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Design criteria and data analysis for cold water demand in buildings
Build. Sci. Vol. 9, pp. 143-150. Pergamon Press 1974. Printed in Great Britain
I 384 I
I
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Design Criteria and Data Analysis for Cold Water Demand in Buildings E. M. McKAY*
The sizing of pipework and tanks for water supply systems in buildings is still largely based on estimated or assumed demands. Clearly more demand data is required but perhaps less obviously there is need also for improved design procedures. This paper considers some appropriate criteria and proposes generalised design procedures for flowrates and storage.
context of water demand). It seems clear however that the acquisition of field data will continue to be on a very limited scale until more thought is given to the ways in which it may best be used.
1. INTRODUCTION SIZING the components of any supply and distri= bution system is implicitly an exercise in weighing risk of inadequacy against system cost. Thus any sound design will be based on a consideration of the likelihood with which given demands occur. Only in this way can the cost of meeting high but infrequent demands be assessed. During a study just completed on the demand for cold water in university chemistry laboratories (to be published) it became apparent that the absence of demand data and of appropriate design criteria were closely linked. Sizing has often been based, of necessity, on rather gross assumptions but this is not to say that such designs will automatically be wrong only that in many situations a more relevant approach is possible. Ideally, generalised procedures are required which lead to (1) a design flowrate for each point in a supply system and (2) the design relationship linking the volume and input flowrate of the supply tank, when used. Now the dictates of economy normally imply that a system will be inadequate to meet the demand on occasions. It is therefore most important that the relevant design criteria are made explicit so that the consequences (in terms of "frequency of inadequency") may be understood. In this paper we consider the choice of appropriate criteria and also the consequent procedures for data analysis so as to allow design values to be formulated. Proposals are made which have been applied successfully in the study of demand referred to above; these are recommended for wider application. In view of the variety of types of buildings and their occupants it is possible that there will never be a single set of design criteria and procedures capableof universal application (in the
2. D E S I G N F L O W R A T E
The demand flowrate, q, at some given point in a system will typically vary with time. A design flowrate is required which if used as the basis for pipe sizing ensures satisfactory operation "most of the time". The usual approach to this problem leads to a design flowrate, Q(P) say, such that it will be exceeded in the long run with a probability P. Now P is equated empirically with the long term fraction of time for which q exceeds Q(P). Further to allow for systematic variations with time of day these design values refer specifically to demand within a so called busy period. Using this definition it can be shown that the probability P is equal to the mean value of the fraction of time within each day when Q(P) is exceeded. It therefore follows that if this time fraction of excess demand is distributed symmetrically between days then the system performance is worse than that assumed on 50 per cent of the days. Also and more important the degree of overloading on these days is unspecified. To overcome the difficulty arising from the use of a single probability the following definition is proposed: The design flowrate QD(0~,fl) is that which, within some defined part of the day, will be exceeded for a fraction of time ~ on no more than a fraction fl of days. An illustration of the relative frequency distributions is given in figure 1. An important consequence of this choice of definition is that whilst one pair of parameters (a, fl) leads to a unique value of QD the converse is not true. This may seem to be a disadvantage but in fact it underlies the most significant advantage of the definition
*Building Research Establishment, Garston, Watford, Hefts. 143
14'4
~
,,E. M. McKay.
O~[ = flowrete on i~doy exceeded for ¢ime ~ 6frac'tion : a
/
QD (a,~) = flowrete which on l in ~ days is exceeded for ¢irne fracfion a
d
.
volumes, to be discussed later. It should he added that the number of users, although variable with time, may for our purpose be taken as the number of occupants, as this term is normally applied. 3. DESIGN P R O C E D U R E
q~ (a) .......
•
/
el
Qa~
(b)
QD(e ~1
F i g . 1.
viz. suitably prepared data enables a more complete picture to be obtained of the interaction between fractional excess demand time within a day and its expected frequency of occurrence. To illustrate the greater understanding provided by the new definition consider the following: ASsume from the primary flowrate data we have obtained those flowrates which within each daily period have been exceeded for the time fraction c¢. Assume ats0 that we have used various values of leading to their relative frequency distributions . f ( Q ~ I ) , f ( Q ~ 2 ) . . . . Then it might be found for example that the design flowrate for s t = 0.01, fit = 0 . 0 5 i s identical with that for ~2 = 0"10, flz = :0.002, i.e. it will also give rise to dissatisfaction for l o p e r cent of the time on 1 day in 500! This is illustrated in figure 2. Similarly the expected frequency of any other excess demand time fraction may be obtained.
d I = 0.05
//% Qo.ol
Qo.~0
I1~ Q D(0"01,0"05]
\ QD(0.10,0.002)
F i g . 2.
