Comparison of two FEA models for calculating stresses

in shell-and-tube heat exchanger

Weiya Jin*, Zengliang Gao, Lihua Liang, Jinsong Zheng, Kangda Zhang

Institute of Process Equipment and Control Engineering, Zhejiang University of Technology, Hangzhou 310032, China

Abstract

Two finite element analysis models of tubesheet of shell-and-tube heat exchangers are highlighted. Traditional theory of elastic foundation

model is used for tube to tubesheet interaction in model I. Pipe elements are used to represent actual interaction between tube and tubesheet in

model II. By the comparison of model I and model II results, it is confirmed that the distributions of the deformations and stress intensities for

both models have very little differences under complicated mechanical and thermal loads. Model I is suitable for FEA of shell-and-tube heat

exchangers, because model I is enough accurate and model II is more complicated and it takes more time and memory spaces of computer.

The axial forces at tube-to-tubesheet for two models are nearly the same and the axial forces generated by bending moments are very small.

The elastic foundation theory of the standards of design is suitable.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Heat exchangers; Tubesheet; Finite element analysis models

1. Introduction

Shell-and-tube heat exchangers are widely used in

process industry. Tubesheet is the main part of the

exchanger. Various researchers in many countries have

done a lot of work for the calculation and design of the

tubesheets. Several countries have drawn up the standards

for design of the shell-and-tube heat exchangers [1–6].

Because a shell-and-tube heat exchanger is a complex

elastic system, which consists of tubesheets, heads, shell,

flanges and bolts, it is difficult to calculate deformations

and stresses accurately. The different simplified methods

are used in those standards according to the different

structures. For fixed tubesheet heat exchangers the

tubesheet is considered as a circular plate on which shell

side pressure ps and tube side pressure pt are acted as

shown in Figs. 1 and 2 [7,8]. Mm;Mp;MR þ MoR and VR are

acted on the connection of shell and plate, flange and plate.

Mm is moments associated with flange pretighting force.

The moment Mp is from hermetization pressure. Moment

MR þ MoR and shear force VR are resulted from the

constraint between the circular flat plate and shell.

Because the perforated plate is weakened by tube holes,

it is simplified as a solid plate which has the same diameter

and thickness as the perforated plate with an effective

Young’s modulus and an effective Poisson’s ratio. The area

outside of the perforated plate is considered as the annular

solid plate which has normal Young’s modulus and

Poisson’s ratio.

When the exchanger is in service, the temperature stress

generated by the temperature difference between tubes and

the shell should be considered.

In addition, it is considered that the tube sheet is

enhanced by the tubes. For this enhancement, the tubesheet

is treated as an equivalent solid plate, which placed on

elastic foundation. When the ratio of the tube diameter d to

the tubesheet diameter Di is very small and there are many

tubes distributed in the tubesheet evenly, this simplification

is reasonable. The tubesheet stresses are declined and the

stress distribution of the tubesheet is changed because of

the enhancement of the tubes for the large heat exchangers.

The enhancement generated by the tube’s axial compressive

stiffness and bending stiffness can affect the deflection and

slope of the tubesheet. The enhancement is related to many

factors, so it is difficult to obtain the exact theoretical

solution. Simplifications are made and some coefficients are

used for easy calculations for the tubesheet design in many

standards of heat exchangers.

The operating conditions and structures of the exchan-

gers are variable for actual exchangers. The temperatures

of each tube passes are different from each other for

0308-0161/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijpvp.2004.02.003

International Journal of Pressure Vessels and Piping 81 (2004) 563–567

www.elsevier.com/locate/ijpvp

* Corresponding author.

E-mail address: ipece@zjut.edu.cn (W. Jin).

the exchangers with several tube passes. The thermal

stresses result from the temperature differences not only

between shell and tubes but also between tubes of different

passes. The stress distribution of tubesheet is more complex.

It is impossible to use the present heat exchanger standards

to solve this problem accurately.

In this paper, finite element analysis (FEA) method is

applied to calculate these complex stresses of the tubesheet

of fixed tubesheet exchangers. The modeling of FEA and the

calculation results are studied.

2. Modeling of FEA

Tubesheet consists of a elastic system with the shell,

flanges and tubes. It is simplified as an equivalent solid plate.

