5.16.1. Appendix 5.1: IHX Matrix Coefficients¶
The coefficients in Eq. (5.4-44) for the \(j\)-th vertical section of the shell in terms of the quantities defined in Section 5.4.2.2 are as follows:
(5.16-1)¶\[a_{1} \left( j \right) = \left( \rho c \right)_{\text{SH}} d_{\text{SH}} + \theta_{2\text{S}} \Delta t H_{\text{S}}\left( j \right) + \theta_{2\text{S}} \Delta t \frac{\left( hA \right)_{\text{snk}}}{P_{\text{S}}}\]
(5.16-2)¶\[a_{2} \left( j \right) = - \frac{1}{2} \theta_{2\text{S}} \Delta t H_{\text{S}}\left( j \right)\]
(5.16-3)¶\[a_{3} \left( j \right) = - \frac{1}{2} \theta_{2\text{S}} \Delta t H_{\text{S}}\left( j \right)\]
(5.16-4)¶\[a_{4} \left( j \right) = - \Delta t H_{\text{S}}\left( j \right) T_{\text{SH}3}\left( j \right) + \Delta t H_{\text{S}}\left( j \right) {\overline{T}}_{\text{CS}3}\left( j \right)
+ \frac{\Delta t\left( hA \right)_{\text{snk}} \left\lbrack T_{\text{snk}} - T_{\text{SH}3}\left( j \right) \right\rbrack}{P_{\text{S}}}\]
(5.16-5)¶\[{\overline{T}}_{\text{CP}3} \left( j \right) = \frac{1}{2} \left\lbrack T_{\text{CS}3}\left( j \right) + T_{\text{CS}3}\left( j + 1 \right) \right\rbrack\]
where
\(\theta_{2\text{S}}\) = the degree of implicitness for the shell-side coolant channel
\(\Delta t\) = the time interval
The coefficients in Eq. (5.4-45) for the \(j\)-th vertical section of the shell-side coolant for normal flow (downward) are:
(5.16-6)¶\[\begin{split}e_{1} \left( j \right) &= \frac{1}{2} A_{\text{c}} {\overline{\rho}}_{\text{CS}}\left( j \right) {\overline{c}}_{\text{CS}}\left( j \right) + \Delta t \frac{{\overline{c}}_{\text{CS}}\left( j \right)}{\Delta z\left( j \right)} \theta_{2\text{S}} \left| w_{\text{S}4} \right| \\
&+ \Delta t P_{\text{S}} H_{\text{S}}\left( j \right) \frac{1}{2} \theta_{2\text{S}} + \Delta t S P_{\text{ST}} H_{\text{ST}}\left( j \right) \frac{1}{2} \theta_{2\text{S}}\end{split}\]
(5.16-7)¶\[e_{2}\left( j \right) = 0\]
(5.16-8)¶\[e_{3}\left( j \right) = - \Delta t P_{\text{S}} H_{\text{S}}\left( j \right) \theta_{2\text{S}}\]
(5.16-9)¶\[e_{4}\left( j \right) = 0\]
(5.16-10)¶\[e_{5}\left( j \right) = - \Delta t S P_{\text{ST}} H_{\text{ST}}\left( j \right) \theta_{2\text{S}}\]
(5.16-11)¶\[e_{6}\left( j \right) = 0\]
(5.16-12)¶\[\begin{split}e_{7}\left( j \right) = \frac{1}{2} A_{\text{CS}} {\overline{\rho}}_{\text{CS}}\left( j \right) {\overline{c}}_{\text{CS}}\left( j \right) - \Delta t \frac{{\overline{c}}_{\text{CS}}\left( j \right)}{ \Delta z\left( j \right)} \theta_{2\text{S}} \left| \ w_{\text{S}4} \right| \\
+ \Delta t P_{\text{S}} H_{\text{S}}\left( j \right) \frac{1}{2} \theta_{2\text{S}} + \Delta t S P_{\text{ST}} H_{\text{ST}}\left( j \right) \frac{1}{2} \theta_{2\text{S}}\end{split}\]
(5.16-13)¶\[\begin{split}e_{8}\left( j \right) = - \Delta t \frac{{\overline{c}}_{\text{CS}}\left( j \right)}{ \Delta z\left( j \right)} \left\{ \theta_{1\text{S}} \left| \ w_{\text{S}3} \right| + \theta_{2\text{S}} \left| w_{\text{S}4} \right| \left\lbrack T_{\text{CS}3}\left( j \right) - T_{\text{CS}3}\left( j + 1 \right) \right\rbrack \right\} \\