So far nothing has been said about the quantity or quantities on which our design flowrate is to be based. Generally the approach has been to take the level of provision of outlets i.e. the design fiowrate ~s obtained starting with the number of outlets of each type. However the number and usage of outlets will in turn be determined by the number and activities of the user population. It therefore seems relevant to derive design flowrate data directly as a function of the numbers and types of users. This course has been adopted here and also for design
FOR F L O W R A T E
Having defined our design flowrate we will now consider the ways in which it may be obtained. Initially we assume that the distribution of Q~ for a specified population of users has been previously determined. And further since any skew distribution may, theoretically at least, be transformed into a normal distribution we will assume, for the purpose of explanation, that Q~ is distributed normally. In this case and given Prob (Q~ > Qo) = fl we have the relation: Qo = # + K F r
(1)
where Kp is the unit normal •deviate associated with a probability of excess fl and # and a are the population mean and standard deviation respectively of the variable Q~. Thus we require to know only the distribution parameters for Q~ in order to find the design flowrate which is less than Q~ on no more than a fraction fl of days. Where the variable Q~ has to be transformed to produce a normal distribution equation (1) relates to the transformed variable and the inverse transformation for Q, is required to regain the design flowrate. The objection often raised to the use of a normal distribution for purposes such as ours is that it permits negative values of the variable. In the present context these would be meaningless. However assuming adequate tests have been applied to show that normality is a valid approximation no serious error will result so long as there is only a small fraction of the distribution to the left of Q, = 0. This amounts to saying that the coefficient of variation is small, say less than 1/3 approximately. In this case only about one thousandth part of the distribution is negative. Alternatively the objection may be overcome by using an appropriate transformation e.g. the log normal distribution. Finally it is important to realise that even with a non normal distribution so long as it has some characteristic shape the use of a relation in the form of equation (1) may still be used even with the untransformed variable. In Such cases the difference is only that the relationship between fl and Ka is not exactly as given in standard normal tables. It must be remembered that an acceptable probability is clearly something of an arbitrary decision in many cases and can easily be over emphasised. For a good discussion on this point see Jennet and Welch[l].
Design Criteria and Data Analysis for CoM Water Demand in Buildings 4. DESIGN FLOWRATE BASED ON EXPERIMENTAL DATA
Generally the parameters of the distribution for Q~ are unknown. In their place we will normally have the sample mean and standard deviation, obtained say from N days records of the demand set up by a known number of users. We may now employ one of two methods for our design value. The first is based on the adoption of working values for /* and or: the second makes use of tolerance limit theory much used for quality control based on the inspection of samples. Both methods will however lead to a design value which is essentially based on the sample statistics obtained. It has been mentioned that our design procedure to be of the most general application must enable us to predict design values for a composite population of users comprising U1, of type A1, U2 of A2 etc: this has been an important consideration in choosing between alternatives. In what follows we continue to assume the variable Q~ to be distributed normally.
4.1 The use of upper confidence limits for the mean and standard deviation Assume we have a set of N daily records of demand for an observed number of users Uo of a single type. We first wish to find the design flowrate which for chosen parameters ~ and [3 corresponds to any number of users U, of the same type. To do this we first make allowance for the sampling variations of the mean and standard deviation by taking upper confidence limits of both. Secondly, we make use of the property of the normal distribution which allows it to be summed with others of the same family. Thus for our upper confidence limits at a confidence level of p, we use the standard formulae (see for example Mullholland and Jones[2]). Upper Conf Lt of Mean = if+ tp(v).s Upper Conf Lt of SD
= X / Z ~ .s
(2)
where ff and s are the sample statistics and t and Z z are found from the relevant statistical tables for the number of degrees of freedom v = N - 1. In this way we find the working values for the distribution parameters of Q, corresponding to the observed number of users Uo which lead to a design value for Uo given by
ao(Uo; o:, [3) =/*(Uo)+Kfr(Uo)
(3)
To find the design value for any number of users of a single type, or of a number of types, we make use of an important theorem and its converse, see
145
Cramer[3]. This states that the sum of any n u m b e r of normally distributed variates is itself normally distributed and has the parameters /*(Sum)
=/.1+/.2+ . . . .
var (Sum) = varl + var2 + . . . . a (Sum)
~/*~ ~vari
= a / ( ~ a2) •
(4)
And from the converse of the theorem we have the parameters of the distribution which corresponds to a single (if hypothetical) user given by /*(1) = / * ( U o ) / U o
o-(1) = a(Uo)/~/U o
(5)
We may now build up the distribution parameters for any value of U by means of the relations, ~(Uo) ~(u) = ~ . v Uo
a(U) =
~(Uo) V'Uo
• M"U
(6)
Hence the design flowrate for U users of a single type may be derived from a study of demand set up by U o users and gives Qo(U; ~, [3) =
/*(Uo)_
_ ~(Uo)
Uo.. u + K a ~
~/U
(7)
It may readily be shown that the method also provides a solution with a composite population of users signified by ~ U i ) . This gives in its simplest form
Qo(~(Ui); ~, [3) = ~al. Ui)+(Zc, Ui) ~
(8)
where al = U~(1) and e i = K#2o'/2(1) Note: where the user population is homogeneous equation (8) may be replaced by a linear approximation in the form QD(U; ~, [3) = A + B U where A and B are simple functions of a and c, and the range over which the approximation is acceptable.
