thermal expansion of a tube

[mielke]

when calculating thermal expansion for a tube with an outside temperature significantly higher than the tube inside temperature, what temperature is used? the average temp, the higher, or the lower?

 

[desertfox]

Hi mielke

How accurate do you want an answer? because if you have thermal gradients across the tube you will have stresses induced between the inside and outside diameters, if you just want a rough estimate of expansion I would use the largest temperature figure.
That said if you can give more detail ie numbers, size of pipe,materials, application, temperatures and any restraints to prevent pipe expansion other than temp gradients you might get a better answer.

desertfox

 

[mielke]

we're looking at 304 SS tubes with water inside at 200F. these tubes are connected by two fixed tube sheets on either end. now the air (flowing over tubes) at the begining will be about 1000F and leaving at 200F.

 

[MintJulep]

If you want to calculate the expansion of the tubes you need to know the temperature of the tubes, not the temperature of the fluids.

 

[dbecker]

mielke,

You are saying that the air flowing over the tubes gets to the temperature of the water. That is a dangerous assumption, because that means there is almost 100% perfect heat transfer there. The bulk temperature of the tube can be assumed if you calculated your Biot number to be less than 0.1

Also, the heat transfer coefficients of both the water and air will have a large effect on the overal bulk temp of the tube. Your tube has water in it, and water is much better at moving heat. Therefore you should know your water velocity and calculate the Heat transfer coefficient using the Dittius-Boelter correlation assuming fully developed internal flow.

For the outside, the air is at 1000F, if the air is not moving fast enough than the bulk temp wil be closer to that of the water and the air will not decrease in temperature a whole lot.

If the air is moving fast enough (locally over the tubes) then you will see much greater heat transfer, and depending on your geometry and flow gradient it may be high enough to get the air temp down. But 200F air output is hard to believe unless this heat exchanger is enormous.

Hope this helps, sorry for the long response.


P.S. desertfox is right, you will see induced stresses no matter what you do, there is an introduced thermal gradient.

 

[mielke]

please keep in mind my original question as to what reference temperature should be used in determining thermal expansion of the tube, assuming the tube outside surface is at about 1000F and the tube inside surface is at about 200F.

 

[MintJulep]

Those assumptions are wrong.

 

[desertfox]

hi mielke

I asked what sort of accuracy that you were looking for?
Also if you wanted a rough ball park figure use the highest temp, but in doing that you ignore any stresses that wiil occur across the diameter.
Once you tell us clearly exactly what you trying to achieve and what is important then you might get better answers.

desertfox

 

[davefitz]

If the temp diff exceeds 200 F, then you have exceeded yield stress- there will be a high axial shear and the yielding side would likely be the hotter side.

s= E*a*DT/ ( 1-pr), s= stress, E= youngs mod , a= coef of thermal expansion, DT = temp diff, pr= poisson's ratio

The correct response would be to calculate the temperature distribution thru the tube using heat transfer relationships.

If you exceed yield , then the cold side governs , whic would be in compression, and for a long tube, it will buckle and relieve the axial stress accordingly. The hot side will yield and ratchet.

 

[corus]

The assumptions are clearly wrong as how else does the air lose heat?
Still, if you assume a temperature distribution T(r) then the mean thermal expansion will be based upon the mean temperature of the tube. You could take the mid thickness temperature but strictly speaking you need to take the integral of 2T(r).r.dr/(r2^2-r1^2) between r1 and r2.

ex-corus (semi-detached)

 

[zekeman]

Before you tackle the thermal expansion problem, you had better get a solution to the heat trnsfer problem and then state the reason for your concern on the expansion;for example you might want the longitudinal expansion of the piping to to get a handle on induced piping stresses or you may want the radial stresses. Both are very tractable.

 

[mielke]

all i want to know if it is customary to reference the longitudinal thermal expansion of the tube to the mean tube temp, lower temp (inner or outer), higher temp (inner or outer), or other. Im looking for a practical solution not necessarily and exact theoretical solution.

Im assuming a bulk temperature and properties based on 200F inside tube and 1000F outside tube in calculations. Based on this it is my understanding that the conductive heat transfer through the wall is independent of the heat transfer inside and outside of the tube.

Given that heat conductive transfer through tube wall
Q=(Tbo-Tbi)/R
where R =ln(ro/ri)/(2 pi l k)

Once again all i want to know if it is customary to reference the longitudinal thermal expansion of the tube to the mean tube temp, lower temp (inner or outer), higher temp (inner or outer), or other. Im looking for a practical solution not necessarily and exact theoretical solution.

