A Mathematical Model For Forecasting Of Water Environmental Sciences Essay

Published: 2021-07-06 06:30:03
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Abstract
Substantial social changes over the past twenty years have had an impact on the demand for urban water. These include, changes in demographics, land use, types of water-using appliances and trends toward lower occupancy households and apartment living, particularly in inner city areas. At the same time, pressure on urban water supplies has increased, owing to declining yield of systems and increasing demands for water allocations to the environment. Therefore, the importance of and interest in demand management strategies has increased. In this project investigation into factors that influence demand and demand management programs that have been undertaken by the UK water industry, particularly United Utilities will be studied. Several population forecasting techniques are used to predict the population of Manchester over a 10 year period so as to identify the best mathematical models for water demand forecasting.
Acknowledgements
Table of contents
CHAPTER 1: INTRODUCTION
Rationale of study
Water is increasingly becoming an important resource all over the world mainly due to an increase in world population and urbanization. However it is sometimes difficult to accept the harsh realities and facts with regard to water without actual figures. Studies carried out by Waterwise (2007) show that less than 1% of the world water resource is fresh and can be used for human consumption. This goes ahead to show how little of fresh water exists in the world. To compound this they further claim that out of this 1% a significant percentage is lost through pollution and is unevenly distributed around the globe.
Environment Agency UK (2013) in the investigations have found that the average daily consumption of water per person in England and Wales is 150 liters. They project the daily consumption of the region to have risen to 800 million by the year 2020. A probe in to the exponential growth of water consumption in the UK can be likened to the advancement of technology and the use of appliances such as heated showers, washing machines, bath tubs etc. Apart from this a significant wastage of water resource is incurred as a result of washing cars, watering gardens and through beautification.
According to Welsh Water (2012) a third of UK’s household consumption goes to bathrooms and the other third is used in flushing toilets. This shows that bathrooms and toilets consume up to 66% of the total national household water expenditure. They further show that out of this less than 3% is used for drinking purposes. These figures go ahead to show how much water is wasted in the UK and in the developed world in general. An article on the Guardian (2008) writes about the massive scale consumption of water in the UK. The article goes ahead to write about how the UK imports as much as 62% of its water resources and is the world’s 6th largest importer of water. According to this article each UK citizen uses around 4600 litres of water on a daily basis comprising of direct and indirect consumption data.
It is evident that the UK is a heavy water consumer and this raises the question of whether the country is at risk of water shortage. In the year 2012 Britain faced a water crisis mainly due to change in climate that is likened to global warming. Welsh water (2012) notes that most of the world’s population faces a problem of acute water shortages. This problem is magnified by data that shows that 1/6 of the world’s total population does not have access to clean and safe drinking water with this situation being made worse by a rogue climate.
Through a system of water footprints environmental conservationists have been able to show how water resources are embedded in agricultural products, industrial goods and in our lifestyles. This provides a general outlook of the equation and only goes ahead to show that there is need for proactive water conservation practices. How do water conservation and water security get tied in with water demand forecasting and to the topic of this study?
In an age of dwindling resources, there is a need to forecast water usage and plan accordingly. WRDMAP (2010) shows that due to an increase in water demand within urban centers and the resulting unpredictability of water supply call for proper water demand forecasting that eventually facilitates sound water management practices.
Water demand forecasting is defined as a planning activity that aims at establishing the quantity of water that will be used over a certain period of time in future (WRDMAP 2010) Demand management on the other hand is specified as the use and allocation of water resource so that they meet the demand of the society without exceeding the environment’s ability to replenish itself. However the success of any demand management activity depends on the accuracy of demand forecasts. Therefore water forecasting is a critical and key activity to maintaining water security in urban and industrial settings not only in the UK but across the globe.
This dissertation will assess the substantial social changes over the past twenty years have had an impact on the demand for urban water. These include, changes in demographics, land use, types of water-using appliances and trends toward lower occupancy households and apartment living, particularly in inner city areas. Water forecasting and factors that affect water demand will be studied so as to equip planning agencies and prepare them to tackle water scarcity, global warming and water management issues effectively.
