Finite Element Analysis Heat Transfer

Non uniform temperature distribution leads to thermal stresses. FEA heat transfer is a useful method to determine these temperatures, and the results can be fed directly into a stress analysis. A sample study shows explores this in a heat exchanger.

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Heat Exchanger FEA with Thermal Loads Sample

File: PVE-3520, Last Updated: June 4, 2013, By: LB

Why use FEA on Heat Exchangers?

FEA Analysis

ASME UHX VIII-1 and 2 rules cover the design of tubesheets, tubes and the shell next to the tubesheet. The rules cover multiple failure modes and provide considerable insight into the safety of the complete exchanger allowing design optimization. But the UHX rules are limited to designs with uniform hole patterns that cover the complete tubesheet. What if the hole pattern is not uniform, or in the case of this sample, the holes are not a uniform size?

Burst testing is an economical way to validate inexpensive products. However burst testing provides more conservative pressure rating than code calculations and it may be unreasonable to use to validate costly or large heat exchangers. Burst testing provides a failure mechanism and a pressure rating but does not provide deep insight into the safety of the whole object in areas that did not fail. Burst testing highlights the weakest area, it does not help optimize the whole design.

Finite Element Analysis (FEA) can be used to obtain the insight into safety as provided by the UHX code rules but for geometries not calculable by the UHX rules. The deflection plots provide an in depth understanding of how the exchanger deforms in response to the thermal and pressure loads. The stress plots show how well the exchanger can handle the loads and deflections; information is provided that allows design optimization. As an added bonus, the FEA provides stress levels for permissible cycle life evaluation.

Using FEA to Replace ASME Code Rules

The ASME code rules must be used if they are applicable. In this sample, the tubesheet has multiple tube sizes eliminating the possibility of using UHX. Standard code rules would still apply to all other areas of the exchanger – The scope of the FEA analysis would be tubesheet design, tubesheet to shell junction and the tube load calculations. Stress limits for the analysis are obtained from ASME VIII-1 the same as for the rest of the exchanger. The rules of ASME VIII-2 are used to determine how these limits are applied when interpreting the stress results.

The UHX rules account for three stresses in the design of an exchanger:

  • the tubesheet stress caused by pressure and thermal loading
  • the shell stress next to the tubesheet caused by tubesheet rotation
  • the tube compression or tension loading caused by tubesheet displacement and thermal expansion.

The stress limits for FEA are the same as used in UHX analysis. Per UHX rules these stresses are analyzed for the following seven load cases in fixed tube exchangers:

  • Tube side pressure only
  • Shell side pressure only
  • Tube + shell side pressure
  • Thermal loads only
  • Tube side pressure + thermal loads
  • Shell side pressure + thermal loads
  • Tube side pressure + shell side pressure + thermal loads

For a finite element analysis to successfully replace the UHX rules for a fixed tubesheet exchanger the three stresses need to be studied for the seven load cases.

Sample Study

The report available at the end of this article provides an in depth analysis of thermal and pressure stresses on an exchanger. Some illustrations from the report are shown here. The exchanger is symmetrical at both ends allowing only half to be modelled and studied.

The tubesheet and part of the shell are solid modelled. The rest of the shell, the head and tubes are shell modeled. The shell portions are less computer intensive to analyze, but provide less information especially at connections and joints. Here shell elements are only used in areas that will not be studied.

A mesh has been applied to both the solid and shell modelled sections. The mesh is reduced in size at locations of interest such as the tubesheet, the tubesheet to tube junction, and the adjacent shell to get more detailed results. The mesh in other areas does not significantly affect the results and has been left coarser.

All thermal and pressure loads are applied to the model. Shown below is the applied pressure load from load case 2 – shell side pressure only. In total seven different cases are run as shown in the report.

Close up of the mesh used in the sample study

Close up of the mesh used in the sample study.

Load case 2 - shell side pressure only

Load case 2 – shell side pressure only.

The sample FEA report walks through all seven load cases and checks all three stresses for each case. Each stress is compared to the ASME allowable stress to determine pass/fail for each load case. The shots below show the tube to tubesheet interaction. The tubesheet dishes under load creating a bending stress in the adjacent shell.

Deformation plot with the displacements magnified 100x.

Deformation plot with the displacements magnified 100x.

Deformation plot with the displacements magnified 100x. The rotation of the tubesheet with the adjacent shell bending it causes can be seen.

Stress plot with the displacements magnified 100x.

Summary

We have successfully used this FEA method to provide reports justifying heat exchanger designs reviewed by Authorized Inspectors and review engineers.FEA can be used to address ASME code rules where calculations cannot be applied. It is an excellent, and in some cases the only option to validate a design. It can be cost effective, reduce lead time and expedite registration.

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2013 Postscript

This report was first written in 2009. The combined shell and solid model was created to reduce the computing time especially important with the required seven runs. With the increasing speed of modern computers we usually do not simplify geometry to shells. The increased modelling effort is no longer justified in saved run times. We have also developed methods of replacing the tubes with springs.


Simulating Heat Exchanger Tubes with Springs

File: File:PVE-4473, Last Updated: Oct 4, 2010, By: CBM

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 heat exchanger model with tubes

The tubes have been removed and will be simulated with springs

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)

Where:

	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

Preload:

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.

Applied spring loads

The information required to define a spring connector is now available. A distributed option has been selected and is applied to all tube holes.


Thermal Analysis

File: File:PVE-4437, Last Updated: Aug 23 2010, By: BTV

Transient and Steady State Analyses

Transient and Steady State Analyses

Heat entering a part in a mold cavity

Thermal analyses are used to study thermal loadings and their resulting temperatures, heat transfer rates, displacements and stresses. These analyses are broken into two main types, steady state and transient. Steady state analysis will determine the energy balanced state at an infinite period in time without any detail on what happens while progressing to this point. Transient thermal analysis is able to analyze the heat flow through a body on a step by step basis allowing temperature effects to be observed over time.

Steady State Analysis

Steady state analysis is used to observe the effects of thermal loadings once the object in question has reached a constant, or steady state. This is useful to determine sustained temperatures, heat transfer rates, displacements and stresses. Steady state analysis is also useful to determine thermal loads and material properties to obtain a final desired result. As a steady state analysis only provides a final continuous result it only requires a single computation making it a very efficient solver.

Transient Analysis

Transient Temperature Graph

Temperature vs time for various locations in the cavity

A transient analysis is used to observe the effects of thermal loadings over time. It allows the user to view the changing temperature gradient through a component from initial though to a steady state condition. Transient thermal impacts are important to analyze as thermal loadings may result in peak stresses prior to reaching a steady state. It is also useful to answer questions such as how long will a component take to reach a desired temperature. As a transient analysis provides solutions for a defined number of time steps many computations are required resulting in a much more complex analysis.

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