The development of a generalised equation for the annual water savings through rainwater tanks under different climatic conditions for southwest Brisbane (Australia)

Optimisation of rainwater tank outcomes has been a challenge for the users, as the water savings and reliability depend on many factors such as rainfall amount, rainfall distribution, tank size, roof area and rainwater demand. Among the analysis methods, daily water balance modelling is the most reasonable and acceptable approach. However, general end-users can hardly interpret/grasp the outcomes of such analyses. To overcome this hurdle, this paper presents the development of a generalised equation for calculating expected water savings through rainwater tanks with major contributing factors like rainfall, rainwater demand, tank size and roof area for southwest Brisbane (Australia). Daily rainfall data was collected for the station, Oxley which is in southwest Brisbane from the Australian Bureau of Meteorology website. The expected annual water savings for different combinations of rainfall, tank size, roof area and demand are presented in the form of charts that were converted to a generalized equation having independent variables of rainfall, tank size, roof area and demand. Results from the developed equation were seen matching with the model (eTank) simulated results. Such an equation can be easily incorporated into mobile apps or computer programs to help quick calculations for expected water savings.


Introduction
Water is a precious resource available on earth. Many countries around the world face an acute shortage of water. The situation in Australia is also alarming. It experiences mean annual rainfall less than 600mm for more than 80 per cent of its area [1]. On top of that, the pressure of constantly increasing population in conjunction with global warming and climate change has led to an increase in water consumption. This has posed great stress on water authorities which are struggling to fulfil the water demand of the people. The water demand is of two types, potable water demand such as water required for drinking, cooking, bathing, etc and non-potable water demand such as water required for toilet flushing, laundry, gardening, car washing, etc. As the non-potable water demand does not require water to be of high quality, therefore, the non-potable water demand can be easily met by adopting water-saving techniques like greywater recycling, wastewater recycling and stormwater harvesting. Since the rainwater is relatively clean and clear from impurities, it is preferred more than all other alternatives. Although, a communal rainwater tank was used in southeast Queensland to meet the potable demand of a small urban IOP Publishing doi: 10.1088/1757-899X/1067/1/012042 2 development, but the energy used for treating the water was four times higher than the energy used in a centralised system [2]. A rainwater harvesting system (RWHS) not only saves the potable water supplied by the water authorities but also reduces the volume of surface runoff into the sewer networks and solves the problem of traffic congestion during rainfall. A set of dimensionless curves were developed to determine the required storage capacity of the rainwater tank based on the desired level of performance [3]. It was showed that the peak flow in a sewer system can be reduced if properly designed rainwater tanks are installed at a vast scale [4]. A researcher suggested that the unit cost of rainwater can be reduced by increasing the consumption rate [5]. In Brazil, rainwater captured from the roof of the fuel station was utilised to wash the vehicles at the fuel station and it managed to save 32 per cent potable water [6]. It was found that a rainwater tank of 40 m 3 can provide water savings ranging from 30 to 60 per cent depending on the water demand [7]. Some studies related to financial benefits and payback period were also carried out for shopping centre [8,9], office building [10] and residential buildings [11,12,13,14,15].
Rainfall does not happen uniformly at every location. Some locations receive a high amount of rainfall whereas other nearby locations receive it in low amount. Moreover, it also varies from year to year at the same location. Hence, an average rainfall value adopted for designing a rainwater harvesting system may not give realistic results. To illustrate this, Imteaz et al., [16] investigated the spatial and climatic variation in annual water savings in Melbourne and concluded that difference in water savings is significant except for low demand and large roof area scenarios. A similar study was also carried out for Adelaide [17], Sydney [18] and Kathmandu [19]. It was also found that the size of the storage tank is overestimated if mean monthly rainfall data is used instead of daily rainfall data [20].
The initial investment required to adopt a rainwater harvesting system in a house is a matter of concern for many people. Therefore, to encourage its acceptance, the government provides incentives and rebates to homeowners for its installation. An investigation was carried out for 10 different locations in greater Sydney and it was found that a high rainfall is strongly correlated with high water savings [21]. It was also concluded that the existing rebate provided by the government is unable to make the benefit-cost ratio greater than or equal to unity for the homeowners [21]. On the other hand, a variable rebate scheme was proposed depending on the annual water savings at different locations within the Sydney metropolitan area [22]. It is certainly an attractive scheme for homeowners and probably an optimisation of the government's spending on subsidies for promoting RWHS installation.
Some studies related to the optimum size of the tank were also carried out in different parts of the world. A set of dimensionless curves were developed that were useful in finding the optimum size of a rainwater tank for the Greater Melbourne area [23]. A mathematical model based on a linear programming approach was used to evaluate the optimum size of the tank in Northern Cyprus [24]. A simple spreadsheet-based daily water balance model was developed to design the size of a rainwater tank connected to a large roof [25]. A user-friendly regional regressive model was developed to estimate the water savings from RWHS in the region [26]. In one study, different detailed methods were analysed to size a rainwater tank and found that the 80% Efficiency criteria provide the best benefit-cost ratio [27]. The potential for potable water savings was investigated for 195 cities in southeastern Brazil which were observed to range from 12% to 79% per year [28]. It was suggested that a proper design and evaluation method is needed to amplify the performance of the RWHS [29].
Imteaz et al., [30] found that a 100% reliability is unachievable in a dry year for a relatively small roof (100m 2 ) connected to a very large tank (10,000L). The performance of RWHS in 3 cities of Iran was evaluated and it was concluded that the annual water savings are directly proportional to the amount of annual rainfall [31]. The water-saving potential of RWHS was investigated for multi-unit buildings in 3 Australian cities and reported that a large tank can provide significant water savings even in dry years [32]. Although the installation of RWHS makes sure that some water is saved by the end of the day but a reasonable payback period is also desired. The potential of RWHS for high rise buildings in 4 Australian cities was studied and found Sydney to have the shortest payback period [33]. As usual, like other systems, RWHS also requires periodic inspection and maintenance for its efficient working and long life [34]. Most of the studies that were carried out to calculate the potential water savings involved mathematical/probabilistic/statistical analyses which are difficult to understand or interpret by a general end-user. Although, some equations were developed to calculate the expected water savings through rainwater tanks, however, these equations were only valid for a particular climatic condition [35,36]. As such, it becomes complicated for a general end-user to use different equations for different climatic conditions for one particular region. Contrary to it, a more versatile and user-friendly equation is required to make it easier and a better option for a general end-user.
Although Brisbane is the third most populous city in Australia, such equation has never been developed for this city. This paper presents the development of a generalised equation to calculate the expected annual water savings through rainwater tanks under five different climatic conditions with major contributing factors as rainfall, tank size, demand and roof area for southwest Brisbane. The expected annual water savings for different rainfall, tank size, demand and roof area are presented in the form of charts that were converted to a generalized equation having independent variables of rainfall, tank size, demand and roof area. Results from the developed equation were seen matching with the model (eTank) simulated results. Such an equation can be used by a general end-user who would like to calculate the expected annual water savings through rainwater tanks by merely using a calculator or a spreadsheet. This equation can also be incorporated into mobile apps or computer programs to help quick calculations for expected annual water savings.

