TANK SIZING FROM RAINFALL RECORDS FOR RAINWATER HARVESTING UNDER CONSTANT DEMAND

BY

JACQUELINE ELSA ALLEN

920407518

A dissertation submitted to the Faculty of Engineering and the Built Environment in partial fulfilment of the requirements for the degree

MAGISTER INGENERIAE

IN

CIVIL ENGINEERING SCIENCE

in the

Faculty of Engineering and the Built Environment

AT

UNIVERSITY OF JOHANNESBURG

(AUCKLAND PARK KINGSWAY CAMPUS)

STUDY LEADER: PROF J. HAARHOFF

AUGUST

2012

Abstract

In recent years, there has been an international trend towards installing rainwater tanks in

an attempt to save water. However, there are no clear guidelines for determining the

optimal size of such a tank in South Africa. This study investigates the possibility of

simplifying the process of sizing a rainwater tank for optimal results. It utilises daily data

from four rainfall stations, namely Kimberley, Mossel Bay, Punda Maria and Rustenburg,

obtained from the South African Weather Services. The water use is considered to be for

indoor purposes only, therefore assuming a constant daily demand to be extracted from the

tank. The required size of a rainwater tank is influenced by the MAP, the area of the roof

draining into the tank, the water demand (both the average demand and seasonal

variations), the desired reliability of supply, and the rainfall patterns. The first step in

simplifying the process is to consolidate the above variables. The tank volume is expressed

as the number of days it could supply the average daily water demand. Another variable is

created which provides the ratio of the total water volume which could theoretically be

harvested from the roof in an average year, to the total water demand, from the tank, for a

year. This has the effect of consolidating the MAP, the roof area, the water demand and the

tank volume into two variables only and eliminates the need to consider numerous demand

values. Using simulations over 16 years for each location, the relationships between these

variables were determined to ensure 90%, 95% and 98% assurance of supply. These

relationships were generalised with the equation DD=a(DAU-1)b+c, which was found to

produce a good fit, taking DD as the number of days of storage and DAU as the ratio of total

amount of water available to the total amount of water demand. This model allows the

rapid determination of the required tank size for a given roof area, once the location and

assurance of supply is provided. The procedure presented, provides a reasonable correlation

with the simulated results, but, at times, does over-estimate the required size. The study

suggests that the proposed method is feasible, however further research is required in

order to refine the method.

i

Table of Contents

List of Figures ............................................................................................................................ iv

List of Tables .............................................................................................................................. v

List of Abbreviations and Acronyms .......................................................................................... v

1 Problem Statement ............................................................................................................ 1

2 Introduction ....................................................................................................................... 2

2.1 Scope and Limitations ................................................................................................. 3

3 Literature Review ............................................................................................................... 5

3.1 Worldwide Use of Rainwater Tanks in the Urban Environment ................................. 5

3.2 Use of Rainwater Tanks in South Africa ...................................................................... 9

3.2.1 Legal provisions concerning rainwater harvesting in South Africa ..................... 9

3.2.2 Domestic use of rainwater tanks in South Africa ................................................ 9

3.2.3 Agricultural use of rainwater tanks in South Africa ........................................... 11

3.2.4 Suitability of rainwater tanks in South Africa .................................................... 12

3.3 Sizing of Rainwater Tanks .......................................................................................... 13

3.3.1 Sizing methods ................................................................................................... 13

3.3.2 Simulation of a rainwater tank .......................................................................... 15

3.3.3 Expressing the reliability of a rainwater tank .................................................... 17

3.3.4 Selecting the operating rule ............................................................................... 18

3.3.5 The time-step ..................................................................................................... 19

3.3.6 Selecting the length of the simulation period ................................................... 21

3.3.7 Selecting the initial condition ............................................................................ 22

3.3.8 Ratio between rainwater demand and rainwater availability ........................... 22

4 Method ............................................................................................................................ 23

4.1 Rainwater Tank Simulation ....................................................................................... 23

ii

4.1.1 Operating rule .................................................................................................... 23

4.1.2 Conservation of volume ..................................................................................... 23

4.1.3 Time-step ........................................................................................................... 26

4.1.4 Rainfall stations selected ................................................................................... 26

4.1.5 Length of simulation period ............................................................................... 27

4.1.6 Initial tank condition .......................................................................................... 27

4.1.7 Effect of demand ................................................................................................ 28

4.1.8 Describing the reliability of the tank ................................................................. 29

4.2 Simplifying the Process ............................................................................................. 31

4.2.1 Reducing the variables ....................................................................................... 31

4.2.2 Calibrating the model ........................................................................................ 32

4.2.3 Directly predicting the modelling constants ...................................................... 32

4.3 Verification of the Time-Step .................................................................................... 33

5 Results .............................................................................................................................. 36

5.1 Rainwater Tank Simulation ....................................................................................... 36

5.1.1 Graphs obtained ................................................................................................ 36

5.1.2 Range of possible tank sizes .............................................................................. 39

5.1.3 Local design curves ............................................................................................ 40

5.2 Generalising the Modelling Constants ...................................................................... 45

5.2.1 Parameters considered ...................................................................................... 45

5.2.2 Effect of MAP on tank size ................................................................................. 46

5.2.3 Best correlation for a ......................................................................................... 48

5.2.4 Best correlation for b ......................................................................................... 49

5.2.5 Best correlation for c ......................................................................................... 50

5.3 Proposed Sizing Method ........................................................................................... 51

5.4 Testing the Validity of the Time-step ........................................................................ 53

iii

5.5 Testing the Accuracy of the Design Curves ............................................................... 55

5.5.1 Local design curves (i.e curves for Kimberley, Mossel Bay, Punda Maria and

Rustenburg) ...................................................................................................................... 55

5.5.2 Generalised design curves (i.e. curves obtained using the DSM) ...................... 56

6 Conclusion ........................................................................................................................ 61

7 Recommendations for Future Research .......................................................................... 64

8 References ....................................................................................................................... 66

iv

List of Figures

Figure 3-1: Distribution of rainwater tanks in South Africa, in absolute numbers (Mwenge

Kahinda, et al., 2010) ............................................................................................................... 10

Figure 3-2: Percentages of rural household use of water sources (DWAF, 2007) .................. 11

Figure 3-3: In-field RWH suitability map .................................................................................. 12

Figure 3-4: The Yield-Before-Spillage operating rule for a single time-step (Mitchell, 2007) . 18

Figure 3-5: The Yield-After-Spillage operating rule for a single time-step (Mitchell, 2007) ... 19

Figure 4-1: Locations of the four rainfall stations.................................................................... 27

Figure 4-2: Comparison of constant and variable demand ..................................................... 29

Figure 4-3: Plot showing the valid interval for a daily time-step ............................................. 35

Figure 5-1: Simulation curves for Kimberley............................................................................ 36

Figure 5-2: Simulation curves for Mossel Bay.......................................................................... 37

Figure 5-3: Simulation curves for Punda Maria ....................................................................... 37

Figure 5-4: Simulation curves for Rustenburg ......................................................................... 38

Figure 5-5: Graph showing DD=a(DAU-1)b+c, DD=c and DAU =1............................................ 42

Figure 5-6: Local design curves for Kimberley ......................................................................... 43

Figure 5-7: Local design curves for Mossel Bay ....................................................................... 43

Figure 5-8: Local design curves for Punda Maria ..................................................................... 44

Figure 5-9: Local design curves for Rustenburg ....................................................................... 44

Figure 5-10: Graph of a vs. MAP .............................................................................................. 46

Figure 5-11: Graph of b vs. MAP .............................................................................................. 47

Figure 5-12: Graph of c vs. MAP .............................................................................................. 47

Figure 5-13: Best correlation for a ........................................................................................... 49

Figure 5-14: Best correlation for b (note that there is no constant progression from (90% to

98%) ......................................................................................................................................... 50

Figure 5-15: Best correlation for c ........................................................................................... 51

Figure 5-16: Validity of time-step for 90% volumetric reliability ............................................ 54

Figure 5-17: Validity of time-step for 95% volumetric reliability ............................................ 54

Figure 5-18: Validity of time-step for 98% volumetric reliability ............................................ 55

v

Figure 5-19: Comparison of simulation to DSM for Kimberley................................................ 58

