**2. Formulation of the problem**

Heat exchangers have versatile industrial applications. They are widely used in the food and chemical industry and in energy and power generation, etc. The heat exchanger very often stops to be used as a separate device, designed solely for the heat exchange and its role is connected with other tasks as well. In such a case, the heat exchanger is a part of a facility used for some technological processes. A versatility of applications of heat exchangers is followed by a high diversity in their designs. Various conditions of the process and properties of the substances heated up or cooled, condensed or evaporated result in a necessity to select the material strength properties and the design itself in a proper way. The analysis of an effect of the perforated plate on structural elements of the heat exchanger is an important issue in designing and operation of these devices Hobler (1986), Horak (2005). Working temperatures of the media flowing in the heat exchanger have a decisive impact on the design. In the initial phase of designing, the designer is interested in values of applied temperatures and differences in temperatures between individual elements of the device. The magnitude of the applied temperatures decides about the selection of materials used in the designed structure (due to their strength properties). With an increase in the temperature, "volume" of the materials applied for elements of the heat exchanger increases. The dimensions of a jacket, perforated plates, heating cartridge pipes, bottoms, as well as any other part of the heat exchanger are subject to alternations. This phenomenon is referred to as heat dilatation. Differences in temperatures of the media working in the heat exchanger cause various heat dilatations, for instance, pipes elongate in a different way than the jacket does, a diameter of the perforated bottom changes differently than the outer wall, etc. Heat dilatations are the most cumbersome problem for the designer of heat exchangers as they can cause considerable stresses in the material that can lead to plastic strains or even a failure of the device. The jacket attains the temperature close to the temperature of the flowing medium with which it comes into contact, pipes have a different temperature outside and inside. A difference in these temperatures, as well as in pressure of the flowing media that results from the task the heat exchager is to fulfill can be considerable. In strength calculations, the designer considers the creep strength limit or the creep limit. They both are time functions. Design difficulties very often arise when a difference in working temperatures in various parts of the heat exchanger is accounted for. Significant changes in strength characteristics of the material occur along with alternations in its temperature, namely: an increase in temperature makes the material more plastic, the tensile strength *Rm* and the immediate yield point *Re* become lower. Strength tests have shown that at higher temperatures, the duration of stress affects strongly the strength and the yield point. Then, the plastic strain of the element depends on: stress, time and temperature. The designer decides thus what material to choose for the elements of the structure to postpone possible plastic strains or a structural failure beyond the predicted operating life. The issues related to computations of circular-symmetrical disks are very important problems in the theory of elasticity and plasticity. Many researchers have investigated also the problem of assembly of pipes in holes of the perforated bottom Ryś (2003). The problem of strength and tightness of such a connection gives rise to serious difficulties to heat exchanger manufacturers. 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 (2002,2008).

#### **2.1 Aim of the investigations**

166 Heat Exchangers – Basics Design Applications

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

Heat exchangers have versatile industrial applications. They are widely used in the food and chemical industry and in energy and power generation, etc. The heat exchanger very often stops to be used as a separate device, designed solely for the heat exchange and its role is connected with other tasks as well. In such a case, the heat exchanger is a part of a facility used for some technological processes. A versatility of applications of heat exchangers is followed by a high diversity in their designs. Various conditions of the process and properties of the substances heated up or cooled, condensed or evaporated result in a necessity to select the material strength properties and the design itself in a proper way. The analysis of an effect of the perforated plate on structural elements of the heat exchanger is an important issue in designing and operation of these devices Hobler (1986), Horak (2005). Working temperatures of the media flowing in the heat exchanger have a decisive impact on the design. In the initial phase of designing, the designer is interested in values of applied temperatures and differences in temperatures between individual elements of the device. The magnitude of the applied temperatures decides about the selection of materials used in the designed structure (due to their strength properties). With an increase in the temperature, "volume" of the materials applied for elements of the heat exchanger increases. The dimensions of a jacket, perforated plates, heating cartridge pipes, bottoms, as well as any other part of the heat exchanger are subject to alternations. This phenomenon is referred to as heat dilatation. Differences in temperatures of the media working in the heat exchanger cause various heat dilatations, for instance, pipes elongate in a different way than the jacket does, a diameter of the perforated bottom changes differently than the outer wall, etc. Heat dilatations are the most cumbersome problem for the designer of heat exchangers as they can cause considerable stresses in the material that can lead to plastic strains or even a failure of the device. The jacket attains the temperature close to the temperature of the flowing medium with which it comes into contact, pipes have a different temperature outside and inside. A difference in these temperatures, as well as in pressure of the flowing media that results from the task the heat exchager is to fulfill can be considerable. In strength calculations, the designer considers the creep strength limit or the creep limit. They both are time functions. Design difficulties very often arise when a difference in working temperatures in various parts of the heat exchanger is accounted for. Significant changes in strength characteristics of the material occur along with alternations in its temperature, namely: an increase in temperature makes the material more plastic, the tensile strength *Rm* and the immediate yield point *Re* become lower. Strength tests have shown that at higher temperatures, the duration of stress affects strongly the strength and the yield point. Then, the plastic strain of the element depends on: stress, time and temperature. The designer decides thus what material to choose for the elements of the structure to postpone possible plastic strains or a structural failure beyond the predicted operating life. The issues related to computations of circular-symmetrical disks are very important problems in the theory of elasticity and plasticity. Many researchers have investigated also the problem of assembly of pipes in holes of the perforated bottom Ryś (2003). The problem of strength and tightness of such a connection gives rise to serious difficulties to heat exchanger manufacturers.

paper.

