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

Agnieszka A. Chudzik

*Technical University of Lodz/Department of Machine Dynamics Poland* 

#### **1. Introduction**

164 Heat Exchangers – Basics Design Applications

turbopump must be done for practical application of the graphite coating on the expander

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Research and Development, 660-661, NTC Inc. ISBN4-900830-76-3, Tokyo, Japan (

*Ai* Area indicated in the results of Gas Chromatograph for species *i mcarbon* Mass flow rate of carbon contained in supplied test gas [ kg/sec ]

*m*sup *ply* Mass flow rate of supplied test gas [ kg/sec ] *Mi* Molecular weight of species *i* [ kg/kmol ]

*R*ˆ Universal gas constant ( = 8.3143 [J/(mol K)] )

Cycle Rocket Engine, *AIAA Paper-1998-3674* 

Vol.57, No.670, pp.445-452. ( in Japanese )

Smith, G. P. et. al. : http:*//*www.me.berkeley.edu*/*gri\_mech*/*

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*Q* Volumetric flow rate of gas [ m3/sec ]

*<sup>i</sup>* Heat transfer rate of species *i* [ J/(m K)]

*AIAA Paper 2004-4210.*

in Japanese )

cycle LNG rocket engine.

**5. Nomenclature** 

*P* Pressure [ Pa ]

*t* Time [ sec ] *T* Temperature [ K ]

**6. References** 

A demand for improved computational methods of complex systems used in modern structures has been followed by development of theory and analysis in the field of stability of shell structures. Nonlinear problems, in which, for instance, shape imperfections, complex loads, nonelastic properties of the material used in the structure are accounted for, are of deep interest. Thanks to an advance in computer technology and numerical methods, a possibility to conduct more precise analysis which corresponds better to actual structures of mathematical models arises. In the linear and nonlinear analysis of structures, stability occupies a special place. To test the load carrying capacity of the structure, the phenomena that occur during a stability loss and after it should be recognized. An application of thinwall elements results in advantages such as light weight, a possibility to carry high loads, thermo-insulating properties, etc. Here, analysis and recognition of transfer phases since an appearance of plastic strains up to a complete reduction of the load carrying capacity is essential. A demand for such complex analysis that includes stability and leads to more actual evaluation of the structure safety has been observed in many disciplines of technology, e.g., in designing of ships, airplanes, pressure chemical apparatus, in modern construction industry and in power and heat generation. Heat exchangers that are widely applied in, e.g., power and heat generation, operate under very high temperatures. The principal elements of these devices are perforated walls – perforated plates in which heating cartridge pipes are fixed. The number of pipes, i.e., of holes in the plate, is very high and these holes are separated from one another by a thin bridge. The issue of differences in temperatures in individual parts of the heat exchanger and in various media that flow in the device is a source of considerable design difficulties. Heat exchangers usually operate under pressure or in vacuum. Independently of the fact that not only the knowledge of material strength properties is needed for computations of heat exchangers, there are legal regulations that standardize the calculations of the devices operating under pressure. They define the way the basic parts are calculated, providing thus hints concerning the structure of the devices under control. A decrease in the safety factor due to stability, i.e., a rapprochement to the real state of stress up to the critical one, is the way the modern engineering structures can be characterized by. Therefore, the calculations of stability, stiffness of thin-walled structural elements are becoming more and more important nowadays in designing and performance of many devices. A heat exchange is a common phenomenon in technology and nature – it occurs when there are differences in

Numerical Analysis of the Structural Stability of Heat Exchangers – The FEM Approach 167

Numerous studies quote results of the analysis of mathematical models. These studies include, however, significant simplifications as regards actual operating conditions, shapes of the calculated elements, the manufacturing technology of connections, and the behavior of materials under operating conditions. Analytical methods consist in a separation of the fragment from the perforated plate among the surrounding heating pipes, assuming the boundary conditions for the operation of the cut circular plate and applying the pressure. The elastic behavior of the background (pipes) is usually not accounted for, the plate edge is treated as fixed (or other boundary conditions), which is very far from the reality. These engineering simplifications of assumptions result in considerable differences between values of actual stresses and those obtained experimentally. Calculations made with the FEM can be the way the majority of the above-mentioned factors are accounted for Chudzik

Differences in the temperatures heat exchangers operate in can result in a remarkable difference in heat displacements. These displacements can give rise to high stresses and strains in parts of heat exchanger such as pipes, jackets and bottoms. An increase in stresses and strains is especially dangerous in case of a failure. It can lead to loads that can result in a stability loss of the perforated bottom together with heating cartridge pipes. A stability loss of the pipe does not have to be followed by damage, but the effects it will cause in the structure depend on the kind and the nature of buckling. A deflection or a shortening of the pipe axis (global stability) that can result in exceeding inconsiderably the critical force may lead to a rapid increase in stresses. When the pipe in the complex structure is buckled, it looses its stability, which can lead to a stability loss of the whole structure. In thin-walled pipes, a new phenomenon has occurred, i.e., local stability. As opposed to global stability, it consists in the fact that the cross-section of the pipe deforms and the rod axis remains straight. Critical stresses under the local stability loss are calculated on the basis of the theory of plates and shells. Analytical calculations of the above-mentioned phenomena give rise to some difficulties. The FEM enables an accurate reproduction of the structure, as well as of the manufacturing technology and the assembly of the heat exchanger. Thus, the calculations of the heat exchanger as regards its stability loss can be carried out. A one-cycle heat exchanger working as a water heater is chosen for our analysis. The perforated bottom of these heat exchangers is an expensive and difficult to manufacture element and, therefore, it is economically justified to decrease its mass. That is why the FEM calculations aimed at a more accurate analysis whose results could affect possible alternations in the heat exchanger design, e.g., through a decrease in the perforated bottom thickness, have been conducted. The calculations are a continuation of the investigations carried out formerly

First studies devoted to stability loss issues were already published more than seven decades ago Zielnica (2001) and they dealt with elastic-plastic shells. An intensive development has been observed since 1955, when Gerard (1956), Lee (1961,1962), Grigoluk (1957) and other researchers published their works. For instance, Sewell (1972), Hutchinson (1972,1973) proposed the methods and the results of solutions to problems of elastic-plastic

(2002,2008).

**2.1 Aim of the investigations** 

(Chudzik, 2002; 2008).

**3. Stability – a literature survey** 

temperatures. A trial to determine the conditions under which a stability loss will occur in the elements of the heat exchanger that are most exposed to this threat is presented in this paper.