Aims and Objectives
Investigate factors that affect water demand in urban areas
Find out how water demand factors relate to each other
Define measures that ought to be adopted for sustainable water management
Formulate a mathematical model for water demand forecasting
Structure of study
This study will have 5 chapters namely the introduction, literature review, methodology, discussion and conclusion. The literature review chapter will take a look at existing scholarly works on the water demand forecasting with reviews on factors that affect water demand. Water management programs their importance and how they relate to water forecasting will also be addressed so as to provide the required knowledge that would help to come up with functional mathematical models. The methodology chapter will cover the steps taken so as to collect, analyse and interpret data for the study. The fourth chapter, the discussion chapter will attempt to meet the aims and objectives of the study by analyzing data collected in Chapter three. The last chapter will be the conclusion which will summarize the findings of the study as well as make recommendations based on the outcome.
CHAPTER 2: LITERATURE REVIEW
2.1 Water management in urban areas
Many cities in the UK are noted to have made policy decisions to effect change towards sustainable water management practices (Green, 2012). In order to understand the concept of sustainable water management we must understand how cities function and how water resources are handled. According to Green, the use of water in urban areas is non-consumptive in the sense that cities import treated water use it and then export the resultant wastes. This implies that the logistical (pumping, piping, control) aspect of bringing in water and taking out an almost equal amount of waste water from the city expends massive amounts of energy.
Big cities in the UK particularly face numerous challenges when it comes to water management due to the fact that the existing infrastructure was designed and installed between the year 1800 and 1920 (Halliday, 1999). The initial piping systems which were made of wood proved to be ineffective due to leakages and were later replaced with iron pipes that were laid down in the late 1800s. Ellis et al. (2004) further notes that there is a rampant misconnection between households and sewer lines in major cities with at least 300,000 British homes being included in the statistic.
Water management practices were privatized in the London in the year 1989. This was due to the inefficiency and unreliability of government agencies at doing this while at the same time raise funds in order meet EU’s prevailing conditions A case study of London’s systems shows that it is made up of a mix of water and sewerage companies and water only companies (WaSC & WoC). Challenges faced by the privatized management of water in Britain have shown that it is important to define responsibilities and provide incentives for various stakeholders to cooperate.
Privatization has also lead to a problem in the identification of boundaries particularly when these are identified based on data of construction. This has led to problems at identifying what fall under the mandate of which organization or company and has in general complicated water management processes.
Figure 1: Scheme for water infrastructure management in London
The United States Environmental protection Agency (2005) notes that water management must be done in a way that ensures that the environment and local ecosystems are protected. This could lead to improved public health, quality of life, reduction of energy expenditure and contributions to the environment through accrued savings. It may be questionable why sustainability always seems to come to the front of water management from an engineering perspective The reason for this is because efficient water use leads to reduction of energy spent, it makes the work of water engineers easier, prevents pollution by lowering the volume of waste water and generally reduce pressure on water and sewerage systems. This would in turn lead to a chain of benefits that would have otherwise made urban water management and planning difficult.
Planning is a key aspect of water management in any urban center and this is done in order to ensure that there is adequate and clean water as well as proper removal of waste water with a key emphasis on public health (United States Environmental Protection Agency 2005). This involves many stakeholders that provide projected data on expected rainfall, population patterns, migrations, lifestyle patterns, electronic household devices etc.
http://www.hunterwater.com.au/Resources/Images/Diagrams/Sustainable-Urban-Planning.jpg
Figure 2: Sustainable and green water management scheme
Scholarly ideas on sustainable water management are truly diverse. This is due to the fact that it is a relatively new concept and there are different opinions as to what such a system should entail. Lawrence et al (1999) have won acclaim for their green scenario that proposes for complete and total water management based on cycles. The green cycle encompasses 1) The recycling of waste water 2) Waste water, storm water, ground water and water supply integrated management and 3) Conservative use of water in industrial and urban settings. To implement the green scenario radical measures ought to be taken so as to recycle waste, harvest water and come up with innovative ways of water and waste transmission.
Despite the hype behind sustainable water management and the green cycle, skeptics have raised several points that question the possible success and viability of these so called sustainable systems (Chocat, 2002). Some scholars argue that there is no substantial information on how safe proposed alternatives such as harvested rain water may be to the human body and what their economic implications may be in the long term. Secondly they state that the green cycle may be exploited by local authorities and management bodies and used as an excuse to neglect the maintenance of water infrastructure. The last concern expressed by skeptics is that decentralizing the existing water infrastructure may be a costly and time consuming affair. This is because the implementation of a green cycle would lead to existing infrastructure becoming obsolete as well as requires an intensive logistical plan in order to put in to place.
