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UNITED STATES DEPARTMENT OF COMMERCE • Jesse H. Jones, Secretary

\ NATIONAL BUREAU OF STANDARDS • Lyman J. Briggs, Director

BUILDING MATERIALS

and STRUCTURES

REPORT BMS6S

1 Methods of Estimating Loads in Plumbing Systems

by

ROY B. HUNTER

ISSUED DECEMBER 16, 1940

The National Bureau ol Standards is a tact-finding organization;

it does not "approve" any particular material or method of construction. The technical findings in this series of reports are to be construed accordingly.

UNITED

FOR SALE

STATES GOVERNMENT PRINTING OFFICE • WASHINGTON

BY THE SUPERINTENDENT OF DOCUMENTS, WASHINGTON, D. C. • PRICE

• 1940

10 CENTS
Foreword

For many years, diflPerences in the plumbing regulations or pluiiibing-code requirements in different localities have been a source of annoyance to building owners, builders,

and public health officials and have hindered the standardiza tion, on a general or national

scale, of plumbing materials and equipment, and of construction design. The principal reasons for the existence of phmrbing codes are the protection of the health of the

occupants of a building and of the inhabitants of the community or city in which the

building is located and the protection of property from damage by water or sewage.

The results sought by phmibing requirements and health regulations applying to plumbing are or should be the same for low-cost or expensive and for small or large

buildings, and may be covered by general regulations applying to all plumbing. Any

simplified or standardized form of plumbing construction which gives these results

that is, which complies with necessary general regulations pertaining to sanitation or health—cannot be prohibited without violating the inherent rights of building owners and of the public.

Because of the above, this report has been written, as succeeding reports on plumbing in the Building Materials and Structures series will be written, first from the standpoint of essential general requirements and second from that of permissible construction

within the general requirements—both general requirements and permissible construction

being based on sound physical principles.

This report deals with one of the factors which must be considered in the selection

of adequate yet economical sizes of pipes for plumbing systems—namely, the load to be expected from a given number and kind of plumbing fixtures. It is expected that it

will be followed by other reports in the Building Materials and Structures series dealing

with other aspects of plumbing problems.

Where the exercise of judgment is necessary in the choice of the numerical value of factors used in developing and illustrating the application of the method of estimating

loads as described in this report, it will be understood that such numerical values, when

not the actual results of Bureau tests or experiments, represent the author's judgment in regard to the most suitable factor to use in the application of the method, and that these

are not to be regarded as standard values, unless later approved as such by a repre- sentative and authoritative body.

Lyman J. Briggs, Director.

[II]
Methods ot Estimating Loads in Plumbing Systems

by

ROY B. HUNTER

CONTENTS

Page

Foreword ii

I. Introduction 1 II. Purpose 1 III. Definitions and symbols 2

IV. Basis of selection of pipe sizes 3

V. Character of flow in plumbing systems. 3

1. Cases of loading 5 VI. Statement of the problem 5 VII. Developmentof the probability function. 6

VIII. Interpretation of the probability function 7 IX Proposed uses of the probability function for making load estimates 7

1. Pertinent information obtainable from the probability function^ _ 7

Page IX. Proposed uses of the probability function for making load estimates—Con.

2. Values of i, T, t, and q 8

3. Presentation of probability data 11

4. Derivation of fixture weights (fix- ture units) 13

5. Relative load weights for different conditions of service 16

6. Application of load chart and

weight table 18 X. Adequacy of the method 19 XI. Estimates of sewage loads 20

XII. Discussion 21 XIII. References 21

ABSTRACT

This report describes a method of estimating the demand and sewage loads for which provision should be made in designing plumbing systems in order that the

service may be satisfactorv. The characteristics of flow through a plumbing system and of the operation of supply valves and plumbing fixtures are described, and

their influence on the method of estimating the load to

be expected is discussed. The relative load-producing

values of different kinds of commonly used plumbing

fixtures are anah zed, and a table is developed giving

relative load weights in terms of a load factor called the

"fixture unit." An estimate curve developed by means of the probability function is given, and its use

in conjunction with the table of fixture units is illus- trated.

I. INTRODUCTION

Aside from the choice of materials and equipment as regards quality, style, and quantity of

fixtures, simplification and standardization of

piping layouts offer the principal opportunities

for reduction in cost of plumbing systems.

Simplification and standardization in this respect must comply with accepted health

regulations and minimum requirements for plumbing, which in turn should be based on

scientific principles. This is true whether the

building is small and "'low-cost'' or large and

expensive. Few, if any, existing plumbing

codes are based entirely on scientific principles;

and they do not in general permit the simplification of plumbing piping layouts that could

be accomplished well within the requirements

for sanitation, and in many cases with more

satisfactory operation than can be obtained by

installations imder existing codes.

This report deals with one of the factors on which minimum requirements should be based,

the maximum load to be provided for in plumbing systems. Other reports are planned, dealing with the principles of water supply and

water distribution in buildings, the principles

of building drainage, and the principles of

venting.

II. PURPOSE

The purpose of this series of papers is to

collect in an organized form the mass of information obtained by the author over a number

of years, beginning with the investigation in 1921 of plumbing of small dwellings, and including the current research (1937-40) on plumbing

for low-cost housing, together with the results

of intervening experiments related to plumbing

requirements, and to interpret the results of

these investigations in a form suitable for direct

[1]
and practical application. It is hoped that

this series of papers will supply the logical answer to many of the controversial questions

pertaining to pipe sizes and design of plumbing

construction.

