I.2 Fixture units at choices of reference design flow rates for simultaneous demand problems of larger water supply systems of Hong Kong

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1 I.2 Fixture units at choices of reference design flow rates for simultaneous demand problems of larger water supply systems of Hong Kong L. T. Wong 1, K. W. Mui Department of Building Services Engineering, The Hong Kong Polytechnic University, Hong Kong, China 1 Tel: (852) ; 1 beltw@polyu.edu.hk Abstract Fixture unit approach used for estimating the probable maximum simultaneous demands in building water supply systems is based on a fact that a given simultaneous reference design flow rate may be produced by different numbers of identical appliances characterized by the appliances discharging flow rates and discharge probabilities. Each appliance is represented by a fixture unit value, which indicates the appliance associated with the same simultaneous demand of a number of base case appliances characterized by the base case discharging flow rate and discharge probability. The validity of the selected reference design flow rate and its sensitivity to the probable maximum simultaneous demand for water systems in high-rise residential buildings are examined in this paper. In particular, fixture units and the estimated probable maximum simultaneous demands due to appliances attributed by discharge probabilities and discharging flow rates ranged from 1/8 to 8 times the based case attributes are considered. Estimated demands from the fixture unit approach are compared with computational results for an example water supply installation by Monte-Carlo simulations. The results showed that the existing choice of a reference flow rate at 10 Ls 1 for the fixture unit approach would be sufficient in determining the probable maximum simultaneous demands not to exceed a probable failure rate of 1% for

2 pairs of WC-and-washbasin installations in residential buildings. An increased reference design flow rate would be required for the applications of the fixture unit approach in demand analysis of larger water installations in similar densely built environment. Keywords Demand analysis, water supply system, fixture unit approach, reference design flow rate Introduction Probabilistic approaches for estimating the usage patterns of water appliances and their associated instant demands at any points of a water supply system have been adopted in many practical building installations [1]. The simultaneous demand problems of a water supply system were addressed from the binomial theory for frequency analyses of usages [2]. Actuations of an appliance in the installation occur randomly and intermittently with variable magnitudes and they can be described by the probability for appliance discharging events. The analysis provided a means for quantifying a probable not to exceed failure rate in fulfilling certain instant demands. The design approach is practical because water supply main is very unlikely to address the simultaneous demands of all installed appliances. The installations may be overloaded with certain number of appliances operating simultaneously where a small failure probability to the theoretically maximum demand is allowed [3]. The validity of the allowable maximum failure rate can be investigated through field measurements of in-use installations. It was reported that the observed maximum demands in some water supply systems did not exceed a failure rate of 1% derived from geometric demand patterns of observed demands in a study [4]. A fixture unit approach was used to evaluate the probable maximum simultaneous demand problems in building water supply systems. This approach is based on the fact that a given simultaneous reference design flow rate can be produced by different numbers of identical appliances characterized by the appliances discharge flow rates and discharge probabilities [5]. The probable discharge flow rate of an appliance can be equivalent to a number of base case appliances and assigned appliances with fixture units. The choice of this reference design flow rate is an assumption needed to be studied in detail for large water supply systems. Apart from solving the problem with the fixture unit approach, Monte-Carlo simulations can also be used to determine the probability density function of system failures to meet the instant water demands [6]. A stochastic model for estimating the instant water demands in a water supply system was developed where modelling parameters were 27

