5.16.3. Appendix 5.3: Cover Gas Flow and Pressure Algorithm

The main steps in the cover gas treatment in Section 5.7 may be summarized in the following algorithm:

1. Initialize. Set $I_{\text{n}} = 0$.

2. Start a loop on $I_{\text{o}}$ for $I_{\text{o}}$ = $\text{ICV}1$, …, $\text{ICV}2$. Set up $\text{INEW} \left( I_{\text{o}} \right)$, $\text{IOLD} \left( I_{\text{n}} \right)$, where $I_{\text{o}}$ is the old compressible volume number and $I_{\text{n}}$ is the new compressible volume number for the compressed arrays.

3. Is there gas in this compressible volume? Is $\text{ITYPCV} \left( I_{\text{o}} \right)$ = 2,3,4,5, or 8? If so, go to step 5.

4. No gas in this compressible volume. $\text{INEW} \left( I_{\text{o}} \right) = 0$,

   $p_{4} \left( I_{\text{o}} \right) = p_{3} \left( I_{\text{o}} \right)$,

   $T_{4} \left( I_{\text{o}} \right) = T_{3} \left( I_{\text{o}} \right)$,

   $m_{4} \left( I_{\text{o}} \right) = 0$. Go to the end of the loop, step 6.

5. Gas in this compressible volume.

   $I_{\text{n}} = I_{\text{n}} + 1$

   $\text{INEW} \left(I_{\text{o}} \right) = I_{\text{n}}$

   $\text{IOLD} \left(I_{\text{n}} \right) = I_{\text{o}}$

   ${p'}_{3} \left( I_{\text{n}} \right) = p_{3} \left( I_{\text{o}} \right), {m'}_{3} \left( I_{\text{n}} \right) = m_{3} \left( I_{\text{o}} \right)$ ,

   ${V'}_{3} \left( I_{\text{n}} \right) = V_{3} \left( I_{\text{o}} \right), {V'}_{4} \left( I_{\text{n}} \right) = V_{4} \left( I_{\text{o}} \right)$ ,

   $\tau' \left( I_{\text{n}} \right) = \tau \left( I_{\text{o}} \right)$

6. End of loop on $I_{\text{o}}$. Loop back to step 3 for the next value of $I_{\text{o}}$.

7. $I_{\text{nmax}} = I_{\text{n}}$

8. Start a loop on $I_{\text{n}}$ for $I_{\text{n}}$ = 1, …, $I_{\text{nmax}}$

Compute the adiabatic expansion and heat flow, as well as a temporary array $c \left( i \right)$ (see below).

9. Re-calculate ${p'}_{3} \left( I_{\text{o}} \right)$ giving.

   ${p'}_{3} \left( I_{\text{n}} \right) = {p'}_{3} \left( I_{\text{n}} \right) \left\{ 1 - \gamma \left\lbrack \frac{{V'}_{4} \left( I_{\text{n}} \right) - {V'}_{3} \left( I_{\text{n}} \right)}{{V'}_{4} \left( I_{\text{n}} \right)} \right\rbrack \right\}$

   Calculate ${T'}_{3} \left( I_{\text{n}} \right)$.

   Set $c \left( I_{\text{n}} \right) = \frac{{p'}_{3} \left( I_{\text{n}} \right)\gamma \Delta t_{\text{s}}}{{m'}_{3} \left( I_{\text{n}} \right) {T'}_{3} \left( I_{\text{n}} \right)}$

   Also set $c \left( I, J \right) = 0$, with $c \left( I_{\text{n}}, I_{\text{n}} \right) = 0$,

   and $d \left( I_{\text{n}} \right) = 0$.

10. End of loop on $I_{\text{n}}$. Loop back to step 9 for the next value of $I_{\text{n}}$.

11. Is there more than one gas compressible volume? Is $I_{\text{nmax}} > 1$? If so, go to step 13.

12. Go to step 25.

13. Start loop on I for I = 1, …, $I_{\text{nmax}}-1$.

Initialize $e_{\text{ij}}$, $F_{1\text{ij}}$, $F_{2\text{ij}}$.