III. DEFINITIONS AND SYMBOLS

A number of terms employed in the plumbing

industry, and a few that are now introduced

for the first time, are hei-e defined in the sense

in which they are to be used in this and later papers of this series. Insofar as they are adequate and applicable, the defuiitions from

Recommended Minimum Requirements for Plumbing [1]^ will be utilized. The plumhing system of a building includes

the water-supply distributing pipes, the fixtures

and fixture traps, the soil, waste, and vent

pipes, the building drain and building sewer, and the storm-water drainage pipes, together

with their devices, appurtenances, and connections within or adjacent to the building.

The building main is the pipe from the street water main or other source of supply to the

building served.

The water-distributing system is the piping

by which the water is conducted from the

building main to its various places of use within

and adjacent to the building, consisting of

laterals, risers, and branches.

The sanitary system of a building is the piping

system, including soil, waste, and vent pipes,

that conducts waste water and water-borne

wastes from the plumbing fixtures to the street sewer or other place of sewage disposal.

A plumbing fixture is any receptacle through

which waste water or water-borne waste is dis- charged into the sanitary system.

A drain is any pipe that carries waste water

or water-borne waste in either the storm or sanitary system.

A fixture drain is the single drain from a

fixture-trap outlet to its junction with another

drain.

A waste pipe is any drain that carries the

waste water from one or more fixtures other

than water closets.

> Figures in brackets indicate the literature references at the end of

this report.

A soil pipe is any drain that carries the waste

water from one or more water closets with or without the waste water from other fixtures. A stack is a vertical section of pipe extending

two or more stories with branches therefrom to serve the stories through which it passes, as waste stack, soil stack, or vent stack.

A horizontal branch is a drain, extending

laterally from a soil or waste stack, with or without vertical sections, to which the fixture

drains in the same story are connected.

A building drain is that part of a sanitary

system which receives the sewage from soil and waste stacks and conducts it to the building

sewer. A building sewer is the main trunk of a sanitary, storm, or combined system from a point 5

feet outside the uiner face of the building waU

to the street sewer. A primary branch is the single drain line leading from the base of soil or waste stack to its point of connection with the building drain or another branch thereof.

A secondary branch is any branch of the building drain other than a primary branch.

A vent or vent pipe is any pipe installed ia a j sanitary system for the pmpose of permitting a

circulation of air through or within the system.

A vent stack or main vent is a vent pipe paralleling a soil or waste stack to which the branch

vents from the stack and its horizontal branches

are connected.

Demand load is the volume rate of demand

for water imposed on the water-distributing

system or any branch thereof by the use of

water, as by plumbing fixtures. Sewage load is the load in volume or volume

rate imposed on the sanitary system or any

branch thereof by the use of plumbing fixtures. Charging load is the volume inflow into the

sanitary system from one or more plumbing

fixtures, applicable when the capacity of the

pipe receiving the load depends mainly on the

mternal volmne (cross section times length) of

the pipe.

Receiving capacity is the internal volume of a section (drain) of the sanitary system, applicable when the total sewage load is an additive

fimction of the separate volumes but not of the

separate rates. Terminal velocity is the constant velocity of

[2]
flow that exists under fixed conditions when

equihbrium between the forces (gravitational)

producing and the forces (frictional) opposing

flow has been estabhslied.

Design factor m is the particular value of r out of n fixtm-es that will be found in operation

a selected fraction of the time under the as- sumed conditions of use. Fixture unit, or load factor, is a numerical

factor which measures on some arbitrary scale the load-producing effect of a single plumbing

fixture of a given kind. The use of the fixture

unit makes it possible to reduce the loadproducing characteristics to a common basis. n =the total number of fixtures or supply

openings of a given kind in the system.

7'=the number of fixtures out of a total of n which at any given instant of observation are foimd operating to impose a demand load on the supply

system, or a sewage load on the drainage system.

m=the design factor (see definition above).

g=the average volume rate of flow, in

gallons per minute, to or from a plumbing fixture during actual operation. Q=the total volume of water in gallons that

flows to or is discharged by a fixture at each use. t=the average duration of flow in seconds

for a given kind of fixture for one use. 7"= the average time in seconds between

successive operations of any given

fixture of a particular kind.

T =a time interval in seconds such that the

event in question (for example, ex- actly r fixtures will be found operating) will occur for an aggregate of 1 second.

(7" =the number of combinations of n things

taken r at a time.

p" = the probability of exactly r fixtures out

of a total of n fixtures being found in

operation at an arbitraiy instant of observation.

r=n y^,p " =the probability that some number of

fixtures between r=m and r=n, in- clusive, will be found operating at an

arbitrary instant of observation.

IV. BASIS OF SELECTION OF PIPE SIZES

In order to select from a series of available or stock sizes the minimum size of pipe adequate

to convey a given flow of a liquid, it is necessary

to know accurately the capacities of the differ- ent sizes of pipe for the conditions under which

the pipe is to be used and to know accurately

the load the pipe will be called upon to carry, capacity and load being expressed in the same

units or in convertible units.

