.. _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