Finite element analysis of steady-state heat conduction

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In the analysis of engineering problems, it is often necessary to understand the internal temperature distribution of the workpiece, such as the operating temperature of the engine, metal workpiece in the heat treatment process of temperature change, fluid temperature distribution. The temperature distribution within an object depends on the heat exchange within the object and the heat exchange between the object and the external medium, which is generally considered to be time-dependent. The heat exchange inside the object is described by the following heat conduction equation (Fourier equation),


The formula is density, kg/m3; For specific heat, for thermal conductivity, T for temperature, ℃;t for time, s; for inner heat source density, w/m3.

For isotropic materials, the thermal conductivity equation can be written in the form of the same heat conduction coefficient in different directions,


In addition to the heat conduction equation, it is necessary to specify initial conditions and boundary conditions for calculating the temperature distribution inside the object. Initial conditions refer to the initial temperature distribution of an object,


The boundary condition refers to the heat exchange of the outer surface of the object and the surrounding environment. In the heat transfer, boundary conditions are generally divided into three categories.

    1. The temperature on the boundary of a given object is called the first class boundary condition.

The temperature or temperature function on the surface of the object is known,

or (6-4)

    1. The thermal input or output on a given object boundary is called the second class boundary condition.

      The heat flux density on the surface of the known object,

or (6-5)

    1. The given convective heat exchange condition is called the Third kind of boundary condition.

      The convection heat transfer coefficient between the object and the fluid medium in contact with it and the temperature of the medium are known.


Where h is the heat transfer coefficient, w/(M2 K), is the temperature of the object surface, is the medium temperature.

If the heat transfer condition on the boundary does not change with the time, the heat source inside the object does not change with the time, after a certain period of heat exchange, the temperature of the object will not change with the time, namely

This type of problem is known as the stationary (Steady state) heat conduction problem. The steady-state heat conduction problem is not that the temperature field does not change with time, but refers to the stable state of temperature distribution, we do not care about how the temperature field inside the object transitions from the initial state to the last stable temperature field. The transient (Transient) heat conduction equation with time change is degenerate into the steady state heat conduction equation, and the steady-state heat conduction equation of three-dimensional problem is,


For isotropic materials, the following equations can be obtained, called Poisson equations,


Considering that the object does not contain an internal heat source, the temperature field in the isotropic material satisfies the Laplace equation,


In the analysis of steady-state heat conduction problems, it is not necessary to consider the influence of the initial temperature distribution of the object on the final stable temperature field, so the initial conditions of the temperature field should not be considered, but only the heat transfer boundary conditions should be considered. The calculation of steady state temperature field is actually the boundary value problem of solving partial differential equation. The temperature field is a scalar field, and after the object is dispersed into a finite element, there is only one temperature unknown on each node, which is simpler than the elastic mechanics problem. In the calculation of the temperature field, the shape function of the finite element is exactly the same as that of the elastic mechanics problem, and the temperature distribution inside the unit is determined by the temperature on the element node. Because of the complexity of the heat exchange boundary conditions in practical engineering problems, it is difficult to measure in many cases, how to define the correct heat exchange boundary condition is a difficult point in the calculation of temperature field.

6.2 General Finite element method for steady-State temperature field analysis

In the previous we have introduced the finite element method can be used to analyze the field problem, the steady-State temperature field calculation is a typical field problem. We can establish the finite element scheme of elastic mechanics problem analysis by using the virtual force equation, and the element stiffness matrix deduced has definite mechanical meanings. In this paper, we introduce how to establish the finite element method of steady-State temperature field analysis by using the weighted residuals (Weighted residual methods).

The boundary value problem of a differential equation can generally be expressed as an unknown function u satisfies a differential equation set,

(within the domain) (6-10)

The unknown function u also satisfies the boundary condition,

(On the Border) (6-11)

If the unknown function u is the exact solution to the above boundary value problem, you satisfy the differential equation (6-10) at any point in the field, and the boundary condition (6-11) is satisfied at any point in the boundary. For complex engineering problems, such exact solutions are often difficult to find and need to be managed to find approximate solutions. The approximate solution selected is a family of known functions with a pending parameter, which is generally expressed as


Which is the undetermined coefficient, known as the function, is called the heuristic function. The heuristic function is derived from the complete sequence of functions and is linearly independent. Since the heuristic function is a complete sequence of functions, any function can be represented by this sequence.

This kind of approximate solution can not exactly satisfy the differential equation and boundary condition, and the error is called the margin.

The remainder of the differential equation (6-10) is,


The margin of the boundary condition (6-11) is,


Select a family of known functions, so that the weighted integral of the margin is zero, forcing an approximate solution to produce a margin in an average sense equal to zero,


Known as the weight function, the formula (6-15) allows you to select a pending parameter.

The method of calculating the approximate solution of the differential equation by using the weighted integral of the margin as zero is called the weighted residual method. Different weights are obtained for different weights, and the common methods include point method, subdomain, least squares method, moment Method and Galerkin method (Galerkin). In many cases, the coefficients matrix of the equations obtained by the Galerkin method is symmetrical, and the general finite element method of the steady-State temperature field analysis is also established by using Galerkin. In the Galerkin method, the heuristic function sequence is directly used as the weighted function, which is taken.

