.. _section-A5.1:

Appendix 5.1: IHX Matrix Coefficients
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The coefficients in :eq:`eq-5.4-32` for the :math:`j`-th vertical section of the
shell in terms of the quantities defined in :numref:`section-5.4.2.2` are as
follows:

.. math::
	:label: A5.1‑1

	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}}}

.. math::
	:label: A5.1‑2

	a_{2} \left( j \right)  =  - \frac{1}{2} \theta_{2\text{S}} \Delta t H_{\text{S}}\left( j \right)

.. math::
	:label: A5.1‑3

	a_{3} \left( j \right)  =  - \frac{1}{2} \theta_{2\text{S}} \Delta t H_{\text{S}}\left( j \right)

.. math::
	:label: A5.1‑4

	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}}}

.. math::
	:label: A5.1‑5

	{\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

:math:`\theta_{2\text{S}}` = the degree of implicitness for the shell-side
coolant channel

:math:`\Delta t` = the time interval

The coefficients in :eq:`eq-5.4-33` for the :math:`j`-th vertical section of the
shell-side coolant for normal flow (downward) are:

.. math::
	:label: A5.1‑6

	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}}

.. math::
	:label: A5.1‑7

	e_{2}\left( j \right) = 0

.. math::
	:label: A5.1‑8

	e_{3}\left( j \right)  =  - \Delta t P_{\text{S}} H_{\text{S}}\left( j \right) \theta_{2\text{S}}

.. math::
	:label: A5.1‑9

	e_{4}\left( j \right) = 0

.. math::
	:label: A5.1‑10

	e_{5}\left( j \right)  =  - \Delta t S P_{\text{ST}} H_{\text{ST}}\left( j \right) \theta_{2\text{S}}

.. math::
	:label: A5.1‑11

	e_{6}\left( j \right) = 0

.. math::
	:label: A5.1‑12

	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}}

.. math::
	:label: A5.1‑13

	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\}

The same coefficients for reversed flow (upward) in the shell-side
coolant channel are:

.. math::
	:label: A5.1‑14

	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}}

.. math::
	:label: A5.1‑15

	e_{2}\left( j \right)  =  - \Delta t P_{\text{S}}H_{\text{S}}\left( j - 1 \right) \theta_{2\text{S}}

.. math::
	:label: A5.1‑16

	e_{3}\left( j \right) = 0

.. math::
	:label: A5.1‑17

	e_{4}\left( j \right) = - \Delta t S P_{\text{ST}} H_{\text{ST}}\left( j - 1 \right) \theta_{2\text{S}}

.. math::
	:label: A5.1‑18

	e_{5}\left( j \right) = 0

.. math::
	:label: A5.1‑19

	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}}

.. math::
	:label: A5.1‑20

	e_{7}\left( j \right) = 0

.. math::
	:label: A5.1‑21

	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\}

The terms :math:`e_{9} \left( j \right)` and :math:`e_{10} \left( j \right)` have been added
to :eq:`eq-5.4-33` 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

.. math::
	:label: A5.1‑22

	e_{1}\left( jm \right) = 1; \quad e_{2,3,4,5,6,7} \left( jm \right) = 0; \quad e_{8}\left( jm \right) = \Delta T_{\text{CS}}\left( jm \right)

For reversed primary channel flow, they are

.. math::
	:label: A5.1‑23

	e_{1}\left( 1 \right) = 1; \quad e_{2,3,4,5,6,7}\left( 1 \right) = 0; quad e_{8}\left( 1 \right) = \Delta T_{\text{CS}}\left( 1 \right)

and for both cases, they are

.. math::
	:label: A5.1‑24

	e_{2}\left( 1 \right) = 0; \quad e_{4}\left( 1 \right) = 0; \quad e_{6}\left( 1 \right) = 0 \\
	e_{3}\left( jm \right) = 0; \quad e_{5}\left( jm \right) = 0; \quad e_{7}\left( jm \right) = 0

The coefficients in :eq:`eq-5.4-34` for the :math:`j`-th vertical section of the
tube are:

.. math::
	:label: A5.1‑25

	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)

.. math::
	:label: A5.1‑26

	c_{2}\left( j \right) = - \frac{1}{2} \Delta t \theta_{2\text{S}} P_{\text{ST}} H_{\text{ST}}\left( j \right)

.. math::
	:label: A5.1‑27

	c_{3}\left( j \right) = - \frac{1}{2} \Delta t \theta_{2\text{S}} P_{\text{ST}} H_{\text{ST}}\left( j \right)

.. math::
	:label: A5.1‑28

	c_{4}\left( j \right) = - \frac{1}{2} \Delta t \theta_{2\text{T}} P_{\text{TT}} H_{\text{TT}}\left( j \right)

.. math::
	:label: A5.1‑29

	c_{5}\left( j \right) = - \frac{1}{2} \Delta t \theta_{2\text{T}} P_{\text{TT}} H_{\text{TT}}\left( j \right)

.. math::
	:label: A5.1‑30

	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

The coefficients in :eq:`eq-5.4-35` for the :math:`j`-th vertical section of the
tube-side coolant for normal flow (upward) are:

.. math::
	:label: A5.1‑31

	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}}

.. math::
	:label: A5.1‑32

	f_{2}\left( j \right)  =  - \Delta t P_{\text{TT}} H_{\text{TT}}\left( j - 1 \right) \theta_{2\text{T}}

.. math::
	:label: A5.1‑33

	f_{3}\left( j \right) = 0

.. math::
	:label: A5.1‑34

	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}}

.. math::
	:label: A5.1‑35

	f_{5}\left( j \right) = 0

.. math::
	:label: A5.1‑36

	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\}

The same coefficients for reversed flow (downward) in the intermediate
coolant channel are:

.. math::
	:label: A5.1‑37

	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}}

.. math::
	:label: A5.1‑38

	f_{2}\left( j \right) = 0

.. math::
	:label: A5.1‑39

	f_{3}\left( j \right)  =  - \Delta t P_{\text{TT}} H_{\text{TT}}\left( j \right) \theta_{2\text{T}}

.. math::
	:label: A5.1‑40

	f_{4}\left( j \right) = 0

.. math::
	:label: A5.1‑41

	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}}

.. math::
	:label: A5.1‑42

	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\}

The terms for :math:`f_{7} \left( j \right)` and :math:`f_{8} \left( j \right)` have been
added to :eq:`eq-5.4-35` 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

.. math::
	:label: A5.1‑43

	f_{1}\left( 1 \right) = 1; \quad f_{2,3,4,5}\left( 1 \right) = 0; \quad f_{6}\left( 1 \right) = \Delta T_{\text{CT}}\left( 1 \right)

For reversed tube-side channel flow, they are

.. math::
	:label: A5.1‑44

	f_{1}\left( j\max \right) = 1; \quad f_{2,3,4,5}\left( j\max \right) = 0; \quad f_{6}\left( j\max \right) = \Delta T_{\text{CT}}\left( j\max \right)

and for both cases, they are

.. math::
	:label: A5.1‑45

	f_{2}\left( 1 \right) = 0; &\quad f_{4}\left( 1 \right) = 0; \\
	f_{1}\left( j\max \right) = 0; &\quad f_{5}\left( j\max \right) = 0