A. Appendix

The transportation parameters for air are expressed by the following equations:

<sup>N</sup> = 2.69 � 1019 cm�<sup>3</sup> indicates the number of gas molecules per unit volume, and <sup>E</sup> is the local field in V/cm

$$\mathcal{W}\_{\rm e} = \begin{cases} -(\rm E/|E|) \left(7.4 \times 10^{21} |E|/N + 7.1 \times 10^{6}\right) & |E|/N > 2.0 \times 10^{-15} \\ -(\rm E/|E|) \left(1.03 \times 10^{22} |E|/N + 1.3 \times 10^{6}\right) & 10^{-16} \leq |E|/N \leq 2.0 \times 10^{-15} \\ -(\rm E/|E|) \left(7.2973 \times 10^{21} |E|/N + 1.63 \times 10^{6}\right) & 2.6 \times 10^{-17} \leq |E|/N < 10^{-16} \\ -(\rm E/|E|) \left(6.87 \times 10^{22} |E|/N + 3.38 \times 10^{4}\right) & |E|/N < 2.6 \times 10^{-17} \end{cases} \tag{A1}$$

$$\mathcal{W}\_{\rm n} = \begin{cases} -2.7E & |E|/N > 5.0 \times 10^{-16} \\ -1.86E & |E|/N \le 5.0 \times 10^{-16} \end{cases} \tag{A2}$$

$$W\_{\mathbb{P}} = 2.34E \tag{A3}$$

(1) to obtain the low-order flux FL

GL i,jþ<sup>1</sup>

FL iþ1

(2) to obtain high-order flux F<sup>H</sup>

FH iþ1

<sup>2</sup> ¼ SiΔt

(3) to define antidiffusion flux

(4) to obtain the temporary solution

wtd i,j <sup>¼</sup> wn

� 1

GH i,jþ<sup>1</sup> iþ1 2,j , G<sup>L</sup> i,jþ<sup>1</sup> 2

<sup>2</sup> <sup>¼</sup> <sup>Δ</sup>tSið Þ Wz i,jþ<sup>1</sup>

ð Þ Wr <sup>i</sup>þ<sup>1</sup>

ð Þ Wz i,jþ<sup>1</sup> 2

iþ1 2,j , G<sup>H</sup> i,jþ<sup>1</sup> 2

<sup>840</sup> <sup>f</sup> <sup>i</sup>þ3,j <sup>þ</sup> <sup>f</sup> <sup>i</sup>�2,j

� � � <sup>139</sup>

Aiþ<sup>1</sup>

Ai,jþ<sup>1</sup>

i,j � <sup>Δ</sup>V�<sup>1</sup>

<sup>2</sup>,j � FH

<sup>2</sup> � GH

i,j FL iþ1 <sup>2</sup>,j � <sup>F</sup><sup>L</sup> i�1 <sup>2</sup>,j <sup>þ</sup> <sup>G</sup><sup>L</sup> i,jþ<sup>1</sup> 2 � <sup>G</sup><sup>L</sup> i,j�<sup>1</sup> 2 h i (A22)

iþ1 <sup>2</sup>,j � FL iþ1

i,jþ<sup>1</sup> <sup>2</sup> � <sup>G</sup><sup>L</sup>

<sup>2</sup>,j ¼ ð Þ 2i þ 1 πΔzΔrΔt

<sup>840</sup> gi,jþ<sup>1</sup> <sup>þ</sup> gi,j

<sup>280</sup> gi,jþ<sup>4</sup> <sup>þ</sup> gi,j�<sup>3</sup>

þ 29

" 533 2,j w

(

Si ¼ 2iπΔr

<sup>2</sup>,j <sup>¼</sup> ð Þ Wr i,j <sup>þ</sup> ð Þ Wr <sup>i</sup>þ1,j

<sup>¼</sup> ð Þ Wz i,j <sup>þ</sup> ð Þ Wz i,jþ<sup>1</sup>

<sup>840</sup> <sup>f</sup> <sup>i</sup>þ1,j <sup>þ</sup> <sup>f</sup> i,j

<sup>840</sup> gi,jþ<sup>2</sup> <sup>þ</sup> gi,j�<sup>1</sup> � � <sup>þ</sup>

� � � <sup>139</sup>

<sup>280</sup> <sup>f</sup> <sup>i</sup>þ4,j <sup>þ</sup> <sup>f</sup> <sup>i</sup>�3,j

� �# (A17)

i,jþ<sup>1</sup>

2 w

where i and j are the sequence number of node along r and z directions, respectively.

