A computational algorithm and the method of determining the temperature field along the length of the rod of variable cross section

1 Introduction

A complex thermo-stress-strained state appears in the bearing components of power plants, internal combustion engines and hydrogen engines. Therefore, the definition of the temperature distribution law along the length of the rod elements is relevant. In this regard, many eminent scientists are engaged in this field [1,2,3]. In [4, 5] based on the finite element method the law of temperature distribution along the length of the rod with limited length and constant cross section was defined. In work [5] the solution of the steady problem of determination of the temperature field along the length of the horizontal rod with insulated side surface, limited length and a constant cross-section was discusses. Thus the cross-sectional area of the left end is under heat flux with a constant intensity and the right end is open to a convective heat transfer with the environment. The heat transfer coefficient and ambient temperature are considered constant. In [6] the effect of temperature on the deformation of the investigated element was considered. There also the law of temperature distribution along the length of the rod when the lateral surface is insulated and the left end is under the the heat flux, the right end is open to heat exchange with the environment was analytically derived. In addition, we consider the problem of determining the temperature field along the length of the horizontal rod with a constant cross section. While the left end obtains a constant temperature, and the remaining surfaces of the rod is undergoes heat transfer to the environment. The results obtained in this work show the same result as in [7,8,9]. In [10] based on the energy conservation law by variational method the heat transfer between the deformable shell and the surrounding fluid was determined. In [11] based on the finite element method the process of heat conduction in the rod elements of nuclear power plants was investigated. They also gave a description of a software package which was developed based on modern tools of programming to solve the considered problem. The developed program is, indeed, universal and user-friendly. In [12] the unsteady field of the temperature distribution in cylindrical rods which were under laser heat sources was investigated. The obtained results can be used in the study of nonstationary thermal processes in the rod with the laser heat source. In [13] the computational methods, algorithms and software package for the study of steady-state thermal stress - strain state of a rod with limited length and constant cross section under the influence of local heat flows, temperatures, heat exchange with consideration of local insulation were considered. For each of the considered problems corresponding regularities have been identified. For some problems the steady-state temperature fields, the components of strain and stress and the displacement field were determined. Also the expressions to calculate the elongation and axial compressive force were obtained. In addition the convergence of the method and the accuracy of the numerical results were studied. In contrast to the above mentioned works, in this paper we considered the development of methods, computational algorithms and programs based on the energy conservation law to study the steady thermal stress and strain state of a horizontal bar with a variable cross section in the form of a circle. The radius of the cross section decreases linearly with the length of the rod starting from the left end. The lateral surface of the rod is thermoinsulated. The cross-sectional area of the left end of the rod is under heat flux with a constant intensity and at the right end is open to convective heat transfer with the environment, where the heat transfer coefficient and ambient temperature are considered constant. For this problem first the temperature distribution law along the length of the rod should be defined. Further, if one end is firmly fixed and the other is free the elongation should be calculated depending on existing heat sources, physical and geometric characteristics of the rod taking into account the presence of insulation. In the case of pinching both ends of the studied rod the value of the axial compressive force should be calculated, taking into condideration the real factors. This also determines the distribution of all components of strains, stresses and the displacement field. The study revealed some patterns of the process. It should be noted that the developed Python programs proved to be effective and user-friendly.

2 Formulation of the problem

Let’s consider a horizontal bar with variable cross-section. Axis Ox directs from the left to the right on the axis of the rod. Assume that the cross section of the rod is a circle. The radius of the rod’s cross section varies linearly along its length, thus r(x) = ax + b, 0 ≤ x ≤ l, where l [cm] − the length of the rod. a, b = const. The radius of the cross-sectional area of the left end denoted by b[cm], thus r(x = 0) = a · 0 + b = b [cm]. Then the radius of the cross section of the right end will be equal to r(x = l) = a · l + b [cm²]. The cross-sectional area of the rod along its length changes according to the quadratic law, thus F(x) = πr² = π(ax + b)², 0 ≤ x ≤ l. Further, assume that the lateral surface of the rod is insulated. The cross-sectional area of the left end is under heat flow q[Bmcm2] with a constant intensity, while the cross-sectional area of the right end is under convective heat transfer with the environment. The temperature of the environment is constant, thus T_oc [°C] = const. The heat transfer coefficient between the rod’s material and the environment is h[Bmcm2⋅∘C]. The thermal properties of the material of the rod is characterized by the coefficient of thermal conductivity kxx[Bmcm2⋅∘C]. Under such impacts it is required to determine the temperature distribution law along the length of the rod with variable cross section. Calculation scheme of the considered problem is shown in Figure 1.

Figure 1

Calculation scheme of the task.