4.2. The use of tolerance limits As an alternative to the method described in 4.1 we may base our design procedure on the theory of proportion defective control by lot sampling, assuming again that the initial or transformed variable is normally distributed. As a basic statistical tool it has had widespread use in quality control but to the author's knowledge it has been used for water supply only by Maver[4] when it was applied to a study of volumetric demand in
. E::M. McKay
146
hospitals. Essentially the method if applied here would give a design value in the form of
XD = 2 + k . s
(9)
where k is a complex function of the probability between days, the level of confidence and sample size. Note only one level of confidence is used for the design value and to this extent it has an advantage over the working parameter approach. The derivation of k may be found in the literature, e.g. see Hald[5]; values of k have been tabulated by Bowker-and Liebermann[6] and Aitchison and Scunthorpe[7]. Assuming a single type of user the derivation given by Hald may be modified for a variable user population to give an equation similar in form to that of equation (7) although it is no easy matter to relate the levels of confidence involved. Finally, whilst the method has greater statistical justification where the users are of a single type, at present no solution has been found to the problem of treating mixed populations. Thus unless the users are, or can be "defined" as being of a single type this approach lacks the general application of that given in 4.1. 5. DESIGN PARAMETERS FOR A
TANKED SUPPLY When we speak of a storage tank it is important to remember that generally it is serving more than one purpose, in what followswe are concerned only with those requirements for (a) meeting the variable demand within the day under normal conditions of supply and (b) the provision of protection against failure of the tank input. Further we restrict the analysis to the case of a supply tank fed from a direct action ball valve i.e. one without delay. It must be stated at the outset that in neither case can the tank capacity, Vc, be specified independently of its rated input Qi, where Q~ is taken as that flowrate which occurs when the ball-valve is fully open. A strong relation will be shown to exist between Vc and Q~ for condition (a) above and one that is less strong but still significant for (b). As there is only one tank for both purposes it is clear that there will be interaction between the two requirements. 5.1. Variable demand in normal operation The problem to be solved under this heading may be stated as follows: how do we determine the requisite tank capacity and its rated input to ensure continuous operation under normal conditions of supply? Now a completely generalised approach which will deal with a composite population of
users, and would also be acceptable to designers, is not yet available. However a new method has been developed which should give adequate results in most situations even with a composite population of users. It will be shown below how the supply tank may be sized given the primary data in the form of volume used, V(t) say, against time of day. However, for any given time t, V(t) will vary from day to day. We therefore consider for each value of t the distribution of V(t) between days and predict a corresponding value V(t, p) which will be exceeded on no more than the fraction p of days. These values V(t, p) are then taken together to form what will be called the design day's demand, which it should be emphasised is not the demand of any one complete day actual or predicted. In fact it is a synthesis of demand variation both within and between days. The computation of V(t, p) for each chosen time t may be carried out in a manner similar to that previously described for flowrates in Section 4; in this case the variable V(t) replaces Q~. Composite populptions of users may be dealt with using an equation in the form of equation (8) with the meanings of the symbols appropriately altered. How many values of t need to be taken will depend on the particular case but it is unlikely to exceed five. Finally by virtue of the greater penalties arising from inadequate storage more stringent design probabilities will need to be applied compared with those for design flowrates. 5.2. The supply tank characteristics for the design day Assume we have the cumulative volume function V(t, p) for the design day. We wish to obtain the appropriate relation between the tank capacity, Vc and Qi to ensure that the tank is unlikely to be emptied whilst the supply exists. Consider any interval within the day of duration j hours and start-time t. Then to satisfy the endpoint condition, i.e. for t e = t+j, we have the relation
V~(t+j) = V~(t)+~tt +i (q,-qo) dt > 0
(lo) where
V~(t) is the volume stored at the time t ql
is the input flowrate
q0
is the demand flowrate
Note qz is a variable lying in the range 0 = q~ < Q~. Without significant loss of accuracy we may take
qi = Q i when V~(t) < Vc and qi = qo when qo < Qi and V , ( t ) = Vc.
Design Criteria and Data Analysis for Cold Water Demand in Buildings
147
5.2 Example of the use of the supply tank
Note: as used here Vc is in fact a variable and slightly less than the tank capacity. N o w equation (10) relates to the end-point condition only and to a given start-time. A satisfactory system will clearly have to meet this requirement for all t and all j. Thus we are required to satisfy simultaneously all possible relations typified by
characteristic To illustrate the method just described we will take a typical example of a design day's demand. Let Table 1 be a set of values for the cumulative demand function V(t), the maxima V s and the corresponding input flowrate maxima Q~(max)=
vj/j.
Vs(t) >Stt +~ ( q o - q i ) dt
The supply tank characteristic obtained will be as in figure 3 simplified to show the accuracy which
in order to ensure that Vs(t ) is always positive. This set of inequalities is clearly infinite if t and j take all possible values. There is however considerable redundancy in the set which may be eliminated. It is also possible to simplify matters by considering the effect of a variable q~. To obtain the supply tank characteristics it is found necessary only to consider q~ at its rated (maximum) value Qi essentially because we are concerned with the capability of the system within each interval t to (t+j). Storage is not being challenged when q~ < Q~ since under these conditions Vs(t) = V~, within the limits previously stated. Our required condition then reduces to the set
150
. , 180 rnmj = --ff~- m3/hr
>: o~3 I00 O_
Vs(t) > ftt+J qodt-jQi
O
Finally for each value o f j there will be a maximum value for the integral (which may of course occur for more than one start-time t). The variable t may now be eliminated and writing Vj for the maximum volume used over the interval j hours we have for the corresponding storage volume Vsj
V~j > V s - j Q ,
1 ~ \ ~ / 9 ) Region for safisfocfory ''~'N~,, \x N~erofi°n
[
50
"~ "~ Possible operofing ~ ~o,nts
\ I
\
\ \ \
\
\\
I
"~
I0 20 Rated input flowrate Qi,
(11)
Thus j is the only independent variable remaining. The supply tank characteristic may now be obtained using equation (11) only. We simply plot a straight line using Vc = V s - j Q ~ for each pair (L Vs). The resulting envelope represents the boundary condition which just satisfies the demand on one design day and it will rarely be necessary to plot more than three or four lines to give an adequate result. The final step is now to set a lower limit to Q,. To ensure that the tank is fully recharged before the start of the next day we set Qi(min) at the design day's total demand averaged over 24 hours.