Putting all the varying ideas aside this study will investigate factors that affect the consumption of water in the Manchester area as well as represent these in a mathematical model. Based on the model the most practical and workable means of reducing water expenditure shall be proposed so as to promote sustainable water management that not only works in theory but that could be effected in real life.
2.2. Factors that affect urban water demand
In order to forecast the demand of water effectively we must first of all understand the factors that affect the demand and supply of water in an urban setting. This becomes a daunting task bearing in mind the complex nature and dynamic nature of water use which demands the consideration of numerous variables and empirical values (WRDMAP, 2010).
2.2.1 Population
The first factor that affects water demand is the population factor of an urban center (Toronto Water Efficiency Plan, 2010). This is because population size is directly related to household sizes and pressure on the existing water systems. Households are a major user of water resources and when there are more households then the demand for water obviously has to go up.
http://newsimg.bbc.co.uk/media/images/44265000/gif/_44265507_uk_pop416x299.gif
Figure 3: UK population projections from 2011 to 2081
The above chart shows the projected population of UK from the year 2011 to the year 2081. The gradient and steep rise of the graph has been extrapolated from population growth values collected between the year 1981 and 2011. The graph is not only alarming but exposes the possible and looming resource crisis that the UK could face in the near future. Current estimates propose that one person uses at least 150 liters / day. If the projected population values are true and if we assume that the UK continues to experience high fertility, high migration and high life expectancy then the daily water requirement is expected to rise to 15 billion liters of water per day. Based on these values there may be a need to double if not triple the current water supply and waste removal capacity of the nation. Therefore population is and will remain a key factor that affects urban water demand. In order to predict and project how much water is needed in an urban center, there must first of all be a keen understanding of household sizes and demographics.
2.2.2 Household fixtures and stock
http://www.waterwise.org.uk/data/Waterwise_Images/Charts/water_usage_in_the_home.jpg
Figure 4: Water consumption in UK households
The above chart shows how water is used within households in the UK. According to Waterwise (2013) 30% of water is used in flushing toilets, 13% is used in the washing of clothes, 12% in showers and another 21% in taps and baths. The mentioned percentages show that up to 76% of water in households is used through machines and other housing stock. This goes ahead to provide that housing stock are indeed a great determinant of urban water consumption since they have an influence on up 76% of all household water expenditure. Therefore the age and efficiency of certain household fixtures such as toilets, showers, tubs, sinks etc. have an overall effect on water consumption - particularly evident when the cumulative effect of household losses is examined. There are an estimated 45 million toilets in the UK with 15% of these being old style toilets that require up to 13 litres of water per flush. A new requirements passed in 2001 however lowered this to a maximum of 6 liters per flush and stimulated the use of low volume toilets in UK households (Green Building Store, 2012)
Based on the Telegraph (2011) Lord Krebs acknowledges the need for efficient low volume toilets, shower heads, washing machines and dripping taps on order to help reduce the UKs water consumption and help it combat global warming as well as reduce its annual carbon footprints. Technology could therefore be married together with the aspect of household fixtures and stock. This is because each of these has an effect on the average household’s water consumption. However there is still room for improvement and development of better and more efficient household systems such as waterless toilets and grey water recycling systems. These could impact greatly upon urban water demand.
2.2.3 Public attitude
Public attitude is another factor that influences water demand. Ballweg (1972) notes that the success of water conservation efforts is greatly dependent upon the attitudes and the perception of the general public. MVA Consultancy (2006) carried out a survey on the opinions of UK citizens towards water conservation and the environment. Their study was able to establish that there was general awareness among citizens on the relationship between the quality of water and the ‘health’ of the environment. Many respondents included in the survey are keen to note that the existing sources of water such as lakes, rivers and rainfall were the major sources of water and that these would be at increasing pressure in the near future.