3 obtained by Monte-Carlo sampling technique without an assumption of the reference design flow rate [7]. However, the fixture unit approach is simple to use and many designs employed the Hunter s probabilistic method for practical water pipe and plant sizing [8]. From the actual building usage patterns, data, and extended laboratory research results, the piping requirements using the probabilistic approach can be applied for both water supply and discharge systems in buildings with tabulations and design curves as specified in some design guides [9]. Codification was resulted from fixture units for probable instances in building water pipe/plant sizing. Water supply loading tables in plumbing design applications were based upon loads in fixture units for practical applications. In this study, the variability of the probable maximum simultaneous demands in water supply installations due to the choice of various reference design flow rates is investigated. Estimated demands from the fixture unit approach are compared with the computational results by Monte-Carlo simulations. Appropriate choices of water supply systems in high-rise buildings of Hong Kong are recommended. Simultaneous demands and fixture unit approach For a base case appliance having repeated cycles of discharge operation with a mean discharge period d (s) and the mean time interval between discharges w (s), the probability of the appliance discharge p at any time is [2], p d (1) w Assume the appliance operations are binomially distributed, and the probability p of N base case appliances operating out of M identical base case appliances installed in the installation, Mp N is given by, where, (1p) is the probability of the appliance not M operating and C is the binomial coefficient, N M p N MN 1 p M N C N p ; C M N M! (2) N! M N! In some water supply system designs, piping systems are designed for a maximum acceptable risk of failure in order to minimize the cost of the system with design number of N (out of M installed, say, M>30) base case appliances operating simultaneously. This design implies that the plants and piping systems might be overloaded when serving all the M appliances operating simultaneously, i.e., the theoretical maximum simultaneous flow rate. When more than N appliances are operating, the acceptable level of the system in terms of reliability is defined as 28

4 engineering unsatisfactory (i.e., the occurrence of failure ). The failure rate is determined by the sum of the probabilities that more than N appliances are operating simultaneously, M p i i N1 N 1 pn 2... pm 1 pm p ; N < M (3) The number of appliances N that are operating simultaneously can be determined by the probability p at an acceptable failure rate, which would be approximated by the Sterling s formula for an engineering acceptable limiting failure rate. The probable number of appliances operating simultaneously can be expressed by Equation (4) with z = [10] for = 1%, which is recommended in some designs, N Mp z 2Mp 1 p (4) The corresponding probable maximum simultaneous demand q d (Ls 1 ) due to the installations of M appliances is then determined by Equation (5), where q (Ls 1 ) is the discharging flow rate of the base case appliance, q d Nq q Mp z 2Mp 1 p (5) Equation (5) can be used to determine the design flow rate of an installation consisted of 2 or more appliance types using the fixture unit approach. The fixture unit approach is established for estimations of the probable maximum simultaneous demands in plumbing and drainage systems in buildings [9]. Specifically, the reference simultaneous flow rate of an installation due to a number of installed identical appliances, say q ref = 10 Ls 1, would be produced by a number of the base case appliances with the base case usage characteristics. Each appliance is then determined with a fixture unit value, which indicates the appliances associated with the same simultaneous demand of base case appliances. The level of reference design flow rate q ref (Ls 1 ) was determined by professional judgement and its sensitivity to the probable maximum simultaneous demand is evaluated in this study. It is noted that each appliance is attributed by the discharge probability and the discharging flow rate, i.e. A b (p b,q b ) and A i (p i,q i ). The same reference design flow rate q ref (Ls 1 ) would be produced by an installation of M i number of appliances A i or M b number of the base case appliances A b. The fixture unit U i at the choice of the reference design flow rate q ref (Ls 1 ) for the appliance type A i is given by, taking the fixture unit of the base case appliance U b =1, M i Ui (6) Mb q ref 29

5 Figure 1 illustrates the idea of using a base case appliance characteristics A b (p b,q b ) with the base case discharge probability p b and the base case discharging flow rate q b (Ls 1 ) as shown in Figure 1(i) to approximate an appliance A i with the discharge probability and the discharging flow rate A i (p i =2p d,q i =q d ) or A i (p i =p d,q i =2q b ). Ideally, the 2 base case appliances should be operated without simultaneous discharging or simultaneous discharging exactly in phase in order to approximate a single operation of the appliance A i as shown in Figure 1(ii), i.e. the ideal cases of approximation. However, probable cases in random discharge patterns of a number of base case appliances were not excluded in the fixture unit approach as shown in Figure 1(iii), i.e. the non-ideal cases of approximation. Indeed, the fixture unit was not only dependent on the attributes (p i and q i ) of an appliance A i but also the choice of reference design flow rate q ref. Preferred approximation Other possibilities qb + qb p b p b 2qb p b qb 2p b qb (i) Base case (ii) Appliance i p b + p b qb 2qb <2p b >2p b <2qb p b : base case discharge probability (iii) Model cases Figure 1: Models of discharging appliance A i (p i,q i ) using a base case appliance A b (p b,q b ) 30