An effective Young’s modulus E* and effective Poisson’s

ratio n* are used for the perforated part of the tubesheet. The

normal Young’s modulus E and Poisson’s ratio n are used for

the un-perforated part of the tubesheet. Tube side pressure is

applied on tube side pressurized area. Shell side pressure is

applied on shell and shell side pressurized area. Gasket force

is applied on tube side sealing area of the tubesheet. Bolting

force is applied on the outside ring of shell side of the tube

sheet. The temperature distributions of the shell, tubes and

tubesheet are considered.

Two methods are used for the modeling of FEA to

consider the tube enhancement to the tubesheet. Half of the

exchanger is considered for both models.

Model I (shown in Fig. 3): Tubesheet is modeled as an

equivalent solid plate on elastic foundation. Axial forces are

considered at tube-to-tubesheet joints (shown in Fig. 4). Bar

elements (spring effect) for tubes coupled with 3D solid

element for tubesheet are used. Nodes generated on

Fig. 1. Shell side pressure and flange moment acted on tubesheet.

Fig. 2. Tube side pressure and flange moment acted on tubesheet.

Fig. 3. Model I.

Fig. 4. Model I loads acted on bar element-to-tubesheet joints.

Fig. 5. Model II.

Fig. 6. Model II loads acted on pipe element-to-tubesheet joints.

W. Jin et al. / International Journal of Pressure Vessels and Piping 81 (2004) 563–567564

the lower surface of the tubesheet are more than the number

of the holes on the tubesheet. On every such node, a bar is

coupled to the 3D solid modeled tubesheet. There are a lot

of bars in the model, which represent tube action to

the tubesheet. The real constant (cross-section area) of the

bar elements is the equivalent area of the actual tube cross-

section area. An effective shell side pressure is acted on

tubesheet shell side, because there is no pressure on the part

of tubesheet tubes connected.

Model II (shown in Fig. 5): Model II used pipe elements

for tubes coupled with 3D solid element for the tubesheet.

This is a one to one tube to tube sheet model which

represents actual interaction between tubes and the

tubesheet. Number of the pipe elements is equal to the

tube number. The displacement and slope of pipe element

nodes must be the same as those of the tubesheet at the tube

to tubesheet joints. Axial forces and bending moments are

considered at tube-to-tubesheet joints (shown in Fig. 6). The

real outside diameter and thickness of the tubes are input for

the real constant of the pipe elements.

3. Example

Models I and II are used in a large two tube pass

shell-and-tube heat exchanger. The exchanger’s major

Table 3

Two operating conditions used for the analysis

Case 1 Case 2

Pressure (MPa) ps 2.01 1.86

pt 0.34 0.18

Temperature (8C) usq 69.95 88

u Pass1 35.35 93

t Pass2 35.35 44

Tubesheet surface

temperature (8C)

Shell side uTs 39.59 52.68

Tube side uTt 31.69 37.33

Diagram

Table 2

Major components’ material properties

Tube sheet Shell Tube

Material code SA226 SA515 SB338

Coefficient

of thermal

expansion

a (1/8C)

21.1 8C 11.52 £ 1026 11.52 £ 1026 8.28 £ 1026

93.3 8C 12.42 £ 1026 12.42 £ 1026 8.46 £ 1026

Poisson’s

ratio n

0.3 n* ¼ 0:4094 0.3 0.3

Young’s

modulus E

21.1 8C 2.014 £ 105

E* ¼ 4:06 £ 104

2.01 £ 105 1.07 £ 105

93.3 8C 1.97 £ 105

E* ¼ 3:96 £ 104

1.97 £ 105 1.03 £ 105

E* ; n* are effective Young’s modulus and effective Poisson’s ratio for

perforated plate.

Table 1

Exchanger major dimension and symbol

Symbol Description Unit mm Value

Di Shell inside diameter mm 1452

ts Shell thickness mm 23

d0 tube outside diameter mm 19.05

tt tube thickness mm 1.2

L Length of tubes between inner

surfaces of tube sheets

mm 7274

T Tube sheet thickness mm 100

N Total tube number mm 2278

P Tube pitch with triangle arrangement mm 25

DT Tube sheet outside diameter mm 1676

Db Circle diameter of bolt arrangement mm 1622

Fig. 7. Model I tube sheet case 1 actual stress intensity diagram.

Fig. 8. Model II tube sheet case1 actual stress intensity diagram.

W. Jin et al. / International Journal of Pressure Vessels and Piping 81 (2004) 563–567 565

dimensions and symbols are listed in Table 1. The properties

of major component’s materials are shown in Table 2. Two

operating conditions for FEA analysis are shown in Table 3.