14. Start loop on $J$ for $J = I_{\text{n}} + 1$, …, $I_{\text{nmax}}$

15. Set $c \left( I,J \right) = 0.0$, $c \left( J,I \right) = 0.0$, $F_{1} \left( I,J \right) = 0.0$, $F_{1} \left( J,I \right) = 0.0$, $F_{2} \left( I,J \right) = 0.0$, $F_{2} \left( J,I \right) = 0.0$

16. End of loop on $J$. Loop back to step 11 for additional values of $J$.

17. End of loop on I. Loop back to step 14 for additional values of $I$.

18. Are there any gas segments? Is $\text{ISG} 1 > 0$?

If not, go to step 25.

19. Start loop on $I_{\text{s}}$ for $I_{\text{s}}$ = $\text{ISG}1$, …, $\text{ISG}2$.

Compute gas flow between compressible volumes.

20. Find $I_{\text{ni}}$ and $I_{\text{no}}$, the inlet and outlet compressible volumes for gas segment $I_{\text{s}}$.

    $I_{\text{ni}} = \text{INEW} \left( \text{JNODG} \left( 1,I_{\text{s}} \right) \right)$

    $I_{\text{no}} = \text{INEW} \left( \text{JNODG} \left( 2,I_{\text{s}} \right) \right)$

21. Iterate routine to obtain ${F'}_{\text{o}},\ {F'}_{1},\ {F'}_{2},\ T_{\text{ij}}$ (See remarks after Eq. (5.7-12)).

22. $F_{1} \left( I_{\text{ni}},I_{\text{no}} \right) = F_{1} \left( I_{\text{ni}},I_{\text{no}} \right) + {F'}_{1}$

    $F_{1} \left( I_{\text{no}},I_{\text{ni}} \right) = F_{1} \left( I_{\text{no}},I_{\text{ni}} \right) + {F'}_{1}$

    $F_{2} \left( I_{\text{ni}},I_{\text{no}} \right) = F_{2} \left( I_{\text{ni}},I_{\text{no}} \right) + {F'}_{2}$

    $F_{2} \left( I_{\text{no}},I_{\text{ni}} \right) = F_{2} \left( I_{\text{no}},I_{\text{nio}} \right) + {F'}_{2}$

    $c \left( I_{\text{no}},I_{no} \right) = c \left( I_{\text{no}},I_{\text{no}} \right) - e\left( I_{\text{no}} \right) T_{\text{ij}} {F'}_{1}$

    $c \left( I_{\text{no}},I_{\text{ni}} \right) = c \left( I_{\text{no}},I_{\text{ni}} \right) - e\left( I_{\text{no}} \right) T_{\text{ij}} {F'}_{1}$

    $c \left( I_{\text{ni}},I_{\text{ni}} \right) = c \left( I_{\text{ni}},I_{\text{ni}} \right) - e\left( I_{\text{ni}} \right) T_{\text{ij}} {F'}_{2}$

    $c \left( I_{\text{ni}},I_{\text{no}} \right) = c \left( I_{\text{ni}},I_{\text{no}} \right) - e\left( I_{\text{ni}} \right) T_{\text{ij}} {F'}_{2}$

    $d \left( I_{\text{no}} \right) = d \left( I_{\text{no}} \right) + e\left( I_{\text{no}} \right) T_{\text{ij}} {F'}_{0}$

    $d \left( I_{\text{ni}} \right) = d \left( I_{\text{ni}} \right) - e\left( I_{\text{ni}} \right) T_{\text{ij}} {F'}_{0}$

    $F_{\text{os}} \left( I_{\text{ni}} \right) = F_{\text{os}} \left( I_{\text{no}} \right) + {F'}_{0}$