However, this raises the questions as to whether this awareness could be linked to actual willingness and/ or action from the respondents interviewed towards water conservation. If a link is established then it could be argued that public attitude does indeed affect water demand in urban areas. The results are alarming in the sense that most respondents interviewed participated in energy saving activities rather than water saving activities. A great majority were seen to relate energy conservation to the environment more than they did for water conservation and the environment. Therefore the study was able to prove that UK citizens are willing to conserve their environment, but there is general opinion that lakes, rivers and water catchment areas could be protected by activities such as switching off the lights when leaving a room, and the recycling of plastic bags rather than taking water conservation measures. There was general willingness among the respondents to conserve water if water saving devices are made cheaper, there is more information on water conservation, they are provided with assistance on installation of these devices, energy saving devices are made easily available, and if they felt that their water company was also putting in efforts (MVA Consultancy, 2006).
HMRC (2011) notes that water could be saved by altering the type of gardens, homes and offices that we build, our transport sector, how we manufacture our goods, how we produce our goods and how we carry out our tourism. It is evident that water demand could be influenced by the general attitudes and opinions of people. This is because through attitude change the culture and lifestyles of people could be altered so that this ultimately affects water demand in an urban center. Despite this the mentioned factor would pose a great challenge to this study in line with statements by Robin (2007) who states that there is difficulty in quantitating and measuring attitudes. This would resultantly lead to problems in representing this factor of urban demand in a mathematical model.
2.2.4 Water rates
Symmonds (2011) notes that water is increasingly becoming a scarce resource and sustainable water use can only be effected through sustainable practices and sustainable behavioural change. The author addresses the ability of water rates and incentives as tools for behavioural change that could be used together with government policy and technology in order to conserve water resources.
Hempling (2009) notes that water rates are bound to increase in the future due to the fact that the demand for water is on the increase, the introduction of more stringent water quality requirements, the reduction in available water resources, the rise in water treatment costs, the deterioration of existing infrastructure and climate change. The scholar therefore states that a rise in water prices is inevitable and artificially frozen rates would be difficult to sustain in the long run.
This raises curiosity as to whether water rates affect demand and as a matter of fact are indirectly proportional. Some analysts claim that water is price inelastic and that consumers do not cut their use of basic resources such as water even when prices are hiked. However a study by Olsmtead (2006) shows that water is price elastic within a certain range. He notes that when the prices of water are low and when rises in water prices are insignificant then there is an impression of false inelasticity. Olmstead’s studies also show that water consumers do respond to changes in water prices but different across the short and the long term scenario. The scholar notes that an increase in 10% of water prices results in a reduction of household consumption by 4% in the short term and by 6% in the long term. Despite the varying rates of response, one thing is for sure from his study: water prices affect water demand.
The Pacific Research Institute (2011) takes a look at how water rates are used in the forecasting of demand for water and finds that many forecasts fail to take consideration of water rates in their computations. The study blames this on certain assumptions with regard to water demand which believe that per capita water demand is of a constant value and that it does not change in value over time. Apart from the mentioned findings the study is able to show that consumers are able to reduce outdoor water use more easily and at a much shorter notice compared to indoor use as a response to price changes.
This study will attempt to address how water rates affect water demand because this has been neglected by many prediction models yet represents an important factor that could greatly influence the amount water used in an urban setting.
2.2.5 Income
Huby and Bradshaw (2012) show that the income levels of people do have an effect on the amount of water that they consume. The scholars investigate this phenomenon and are able to find out that people with higher incomes tend to spend more money on water compared to those with lower incomes. They likened this to increased water usage on watering lawns and gardens as well as a generally higher household consumption due to the use of luxury stock and devices. They further assess the ratio of incomes spent across different income brackets and this is represented in the chart below:
Figure 5: Water cost as a ratio of income
The above chart is able to show that the low income households tend to spend a larger percentage (up to 11%) of their total income on water compared to the upper income brackets that spend less than 2% of their income on water. This implies that the upper income levels would afford to consume more water unlike the later. Data from the FRS (2010) shows that up to 20% of all households in the UK are water poor and struggle to foot their water bills. It goes without saying that such populations would be willing to conserve water so as to save on their annual household costs. Therefore it is also expected that high income areas within the city would experience a generally higher amount of water consumed per capita compared to low income areas. If this holds true, then it would be possible to predict the demand for water based on the projected incomes of people. From this paper, an attempt shall be made to formulate a model whereby the income levels of the occupants within a certain region shall be used in order to forecast demand per unit time.
2.3. Approaches towards water demand forecasting
Current water demand forecasting techniques do exist and are in current use within water regulation bodies in the UK. However there still is no perfect model that could be able to predict water demand without an inherent risk of error. We shall attempt to study the various water forecasting techniques so as to gain a full understanding of how these operate and what their possible strengths and weaknesses could be.