\
"--. I \ "r--..\,
i
30 m3/hr
Fig. 3. might still be achieved. Taking two points by way of illustration we may then superimpose the corresponding cumulative inputs on the cumulative demand as shown in figure 4. It will be seen that the stored volume just falls to zero in each case. Note also the method has allowed for the fact that q~ < Q~ for part of the working day. The example shows the power of the method in cases of widely varying demand within the day. The value o f j has been allowed to increase in steps of one hour but
Tab& 1. t
V(t) j Vj VJ~
0800
0900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
0 1 38 38
2 2 68 34
10 3 92 30"7
25 4 104 26
57 5 115 23
67 6 139 23"2
72 7 154 22
110 8 166 20-8
140 9 174 19"3
164 10 177 17-7
176 11 179 16"2
179 12 180 15
180
14s
E:
.:Mc ay
/
240
.i
220
!
200
IV ....
-
Vs(t)min = Vc--ltto in (qo-qi) dt
//
d
160
o
140
o
120
E
-
O,,mo,o., I'l/i/i e input (2)_ I
100
E
......
._~/
J
/
~
[
(12)
where to is a reference time, taken to be the start of the working day, such that Vs(to) = V~ i.e. the stored volume is at its rated maximum. Also we take Day 1 to refer to the day in which breakdown occurs, Day 2 is the day after, tB = time of breakdown, t' = t +24 hours, and tPd = average daily consumption.
180
~E
the minimum stored volume with supply connected, i.e. normal operation, to occur at t = tmin i.e.
Then for D a y 1 and t > tB we have
demand Vo(i')
V~(t) = V~ - It,~ (qo -qz) d t - Itn, qodt
B0
i/ 60
40
-
C u m u l a t i v e | # "
--.
input(1).../// -- _ j l I [
(I) Vc:32m3, Qi:2Om31hr (2) Vc:86m 3, Oi--lOm31hF
20
Thus Vs(t) goes on decreasing throughout the working day which without loss of generality can be the full 24 hours. For Day 2 and t' > t'~ (time of restoration)
V~(t) = Vc-I',~ (qo -qi) dt-~'t'~ qodt
l 4
J l 12 16I ~0i 8 Time of day, hr
i 24
-Jtt'~' (qo - Qi) dt --
t
~- V c - V a - j t o ( q o - q ~ )
Fig. 4.
dt
since figure 3 illustrates the point that larger increments would have sufficed. We note also that increments greater than one hour Could have been used for the design day's demand.
5.3. Designing for protection against supply failure It must be remembered that we are now dealing with the comparatively rare event of supply failure and also that such failure will not in general be related to the demand of the building in question. It seems reasonable therefore to take the demand on the day of failure to be that of the mean (strictly the median) day and defined in similar terms to those used for the design day given earlier. Also we define the term storage Design Period, Tp as that period of time for which the supply may be removed under the worst conditions, during the mean day; it allows for the demand to be met during this period and continuously after restoration is effected. The completely general case which allows for all Te, all rated inputs, Q~ and further takes account of the variable duration within the day when the stored volume is at its rated maximum is still unsolved. We may however using the following procedure give good estimates for particular cases. This will be done for Tp = 24 hours. First we take
5~toqo dt ___Jtt:o qo dt
and
~tt~(qo-q~) dt
~- S~:,,( q o - Q~) dt Note for the latter approximation we err on the side of safety. Now it may be shown that the critical range for the time of breakdown is given by to < t B < tml, and this gives rise to the minimum stored volume in Day 2 at t' = tmln' whence
gs(tmln t'm,~ (qo-qi) dt ' ') = V c - Va-Sto, = v,-(e~+ v/)
Where V j is obtained from the mean day's supply tank characteristic using the value of Q~ previously chosen to satisfy normal operation. We may therefore write
vc >= ¢~ + v / for the tank volume which ensures that the demand may be satisfied continuously throughout, and beyond, a 24 hour breakdown. Acknowledgement The work described has been carried out as part of the research programme of the Building Research Establishment of the Department of the Environment and this paper is published by permission of the Director.
Design Criteria and Data Analysis for CoM Water Demand in Buildings REFERENCES
1. W.J. JENNETTand B. L. WELCH,The control of proportion defective as judged by a single quality characteristic varying on a continuous scale. Suppl. J. Roy. Stats. Soc, 6, 80 (1939). 2. H. MULLHOLLANOand C. R. JONES. Fundamentals of Statistics. Butterworth (1968). 3. CRAMrR, Mathematical Methods of Statistics, Princeton (1946). 4. T.W. MAYER,A study of water consumption in ward units. Tech. Paper No. 2, Building Services Research Unit (1964). 5. A. HALO, Statistical Theory with Engineering Applications, Wiley (1952). 6. A.H. BOWKERand G. J. LIEBERMANNEngineering Statistics, Prentice Hall (1959). 7. J. AITCmSONand D. SCtmTHORPE.Design Value Tables. Building Services Research Unit (1964). La grandeur de tuyaux et de citernes pour les syst~mes d'alimentation en eau de bgttiments est encore largement fond6e sur les demandes estim6es ou suppos6es. I1 est 6vident que des donn6es suppl6mentaires sur les besoins sont n6cessaires, mais peut atre moins en 6vidence est la n6cessit6 de proc6dures de design am61ior6es. Ce texte consid~re quelques crit6res appropri6s et propose des proc6dures de design pour les d6bits et l'emmagasinage.
Die Gr6Benbemessung von Rohranlagen und Behgltern fur Wasserversorgungssysteme in Geb/iuden griindet sich im allgemeinen auf gesch/~zten oder angenommenen Bedarf. Es werden offensichtlich mehr Daten fiir Bedarf ben/Stigt, aber, vielleicht weniger offensichtlich, ist eine Verbesserung der Entwurfsprozedur n6tig. Dieser Bericht befal3t sich mit einigen entsprechenden Kriterien und empfiehlt verallgemeinerte Entwurfsprozeduren far FlieBgeschwindigkeiten und Speicherung.