**2. Formulation of the problem** 

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 (Chudzik, 2002; 2008).

#### **3. Stability – a literature survey**

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

Numerical Analysis of the Structural Stability of Heat Exchangers – The FEM Approach 169

The first three phases are of fundamental importance to designers. In order to determine boundaries between these phases, the following should be known: the critical load, the load at which first plastic deformations occur and the limit load understood as the maximum

Phase I, the prebuckling phase, occurs when the behavior of the element is described with the classical linear theory. A solution (analytical or numerical) to this problem does not give rise to serious difficulties. Phase II is characterized by high deflections. Due to this fact, the problem becomes geometrically nonlinear whereas physically it is still linear, and its solution allows one to analyze the behavior of the structure until the limit of proportionality is attained. Different modes of buckling can occur in this phase: local and global. An appearance and development of plasticized regions are characteristic of phase III. An analysis and knowledge of the structure in this phase is very useful and needed in the designing process as the load of the structure attains the maximal value in this phase. The determination of the value of this load, referred to as the limit load carrying capacity, requires a physical and geometrical solution to the nonlinear problem of stability. A full analysis in the elastic-plastic range consists in taking into account numerous phenomena that occur in real structures (initial imperfections, interactions of different modes of buckling, the "shear lag" phenomenon, etc.). Despite such a complex issue, many problems related to, e.g., plates subject to simple loads, rectangular plates under uniform compression, have already been solved and the results are supported by the corresponding experimental investigations. However, there is a large group of structures subject to complex loads, whose structure is more sophisticated and for which the determination of the limit load carrying capacity via solution to the stability problem under high deflections in the elastic-plastic range is very complicated even if up-to-date computational numerical methods are employed. Therefore, modern engineering structures during the designing process are based on theories of elasticity on the assumption of high deflections which allow for determination of the load at which plasticization of the structure begins. This state is referred to as the limit state and it is treated as a sort of failure criterion that allows one to design a safe structure although sometimes not an economically favorable one Królak at al

The calculations were made for a one-cycle heat exchanger, Fig. 1, whose model has been developed on the basis of the technical documentation of the Py-100-020 decarbonized water

The elements that were taken into account in the 3D model of the heat exchanger are as follows: perforated plates (1), bottoms (2), a jacket (3), heating cartridge pipes (8), heating cartridge gaskets (10) shown in Fig.1. Perforated plates are fixed to bottoms with screw fasteners (9). Connector pipes (6) and (7) supply and take off the steam. Detail B shows dimensions of the hole in the perforated bottom before rolling out and a view of the pipeperforated bottom connection after rolling out. The perforated plates have a hexagonal system of holes that make perforations. The materials used in the structure are listed in

load after which the damage of the structure will start.

(1990).

heater.

Table 1.

**4. The model under investigation** 

stability of structures made of pipes, plates and shells, Nowak and Życzkowski (1963) are responsible for a survey of shell structures. Iiiuszyn (1944) developed the fundamentals of the stability theory of thin-walled shells made of an incompressible material with arbitrary characteristics of reinforcement beyond the elastic limit. Zielnica (1969), Bijlaard (1950), Murphy and Lee (1971), Gellin (1979), Hardig (1978), Sobel and Newmann (1982) – these are the researchers who blazed a trail in the analysis of shell stability to follow as they dealt with:


In 1947 Shanley presented a new concept of the stability loss in rods, referred to as buckling under increasing load. The stability investigations of the jointly supported rod under compression showed that the rod was subject to buckling and the axial force compressing the rod grew simultaneously after it reached the critical load. Shanley's concept of buckling under critical load enabled to simplify the stability equations as in that case the limit that separates the active process zone from the deloading zone does not have to be established.

In the uniaxial state of stresses (rod systems), the stability analysis can be based on the actual material characteristics obtained from, e.g., the uniaxial elongation test.

In the biaxial state of stresses (shells, plates), the knowledge of material characteristics is not enough as in the case discussed above. To determine the dependence between components of the tensor of the stress and strain state beyond the elastic limit, one should go into detail in the theory of plasticity. Nowadays there are numerous publications devoted to this issue. The majority of them include solutions to comparatively simple models exposed to a stability loss. If the literature survey were to be further discussed here, it could take a form a separate book. The author's task was, however, to conduct a numerical analysis of the more complex device such as the heat exchanger is. A demand for such calculations result from a lack of studies presenting the calculation methods, numerical computations, results of the calculations of, e.g., more complex experimental structures as regards their stability loss. Experimental investigations to compare the results would be very expensive. The results of the experiment will be followed by the need to lead the device to a failure. A mathematical analysis of such a device also gives rise to many difficulties to researchers due to complexity of the problem and the device.