WRDMAP (2012) notes that the selection of a water forecasting techniques is mainly dependent on three aspects: the accuracy and the reliability of the method in question, the overall cost of implementing the technique and the possible benefits of achieving highly accurate results. Cost is ultimately a determining factor in the sense that the method must fit within the sponsoring organisation’s budget plan and is economically feasible in the long run.
2.3.1. Judgmental forecasts
This type of approach relies on the personal judgment of the water experts and engineers involved. This type of forecasting is not empirical and subjective in nature and is largely prone to biases such as professional bias, personal bias, seasonal bias and spatial bias. This according to Billings and Jones (2008) is often used as a supplementary technique in other forecasting approaches particularly for values that do not have the ability to affect the outcome greatly.
2.3.2. Extrapolation of historical data
The past most often informs about the future. Therefore water demand values from the past and their trend over a certain period of time could be used to predict what could happen in future. However most extrapolations predict the expected behavior of a variable with respect to time only and other factors that could affect the dependent variable for example, population, prices, technology, attitudes, income and household stock are assumed to be subject to and represented by time (Fraunhofer Application Center System, 2012). Despite the advantages of this technique such as its low cost, low data requirements there are many shortcomings that limit its implementation for example its untrue premise that water demand patterns remain steady over time, errors in the past are magnified and extrapolated in to the future and the fact that the technique focuses on demand rather than factors that affect demand (Billings and Jones, 2011).
2.3.3. Time series approach
The time series approach makes use of the least squares regression technique in order to determine a trend line in historical water demand. Since the technique makes use of auto regression values are acquired from the past and are used to compute those of the future. This is considered as an international water demand forecasting technique and can predict water demand based on the values of population and water use per capita (WRDMAP, 2012). The time series approach is preferred by some analysts due to its relatively low cost and low data volumes required to come up with predictions. However the approach fails when it assumes that factors that affect demand remain constant over time and relies on past data to come up with future data which may lead to propagation of errors (McLeod and Hipel, 1994)
2.3.4 Econometric Analysis
The end use forecasting approach constitutes of multiple linear and non-linear regression computations. An advantage of this technique is due to the fact that water consumption and /or demand could be related to a number of variables such as GDP, income, water rates etc. as it facilitates the setting up of an analysis of multiple variable relationships. Through this technique the response of consumers towards changes in water prices, income levels and GDP can be established through a combination of statistical and economic operations.
2.3.5 Component analysis approach
This approach forecasts the consumption of water based on an individual variable for example shower heads, toilet flushing mechanism. Despite the seeming ease and simplicity of this technique, it requires a huge amount of data and information on specific variables compared to other techniques. This approach forecasts water demand for individual components and then aggregates this in order to simulate the total expected consumption (WRDMAP, 2012).
2.4 Relationship between population forecasting and water demand forecasting
Population is inevitably the greatest influencer of water consumption. This is because all the other factors of water demand such as attitude, technology, water rates and income are simply functions of population and are greatly dependent upon it (City of Olympia, 2009). Therefore this study will attempt to predict the water demand of Manchester city based on the projected population of the city while at the same address the question of which the best population forecasting technique with regard to water demand forecasting.
However to accurately predict water demand using population projections there is a need for careful and accurate profiling of the existing population. This involves collection of data on the household sizes, household incomes, water demand per household, age of members of households, peak usage hours and seasonal change in use of the population. Such computations are comple but remove the margin of error that could exit from blanket assumptions (Billings and Jones 2011, pp. 80-81). Such computations may exceed the scope of this study and future demand shall be projected based the cohort component method, the ratio method, the arithmetic growth method and the declining growth technique.
2.4.1 Cohort component method
This technique involves the separation of people within the population in to segments which could include age and sex and then predict future changes in trends of these segments based on mortality, fertility and migration. This technique is based upon 5 year page segmentations and 5 year projections but could also be used for long term projections.