Methodology
An earlier developed daily water balance model, eTank [37] was used for the evaluation of rainwater tank outcomes. The model considers the daily rainfall, tank size, rainwater demand (indoor and outdoor), roof area and the losses that occur while the rainwater goes into the rainwater tank. For this study, a 15% loss was considered which included spillage, evaporation, first flush and leakage [38]. This model can calculate the annual water savings, town water use, overflow, reliability and the outdoor use of water. However, the development of a generalised equation for annual water savings was the objective of this study. Hence, other outcomes like town water use, overflow, reliability and the outdoor use of water are not taken into consideration.
For each climatic condition (dry, mild dry, average, mild wet and wet), expected annual water savings were calculated for different combinations of input conditions like tank size (2500L, 5000L, 7500L and 10,000L), rainwater demand (200L/day, 300L/day, 400L/day and 500L/day) and roof area (150m 2 , 200m 2 , 250m 2 and 300m 2 ). It should be noted that these are the standard tank sizes used in Australia and the typical values of rainwater demand and roof area for an Australian household. Following this, for each tank size, a set of curves were produced between annual water savings and roof area for different demand scenario. From these curves, a generalised equation was developed for each climatic condition using the best fit technique. Ultimately, from these equations, a single equation was developed for the station under consideration. The process of the development of a generalised equation is described as follows: -After drawing the curves between annual water savings and roof area for different demand scenarios, it was observed that each curve follows a logarithmic pattern and hence it can be represented as follows:-= 1 * ln( ) ± 1 (1) Where AWS denotes the annual water savings, RA is the roof area, x1 and y1 is the coefficient and intercept respectively. As four rainwater demands were considered, therefore, four such equations were derived using the best fit technique yielding four different values for x1 and y1.