Figure 5-20: Comparison of simulation to DSM for Mossel Bay .............................................. 58

Figure 5-21: Comparison of simulation to DSM for Punda Maria ........................................... 59

Figure 5-22: Comparison of simulation to DSM for Rustenburg ............................................. 59

List of Tables

Table 4-1 Effect of a changing a constant demand from 100 l/day to 150 l/day .................... 28

Table 5-1: Range of tank sizes obtained .................................................................................. 40

Table 5-2: Table of coefficients of determination for trend lines ........................................... 41

Table 5-3: Constants of local design curves ............................................................................. 45

Table 5-4: Difference in required tank size determined by simulation and local design curves

or generalised design curves ................................................................................................... 56

Table 5-5: Coefficient of determination for the DSM when compared to the simulated

results ....................................................................................................................................... 60

List of Abbreviations and Acronyms

DAU: Dimensionless area units

DD: Days of demand

DSM: Dimensionless sizing method

MAP: Mean annual precipitation

RH: Rainwater harvesting

RHS: Rainwater harvesting system

1

1 Problem Statement

Internationally there has been a trend towards implementing rainwater harvesting as a form

of water conservation as well as a flood control strategy. However in South Africa,

implementation, especially in urban areas has been limited. In urban areas, rainwater

harvesting systems are generally installed from necessity caused by drought or an

inadequate water supply from a piped water distribution system. Implementation of

rainwater harvesting for stormwater management is unusual. Possible reasons for the lack

of implementation include a lack of financial incentive to install rainwater harvesting

systems and the fact that no design guidelines for rainwater harvesting systems exist for

South Africa. In order to address an aspect of the current lack of design guidelines, this

study investigates the feasibility of a simplified sizing method for rainwater tanks in South

Africa. The study uses rainfall data, supplied by the South African Weather Services, to first

simulate the behaviour of a rainwater tank and then attempts to develop a simplified sizing

method from the simulation results.

2

2 Introduction

Rainwater harvesting (RH) is a general term used to refer to the collection, storage and use

of rainwater for both domestic and agricultural purposes. In recent years there has been an

international trend towards the utilisation of rainwater harvesting systems (RHS) both as a

water conservation technique and a form of stormwater management. In a number of

countries, this has been encouraged by government, either through subsidies or by the

introduction of laws which make the implementation of such systems mandatory

(Herrmann, et al., 1999; GDRC, 2007; Gold Coast City Council, 2005; Texas Water

Development Board, 2005).

Current implementation of RHSs in South Africa is generally due to necessity with

implementation for water conservation or stormwater management being unusual (Jacobs,

et al., 2011; Mwenge Kahinda, et al., 2010). Most RHSs in South Africa are found in rural

areas and are used as an alternative source of water to augment poor service supply. In

urban areas, implementation is generally related to drought conditions, with households

using rainwater to supplement the municipal water supply when water restrictions are in

place.

The lack of large scale implementation of RHSs in South African urban areas may be related

to the fact that to have a significant impact, even for individual households, large tanks are

required. These large tanks are aesthetically unappealing and require a large initial capital

expenditure, making them uneconomical. In conjunction with this problem is the fact that

there are currently no guidelines or subsidies provided for sizing or implementation of such

systems in South Africa. Research is being done on the feasibility of RH, especially for small

scale agricultural use, but there is no significant research into the sizing of RHSs based on

domestic demand in South Africa.

3

The sizing of the tank of a RHS is complicated due to the large number of variables that play

a role in determining the optimum tank size, which means that every tank must be

individually sized. The most common approach used for sizing the tank of a RHS is a water

balance approach in which the behaviour of the tank is simulated over time. In order to

obtain an accurate result the simulation must be run over a period of at least 10 years with

a relatively small time step (generally a daily time-step) and thus requires a relatively

detailed and complete rainfall record, often difficult to obtain.

This study investigates the development of a simplified method for sizing the tank of a RHS

in South Africa. The water balance approach is used to simulate the behaviour of tanks at

four locations in South Africa. The results of the simulations are then used to identify

correlations between parameters obtained from the rainfall records and the required tank

size. These correlations provide a simplified method for sizing a rainwater tank, the results

of which are compared to the simulated results in order to determine the accuracy of the

method.

2.1 Scope and Limitations

The current study is a preliminary study on the feasibility of a simplified rainwater tank

sizing method which can be applied at any location within South Africa and is suitable for all

roof sizes. As such data from all climatic regions were used in order to develop a possible

sizing method and simulations for large roof areas or large tank sizes were included in the

study.

The study considers a constant demand intended for indoor use. The harvested water would

thus be used for toilet flushing and laundry purposes, but not for garden irrigation. This

assumed usage requires a high reliability of supply, as such the study is conducted for 90%,

95% and 98% reliability.

4

As the study is a preliminary study, the quality of the harvested rainwater was not

considered. This means that no first flush system was included in the study. The efficiency of

the harvesting system was also neglected for the purpose of the study. If the results of the

study show that the developed sizing method shows promise, future studies would need to

incorporate such considerations.

5

3 Literature Review

3.1 Worldwide Use of Rainwater Tanks in the Urban Environment

In recent years, the trends towards sustainable development and Integrated Urban Water

Management have led to worldwide encouragement of water re-use and RH. A number of

countries have introduced subsidies for installing rainwater tanks and, in some cities, the

installation of rainwater tanks has become a mandatory requirement for any new building.

A number of these cities have found that the rainwater tanks provide significant reductions

in the amount of water supplied through the citys water supply system and some have also

found that rainwater tanks decrease the stormwater runoff and thus assist in stormwater

management (Hamdan, 2009). In this section, selected cases from both developed and

developing countries are reviewed.

In Berlin, Germany, a basement rainwater tank was installed in a large scale urban

development (Daimler Chrysler Potsdamer Platz) in 1998. This tank collected water from the

roofs of 19 buildings. The purpose of the system was to decrease the stormwater in order to

prevent flooding, save water and also to improve the micro-climate of the area. Another

system in Berlin collected rainwater from streets and parking areas as well as roofs and

diverted it to a 160 m3 tank. This water was then treated and used for flushing toilets and

watering gardens. An estimated 58% of the stormwater runoff from the area was retained

by this system (GDRC, 2007). During the 1990s more than 100 000 rainwater storage tanks,

providing water for non-potable use, were installed in Germany. A number of city councils in

Germany provided financial incentives to promote the use of rainwater tanks as an

alternative water supply. For instance, if a RHS was installed and used, the storm water

taxes did not need to be paid (Herrmann, et al., 1999).

Another example from the developed world is Tokoyo (Japan). In Tokyo, both the Ryogoku

Kokugikan Sumo-wrestling Arena and the Sumida City Hall installed systems which diverted

6

rainwater into underground storage tanks. The rainwater was then used for flushing toilets

and also in the air conditioning system of the buildings. After the successful implementation

of these systems, a number of other public facilities introduced similar systems. By 2007,

approximately 750 buildings in Tokyo, both public and private, had introduced RHSs (GDRC,

2007).

Bangladesh is an example of a developing country. Contamination of the groundwater

supply with arsenic in parts of Bangladesh has led to the use of RHSs to provide drinking

water. Studies have shown that this water is safe for drinking and can be stored for up to 5

months without fear of bacterial contamination. Starting in 1997, approximately 1000 RHSs

were installed in the country and the Forum for Drinking Water Supply & Sanitation has

been promoting rainwater as a safe water supply in urban areas (GDRC, 2007).

Since 2000, more than 2 million rainwater tanks have been installed in Gansu Province in

China. These tanks have a combined storage volume of more than 73 million m3, supply

drinking water to 1.97 million people and help to irrigate 236 400 ha of land in the province.

Seventeen provinces in China have adopted RH as an alternative water source and a total of

5.6 million rainwater tanks have been installed in the country (GDRC, 2007).

RHS also offers great, but as yet unrealised, potential for households in Barcelona, Spain. It

has been found that a small tank could supply a family in Barcelona with enough water to

flush their toilets and do their washing without using the municipal supply. It has also been

found that increasing the tank size would allow for a substantial portion of the familys

irrigation requirements to be met by the tank. The study found that large scale

implementation of such systems would be extremely beneficial in Barcelona (Domenech, et

al., 2011).

