4. Cycle-averaged flows and salinities

Based on the volumes per locking cycle we now can, for each of the locking heads, determine the total transported volumes with their corresponding salinities. From these volumes the cycle-averaged flows can be determined.

4.1. Fresh side

The combined equation for the fresh side gives the total transport of the entire locking cycle. This equation is as follows:

(1)\[M_F = M_{F,1} + M_{F,2} + M_{F,4}\]

Aside from that we have information about the amount of water that is withdrawn from the fresh side

(2)\[V_F^- = V_{Level,LT} + V_{Ship,Up} + V_{U,F} + Q_{flush} \cdot 2 \cdot T_{open}\]

and the volume that is discharged to the fresh side

(3)\[V_F^+ = V_{Level,HT} + V_{U,F} + V_{Ship,Down}\]

By dividing both volumes by the time spent on a total locking cycle, we can determine the cycle-averaged flows. Each of these flows has a corresponding discharge, and can be connected to cells in a far-field model as a discharge or withdrawal:

  • Withdrawal from the fresh side, with the prevailing salinity \(S_F\):

(4)\[Q_F^- = \frac{ V_F^- }{ T_{cycle} }\]

  • Discharge to the fresh side with a to-be-determined average salinity:

(5)\[Q_F^+ = \frac{ V_F^+ } { T_{cycle} }; S = S_F^+\]

The average salinity for the water discharged to the fresh side is determined from the mass and volume transports:

(6)\[S_F^+ = -\frac{ \left( M_F - V_F^- S_F \right) }{ V_F^+ }\]

In case of stand-alone application, but also to compare with other locks or salt intrusion measure configurations, it can be useful to express the salt transport as a mass flux:

(7)\[{\dot{M}}_F = \frac{ M_F }{ T_{cycle} }\]

4.2. Salt side

The combined equation for the salt side is:

(8)\[M_S = M_{S,2} + M_{S,3} + M_{S,4}\]

Again, we can write down the volumes going to and from the salt side. For the withdrawal that is:

(9)\[V_S^- = V_{Level,HT} + V_{Ship,Down} + V_{U,S,Flush}\]

and the volume that is discharged to the salt side

(10)\[V_S^+ = V_{Level,LT} + V_{U,S,Flush} + V_{Ship,Up} + Q_{flush} \cdot 2 \cdot T_{open}\]

By dividing both volumes by the time spent on a total locking cycle, we can determine the cycle-averaged flows. Each of these flows has a corresponding discharge, and can be connected to cells in a far-field model as a discharge or withdrawal:

  • Withdrawal from the salt side, with the prevailing salinity \(S_S\):

(11)\[Q_S^- = \frac{ V_S^- }{ T_{cycle} }\]

  • Discharge to the salt side with a to-be-determined average salinity:

(12)\[Q_S^+ = \frac{ V_S^+ } { T_{cycle} }; S = S_S^+\]

The average salinity for the water discharged to the salt side is determined from the mass and volume transports:

(13)\[S_S^+ = \frac{ \left( M_S - V_S^- S_S \right) }{ V_S^+ }\]

In case of stand-alone application, but also to compare with other locks or salt intrusion measure configurations, it can be useful to express the salt transport as a mass flux:

(14)\[{\dot{M}}_S = \frac{ M_S }{ T_{cycle} }\]

For an equilibrium state, with the lock operating with constant operation for long periods of time, it obviously holds that

(15)\[{\dot{M}}_S = {\dot{M}}_F\]