Pipe formulas, given in terms of the velocity

of flow and factors on which the velocity depends, are the usual means employed in esti- mating capacities of pipes. It is an easy step

to convert a formula giving the velocity of flow

in a pipe into one giving the volume rate of flow. As previously indicated, the applicability of pipe formulas to the conditions existing in plumbing systems in service will be treated in

other papers. However, since it is necessary to express capacity and load in the same units in

selecting the proper size of pipe for a given use,

it becomes necessary to consider the units in

which both the expected load and the capacity

of the pipes involved can be expressed before

choosing the units in which load estimates

should be expressed. A pipe-flow formula is merely an equation of dynamic equilibrium and,

as such, applies only to the particidar conditions—namely, the condition of uniform con- tinuous flow in the pipe. Hence the conventional pipe formulas apply to the irregular and

intermittent flows that occur in plumbing sys- tems only during that time (usually very short)

and in that section of pipe in which the variable

factors involved (velocity or volume rate of

flow, pressure or hydraulic gradient, and hydraulic radius) are constant.

V. CHARACTER OF FLOW IN

PLUMBING SYSTEMS

Probably the best conception of the limitations of the applicability of a pipe formula for adequately estimating the capacities of plumbing pipes, and the best indication of the logical

units to apply in expressing capacity and load

in selecting the proper size of pipe to use in

particular parts of the system, will be given

by considering the changing character of flow

[3]
as the water passes through the system from

the som'ce of supply to the building sewer. Water closets with flush-valve control of the

supply are chosen for this illustration of varying flow characteristics, partly because critical loading conditions in the various parts of the

plumbing system are usually produced by

these fixtures and partly because the factors

that determine the load for an individual

fixture are more nearly constant and can be more readily evaluated than for other fixtures. Assume that each water closet demands for each flush Q =4 gallons in the average time

^=9 seconds, giving an average rate of supply

to the closet of approximately g=27 gpm, and

that each of these demands will occur on the

average of once in T=5 minutes =300 seconds.

Since the water-supply pipes are flowing full under pressure, there will be continuity of flow

in both time and quantity in any part of the

supply system through which water flows to the water closet or to a group of closets; and

therefore the peak load at any time will be the

sum of the separate demands, r, occurring at that time, or rq gpm, where g is the average

flow per fixture. In this case a pipe formula

is applicable for computing limiting capacities on the basis of diameters, lengths, roughness,

and available head. The difficult part of the

problem is to estimate the maximum r out of a

total of n water closets supplied by the system

for which provision should be made.

In the drainage system the flow fi'om the

water closet through the fixture drain (closet

bend) into a horizontal branch will be at about

the average rate of 27 gpm for a period of from

8 to 10 seconds. However, the flow will be

less nearly uniform than the flow into the bowl,

rising to a rate of from 30 to 40 gpm temporarily,

as siphon or jet action comes into play in the

bowl, and dropping immediately following the

peak siphon action to a rate approximately

equal to that of the supply at the time. In the horizontal branch the temporary peak

flow will flatten out, and the stream will tend

to assume the terminal velocity (see deflnition)

for the diameter and slope of the drain and the

existing volume rate of flow. As the stream

enters the vertical stack, the sudden change in

direction piles the water up, tending to form a

slug of water at the entrance. In fact, the

surge from a single water closet entering a 3- , or 4-inch vertical stack through a short hori- ) zontal branch of the same diameter will almost

fill the stack at this point. Immediate accelera-

, tion takes place in the vertical stack and con-

' tinues throughout the length of the stack or , until the terminal velocity of the stack is , reached for the volume rate of flow.

^ From the stack the stream will enter the , sloping primary branch at a velocity much j greater than the terminal velocity for that \ branch (shooting flow) ^ and will be deceler- , ated until the terminal velocity for that volume j rate of flow is approximately reached. If the , volume rate of flow, the entrance velocity, and

the slope, diameter, and roughness of the pri- j mary branch are in the proper relation, the , transition from shooting flow at the entrance \ to gravity flow will occur in a hydraulic jump, j a phc>nomenon which frequently occurs in the \ primary branches of building drains. j Throughout the drainage system in the direc- j tion of flow there is a continuous decrease j in the volume rate of flow from a single closet | or a group of closets flushing at the same time, || a corresponding increase in the time of flow,

jj and a continuous decrease in the peaks pro- i duced by siphon or jet action in the bowl or | by the overlapping of the flows from diflerent | closets. The conception of flow in plumbing drains as described has been verified by numerous ex- periments in the laboratory and by numerous

observations and depth measurements made on

the building drains of large systems in actual

service. In plumbing drains the peak rate of flow from a water closet at the point of entrance

to the horizontal branch is usually reduced to I about one-half that rate at the point of transi- e tion from shooting flow to gravity flow in a j primary branch, and the volume rate of flow e continues to decrease from that point on. t If a single water closet flushes on the average J at intervals of 5 minutes and delivers 4 gallons i for each flush, the stream in the drain tends i toward a continuous flow of Q/T=4/5 =0.8 gpm | as the distance from the water closet increases, ii Likewise for a very large number of water

2 Shooting flow here refers to the high-velocity constantly decelerating flow that occurs in the primary branch of a building drain between its juncture with the soil stack and the hydraulic jump in the primary

branch. <

[4]
closets, /i, each used at that average rate, ami

a very long building drain, the flow tends

toward a uniform value 0.8 n gpm. If n is greater than Tjt, it is obvious that flow in the

main building drain will be continuous for the

greater part of the time that each of the closets

is being used at the average rate of once in T

seconds, and that the tendency toward uniform

continuous flow will increase as the size of the

plumbing system increases—that is, as n and

the length of the drains increase. Hence in a very large system in which all n closets are in continuous use, the outflow will be continuous

and the rate will fluctuate around 0.8 n gpm.