149
I 384 I
I
I(A3f)I
Design Criteria and Data Analysis for Cold Water Demand in Buildings E. M. McKAY*
The sizing of pipework and tanks for water supply systems in buildings is still largely based on estimated or assumed demands. Clearly more demand data is required but perhaps less obviously there is need also for improved design procedures. This paper considers some appropriate criteria and proposes generalised design procedures for flowrates and storage.
context of water demand). It seems clear however that the acquisition of field data will continue to be on a very limited scale until more thought is given to the ways in which it may best be used.
1. INTRODUCTION SIZING the components of any supply and distri= bution system is implicitly an exercise in weighing risk of inadequacy against system cost. Thus any sound design will be based on a consideration of the likelihood with which given demands occur. Only in this way can the cost of meeting high but infrequent demands be assessed. During a study just completed on the demand for cold water in university chemistry laboratories (to be published) it became apparent that the absence of demand data and of appropriate design criteria were closely linked. Sizing has often been based, of necessity, on rather gross assumptions but this is not to say that such designs will automatically be wrong only that in many situations a more relevant approach is possible. Ideally, generalised procedures are required which lead to (1) a design flowrate for each point in a supply system and (2) the design relationship linking the volume and input flowrate of the supply tank, when used. Now the dictates of economy normally imply that a system will be inadequate to meet the demand on occasions. It is therefore most important that the relevant design criteria are made explicit so that the consequences (in terms of "frequency of inadequency") may be understood. In this paper we consider the choice of appropriate criteria and also the consequent procedures for data analysis so as to allow design values to be formulated. Proposals are made which have been applied successfully in the study of demand referred to above; these are recommended for wider application. In view of the variety of types of buildings and their occupants it is possible that there will never be a single set of design criteria and procedures capableof universal application (in the
2. D E S I G N F L O W R A T E
The demand flowrate, q, at some given point in a system will typically vary with time. A design flowrate is required which if used as the basis for pipe sizing ensures satisfactory operation "most of the time". The usual approach to this problem leads to a design flowrate, Q(P) say, such that it will be exceeded in the long run with a probability P. Now P is equated empirically with the long term fraction of time for which q exceeds Q(P). Further to allow for systematic variations with time of day these design values refer specifically to demand within a so called busy period. Using this definition it can be shown that the probability P is equal to the mean value of the fraction of time within each day when Q(P) is exceeded. It therefore follows that if this time fraction of excess demand is distributed symmetrically between days then the system performance is worse than that assumed on 50 per cent of the days. Also and more important the degree of overloading on these days is unspecified. To overcome the difficulty arising from the use of a single probability the following definition is proposed: The design flowrate QD(0~,fl) is that which, within some defined part of the day, will be exceeded for a fraction of time ~ on no more than a fraction fl of days. An illustration of the relative frequency distributions is given in figure 1. An important consequence of this choice of definition is that whilst one pair of parameters (a, fl) leads to a unique value of QD the converse is not true. This may seem to be a disadvantage but in fact it underlies the most significant advantage of the definition
*Building Research Establishment, Garston, Watford, Hefts. 143
14'4
~
,,E. M. McKay.
O~[ = flowrete on i~doy exceeded for ¢ime ~ 6frac'tion : a
/
QD (a,~) = flowrete which on l in ~ days is exceeded for ¢irne fracfion a
d
.
volumes, to be discussed later. It should he added that the number of users, although variable with time, may for our purpose be taken as the number of occupants, as this term is normally applied. 3. DESIGN P R O C E D U R E
q~ (a) .......
•
/
el
Qa~
(b)
QD(e ~1
F i g . 1.
viz. suitably prepared data enables a more complete picture to be obtained of the interaction between fractional excess demand time within a day and its expected frequency of occurrence. To illustrate the greater understanding provided by the new definition consider the following: ASsume from the primary flowrate data we have obtained those flowrates which within each daily period have been exceeded for the time fraction c¢. Assume ats0 that we have used various values of leading to their relative frequency distributions . f ( Q ~ I ) , f ( Q ~ 2 ) . . . . Then it might be found for example that the design flowrate for s t = 0.01, fit = 0 . 0 5 i s identical with that for ~2 = 0"10, flz = :0.002, i.e. it will also give rise to dissatisfaction for l o p e r cent of the time on 1 day in 500! This is illustrated in figure 2. Similarly the expected frequency of any other excess demand time fraction may be obtained.
d I = 0.05
//% Qo.ol
Qo.~0
I1~ Q D(0"01,0"05]
\ QD(0.10,0.002)
F i g . 2.