7

In Bermuda, limestone glides have been added to roofs to divert rainwater into storage

tanks, which are generally underground. These systems are regulated by a Public Health Act

requiring the roof area to be painted with a non-toxic paint, free from metals. The Act also

required owners to keep their roofs, gutters and pipes clean and to clean the tank at least

once every six years (GDRC, 2007).

After the tsunami of 2004, the general population was left without a water supply system in

Banda Aceh (Indonesia). Initially the people relied on shallow wells that had been dug by

individual households. It was later discovered that a number of these wells were

contaminated and the water was not safe for drinking. A project was initiated in 2007, in

which a number of RHSs were installed and monitored. The study determined that RH is the

most suitable method of supplying the community with water. Indonesia is an ideal place

for RH due to the high annual rainfall combined with the fact that the rain is evenly

distributed throughout the year, thus eliminating the need for large tanks to provide enough

water throughout a dry season (Teh, et al., 2009).

RHS was a prerequisite for obtaining a residential building permit in St. Thomas, US Virgin

Islands. A single-family house was required to have a system which drained an area of at

least 112 m2 into a tank which had to provide at least 45 m3 in storage volume. Although no

limits have been put in place on the materials used for the system, the usage of the water

has been limited to non-potable uses unless treatment is provided (GDRC, 2007).

In Seoul, South Korea, a major real-estate development known as Star City implemented a

RHS as a flood control measure in 2007. The development consisted of 4 high-rise buildings

located on a 5ha site. This RHS was found to adequately control a 50-year flood. The system

included a network which monitors the level of water in the tanks at the Central Disaster

Prevention Agency in the City Office. Depending on the level in the tanks and the amount of

rain expected, it was possible for the agency to issue an order for the building owners to

8

empty their tanks, so as to ensure that the system provided adequate protection against the

predicted flood (Mun, 2011).

A number of RH guidelines have been developed for particular cities or areas around the

world. Examples of these guidelines are the Texas Manual on Rainwater Harvesting,

released by the Texas Water Development Board (2005) in the USA and the Interim

Rainwater Tank Guidelines released by the Gold Coast City Council (2005) in Australia. Both

of these documents contain information on the functional requirements of RHSs, the basic

design requirements and the method which should be used to design such a system.

Unfortunately, very few authorities have compiled such guidelines and there are none

available for most of the worlds cities. Furthermore, the water laws in most countries do

not encourage the use of such systems (Hamdan, 2009).

In both Australia and Poland, life cycle costing determined that RHSs are not a financially

viable option for individual home owners. A 2010 study by Rahman et al, conducted in

Australia, found that it would take a typical homeowner approximately 30 years to recoup

the cost of a tank without government subsidies. In a number of countries where rainwater

tanks are utilised on a large scale, it is done because of necessity, due to lack of adequate

water supply, or due to government intervention - either by introducing regulations making

RH mandatory or by providing individual homeowners with substantial financial incentives

to install such systems (Slys, 2009; Rahman, et al., 2010).

A number of cases where successful implementation of RH by utilising rainwater tanks can

be found internationally. This suggests that large scale deployment of rainwater tanks could

have a significant effect on the water volume supplied to urban areas by municipal supply

systems in South Africa. However further investigation is required to determine the effect of

such an implementation in South Africa.

9

3.2 Use of Rainwater Tanks in South Africa

3.2.1 Legal provisions concerning rainwater harvesting in South Africa

The legal provisions pertaining to RH in South Africa are contained in the National Water Act

(NWA) (Act No 36 of 1998), with the Water Services Act (WSA) (Act No 108 of 1997) playing

a limited role in determining the legal requirements for utilising such a system. Under the

NWA any taking of water from a resource requires a licence, unless stipulated in section

22 of the act. Section 22 of the act allows a user to take water directly from any water

resource to which the user has lawful access for reasonable domestic use and specifically

allows the storage and use of runoff water from a roof. However it does state that the use

of water for commercial purposes (such as farming) is not covered by this section. This

means that the use of a domestic RHS without a licence can be deemed legal in South Africa,

unless the local municipality has by-laws enforcing the registration of such a system

(Mwenge Kahinda, et al., 2007; Jacobs, et al., 2011).

3.2.2 Domestic use of rainwater tanks in South Africa

The use of rainwater tanks in South African urban areas is relatively uncommon. This is

mainly due to the high cost of installing a rainwater tank, the fact that they are aesthetically

unappealing and the limited saving (in monetary terms) that they can offer. The potential

yield of a typical RHS in Johannesburg would only be able to supply approximately 36% of

the water required for garden irrigation (Jacobs, 2010). In addition to this, the seasonal

rainfall which prevails over most of South Africa means that a very large tank is required to

capture enough water during the rainy season to provide any significant supply during the

dry season. This entails a large initial capital cost with relatively small financial savings,

making the installation of such a tank uneconomical for the individual home owner (Jacobs,

et al., 2011; Mwenge Kahinda, et al., 2008).

10

The use of rainwater tanks in rural areas is more common and, in some cases, rainwater

tanks are relied upon as the primary source of drinking water. This is especially true of areas

in the Eastern Cape and KwaZulu-Natal. Figure 3-1 shows the number of rainwater tanks

found in each of the provinces of South Africa in 2010. However, according to the 2006

water services coverage as shown in Figure 3-2, released by the Department of Water

Affairs and Forestry, domestic rainwater harvesting (DRWH) is one of two least used water

sources in rural communities, with only 0.3% of rural households using domestic RH as a

source of water (Mwenge Kahinda, et al., 2010).

Figure 3-1: Distribution of rainwater tanks in South Africa, in absolute numbers (Mwenge Kahinda, et al., 2010)

11

Figure 3-2: Percentages of rural household use of water sources (DWAF, 2007)

3.2.3 Agricultural use of rainwater tanks in South Africa

The use of rainwater tanks as a water supply for irrigation supply is well established and

rainwater tanks can be found on commercial farms. However, the use of RH by small-scale

farmers is negligible and should be encouraged as it has the potential to cater for the

irrigation needs of the population in areas of the country where there is no readily available

water source (Mwenge Kahinda, et al., 2011).

The influence of implementing field RH for small scale irrigation purposes on catchment

runoff was investigated in two catchments and found to be negligible (Mwenge Kahinda, et

al., 2009). Another study stated that the implementation of RH in the Modder River basin

has decreased the annual runoff by approximately 26%, but that further research must be

done to determine what the impact of this decrease will be on downstream areas

(Tetsoane, et al., 2008).

12

3.2.4 Suitability of rainwater tanks in South Africa

In South Africa, the use of rainwater tanks could be used as an alternative water source for

people in areas where bulk supply lines are not available. This would not provide savings in

terms of monetary gains to the households, but would provide savings by obviating the

need for a reticulation system. It would also have a social impact. Amongst other things, it

would decrease the time individuals take to carry water from the source to their homes.

This suggests that the feasibility of installing a rainwater tank for the individual should be

determined by considering the social impact as well as the installation costs. A set of

suitability maps for field RH in South Africa, which include the social impact have been

developed. The maps were based on aridity zones, rainfall, land cover, soil cover, ecological

sensitivity and socio-economic aspects. An example of such a map can be seen in Figure 3-3

below. As can be seen in the figure, most of the country falls into either the moderate or

high suitability zones (Mwenge Kahinda, et al., 2008).

Figure 3-3: In-field RWH suitability map

13

3.3 Sizing of Rainwater Tanks

3.3.1 Sizing methods

Methods used to size rainwater tanks can be categorised as simplified methods (including

graphical methods), matrix methods, statistical methods and simulation methods. The

simplified methods should only be used as preliminary design methods: they require little

calculation, but are not very accurate. The most commonly used method is a simulation

method based on the conservation of volume, as this method provides accurate results and

also provides continuous results for the inflow, outflow and volume of water available in the

tank at a given time (Palla, et al., 2011; Liaw, et al., 2004).