3 The solution to the problem using the energy conservation law

To solve this problem in accordance with the fundamental law of energy conservation it the functional of total thermal energy for the considered problem is needed to write

where S(x = 0) - the cross-sectional area of the left end of the rod which is under heat flux q; S(x = l) - the cross-sectional area of the right end of rod which is open to convective heat transfer. S(x = l) – the volume of the test rod. Now, considering that the studied process is established and the lateral surface of the rod is fully insulated, also q, h, T_oc = const, the field of temperature distribution along the length of the rod is approximierted by the full second-order polynomial.

where a, b, c = const;

Using (2-3) it is possible to determine the temperature gradient.

Besides, the function (2) must give a minimum to the functional (1) of the full thermal power of the rod. First calculate the integrals in (1) consedering (2).

where F₀ = F(x = 0) = π(a · 0 + b)² = πb² – the cross-sectional area of the left end of the rod which is under heat flux – q.

where F_l = a · l + b.

Substituting (4-6) into (1) we find the integral form of the functional of total thermal energy for the considered problem.

In this expression I = I (T_i, T_j, T_k). Minimizing I by T_i, _Tj and T_k we can get a system of equations with the natural boundary conditions.

After a little simplification, system (8) can be rewritten in the following form.

where a₁ = (3al + 14b); a₂ = (al + 4b); a₃ = (al + 2b); a₄ = 3al + 4b;

Further, solving the system (9) define nodal temperature value T_i, T_j and T_k.

where

Then the law of temperature distribution along the length of the rod can be defined as:

4 Analysis of the obtained results

Let’s consider the solution of the problem with the following initial data:

In this case, the temperature distribution along the length of the rod according to (13) has the following form:

Then:

Then the graph of the temperature distribution field along the length of the rod under the assumed initial data will look like Figure 2.

Figure 2

The field of temperature distribution along the length of the rod.

The figure shows that the distribution of temperatures has a slightly parabolic character. This is due to the variability of the cross-section of the rod along its length that varies according to the square law, and the radius which varies according to the linear law.

If one of the ends is firmly fixed and the other is free, due to the heat source it becomes longer. The value of elongation can be determined on the General laws of thermophysics:

Here α = 0.0000125 (1∘C) - coefficient of thermal expansion of the material of the rod.

If both ends are firmly fixed, then axial compressive force in the rod due to thermal expansion occurs. Its value can be determined by using the compatibility conditions of deformation [13]. In our case, according to [13] the value of the axial force is determined by the next formula:

Because E = 2 * 10⁶F_cp = 9, then R = −3062209.933. In this case the field distribution of thermo-elastic stress in the cross-sections of the studied rod occurs, which is defined as:

The law of deformation component of thermoelastic distributiom is determined based on the laws of physics [13]:

Using the found temperature distribution law (14) we can calculate the distribution law of thermal component of deformation:

Then according to the generalized Hooke’s law we can determine the distribution law of thermal component of stress:

Further, it’s possible to determine the distribution law of elastic component of deformation:

Further, based on generalized Hooke’s law we can determine the distribution law of elastic component of stress:

Figure 3 shows the distribution of the three strain components along the length of the rod with variable cross-section, firmly fixed at both ends.

Figure 3

The distribution law of deformation: 1 - ϵ(x); 2 - ϵ_T (x) u3 - ϵ_x (x).

The graph shows that the temperature and thermo - elastic components of deformation are compressive in nature throughout the length of the rod. When the elastic component of deformation on the length of the rod 0 ≤ x ≤ l = 67 cm has stretchable nature, on the length 67 ≤ x ≤ l = 90 cm – compressive nature. This process caused by the presence of a large heat flow (q=−500Bmcm2) on the left end of the rod, where the cross-sectional area at the left end is four times more than on the right end.

Figure 4 shows the distribution of three stress components along the length of the rod with variable cross-section area and firmly fixed at both ends.

Figure 4

The laws of stress distribution: 1 - σ(x); 2 - σ_T(x) u 3 – σ_x (x).

From the figure it is evident that all stress components have the same character as the deformation.

Figure 5 shows the distribution of displacement of cross-sections of the rod.

Figure 5

The distribution law of displacement of cross-sections of the rod.

The figure shows that all cross sections move from left to right. It is also due to the large heat flux at the left end of the rod. The largest displacement is on the cross section of the rod with coordinates x=51 cm, as the displacement on both ends of the rod is equal to 0.

5 Conclusions

The developed computational algorithms and methods based on the fundamental energy conservation law allows us to solve a variety of applied engineering problems with high accuracy. Also, they provide an opportunity to solve many applied engineering problems of thermal conductivity taking into account the presence of various types of local heat sources.

It should be noted that the proposed algorithm and method can simultaneously determine the distribution laws of the temperature, three components of strain and stress, and also displacment. Along with that was defined the value of the elongation and the resultant axial force. The obtained results allow to carry out deep analysis of thermo - stressed rod with variable cross section and limited length. Using the developed approach it is possible to explore the emerging and complex thermo-stress-strain state of load-bearing elements of power plants, internal combustion engines, jet engines and the hydrogen, and oil heating stations used in transportation of highly waxy oil through the pipeline.