5.16.2. Appendix 5.2: IHX Matrix Solution Algorithm

The solution of the matrix represented by Eq. (5.4-44) through Eq. (5.4-47) and whose coefficients are given in Section 5.16.1 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.

$\text{JMAX}$ = the number of nodes in the primary and intermediate coolant

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

1. Set $j$ = 1.

2. Multiply Eq. (5.4-44) by $\frac{1}{a_{1} \left( j \right)}$

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

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

   $e_{7}\left( j \right) \rightarrow e_{7}\left( j \right) - \ e_{3}\left( j \right)a_{3}\left( j \right)$;

   $e_{8}\left( j \right) \rightarrow e_{8}\left( j \right) - e_{8}\left( j \right)e_{4}\left( j \right)$;

   $e_{3}\left( j \right) \rightarrow 0$;

4. Multiply non-zero coefficients in Eq. (5.4-45) by ${1}{e_{1} \left( j \right)}$.

   $e_{5}\left( j \right) \ \longrightarrow \ \ \frac{e_{5}\left( j \right)}{e_{1}\left( j \right)} \ ;\ \ e_{7}\left( j \right) \longrightarrow \ \ \frac{e_{7}\left( j \right)}{e_{1}\left( j \right)} \ ;$

   $e_{8}\left( j \right) \ \longrightarrow \ \ \frac{e_{8}\left( j \right)}{e_{1}\left( j \right)} \ ;\ \ e_{10}\left( j \right) \longrightarrow \ \ \frac{e_{10}\left( j \right)}{e_{1}\left( j \right)}$

   $e_{1}\left( j \right) \ \longrightarrow \ \ 1$

5. $c_{1}\left( j \right) \ \longrightarrow \ \ c_{1}\left( j \right) \ - c_{2}\left( j \right) e_{5}\left( j \right)$

   $c_{3}\left( j \right) \ \longrightarrow \ \ c_{3}\left( j \right) \ - c_{2}\left( j \right) e_{7}\left( j \right)$

   $c_{4}\left( j \right) \ \longrightarrow \ \ c_{4}\left( j \right) \ - c_{2}\left( j \right) e_{10}\left( j \right)$

   $c_{6}\left( j \right) \ \longrightarrow \ \ c_{6}\left( j \right) \ - c_{2}\left( j \right) e_{8}\left( j \right)$

   $c_{2}\left( j \right) \ \longrightarrow \ \ 0$

   $f_{1}\left( j \right) \ \longrightarrow \ \ f_{1}\left( j \right) \ - f_{7}\left( j \right) e_{10}\left( j \right)$

   $f_{3}\left( j \right) \ \longrightarrow \ \ f_{3}\left( j \right) \ - f_{7}\left( j \right) e_{5}\left( j \right)$

   $f_{6}\left( j \right) \ \longrightarrow \ \ f_{6}\left( j \right) \ - f_{7}\left( j \right) e_{8}\left( j \right)$

   $f_{8}\left( j \right) \ \longrightarrow \ \ f_{8}\left( j \right) \ - f_{7}\left( j \right) e_{7}\left( j \right)$

   $f_{7}\left( j \right) \ \longrightarrow \ \ 0$

   $e_{1}\left( j + 1 \right) \ \longrightarrow \ \ e_{1}\left( j + 1 \right) \ - e_{6}\left( j + 1 \right) e_{7}\left( j \right)$

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

   $e_{8}\left( j + 1 \right) \ \longrightarrow \ \ e_{8}\left( j + 1 \right) \ - e_{6}\left( j + 1 \right) e_{8}\left( j \right)$

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

   $e_{6}\left( j + 1 \right) \ \longrightarrow \ \ 0$

6. Multiply non-zero coefficients in Eq. (5.4-46) by $\frac{1}{c_{1} \left( j \right)}$.

   $c_{3}\left( j \right) \ \longrightarrow \ \ \frac{c_{3}\left( j \right)}{c_{1}\left( j \right)} \ ;\ \ c_{4}\left( j \right) \longrightarrow \ \ \frac{c_{4}\left( j \right)}{c_{1}\left( j \right)}$

   $c_{5}\left( j \right) \ \longrightarrow \ \ \frac{c_{5}\left( j \right)}{c_{1}\left( j \right)} \ ;\ \ c_{6}\left( j \right) \longrightarrow \ \ \frac{c_{6}\left( j \right)}{c_{1}\left( j \right)}$

   $c_{5}\left( j \right) \ \longrightarrow \ \ \frac{c_{5}\left( j \right)}{c_{1}\left( j \right)} \ ;\ \ c_{6}\left( j \right) \longrightarrow \ \ \frac{c_{6}\left( j \right)}{c_{1}\left( j \right)}$

7. $f_{1}\left( j \right) \ \longrightarrow \ \ f_{1}\left( j \right) \ - f_{3}\left( j \right) C_{4}\left( j \right)$

   $f_{8}\left( j \right) \ \longrightarrow \ \ f_{8}\left( j \right) \ - f_{3}\left( j \right) C_{3}\left( j \right)$

