**6. Analysis of the results**

182 Heat Exchangers – Basics Design Applications

Fig. 20. Total strains in pipes subject to the maximal strain; magnified deformation of pipes

Fig. 21. Strain path in the pipe along its length [mm] - an example (the pipe is mounted in

the position closest to the geometrical axis of the heat exchanger).

[mm] – case 2.

A shell-solid model of the heat exchanger - a water heater - was subjected to the numerical calculations. The author's task was to conduct a possibly full analysis of stresses and strains occurring in structural elements of the heat exchanger, especially under extreme conditions that appear during, e.g., emergency operation. The calculations were carried out for altered thickness of the plate (in the first case by 15%, in the second one by 50%). This change was dictated by economic reasons. On the basis of the calculation results and the analysis of distributions of stresses and strains, one can conclude that the perforated bottom (the perforated plate) is not the place where stresses concentrate. In the first case, the reduced stresses in the bottom were equal to approx. 30 MPa (Fig.10), and in the second case they increased to approx. 50 MPa (Fig.18). The analysis of local zones of stresses in the whole heat exchanger has allowed us to find out that the maximal stresses in both cases appeared in the connections between the collar with the jacket and they were equal to, respectively, 198MPa, and in the second case they increased by approx. 10%. As opposed to the values of stresses, the total strains differ considerably. The maximal strains occurring in the heat exchanger were of the magnitude of 0.07 mm in the first case (Fig.5), whereas in the second case they increased up to 4.45 mm (Fig.13). Some exemplary maps of strains in the selected pipes mounted, correspondingly:


Numerical Analysis of the Structural Stability of Heat Exchangers – The FEM Approach 185



The analysis of the obtained results has allowed us to observe the places that are most prone to a stability loss, where a concentration of stresses occurs and to determine possible alternations in the thickness of perforated plates (which would be justified economically).

Achtelik H., Gasiak G., Grzelak J. (2005). *Strain and load carrying capacity of perforated plates* 

Chudzik A.A.( 2002). *Analysis of the state of stress in perforated plates of heat exchangers, including effects of elastic and plastic zones*, PhD Dissertation [in Polish] Chudzik A.A. (2008). *Preliminary analysis of inelastic buckling of the heat exchangers,* Journal Of

Gellin S. (1979). *Effect of an axisymmetric imperfection on the plastic buckling of an axially* 

Gerarad G. (1956). *Compressive and torsional buckling of thin-wall cylinders in the yield region,* 

Grigoluk E.I. (1957). *O vypucivanii tonkich oblocek za predelom uprugosti,* Izv. AN SSSR, Otd.

Grigoluk E.I. (1957). *Cisto plasticeskaja proteja ustojcivosti tonkich oblocek,* Prikl. Matem. i mech.,

Harding J.E., (1978). *The elastic-plastic analysis of imperfect cylinders,* Proc. Ins. Civ. Eng, Part 2 Hobler T. (1986), *Heat Transfer and Heat Exchangers,* Wydawnictwo Naukowo-Techniczne*,*

Horak J., Lord G.J., Peletier M.A. (2005), *Cylinder buckling: the mountain pass as an organizing* 

Hutchinson J.W. (1972). *On the postbuckling behaviour of imperfection-sensitive structures in the* 

Hutchinson J.W. (1973*). Imperfection sensitivity in the plastic range*, J. Mech. Phys.Solids, Vol 21 Iljussin A.A. (1944). *Ustojcivost plastinok I oblocek za predelom uprugosti*, Prikl. matem. i mech.,

Lee L.H.N. (1961). *Inelastic buckling of cylindrical shells under axial compression and internal* 

Lee L.H.N. (1962). *Inelastic buckling of initially imperfect cylindrical shells subject to axial* 

Lee L.H.N. (1962). *Inelastic buckling of cylindrical shells under axial compression and internal* 

Murphy L. M., Lee L.H.N. (1971). *Inelastic buckling process of axially compressed cylindrical* 

*pressure,* w: Proc.4thU.S. Nat. Congr. Appl. Mech., Berkley, Calif.

*shells subject to edge constrains,* Int. J. Solids Struct., 7,8

*under axial – symmetrical load*, Oficyna Wydawnicza, Opole, Poland [in Polish]

collar,

exchanger.

**8. References** 

User'

s Guide ANSYS 12

NACA Techn. Note, No.3276

*center*, *arXiv:math. AP/0507 263*, 1

*plastic range,* J Appl. Mech., Vol.39

*pressure,* Developments in Mechanics, 1

*compression,* J. Aerospace Sci., Vol.29, No.1

Tech. N., 10

Warsaw [in Polish]

21, No 6

8, No. 5

Bijlaard P.P. (1950). *On the plastic buckling of plates,* J. Aeron. Sci., Vol.17

Theoretical and Applied Mechanics, Warsaw [in Polish] *Concession documentation of the Py-100-020 decarbonized water heater* [in Polish]

*compressed cylindrical shell*, J. Appl. Mech., Vol. 46

The analysis of the calculation results has shown that the pipes mounted in the outer part are subject to higher deformations that the pipes mounted inside the central part of the plate that are subject more to elongation than to deflection.

In the general case, a stability loss of the elastic structure is a nonlinear problem that can be investigated with the iterative or incremental analysis of large displacements. This issue was dealt with by Arygyris (1977). With respect to the critical value of load, an influence of geometrical nonlinearity remains insignificant in numerous cases. When it is exceeded, a radical change in the configuration that corresponds to the equilibrium state occurs. If we are interested in the value Pkr and the postcritical state, it is enough to apply an elastic analysis. The task is not so easy in the heat exchanger under investigation. The structure of the device is complex and loaded with pressures that follow from the operation, therefore the FEM calculations were proposed. The investigations of the deformation nature of the heat exchanger and the analysis of strains and displacements are very important tasks which would allow designers to evaluate the correctness of the structure in order to avoid considerable differences in displacements, e.g., through a change in the connection of the perforated bottom with the jacket or an application of a pipe compensation, etc.

The conducted here analysis of strains in individual structural elements of the heat exchanger allows researchers to foresee the places of stress concentration. Thus, respective changes can be introduced in order to decrease a number of failures of the structure.
