.. _section-A5.2:

Appendix 5.2: IHX Matrix Solution Algorithm
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The solution of the matrix represented by Eqs. :eq:`eq-5.4-32` through :eq:`eq-5.4-35` and
whose coefficients are given in :numref:`section-A5.1` is accomplished by Gaussian
elimination, making use of the zeros present in the matrix. It is
presented as an algorithm as it is coded in subroutine TSIHX. The arrows
in the following mean replacement of what is on the left by the
expression on the right.

:math:`\text{JMAX}` = the number of nodes in the primary and intermediate coolant

:math:`\text{JMAX} - 1` = the number of nodes in the shell and tube

1. Set :math:`j` = 1.

2. Multiply :eq:`eq-5.4-32` by :math:`\frac{1}{a_{1} \left( j \right)}`

:math:`a_{2}\left( j \right) \rightarrow \frac{a_{2}\left( j \right)}{a_{1}\left( j \right)};a_{3}\left( j \right) \rightarrow \frac{a_{3}\left( j \right)}{a_{1}\left( j \right)} ;\ a_{4}\left( j \right) \rightarrow \frac{a_{4}\left( j \right)}{a_{1}\left( j \right)};a_{5}
= \frac{a_{5}\left( j \right)}{a_{1}\left( j \right)};\ a_{1}\left( j \right) \rightarrow 1`;

3.
:math:`e_{1}\left( j \right) \rightarrow e_{1}\left( j \right) - e_{3}\left( j \right)a_{2}\left( j \right)`

:math:`e_{7}\left( j \right) \rightarrow e_{7}\left( j \right) - \ e_{3}\left( j \right)a_{3}\left( j \right)`;

:math:`e_{8}\left( j \right) \rightarrow e_{8}\left( j \right) - e_{8}\left( j \right)e_{4}\left( j \right)`;

:math:`e_{3}\left( j \right) \rightarrow 0`;

4. Multiply non-zero coefficients in :eq:`eq-5.4-33` by :math:`{1}{e_{1} \left( j \right)}`.

:math:`e_{5}\left( j \right) \  \longrightarrow \ \ \frac{e_{5}\left( j \right)}{e_{1}\left( j \right)} \ ;\ \ e_{7}\left( j \right)  \longrightarrow \ \ \frac{e_{7}\left( j \right)}{e_{1}\left( j \right)} \ ;`

:math:`e_{8}\left( j \right) \  \longrightarrow \ \ \frac{e_{8}\left( j \right)}{e_{1}\left( j \right)} \ ;\ \ e_{10}\left( j \right)  \longrightarrow \ \ \frac{e_{10}\left( j \right)}{e_{1}\left( j \right)}`

:math:`e_{1}\left( j \right) \  \longrightarrow \ \ 1`

5.

:math:`c_{1}\left( j \right) \  \longrightarrow \ \ c_{1}\left( j \right) \  - c_{2}\left( j \right) e_{5}\left( j \right)`

:math:`c_{3}\left( j \right) \  \longrightarrow \ \ c_{3}\left( j \right) \  - c_{2}\left( j \right) e_{7}\left( j \right)`

:math:`c_{4}\left( j \right) \  \longrightarrow \ \ c_{4}\left( j \right) \  - c_{2}\left( j \right) e_{10}\left( j \right)`

:math:`c_{6}\left( j \right) \  \longrightarrow \ \ c_{6}\left( j \right) \  - c_{2}\left( j \right) e_{8}\left( j \right)`

:math:`c_{2}\left( j \right) \  \longrightarrow \ \ 0`

:math:`f_{1}\left( j \right) \  \longrightarrow \ \ f_{1}\left( j \right) \  - f_{7}\left( j \right) e_{10}\left( j \right)`

:math:`f_{3}\left( j \right) \  \longrightarrow \ \ f_{3}\left( j \right) \  - f_{7}\left( j \right) e_{5}\left( j \right)`

:math:`f_{6}\left( j \right) \  \longrightarrow \ \ f_{6}\left( j \right) \  - f_{7}\left( j \right) e_{8}\left( j \right)`

:math:`f_{8}\left( j \right) \  \longrightarrow \ \ f_{8}\left( j \right) \  - f_{7}\left( j \right) e_{7}\left( j \right)`

:math:`f_{7}\left( j \right) \  \longrightarrow \ \ 0`

:math:`e_{1}\left( j + 1 \right) \  \longrightarrow \ \ e_{1}\left( j + 1 \right) \  - e_{6}\left( j + 1 \right) e_{7}\left( j \right)`

:math:`e_{4}\left( j + 1 \right) \  \longrightarrow \ \ e_{4}\left( j + 1 \right) \  - e_{6}\left( j + 1 \right) e_{5}\left( j \right)`