6 Results and discussions A base case appliance in an existing design guide attributed by the discharge probability p b = and the discharging flow rate q b =0.15 Ls 1 was used for discussion, i.e. A b (p b,q b )~[0.0282, 0.15]; and the corresponding base case fixture unit was U b =1 at the base case reference design flow rate q ref =10 Ls 1 [5]. In order to illustrate the sensitivity of the fixture units due to the choice of the reference design flow rate q ref (Ls 1 ), appliances attributed by discharge probabilities and discharging flow rates ranged from 1/8 to 8 times the based case attributes were considered, i.e. A i =A i (p i,q i ), where p i =kp b, q i =kq b and k[0.125, 8] respectively. Values of the fixture units were evaluated at various reference design flow rates q ref (Ls 1 ). Fixture unit Ui k= Reference design flow rate q ref (Ls 1 ) p i =kp b ; q i =kq b Figure 2: Fixture units of appliances references to a base case appliance of discharge probability of and discharge flow rate of 0.15 Ls 1 Figure 2 shows the fixture units U i of appliance A i with reference to A b at reference design flow rates q ref between 1 Ls 1 and 1000 Ls 1. It was noted that a unity based case fixture unit U b was defined for all reference design flow rates. Fixture units U i of appliances A i (kp i,kq i ) at k=0.125, 0.25, 0.5, 1, 2, 4, 8 were 0.009, 0.04, 0.193, 1, 5.6, 33, 200 at a reference design flow rate q ref =1 Ls 1 ; U i =0.013, 0.054, 0.229, 1, 4.5, 21, 101 at q ref =10 Ls 1 ; and U i =0.015, 0.06, 0.243, 1, 4.2, 18, 74 at q ref =100 Ls 1, respectively. It was observed that the reference design flow rates q ref had some influences on the fixture units of A i. The fixture unit ratio i,q ref indicates the variations of the values of fixture units U i of an appliance i at a selected reference design flow rate q ref as compared with the base case reference design flow rate q ref =10 Ls 1 and is expressed by an equation below. Ideally, 31

7 the fixture unit ratio i,q ref of an appliance is ideally close to unity over a range of q ref (Ls 1 ), which the selected reference design flow rate is insensitive to the fixture units. U i,q i,q ref (7) ref Ui,10 The results showed that the fixture unit ratio of appliances of k=0.125, 0.25, 0.5, 1, 2, 4, 8 times the base case attributes were =0.67, 0.74, 0.84, 1, 1.24, 1.57, 1.98 at a i,q ref 1 reference design flow rate q ref =1 Ls 1 ; and =1.14, 1.11, 1.06, 1, 0.92, 0.83, 0.74 i,q ref 100 at q ref =100 Ls 1, respectively. Apparently, the choice of a smaller reference design flow rate, e.g. at q ref =1 Ls 1, resulted a larger variation of i. (a) Fixture unit ratio i max k= p i =kp b ; q i =kq b min Reference design flow rate q ref (Ls 1 ) Fixture unit ratio i 2 1 p i =kp b ; q i =q b k[0.125, 8] (b) Reference design flow rate q ref (Ls 1 ) 32

8 (c) Fixture unit ratio i k=8 4 p i =p b ; q i =kq b Reference design flow rate q ref (Ls 1 ) Figure 3: Fixture unit ratios i for appliances A i (kp i,kq i ), k[0.125, 8] Figure 3 shows the fixture unit ratios for appliances A i =A i (p i,q i ), p i =kp b, q i =kq b, k[0.125, 8], grouped into 3 cases in (a), (b) and (c). It was noted that the maximum and minimum fixture unit ratios max, min and ranges of p i and q i were shown in the figure. Fixture unit ratios determined from the results presented in Figure 2 were showed in Figure 3(a) for p i =kp b, q i =kq b. It confirmed that a smaller variation of was found when a larger reference design flow rate (e.g. q ref 100 Ls 1 ) was selected as compared with a smaller reference design flow rate. Figure 3(b) showed that the fixture unit ratios were less sensitive to the discharge probability range p i =kp b for an appliance at the base case discharging flow rate q i =q b ; the corresponding discharge unit ratios i were between 0.94 and However, the ratios were sensitive to the discharging flow rates q i =kq b for an appliance operating at the base case discharge probability p i =p b as shown in Figure 3(c); the corresponding discharge unit ratios were from i =0.67 to 2.17 at q ref =1 Ls 1, i =0.69 to 1.15 at q ref =100 Ls 1 and i =0.61 to 1.20 at q ref =1000 Ls 1. Comparison with a stochastic model The probable maximum simultaneous demand of an installation with the number of appliances can be evaluated by a stochastic model [7]. This model was applied to evaluate the probable maximum simultaneous water demands of domestic washrooms at complex usage patterns, where the appliances in the same washroom would or would not operate simultaneously. The model parameter can be identified from some descriptive distribution functions. 33