+ \Delta t P_{\text{S}} H_{\text{S}}\left( j \right) \left\{ T_{\text{SH}3}\left( j \right) - \frac{1}{2} \left\lbrack T_{\text{CS}3}\left( j \right) + T_{\text{CS}3}\left( j + 1 \right) \right\rbrack \right\} \\
+ \Delta t S P_{\text{S}} H_{\text{ST}}\left( j \right) \left\{ T_{\text{TU}3}\left( j \right) - \frac{1}{2} \left\lbrack T_{\text{CS}3}\left( j \right) + T_{\text{CS}3}\left( j + 1 \right) \right\rbrack \right\}\end{split}\]
The same coefficients for reversed flow (upward) in the shell-side coolant channel are:
(5.16-14)¶\[\begin{split}e\left( j \right) = \frac{1}{2} A_{\text{CS}} {\overline{\rho}}_{\text{CS}}\left( j - 1 \right) {\overline{c}}_{\text{CS}}\left( j - 1 \right) + \Delta t \frac{{\overline{c}}_{\text{CS}}\left( j - 1 \right)}{\Delta z \left( j - 1 \right)} \theta_{2\text{S}} \left| w_{\text{S}4} \right| \\
+ \Delta t P_{\text{S}} H_{\text{S}}\left( j - 1 \right) \frac{1}{2} \theta_{2\text{S}} + \Delta t S P_{\text{ST}} H_{\text{ST}}\left( j - 1 \right) \frac{1}{2} \theta_{2\text{S}}\end{split}\]
(5.16-15)¶\[e_{2}\left( j \right) = - \Delta t P_{\text{S}}H_{\text{S}}\left( j - 1 \right) \theta_{2\text{S}}\]
(5.16-16)¶\[e_{3}\left( j \right) = 0\]
(5.16-17)¶\[e_{4}\left( j \right) = - \Delta t S P_{\text{ST}} H_{\text{ST}}\left( j - 1 \right) \theta_{2\text{S}}\]
(5.16-18)¶\[e_{5}\left( j \right) = 0\]
(5.16-19)¶\[\begin{split}e_{6}\left( j \right) = \frac{1}{2} A_{\text{CS}} {\overline{\rho}}_{\text{CS}}\left( j - 1 \right) {\overline{c}}_{\text{CS}}\left( j - 1 \right) - \Delta t \frac{{\overline{c}}_{\text{CS}}\left( j - 1 \right)}{\Delta z\left( j - 1 \right)} \theta_{2\text{S}} \left| w_{\text{S}4} \right| \\
+ \Delta t P_{\text{S}} H_{\text{S}}\left( j - 1 \right) \frac{1}{2} \theta_{2\text{S}} + \Delta t S P_{\text{ST}} H_{\text{ST}}\left( j - 1 \right) \frac{1}{2} \theta_{2\text{S}}\end{split}\]
(5.16-20)¶\[e_{7}\left( j \right) = 0\]
(5.16-21)¶\[\begin{split}e_{8}\left( j \right) = - \Delta t \frac{{\overline{c}}_{\text{CS}}\left( j - 1 \right)}{\Delta z\left( j - 1 \right)} \left\{ \left( \theta_{1\text{S}} \left| w_{\text{S}3} \right| + \theta_{2\text{S}} \left| w_{\text{S}4} \right| \right) \left\lbrack T_{\text{CS}3}\left( j \right) - T_{\text{CS}3}\left( j - 1 \right) \right\rbrack \right\} \\
+ \Delta t P_{\text{S}} H_{\text{S}}\left( j - 1 \right) \left\{ T_{\text{SS}3}\left( j - 1 \right) - \frac{1}{2} \left\lbrack T_{\text{CS}3}\left( j - 1 \right) + T_{\text{CS}3}\left( j \right) \right\rbrack \right\} \\
+ \Delta t S P_{\text{ST}}\left( j - 1 \right) \left\{ T_{\text{TU}3}\left( j - 1 \right) - \frac{1}{2} \left\lbrack T_{\text{CS}3}\left( j - 1 \right) + T_{\text{CS}3}\left( j \right) \right\rbrack \right\}\end{split}\]
The terms \(e_{9} \left( j \right)\) and \(e_{10} \left( j \right)\) have been added to Eq. (5.4-45) because they appear during the solution of the simultaneous equations. These arrays are set to zero before the solution is begun.