:math:`e_{8}\left( j + 1 \right) \  \longrightarrow \ \ e_{8}\left( j + 1 \right) \  - e_{6}\left( j + 1 \right) e_{8}\left( j \right)`

:math:`e_{9}\left( j + 1 \right) \  \longrightarrow \ \ e_{9}\left( j + 1 \right) \  - e_{6}\left( j + 1 \right) e_{10}\left( j \right)`

:math:`e_{6}\left( j + 1 \right) \  \longrightarrow \ \ 0`

6. Multiply non-zero coefficients in :eq:`eq-5.4-34` by :math:`\frac{1}{c_{1} \left( j \right)}`.

:math:`c_{3}\left( j \right) \  \longrightarrow \ \ \frac{c_{3}\left( j \right)}{c_{1}\left( j \right)} \ ;\ \ c_{4}\left( j \right)  \longrightarrow \ \ \frac{c_{4}\left( j \right)}{c_{1}\left( j \right)}`

:math:`c_{5}\left( j \right) \  \longrightarrow \ \ \frac{c_{5}\left( j \right)}{c_{1}\left( j \right)} \ ;\ \ c_{6}\left( j \right)  \longrightarrow \ \ \frac{c_{6}\left( j \right)}{c_{1}\left( j \right)}`

:math:`c_{5}\left( j \right) \  \longrightarrow \ \ \frac{c_{5}\left( j \right)}{c_{1}\left( j \right)} \ ;\ \ c_{6}\left( j \right)  \longrightarrow \ \ \frac{c_{6}\left( j \right)}{c_{1}\left( j \right)}`

7.

:math:`f_{1}\left( j \right) \  \longrightarrow \ \ f_{1}\left( j \right) \  - f_{3}\left( j \right) C_{4}\left( j \right)`

:math:`f_{8}\left( j \right) \  \longrightarrow \ \ f_{8}\left( j \right) \  - f_{3}\left( j \right) C_{3}\left( j \right)`

:math:`f_{5}\left( j \right) \  \longrightarrow \ \ f_{5}\left( j \right) \  - f_{3}\left( j \right) C_{5}\left( j \right)`

:math:`f_{6}\left( j \right) \  \longrightarrow \ \ f_{6}\left( j \right) \  - f_{3}\left( j \right) C_{6}\left( j \right)`

:math:`f_{3}\left( j \right) \  \longrightarrow \ \ 0`

:math:`e_{1}\left( j + 1 \right) \  \longrightarrow \ \ e_{1}\left( j + 1 \right) \  - e_{4}\left( j + 1 \right) c_{3}\left( j \right)`

:math:`e_{9}\left( j + 1 \right) \  \longrightarrow \ \ e_{9}\left( j + 1 \right) \  - e_{4}\left( j + 1 \right) c_{4}\left( j \right)`

:math:`e_{10}\left( j + 1 \right) \  \longrightarrow \ \ e_{10}\left( j + 1 \right) \  - e_{4}\left( j + 1 \right) c_{5}\left( j \right)`

:math:`e_{8}\left( j + 1 \right) \  \longrightarrow \ \ e_{8}\left( j + 1 \right) \  - e_{4}\left( j + 1 \right) c_{6}\left( j \right)`

:math:`e_{4}\left( j + 1 \right) \  \longrightarrow \ 0`

:math:`f_{1}\left( j + 1 \right) \  \longrightarrow \ \ f_{1}\left( j + 1 \right) \  - f_{2}\left( j + 1 \right) c_{5}\left( j \right)`

:math:`f_{4}\left( j + 1 \right) \  \longrightarrow \ \ f_{4}\left( j + 1 \right) \  - f_{2}\left( j + 1 \right) c_{4}\left( j \right)`

:math:`f_{7}\left( j + 1 \right) \  \longrightarrow \ \ f_{7}\left( j + 1 \right) \  - f_{2}\left( j + 1 \right) c_{3}\left( j \right)`

:math:`f_{6}\left( j + 1 \right) \  \longrightarrow \ \ f_{6}\left( j + 1 \right) \  - f_{2}\left( j + 1 \right) c_{6}\left( j \right)`

:math:`f_{2}\left( j + 1 \right) \  \longrightarrow \ \ 0`

8. Multiply non-zero coefficients in :eq:`eq-5.4-35` by :math:`\frac{1}{f_{1} \left( j \right)}`.

:math:`f_{5}\left( j \right) \  \longrightarrow \ \ \frac{f_{5}\left( j \right)}{f_{1}\left( j \right)} \ ;\ \ f_{6}\left( j \right)  \longrightarrow \ \ \frac{f_{6}\left( j \right)}{f_{1}\left( j \right)} ;`

:math:`f_{8}\left( j \right) \  \longrightarrow \ \ \frac{f_{8}\left( j \right)}{f_{1}\left( j \right)} \ ;\ \ f_{1}\left( j \right)  \longrightarrow \ 1`

9.