3.3.1.1 Simplified sizing methods

Demand Side Approach

When using the demand side approach, the storage requirement is assumed to be equal to

the largest demand which must be supplied by the tank. This is simply calculated by

determining the daily water demand and multiplying by the number of days in the average

longest dry period. This method assumes that the roof is large enough to capture the

required amount of rain. It does not account for the possibility that the rainfall patterns may

not provide enough rain to fill the tank immediately before the dry period. This approach is

best suited to areas with year-round rainfall. Due to the assumptions this approach entails,

the result can lead to a less reliable water supply than expected (Centre for Science and

Environment). Due to the fact that the method ignores important parameters in the sizing of

rainwater tanks it is not comparable to data driven methods.

A variation on this method for sizing rainwater tanks has been developed in the United

Kingdom (Fewkes, et al., 2000). The method requires the calculation of an input ratio, which

is defined as A*(MAP)/D, where A is the catchment area, MAP is the MAP and D is the

average annual demand. This ratio is then used along with a desired performance level to

determine the number of days storage that ought to be provided by means of a set of

14

design curves. The design curves are curves showing the storage period versus the

volumetric reliability for different values of the input ratio. The method uses a single curve

for a given input ratio, as it was found that in the United Kingdom, very little difference

exists from one location to another (Fewkes, et al., 2000).

Supply Side Approach

The supply side approach is generally used for areas which have a long dry period. The

required tank size is assumed to be large enough to capture the maximum amount of rain in

the wet season. This assumption makes it easy to over-size the tank and so this method can

lead to an unnecessarily expensive design (Rees, et al., 2002). The supply side approach

provides a maximum limit for the tank size and would provide a required tank size which is

larger for a large roof area than that obtained when using a smaller roof size, which

contradicts the resulted obtained when simulating the behaviour of a rainwater tank. The

supply side approach does not consider the effect of demand on the required tank size and

is thus, not comparable to a data driven process, such as simulating the behaviour of a tank.

The United Kingdoms Environment Agency recommends a method, which combines the

Supply and Demand Side Approaches. The method determines the required size of the tank

as being either a user-defined percentage of the MAP (5% is suggested for the UK), or of the

average annual demand, whichever is smaller (Ward, et al., 2010). Ward, Memon and Butler

(2010) state that the size is then calculated as:

=

Where: P= user defined percentage

A= area of the roof

Cf= runoff coefficient

F = system filter efficiency

R= annual rainfall

R would be replaced by the annual demand (D), if D

15

3.3.1.2 Computer simulation methods

The most accurate method for determining the optimal tank size is to simulate the

behaviour of the tank by using a computer model. A number of computer models which are

capable of performing such a simulation are available. These include the Model for Urban

Stormwater Improvement Conceptualisation (MUSIC), Aquacycle and Raincycle (Ward, et

al., 2010).

Studies conducted by Ghisi (2010) and Ward (2010) compared results from various methods

as well as considering the possibility of using a simplified sizing method. Both studies

determined that each rainwater tank must be individually sized based, on the specific

location, catchment size and water demand. The study by Ward (2010) also found that the

use of a single average rainfall value was not adequate, and recommended using a

simulation process to obtain an accurate size.

3.3.2 Simulation of a rainwater tank

Sizing a rainwater tank can become rather complicated as the required size is dependent on

a number of factors. The optimal tank size depends on the collection system, the area of the

roof from which water will be collected, the volume of water that the tank will be required

to supply, whether the demand will be constant or variable, the required reliability of the

water supply from the tank, the depth of precipitation which can be expected and the

rainfall pattern of the area where the tank will be situated (Ghisi, 2010). The most common

method for determining the correct size for a rainwater tank is thus to do a continuous

simulation of the tank behaviour for a given rainfall record. The continuous simulation of a

rainwater tank is popular, because the mathematics required is very easily understood, it is

relatively easy to develop a model or spreadsheet which will do such a simulation, the

behaviour of the system is accurately mimicked and it is easy to incorporate seasonal

changes in rainfall or water demand (Fewkes, et al., 2000).

16

3.3.2.1 Conservation of volume approach

The conservation of volume approach simply ensures the volume of water in the tank at the

end of a given time-step must be equal to the volume of water in the tank at the beginning

of the time-step (end of the previous time-step) plus the amount of water that has entered

the tank during the given time-step minus the water which has left the tank during the time-

step and is given by the following equation (Male, et al., 2006; Su, et al., 2009).

= 1 + 3-1

This is subject to the constraint that the maximum volume of water in the tank is limited to

the size of the tank i.e. 0 .

Where: Vt = volume of rainwater in the tank at the end of time interval t

Qt = volume of rainwater that enters the tank during time interval t

Dt = demand (volume of water that is removed from the tank) during

time interval t

S = maximum storage capacity.

3.3.2.2 Yield-after-spill and yield-before-spill operating rules

Inflow and outflow from a tank is a continuous process in time. By approximating this

process with discrete time intervals, the inflow and outflow over the time interval has to be

reduced to a point inflow and point outflow.

The literature suggests two different approaches that can be used to overcome the

problem. The first option is the yield-after-spill (YAS) approach. For this approach, the

rainwater is added to the tank and the spillage is immediately removed by limiting the tank

volume. At the end of the time-step, the water demanded from the tank is subtracted from

a volume which can never exceed the maximum storage volume. In reality the water could

be removed and rainwater added to the tank simultaneously. This could mean that the

maximum limit would not be reached in reality even though it is reached computationally.

When spillage does occur, this option effectively decreases the tank capacity and thus

underestimates the amount of water which can be supplied by the tank. For this approach

the following equations can be used to determine the yield from the tank and the volume of

water in the tank (Fewkes, et al., 2000; Liaw, et al., 2004; Palla, et al., 2011).

17

= ; 1 3-2

= 1 + ; 3-3

Where Yt= yield from the tank during the time interval t

Vt = volume of rainwater in the tank at the end of time interval t

Qt = volume of rainwater that enters the tank during time interval t

Dt = demand (volume of water that is removed from the tank) during

time interval t

S = maximum storage capacity.

The second approach, the yield-before-spill (YBS) approach, effectively does not allow the

tank to spill during the time step. It collects all the inflow, and satisfies the full demand. If

any surplus remains at the end of the time step, this surplus is spilled. This has the opposite

effect on the tank capacity and thus overestimates the water which can be supplied by the

tank. This approach is described by the following equations (Fewkes, et al., 2000; Liaw, et

al., 2004; Palla, et al., 2011).

= ; 1 + 3-4

= 1 + ; 3-5

Where the symbols are as described above.

3.3.3 Expressing the reliability of a rainwater tank

The performance of a rainwater tank could be assessed by either the volumetric reliability

(water-saving efficiency) or a time-based reliability. The volumetric reliability is calculated by

dividing the total volume of rainwater supplied by the tank by the total volume of water

which is required during the simulation period (total demand). The time based reliability is a

measure of the amount of time for which the tank provides the total amount of water

18

demanded. It can be calculated by dividing the number of time-steps when the full demand

is met by the total number of time-steps (Liaw, et al., 2004; Palla, et al., 2011).

The volumetric reliability method is not as greatly influenced by the choice of time-step. It

also has the advantage of taking into account all of the water that the tank provides. The

time-based reliability disregards any water supplied by the tank when the demand is only

partially met and, consequently, the time-based reliability will be zero for all tank volumes

below a certain size, whereas the volumetric reliability will reflect values which continue to

decrease with tank size (Liaw, et al., 2004; Mitchell, 2007; Palla, et al., 2011).

3.3.4 Selecting the operating rule

Figures 3-4 and 3-5 show graphical representations of the computational process for a single

time-step for the yield-before-spillage and yield-after-spillage operating rules respectively.

The figures show the temporary storage which is created by using the yield-before-storage

operating rule, as well as the reduced effective capacity that results from using the yield-

after-spillage operating rule. By comparing the position of the final storage level in the two

figures it can be seen that the yield-after-spillage operating rule will provide storage

volumes which are lower than those obtained by using the yield-before-spillage operating

rule. The yield-after-spillage operating rule thus provides larger values for the required tank

size and is a more conservative operating rule than the yield-before-spillage operating rule.