So far nothing has been said about the quantity or quantities on which our design flowrate is to be based. Generally the approach has been to take the level of provision of outlets i.e. the design fiowrate ~s obtained starting with the number of outlets of each type. However the number and usage of outlets will in turn be determined by the number and activities of the user population. It therefore seems relevant to derive design flowrate data directly as a function of the numbers and types of users. This course has been adopted here and also for design
FOR F L O W R A T E
Having defined our design flowrate we will now consider the ways in which it may be obtained. Initially we assume that the distribution of Q~ for a specified population of users has been previously determined. And further since any skew distribution may, theoretically at least, be transformed into a normal distribution we will assume, for the purpose of explanation, that Q~ is distributed normally. In this case and given Prob (Q~ > Qo) = fl we have the relation: Qo = # + K F r
(1)
where Kp is the unit normal •deviate associated with a probability of excess fl and # and a are the population mean and standard deviation respectively of the variable Q~. Thus we require to know only the distribution parameters for Q~ in order to find the design flowrate which is less than Q~ on no more than a fraction fl of days. Where the variable Q~ has to be transformed to produce a normal distribution equation (1) relates to the transformed variable and the inverse transformation for Q, is required to regain the design flowrate. The objection often raised to the use of a normal distribution for purposes such as ours is that it permits negative values of the variable. In the present context these would be meaningless. However assuming adequate tests have been applied to show that normality is a valid approximation no serious error will result so long as there is only a small fraction of the distribution to the left of Q, = 0. This amounts to saying that the coefficient of variation is small, say less than 1/3 approximately. In this case only about one thousandth part of the distribution is negative. Alternatively the objection may be overcome by using an appropriate transformation e.g. the log normal distribution. Finally it is important to realise that even with a non normal distribution so long as it has some characteristic shape the use of a relation in the form of equation (1) may still be used even with the untransformed variable. In Such cases the difference is only that the relationship between fl and Ka is not exactly as given in standard normal tables. It must be remembered that an acceptable probability is clearly something of an arbitrary decision in many cases and can easily be over emphasised. For a good discussion on this point see Jennet and Welch[l].
Design Criteria and Data Analysis for CoM Water Demand in Buildings 4. DESIGN FLOWRATE BASED ON EXPERIMENTAL DATA
Generally the parameters of the distribution for Q~ are unknown. In their place we will normally have the sample mean and standard deviation, obtained say from N days records of the demand set up by a known number of users. We may now employ one of two methods for our design value. The first is based on the adoption of working values for /* and or: the second makes use of tolerance limit theory much used for quality control based on the inspection of samples. Both methods will however lead to a design value which is essentially based on the sample statistics obtained. It has been mentioned that our design procedure to be of the most general application must enable us to predict design values for a composite population of users comprising U1, of type A1, U2 of A2 etc: this has been an important consideration in choosing between alternatives. In what follows we continue to assume the variable Q~ to be distributed normally.
4.1 The use of upper confidence limits for the mean and standard deviation Assume we have a set of N daily records of demand for an observed number of users Uo of a single type. We first wish to find the design flowrate which for chosen parameters ~ and [3 corresponds to any number of users U, of the same type. To do this we first make allowance for the sampling variations of the mean and standard deviation by taking upper confidence limits of both. Secondly, we make use of the property of the normal distribution which allows it to be summed with others of the same family. Thus for our upper confidence limits at a confidence level of p, we use the standard formulae (see for example Mullholland and Jones[2]). Upper Conf Lt of Mean = if+ tp(v).s Upper Conf Lt of SD
= X / Z ~ .s
(2)
where ff and s are the sample statistics and t and Z z are found from the relevant statistical tables for the number of degrees of freedom v = N - 1. In this way we find the working values for the distribution parameters of Q, corresponding to the observed number of users Uo which lead to a design value for Uo given by
ao(Uo; o:, [3) =/*(Uo)+Kfr(Uo)
(3)
To find the design value for any number of users of a single type, or of a number of types, we make use of an important theorem and its converse, see
145
Cramer[3]. This states that the sum of any n u m b e r of normally distributed variates is itself normally distributed and has the parameters /*(Sum)
=/.1+/.2+ . . . .
var (Sum) = varl + var2 + . . . . a (Sum)
~/*~ ~vari
= a / ( ~ a2) •
(4)
And from the converse of the theorem we have the parameters of the distribution which corresponds to a single (if hypothetical) user given by /*(1) = / * ( U o ) / U o
o-(1) = a(Uo)/~/U o
(5)
We may now build up the distribution parameters for any value of U by means of the relations, ~(Uo) ~(u) = ~ . v Uo
a(U) =
~(Uo) V'Uo
• M"U
(6)
Hence the design flowrate for U users of a single type may be derived from a study of demand set up by U o users and gives Qo(U; ~, [3) =
/*(Uo)_
_ ~(Uo)
Uo.. u + K a ~
~/U
(7)
It may readily be shown that the method also provides a solution with a composite population of users signified by ~ U i ) . This gives in its simplest form
Qo(~(Ui); ~, [3) = ~al. Ui)+(Zc, Ui) ~
(8)
where al = U~(1) and e i = K#2o'/2(1) Note: where the user population is homogeneous equation (8) may be replaced by a linear approximation in the form QD(U; ~, [3) = A + B U where A and B are simple functions of a and c, and the range over which the approximation is acceptable.