Therefore an age group of between the ages of10 -15 in the year 2010 is projected as the age group between the ages of 15-20 in the year 2015. This continues perpetually with the application of mortality, fertility and migration factors so as to forecast the expected future population. Freeke (2012) acknowledges the usefulness of the cohort forecasting method in predicting population to the housing sector mainly because it is focused on the population and provides a better understanding of household formations, net flow of population and the effects of natural population changes. In addition to this the scholar adds that this method provides adequate information on the affordability of households and age characteristics of the population. However this technique is critiqued for its complexity and is at risk of being erroneous due to inaccuracies in mortality and fertility rate data (O’Neill, 2001)
Figure 6: Cohort component population 5 year step technique
2.4.2. Ratio technique
The population ratio forecasting technique is based on the premise that the population of people within an area increases in an even manner until the area upgrades to a larger entity. Eberhardt (1987) covers the ratio method of population forecasting and notes that the despite the correlation that takes place in the technique due to the appearance of the same observation as a numerator and denomerator in various ratios it is provides an effective way of dealing with removals within a data set.
The ratio method can be summarized as that which projects the population of a sample based on the application of mathematical functions on the population figure and not individual components. This method is also known for adjusting the distribution to a total value that is proportionate to the frequencies in the distribution.
In most instances the ratio projection technique is used for areas where larger and inclusive projections do exist. As noted by Hofe and Wang (2007, pp. 89) a major limitation of this technique is that is based on the assumption that smaller units or an area will grow at the same rate as the larger unit as a whole.
2.4.3 Arithmetic growth method
This is the simplest method of population forecasting as it involves an even increment of general population year after year. Some analysts argue that despite the seemingly inaccurate nature of the technique it can prove effective in some instances due to the fact that over projections and under projections of population values over successful years tend to cancel each other out (Hofe and Wang 2007, pp 120). This can be represented in the below equation
dp / dt = Ka
whereby
dp – represent the change in population
dt – represents the aspect of time
Ka – represents the constant gradient of change
CHAPTER 3: METHODOLOGY
3.1 Research method
This study will be based on primary research where first hand data shall be collected from public utility services in Manchester. The main reason why primary research was chosen for this study is due to the nature of the investigation where data has to be collected from the field and this related to the localized scenario of Manchester. Despite this, there are numerous advantages of primary research which include the ability to collect highly specific and relevant information, ease of data interpretation, ability to collect latest data, and in order add to existing something of value to existing knowledge (VanderMeyer and Rys,2009, pp 66) . However at the same time it is worth noting some disadvantages of this method of research namely: highest associated costs and the time consuming nature.
Secondary research as a main option for this study was not considered due to the relatively little information that exists on water demand forecasting in Manchester city and, vast nature of the topic and the high number of possible models and techniques. Therefore it would be rather challenging to find existing studies that have matching characteristics to those required for the study. Despite this fact the primary research done in the study is informed by the outcome of the extensive literature review that provides the necessary background information on weather and population forecasting techniques. This is important as it provides the researcher with an understanding of the existing options which allows them to craft customized solutions for the problems being tackled in the study.
3.2 Analysis technique
Mathematical models for population forecasting and water demand forecasting shall be used so as to process raw data and hopefully meet the aims and objectives of the study. Four population forecasting models shall be selected from popular techniques and then be used to predict the population of Manchester based on demographic data provided by United Utilities. Each of these models shall then be used with available water data so as to predict the demand for water in the select area for 10 years using a component analysis approach – population as a component of water demand. The study does not use any analytical and prediction software but bases the outcome of mathematical models only.
The models used in this study could be simplified by the equation as shown below:
Current Per Capita Consumption = Average Consumption / Current Population
Future Consumption = Future population * Current Per Capita Consumption
Where, the future population is predicted using four techniques which include the ratio, cohort component, arithmetic growth, and decline growth method.
3.2 Data selection
United Utilities which a water company based in Manchester will be main source of data for this study. Data will be collected from the company’s records in order to study the average water consumption of its clients and then relate this to the population of the residents. Several population forecasting techniques shall be used together with the data collected at United Utilities so as to define the best population forecasting technique that ought to be incorporated in to water demand forecasting modeling.
I chose to analyse data from the Manchester area mainly because I reside in this area. This implies that it would be convenient to collect data due to easy logistics and access. The environment and its conservation is of key importance to Manchester City (Manchester City, 2013). Therefore it is evident that water and environmental conservation is important to the city and I hope to add value to the Manchester Green City plan through this study.