In addition, the boundary conditions for normal shell-side coolant channel flow are
(5.16-22)¶\[\begin{align}
e_{1}\left( jm \right) = 1; && e_{2,3,4,5,6,7} \left( jm \right) = 0; && e_{8}\left( jm \right) = \Delta T_{\text{CS}}\left( jm \right)
\end{align}\]
For reversed primary channel flow, they are
(5.16-23)¶\[\begin{align}
e_{1}\left( 1 \right) = 1; && e_{2,3,4,5,6,7}\left( 1 \right) = 0; && e_{8}\left( 1 \right) = \Delta T_{\text{CS}}\left( 1 \right)
\end{align}\]
and for both cases, they are
(5.16-24)¶\[\begin{split}\begin{align}
e_{2}\left( 1 \right) = 0; && e_{4}\left( 1 \right) = 0; && e_{6}\left( 1 \right) = 0; \\
e_{3}\left( jm \right) = 0; && e_{5}\left( jm \right) = 0; && e_{7}\left( jm \right) = 0
\end{align}\end{split}\]
The coefficients in Eq. (5.4-46) for the \(j\)-th vertical section of the tube are:
(5.16-25)¶\[\begin{split}c_{1}\left( j \right) = \left( \rho c \right)_{\text{TU}} \frac{1}{2} \left( P_{\text{ST}} + P_{\text{TT}} \right) d_{\text{TU}} + \Delta t \theta_{2\text{S}} P_{\text{ST}} H_{\text{ST}}\left( j \right) \\
+ \Delta t \theta_{\text{ST}} P_{\text{TT}} H_{\text{TT}}\left( j \right)\end{split}\]
(5.16-26)¶\[c_{2}\left( j \right) = - \frac{1}{2} \Delta t \theta_{2\text{S}} P_{\text{ST}} H_{\text{ST}}\left( j \right)\]
(5.16-27)¶\[c_{3}\left( j \right) = - \frac{1}{2} \Delta t \theta_{2\text{S}} P_{\text{ST}} H_{\text{ST}}\left( j \right)\]
(5.16-28)¶\[c_{4}\left( j \right) = - \frac{1}{2} \Delta t \theta_{2\text{T}} P_{\text{TT}} H_{\text{TT}}\left( j \right)\]
(5.16-29)¶\[c_{5}\left( j \right) = - \frac{1}{2} \Delta t \theta_{2\text{T}} P_{\text{TT}} H_{\text{TT}}\left( j \right)\]
(5.16-30)¶\[\begin{split}c_{6}\left( j \right) = - \Delta t \left\lbrack P_{\text{ST}} H_{\text{ST}}\left( j \right) + P_{\text{TT}} H_{\text{TT}}\left( j \right) \right\rbrack T_{\text{TU}3}\left( j \right) \\
+ \Delta t P_{\text{ST}} H_{\text{ST}}\left( j \right) \frac{1}{2}\left\lbrack T_{\text{CS}3}\left( j \right) + T_{\text{CS}3}\left( j + 1 \right) \right\rbrack \\
+ \Delta t P_{\text{TT}} H_{\text{TT}}\left( j \right) \frac{1}{2}\left\lbrack T_{\text{CT}3}\left( j \right) + T_{\text{CT}3}\left( j + 1 \right) \right\rbrack\end{split}\]
The coefficients in Eq. (5.4-47) for the \(j\)-th vertical section of the tube-side coolant for normal flow (upward) are:
(5.16-31)¶\[\begin{split}f_{1}\left( j \right) = \frac{1}{2} A_{\text{CT}} {\overline{\rho}}_{\text{CT}}\left( j - 1 \right) {\overline{c}}_{\text{CT}}\left( j - 1 \right) + \Delta t \frac{{\overline{c}}_{\text{CT}}\left( j - 1 \right)}{\Delta z\left( j - 1 \right) S} \left| w_{\text{T}4} \right| \theta_{2\text{T}} \\
+ \Delta t P_{\text{TT}} H_{\text{TT}}\left( j - 1 \right) \frac{1}{2} \theta_{2\text{T}}\end{split}\]
(5.16-32)¶\[f_{2}\left( j \right) = - \Delta t P_{\text{TT}} H_{\text{TT}}\left( j - 1 \right) \theta_{2\text{T}}\]
(5.16-33)¶\[f_{3}\left( j \right) = 0\]
(5.16-34)¶\[\begin{split}f_{4} \left( j \right) = \frac{1}{2} A_{\text{CT}} {\overline{\rho}}_{\text{CT}}\left( j - 1 \right) - \Delta t \frac{{\overline{c}}_{\text{CT}}\left( j - 1 \right)}{\Delta z \left( j - 1 \right) S} \left| w_{\text{T}4} \right| \theta_{2\text{T}} \\
+ \Delta t P_{\text{TT}} H_{\text{TT}} \left( j - 1 \right) \frac{1}{2} \theta_{2\text{T}}\end{split}\]