4.2. The use of tolerance limits As an alternative to the method described in 4.1 we may base our design procedure on the theory of proportion defective control by lot sampling, assuming again that the initial or transformed variable is normally distributed. As a basic statistical tool it has had widespread use in quality control but to the author's knowledge it has been used for water supply only by Maver[4] when it was applied to a study of volumetric demand in
. E::M. McKay
146
hospitals. Essentially the method if applied here would give a design value in the form of
XD = 2 + k . s
(9)
where k is a complex function of the probability between days, the level of confidence and sample size. Note only one level of confidence is used for the design value and to this extent it has an advantage over the working parameter approach. The derivation of k may be found in the literature, e.g. see Hald[5]; values of k have been tabulated by Bowker-and Liebermann[6] and Aitchison and Scunthorpe[7]. Assuming a single type of user the derivation given by Hald may be modified for a variable user population to give an equation similar in form to that of equation (7) although it is no easy matter to relate the levels of confidence involved. Finally, whilst the method has greater statistical justification where the users are of a single type, at present no solution has been found to the problem of treating mixed populations. Thus unless the users are, or can be "defined" as being of a single type this approach lacks the general application of that given in 4.1. 5. DESIGN PARAMETERS FOR A
TANKED SUPPLY When we speak of a storage tank it is important to remember that generally it is serving more than one purpose, in what followswe are concerned only with those requirements for (a) meeting the variable demand within the day under normal conditions of supply and (b) the provision of protection against failure of the tank input. Further we restrict the analysis to the case of a supply tank fed from a direct action ball valve i.e. one without delay. It must be stated at the outset that in neither case can the tank capacity, Vc, be specified independently of its rated input Qi, where Q~ is taken as that flowrate which occurs when the ball-valve is fully open. A strong relation will be shown to exist between Vc and Q~ for condition (a) above and one that is less strong but still significant for (b). As there is only one tank for both purposes it is clear that there will be interaction between the two requirements. 5.1. Variable demand in normal operation The problem to be solved under this heading may be stated as follows: how do we determine the requisite tank capacity and its rated input to ensure continuous operation under normal conditions of supply? Now a completely generalised approach which will deal with a composite population of
users, and would also be acceptable to designers, is not yet available. However a new method has been developed which should give adequate results in most situations even with a composite population of users. It will be shown below how the supply tank may be sized given the primary data in the form of volume used, V(t) say, against time of day. However, for any given time t, V(t) will vary from day to day. We therefore consider for each value of t the distribution of V(t) between days and predict a corresponding value V(t, p) which will be exceeded on no more than the fraction p of days. These values V(t, p) are then taken together to form what will be called the design day's demand, which it should be emphasised is not the demand of any one complete day actual or predicted. In fact it is a synthesis of demand variation both within and between days. The computation of V(t, p) for each chosen time t may be carried out in a manner similar to that previously described for flowrates in Section 4; in this case the variable V(t) replaces Q~. Composite populptions of users may be dealt with using an equation in the form of equation (8) with the meanings of the symbols appropriately altered. How many values of t need to be taken will depend on the particular case but it is unlikely to exceed five. Finally by virtue of the greater penalties arising from inadequate storage more stringent design probabilities will need to be applied compared with those for design flowrates. 5.2. The supply tank characteristics for the design day Assume we have the cumulative volume function V(t, p) for the design day. We wish to obtain the appropriate relation between the tank capacity, Vc and Qi to ensure that the tank is unlikely to be emptied whilst the supply exists. Consider any interval within the day of duration j hours and start-time t. Then to satisfy the endpoint condition, i.e. for t e = t+j, we have the relation
V~(t+j) = V~(t)+~tt +i (q,-qo) dt > 0
(lo) where
V~(t) is the volume stored at the time t ql
is the input flowrate
q0
is the demand flowrate
Note qz is a variable lying in the range 0 = q~ < Q~. Without significant loss of accuracy we may take
qi = Q i when V~(t) < Vc and qi = qo when qo < Qi and V , ( t ) = Vc.
Design Criteria and Data Analysis for Cold Water Demand in Buildings
147
5.2 Example of the use of the supply tank
Note: as used here Vc is in fact a variable and slightly less than the tank capacity. N o w equation (10) relates to the end-point condition only and to a given start-time. A satisfactory system will clearly have to meet this requirement for all t and all j. Thus we are required to satisfy simultaneously all possible relations typified by
characteristic To illustrate the method just described we will take a typical example of a design day's demand. Let Table 1 be a set of values for the cumulative demand function V(t), the maxima V s and the corresponding input flowrate maxima Q~(max)=
vj/j.
Vs(t) >Stt +~ ( q o - q i ) dt
The supply tank characteristic obtained will be as in figure 3 simplified to show the accuracy which
in order to ensure that Vs(t ) is always positive. This set of inequalities is clearly infinite if t and j take all possible values. There is however considerable redundancy in the set which may be eliminated. It is also possible to simplify matters by considering the effect of a variable q~. To obtain the supply tank characteristics it is found necessary only to consider q~ at its rated (maximum) value Qi essentially because we are concerned with the capability of the system within each interval t to (t+j). Storage is not being challenged when q~ < Q~ since under these conditions Vs(t) = V~, within the limits previously stated. Our required condition then reduces to the set
150
. , 180 rnmj = --ff~- m3/hr
>: o~3 I00 O_
Vs(t) > ftt+J qodt-jQi
O
Finally for each value o f j there will be a maximum value for the integral (which may of course occur for more than one start-time t). The variable t may now be eliminated and writing Vj for the maximum volume used over the interval j hours we have for the corresponding storage volume Vsj
V~j > V s - j Q ,
1 ~ \ ~ / 9 ) Region for safisfocfory ''~'N~,, \x N~erofi°n
[
50
"~ "~ Possible operofing ~ ~o,nts
\ I
\
\ \ \
\
\\
I
"~
I0 20 Rated input flowrate Qi,
(11)
Thus j is the only independent variable remaining. The supply tank characteristic may now be obtained using equation (11) only. We simply plot a straight line using Vc = V s - j Q ~ for each pair (L Vs). The resulting envelope represents the boundary condition which just satisfies the demand on one design day and it will rarely be necessary to plot more than three or four lines to give an adequate result. The final step is now to set a lower limit to Q,. To ensure that the tank is fully recharged before the start of the next day we set Qi(min) at the design day's total demand averaged over 24 hours.
