Weight Function Distribution for a Full Circular Pipe

Consider a full circular conductive cross sectional area with radius as illustrated in Figure 3.1. Assume the magnetic field is in the negative y-direction, i.e. = − , and that a unit current is injected at an angle of and the unit current leaves the circumference of the pipe at an angle of .

y

I

I

x B

Y

Y

R

Figure 3.1: Flow in a full circular pipe with radius

In the absence of the magnetic field and flow, the virtual current density is given by [122]

∙ = 0 Eq. 3-1

Assuming uniform conductivity , the virtual current density is related to the virtual potential gradient by the following equation

= Eq. 3-2

Substituting Eq. 3-2 into Eq. 3-1 gives the equation for the virtual potential which is Laplace’s equation, i.e.

= 0 Eq. 3-3

The equations stated above were previously discussed in Section 2.5.5 and stated here for convenience. Laplace’s equation in Eq. 3-3 is a classic problem encountered in many fields of engineering. It can be used to describe the steady-state of temperature, potential, stress or flow distributions [159]. The general solution is obtained using the method of separation of variables and is provided in the literature [160, 161]. The general solution for the virtual potential in polar coordinates( , ) takes the form

( , ) = + ( + ) Eq. 3-4

where is an arbitrary constant, and and are found by applying appropriate boundary conditions.

3.2.1 Boundary Condition Assuming Dirac 1D Delta Function

The boundary condition applied was the Neumann boundary condition using Dirac delta function⁴ [162]. Referring to Figure 3.1, the Neumann boundary condition ( ) at the circumference in polar coordinates using Dirac function and for a unit radius, i.e.

= 1 (using polar coordinates), can be written as follows

∙ | = = ( − ) − ( − ) Eq. 3-5

where is a unit vector of the circumferential surface, and is the delta function (impulse response) which is applied to express a unit current entering a surface at an angle and leaving at an angle of . The angles of and , in practice, refer to the locations of the electrodes across which the potential difference is measured.

4 I am grateful to Dr. Laszlo Kollar and Prof. Gary Lucas for their work in the mathematical modelling of

To include the effect of the radius , consider two circular pipes with different radii, and as shown in Figure 3.2. Assume a pipe with a unit length so that the problem can be simplified to a two-dimensional one. Referring to Figure 3.2, the unit current passing through the length inside the section of the pipe with radius is the same as the unit current passing through the length inside the pipe section with radius , if the ratio of / is equal to the ratio / . Mathematically, this can be written as

= → = Eq. 3-6

Hence, the current in both cross sections is given by

= = Eq. 3-7

d2

R2

Ix

=1

Ix I_x I_x

d

R1

Figure 3.2: Two circular cross sections with different radii

From Eq. 3-6 and Eq. 3-7, the ratio of the virtual current densities and , in the two pipes can be expressed as

= = Eq. 3-8

According to Eq. 3-8, it can be noted that the virtual current density is proportional to 1/ . Hence, the boundary condition in Eq. 3-5 at the circumference, where the electrode is placed on a surface with curvature1/ , is modified as follows

∙ | = = 1

( ( − ) − ( − )) Eq. 3-9

Eq. 3-9 is the boundary condition that will be applied to find the solution to Laplace’s equation given in Eq. 3-3 to find the virtual potential ( , ). Note that for a given size pipe and uniform conductivity, the virtual current density distribution will be the same regardless of the actual fluid conductivity. Therefore, for this analysis which follows, it is satisfactory to assume that = 1.

3.2.2 Solution to Laplace’s Equation to Obtain the Virtual Potential

Applying the boundary condition in Eq. 3-9 to the general solution of Laplace’s equation, i.e. Eq. 3-4 gives the following

∂

∂r = ( cos + sin ) = 1

( ( − ) − ( − )) Eq. 3-10

Multiplying both sides in Eq. 3-10 by cos and integrating with respect to from 0 to 2 gives

From the orthogonality of trigonometric functions, i.e.

1

0

Equation 3-11 can be simplified to

= 1

(cos − cos ) Eq. 3-14

By rearranging Eq. 3-14, the value for is given by

=cos − cos

Eq. 3-15 Similarly, multiplying both sides in Eq. 3-10 bysin and integrating with respect to

from 0 to 2 gives

From the orthogonality of trigonometric functions, i.e.

1

and also Eq. 3-13, Eq. 3-16 can be simplified to

= 1(sin − sin ) Eq. 3-18

By rearranging Eq. 3-18, the value for is given by

=sin − sin

Eq. 3-19

Since is an arbitrary constant and the virtual current density is the derivative of the virtual potential thus, the term will not appear, i.e. = 0 in Eq. 3-4.

Next, substituting Eq. 3-15 and Eq. 3-19 into Eq. 3-4, the solution to the Neumann boundary value problem is obtained as follows

( , ) = 1

(cos cos − cos cos

+ sin sin − sin sin )

Eq. 3-20

Using trigonometric identities, Eq. 3-20 can be simplified to

( , ) =1 1

(cos ( − ) − cos ( − )) Eq. 3-21

3.2.3 Virtual Current Density and Weight Function

The solution to the virtual current density is obtained from the virtual potential equation given in Eq. 3-21 thus; the virtual current density can be found from calculating the derivatives of the virtual potential as follows

= = 1

The local weight function is defined as [122]

= × Eq. 3-23

Assuming a uniform and transverse magnetic flux density in the negative y-direction, i.e. = [0, − , 0], the weight vector simplifies to a 2D problem and since = 0, the only non-zero component is (Bevir’s conclusion [125]) which is given by

= − Eq. 3-24

Substituting the virtual current density (Eq. 3-22) into the weight function expression in Eq. 3-24 gives the following

= [cos ( − ) cos − cos ( − ) cos

− (sin ( − ) sin − sin ( − ) sin )]

Eq. 3-25

Simplifying Eq. 3-25 using trigonometric identities gives the following local weight function distribution for a circular pipe full of conductive fluid.

= [(cos( − ( − 1) ) − cos( − ( − 1) )] Eq. 3-26

The angles of and in practice correspond to the angular position of the electrodes across which the potential difference is measured as will be shown below.