# Thermal expansion of a pipe with wall temperature gradient

• bjarneEng
In summary, the outside ambient temperature affects the stresses in the pipe. However, the stress at the inner diameter is still significant.
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.

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 ?

What would be the calculation method?

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

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 ?

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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.

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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?

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.

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?

bjarneEng said:
an easy constraint in order to solve the differential equation

Have you got as far as deriving the differential equation ?

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Chestermiller said:
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.

Nidum said:
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.

bjarneEng said:
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.

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

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I will post more questions if I get the time to solve my problem :D

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.

## 1. What is thermal expansion?

Thermal expansion is the tendency of materials to expand or contract in response to changes in temperature. This phenomenon occurs when the temperature of a material increases or decreases, causing its molecules to either move further apart or closer together, resulting in a change in size or volume.

## 2. How does thermal expansion affect a pipe with a wall temperature gradient?

When a pipe is exposed to a temperature gradient, with one side being heated and the other side being cooled, it will expand on the heated side and contract on the cooled side. This can cause the pipe to bend or deform, potentially leading to leaks or other structural issues.

## 3. What factors can influence the thermal expansion of a pipe?

The thermal expansion of a pipe can be influenced by several factors, including the material of the pipe, the temperature difference across the pipe, the length and diameter of the pipe, and the coefficient of thermal expansion of the material. The direction of the temperature gradient and any external forces acting on the pipe can also play a role.

## 4. How is the thermal expansion of a pipe calculated?

The thermal expansion of a pipe can be calculated using the formula: ΔL = L₀αΔT, where ΔL is the change in length, L₀ is the initial length, α is the coefficient of thermal expansion, and ΔT is the temperature difference. This formula can be used to determine the expected expansion or contraction of a pipe based on the given parameters.

## 5. What are some potential solutions to minimize the effects of thermal expansion on a pipe?

Some potential solutions to minimize the effects of thermal expansion on a pipe include using materials with a lower coefficient of thermal expansion, installing expansion joints or flexible connectors, and implementing proper installation techniques such as leaving room for expansion and contraction. It is also important to consider the thermal properties of the surrounding environment when designing and installing a pipe system.

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