Since the stream in the main building drain

and building sewer of large biiildings approaches

uniform flow, and since these drains are ordinarily laid with a uniform slope, load and

capacity can be most adequately expressed in

volume rate of flow and a pipe formula for imiform flow may be used in estimating limits

of capacity for building drains. However, it should be kept in mind that the volume rate of flow in any particular section of the drainage

system is not an additive function of the separate volume rates of flow into the system.

There is another consideration that has a

direct beaiing on the method chosen for esti- mating the loads to be provided for in a building drainage system and on the choice of units

such that load and capacity may be expressed in the same units. The horizontal branches, the

entrance fittings to soil and waste stacks, and

primary branches are parts of the drainage

system in which critical loading or possible overloading are most likely to occur. In horizontal

branches which lie near the fixtures, and to a

lesser degree in the stack fittings and horizontal branches, the determinate factors in the

selection of adequate drain pipes are the charging load and the receiving capacity. In esti- mating the charging load to be provided for,

the problem is again to obtain a reasonable

estimate of the value of r to be provided for in relation to n. The details of the method re- lating to receiving capacity will be completed

in the papers dealing with capacities.

1. Cases of Loading

From the character of the flow in the watersupply system and in the drainage system as

descj-ibed, it is obvious that there &vv, three

distinct cases of loading to be consideretl

(1) One applying to the supply (the demand

load) which will be measui'cd by 2/(2, where r

is the number of separate like demands at the

time for one kind of fixture and g[ the average

volume rate of flow per demand, rq the total flow for r fixtures of one kind, and Zrg the sum

or total flow for fixtures of difl'erent kinds;

(2) one applying to hoi'izontal branches (the

charging load) which will be measured by 2rQ,

hi which r is the number of like fixtures flowing

at the time and Q is the average volume of

sewage per fixture introduced into the drain

within the time necessary to consider in relation

to a particular location of the drain; and (3) a

third case applying as a limit for building drains

and building sewers which will be measured in terms of "LnQJT, in which n is the total number

of any one khid of fixtm-e installed in the system,

Q the average volume used per fixture, and T

the average time between uses. In cases 1 and

2, the load rg to be expected and provided for in the design of the plumbing system depends,

with a few exceptions, on approximately the same time factors, t and T, and hence bears

approximately the same relation to the number

of fixtin-es, n. (See page 20.)

VI. STATEMENT OF THE PROBLEM

As stated in describing the conditions of

flow in plumbing systems for which load esti- mates are sought, the most difficult part of the

problem is to estimate the proportional part of the total possible load that should be provided

for—that is, the number of demand or charging

loads, r, bearing some logical relation to the

total number of fixtures, n. In an earlier paper [1] the authors suggested the relation

between r and n expressed by a probability

function as a possible means of estimating

loads to be expected in plumbing systems and

developed the function applying to certain

special cases.

It now appears that the possibilities of the

pi'actical application of this function and the

limitations of the method under certain conditions have not been understood clearly. It seems advisable, therefore, to consider the

probability function in its more general form.

[5]
I

I

to explain more fully the suggested method of employing the function in making the estimates

in question, and to point out certain limitations

in its practical application. The problem

may be stated as follows: Assuming that there are n fixtures in a system, each operated once

in T seconds on the average, and that each

operation is of t seconds' average duration,

what is the probability that r fixtures will be

found operating simultaneously at any arbitrarily chosen instant of observation?

It is necessary to define the expression,

"operating simultaneously," in order to completely define a particular event of "r fixtures

operating simultaneously." In the following

development of the theory, this event will be

considered as occurring when r, and only r, fixtures are found flowing at the instant of

observation; and hence the r fixtures found

flowing will include all those, and only those,

which began their operation during the f-second

interval immediately preceding the instant of

observation.

VII. DEVELOPMENT OF THE PROBABILITY FUNCTION

By the generally accepted concept of probability, the probability that a particular fix- ture out of a number, n, will be found operating

at any arbitrarily chosen instant of observation

is tjT, where t has been defined as the duration

of each operation and T as the time between

operations of each fixture. In the same manner, the probability that the particular fixture

will not be in operation at the instant of observation is 1 —i / T or (T—t)/T.

A law of combinations that applies to the

composite event of which the probability is sought in this problem may be stated as fol- lows: The number of ways in which two or more independent events can occur together is the product of the ways each can occur separately. A similar law of probability may be

stated as follows: The probability of two or more independent events occurring together, in

this case at the same instant, is the product of

the probabilities of their separate occurrence. By the law of combinations, the probability

that none of the remaining n—1 fixtures will be operating at the instant of observation is

The probability that a particular

one of ?i fixtures and none of the remaining ?? — 1 fixtures will be operating at the instant is

But since there are n fixtures, there are n ways ' in which the event, one and only one fixture - found flowing, can occur, and hence the probaj bility of this event being found occurring at the instant of observation is

|

This can be generalized fox any group of r, j and r only, constituting the particular event I for which the probability is sought, by again t applying the definitions and laws. 'i In any group of r fixtures the probability a that any particular one will begin to operate J within a given t seconds is t/T, and since there i are r cases in the group, the probability, that ^

all r fixtures will begin operation within the same f seconds is (t/Ty. Also, of the n fix-

'

tures, any one of

n{n—l) {n—2) to r factors_ nl ,

different groups is equallj^ likely, and hence by ' the law of combinations the probability that

' some one of the equally likely groups of r fixtures

will be in operation at the instant of observation will be ^"(^Y'^r- As before, the probability that the remaining 71 —r flows will fall

outside tins particidar t seconds is ( j, 1 •

.

