Novel numerical algorithm for solving temperature distribution of annular region

Abstract

The invention belongs to the field of heat conduction equation solving, and particularly relates to a novel calculation method for solving internal temperature distribution of an annular region, which has the characteristic of rapid calculation. The main thought of the numerical calculation method is as follows: 1, reducing the order of a differential equation by using two new unknown numbers; 2, deriving the relationship of the temperatures of four adjacent nodes through integration; 3, deriving the relationship of the temperatures of any node and a boundary; and 4, solving the temperature of each node by using a boundary condition. The method can perform global solution, has the characteristics of small calculation amount and small occupied computer memory while ensuring the calculation precision, and can quickly obtain the internal temperature distribution condition of the annular region.

Description

Technical Field

[0001] The invention belongs to the field of solving the temperature distribution of annular areas, and particularly relates to a new calculation method with small memory usage and fast calculation characteristics. Background Art

[0002] In various engineering fields, there are a large number of annular and circular components. Taking nuclear power plants as an example, it is of great significance to detect the temperature distribution of the pipes in real time. However, in the past, when solving the temperature distribution of the annular area, it is usually necessary to construct a large-scale matrix according to different difference formats to calculate and obtain the calculation results. Large-scale matrices require more memory, which leads to longer calculation time. Therefore, a new fast calculation method was developed to solve this problem.

[0003] This method requires special processing of boundary conditions and can quickly and accurately calculate and solve the temperature distribution in the annular area. Summary of the invention

[0004] Purpose of the invention: A new calculation method for solving the heat conduction problem in annular area, which does not require the construction of a large-scale matrix, but only needs to process the third-order matrix and has the characteristics of fast calculation. The main purpose is to quickly obtain the temperature distribution in the annular area.

[0005] Technical solution: In order to achieve the above-mentioned purpose, the present invention adopts the following technical solution.

[0006] A new calculation method for simulating the solution of the heat conduction equation in an annular region is characterized by introducing a gradient variable to reduce the order of the differential equation, reducing the heat conduction equation in a two-dimensional polar coordinate system to a system of three first-order partial differential equations, and deriving the matrix relationship between two variables between two adjacent nodes, and then deriving the relationship between the node at the boundary and any node, and using the known conditions obtained by measurement to achieve the solution. The method is implemented in the following steps in the calculation process: Step 1, abstract the solution of the temperature distribution of a certain pipeline into the solution of the heat conduction equation in the annular area; Step 2: Use the newly introduced gradient variable to reduce the order of the heat conduction equation in the two-dimensional polar coordinate system, and transform a second-order partial differential equation with one dependent variable and two independent variables into a first-order partial differential equation system with two dependent variables and two independent variables, as shown below: Where k is the thermal conductivity, r is the radial position, V is the heat flux density in the radial direction, W is the heat flux density in the circumferential direction, T is the temperature, and S is the source term. Step 3, integrate the three equations with respect to space to obtain the spatial relationship between the two variables: determine the time and space steps Δx and Δy, and use the subscripts i, i+1, i+2... and j, j+1, j+2... to represent the spatial positions of the temperature at the node in the radial and circumferential directions respectively; Step 4: By integrating the temperature and heat flow of adjacent nodes, we can get the relationship between the concentration and concentration gradient at adjacent nodes at time m+t: in, B=[B (i-1,j) ,B (i,j-1) ,B (i-1,j-1) ] C=[C (i-1,j) ,C (i,j-1) ,C (i-1,j-1) ] T in, Step 5, after obtaining the relationship between the temperature and heat flux density at four adjacent nodes, the relationship between the temperature and heat flux density at any node and the boundary node can be further obtained through such relationship.

[0007] The beneficial effects of the present invention are: (1) There is no need to process boundary conditions during the solution process. Boundary conditions can be directly applied to achieve the solution, saving calculation time; (2) In the process of solving two-dimensional problems, the order of the largest matrix involved does not increase with the increase of the number of grids and is always three-order, so the calculation amount is small and the calculation time is short; (3) Applicable to situations where physical parameters change; (4) There is no need to retain too many variable values ​​during the solution process, and the computer memory is small. BRIEF DESCRIPTION OF THE DRAWINGS

[0008] Figure 1 The basic steps for solving the new method; Figure 2 It is a simplified two-dimensional model of the pipeline in the example; Figure 3 This is the result obtained when using the new calculation method. DETAILED DESCRIPTION

[0009] The present invention provides a method for solving the temperature distribution on an annular region according to a novel numerical calculation method. The present invention is further described below with reference to the accompanying drawings and specific examples.

[0010] Examples For the temperature distribution problem in a two-dimensional annular region, Figure 2 As shown, the parameters of the problem are dimensionless. It is assumed that the inner diameter length of the annular region is 1, the outer diameter length is 2, and the thermal conductivity is 1; boundary conditions: the temperature on the inner diameter boundary is 0, and the heat flux density along the radial direction on the outer diameter boundary is also 0; Source term: -12r 2 cos2θ. Find: The temperature distribution inside the annular region and compare it with the exact solution Compare.

[0011] Solution: By integrating and substituting various values, we can obtain the change of pollutant concentration over time at the upstream of the river, that is, at x = 1. The result is basically consistent with the precise conditions. The error calculation formula is: Where T(r,θ) is the calculation result of the half-boundary method, For an accurate solution, it is calculated that the maximum error is 0.9901%.

Claims

1. A new calculation method for calculating the temperature distribution in the annular region is to reduce the order of the heat conduction equation in the two-dimensional polar coordinate system by using the newly introduced unknowns, which is characterized by: The steps of the new calculation method for calculating the temperature distribution in the annular area are as follows: (1) The inner diameter of the annular region is taken as the inner boundary, and the outer diameter is taken as the outer boundary; (2) Using the newly introduced unknowns, the heat conduction equation in the two-dimensional polar coordinate system is reduced to a system of equations containing only first-order partial differential forms. (3) Integrate the obtained set of equations with respect to space to obtain the relationship between the three variables in the radial and circumferential directions. Determine the time and space steps Δx and Δy, and use the subscripts i, i+1, i+2... and j, j+1, j+2... to represent the spatial position of the temperature at the node in the radial and circumferential directions respectively; in this way, the relationship between any point in space and the temperature and heat flux density on the inner boundary is obtained through integration.

2. The novel calculation method for calculating the temperature distribution in the annular region according to claim 1 is characterized in that The newly introduced variables (V) and (W) in step (2) can transform the second-order derivative of the field variable (T) in the equation into the first-order derivative expression of the newly introduced variable, and the variables (V) and (W) have physical meanings, which makes the new method have the characteristics of global solution.

3. The novel calculation method for calculating the temperature distribution in the annular region according to claim 1 is characterized in that After the relationship between the two points is determined in step (3), the value of the unknown quantity at each node can be calculated one by one through the relational expression by recursion.

4. The novel calculation method for calculating the temperature distribution in an annular region according to claim 1 is characterized in that In the process of solving step (3), the unknown quantity is only related to half of the boundary conditions and has nothing to do with the number of divisions in the distance.

5. The novel calculation method for calculating the temperature distribution in an annular region according to claim 1, characterized in that When solving the unknown quantities at each node, the matrix equations calculated by the computer are always third-order, and the calculation speed is fast.

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