1. Electromagnetic flowmeter measurement theory
The integral of the electromagnetic flowmeter is given by Bevir in 1970:
Where: U2-U1 is the potential difference between the two electrodes; A represents the integral of all the spaces; It is called a vector weight function and is an amount determined only by the structure of the electromagnetic flowmeter itself. Its expression is:
and:
Where: G and F are with
The scalar potential, they satisfy the Laplace equation:
It can be seen from the above analysis that the measurement of the potential difference is not affected by changes in the temperature, pressure, density, and conductivity (above a certain threshold) of the fluid, and has great advantages.
2. Theoretical calculation of plug-in electromagnetic flowmeter
A typical plug-in flow meter structure is shown in Figure 1. The electrodes are inserted into the tube and the magnetic poles remain outside the tube, creating a local magnetic field around the electrodes.
Figure 1 Schematic diagram of the plug-in flowmeter
Figure 2 simplified physical model
The physical model is established as shown in Fig. 2: e1 and e2 are two electrodes inserted into the pipe, and the electrode position is determined by the insertion depth b and the electrode opening angle θ0. It is the magnetic field generated by the external magnetic pole. Based on this model, the distribution of G, F, and W is calculated.
2.1, calculation of the virtual potential G
Since there is an inserted electrode in the pipe, the Laplace equation of equation (4) cannot be directly used to solve the virtual potential. We can think of the model's virtual potential distribution as the superposition of the virtual potentials generated by only the electrodes and the boundary, ie, G=G0+Gr.
2.1.1. Only the electrode's virtual potential distribution
Assuming the boundary is infinite, according to the definition of the virtual current:
Gauss's law in ordinary electric fields:
According to the principle of symmetry, the virtual potential should have a form similar to the potential, namely:
According to the geometric relationship of Figure 2, it is not difficult to find the analytical expression of G0:
2.1.2. Virtual potential distribution with only boundaries
Boundary conditions due to insulation of the measuring tube wall =0, ie:
Equation (9) is the boundary condition of Gr. Since the virtual potential is determined only by the boundary at this time, there is an equation:
This is a Laplace equation for the solution of the condition, which can be solved using the separation variable and the Fourier coefficient formula. Since it is difficult to obtain the analytical solution of the boundary condition, we use the difference method in the radial direction to obtain the boundary condition of Gr to obtain the numerical solution of Gr.
3.2, calculation of magnetic potential F
Since the insertion depth of the electrode is generally only 10% to 12.5% ​​of the diameter of the pipe, it is assumed that the magnetic induction intensity near the electrode is uniform, that is:
3.3, calculation of weight function W
The definition of the gradient can be obtained:
Since the magnetic field is uniform, it is not difficult to get:
3.4, calculation of output potential difference
Assuming that the flow in the pipeline is a fully developed turbulent flow, we use a classical turbulence model whose flow field distribution is:
The output potential difference U is obtained from the obtained W.
3, programming calculation
As can be seen from the above discussion, the key to the problem lies in the calculation of the virtual potential function G, which uses a discrete method to calculate G in consideration of accuracy requirements and resource consumption. The specific implementation steps are as follows:
1), the region of interest is meshed on the two-dimensional rectangular coordinates, and the G0 value on each micro-element is obtained using equation (8);
2) Calculate the G0 normal direction partial derivative value of the mesh at the boundary in equation (9) using the difference method as the boundary condition for calculating Gr;
3), by separating the variables, using the Fourier coefficient formula, and the discrete Simhson integral method to calculate the semi-analytical expression of Gr (10), calculate the Gr value of each grid, and synthesize G;
4), calculate the difference of G in the x direction according to formula (13), and obtain the W value of each grid;
5) Calculate the output voltage by combining the flow field model of equation (14).
Write a program to calculate the output voltage of different flow fields and different electrode positions, and draw the equipotential distribution map of G and W.
4. Results and analysis
4.1, the distribution of the virtual potential G (take the electrode spacing is 0.1R)
Take b = 0.9R (R is the pipe radius), θ = 0.0555 rad, draw the G distribution and enlarge the area near the electrode as shown in Figure 3.
Figure 3 G distribution and partial enlargement of b = 0.9R
The black dots in Fig. 3 are electrodes, and it can be clearly seen that G is mainly distributed around the electrodes and a significant change occurs in the distribution at the boundaries.
4.2, weight function W distribution (take the electrode spacing is 0.1R)
Take b = 0.9R, θ = 0.0555 rad, and plot the W distribution as shown in Fig. 4.
Figure 4 W distribution and partial enlargement of b = 0.9R
It can be seen from Fig. 4 that W is mainly distributed in the vicinity of the electrodes and is symmetrically distributed.
4.3, output potential difference
It can be found by calculation that the weight function W is mainly distributed near the electrodes. Choose b=0.752R, right Perform full-space integration and obtain the output potential difference U=0.1475V (for the sake of seeing together, assume vmax=1m/s, R=1m, B=1T at the electrode); integrate the annular region of the circumference 0105R of the electrode. Find the output potential difference U = 0.1231. Therefore, the main effect on the final output potential difference is the flow field near the electrode. Explain that our assumed magnetic field model is available.
Select the turbulence model commonly used in simulation calculations
The calculation is performed, and vmax=1 is taken, and different turbulence coefficients n are solved at different insertion depths, and the results are shown in Table 1.
Table 1 Output potential difference at different electrode positions and different turbulence coefficients
The turbulence coefficient-output potential difference curve is plotted as shown in Fig. 5.
Figure 5 Turbulence coefficient - output potential difference fit curve
Perform a least squares fit on each set of data, and calculate the slope and linearity as shown in Table 2.
Table 2 Potential difference at different electrode positions fitted to the slope and linearity of the line
It can be seen from Fig. 5 that vmax=1, that is, under the same flow rate, different turbulence coefficients n correspond to different output voltages. However, when b=0.752R, which is also the average flow point position, the output potential difference U value is basically unchanged. Therefore, as long as the electrode is inserted into this position, it can be used to measure the flow rate. In order to study the measurement error caused by the insertion depth deviating from the average flow point, assuming that the output potential difference at the average flow point position is the standard value, it is calculated that the deviation between the insertion depth and the average flow point is in the range of 011R, and the relative error between the output potential and the standard value. It is about 1% to 2%.
5 Conclusion
This article has done the following work:
1) Establish a physical model of the plug-in electromagnetic flowmeter, and write a program to calculate the numerical solution of the virtual potential and the weight function to guide the actual production and operation of the plug-in electromagnetic flowmeter;
2) Introduce the classical turbulence model, simulate the output voltage of different turbulence coefficients and different electrode positions, give the relationship curve, and theoretically give the optimal working position of the electrode.
It is hoped that in the further work, the physical object of the plug-in flowmeter can be processed and processed, and the theoretical analysis result can be verified by the flow calibration experiment.
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