Therefore, the probability that exactly r fix- tures will be operating at a particular instant

^ of observation in the maiiner defined is

. . . (2)

This is the general term of a series, expressing ^ the probability that r out of n fixtures will be | foimd flowing at any arbitrary instant of ' observation. This series may be represented | as '

[6]
(3)

r=(j r=0 J- Equation 3 is equivalent to the binoniial

/ t T— 1\

expansion of ( j""!" 7^ ) tlie

conventional expi'ession for certainty, since

either no fixtures, r — Q, or some niunber of

fixtures from r=f to r = n must be operating

at the instant of observation.^

VIII. INTERPRETATION OF THE PROBABILITY FUNCTION

As previously stated, the probability function,

Pr, eq 2, gives the probability that exactly r

fixtiires out of a total of n will be found operating at an arbitrary instant of observation, provided that all n fixtures are in continuous use

at the assumed rate. The probability, ^"i may also be interpreted as the percentage or

fraction of the time in the long run that r flows

will occur in the manner defined, since the

fraction

time of r fixtures operating

total time

is also the probability of the occurrence. Hence for any given values of n and r, if the

probability function as developed is multiplied

by a time t in seconds such that

rjp''=rC"l~^(^y-'=l second, ... (4) or T r

the equation signifies that / fixtures will be in simultaneous operation for an aggregate of 1 second out of every r seconds that all n fixtures are in use at the assumed rate. Likewise, the

condition that a chosen design factor, r=m,,

will not be exceeded more than a given fraction

of the time 1/r, is expressed by

r = 77!+l (5)

• ' Equations 2 and 3 may be developed by other methods and in dif- ferent forms. However, the results will be identical or may be reduced

to an identity, as evidenced by the function developed by Fry [2] for the analogous problem of busy lines in telephone systems (see eq 156, p. 335,

of the reference), which differs from eq 2 only in the form to which it is reduced and in the notation employed.

Equations 4 and 5 are based on the assumption that all n fixtures will be in continuous

use over the entire time r at the average rate

of once in T seconds. The time r may be

reduced to days or years on the basis of the

daily period of peak use, by assuming or determining this period and computing on the

basis of a day of that length.

IX. PROPOSED USES OF THE PROBABILITY FUNCTION FOR MAKING

LOAD ESTIMATES

1. Pertinent Information Obtainable From

THE Probability Function

It may be helpful in judging the reasonableness of the proposed application of the probability function to summarize the information

obtainable from the probability function before

proceeding with the selection of the time

factors applicable to practical cases. The following pertinent information can be

obtained from the equations developed:

(a) The probability that a given number of

fixtures, m, out of a total of n, will be operating

at an arbitrary instant of observation, determined by eq 2. (b) The fraction of the time that m and only

m fixtures will be operating at the same instant, a second interpretation of eq 2.

(c) The fraction of the total time that any

number of fixtures greater than the design

number, m, will be operating at the same

instant, determined by eq 5. (d) The ratio of any two successive terms in

the series of eq 2; for example, the ratio of the

fraction of the time r-|-l to the fraction of the

time r fixtures will be operating at the same

instant, determined by

TJ \ T

(6)

There are certain characteristics of the

probability series given in eq 2 which will be of

material aid in determining the conditions on

which the design factor m is based. In the

series, eq 2, the value of successive terms in- creases as /' increases up to the most probable

272131°—40 2 [7]
value or values and then decreases. Also, the

ratio of any term to the preceding term of the

series decreases continuously as r increases

from r=l to r—n . These characteristics make

it possible to determine the value m that will not be exceeded more than the fraction l/r of the time without completing the entire summation from the w + 1 to nth term.

If pm^'i-lT, and if the ratio (eq 6) n — m t ^1

r = n the summation y^,p" must be less tlian 1/t, r=m+l

because the finite series

which results from the series pz+i+pm+2+ +p:,

by multiplying and dividing this last series by

Pm, substituting the value 1/2 for the ratio

Pm+i/Pm, and replacing the factor p^ by l/r, is less, term by term, than the infinite series e •

• ]

which is known to approach the value l/r as the number of terms increases without limit. When the series satisfies this condition, it can

be positively stated, without summation, that

a design factor r—m will be exceeded less than

l/r of the time.

ft — I' t If —r—rXTfr—;~>l/2, sunnnatioii is necessarv r+ 1 i — t

in order to determine a value of r=m that will not be exceeded more than l/r of the time, but

it is unnecessary to carry the summation beyond

n —T t the term where —r^'Kyf—j<l/2.

r + 1 I —t

2. Values or ^ T, r, and q In applying the probability fimction for estimating the design load mq, it is necessary to

select values of t, T, and r from which to compute the value of m and to select a value of q, the factors excepting r pertaining to a particular

kind of fixture and service. The actual values

selected in any case are largely a matter of engineering judgment. In this connection, it

is to be understood that in the following

development and illustrative examples, the

values selected represent the author's judgment

in regard to the appropriate values for producing satisfactory service and are based on

the author's interpretation of the information

available.