Figure 3-4: The Yield-Before-Spillage operating rule for a single time-step (Mitchell, 2007)

19

Figure 3-5: The Yield-After-Spillage operating rule for a single time-step (Mitchell, 2007)

Liaw and Tsai (2004) investigated the effect of the operating rule on both the volumetric

and time based reliability of a rainwater tank. They found that the reliabilities determined

using the yield-before-spillage operating rule exceeded those calculated using the yield-

after-spillage operating rule. They also found that when using the yield-after-spillage

operating rule for tanks where the ratio of demand for a given time-step to storage capacity

was greater than 0.5, the time based reliability fell rapidly to zero. They also found that

there was a minimum storage capacity required for the time based reliability to exceed zero

and for these reasons they recommended using the yield-before-spillage operating rule.

Similarly, a study by Mitchell (2007) found that the yield-before-spillage operating rule

provided an overestimate of the volume of water which could be supplied by a tank. The

study also determined that the effect of the temporary storage or reduced effective

capacity was increased for a larger time-step. The author suggested using both of the

operating rules and taking the average of the two sets of results as the final result.

3.3.5 The time-step

The simplified sizing methods for rainwater tanks generally use a single average rainfall

value to estimate the required size of a rainwater tank. However, the use of an average

value disregards the variability of the rainfall and thus does not provide an accurate result.

20

As such, it is suggested that continuous simulations using smaller time-steps should be used

in order to obtain a realistic and appropriate tank size (Male, et al., 2006; Ward, et al.,

2010).

Fewkes and Butler (2000) determined that the selection of an appropriate time-step for

simulating a given rainwater tank should be based on the storage fraction of the given tank.

The study found that larger time-steps could be used for larger storage tanks, without

compromising the accuracy of the results obtained. The appropriate intervals for given time-

steps are given below.

= =

=

1000

3-6

Where S and A are as previously defined

Hourly models: Sf 0.01 (small storage volumes)

Daily models: 0.01

21

For yield-after-spillage operating rule: Constant demand < 0.7 3-7

Seasonal demand < 0.35 3-8

For yield-before-spillage operating rule: Constant demand < 0.18 3-9

Seasonal demand < 0.09 3-10

If the ratio between the demand in a given time-step and the storage volume of the tank is

larger than the appropriate value given in Equations 3-7, 3-8, 3-9 and 3-10, selecting a

smaller time-step, will decrease the demand within a given time-step and thus decrease the

value of the ratio, if the maximum storage volume of the tank (S) remains constant.

3.3.6 Selecting the length of the simulation period

The length of the simulation period is determined by two considerations, namely the effect

on tank size relating to the volumetric reliability and possibly an effect on the required time-

step.

The first consideration was investigated by Liaw and Tsai (2004). The study found that the

accuracy of the calculated volumetric reliability improved as the length of the simulation

period increased. As such they suggested using a record length of 50 years.

Mitchell (2007) further investigated the effect of an increased simulation period and found

that a 10-year simulation period provided results within 3% of those obtained by using a 50-

year simulation period. Consequently, the author suggested using a 10-year simulation

period, which was representative of the long-term rainfall trends at the site in question, as

this required less computational effort and was also easier to obtain than 50-year records,

but did not significantly compromise the accuracy of the results.

Mitchell (2007) also showed that the 10-year simulation period had the same level of

sensitivity to the selected time-step as the 50-year simulation period. Thus, the length of the

simulation period did not affect the selection of an appropriate time-step in any way. This

means that the length of the simulation can be selected independently of the time-step.

22

3.3.7 Selecting the initial condition

Mitchell (2007) determined that starting the simulation with an empty tank provided more

accurate long term estimates regardless of the period over which the simulation was run.

The author also commented that it was likely that a tank would be used as soon as it was

installed, rather than waiting for it to be filled first. Consequently, he suggested that the

simulation be started with an empty tank.

3.3.8 Ratio between rainwater demand and rainwater availability

Ghisi (2010) attempted to find a correlation between the ratio of rainwater demand to

availability on the one hand and the tank capacity on the other. Ghisi referred to the ratio as

factor F which is the inverse of the input ratio described by Fewkes and Warm (discussed in

section 3.3.1.1, above). No correlation between the ratio of rainwater demand to availability

on the one hand and the tank capacity on the other could be found. However, it was found

that the range of possible tank sizes is smaller for a higher value for factor F.

=( ) (1000)(365)

() 3-11

23

4 Method

4.1 Rainwater Tank Simulation

4.1.1 Operating rule

As discussed in section 3.3.4 above, two possible operating rules can be used to simulate

the behaviour of a rainwater tank, namely the yield-before-spill and the yield-after-spill

operating rules. This study uses the suggestion made by Mitchell (2007) that a simulation

should be run using both operating rules and then the average of the two sets of results

should be taken as the final result. This should provide more accurate results as the yield-

before-spillage operating rule has been found to underestimate the required tank size and

the yield-after-spillage has been found to overestimate the required size. It is assumed that

taking the average of the two will balance the errors of the two rules and thus provide a

tank size which is closer to the actual size required.

4.1.2 Conservation of volume

The first step in simulating the behaviour of a rainwater tank was to set up the equations so

as to ensure conservation of volume throughout the simulation. Microsoft Excel was used to

perform the following sequence of calculations for each time-step.

4.1.2.1 Yield-after-spill

First the volume of rain water which can be collected from the tank is calculated

=

1000 4-1

24

The volume of rainwater which is added to the tank during the time-step is then the minimum of the water that can be collected or the volume of empty space remaining in the tank, as shown in Equation 4-2.

= ; 1 4-2

The volume of spillage, or overflow from the tank is then calculated as the difference

between the amount of rainwater which could be collected during the time-step and the

volume of water which was added to the tank, or zero, if this is less than zero.

= + 1 ; 0 4-3

The yield from the tank during the time-step is then equal to the demand during the time-

step, or the amount of water in the tank at the beginning of the tank plus the amount of

water which has been added to the tank, whichever is smaller

= ; 1 + 4-4

Finally the volume of water in the tank at the end of the time-step can be calculated as the volume of water in the tank at the beginning of the tank plus the volume of water which is added to the tank during the time-step minus the yield from the tank during the time-step

= 1 + 4-5

Where:

Vp = Volume of precipitation on catchment (m3)

A = Area of catchment (m2)

Pt = Precipitation during time-step t (mm)

Qt = Volume of water added to tank during time-step (m3)

S = Maximum storage volume of tank (m3)

Vt-1 = Volume of water in tank at the beginning of time-step (m3)

25

Vs = Volume of spillage (m3)

Yt = Yield during time-step (m3)

Dt = Water demand during time-step (m3)

Vt = Volume of water in the tank at the end of the time-step (m3)

4.1.2.2 Yield-before-spill

As for the yield-after-spill operating rule, the volume of rain water which can be collected

from the tank is calculated

=

1000 4-6

In the case of the yield-before-spill the yield during the time-step is calculated next. This is

the minimum of the demand during the time-step or the volume of water in the tank plus

the volume of water which is added to the tank during the time-step

= ; 1 + 4-7

Next the volume of spillage or overflow from the tank is calculated as the maximum of the

volume of rainfall which can be collected plus the volume of water that was in the tank at

the beginning of the time-step minus the maximum storage volume of the tank and minus

the yield from the tank during the time-step

= + 1 ; 0 4-8

26

Finally the volume of water in the tank at the end of the time-step is then calculated as the

volume of water in the tank at the beginning of the time-step plus the volume of water

added to the tank during the time-step minus the yield from the tank and minus the volume

of spillage

= 1 + 4-9

Where the symbols are as listed above.

4.1.3 Time-step

Studies by Fewkes and Butler (2000) and Mitchell (2007) provided methods for determining

the appropriate time-step. Both methods required using the tank volume to determine the

time-step. This created a problem, as the current study required a selected time-step to

determine an appropriate tank volume. A daily time-step was used as starting point and its

validity was subsequently checked once the tank volume had been determined.

