A 50mm diameter strut that supports the heat exchanger in a nuclear power station is rigidly fixed at both ends. If the temperature increases from an initial temperature of 25?C to 300?C when the generator is at full power, calculate the force induced in the strut due to this temperature change. Assume the coefficient of linear expansion for the strut material is 14x10-6 / ?C and Young’s Modulus (E) is 210 GPa.
Would anyone know how to go about solving this?
Thanks a lot :)..Thermodynamics question?
Young’s Modulus is an extension of Hooke's law and relates stress to strain in materials.
Young’s Modulus E = Stress / Strain
Strain is the ratio of change of length / length and so is dimensionless, hence Young’s Modulus has units of stress, which is the same as pressure.
Our strut is increased in temperature by 300 - 25 = 275°C
The coefficient of linear expansion for the strut material is 14x10^-6 / °C
So the strut wants to extend by 275 x 14x10-6 = 3,850 x 10^-6 or 3.85 x 10^-3
But 3.85 x 10^-3 what? We have multiplied a coefficient with units of %26quot;per degree%26quot; by degrees, so we we have ended up with a number without any dimensions. And as we aren't told the length of the strut, we can't work out exactly how much it expands.
The key to this question is to realise that the coefficient of linear expansion is a ratio, like strain. So the strut wants to expand by 3.85 x 10^-3 metres per metre, or feet per foot if you prefer. It doesn't matter what the unit of measurement is, or the length of the strut, we have got the expansion ratio.
But the strut is rigidly fixed at both ends: it can't expand. So there must be a force at each end point retaining it at, or compressing it back to, its original length. So rather than expanding by 3.85 x 10-3 m/m, it is being compressed by the same ratio, which is strain.
Young’s Modulus E = Stress / Strain
Stress = E x Strain = 210 10^9 x 3.85 x 10^-3 = 808.5 x 10^6 Pascals or Newtons/m2
We are asked for the force induced in the strut due to the temperature change, so we need to turn this stress into the force on the strut.
The area of the strut is π d2 / 4= π 502 / 4 = 1,963.5 mm2 or 1.963 x 10^-3 m2 (1mm2=10^-6m2)
Newtons = Newtons/m2 x m2 = 808.5 x 10^6 x 1.963 x 10^-3 = 1,587 x 10^3 or 1.587 x 10^6 N
Answer: Force induced in the strut due to the temperature change = 1.587 x 10^6 N