For the purposes of this discussion satisfactory

service is defined in a relative sense as that in

which interruption in service because of controllable factors, such as the sizes and arrangement of pipes, is infrequent and is of sufficiently

short duration to cause no inconvenience in

the use of fixtures or any unsanitary condition

in the plumbing system. Attainment of satis- factory service will depend on the selection of the design factor m, or more specifically on the

value of r from which the value of m is computed. The value of r selected for illustrating

the proposed application is 100 seconds, which

provides for wholly satisfactory service 99 per- cent of the time and for reasonably satisfactory

service all of the time if the design load mq gpm

is not greatly exceeded. In this connection,

it will be observed that if m is exceeded in actual service it is most likely to be exceeded

by one fixture only and is progressively less likely to be exceeded by two, by three, or more. Obviously t and q bear a direct relation to Q

in respect to the values necessary to provide

satisfactory service if m fixtures are in operation simultaneously. Since there is a considerable range in the values of t and Q on which

the value of q depends for any particular fixture,

it will be very helpful to the engineer in determining reasonable values to be used for a

particular kind of fixture to consider the characteristics of operation of that kind of fixture.

It is a characteristic of water closets that

they will operate more or less effectively under

any average rate of supply from about 15 gpni

up to rates of about 30 gpm or more delivered

in any time ranging from about 6 seconds up. For each type and design of water-closet bowl

there is an intermediate smaller range of aver- age rate of supply within which there is no

detectable dift'erence in the effectiveness of the flush in emptying the bowl of its contents. There is likewise a range in time of flow within

[8]
which the bowl will he effectively emptied of

its contents with an average rate of supply

anywhere within the smaller range referred to. From the evidence of experiments reported

in the earlier paper [1], the Subcommittee on Plumbing of the United States Department of Commerce Building Code Committee agreed

upon an average rate of 30 gpm for 10 seconds

as a reasonable and safe basis for estimating

loads to be expected in building drainage

systems. The experiments referred to were

designed to obtain the maximum loads per

closet that could be delivered to the drains

within the operational range of water closets,

and no attempt was made to determine either

the most effective rate of supply for a particular

type of bowl or an average rate that would produce a satisfactory flush in all types of closet bowls. It is to be expected, as has been the

case, that overestimates would result when the

maximum values of all load factors involved

are employed in estimating. More recent ex- periments [3] by Camp give rates of supply for

safe and economical flushes ranging from 20

gpm to 29 gpm for different types of closet bowls and. times of flush ranging from 7.5 to 9 seconds, omitting data for one bowl with spiral

flow action. The averages for six difl'erent bowls are 25.9 gpm and 8.2 seconds. Unpublished data from still more recent experiments

[4] at the National Bureau of Standards indi- cate that the most effective removal of the contents of a closet bowl occurs with rates of supply ranging from about 20 to 24 gpm and

that the complete contents of the bowl are removed, if removable by any rate of flow, in times ranging from about 6 to about 10 seconds

for different types of bowls.

In both cases the data were obtained with

approximately uniform flow through the bowl,

and that average rate of flow which maintained

continuous siphon action through the bowl was

taken as the criterion for the most efl'ective flushing rate. Considering the wide ranges in rate and time of supply within which a closet bowl will operate effectively, Camp's and

Golden 's data may be regarded as in substantial agreement. However, none of the

experiments cited closely simulates service

conditions, for the rate of supply in actual

service is not luiiform, whether supplied by

flush valve oi' flush tank: it I'iscs to ii mnxininiu

at or near the beginning iuid griiduiilly decreases as the valve closes or as the tank

empties. The characteristics of an effective flush may be summarized as follows:

A quick priming of the siphon; a continuous

siphon action for sufficient time to clear the

bowl of its contents and carry them through

the trapway; and a breaking of the siphon

action before the flow ceases, in order to refill the trap. Wliether by design or chance, the

characteristic operation of conventional supply

devices (flush valves and flush tanks), if pro- perly adjusted for volume and time, controls

the rate of supply in a manner that meets

these flushing requirements admirably. In

figure 1, curve 1 is the reproduction from

Camp's report of a typical rate curve for a

flush valve, representing a flow of 3.8 gallons

in 8.5 seconds giving a peak rate at the end of two seconds of approximately 37 gpm and an average rate of 26.9 gpm over the time the

valve is in operation. Curve 2 is the rate curve for a flush tank plotted from Golden 's data for a flow of 4 gallons in 9.1 seconds giving

a peak rate about 31.5 gpm at the end of 1 second and an average rate of 26.4 gpm. In

both cases the flow reaches a rate within the

first second amply sufficient to prime the siphon

of any standaxd type of closet bowl and thereafter is maintained for 6 or 7 seconds within

the range required to produce continuous

40

<

/ /

/ /

-Tf1 / +7—

1/

li

1. FLUSH VALVE

2. FLUSH TANK

\

i;

1

I i

I A—

\ \ \

0 2 4 6 8 TIME -SECONDS

Figure 1.

Typical time-supply m,rves for water-closet bonds.

\9]
siphon action. The total volume, the average

rate of flow, and the duration of the flush in

these examples are all greater than necessary

for an effective flush in the types of bowls most

commonly used—washdown, reverse trap, and

siphon-jet bowls. It is impractical to attempt

to estimate individual variations either in rates of supply or in rates of use of fixtures. Hence, it is advisable to set the factors chosen

for making estimates high enough to allow some leeway in adjusting down to the volume

and rate of supply needed for satisfactory

service in particular cases, since it is impossible

to adjust for greater rates of supply than the

supply pipes are capable of delivering.