\
"--. I \ "r--..\,
i
30 m3/hr
Fig. 3. might still be achieved. Taking two points by way of illustration we may then superimpose the corresponding cumulative inputs on the cumulative demand as shown in figure 4. It will be seen that the stored volume just falls to zero in each case. Note also the method has allowed for the fact that q~ < Q~ for part of the working day. The example shows the power of the method in cases of widely varying demand within the day. The value o f j has been allowed to increase in steps of one hour but
Tab& 1. t
V(t) j Vj VJ~
0800
0900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
0 1 38 38
2 2 68 34
10 3 92 30"7
25 4 104 26
57 5 115 23
67 6 139 23"2
72 7 154 22
110 8 166 20-8
140 9 174 19"3
164 10 177 17-7
176 11 179 16"2
179 12 180 15
180
14s
E:
.:Mc ay
/
240
.i
220
!
200
IV ....
-
Vs(t)min = Vc--ltto in (qo-qi) dt
//
d
160
o
140
o
120
E
-
O,,mo,o., I'l/i/i e input (2)_ I
100
E
......
._~/
J
/
~
[
(12)
where to is a reference time, taken to be the start of the working day, such that Vs(to) = V~ i.e. the stored volume is at its rated maximum. Also we take Day 1 to refer to the day in which breakdown occurs, Day 2 is the day after, tB = time of breakdown, t' = t +24 hours, and tPd = average daily consumption.
180
~E
the minimum stored volume with supply connected, i.e. normal operation, to occur at t = tmin i.e.
Then for D a y 1 and t > tB we have
demand Vo(i')
V~(t) = V~ - It,~ (qo -qz) d t - Itn, qodt
B0
i/ 60
40
-
C u m u l a t i v e | # "
--.
input(1).../// -- _ j l I [
(I) Vc:32m3, Qi:2Om31hr (2) Vc:86m 3, Oi--lOm31hF
20
Thus Vs(t) goes on decreasing throughout the working day which without loss of generality can be the full 24 hours. For Day 2 and t' > t'~ (time of restoration)
V~(t) = Vc-I',~ (qo -qi) dt-~'t'~ qodt
l 4
J l 12 16I ~0i 8 Time of day, hr
i 24
-Jtt'~' (qo - Qi) dt --
t
~- V c - V a - j t o ( q o - q ~ )
Fig. 4.
dt
since figure 3 illustrates the point that larger increments would have sufficed. We note also that increments greater than one hour Could have been used for the design day's demand.
5.3. Designing for protection against supply failure It must be remembered that we are now dealing with the comparatively rare event of supply failure and also that such failure will not in general be related to the demand of the building in question. It seems reasonable therefore to take the demand on the day of failure to be that of the mean (strictly the median) day and defined in similar terms to those used for the design day given earlier. Also we define the term storage Design Period, Tp as that period of time for which the supply may be removed under the worst conditions, during the mean day; it allows for the demand to be met during this period and continuously after restoration is effected. The completely general case which allows for all Te, all rated inputs, Q~ and further takes account of the variable duration within the day when the stored volume is at its rated maximum is still unsolved. We may however using the following procedure give good estimates for particular cases. This will be done for Tp = 24 hours. First we take
5~toqo dt ___Jtt:o qo dt
and
~tt~(qo-q~) dt
~- S~:,,( q o - Q~) dt Note for the latter approximation we err on the side of safety. Now it may be shown that the critical range for the time of breakdown is given by to < t B < tml, and this gives rise to the minimum stored volume in Day 2 at t' = tmln' whence
gs(tmln t'm,~ (qo-qi) dt ' ') = V c - Va-Sto, = v,-(e~+ v/)
Where V j is obtained from the mean day's supply tank characteristic using the value of Q~ previously chosen to satisfy normal operation. We may therefore write
vc >= ¢~ + v / for the tank volume which ensures that the demand may be satisfied continuously throughout, and beyond, a 24 hour breakdown. Acknowledgement The work described has been carried out as part of the research programme of the Building Research Establishment of the Department of the Environment and this paper is published by permission of the Director.
Design Criteria and Data Analysis for CoM Water Demand in Buildings REFERENCES
1. W.J. JENNETTand B. L. WELCH,The control of proportion defective as judged by a single quality characteristic varying on a continuous scale. Suppl. J. Roy. Stats. Soc, 6, 80 (1939). 2. H. MULLHOLLANOand C. R. JONES. Fundamentals of Statistics. Butterworth (1968). 3. CRAMrR, Mathematical Methods of Statistics, Princeton (1946). 4. T.W. MAYER,A study of water consumption in ward units. Tech. Paper No. 2, Building Services Research Unit (1964). 5. A. HALO, Statistical Theory with Engineering Applications, Wiley (1952). 6. A.H. BOWKERand G. J. LIEBERMANNEngineering Statistics, Prentice Hall (1959). 7. J. AITCmSONand D. SCtmTHORPE.Design Value Tables. Building Services Research Unit (1964). La grandeur de tuyaux et de citernes pour les syst~mes d'alimentation en eau de bgttiments est encore largement fond6e sur les demandes estim6es ou suppos6es. I1 est 6vident que des donn6es suppl6mentaires sur les besoins sont n6cessaires, mais peut atre moins en 6vidence est la n6cessit6 de proc6dures de design am61ior6es. Ce texte consid~re quelques crit6res appropri6s et propose des proc6dures de design pour les d6bits et l'emmagasinage.
Die Gr6Benbemessung von Rohranlagen und Behgltern fur Wasserversorgungssysteme in Geb/iuden griindet sich im allgemeinen auf gesch/~zten oder angenommenen Bedarf. Es werden offensichtlich mehr Daten fiir Bedarf ben/Stigt, aber, vielleicht weniger offensichtlich, ist eine Verbesserung der Entwurfsprozedur n6tig. Dieser Bericht befal3t sich mit einigen entsprechenden Kriterien und empfiehlt verallgemeinerte Entwurfsprozeduren far FlieBgeschwindigkeiten und Speicherung.
149