Data
This study was carried out for southwest Brisbane. Based on the direction from the Central business district and the availability of data, a station was selected. The details regarding the station and the climatic years are described in Table1. The good quality rainfall data is available at the bureau of meteorology website (http://reg.bom.gov.au/climate/data/) and was downloaded for the selected station. Five different climatic conditions namely dry, mild dry, average, mild wet and wet years were adopted after doing the statistical analysis of the rainfall data. These climatic years were categorised based on the percentile value of the annual rainfall [30]. For instance, a year having annual rainfall value closer to 10 percentile value denotes a dry year. Similarly, a year having annual rainfall value closer to 25 percentile value denotes a mild dry year, a year having annual rainfall value closer to mean value denotes an average year, a year having annual rainfall value closer to 75 percentile value denotes a mild wet year and lastly, a year having annual rainfall value closer to 90 percentile value denotes a wet year. A single year may represent an unrealistic rainfall pattern (i.e., sporadic bursts and/or longer dry periods). Therefore, 4 additional years were selected for each climatic condition. They were selected such that 2 years have a rainfall value immediately higher and 2 years have a rainfall value immediately lower than the rainfall value of the year that is taken into consideration. The daily rainfall data of different climatic years were inserted in the model (eTank). Expected annual water savings were then calculated for different combinations of tank size, rainwater demand and roof area.

Development of the generalised equation
Annual water savings were calculated using the model (eTank) for a dry year and tank size of 2500L for different combinations of rainwater demand and roof area. Figure 1 shows the relationship between annual water savings and roof area for different demand scenario. Each curve in figure 1 follows a logarithmic pattern and hence can be expressed as a single logarithmic equation using the best fit technique.
For 300L/day demand, = 11.37 * ln( ) − 7.72 (18) Where 'AWS' represents the expected annual water savings in kL and 'RA' represents the roof area in m 2 . In the above equations, the coefficients (5.64, 11.37, 16.6 and 20.27) and the intercepts (-14.16, 7.72, 30.72 and 47.31) are correlated with the demand. Therefore, using the best fit technique, the abovementioned coefficients and the intercepts are expressed as equation (21) and equation (22) The coefficients and the intercepts of the first term in equations (23)- (26) can be replaced with equation (27) and equation (28) respectively. Again, the coefficients and the intercepts of the second term in equations (23)- (26) can be replaced with equation (29) and equation (30) respectively. As such, a single equation is obtained as follows to calculate the expected annual water savings for the dry year that contains independent variables like roof area, rainwater demand and tank size: - Similarly, four more equations were developed for four remaining climatic years i.e., mild dry, average, mild wet and wet year.
All the above mentioned climatic years are categorised according to the percentile value of annual rainfall. The coefficients (22.76, 23.5, 22.55, 13.86 and 18.83) and the intercepts (162.1, 165.1, 159.4, 90.9 and 136.9) of the first term are correlated with the annual rainfall and can be expressed as equation (36) and equation (37)

Conclusions
Most of the studies related to rainwater harvesting have been carried out using advanced tools or programs. These tools are usually not user friendly and require some level of expertise to handle them. A general end-user may not possess this level of expertise and hence may not be able to work out the amount of annual water savings. To overcome this hurdle, this paper presents the development of a generalised equation for calculating the expected water savings through rainwater tanks with major contributing factors like rainfall, rainwater demand, tank size and roof area for southwest Brisbane (Australia). Results from the developed equation were seen matching with the model (eTank) simulated results in the figures (2)-(6). Such an equation is very useful for a general end-user as it makes it convenient for a person to calculate the annual water savings by merely using a calculator or a spreadsheet. The equation can even be used in making a mobile application to help quick calculations. It should be noted that this equation is only valid for a particular region i.e., south-west Brisbane. The development of similar equation for other regions can be a target of future research.