Hunter‘s Probability Method Applied to Determine Water Supply Design Second Flow in Buildings of Residents

Zhang Minghua Gao Yufei
Xi‘a(chǎn)n University of Architeture and Technology School of Environment ＆ Municipal Engineering
Add:No.13 YanTaRd, Xi‘a(chǎn)n, Code:710055

ABSTRACT This article discusses the calculation of design second flow in the buildings of residents, and studies two important factors influencing design flow step by step.
KEYWORDS design second flow in the buildings of residents , frequency of fixture use, water supply reliability, program operation

1、 At present the calculation about design second flow in the buildings of residents and some problems

According to the current norm《water supply and water drainage of buildings design norm》(GBJ15-88)(is called《design norml》for short as below).The calculation about design second flow in the buildings of residents is given as below:

qg=0.2α（Ng)^1/2+KNg (1)

here qg is the design flow of the calculated pipe line;
Ng is the equivalent of sanitary fixtures of the calculated pipe line;
α=1.02,K=0.0045;
This formula comes from original norm 《water supply、 water drainage and hot water supply design norm》(TJ15-74),where the formula about design flow is qg=0.2α（Ng)^1/2+KNg,which is so-called the law of square root. Eq.(1)is based on the formula in 《design norm》(TJ15-74)according to former measured data. Eq. (1)is established by probability analysis, which is the first formula according to our country‘s actual situation, but it is also subordinate to the scope of the law of square root. As the accidental chance of measured data, and at that time the limitation of life standard, this formula is not highly reliable in theory and practice, which can‘t meet actual demands to a certain extent.
Some main problems exists in Eq.(1):
(1)It considers only non-uniformity of water usage, while doesn‘t embody the random of water usage.
(2)It can‘t reflect various factors influencing water consumption such as the number of people served by fixtures, various combination forms of different fixtures, water supply system.
(3)It can‘t reflect the design flow compared to different water consumption quota.
(4)It is about the value of α K at 3 different situations listed by 《design norm》(GBJ15-88).

1. Hunter‘s probability method‘s theoretical deference and some problems needed to resolve in our country

When calculating the design second flow in the buildings of residents, we need analyze all possible variable, then get the optimum system meeting the demand. Although the number of people engaged in studies of water supply (installations) is not so large, research activity has grown considerably both in Japan and other countries. In the past many developments have taken place and new techniques are being introduced in practice. It has now become mandatory to analyze and protect the environment from sewage and water pollution problems. Keeping this in view, it has become increasingly important to carry out further studies based on new ideas on water supply fixtures and develop a new perspective toward design methods and design data. At present, Hunter‘s probability method is used broadly in America、 Japan and many European countries.
The theory of Hunter‘s probability method is established as shown:
It was put forward by A American Roy. B. Hunter in 1924 and further made more perfect in 1940.He made highly practical charts. Its basic principle is. the use of sanitary fixtures in the system is assumed as an accidental event, which is depicted by the mathematics model of the binomial distribution. It is supposed N sanitary fixtures is linked at a certain pipe line, while every fixture is turned on and turned off independently. The quota of flow of every sanitary fixtures is q0, and the maximum water supply second flow is q0N, the minimum water supply second flow is 0, then the water supply flow at random passing though the calculated pipe line is defined as q (0≤q≤q0N). when designing the system, we have to consider meeting the consumer‘s need and economical feasibility. So we can simply study the extreme situation, exactly water consumption at high hour in high day. It is supposed the frequency of each sanitary fixture use at high hour is p, the the non-frequency of its use is (1-p). So the probability that I fixtures is simultaneously used among N fixtures at maximum is given as shown:

P(x-i)=CⁱNPⁱ(1-P)^(N-1) (2)

According to Hunter‘s definition, the probability value about single system which is consisted of a single sanitary fixture is shown as below:

Here Pm the probability value of simultaneously use of m fixtures at most;
M is the design value of simultaneous use of sanitary fixtures;
P is the use probability of single fixture at water consumption peek time;
Pr is the guarantee value of water supply;
Hunter‘s method is more scientific, while the calculation result with it differs variously compared with Eq. (1). The main reason is:
(1)Although t/T about sanitary fixtures in Hunter‘s method is based on the investigation of hotel and household, as a result of the difference of the character of buildings、life custom district, we need study the value is whether reasonable and whether accords with our country‘s actual situation.
(2)Hunter‘s calculation curve can be applied in flush valve, while Eq.(1) can‘t be used in flush valve in our country.
(3)The style of sanitary fixtures is different . In Hunter‘s curve, water supply flow rate of flush tank and bathtub is 0.25l/s and 0.51l/s separately, which is much larger than 0.1l/sand O.2l/s in our country.
　　(4) The guarantee value of Hunter‘s probability method is O.99, which is a supposed value .It　is needed to study how to determine the specific guarantee value.
It is considered by analysis of Hunter‘s method that theoretically the probability of sanitary fixture　use is the same, the peek time of fixtures use only exits at water consumption peek time by actual　analysis ,so it is needed to determine the time length of peek time of sanitary fixtures use,　not just given 24 hour.
3. two important parameter influencing design second flow
　　(1) probability value of fixtures use The definition formula of probability P is p=t/T, here T is time interval of fixtures used continuously twice, t is time length of every water usage.
We can see from the definition formula:
(1) It is related to water consumption standard .The normal is more high ,the number of water use and time length of water use both increase ,so the value of p becomes large;
(2) It is related to total number of fixtures liked at designed pipe line .With the increasing of total number, the value of P decreases. The researcher in Japan has made actual investigation ,as shown in table 1:

Probability of simultaneous use of various equipment (%) table.l

type of fixtures
probability value 1 2 4 8 12 16 24 32 40 50 70 100 Flush valve 100 50 50 40 30 27 23 19 17 15 12 10 general
equipment 100 100 70 55 48 45 42 40 39 38 35 33

(3) It is related with the number of people served by fixtures .The number is much larger, the value of is larger;
(4) It is concerned with combination form of fixtures.
(5) It is concerned with the style of sanitary fixtures .For examlple ,flush water volume of Flush tank differs from 2L to 12L .At other same conditions, the ratio between maximum probability value and minimum probability value is 6, varying much largely.
(6) It is related to the value of T. Water use time of family is 18h-24h listed in 《designnorm》 .Actually the peek time is commonly at 6AM-10AM, 6PM-1OPM. The probability value is 2-3 much larger than the time used with 18h-24h.
In fact the design flow in buildings of residents is related to sex-age structure of residents、 people‘s behavior . means of water use 、 climate and so on. As the use probability is a multi- factor value, it is impossible to take all influencing factors into consideration. We have to just consider some important factors. At present as a result of measured level 、 research funds、 and understanding about this in our country, we can‘t make systematic and effective investigation(in Japan it needs about 5-6 years). We should not only collect data about this to accumulate experience, but also use the experience of other countries similar to our country in climate and life custom. The bble.2 as below shows the investigation in Japan about water consumption of families, which mainly accords to my investigation. The probability got is based on table.2.
(2) reliability of water supply
As water supply system belongs to the scope of service system, it is required that the value of reliability should be determined by customers‘ demand. In the norm 《Fundamental terms anddefinitoion of reliability》,the definition of reliability R is: the product‘s capacity to accomplish the formulated function at formulated conditions in formulated time.

times of water use in buildings of residents(in summer) table.2

time at homet(h) structure of family numbers t≥0.8×24=19.20.6
0.6×24=14.4≤t＜19.2 t＜14.4 infant, professional housewife, old people child, students, professional housewife college student, husband(employment),wife(employment) t≥19.2
14.4≤t＜19.2
t＜14.4 times of defecation (times/person.weekly) times of urination (times/person.weekly) male female male female number of family infant or not times of bathing(times/per.weekly) times of showering(times/per.weekly) times of water supply of bathing (times/cap.weekly) times of washing(times/cap.weekly) 2w N 　3.0(2.5) 4.0(3.0) 2.8 6.5 　 Y 　3.4(2.5) 3.6(3.0) 3.4 9.5 3人 N 　4.5(3.0) 2.5(3.5) 4.5 11.0 　 Y 　3.8(3.0) 3.2(3.5) 3.8 10.5 4人 Y 　4.0(2.8) 3.0(3.5) 4.0 14.5 　 N 　4.1(3.0) 2.9(3.5) 4.1 13.0 5人 Y 　3.5(3.0) 3.5(3.0) 3.5 17.0

Note:the number in brackets is got by the author in the year of 2000 due to the residents of Xi‘a(chǎn)n Architecture and Technology University.
A ccording to the character of water use, requirement of water supply reliability is determined in economical principle and by the necessity of sanitary condition of buildings and life service normal. In all situations, reliability of the system requires that when function of water supply is destroyed, it should meet people‘s demand without any influence.
Due to municipal water supply system, it is complicated to determined working parameter of system (such as the unit of water use 、time duration of water use 、 time length of water use ). It is feasible to take any effective control of water consumption due to such water supply system as the service system. Design index of system function and the limitation of allowance is determined according to the character of wervice objects, which also determine the grantee value of water supply reliability. But the problem of investment also exists. The grantee value is larger, then the investment is higher, increasing burden of service objects so it dissatisfies service objects. We must take full consideration of grantee value and investment, so the optimum problem exits. Table.3 is qualification disposal of water supply reliability.