Another reason why this study focuses on Manchester is due to the fact that this has been mentioned as a boom city in the UK. An article in the Guardian (2012) addresses the exploding population of Manchester and notes that the city’s population is growing at a rate that is three times that of the nation as a whole. The article further goes ahead to cite that this is due to the migration of young people in to the city. It goes without saying that a rising population is accompanied by pressure on local utilities and infrastructure. This strategically places this study and emphasizes its importance as the outcome would be used to inform future decisions on the design of water supply infrastructure.
3.3 Limitations of study
There are several limitations of this study and the first is due to the fact that population is just but a component of water demand. Therefore despite the fact that numerous population forecasting models are implemented so as to come with a predicted demand, there still is a challenge with regard to the fact that these models are mainly based on the premise that the per capita consumption remains the same with time. This is despite the fact that this could be altered by changes in technology, water rates, incomes etc.
This could lead to errors in water demand forecasts since changes in household equipment and stock could greatly lower the current per capita consumption. Secondly there is a possibility that future technology shall be inclined towards water conservation and green technology. This implies that the per capita consumption should gradually decline as the existing technology progresses. However there are mathematical and practical limitations as to the methods that could be used to predict and factor this in within the model of the study.
CHAPTER 4: RESULTS AND DISCUSSION
4.1 Per Capita Consumption
The per capita consumption of water in Manchester is very important to the study as this value shall be used in the projection of future demand. The average water per capita consumption of the UK based on 2008 - 2009 values is 145 litres / day (Environment Agency UK, 2008). This is a blanket value that has been arrived at as an average of all per capita consumptions from all over the UK. This is with respect to the fact that some areas consume more water compared to others and therefore this value cannot be assumed to also hold true for the Manchester area. Therefore, separate values based on the sample population covered by United Utilities’ data are selected and used to come up with the average water consumption for each city resident.
However during the study it was noted that there is a possibility of the data being skewed as a result of seasonal changes and other climatic conditions. Guardian (2001) addresses the effect of seasons on water consumption and notes that more water is spent during hot and dry seasons compared to when it is rainy. This may be due to the fact that people tend to bath more, wash more when it is dry. People also tend to water their lawns, wash cars and engage in water based recreation activities when it is hot and dry. Therefore isolation of data collection to a particle time of the year or a single year exposes the study to the inherent inaccuracies. Due to this, the average daily water consumption is derived from a 5 year average that is expected to greatly enhance data accuracy and validity of the final outcome.
A 5 year average of water consumption within the greater Manchester is found to be 91520 mega litres. This is equivalent to 91520 * 1000000 = 91,520,000,000 litres. In order to get the daily water consumption this value is then divided by 365 days of the year to get 250,740,000 litres. In order to get the per capita water consumption for Manchester, we must the divide this figure by the total number of current residents in Manchester. This is as represented below:
Current per capita consumption = Average daily consumption over 5 years / Current population
Current per capita consumption = 250, 740, 000 / 2, 682, 500
= 93.47 litres
The study determines that the per capita consumption for Manchester city is 93.47 litres based on a population value of 2.57 million as stated by government data (Manchester 2011). This value has been derived from the United Utilities database and is representative of a five year average which means that the data is free from errors that could have resulted from climatic and seasonal changes.
4.2 Arithmetic growth method
The first method of population growth the ‘Arithmetic growth’ is tested in this part of the study. The principle behind this method is that growth is assumed to be constant based on a gradient value that has been acquired from the past. The population of greater Manchester is estimated as 2,682,500 as of 27th March 2011. This value is taken to be an approximate current value of the UK population despite the time difference (UK Census, 2011). In order to establish a trend in the population growth and define a gradient we must first of all establish a starting and ending point. This study opts to base projections on the gradient of change of population data between the year 2001 and the year 2011. The population of greater Manchester as of the year 2001 is noted to be 2, 516, 200. Therefore during the 10 year span between 2001 and 2011 the population of the city is seen to have grown from 2, 682, 500 to 2, 516, 200 which represents a population increase of 166,300 residents which translates to a percentage of 6.61%.
However since census are done after every 10 years, the rate of change in population varies across the different spans. This is to imply that the percentage of change in population between the year 1981 and the year 1991 could be different from that of the year 1991 and 2001. Therefore there are two ways in which arithmetic growth forecasting could be applied. The first is whereby an average gradient line is mapped based on the historical values of populations while the second is where future population is projected based on the gradient of the last two census undertakings.