#### **3.1 Stability of load carrying elements of the structure**

The behavior of load carrying elements of the structure under increasing loads can be divided into a few phases of their operation, namely: linear in the precritical state, postbuckling elastic under considerable deflections, postcritical elastic-plastic and a failure.

stability of structures made of pipes, plates and shells, Nowak and Życzkowski (1963) are responsible for a survey of shell structures. Iiiuszyn (1944) developed the fundamentals of the stability theory of thin-walled shells made of an incompressible material with arbitrary characteristics of reinforcement beyond the elastic limit. Zielnica (1969), Bijlaard (1950), Murphy and Lee (1971), Gellin (1979), Hardig (1978), Sobel and Newmann (1982) – these are the researchers who blazed a trail in the analysis of shell stability to follow as they dealt

determination of bifurcation loads (eigenvalues) in the linear range under the

 determination of bifurcation points on nonlinear paths or critical loads under geometrical nonlinearities in the precritical state, accounting for bending effects before the stability loss and determination of nonlinear equilibrium paths including

application of the FEM method to determine nonlinear paths of the subcritical

In 1947 Shanley presented a new concept of the stability loss in rods, referred to as buckling under increasing load. The stability investigations of the jointly supported rod under compression showed that the rod was subject to buckling and the axial force compressing the rod grew simultaneously after it reached the critical load. Shanley's concept of buckling under critical load enabled to simplify the stability equations as in that case the limit that separates the active process zone from the deloading zone does not

In the uniaxial state of stresses (rod systems), the stability analysis can be based on the

In the biaxial state of stresses (shells, plates), the knowledge of material characteristics is not enough as in the case discussed above. To determine the dependence between components of the tensor of the stress and strain state beyond the elastic limit, one should go into detail in the theory of plasticity. Nowadays there are numerous publications devoted to this issue. The majority of them include solutions to comparatively simple models exposed to a stability loss. If the literature survey were to be further discussed here, it could take a form a separate book. The author's task was, however, to conduct a numerical analysis of the more complex device such as the heat exchanger is. A demand for such calculations result from a lack of studies presenting the calculation methods, numerical computations, results of the calculations of, e.g., more complex experimental structures as regards their stability loss. Experimental investigations to compare the results would be very expensive. The results of the experiment will be followed by the need to lead the device to a failure. A mathematical analysis of such a device also gives rise to many difficulties to researchers due to complexity

The behavior of load carrying elements of the structure under increasing loads can be divided into a few phases of their operation, namely: linear in the precritical state, postbuckling elastic under considerable deflections, postcritical elastic-plastic and a

actual material characteristics obtained from, e.g., the uniaxial elongation test.

with:

equilibrium.

have to be established.

of the problem and the device.

failure.

**3.1 Stability of load carrying elements of the structure** 

membrane precritical state of stresses,

imperfections (shape imperfections),

The first three phases are of fundamental importance to designers. In order to determine boundaries between these phases, the following should be known: the critical load, the load at which first plastic deformations occur and the limit load understood as the maximum load after which the damage of the structure will start.

Phase I, the prebuckling phase, occurs when the behavior of the element is described with the classical linear theory. A solution (analytical or numerical) to this problem does not give rise to serious difficulties. Phase II is characterized by high deflections. Due to this fact, the problem becomes geometrically nonlinear whereas physically it is still linear, and its solution allows one to analyze the behavior of the structure until the limit of proportionality is attained. Different modes of buckling can occur in this phase: local and global. An appearance and development of plasticized regions are characteristic of phase III. An analysis and knowledge of the structure in this phase is very useful and needed in the designing process as the load of the structure attains the maximal value in this phase. The determination of the value of this load, referred to as the limit load carrying capacity, requires a physical and geometrical solution to the nonlinear problem of stability. A full analysis in the elastic-plastic range consists in taking into account numerous phenomena that occur in real structures (initial imperfections, interactions of different modes of buckling, the "shear lag" phenomenon, etc.). Despite such a complex issue, many problems related to, e.g., plates subject to simple loads, rectangular plates under uniform compression, have already been solved and the results are supported by the corresponding experimental investigations. However, there is a large group of structures subject to complex loads, whose structure is more sophisticated and for which the determination of the limit load carrying capacity via solution to the stability problem under high deflections in the elastic-plastic range is very complicated even if up-to-date computational numerical methods are employed. Therefore, modern engineering structures during the designing process are based on theories of elasticity on the assumption of high deflections which allow for determination of the load at which plasticization of the structure begins. This state is referred to as the limit state and it is treated as a sort of failure criterion that allows one to design a safe structure although sometimes not an economically favorable one Królak at al (1990).