:math:`e_{1}\left( j + 1 \right) \  \longrightarrow \ \ e_{1}\left( j + 1 \right) \  - e_{9}\left( j + 1 \right) f_{5}\left( j \right)`

:math:`e_{10}\left( j + 1 \right) \  \longrightarrow \ \ e_{10}\left( j + 1 \right) \  - e_{9}\left( j + 1 \right) f_{5}\left( j \right)`

:math:`e_{8}\left( j + 1 \right) \  \longrightarrow \ \ e_{8}\left( j + 1 \right) \  - e_{9}\left( j + 1 \right) f_{6}\left( j \right)`

:math:`e_{9}\left( j + 1 \right) \  \longrightarrow \ 0`

:math:`f_{1}\left( j + 1 \right) \  \longrightarrow \ \ f_{1}\left( j + 1 \right) \  - f_{4}\left( j + 1 \right) f_{5}\left( j \right)`

:math:`f_{7}\left( j + 1 \right) \  \longrightarrow \ \ f_{7}\left( j + 1 \right) \  - f_{4}\left( j + 1 \right) f_{8}\left( j \right)`

:math:`f_{6}\left( j + 1 \right) \  \longrightarrow \ \ f_{6}\left( j + 1 \right) \  - f_{4}\left( j + 1 \right) f_{6}\left( j \right)`

:math:`f_{4}\left( j + 1 \right) \  \longrightarrow \ \ 0`

10. Set :math:`j\ \  \longrightarrow \ \ j + 1`

11. If :math:`j < \text{JMAX}`, go to step 2

12. Multiply the non-zero coefficients in :eq:`eq-5.4-35` by :math:`\frac{1}{f_{1} \left( j \right)}`

:math:`f_{6}\left( j \right) \  \longrightarrow \ \ \frac{f_{6}\left( j \right)}{f_{1}\left( j \right)} \ ;\ \ f_{7}\left( j \right)  \longrightarrow \ \ \frac{f_{7}\left( j \right)}{f_{1}\left( j \right)} ;\ \ f_{1} \left( j \right)  \longrightarrow \ 1`

13.

:math:`e_{1}\left( j \right) \  \longrightarrow \ \ e_{1}\left( j \right) \  - e_{10}\left( j \right) f_{7}\left( j \right)`

:math:`e_{8}\left( j \right) \  \longrightarrow \ \ e_{8}\left( j \right) \  - e_{10}\left( j \right) f_{6}\left( j \right)`

:math:`e_{10}\left( j \right) \  \longrightarrow \ \ 0`

14.

:math:`e_{8}\left( j \right) \  \longrightarrow \ \ \frac{e_{8}\left( j \right)}{e_{1}\left( j \right)} \ ;\ \ e_{1}\left( j \right)  \longrightarrow \ 1`

15.

:math:`\Delta T_{\text{CP}}\left( j \right) = e_{8}\left( j \right)`

:math:`\Delta T_{\text{CT}}\left( j \right) = f_{6}\left( j \right) - f_{7}\left( j \right)\Delta T_{\text{CS}}\left( j \right)`

16. Set :math:`j\ \  \longrightarrow \ \ j - 1`

17. If :math:`j < 1`, go to step 23

18.

:math:`\Delta T_{\text{CT}}\left( j \right) = f_{6}\left( j \right) - f_{5}\left( j \right)\Delta T_{\text{CT}}\left( j + 1 \right) - f_{8}\left( j \right)\Delta T_{\text{CS}}\left( j + 1 \right)`

19.

:math:`\Delta T_{\text{TU}}\left( j \right) = C_{6}\left( j \right) - C_{3}\left( j \right)\Delta T_{\text{CS}}\left( j + 1 \right) - C_{4}\left( j \right)\Delta T_{\text{CS}}\left( j + 1 \right) - C_{5}\left( j \right)\Delta T_{\text{CT}}\left( j + 1 \right)`

20.

:math:`\Delta T_{\text{CS}}\left( j \right) = e_{8}\left( j \right) - e_{5}\left( j \right)\Delta T_{\text{TU}}\left( j \right) - e_{7}\left( j \right)\Delta T_{\text{CS}}\left( j + 1 \right) - e_{10}\left( j \right)\Delta T_{\text{CT}}\left( j \right)`

21.

:math:`\Delta T_{\text{SH}}\left( j \right) = a_{4}\left( j \right) - a_{2}\left( j \right)\Delta T_{\text{CS}}\left( j \right) - a_{3}\left( j \right)\Delta T_{\text{CS}}\left( j + 1 \right)`

22. Go to step 16

23. End