The latter has been chosen for this study due to the fact that the change in population between the year 2001 and 2011 could be informed by current events, phenomenon and migratory patterns of people. This is in the sense that data from the year 1971 and the year 1981 (for example) may not reflect what is happening currently and may not reflect current factors that affect population growth. If data from the year 1971 is incorporated to the study it could skew the results away from the real issues on the ground by factoring historical forces that may have long changed.
Therefore if this approach is taken to be true, then the population of greater Manchester is projected to have risen to 2, 859, 813 by the year 2021. This is as plotted in the graph below:
Figure 7: Population projection based on algorithmic growth method
Therefore if the per capita consumption remains constant the daily consumption of greater Manchester is expected to increase to1.0661 * 2, 859, 813 * 93.47 = 267, 306, 745 litres per day The maximum daily consumption is also estimated to be 481,152, 141litres per day based on this method.
The maximum daily per capita consumption is found to be = 1.8 * 93. 47 = 168. 246 litres
4.3 Geometric progression method
The geometric progression method is a little bit different from the arithmetic method in the sense that it is based on the premise that the rate if change of population is proportional to the size of the population being measured. Therefore the present population of Manchester city is taken to be a determining factor of how large the population is expected to grow or decline. However care is taken in the use of this method since it is prone to producing absurd results in instances where the rate of change is high and this is applied over a prolonged period of time. It is further noted that this technique is prone to errors due to complex computations and the use of aids such as logarithms (UN, 1986). Despite this the technique is favoured as it could be used to project population based on the outcome of two census results. Kharagpur (2012) further notes that the geometric progression method is most suitable for new and upcoming centers due to the fact that it predicts relatively higher population values.
However since arithmetic growth projections were done based on the outcome of only to census this forecast shall be done based on data from four census. These are 2011, 2001, 1991, and the year 1981. To aid computations the below chart is plotted
Year
Population
Increment per decade
Incremental increase
% increment per decade
1981
2575441
-
-
-
1991
2569700
-5741
-
-0.223
2001
2516200
-53500
-59241
-2.08
2011
2682500
166300
112800
4.622
Net value
107059
53559
2.319
Average
35686.3
17853
0.773
Average percentage increase after every 10 years = 0. 773
According to this technique population is given by the formula P n = P (1+i/100) n
Therefore the population for the year 2021 :
Population for the year 2021 = Population of 2011 * ( 1 + i / 100 )n
= 2, 682, 500 * ( 1 + 0.773 / 100) 1
= 2, 682, 500 * 1. 00773
= 2, 682, 500. 00773
This implies that the total daily water consumption according to the geometric progression technique is expected to be 250 , 773, 275 . 231 litres which is equivalent to 2, 682, 500 . 00773 * 93. 47. This seems to be a lower value compared to that acquired through arithmetic progression despite numerous literature that claim that it often produces higher values. However a close look at the population values between 1981 and the year 2001 shows a steady decline in population over this period. Due to the nature of the geometric progression technique these values and their averages are factored in and used in the calculation of future values. Therefore the net result is that the average rate of increase per decade is found to be very low, which as a matter of fact can be termed as negligible due to negative increments that occurred during the past.
This raises the question; how accurate and effective is the geometric growth method? Based on the outcome it seems as if the method is prone to inaccuracies that could be caused by past occurrences. For example a war in the past that causes massive deaths or a pandemic means that future estimates would be very low. The only explanation to this would be that it is assumed that the probability of past occurrences happening in future is assumed to be very high. The outcome of this research points at weaknesses of this technique as it has come up with a very low daily water consumption value for the year 2021 which is over 1650 mega litres lower than that generated through arithmetic progression.
4.4 Ratio method
Mehta (2012) notes that the ratio method of population projection is the most effective method of population projection for smaller regions in comparison to larger ones. This method is based on a ratio value of a city or district to that of the nation or regional block. The only challenge with this technique is that national projections have to be acquired so as to calculate the projected city population. However Schmidt (1951) was keen to note that mistakes in the national forecast could be carried forward to the local level through the use of this technique. He also states that this method has been popularized due to its convenience and the fact that data on the national scale is easy to find. They also note that this method is favoured due to the fact that large scale forecasts are more accurate compared to smaller ones and that projections for smaller areas that have been base

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