Heat exchangers are often very large models. In order to simplify them for FEA we remove all of the tubes and simulate using springs. This will significantly reduce the number of elements and allow the model to mesh and solve in a more reasonable time frame.
The heat exchanger model with tubes
The tubes have been removed and will be simulated with springs
In order to use the spring connectors we have to determine the applicable spring rate of the tubes. There is a direct relationship between Hooke's law for springs and the material's modulus of elasticity.
Hooke's law for springs: F = k * d
Axial displacement: d = F * L / (A * E)
F = Axial Force [lb] k = Spring Rate [lb/in] d = Displacement [in] L = Tube Length [in] A = Cross Sectional Area [sq. in] E = Modulus of Elasticity [psi]
Combining these equations we get: d = k * d * L / (A * E)
k = A * E / L
Solidworks Simulation has a distributed spring option which allows a distributed spring rate over a selected area. The equation now becomes:
k = A * E / (L * A)
Therefore: k = E / L
In the case of fixed tubesheets, the differential thermal growth between the tubes and the shell generates a load acting on the tubesheets. The spring connectors must be preloaded to account for this force. We know that the equation for thermal expansion is:
d = ∝ * L * ΔT
where: d = Displacement [in] ∝ = Material's thermal expansion coefficient [ in/in/°F] L = Original tube length [in] ΔT = Change in temperature [°F] from ambient to the operating temperature
The difference in expansion between the shell side and the tube side is:
d = ∝ tube side * L * (Ttube side - 70) - ∝shell side * L * (Tshell side - 70)
This value is substituted into the spring rate equation (F = k * d) to determine the corresponding preload.