 

[boilerone]

I would support Davefitz (the tube never gets to outside gas temperature if is internally cooled and need to get tube outside surface from calculations). Here I am giving an actual example that was used over and over in my previous boiler furnace tube designs:

The typical furnace combustion gas temperature is at 1800F, the water inside the tube is at saturation temperature of around 621F under 1800psi pressure. From simple heat transfer calculations assuming a stable condition it is found that the tube outside surface temperature is going to be, say, 675F. Therefore you are looking at a temperature curve going from 621F to 675F between the tube's inside and outside surfaces. The mean temperature found on this curve should be what you are looking for to get the exact thermal expansion values. In reality, we simply used the tube outside temperatures for the calculations not because it is mathematically time-consuming to get the precise temperature but because of wanting always to be on the conservative sides.

 

[MintJulep]

It is customary to reference thermal expansion calculations for a material to the temperature of the material. Not to temperatures of other things.

If you have a tube with 1000 degree air on one side and 200 degree water on the other side the temperature of the tube is NOT 1000 degrees and it is NOT 200 degrees. It is somewhere in between.

 

[zekeman]

I think this problem has been overblown in that for thin-walled pipe undergoing heat transfer from a gas, the usual assumption is that the water flowing has a film coefficient order od magnitude greater than the gas to pi[e film and the thermal conductivity of a metal pipe is large.
Then a reasonable assumption is that the temperature throughou the cross-section is almost constant, certainly enough tuse 200 F as the "average temperature,
So I think it is a non problem and the only thing I would be interested in is the longitudinal expansion which is simply alpha*L*deltaT

And BTW, it is entirely conceivable that the exit gas temperature could approach 200 if your gas mass flow rate is small enough, even for a short pipe.

 

[speco]

Mielke,

The thermal expansion is related to the average metal temperature. What I would be concerned about is that you referred to the two tubesheets as "fixed". The obvious question is, fixed to what?

If this exchanger has some type of frame or shell, it will be at or near the temperature of the hot gas around it. The tubes will be at a very different average temperature, somewhere between the average tube fluid and the average hot gas temperature. In your case, it's likely that there will be a significant difference in those temperatures, which will probably try to pull the tubes out of the tubesheets. If you don't provide some method of differential expansion, this exchanger is extremely likely to fail.

Regards,

Speco

 

[corus]

I'm not sure what a practical solution is as compared to a theoretical one but for a plane strain problem where the ends of the tube are restrained from rotation then the expansion is based upon the mean temperature of the tube. If you can't do integration then just take the mean as (1000 + 200)/2 = 600C. That'll be good enough.

ex-corus (semi-detached)

 

[davefitz]

Back to the original question.

If the DT is les thant the DT needed for exceeding yield stress, then the mass averaged metal temperature would be used for calculating the linear elastic axial thermal expansion. The mass averaged metal temp is close to, but not exactly equal to , the average of (To + Ti)/2. I am sure a text by Timoshenko would have this as an example.

If you exceed yield stress, which would occur if To > Ti + 200 F, then linear elastic equations no longer work. A mcuh more complicated calculation is required. And if the governing stress is compression , then a long tube will buckle out of plane.

The heat trasnfer problem needs to be solved first, as this determines the temperature distribution thru the tube.

 

[corus]

davefitz,
I don't see how the yield stress comes into the question of determining the mean temperature and thermal expansion of the tube. There is no plastic thermal expansion and as such no elastic thermal expansion, as far as I'm aware. There can be a phase change in the metal at a certain temperature which will affect the coefficient of thermal expansion (which tends to increase with temperature anyway) but for the 'practical' method required by the OP it would appear that a constant value would do anyway. You could also consider creep relaxation too, but that might be too much too.

ex-corus (semi-detached)

 

[moltenmetal]

The question about the tube metal temperature in an exchanger is an interview question we use with potential applicants. You should, as zekeman has done, be able to figure out which heat transfer resistance dominates and hence which temperature the tube metal runs closest to, based on a few basic pieces of information about the exchanger and the two fluids.

There's no point in concerning yourself in this case with thermal expansion due to a mean metal temperature of (1000-200)/2, since the air side of the tube will NEVER get to that temperature or anything close to it unless the water side is EMPTY.