The following is an example of a second-order ordinary differential equation, which illustrates the Galerkin method (see section 3, "Basic principle and numerical method of finite element method", 1.2.)

Solving the second order ordinary differential equation by example

Boundary conditions: At that time, at that time,.

Take two approximate solutions:

By the formula (6-15), two weighted integral equations can be obtained,

After the integration, we can get a two-yuan equation Group,

Approximate solution is,

The exact solution of the equation is,

The results of approximate solutions and exact solutions are shown in table 6-1,

Table 6-1 Comparison of approximate solutions and exact solutions










Assuming that the element's shape function is,

The temperature of the cell junction is,

The temperature distribution inside the unit is

Taking two-dimensional problem as an example, the process of establishing the general finite element format of steady-state temperature field by Galerkin method is illustrated. The steady-state heat conduction equation of two-dimensional problem is,


The first type of heat exchange boundary is


The second type of heat exchange boundary condition is,


The third kind of boundary condition is,


In a unit, the weighted integral formula is


To be integral by division,

Applying the green theorem, the weighted integral formula in a unit is written as


Using the Galerkin method, the selection weight function is,

The temperature distribution function and the heat exchange boundary condition in the unit are replaced by the (6-18) equation, and the weighted integral formula of the Unit is


After the heat exchange boundary condition is replaced, the second kind of heat-exchange boundary term and the third kind of heat-exchange boundary term appear in the (6-19) formula, but there is no entry corresponding to the first kind of heat exchange boundary. This is because, as a function of weight, the first type of heat exchange boundary is automatically satisfied. Written in matrix form there,


The equation (6-20) is an n-linked linear equation group that can determine the temperature of n nodes. The (6-20) is represented by the finite element format,


where the matrix [k]e is the heat conduction matrix of the element or the temperature stiffness matrix, {T}e is the node temperature vector of the element, {P}e is called the temperature load vector of the unit or the thermal load vector (Thermal load vectors). For a particular cell, the element heat conduction matrix [k]e and temperature load vector {p}e elements are, respectively,



If a cell is completely inside the object,

The weighted integral equation on the whole object is the sum of the unit integral equations,


According to the relationship between the local number of the unit node and the whole number, the whole stiffness matrix is obtained directly, and the whole equation Group is

6.3 Finite element columns of triangular elements

Figure 6-1 Triangle Unit

Recalling the contents of the third chapter, we can find that the triangular element used in the calculation of two-dimensional temperature field can use the same shape function as the method used to calculate the plane problem of elastic mechanics.

The temperature distribution in the cell is expressed as a temperature value on the junction,


On the triangular unit, the Galerkin method is used to


The coefficient of thermal conductivity in a cell is assumed to be constant,



The stiffness matrix of the unit is,

Obviously, the heat conduction matrix of the unit is symmetrical.

If the internal heat source of the unit is constant, the temperature load item generated by the internal heat source is


The green formula is available.


For convenience, the heat exchange boundary is uniformly expressed as the third type of heat exchange boundary,


If there is a hot swap on the side of the cell, the boundary heat transfer condition on each edge is generated in the element stiffness matrix with additional items,




The temperature load vectors generated by the boundary heat exchange conditions are,




6.4 Example of temperature field analysis

The square section of the chimney 6-2 shows that the chimney is constructed from concrete, the edge length is 60cm, the channel side length is 20cm, the concrete thermal conductivity is. Assuming that the temperature of the inner surface of the chimney is℃, the outer surface of the chimney is exposed to air, the air temperature is ℃ and the heat transfer coefficient is. Calculates the steady-state temperature field in the Chimney section. (See,finite Element Method theory and application with ANSYS, p279)

Fig. 6-2 Chimney Section 6-3 Finite element model

Figure 6-4 Steady-state temperature distribution

Figure 6-5 Thermal Flow distribution

The steady-State temperature field distribution is independent of the initial state of the object, and is it related to the thermal conductivity of the material? We made some modifications to the chimney model, assuming that the chimney wall consisted of two layers of material. The inner layer material is concrete, the outer surface of the cross-sectional dimension is, the size of the chimney channel is unchanged, still is. The thermal conductivity of the outer material is the same as the cross-sectional dimension of the outer surface and the cross-sectional dimension of the inner surface. The heat exchange boundary condition is unchanged, and the two-layer chimney is shown in the finite element model 6-6.

Fig. 6-6 Finite element model of double chimney

Fig. 6-7 temperature distribution of double chimney

Fig. 6-8 heat flux distribution of double chimney

The temperature distribution and heat flux distribution of two different structure chimneys are compared, and the distribution of steady state temperature field is correlated with the heat transfer coefficient of the material. The thermal conductivity of the outer layer of the two-layer chimney is relatively small, close to the insulation material, heat is quickly passed into the inner chimney, but the heat transfer to the external environment is much slower, so the inner chimney temperature is very high. Compared with the heat flow distribution, the insulation material can effectively prevent heat dissipation. The heating pipes in the northern cities are covered with a layer of insulating material that is based on this principle.

Finite element analysis of steady-state heat conduction

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