9 In order to compare the influences due to the choice of the reference design flow rate in the fixture unit approach on the probable maximum simultaneous demand of an installation, the stochastic model takes a constant discharge probability p i and a constant discharging flow rate q i (Ls 1 ) for a number of appliances A i =A i (p i,q i ) of the same type; i=1 n i in the installation. The discharge operation is described by a random number p * [0,1]. q i 0 ; p p * i * (8) qi ; p pi In each simulation j, the simultaneous discharging flow rate q d,j (Ls 1 ) is determined by, n i q q ; i=1 n i ; (9) d,j i1 i The probable maximum simultaneous demand q * (Ls 1 ) is determined by the d distributions of all simulated simultaneously discharge flow rates ~ (Ls 1 ) from all simulations j=1 n s, where the allowable failure rate of 1% taken in some practices adopting the fixture unit approach, q * d q * d F ; 1 ~ q d dq d (10) 0 The required number of simulations n s can be determined with reference to the improvement on errors by further simulation steps. Two expressions of errors are used; the absolute modelling error a is determined by the modelled number of simultaneous discharging appliances for 99% cases N *, corresponding to =0.01 in Equation (4), * N N a * ; N Mp z 2Mp1 p (11) N And the relative modelling error at n s simulations expressed by the change of model output due to an increment of 1 simulation and is given by, N r 1 (12) N * ns1 * n s Regarding a discharge probability p[0.01,0.05], it was reported that the maximum absolute modelling error a would remain unchanged for simulations n s >10000, the corresponding relative modelling error r was q d 34

10 The fixture unit approach at the reference design flow rate q ref [1,1000] was used to determine the probable maximum simultaneous demands of installations which composed of 2 different appliance types A 1 and A 2, operating at a residential discharge pattern as shown in Table 1. The probable maximum simultaneous demands determined by the fixture unit approach at a reference design flow rate q (Ls 1 ) were then d, ref compared with those q * (Ls 1 ) determined by the stochastic model. The percentage d deviation between the probable maximum simultaneous demands is given by, q d,ref f,ref 1 100% * q (13) d f, ref Table 1: Example fixtures Appliances Discharge probability, p Discharging flow rate, q (Ls 1 ) Fixture unit at q ref =10 Ls 1 Fixture unit at q ref =250 Ls 1 Washbasin WC Figure 4 shows the percentage deviations of the fixture unit approach for an installation size from 100 to washbasin-and-wc pairs in residential buildings. In the figure, a positive value indicates an over-estimate by the fixture unit approach, this overestimation of the probable maximum simultaneous demands would be considered as satisfactory that the design of not-to-exceed the maximum allowable failure rate =1%. The results showed that the choice of reference design flow rates had significant influence on the predicted probable maximum simultaneous demands and hence a wide range of deviations f,ref (Ls 1 ) were reported. The deviations varied between 7% to 5%. Taking the existing practice of using a reference design flow rate of 10 Ls 1 as an example, the results showed that the fixture unit approach would give satisfactory predictions of the probable maximum simultaneous discharge flow rates for installation sizes of 900 residential washbasin-and-wc pairs. Within these range limits, an overestimate by the fixture unit approach at a reference design flow rate of 10 Ls 1 would not be more than 3% as compared with the ones determined by the stochastic model. It is noted that the installation sizes for an 80-storey high-rise residential building in Hong Kong and a housing estate of 40 high-rise residential buildings are about 1200 and 10,000 respectively. An increased reference design flow rate would be required for the 35

