Pump formula (Water Overlay)

Pumps are hydraulic structures can pump water from one location to another.

The direction the pump pumps in is based on the water level at either end of the pump. A pump pumps water from the area with a lower water level to the area with a higher water level. If no (differing) water levels are defined, the surface height is used instead. The direction is determined during the initialization phase of the hydrological model, and does not change during the calculation even when water levels change.

A negative pump speed is also allowed. This will cause the pump to be "draining", rather than "pumping". This will cause water to be moved in the opposite direction.

The amount being pumped or drained is dependent on the water levels on both sides of the pump.

Case 1: Pumping:

First the flow capacities are calculated.

If a positive Pump q is defined:

Q_{p,t}=\Delta t\cdot q_{t}

If a lower threshold T_l,t is defined as well:

Q_{l,t}=min(0,w_{l,t}-T_{l,t})

If an upper threshold T_{u,t} is defined as well:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_{u,t} = max ( 0, T_{u,t} - w_{t,u} )}

If a capacity Q_total is defined:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_c = Q_{total} -\sum_{0..t-1}Q_i}

After calculating the capacities, the actual water pumped upwards is calculated. If any of the terms are undefined, they are not included.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta w = \frac {max( 0 , min( Q_{l,t} , Q_{u,t} , Q_{t} , Q_{c} ) ) }{A}}

Case 2: Draining:
First the flow capacities are calculated.

If a negative Pump q is defined:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_{p,t} = \Delta t \cdot q_t }

If a lower threshold T_l,t is defined as well:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_{l,t} = max ( 0, w_{l,t} - T_{l,t} )}

If an upper threshold T_u,t is defined as well:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_{u,t} = min ( 0, T_{u,t} - w_{t,u} )}

If a capacity Q_total is defined:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_c = Q_{total} - abs ( \sum_{0..t-1}Q_i )}

After calculating the capacities, the actual water pumped upwards is calculated. If any of the terms are undefined, they are not included.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta w = \frac { max( 0 , max( Q_{l,t} , Q_{u,t} , Q_t , -Q_c ) ) }{ A} }

Where:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{t,l}} = The water level at the lower water level entry at time t, relative to datum.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{t,u}} = The water level at the upper water level entry at time t, relative to datum.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{l,t}} = The LOWER_THRESHOLD of the pump at time t.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{u,t}} = The UPPER_THRESHOLD of the pump at time t.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_t} = The PUMP_Q of the pump at time t.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta t} = Computational timestep.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_{total}} = The pump capacity, defined in INLET_CAPACITY of the pump.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_{0..n}} = The calculated previous amounts of flow.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_{l,t}} = The desired amount of water at the entry based on the pump's LOWER_THRESHOLD attribute.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_{u,t}} = The desired amount of water at the exit based on the pump's UPPER_THRESHOLD attribute.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_t} = The PUMP_Q at timestep Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta t} , which is the amount of water the pump can possibly pump (or drain, when negative).
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_c} = The maximum amount of water flow based on the total capacity of the pump.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} = Size of the entry/exit area in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^2} , based on the amount of cells and cell size. Current implementation only has one cell.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta w} = The amount of water flow which takes place.

Pump formula (Water Overlay)

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Pump formula (Water Overlay)

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