4.1.4 Rainfall stations selected

Rainfall data for four rainfall stations (Kimberley, Mossel Bay, Punda Maria and Rustenburg)

obtained from the South African Weather Services was used for simulating a rainfall tank at

each location. The locations of the four stations can be seen in Figure 4-1, shown below. The

rainfall stations selected included three locations which have a long dry season, namely

Kimberley, Punda Maria and Rustenburg. These locations would thus require large tanks in

order to provide water throughout the dry period. Mossel Bay, however, receives rainfall

throughout the year, with no significant dry season, thus providing an indication of the

effect of abundant rainfall on the storage capacity of a rainwater tank.

27

Figure 4-1: Locations of the four rainfall stations

4.1.5 Length of simulation period

Mitchell (2007) suggested that a 10-year simulation period provided a good balance

between computational effort and accuracy. Liaw and Tsai (2004), however, showed that

the accuracy of a simulation would continue to improve, by as much as 3% as the length of

the simulation period increased. As each rainfall station had data for a 16 year period (1994-

2009) readily available and the simulations were run using Microsoft Excel, the

computational effort demanded by a 16-year simulation period was not significantly greater

than that required for a 10 year period. Consequently the period from January 1994 to

December 2009 was used to simulate rainwater tanks at the four stations.

4.1.6 Initial tank condition

The simulations used a tank which was initially empty, as Mitchell (2007) had suggested that

doing so provided the best results and it was also considered to be the most likely scenario,

as the tank would be empty after installation.

28

4.1.7 Effect of demand

Both the DAU and the tank volume in terms of DD are ratios which are dependent on the

daily demand which will be extracted from the tank. The magnitude of the daily demand will

thus have no effect on the simulation results as long as the demand remains constant for

the duration of the simulation period. This can be confirmed by comparing the results of the

simulation for 90% volumetric reliability at Kimberley for 100 l/day to the results obtained

when the demand is changed to 150 l/day, these can be seen in Table 4-1, below. As

expected, there was no difference in the required DD obtained for a given value of DAU.

This clearly showed the advantage of using the dimensionless approach, as it allows the

same design curves to be used for any value of demand.

Table 4-1 Effect of a changing a constant demand from 100 l/day to 150 l/day

100l/d 150l/d

Dimensionless area units Days of Demand Days of Demand

1.5 109.7106 109.7106 2 98.5475 98.5475

2.5 89.2081 89.2081 3 83.5493 83.5493

3.5 80.1511 80.1511 4 77.0900 77.0900

4.5 74.0566 74.0566 5 71.8348 71.8348

5.5 69.6333 69.6333 6 67.4506 67.4506

6.5 65.5462 65.5462 7 63.7439 63.7439

7.5 61.9984 61.9984 8 60.6486 60.6486

8.5 59.3995 59.3995 9 58.2509 58.2509

9.5 57.2067 57.2067 10 56.1673 56.1673

29

Figure 4-2 shows the results of a simulation done using a constant demand and one done

after adjusting the demand to account for the fact that less water will be used when there is

less water available in the tank. In the latter case, the demand was assumed to decrease by

50% when the tank was less than 50% full. From the figure it was clear that the effect of a

variable demand on the simulation results is significant and thus results based upon a

constant demand could not be assumed to hold true for a variable demand. Using a

constant demand is thus a conservative approach, for tanks with DAU >2.

Figure 4-2: Comparison of constant and variable demand

4.1.8 Describing the reliability of the tank

A decision had to be made as to how the tank reliability would be calculated. Based on the

literature review, two possible approaches could be followed:

0

20

40

60

80

100

120

140

160

0 2 4 6 8 10

DD

(d

ays)

DAU

Kimberley (90%)

Constant

Variable

30

The volumetric reliability of a rainwater tank is the ratio of the total volume of water

supplied by the tank to the total water demand over the simulation period.

The time based reliability is a measure of the amount of time for which the tank

provides the total amount of water demanded.

The volumetric reliability is less affected by the time-step which is used for simulation as

the time based reliability ignores any time step where the demand is greater than the

supply. The volumetric reliability also considers all of the water supplied by the tank,

whereas the time based reliability does not take the water supplied during a time-step

where the entire demand cannot be met into account. It is for these reasons, that the

volumetric reliability was used rather than time based reliability.

The volumetric reliability is determined by:

=

4-10

Where RV is the volumetric reliability

Yt is the yield from the tank in a given time-step

Dt is the demand for the time-step.

The simulations calculated the volumetric reliability by using the average result of the yield-

before-spill and the yield-after-spill operating rules.

31

4.2 Simplifying the Process

4.2.1 Reducing the variables

In order to simplify the sizing process, the number of variables was decreased. This was

done by combining the tank volume and the daily demand from the tank to obtain a number

of DD, as can be seen in the equations below. The roof area and MAP can also be combined,

as shown below, to obtain a dimensionless area for the roof size, the dimensionless area

unit (DAU) is the inverse of the factor F, used by Ghisi (2010) and is equal to the input ratio

described by Fewkes and Warm (2000).

The tank volume is described as DD defined in Equation 4-11 below

=

3

3

= 4-11

The DAU could then be described as:

=

2

3

=2

10003653

= 4-12

Where DD Days of Demand

DAU Dimensionless Area Units

The DD is an indication of the tank size and the DAU provides an indication of the theoretical

maximum amount of rain water that will be able to enter the tank. By using Equations 4-11

and 4-12 a number of data points can be obtained for the DD versus DAU for a particular

weather station. The effect of tank size, daily demand, roof size and MAP are all

incorporated into these graphs.

32

The spreadsheet, which was created to simulate the rainwater tank, provided the

volumetric reliability as a percentage of the total water demand which was supplied by the

tank. This was used to plot graphs of DD versus DAU for various percentages of demand

supplied. As mentioned previously, the current study assumed that the tank would be used

for household water supply, such as flushing of toilets and not for garden irrigation, as this is

not a necessity. Thus, a high reliability was required for the supply. Consequently the graphs

were plotted for 90, 95 and 98 percent of the required demand being supplied by the tank.

4.2.2 Calibrating the model

A series of data points was extracted from the time simulations and plotted on a DAU versus

DD plot, thus defining relationships for different supply reliabilities at each location. This

data was further compacted to a few modelling constants by fitting least-square curves

through the data points in order to obtain the local design graphs.

4.2.3 Directly predicting the modelling constants

The modelling constants, described above were compared with numerous simplified

parameters obtained from the daily rainfall records in an effort to find a meaningful

relationships. If reliable relationships could be found, the proposed design procedure could

be applied based on summary statistics. The parameters obtained from the daily rainfall

records included, but were not limited to:

the MAP

the average percentage of the MAP that falls in the driest month

the average percentage of the MAP that falls in the driest 6 months per month

(calculated as the total volume of rainfall in the driest 6 months divided by 6 and

averaged over all of the years)

33

the average longest number of consecutive dry days (calculated as the average of

the longest number of consecutive dry days found each year)

the longest number of consecutive dry days during the simulated period.

4.3 Verification of the Time-Step

After the tank sizes had been calculated, it was possible to check the validity of using a daily

time-step for the simulations. This was done using both guidelines, described in section

3.3.5.

The first guideline, as suggested by Fewkes and Butler (2000) is defined by Equation 3-6 and

the relevant limits as discussed in section 3.3.5.

Introducing the DAU, given by Equation 4-12 in section 4.2.1 the guidelines can be adapted

for use with the graphs of DAU versus DD.

By rewriting Equation 4-12:

1000

=

1

(365) 4-13

Then substituting 4-14 into Equation 3-6:

=

(365)() 4-14

34

Equation 4-11, defining the DD can then be applied to 4-15 to obtain:

=

(365) 4-15

Then applying the limits as stated in section 3.3.5:

=

365 0.01 4-16

Rearranging 4-16:

3.65() 4-17

Using the same technique for the other limits, the following equations are obtained:

Hourly models : 3.65() 4-18

Daily models : 3.65 (DAU) < < 45.625 () 4-19

Monthly models : 45.625 (DAU) 4-20

Mitchell (2007) suggested that in order to use a daily time-step for simulations where a

constant demand is assumed a second set of intervals should be used. The intervals are

defined by Equation 3-7 and Equation 3-9, given in section 3.3.5.