   $f_{5}\left( j \right) \ \longrightarrow \ \ f_{5}\left( j \right) \ - f_{3}\left( j \right) C_{5}\left( j \right)$

   $f_{6}\left( j \right) \ \longrightarrow \ \ f_{6}\left( j \right) \ - f_{3}\left( j \right) C_{6}\left( j \right)$

   $f_{3}\left( j \right) \ \longrightarrow \ \ 0$

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

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

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

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

   $e_{4}\left( j + 1 \right) \ \longrightarrow \ 0$

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

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

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

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

   $f_{2}\left( j + 1 \right) \ \longrightarrow \ \ 0$

8. Multiply non-zero coefficients in Eq. (5.4-47) by $\frac{1}{f_{1} \left( j \right)}$.

   $f_{5}\left( j \right) \ \longrightarrow \ \ \frac{f_{5}\left( j \right)}{f_{1}\left( j \right)} \ ;\ \ f_{6}\left( j \right) \longrightarrow \ \ \frac{f_{6}\left( j \right)}{f_{1}\left( j \right)} ;$

   $f_{8}\left( j \right) \ \longrightarrow \ \ \frac{f_{8}\left( j \right)}{f_{1}\left( j \right)} \ ;\ \ f_{1}\left( j \right) \longrightarrow \ 1$

9. $e_{1}\left( j + 1 \right) \ \longrightarrow \ \ e_{1}\left( j + 1 \right) \ - e_{9}\left( j + 1 \right) f_{5}\left( j \right)$

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

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

   $e_{9}\left( j + 1 \right) \ \longrightarrow \ 0$

   $f_{1}\left( j + 1 \right) \ \longrightarrow \ \ f_{1}\left( j + 1 \right) \ - f_{4}\left( j + 1 \right) f_{5}\left( j \right)$

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

   $f_{6}\left( j + 1 \right) \ \longrightarrow \ \ f_{6}\left( j + 1 \right) \ - f_{4}\left( j + 1 \right) f_{6}\left( j \right)$

   $f_{4}\left( j + 1 \right) \ \longrightarrow \ \ 0$

10. Set $j\ \ \longrightarrow \ \ j + 1$

11. If $j < \text{JMAX}$, go to step 2

12. Multiply the non-zero coefficients in Eq. (5.4-47) by $\frac{1}{f_{1} \left( j \right)}$

    $f_{6}\left( j \right) \ \longrightarrow \ \ \frac{f_{6}\left( j \right)}{f_{1}\left( j \right)} \ ;\ \ f_{7}\left( j \right) \longrightarrow \ \ \frac{f_{7}\left( j \right)}{f_{1}\left( j \right)} ;\ \ f_{1} \left( j \right) \longrightarrow \ 1$

13. $e_{1}\left( j \right) \ \longrightarrow \ \ e_{1}\left( j \right) \ - e_{10}\left( j \right) f_{7}\left( j \right)$

    $e_{8}\left( j \right) \ \longrightarrow \ \ e_{8}\left( j \right) \ - e_{10}\left( j \right) f_{6}\left( j \right)$

    $e_{10}\left( j \right) \ \longrightarrow \ \ 0$

14. $e_{8}\left( j \right) \ \longrightarrow \ \ \frac{e_{8}\left( j \right)}{e_{1}\left( j \right)} \ ;\ \ e_{1}\left( j \right) \longrightarrow \ 1$

15. $\Delta T_{\text{CP}}\left( j \right) = e_{8}\left( j \right)$

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

16. Set $j\ \ \longrightarrow \ \ j - 1$

17. If $j < 1$, go to step 23

18. $\Delta T_{\text{CT}}\left( j \right) = f_{6}\left( j \right) - f_{5}\left( j \right)\Delta T_{\text{CT}}\left( j + 1 \right) - f_{8}\left( j \right)\Delta T_{\text{CS}}\left( j + 1 \right)$

19. $\Delta T_{\text{TU}}\left( j \right) = C_{6}\left( j \right) - C_{3}\left( j \right)\Delta T_{\text{CS}}\left( j + 1 \right) - C_{4}\left( j \right)\Delta T_{\text{CS}}\left( j + 1 \right) - C_{5}\left( j \right)\Delta T_{\text{CT}}\left( j + 1 \right)$

20. $\Delta T_{\text{CS}}\left( j \right) = e_{8}\left( j \right) - e_{5}\left( j \right)\Delta T_{\text{TU}}\left( j \right) - e_{7}\left( j \right)\Delta T_{\text{CS}}\left( j + 1 \right) - e_{10}\left( j \right)\Delta T_{\text{CT}}\left( j \right)$

21. $\Delta T_{\text{SH}}\left( j \right) = a_{4}\left( j \right) - a_{2}\left( j \right)\Delta T_{\text{CS}}\left( j \right) - a_{3}\left( j \right)\Delta T_{\text{CS}}\left( j + 1 \right)$

22. Go to step 16

23. End