Rainwater tank modelling
2.2 generAl concepts underlyIng A rAIn wAter tAnK model
Modelling the water volume inside a rainwater tank would seem straight forward compared to, say, modelling the water balance of an open paddock of grass. A rainwater tank is usually a closed vessel so that evaporative and leakage losses can be considered negligible, and there is no direct entry of rainfall.
The water volume inside the tank could then be determined simply by the amount of roof runoff entering the tank, the amount of water drawn off (yield) to supply demand, and the amount that is lost through tank overflow. The equations needed for a rainwater tank simulation model are built upon this simple mass balance concept.
A rainwater tank system is usually comprised of a collection system (i.e., roof, gutter and downpipe) and a storage system (i.e., tank) as shown in Figure 2.1. The collection system consists of the connected roof area A (m2), representing the area of the roof that drains to the tank, and which may be smaller than the total roof area of the household (often not all of the roof area is connected to the tank). The collection system may also include a first flush device which retains a certain volume FF (m3). A first flush device is installed to discard the first amount of roof runoff as a contaminant mitigation measure. The storage component includes a tank with an active storage S (m3), which excludes the dead storage volume (DS) below the supply pipe, and the area above the overflow pipe. In this closed system, water export is via the yield to household Yt, and tank overflow Ot.
DS A
Et Pt IL
Yt CL
It
FF Ot
Vt S
Figure 2.1 Schematic representation of a rainwater tank system.
The roof area collects rainfall Pt (mm) over a time period t. But not all rainfall that falls on the roof becomes inflow to the tank. Some of the rainfall at the beginning of a rainfall event will be lost due to wetting up of the roof surface (adhesion or detention loss), and this initial loss can be represented using a single storage of capacity IL (mm). The initial storage is also subjected to evaporation (Et). After the initial storage is filled, the roof runoff will be further reduced by a continuing loss factor CL (%) representing a continuous percentage loss due to water being splashed or blown off the roof or gutter, and runoff capture by the first flush system. The initial and continuing losses can be represented by the following equations (Coombes & Barry, 2007; Mitchell et al. 2008):
RRt =max[(Pt + RSTt−1−IL), ]0 (2.1)
RSTt = max[(Pt +RSTt−1)− RRt −Et, ]0 (2.2)
HR A RR CL
t = t −
*
1000 1
* 100 (2.3)
where HRt is the harvestable roof runoff (m3), RRt is the roof runoff in mm, RSTt and RSTt−1 are the roof storage level in mm in the current and previous time-step respectively and Et (mm) is the evaporation over a time period t. If a first flush device is present, the inflow to the tank It is given by:
It = max[(HRt +FFIt−1−FFV), ]0 (2.4)
FFIt = max[(HRt +FFIt−1)−It −LR, ]0 (2.5)
where FFIt and FFIt−1 are the volume of roof runoff captured by the first flush device at time t and t−1, respectively; FFV is the capacity of the first flush device and LR is the leakage rate. The leakage rate represents the volume (m3) at which the first flush device drips in one time-step. If the system does not have a first flush device, the inflow to the tank It is equal to the harvestable roof runoff HRt.
The second part of the model uses a simple water balance to simulate the amount of water stored in the tank (Mitchell et al. 2008):
Vt =Vt−1+It −Ot −Yt (2.6)
where Vt and Vt−1 are the volume of water (m3) in the tank at the end of the current time-step and at the end of the previous time-step, respectively; Ot is the tank overflow (m3); and Yt is the yield or volume of water extracted from the tank during time-step t. The water balance in Equation 2.6 assumes that the rainwater tank is a closed vessel (as mentioned previously) and hence direct rainfall and evaporation are excluded.
To solve the water balance given by Equation 2.6 in a model simulation, overflow and yield must be calculated for each time-step. However, tank inflow, yield and overflow can occur in any order or can occur simultaneously during any time-step. If the time-steps are large, it is also possible that multiple events occur within one time-step, but the model will only be able to simulate the multiple events as one lumped event. The discrete nature of Equation 2.6 means that an order for the inflow, yield and overflow processes must be assumed in the calculation process. If yield is assumed to occur before overflow (spill), the model type is called YBS (Yield Before Spill). If spill is assumed to occur before yield, the model type is called YAS (Yield After Spill). Therefore, for a YBS model, the yield Yt for a time t is given by (Jenkins &
Pearson, 1978; Fewkes & Butler, 2000; Liaw & Tsai, 2004):
Yt = min[D Vt, t−1 +It] (2.7)
Vt =min[S V, t−1+It −Yt] (2.8)
where Dt is the demand for tank water (m3) for a time-step t, S is the active storage (m3) which excludes any dead storage below the water level from which yield is taken.
For a YAS model, the equations are (Jenkins & Pearson, 1978; Fewkes & Butler, 2000; Liaw & Tsai, 2004):
Yt = min[D Vt, t−1] (2.9)
Vt =min[S−Y Vt, t−1+It −Yt] (2.10) As the equations for Vt constrains the tank water volume remaining at the end of the time-step to its maximum volume, the excess water is routed as overflow. Hence overflow Ot is calculated as:
For YBS model:
Ot = max[(Vt−1+ It −Yt)−S, ]0 (2.11)
For YAS model:
Ot = max[(Vt−1+It −Yt) (− S−Yt),0]= max[(Vt−1+ It)−S, ]0 (2.12) Comparison of the yield equations for the YBS and YAS models shows that yield will often be greater in the YBS model in each time-step, since it can include the inflow water as part of the yield to meet household demand. Consequently, the overflow will be potentially smaller in the YBS model in each time-step, since tank yield is first removed from any inflow before the remaining inflow is available for overflow. The implication of this ‘order of calculation’ concept on predicting rainwater tank behaviour will be discussed later.
To simulate the complete rainwater tank behaviour, these calculations of yield, overflow and water volume are simply repeated for a continuous series of time-steps, with the tank water volume from the previous time-step becoming the starting tank water volume for the next time-step. In this way, the rainwater tank model outlined above will predict the partitioning of any roof-impacting rainwater into initial roof loss, continuing roof losses and roof runoff. The runoff may then be further partitioned into loss from a first flush device and tank inflow. Any inflow is then added to the water store within the tank.
Meanwhile, the tank water store may undergo further partitioning with some being extracted as yield to the household. If there is insufficient water in the tank to supply all of the household demand for a particular time-step, then the yield will be less than demand. Where the tank storage capacity is exceeded at the end of the time-step, excess tank water is partitioned to overflow.
The inputs required by this model are the connected roof area, effective tank volume, the depth of the roof storage representing initial loss, the proportion of roof runoff that is continuously lost due to splash, the capacity of the first flush device, and the first flush leakage rate. For each time-step, rainfall, evaporation and household water demand is required. The length of the time-step will determine how much variability of these inputs over time can potentially be captured by the modelling. The amount of variability that needs to be captured will depend on the purpose of the modelling, and this will be discussed later.
The outputs from this model are the predicted yield and overflow over the period simulated for the specified roof area and tank storage volume. The long term yield Y can be calculated and compared with the long term demand D to provide the tank performance measure volumetric reliability VR, the proportion of household demand that is likely to be supplied from the rainwater tank where:
Y Yt D D
where T is the total number of time-steps in the period of simulation.
Having described the general components and concepts underlying a physically rigorous rainwater tank model, we will now explore the modelling approaches that have been used to address particular objectives and issues.