" 533

� � � <sup>1</sup>

(

<sup>w</sup> <sup>¼</sup> <sup>i</sup>ΔrNi,j ð Þ Wr <sup>i</sup>þ<sup>1</sup>

<sup>w</sup> <sup>¼</sup> <sup>i</sup>ΔrNi,j ð Þ Wz i,jþ<sup>1</sup>

<sup>w</sup> <sup>¼</sup> <sup>i</sup>ΔrNi,jþ<sup>1</sup> ð Þ Wz i,jþ<sup>1</sup>

<sup>w</sup> <sup>¼</sup> ð Þ <sup>i</sup> <sup>þ</sup> <sup>1</sup> <sup>Δ</sup>rNiþ1,j ð Þ Wr <sup>i</sup>þ<sup>1</sup>

2 ≥ 0

> 2 < 0

<sup>2</sup>,j <sup>≥</sup> <sup>0</sup>

http://dx.doi.org/10.5772/intechopen.79215

Numerical Modeling of Partial Discharge Development Process

<sup>2</sup> (A13)

<sup>2</sup> (A14)

<sup>2</sup> (A15)

<sup>840</sup> <sup>f</sup> <sup>i</sup>þ2,j <sup>þ</sup> <sup>f</sup> <sup>i</sup>�1,j � �

29

f i,j ¼ iΔrNi,jð Þ Wr i,j (A18)

gi,j ¼ iΔrNi,jð Þ Wz i,j (A19)

wi,j ¼ iΔrNi,j (A23)

� �# (A16)

<sup>840</sup> gi,jþ<sup>3</sup> <sup>þ</sup> gi,j�<sup>2</sup> � �

<sup>2</sup>,j (A20)

<sup>2</sup> (A21)

<sup>2</sup>,j < 0

(A11)

125

(A12)

<sup>2</sup>,j <sup>¼</sup> ð Þ <sup>2</sup><sup>i</sup> <sup>þ</sup> <sup>1</sup> πΔzΔrΔt Wð Þ<sup>r</sup> <sup>i</sup>þ<sup>1</sup>

$$\frac{\alpha}{N} = \begin{cases} 2.0 \times 10^{-16} e^{\frac{-7248 \times 10^{-15}}{|E|/N}} & |E|/N > 1.5 \times 10^{-15} \\ 6.619 \times 10^{-17} e^{\frac{-5.90 \times 10^{-15}}{|E|/N}} & |E|/N \le 1.5 \times 10^{-15} \end{cases} \tag{A4}$$

$$\frac{\eta\_2}{N} = \begin{cases} 8.889 \times 10^{-5} |E|/N + 2.567 \times 10^{-19} & |E|/N > 1.05 \times 10^{-15} \\ 6.089 \times 10^{-4} |E|/N - 2.893 \times 10^{-19} & |E|/N \le 1.05 \times 10^{-15} \end{cases} \tag{A5}$$

$$
\eta\_3/N^2 = 4.7778 \times 10^{-59} (|E|/N)^{-1.2749} \tag{A6}
$$

$$
\eta = \eta\_2 + \eta\_3 \tag{A7}
$$

where η<sup>2</sup> and η<sup>3</sup> are the two-body and three-body attachment coefficients, respectively.

$$
\beta = 2.0 \times 10^{-7} \tag{A8}
$$

$$D = 0.3341 \times 10^9 (|E|/N)^{0.54069} |W\_\epsilon/E| \tag{A9}$$