(5.16-35)¶\[f_{5}\left( j \right) = 0\]
(5.16-36)¶\[\begin{split}f_{6}\left( j \right) = - \Delta t \frac{{\overline{c}}_{\text{CT}}\left( j - 1 \right)}{\Delta z\left( j - 1 \right) S} \left\{ \left( \left| w_{\text{T}3} \right| \theta_{1\text{T}} + \left| w_{\text{T}4} \right| \theta_{2\text{T}} \right) \left\lbrack T_{\text{CT}3}\left( j \right) - T_{\text{CT}3}\left( j - 1 \right) \right\rbrack \right\} \\
+ \Delta t P_{\text{TT}} H_{\text{TT}}\left( j - 1 \right) \left\{ T_{\text{TU}3}\left( j - 1 \right) - \frac{1}{2} \left\lbrack T_{\text{CT}3}\left( j - 1 \right) + T_{\text{CT}3}\left( j \right) \right\rbrack \right\}\end{split}\]
The same coefficients for reversed flow (downward) in the intermediate coolant channel are:
(5.16-37)¶\[\begin{split}f_{1}\left( j \right) = \frac{1}{2} A_{\text{CT}} {\overline{\rho}}_{\text{CT}}\left( j \right) {\overline{c}}_{\text{CT}}\left( j \right) + \Delta t \frac{{\overline{c}}_{\text{I}}\left( j \right)}{\Delta z\left( j \right) S} \left| \ w_{\text{T}4} \right| \theta_{2\text{T}} \\
+ \Delta t P_{\text{TT}} H_{\text{TT}}\left( j \right) \theta_{2\text{T}}\end{split}\]
(5.16-38)¶\[f_{2}\left( j \right) = 0\]
(5.16-39)¶\[f_{3}\left( j \right) = - \Delta t P_{\text{TT}} H_{\text{TT}}\left( j \right) \theta_{2\text{T}}\]
(5.16-40)¶\[f_{4}\left( j \right) = 0\]
(5.16-41)¶\[\begin{split}f_{5}\left( j \right) = \frac{1}{2} A_{\text{CT}} {\overline{\rho}}_{\text{CT}}\left( j \right) {\overline{c}}_{\text{CT}}\left( j \right) - \Delta t \frac{{\overline{c}}_{\text{CT}}\left( j \right)}{\Delta z\left( j \right) S} \left| \ w_{\text{T}4} \right| \theta_{2\text{T}} \\
+ \Delta t P_{\text{TT}} H_{\text{TT}}\left( j \right) \frac{1}{2} \theta_{2\text{T}}\end{split}\]
(5.16-42)¶\[\begin{split}f_{6}\left( j \right) = - \Delta t \frac{{\overline{c}}_{\text{CT}}\left( j \right)}{\Delta z\left( j \right) S} \left\{ \left( \left| w_{\text{T}3} \right| \theta_{1\text{T}} + \left| w_{\text{T}4} \right| \theta_{2\text{T}} \right) \left\lbrack T_{\text{CT}3}\left( j \right) - T_{\text{CT}3}\left( j + 1 \right) \right\rbrack \right\} \\
+ \Delta t P_{\text{TT}} H_{\text{TT}}\left( j \right) \left\{ T_{\text{TU}3}\left( j \right) - \frac{1}{2} \left\lbrack T_{\text{CT}3}\left( j \right) + T_{\text{CT}3}\left( j + 1 \right) \right\rbrack \right\}\end{split}\]
The terms for \(f_{7} \left( j \right)\) and \(f_{8} \left( j \right)\) have been added to Eq. (5.4-47) because they appear during the solution of the simultaneous equations. These arrays are also set to zero before the solution is begun.
Also, the boundary conditions for normal tube-side coolant channel flow are
(5.16-43)¶\[\begin{align}
f_{1}\left( 1 \right) = 1; && f_{2,3,4,5}\left( 1 \right) = 0; && f_{6}\left( 1 \right) = \Delta T_{\text{CT}}\left( 1 \right)
\end{align}\]
For reversed tube-side channel flow, they are
(5.16-44)¶\[\begin{align}
f_{1}\left( j\max \right) = 1; && f_{2,3,4,5}\left( j\max \right) = 0; && f_{6}\left( j\max \right) = \Delta T_{\text{CT}}\left( j\max \right)
\end{align}\]
and for both cases, they are
(5.16-45)¶\[\begin{split}\begin{align}
f_{2}\left( 1 \right) = 0; && f_{4}\left( 1 \right) = 0; \\
f_{1}\left( j\max \right) = 0; && f_{5}\left( j\max \right) = 0
\end{align}\end{split}\]