal resistance of ~~~R1/ft or a total of ~~~R17 to the bottom of your defrost hole.
From one similar situation(but not nearly as deep) I heard about 20 years ago I suggest the time required for the defrost to occur at 17 ft is probably ~~~2-3 months.

A much faster solution would be to drill holes (maybe in a 4ftx4ft matrix(9 holes total))to the 17 ft depth and heat the holes with pumped down hot water,electric resistance heaters or a recirculating glycol pipe heating system. I would fill the holes with water(and keep them filled) so there is good thermal contact between the heaters and the "soil". Potential problem - will the "soil" collapse into the holes (especially with the hot water method)as the melting occurs - solution steel pipe liners with endcaps for the holes? If you have some extra time( ~~a week or two) one center hole would probably do the same thing.

Also keep the surface covered to prevent heat loss during the defrosting. Cheap insulation - 2 ft thick layer of straw with two ft. of fluffy snow on top of the straw.
The surface insulation should extend at least 10 ft beyond the 8x8ft patch.

PS#1. It is unlikely that this is the first time a similar "problem" has occured in the history of mankind ask experienced contractors/government inspectors what has been done in the past.

PS#2. If a person is going to be sent down this hole for some reason make sure all safety codes will be met -steel hole liners etc

 

[sailoday28]

corus (Mechanical)What dimension are you using for infinity and what is the justification of no heat flow at that location? Are you also implying there is no heat flow from/into the sides?

 

[corus]

The expression I used was for one dimensional heat flow and thus assumes no heat flow to the sides. An adaptation of the basic heat flow by including a heat flux for the heat loss to the sides could represent quasi 2D heat flow.
For the enegery to melt the ice I suppose you'd have to consider the change in specific heat/density/conductivity as a function of temperature.
Basically as a hand calculation it would be too difficult to solve and you'd have to use FE techniques, otherwise you'd have to use empiricial data and guesstimate the answer.

corus

 

[sailoday28]

corus (Mechanical) If you are considering 1D in depth, then although harder to solve, why not put in a boundary condition at the 17 ft depth for heat transfer from the soil deeper than 17 ft?

 

[ProcsssDr]

This problem is similar to the melting of permafrost in the Arctic. During the summer, the permafrost thaws only a couple of feet. Of course, the air temperature is not 100F, but I think 17 feet is a long distance to thaw and it should take quite a whil-- months perhaps.

 

[corus]

sailoday28, what boundary condition would you apply at that depth if you don't know the heat loss there and the temperature isn't fixed?
Another option, which I've never used, is to use infinte elements when using finite elements, or use the boundary element method which deals with infinity.

corus

 

[sailoday28]

corus (Mechanical)
The boundary condition at that depth for two different mediums would be:
The temp of the interface of each medium is the same and FLUX, -kdt/dx of top medium=-kdtdx of 2nd medium
where the k of each is know. For the lower medium,you could further assume it is infinite in the 1D.

I haven't researched availability of closed form solutions to above, but a search of Journal of Heat Transfer (ASME) might help.

 

[JKEngineer]

About the question of the boundary condition at the far end of the area of interest: An approach I have used is to extend the calculational domain (in FEA) well beyond the boundary of interest, using coarse element sizes for economy. I extended it until the numerical results in the area of interest were not affected by whether the distant boundary was assumed adiabatic or isothermal. I feel comfortable that that gives a good estimate for the performance of the system in the area of interest.
Jack

Jack M. Kleinfeld, P.E. Kleinfeld Technical Services, Inc.
Infrared Thermography, Finite Element Analysis, Process Engineering

http://www.KleinfeldTechnical.com

 

[sailoday28]

JKEngineer (Chemical) Dec 21, 2004
About the question of the boundary condition at the far end of the area of interest: An approach I have used is to extend the calculational domain (in FEA) well beyond the boundary of interest,

The above approach is basically for a semi-infinite solid for which closed formed solutions are calculable. For a step shock at the top boundary (x=0), the closed form solution suggested by
CORUS, will be in terms of the ERROR FUNCTION. With a varying temp or that with a heat transfer coefficient at x=0, the Duhammel intergal theoem could be used. For that boundary condition, numerical analysis is probably a lot easier.

 

[JKEngineer]

The closed form solution gets more difficult, I think, if the domain of interest is not uniform on its interior. For example if it has inclusions of one sort or another. Esp. if they are not nice geometries.

Jack

Jack M. Kleinfeld, P.E. Kleinfeld Technical Services, Inc.
Infrared Thermography, Finite Element Analysis, Process Engineering

http://www.KleinfeldTechnical.com

 

[sailoday28]

JKEngineer (Chemical) Dec 21, 2004
The closed form solution gets more difficult, I think, if the domain of interest is not uniform on its interior. For example if it has inclusions of one sort or another. Esp. if they are not nice geometries


I agree. Perhaps, FEA would be better suited for the original problem in three dimensions.

 

[JStephen]

I suspect you could come up with a theoretical solution IF you substitute a round surface for a square one (IE, axi-symmetric) and IF you just turned it into the conduction problem, rather than dealing with the ice-melt at the interface. Whether it would be worth the effort to do this is a different issue.

It might also be of interest to solve the steady-state problem where you're heating the 8' square or circle, just to find the heat loss into the environment.

 

[owg]

Based on my recollection of softening tar sand prior to surface mining in winter, and depending on where on earth you are, most of the heating will come from the bottom of the block, that is from the earth. As mentioned earlier, insulating the surface will achieve the objective. However the 100 deg F air will help. Unfortunately I never saw the calculations. The 17 feet depth of freezing is a surprise. Where is this?

HAZOP at

http://www.curryhydrocarbons.ca