Thermal Expansion in Vessel

For simplicity take the diameter of the tank to be at the midpoint of the plate.

You need to know the coefficient of expansion of the material. The rest is easy.

Multiply the diameter by the coefficient of expansion by the number of degrees of temperature change to get the change in diameter.

If you don't know the CTE of your metal, you might look here:

If you don't know what CTE is, good luck.

My friend, when you design the pressure vessel the temperature according to the material was considered in order to get the plate thickness, related to the material choose, so if your vessel was designed by an engineer, it must not have changes in its diameter due to temperature or pressure changes from ambient to process. You could have some linear expansion and that could be calculated as if the vessel is a pipe, must be around 4.6 inches/100 feet, let one side of the vessel supports free for longitudinal movement, do not put anchors in one side.

ALL metals grow with a temperature rise, and ALL vessels grow (especially the diameter) if internal pressure is increased beyond ambient.

Pressure vessels are no different- but the materials and the design take this growth into effect in order to maintain "safe" conditions.

(2600 mm)(300°C - 45°C)(1.17 × 10^-5) ≈ 7.76 mm; heated diameter 2607.76 mm.

"Please suggest any technical reference book"

I would try any mechanics of materials book which covers this topic adequately.

Firstly, why are you interested in knowing the change in diameter?

Any metal expands with increase in temperature. You can use following formula to find out the increase/ decrease in diameter:

DT= Dr*(T-Tr)*(1+Alfa)

where DT = Diameter at temperature T

Dr = diameter at reference temperature Tr

Alfa is coefficient of thermal expansion of the material at mean of T and Tr

Thus knowledge of Dr, Tr and Alfa is very important.

Here you can see that if T increases, DT increases. If T decreases, DT will also decrease.

Now, if you want to know the change in volume of the vessel, you can insert same formula in volume formula (PI*D²*L)

Here L has similar characteristic as D.

Thus formula for volume becomes:

VT= PI*( Dr*(T-Tr)*(1+Alfa))² * (Lr*(T-Tr)*(1+Alfa))

You may solve the equation accurately. But, in general with little approximation it becomes:

VT=Vr*(T-Tr)*(1+3*Alfa)

In above explanation, Lr is length at reference temperature Tr and LT is length at temperature T

&

Vr is volume at reference temperature at reference temperature Tr and VT is volume at temperature T.

Now, normally the ends of vessel are not flat, thus there is no clear simple measurement of L. Besides, diameter also will not be very uniform at different cross sections and there may be some ovality/ Dents(for less thick vessel... almost impossible for your thickness).

But if you measure the volume of the vessel, at reference temperature, you can calculate volume at any other temperature T using last equation for volume.

Here I am assuming that the temperature is uniform across the thickness.

Also note that coefficient of thermal expansion also will vary over the range of 45 deg C to 300 deg C. (e.g. it will be different at say 45 deg C to 60 deg C that what it will be between 285 deg C to 300 deg C).

Thus applying everything depends up on the accuracy you need.

You have not indicated why you need change in diameter of the vessel due to thermal expantion only. Diameter will expand due to working pressure of the process fuid also.

Sir, Thank you for your reply. But the coefficient of expansion used in your calculation is coefficient of linear expansion, can it be used for expansion in diameter? Prashant

Yes, the coefficient of thermal expansion must be linear, and the calculations are based (and must) on a linear expansion which causes an expansion in circumference which lead to an enlargement of diameter.

Yes

C = ∏D

Circumference = Pi times Diameter

So D = C/∏

Clearly the circumference expands according to the normal coefficient of expansion, and, the diameter is proportional to the circumference.

It turns out that this linear relationship holds even in extreme circumstances. For example you might expect a small hole in a sheet of metal to get smaller when the metal heats up because of the expansion "pushing" in on the hole, but, the hole expands as though it was still filled:-

The effect is shown very exaggerated here.

You can confirm that it's true by repeatedly doing the thought experiment: take a huge an infinite sheet of metal ; drill a hole; heat up the sheet; drill a (slightly bigger) hole; cool down the sheet etc.

hello, i just find out this thread, it is helpful for my learning . could you please send the thermal expansion calculation pdf file to me? thank you

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This calculations are right if the vesel is a solid body, if you want to know how much the external diameter of the vessel vary, in case that you forgot to adjust the stress of the material at process or maximum temperature, firs check about the pressure expansion,then you will need to calculate the lineal expansion of the lenght of the cross section of the vessel and then go back to find the new diameter, calculate the difference and thats it.

With 300C you may have a high pressure depending on the content. That pressure may also contribute to your expansion. But it depends on what is inside and its pressure.