Thermal deflection of the inside of a pressure vessel.

[BjarneEng]

I have a motor incased in a pressure vessel.
Inside the casing the motor atmosphere is rated to 80°C. The outside atmosphere can be towards -2°C.
I assume that the outside temperature of the casing is higher than -2° but lower than 80°C so I would get a temperature gradient over the casing wall thickness.

The motor has various pressure compartments for driving cooling flows over the motor that require seal between the motor and the casing.
I would like to know the clearance between the motor and the casing for all temperature scenarios.
I can do this with FEA but it is a cumbersome process so an analytical approach would be preferred.
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.

 

[NRP99]

Change in diameter due to temperature gradient across thickness is what I guess you want to calculate.
Thermal expansion of cylinder diameter=diameter*temperature difference*coefficient of expansion.
The formula remains same for any thermal expansion, diametrical or longitudinal.

 

[BjarneEng]

Thank you nrp99,
If the tempratur was uniform over the thickness i would agree. In my problem the outside dia would constrain the inside dia so im not sure its the right solution in this case.

 

[NRP99]

I guess expansion will be linearly varying from the outside diameter(-2^oC) to inside diameter(80^oC). So you can calculate the inside diameter expansion and outside diameter expansion separately and your expansion of in between diameters will be along line connecting these two expansion values.

 

[1gibson]

The post is really asking "what happens when the ID wants to expand and be bigger than the OD wants to allow, and how do you calculate that?"

 

[NRP99]

1gibson

In my humble opinion, answer is straight forward. The outer diameter will not allow the inner diameter to expand. Still the expansion will be linear as I already stated in earlier post. The inner fibers will get compressed in between contraction of outer fiber and expansion of inner fiber. Net amount of expansion/contraction will depend on the exact values of the OD expansion and ID contraction.