11 design criterion of 1% failure probability allowed for water supply systems in buildings. This study showed the reference design flow rates q ref =100 Ls 1 would be adequate for a residential installation of size up to 10,000. Table 1 gives the example fixture units for appliances at a reference flow rate of 100Ls 1. The existing fixture units used for some buildings are shown for comparison. The results suggested that fixture units can be used for some appliances in high-rise buildings. Percentage deviation f,ref 6% 2% % -6% 5 10 q ref =1 Ls 1-10% % Installation appliances M Figure 4: Percentage deviations f,ref of the design flow rates by the fixture unit approach at reference flow rates q ref Conclusion Fixture unit approach has been used for estimating the probable maximum simultaneous demands in water systems for a lot of buildings for many years based on a reference design flow rate of 10 Ls 1. This paper reported that the selection of the reference design flow rate would have significant influence on the estimated probable maximum simultaneous demand. The existing choice of the reference design flow rate would underestimate the demands of water supply systems in some high-rise buildings, i.e. more than 1% probability for the demands to exceed the estimated probable maximum simultaneous demands. The existing assumption of the reference design flow rate adopted in fixture unit approach would give good estimation for an installation up to 900 WC-and-washbasin pairs in residential buildings. The reference design flow rate would be increased for larger installations in high-rise buildings so that a good estimate can be made. This paper presents useful information in for the application of fixture unit approach in estimating the probable maximum simultaneous demands in water systems 36

12 of high-rise buildings and enables further studies on water supply system designs for similar built environment having a high population density. Acknowledgment The work described in this paper was partially supported by a grant from the Research Grant Council of the HKSAR, China (PolyU5305/06E and ) and by a grant from The Hong Kong Polytechnic University (GU551, GYG53). References 1. Konen T.P. and Goncalves O.M. (1993). Summary of mathematical models for the design of water distribution systems within buildings. Proceedings of CIBW062 International Symposium of Water Supply and Drainage for Buildings. 2. Hunter R.B. (1940). Methods of estimating loads in plumbing systems, Report BMS65. National Bureau of Standards, Washington. 3. Oliveira L.H., Goncalves O.M. and Uchida C. (2009). Performance evaluation of dual-flush WC cistern in a multifamily building in Brazil. Building Services Engineering Research and Technology, February, Mui K.W., Wong L.T. and Yeung M.K. (2008). Epistemic demand analysis for fresh water supply of Chinese restaurants. Building Services Engineering Research and Technology, May, Wise A.F.E. and Swaffield J.A. (2002). Water, sanitary and waste services for buildings (5th ed.). London: Butterworth Heinemann. 6. Courtney R.G. (1972). A Monte-Carlo method for investigating the performance of a domestic water system. Proceedings 1 st International Symposium on Water Supply and Drainage for Buildings CIBW062 (pp ) September, BRE, UK. 7. Wong L.T. and Mui K.W. (2008). Stochastic modelling of water demand by domestic washrooms in residential tower blocks. Water and Environment Journal, June, Plumbing services design guide (2002). The Institute of Plumbing, Essex, UK. 9. Galowin L.S. (2008). Hunter fixture units development. Proceedings - 34 th International Symposium on Water Supply and Drainage for Buildings CIBW062 (pp.58-80) September, The Hong Kong Polytechnic University, Hong Kong. 10. Wong L.T. and Mui K.W. (2007). Modeling water consumption and flow rates for flushing water systems in high-rise residential buildings in Hong Kong. Building and Environment, May,

Engineering, the Hong Kong Polytechnic University. Dr. K. W.

13 Presentation of Authors Dr. L. T. Wong is an associate professor at the Department of Building Services Engineering, the Hong Kong Polytechnic University. Dr. K. W. Mui is an assistant professor at the Department of Building Services Engineering, the Hong Kong Polytechnic University. 38

A survey of the sanitation load for domestic high-rise building estates in Hong Kong

A survey of the sanitation load for domestic high-rise building estates in Hong Kong L. T. Wong and K. W. Mui beltw@polyu.edu.hk behorace@polyu.edu.hk Department of Building Services Engineering, The Hong