Considering the problem of estimates from

all angles, an average rate of supply of 27 gpm

for a duration of 9 seconds, giving a volume of approximately 4 gallons per flush, appears to meet requirements generally as nearly as can be done in round numbers and will be employed

for flush valves for water closets in evaluating

the probability function in the further development of the methods of estimating demand

loads. The rate of supply to flush tanks for water

closets is not related directly to the rate of

supply required by the closet for effective

operation. The only essential for operation

of the fixture is that the tank refill in the interim

between operations. A rate of 4 gpm is ordinarily considered sufficient, which for a volume of 4 gallons gives a value of 60 seconds

for f. The value of T should obviously be the same for flush-valve and flush-tank supply.

In case of faucet-controlled supply for fix- tures, it is not possible to base the time factors or the quantity of water used on the characteristic operation of the fixture as was done for flush valves for water closets, because the manner in which faucets are operated depends

largely on personal habits or preferences.

For such fixtures an arbitrary selection of values, made from a consideration of relative rates of supply and volumes used, appears to be the only recourse. An average rate of supply of 8 gpm would per- mit the drawing of 8 gallons in 1 minute (60

seconds), 16 gallons in 2 minutes, 24 gallons in

3 minutes, etc. Bathtubs, depending on size and style, hold from 25 to 40 gallons when filled

to the overflow. Ordinarily only a fraction,

possible % to H, of these volumes will be used for a bath. The time, T, between uses will include

the time required to draw the water, the time

taken for bathing, the time required to empty

the tub, and any additional time consumed in

the complete bathing operation. This total time, T, between operations of the fixture seems

likely to range from about 15 minutes for hurried baths to about 30 minutes for baths taken more leisurely. It also seems likely that

smaller quantities of water will be used under

the former than under the latter conditions.

Now, assume that 8 gpm is an ample average

rate of supply for a bathtub. This is 60 percent

higher than some [5] and 20 percent lower than

other [6] estimates of a satisfactory average rate

of supply for a bathtub. An average rate of

supply of 8 gpm, an assumed average time of

filling of t=60 seconds, and an assumed time

between operations of T=900 seconds (15

minutes) would provide for an average of 8

gallons per bath. For an average volume of 16 gallons per bath and the same rate of supply, 8 gpm, 120 seconds (t) will be required to draw

the bath. If the rate of operation (average

time taken per bath) is once in 30 minutes, giving 7"= 1,800 seconds, the ratio of t/T is the same in both cases; 60/900=120/1,800=1/15.

Since for a given value of n the value of p" for any value of r is determined by the ratio t/ T, the

probability of a selected design load mq being

exceeded will be exactly the same for the two

cases cited or for any other case in which the

time T is proportional to the volume used and

the same basic rate q is employed. From these

considerations, a design load mq, for bathtubs

in congested service, based on an average rate

of supply of 8 gpm, and a ratio tfT=l/15 ap- pears to insure fairly satisfactory service and is used as the basis of the development and comparisons following.

The values selected for the three fixtures dis- cussed are given in table 1.

Table 1. —Values of t/T, g, and Q

Kind of fixture tIT q in gpm Q in gallons

Flush valves for water closets ... 9/300 60/300

1/15

27

48

4

4

8 to 16 Flush tanks tor water closets Bathtubs

[10]
3. Presentation of Probability Data

Equations 4 and 5 are not satisfied in general

by integral values of r and n and a given value

of 1/t; that is, for occurrence exactly a given

percentage of the time. However, by trial and

error, a value of r for each given value of n

can be computed for which the probability of occurrence is in excess of a given percentage,

for example 1 percent of the time, by the smallest possible fraction. From this probability, a particular value m of r, for which the

probability of occurrence of any number

greater than m is not greater than a given

percentage of the time, may be readily determined by an approximation of 2^?". r=m+

If ip1+x <{l/2)p^, no further computation is necessary as pointed out on page 8. If '^(l/2)p^, it will ordinarily be necessary

to carry the summation to a few terms only to determine the value of m required.

Probability data for flush valves for water

closets, flush tanks for water closets, and bath

tubs are given in table 2, in which r is tlie greatest number of fixtures out of n tliat will be in operation 1 percent of the time and m

is the number of fixtures that will not be ex- ceeded more than 1 percent of the time.

Table 2.

Probability data for three types of fixtures

Flush valves, for f/!r=9/300=3/100 Flush tanks, for t/T=60lm=H5

n r m n r P" m n r P" m

6 2 0. 0119 2 3 2 0. 0960 2 3 2 0. 0124 2

16 3 .0102 3 6 3 .0512 3 8 3 .0118 3 30 4 .0101 4 7 4 .0281 4 15 4 .0126 4 47 5 .0104 5 9 5 .0165 5 22 5 .0107 5 66 6 .0106 6 15 7 . 0138 7 31 6 .0118 6 85 7 .0100 7 25 10 .0118 10 40 7 .0112 7 107 8 .0105 8 40 14 .0115 14 59 9 . 0104 9

129 9 .0104 9 64 20 .0112 20 113 14 .0102 14

151 10 .0101 10 132 36 .0104 37 172 19 .0103 19 199 12 .0102 12 240 60 .0101 67 221 23 .0102 23

299 16 .0101 16 305 74 .0102 78 309 30 .0100 31

Bathtubs, for ^/r=i/i5

It will be observed that for flush valves tjT

is small, and r=m for all values of n in the

range given, and that the greater tjT the smaller is the range in which r= m. Figure 2 shows the curves obtained by plotting m and n for the three fixtures, curve 1 being

100

75

E 50

I. FLUSH VALVES, /'r=O.OI

2. BATH TUBS, 1/1'= 0.01

3. FLUSH TANKS, /'l' = 0.01

4. FLUSH VALVES, l/r = 0.000001

300

Figure 2.