Using Equation 4-11, for the DD:

For yield-after-spillage operating rule: DD > 1/0.7 > 1.429 days 4-21

For yield-before-spillage operating rule: DD > 1/0.18 > 5.556 days. 4-22

The first set of intervals as defined by Equation 4-18, Equation 4-19 and Equation 4-20,

provides an indication of the time-step which should be used depending on both the DAU

35

and the DD. The second set of intervals, defined by Equation 4-21 and 4-22, is only

dependent on the DD. However the second set of intervals also provides a way of checking

the reliability of the two different operating rules. The two sets of intervals provided vastly

different intervals, due to this, both sets of intervals are considered.

To investigate the validity of the daily time-step, the two sets of intervals were plotted on

the same set of axes used for the simulated graphs. The plot obtained can be seen in

Figure 4-3 below. In order for a daily time-step to be the best choice, all of the data points

should fall within the shaded area of the graph.

Figure 4-3: Plot showing the valid interval for a daily time-step

36

5 Results

5.1 Rainwater Tank Simulation

5.1.1 Graphs obtained

The simulation process provided a set of data points (DAU and DD) for each of the

volumetric reliabilities at each of the stations selected. These data points were plotted in

order to obtain a set of simulation curves at each of the four stations. These curves can be

seen in Figures 5-1, 5-2, 5-3 and 5-4 below.

Figure 5-1: Simulation curves for Kimberley

0

50

100

150

200

250

0 2 4 6 8 10

DD

(d

ays)

DAU

Kimberley

98%

95%

90%

37

Figure 5-2: Simulation curves for Mossel Bay

Figure 5-3: Simulation curves for Punda Maria

0

50

100

150

200

250

0 2 4 6 8 10

DD

(d

ays)

DAU

Mossel Bay

98%

95%

90%

0

50

100

150

200

250

0 2 4 6 8 10

DD

(d

ays)

DAU

Punda Maria

98%

95%

90%

38

Figure 5-4: Simulation curves for Rustenburg

The graphs showed that tank sizes must be significantly larger for smaller DAU. This was an

expected result as the DAU provides an indication of how much more water can be captured

than would be required to meet the demand. Thus, at smaller values of DAU, a larger

portion of the water which is available for capturing must be captured. To do this, a larger

tank is required to capture larger volumes. The graph for 98% reliability at Punda Maria was

not included, as the required tank size was considered to be unreasonably large.

The design curves for Mossel Bay (Figure 5-2) showed that the tank volumes required to

meet the demand there were significantly lower than those required for the other stations.

This could be attributed to the fact that Mossel Bay is situated in a year-round rainfall

region, whereas all three of the other stations are located in the summer rainfall region. This

leads to relatively smaller tank sizes being required in Mossel Bay, as there is no significant

dry period. The other stations received relatively lower rainfall during the winter months

and thus, tanks must be large enough to capture enough of the rainfall during the summer

0

50

100

150

200

250

0 2 4 6 8 10

DD

(d

ays)

DAU

Rustenburg

98%

95%

90%

39

months to ensure enough water is available during the winter months. A similar result for

tank sizes would be expected for winter rainfall regions as were obtained for summer

rainfall regions. This shows clearly that rainwater tanks are more effective in year-round

rainfall regions, as would be expected.

5.1.2 Range of possible tank sizes

Table 5-1 shows the range of tank sizes obtained when considering all of the rainfall stations

for 2.5, 5, 7.5 and 10 DAU as well as the difference between the largest and smallest sizes

obtained in all regions. For a given volumetric reliability (90%, 95% and 98%) the difference

between the largest and smallest tank sizes across the four stations decreases as the DAU

increases. This suggests that the rainfall region has somewhat less of an effect on the

required tank size when the area of the roof used as a catchment is large (roof size will

increase the volume of rain which can be captured in a year and will thus increase the DAU).

Conversely, when comparing the difference between the largest and smallest tank sizes for

a given dimensionless area unit, it can be seen that the range increases as the volumetric

reliability increases. From this it can be seen that the effect of rainfall region becomes more

pronounced when a higher volumetric reliability is required.

40

Table 5-1: Range of tank sizes obtained

Dimensionless Area units Largest Tank size (days) Smallest Tank size (days) Difference (days)

90% Volumetric Reliability

2.5 127.5 25.4 102.1

5.0 104.0 19.3 84.6

7.5 95.7 17.6 78.1

10.0 89.0 17.0 78.1

95% Volumetric Reliability

2.5 161.4 35.3 126.1

5.0- 132.8 26.9 105.8

7.5 121.5 24.8 96.7

10.0 114.5 23.8 90.7

98% Volumetric Reliability

2.5 227.3 53.5 173.8

5.0 157.0 36.9 120.1

7.5 149.3 34.8 114.4

10.0 144.7 33.4 111.3

5.1.3 Local design curves

From the data points obtained from the simulations (Figures 5-1, 5-2, 5-3 and 5-4) it can be

seen that all the simulations produce curved results, with the highest tank volumes obtained

at the lowest DAU. In addition to this, all the results tend toward being parallel to the y-axis

for small DAU and parallel to the x-axis for large values of DAU, but they are not asymptotic

to the x and y axes. This suggests that curves with the same general shape could be fitted to

the simulated data.

The general shape of the simulations appear to be similar to that of a power function with a

negative exponent which would have the general equation y = axb, this was confirmed by

fiting trend lines to the data and comparing the coefficients of determination obtained.

Table 5-2 shows the various coefficients of determination, with the largest value for each

41

data set highlighted in yellow. The table shows that the power function provided the best

correlation in all but one case. However due to the fact that the data points are not

asymptotic to the x and y axes the general equation was adapted to DD=a(DAU-1)b+c. Using

the equation DD=a(DAU-1)b+c (where b is always a negative value) a local design curve can

be fitted to the data for each of the reliabilities at each location. These curves are

asymptotic to DD=c (the red line in Figure 5-5), which corresponds to the minimum tank

volume as determined by each simulation. The curves are also asymptotic to DAU=1 (the

green line in Figure 5-5), which corresponds to the minimum dimensionless area unit which

can be used to obtain 100% volumetric reliability (this is where the maximum amount of

water which can be captured per year is equal to the water demand for a year).

Table 5-2: Table of coefficients of determination for trend lines

R2 values

Exponential Linear Logarithmic Polynomial Power

Kimberley

90% 0.9343 0.8772 0.9853 0.9738 0.9983

95% 0.9551 0.9185 0.9933 0.9766 0.9942

98% 0.9246 0.8835 0.9717 0.9461 0.9823

Mossel Bay

90% 0.7276 0.625 0.8177 0.8627 0.9014

95% 0.6945 0.5823 0.7785 0.8272 0.8757

98% 0.6531 0.5649 0.7746 0.8492 0.8502

Punda Maria

90% 0.9323 0.8816 0.9872 0.9858 0.9982

95% 0.8878 0.838 0.9607 0.9509 0.9833

Rustenburg

90% 0.8798 0.8291 0.9631 0.9598 0.9852

95% 0.8282 0.7636 0.9235 0.9336 0.9622

98% 0.7935 0.7569 0.8978 0.9616 0.9232

42

Figure 5-5: Graph showing DD=a(DAU-1)b+c, DD=c and DAU =1

Figures 5-6, 5-7, 5-8 and 5-9 show the fitted curves obtained for each of the stations, with

Table 5-3 showing the values of a, b and c as well as the coefficient of determination for

each of the curves. The values obtained for the coefficients of determination are all close to

1 (0.9447 to 0.9996), consequently the curves can be considered to be reasonably close to

the data obtained from the simulations.