Qualification disposal of reliability table.3

reliability
Pr average possibility of water breaking possibility of water breaking at peek time time of water breaking in a year(h) time of water breaking in a month time of water breaking in d day (s) (min) 0.9875 0.0125 0.0375 109.5 9.125 1080 18 0.99 0.01 0.03 87.6 7.3 864 14.4 0.995 0.005 0.015 43.8 3.65 432 7.2 0.997 0.003 0.009 26.28 2.19 259.2 4.32 0.999 0.001 0.003 8.76 0.76 86.4 1.44

Note:(1) 0.9875 is a grantee value applied in our country‘s design normal
(2)0.99 is a grantee value applied in Russian design normal
(3)0.997 is a grantee value just for determining the extent of influencing
(5)a year is consisted of 365 days; the peek time length of water use is 8h

From table.3, it is shown when the grantee value is 0.995、0.997、0.999 or much larger, it does not influence life of people much bitterly.
4、drawing up program and contrast
This program is drawn up in computer language Foxbase, having the character of simple debugging and great speed. It can be run in Visual Fox, suit for designers. The result of debugging is shown as below:
Attention: Every family consists of 3.5 persons; the volume of flush tank is supposed 7L; the probability of washing use is calculated due to table.2; the time length of water use is 8h. We can get the use probability of bathtub is 0.024;probability of washing basin is 0.022.
Probability of flush tank is 0.045;probability of flush valve is 0.0033;probability of showering is 0.0167;probability of bathtub is 0.024;probability of washing basin is 0.022.
In table.4、table5、table.6 a=1.02 and K=0.0045
(1)3 different grantee value(0.99、0.995、0.999）， contrast of design second flow

contrast of number of washing basin by probability method table.4

number of use Pr=0.99 Pr=0.997 Pr=0.999 number of use Pr=0.99 Pr=0.997 Pr=0.999 1 [1,7] [1,4] [1,2] 9 [162,190] [135,159] [115,137] 2 [8,20] [5,12] [3,9] 10 [191,219] [160,186] [138,260] 3 [21,38] [13,27] [10,20] 11 [220,249] [187,213] [161,188] 4 [39,59] [28,44] [21,35] 12 [250,280] [214,241] [189,213] 5 [60,82] [45,64] [36,52] 13 [281,311] [242,270] [214,240] 6 [83,107] [65,86] [53,71] 14 [312,343] [271,299] [241,267] 7 [108,134] [87,109] [72,92] 15 [344,375] [300,329] [268,295] 8 [135,161] [110,134] [93,114] 16 [376,407] [330,359]

[296,323]

contrast of number of simultaneous use of showering by probability method table.5

number of use Pr=0.99 Pr=0.997 Pr=0.999 number of use Pr=0.99 Pr=0.997 Pr=0.999 1 [1,2] [1] [1] 9 [59,68] [49,58] [43,50] 2 [3,7] [2,5] [2,4] 10 [69,79] [59,67] [51,59] 3 [8,14] [6,10] [5,9] 11 [80,89] [68,77] [60,68] 4 [15,21] [11,16] [10,13] 12 [90,100] [78,87] [69,77] 5 [22,30] [17,23] [14,19] 13 [101,111] [88,97] [78,87] 6 [31,39] [24,31] [20,26] 14 [112,122] [98,108] [88,96] 7 [40,48] [32,40] [27,34] 15 [123,134] [109,118] [97,106] 8 [49,58] [41,48] [35,42] 16 [135,145] [119,129] [107,117]

contrast of number of simultaneous use of washing basin by probability method table.6

number of use Pr=0.99 Pr=0.997 Pr=0.999 number of use Pr=0.99 Pr=0.997 Pr=0.999 1 [1,3] [1,2] [1] 9 [77,90] [65,76] [56,66] 2 [4,10] [3,6] [2,5] 10 [91,104] [77,88] [67,77] 3 [11,18] [7,13] [6,10] 11 [105,118] [89,101] [78,89] 4 [19,28] [14,21] [11,17] 12 [119,132] [102,114] [90,101] 5 [29,39] [22,31] [18,25 13 [133,147] [115,128] [102,114] 6 [40,51] [32,41] [26,34] 14 [148,162] [129,142] [115,127] 7 [52,63] [42,52] [35,44] 15 [163,177] [143,156] [128,140] 8 [64,76] [53,64] [45,55] 16 [178,192] [157,170] [141,154]

Note in bable.4、table.5、table6 [45,55] denotes that the number of sanitary fixtures simultaneous use is the same.
Figure.1、figure.2、figure.3 shows the contrast of design second flow at 3 different grantee value.
(2)contrast of design second flow between probability method and normal formula

By analysis, we can see that grantee value of water supply reliability does not influence design second flow much bitterly. To compare with the result we suppose Pr=0.9875.
The use of mixed fixtures can‘t be simply added by such single fixture. obviously less than the added result. A Japan researcher poses that flow of simultaneous use of different fixtures is the maximum flow of single fixture use added half flow of other single fixture.
According to his research. we can get figure.4 as below.

5、conclusion
We can draw such conclusion by analysis:
(1)The design second flow got by Hunter‘s probability method is much less than what is got by normal formula. The main reason is at an accurate probability value condition , the value of Pr has much influence on design second flow. So it is meaningful to determine the probability meeting actual demand. This needs us to do full and accurate investigation.
(2)Form the result and the figures, we can see when Pr is a fixed value in spite of different number of fixtures, the value of Pm is different (Pm≥Pr). When the number of simultaneous use is the same ,the number of total fixtures is in a value scope, that is to say the value of the design second flow is more separated.
(3)This character does researches at the water use behavior of people and get the more convincing probability of fixtures use . We get the calculation method considering economic and feasibility.
(4)Hunter‘s probability method is more scientific and more reliable than the normal formula.