The temperatures of two tube passes are the same in case 1

but different in case 2.

Elastic foundation theory are used for model I. We

meshed the perforated area as per actual drawing very

finely and generated 5120 nodes. On every node, a bar

was coupled to the 3D solid modeled tubesheet.

There were total 5120 bar representing 2278 tubes

connected to the tubesheet. The cross area of each bar

element was calculated based on equivalent tube area as

following:

A ¼2278

p

4½19:052 2 ð19:05 2 2 £ 1:2Þ2�

5120¼ 29:925 mm2

the effective pressure is calculated below:

p0 ¼ps

p4

14502 2 2278 p4

19:052� �

p4

14502¼ 0:6068ps

Total 87,070 elements and 72,553 nodes are generated in

the model I.

In model II, 2278 pipe elements represent 2278 tubes in

the exchanger. These pipe elements are coupled 3D solid

Fig. 9. Model I tube sheet case2 actual stress intensity diagram.

Fig. 10. Model II tube sheet case2 actual stress intensity diagram.

Fig. 11. Model I tube case 1 axial stress ðSZÞ diagram.

Fig. 12. Model II tube case 1 axial stress ðSZÞ diagram.

Fig. 13. Model I tube case 2 axial stress ðSZÞ diagram.

W. Jin et al. / International Journal of Pressure Vessels and Piping 81 (2004) 563–567566

element for the tubesheet at actual location as per drawing.

Total 2,94,864 elements and 78,151 nodes are generated.

The elements of model II are three times more than those of

model I. The modeling and calculation of model II are more

complex and it takes more time and memory spaces of

computer.

4. Calculation results

The stress distributions in the tubesheet and the tubes

for Cases 1 and 2 calculations are shown in Figs. 7–14.

The maximum and minimum axial stresses of the tubes for

model I at tube-to-tubesheet are listed in Table 4. The

maximum and minimum axial stresses of the tubes at

different angle of model II at different location of tube-to-

tubesheet joints are listed in Table 5.

5. Conclusion

The pipe elements in model II can not only take axial

force but also bending moment at tube to tube sheet joint

Therefore, Model II is nearer to actual condition and results

shall be more actural.

The axial forces at tube-to-tubesheet from two models

are nearly the same. The axial forces generated by the

bending moments are very small. So the elastic foundation

theory of the standards of design is suitable.

The maximum stress intensities and distributions of the

stress intensities for these two models have very little

differences under complicated mechanical and thermal

loads So the FEA analyses for the tubesheet with these

two models are reliable.

By the comparison of model I and model II results,

Model I is suitable for FEA of shell-and-tube heat

exchangers, because model I is enough accurate and

model II is more complicated and it takes more time and

memory spaces of computer.

References

[1] TEMA. Standard of Tubular Exchanger Manufactures Association;

1999.

[2] BS5500. British Standard Specification for Fusion-welded Pressure

Vessels; 1982.

[3] ASME. ASME Boiler and Pressure Vessel Code?Section VIII Division

2, New York, American Society of Mechanical Engineers; 1998.

[4] JIS B 8249-1996. Shell and Tube Heaat Exchangers; 1996.

[5] Germany AD Pressure Vessel Code. B5 Flat end, Flat plate and its

support; 1982

[6] China GB151-1999. Shell and Tube Heat Exchanger; 1999.

[7] Ming-de Xue. The Basis of Tubesheet Design Rules in the

Chinese Pressure Vessel Code. Int J Pressure Vessel Piping 1990;

186:13–20.

[8] Ke-zhi Huang. An advised calculation method for stress of fixed

tubesheet heat exchanger. Mechnical Engineer 1980;2:1–23.

Fig. 14. Model II tube case 2 axial stress ðSZÞ diagram.

Table 5

Comparison of axial stresses of the tubes at tube-to-tubesheet joints

Case 1 Case 2

Model I Max. axial stress (MPa) 34.219 49.081

Model II (for the

tube which has the

max. axial stress at

0 degree)

Axial stress (MPa) 08 36.502 49.359

908 35.065 47.695

1808 35.266 48.051

2708 36.703 49.714

Bending stress (MPa) 1.0259 1.2028

Table 4

Comparison of maximum stress intensities of tubesheet

Maximum stress intensities of the

tubesheet (MPa)

Model I Model II

Case 1 105.83 115.68

Case 2 166.0 170.3

W. Jin et al. / International Journal of Pressure Vessels and Piping 81 (2004) 563–567 567