    $F_{\text{os}} \left( I_{\text{ni}} \right) = F_{\text{os}} \left( I_{\text{no}} \right) - {F'}_{0}$

    $F_{\text{os}} \left( I_{\text{s}} \right) = {F'}_{0}$

    $F_{\text{s}1} \left( I_{\text{s}} \right) = {F'}_{1}$

    $F_{\text{s}2} \left( I_{\text{s}} \right) = {F'}_{2}$

23. End of loop on $I_{\text{s}}$. Loop back to step 20 for the next value of $I_{\text{s}}$.

24. Solve the matrix equation for $\Delta p \left( i \right)$ (see Eq. (5.7-20)).

25. Start loop on $I_{\text{n}}$ for $I_{\text{n}} = 1$, … $I_{\text{nmax}}$

26. ${p'}_{4}\left( I \right) = {p'}_{3}\left( I \right) + \Delta p_{\text{i}}$

    ${m'}_{4}\left( I \right) = {m'}_{3}\left( I \right) + \Delta t_{\text{x}} F_{\text{os}}\left( I \right)$

27. End of loop on $I_{\text{n}}$. Loop back to step 26 for additional values of $I_{\text{n}}$.

28. If $I_{\text{nmax}} = 1$, go to step 34.

29. Start loop on $I$ for $I$ = 1, …, $I_{\text{nmax}}-1$.

30. Start loop on $J$ for $J$ = $I + 1$, $I_{\text{nmax}}$.

31. ${m'}_{4} \left( I \right) = {m'}_{4} \left( I \right) + \Delta t_{\text{s}} \left\lbrack F_{1}\left( J,I \right)\Delta p\left( I \right) + F_{2}\left( J,I \right)\Delta p\left( J \right) \right\rbrack$

    ${m'}_{4} \left( J \right) = {m'}_{4} \left( J \right) + \Delta t_{\text{s}} \left\lbrack F_{1}\left( I,J \right)\Delta p\left( J \right) + F_{2}\left( I,J \right)\Delta p\left( J \right) \right\rbrack$

32. End of loop on $J$. Loop back to step 31 for additional values of $J$.

33. End of loop on $I$. Loop back to step 30 for additional values of $I$.

34. For $I = 1$, …, $I_{\text{nmax}}$ set

    ${T'}_{4} \left( I \right) = \frac{{p'}_{4} \left( I \right) {V'}_{4} \left( I \right)}{{n'}_{4} \left( I \right)R}$

35. Start loop on $I_{\text{n}}$ for $I_{\text{n}} = 1$, …, $I_{\text{nmax}}$

36. $I_{o} = \text{IOLD}\left( I_{n} \right)$

    $p_{3}\left( I_{o} \right) = {p'}_{3} \left( I_{n} \right)$

    $m_{3}\left( I_{o} \right) = {m'}_{3} \left( I_{n} \right)$

    $T_{3}\left( I_{o} \right) = {T'}_{3} \left( I_{n} \right)$

37. End of loop on $I_{\text{n}}$. Loop back to step 36 for additional values of $I_{\text{o}}$.

38. If $\text{ISG}1 = 0$, go to step 40.

39. For $I_{\text{s}} = \text{ISG}1$, …, $\text{ISG}2$

    $I_{1} = \text{INEW}\left( \text{JNODG}\left( 1,I_{\text{s}} \right) \right)$

    $I_{2} = \text{INEW}\left( \text{JNODG}\left( 2,I_{\text{s}} \right) \right)$

    $F_{\text{g}4}\left( I_{\text{s}} \right) = F_{\text{so}}\left( I_{\text{s}} \right) + F_{\text{s}1}\left( I_{\text{s}} \right)\Delta p\left( I_{2} \right) = F_{\text{s}2}\left( I_{\text{s}} \right)\Delta p\left( I_{1} \right)$

40. Optional debugging print-out.

41. Return