0

50

100

150

200

250

0 2 4 6 8 10

DD

(d

ays)

DAU

DD=c and DAU=1

DD=14.63(DAU-1)^-1.08+15.78

DD=c=15.78

DAU=1

43

Figure 5-6: Local design curves for Kimberley

Figure 5-7: Local design curves for Mossel Bay

0

50

100

150

200

250

0 2 4 6 8 10

DD

(d

ays)

DAU

Kimberley

98%

95%

90%

0

50

100

150

200

250

0 2 4 6 8 10

DD

(d

ays)

DAU

Mossel Bay

98%

95%

90%

44

Figure 5-8: Local design curves for Punda Maria

Figure 5-9: Local design curves for Rustenburg

0.00

50.00

100.00

150.00

200.00

250.00

0 2 4 6 8 10

DD

(d

ays)

DAU

Punda Maria

95%

90%

0

50

100

150

200

250

0 2 4 6 8 10

DD

(d

ays)

DAU

Rustenburg

98%

95%

90%

45

Table 5-3: Constants of local design curves

a b c R2

Kimberley

90% 62.3 -0.475 36.738 0.9597

95% 72.606 -0.369 52.84 0.9447

98% 76.241 -0.334 71.70 0.9558

Mossel Bay

90% 14.633 -1.08 15.78 0.9912

95% 21.845 -0.931 20.998 0.9996

98% 33.302 -0.942 28.858 0.982

Punda Maria

90% 104.27 -0.365 31.06 0.9762

95% 112.96 -0.274 54.08 0.9936

Rustenburg

90% 85.627 -0.351 50.94 0.9933

95% 102.39 -0.431 74.709 0.9984

98% 141.73 -0.594 102.53 0.9501

5.2 Generalising the Modelling Constants

5.2.1 Parameters considered

A number of parameters were determined from the rainfall data provided by the South

African Weather Services and compared to the constants in Table 5-2 above in order to find

a correlation. These parameters included: the MAP, the number of months per year in

which less than a given percentage of the MAP was recorded, the average monthly

percentage of MAP recorded in the driest six months and the average of the longest number

of consecutive days without rain per year, to name but a few. In most cases the results were

not useful, as no correlation could be found. The following sections discuss the most

significant results only.

46

5.2.2 Effect of MAP on tank size

Figures 5-10, 5-11 and 5-12 show the plots of a, b and c versus the MAP, respectively. The

plots clearly show that there was no correlation between the MAP and any of the constants.

The coefficient of determination obtained when fitting trend lines to the data ranges from

0.0061 to 0.3753, confirming that there is no correlation between the MAP and any of the

constants. This confirmed that the MAP should not be considered to be the determining

factor when sizing a rainwater tank. Even if a large amount of rain is expected at a given

location, if the rain is seasonal, a large tank will still be required.

Figure 5-10: Graph of a vs. MAP

0

100

200

300

400

500

600

700

0 20 40 60 80 100 120 140 160

MA

P (

mm

/ye

ar)

a

Graph of a vs. MAP

90%

95%

98%

47

Figure 5-11: Graph of b vs. MAP

Figure 5-12: Graph of c vs. MAP

0

100

200

300

400

500

600

700

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2

MA

P (

mm

/ye

ar)

b

Graph of b vs. MAP

90%

95%

98%

0

100

200

300

400

500

600

700

0 20 40 60 80 100 120

MA

P (

mm

/ye

ar)

c

Graph of c vs. MAP

90%

95%

98%

48

5.2.3 Best correlation for a

Figure 5-13 shows the plot of a versus the average percentage of the MAP that fell in the

driest 6 months per month. This provided the best correlation to a. This was calculated as

the total rainfall in the driest 6 months divided by 6 and averaged over all of the years, and

taken as a percentage of MAP. The percentage of rain which fell in the driest 6 months

provides a good indication of the seasonality of the rainfall as 8.3% of MAP would normally

be expected in every month for a location where the rain was evenly distributed. As the

value of the percentage of MAP decreased the driest 6 months would be drier and the other

6 months would be wetter, thus the seasonality of the rainfall must be increased. The figure

shows the general trend that a will be higher when the seasonal variation of rainfall at the

location was greater, which would in turn increase the tank sizes obtained from the

equation. It would also have an effect on how quickly the tank sizes decreased: an increased

value for DAU (a larger a value would cause the graph to curve more, causing the tank size

to drop faster initially, but then stabilise more quickly for larger dimensionless area unit

values). This result clearly shows the impact of rainfall region on the required tank size.

49

Figure 5-13: Best correlation for a

5.2.4 Best correlation for b

The best correlation for b to all of the parameters tested was found when comparing b to

the average longest number of consecutive dry days (calculated as the average of the

longest number of consecutive dry days found each year). However, the parameter did not

correlate closely with b, as can be seen in Figure 5-14 below. The correlation with the

average number of consecutive dry days, once again confirmed the significance of the

rainfall region in determining the required tank size. Parameter b had an effect on the

magnitude of the tank volume. It also had an overwhelming effect on the curved shape of

the graph. When b decreased (all values of b are be negative) the graph would have a

steeper curve and the required tank volume would decrease more quickly and then, once

again, stabilise more quickly.

0

1

2

3

4

5

6

7

0 20 40 60 80 100 120 140 160

Ave

rage

% m

on

thly

rai

nfa

ll in

dri

est

6 m

on

th p

eri

od

a

Graph of a vs average % rainfall per month in driest 6 months

90%

95%

98%

50

Figure 5-14: Best correlation for b (note that there is no constant progression from (90% to 98%)

5.2.5 Best correlation for c

In the case of parameter c the correlation was logically determined by recognising that the

value of c for a 100% volumetric reliability would be equal to the longest number of

consecutive days with no rainfall which occurred during the simulation period. This would

provide the smallest tank size which would contain enough water to meet the demand for

those consecutive dry days. Thus the correlation between c and the longest number of

consecutive dry days would be the line y = x and would pass through the origin.

Consequently, it should not be surprising that the best correlation for c was obtained when

comparing c to the longest number of consecutive dry days and ensuring that the trend lines

passed through the origin, as shown in Figure 5-15. Unlike parameters a and b, parameter c

only had an effect on the magnitude of the required tank size. As c was equal to the

minimum tank size which was required to obtain a given volumetric reliability, it did not

have any effect on the curvature of the graphs. Once again the best correlation was

0

20

40

60

80

100

120

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4

Nu

mb

er

of

day

s

b

Graph of b vs average of longest number of consecutive dry days

90%

95%

98%

51

obtained by comparing the constant to a parameter related to the seasonality of the rainfall,

which once again confirmed the importance of rainfall region in determining required tank

size.

Figure 5-15: Best correlation for c

5.3 Proposed Sizing Method

By using the graphs of the best correlations for a, b and c (Figure 5-13, 5-14 and 5-15), it was

possible to size a tank by reading the values of the constants from the graphs and then using

the equation DD=a(DAU-1)b+c to obtain a generalised design curve relating the tank volume

in DD to the DAU. Once the generalised design curve has been obtained, the roof area can

be used to determine the DAU which correspond to the roof size at the given location and,

from there, the required tank size can be directly calculated.

0

20

40

60

80

100

120

140

160

180

200

0 50 100 150 200

Co

nse

cuti

ve d

ays

wit

h n

o r

ain

fall

c

Graph of c vs Longest number of consecutive days with no rainfall

90%

95%

98%

100%

52

The proposed method for sizing a rainwater tank could be illustrated the following

hypothetical design example for a house with a roof area of 75 m2 in an area where the

MAP is 950 mm/year, the average number of consecutive dry days is 70 days/year and the

driest six months contribute on average, 5% to the MAP. If water were to be withdrawn at a

constant daily rate of 100 l/day (typical for four people in rural South Africa using

25 litre/capita/day), what tank size would be required to assure the supply at 90%, 95% and

98%?

Figures 5-13, 5-14 and 5-15are not verified sufficiently for such general application. Should

future work succeed in fairly robust correlations, then they could be used as demonstrated

below.

DAU = (75*950)/(1000*365*0.1) = 1.952

Reading from Figures 5-14, 5-15 and 5-16:

a = 36 (90%) or 46 (95%) or 54 (98%)

b = -0.42 (90%) or -0.34 (95%) or -0.46 (98%)

c = 18 (90%) or 28 (95%) or 42 (98%)

DD = a(DAU-1)b+c = 54.75days(90%) or 74.77days(95%) or 97.23days(98%)

Tank volume = DD * Daily Demand

Tank volume = 5.475 m3 (90%) or 7.477 m3 (95%) or 9.723 m3 (98%) m3.

The method described above will be referred to as the dimensionless sizing method (DSM)

53

5.4 Testing the Validity of the Time-step

Using the method described in section 4.3, the following equations provided limitations for

using a daily time-step:

From the sugge