Thermal expansion of a pipe with wall temperature gradient

[bjarneEng]

I have a pipe holding liquid at 80°C. The outside atmosphere is -2°C.
The pipe gets a temperature gradient over the wall thickness. The outside fibers will thus restrain the inner fibers from expanding. I would like to know the increase in the inner dia of the pipe.

I have found temperature gradient dependent formulas in Roark’s formulas for the stress in the casing wall but I can’t find any analytical formulas for the diametrical change on the inside (or outside) diameter.

 

[Nidum]

Not a particularly difficult calculation but you do need to know the outside wall temperature of the pipe in order to get an accurate answer .

Is this a thin wall or thick wall pipe ?

 

[bjarneEng]

What would be the calculation method?

The pipe has an inner diameter of 800mm and a wall thickness of 55mm.

 

[Nidum]

Ok .

We need more information . The pipe can exist in three conditions :

(a) As made new in the factory .
(b) As installed with no hot fluid flowing .
(c) As installed with hot fluid flowing .

What do you want to use as your base condition for working out the expansion ? (a) or (b) ?

Is the pipe pressurised when hot fluid is flowing ?

 

[bjarneEng]

Hi,
Condition (a)
But what I really want is to know if there are any analytical approach to this problem.
I have access to FEA tools to calculate the result, but I would like to run the problem through an excel spread sheet or similar in the future.

The pipe is pressurized, but I would assume I am allowed to super position for the forces and deflections.

 

[Chestermiller]

Is the pipe constrained axially, so that the strain in the axial direction is zero? Do you know how to solve for the temperature profile as a function of radial position? Do you know how to express the stress tensor (in cylindrical coordinates) in terms of the radial strain, the hoop strain, and the strain that would have resulted from unconstrained thermal expansion?

 

[bjarneEng]

The pipe is not constrained axially so there is only thermal strain in the axial direction.
The temperature profile I believe would be T(r)=(T.2-T.1)*(ln(r/r.1)/(ln(r.2/r.1)+T.1 ("1" inner face; "2" outside face)
I would assume there is only radial stress and hoop stress in the pipe.

As noted above Roark has formulas for hoop stress at either inner or outer dia's but this is as far as I can find in literature.
I was hoping there was someone who had solved the problem before.
I guess the big problem here is that I don't have an easy constraint in order to solve the differential equation.

 

[Chestermiller]

Do your formulas from Roark include the effect on the stresses of thermal expansion? Regarding the axial strain, what about the Poisson effect associated with the radial and hoop strains?

 

[Nidum]

an easy constraint in order to solve the differential equation

Have you got as far as deriving the differential equation ?

 

[bjarneEng]

Do your formulas from Roark include the effect on the stresses of thermal expansion? Regarding the axial strain, what about the Poisson effect associated with the radial and hoop strains?

Roark includs the stress assosiated with a temprature gradient over a wall.
You are right we need to account for poisson ratio effect.

 

[bjarneEng]

Have you got as far as deriving the differential equation ?

No I have not :D
I was just arguing: if I don't know what the inner or outer dia of the cylinder will be after loading it is difficult to set any constraints for the differential equation.

 

[Chestermiller]

No I have not :D
I was just arguing: if I don't know what the inner or outer dia of the cylinder will be after loading it is difficult to set any constraints for the differential equation.

For applying stress boundary conditions, the changes in inner and outer diameters are insignificant. The stress boundary conditions should be applied as if they are at the original boundaries.

 

[Chestermiller]

What would you surmise the stress boundary conditions on the pipe should be? (This should be your first step in solving this problem)

 

[bjarneEng]

Thank you for your attention.
I will post more questions if I get the time to solve my problem :D

 

[JBA]

In starting to do a thought exercise on this issue I realized that a critical bit of information missing for this analysis is the ambient temperature of the pipe at the time of its manufacture; without that no solution based upon the given conditions is possible.