Probable relation of m. to n from table 1.

fill
500

400

300

a.

200

100

I. FLUSH VALVES

2. BATH TUBS

3. FLUSH TANKS

4. AVERAGE FLOW, FLUSH VALVES

AND FLUSH TANKS

5. AVERAGE FLOW, BATH TUBS

300

Figure 3.Prohahle flow in relation to n.

for flush valves, curve 2 for bathtubs, and

curve 3 for flush tanks. A value of m for flush

valves that would not be exceeded for more

than one-raillionth of the time was also computed, and the results are represented in figure

2 by curve 4. Estimates of pipe capacities will ordinarily

be obtained in flow units, for example in gallons

per minute. Hence it will be more convenient

in practice to have the demand load expressed

in the same imits. This may be obtained by

plotting mc[ and n as shown by figure 3, in which

g is taken as 27 gpm for flush valves, 4 gpm for

flush tanks, and 8 gpm for bathtubs. The

average flow, nQjT, during the time the fixtures are being used at the rate of once in the time T

is also represented in figure 3, for flush valves

and flush tanks by line 4 and for bathtubs by

line 5. Cm-ves 1,2, and 3, in figiu'e 3, show the relation of demand loads to number of fixtures,

based on estimated time factors representing

congested conditions of service—that is, the

maximum practical rate at which fixtures can

be used continuously in actual service. Assuming the correctness of the factors employed in evaluating the probability functions, the curves may be used for estimating the demand loads

for any particular number of fixtures of one

given kind. However, the design load for all kinds of fixtures installed in one system should

not be the sum of the design loads computed

separately for each kind of fixture, even though

the individual curves may be correct. Simultaneous operation of difl^erent kinds of fixtures

is a chance occurrence which would have to be

evaluated by another probability function.

Although such an evaluation is possible on the

basis of the assumed average time factors, the

process is too complicated for convenient appli12]

I
cation in the field or office and, because the

time factors for faucet-supplied fixtures on

which the probability of simultaneous operation

depends cannot be determined in general,

would in the end lead to doubtful results. For

these reasons it is believed that reasonably

satisfactory results, which are much simpler to apply, may be obtained by weighting each

kind of fixture and applying the weighted sum

of the total numi)ers of fixtures of all kinds to a load curve for flush valves or a load curve for

flush tanks, according to wliich type of supply

is to be used.

4. Derivation of Fixture Weights (Fixture

Units)

In Recommended Minimum Requirements

for Plumbing [1], fixtures were weighted in the

scale from 1 to 6, this scale being selected

largely because, of the fixtures installed in the

greatest numbers, the smallest load (lavatory) was estimated to be about one-sixth of the

largest load (water closet). The scale chosen

is purely arbitrary, and it is now suggested

that a decimal scale, 1 to 10, would give a much

more flexible system of estimating.

A comparison of the relative loading eft'ects of the three kinds of fixtures may be made by

means of the curves in figure 3. Obviously

there is no common exact ratio of the loading

effects of any two different kinds of fixtures,

each kind using water at dift'ei'ent volumes, Q,

volume rates, g, and different time intervals, T, over any considerable range in number of

fixtures, n. For example, for a demand of 50 gpm, the number of fixtures from which this load would be occurring not more than 1 percent

of the time is approximately (fig. 3) as 5:34:33,

respectively, for flush valves for water closets, flush tanks for water closets, and bathtub

supply (total). The load-producing weights

are inversely as the number of fixtures pro- ducing a given load; hence, on the basis of

10 for flush valves, the weights for a demand

of 50 gpm are 10:1.47:1.52. Table 3 gives

the relative weights computed in this manner

from a comparison of the three curves for demand loads at increments of 50 gpm from

150 to 300 gpm, the range in which the pro- posed method of estimates will have its greatest application and usefulness.

Table 3.

Relative demand weigldis of fixtures

Demand, gpm

ISO 200

250

300

Flush valves

Number

of fixtures

5693

133 m

Average weight .

Weight

Fhish tanks

Number

of fixtures

133

185 238

292

Relative weight

4.20

5. 03 5.59 6.03

5. 21

Bathtubs

Number

of fixtures

1B7 238 312

398

Relativf weight

3. 35

3. 90

4. 30 4.42

3.99

Neither the possible accuracy in estimating

the demand to be expected in a plumbing sys- tem nor in estimating the capacity of pii)es

selected for supplying the estimated demand

justifies the assignment of weights in the scale

1 to 10 closer than to the nearest integer.

Accordingly, on the basis of average values, the

relative weights of the three fixtures become 10

for flush valves for water closets, 4 for bathtubs

(total hot- and cold-water supply), and 5 for flush tanks.

It will be observed that these fixture weights

have the same significance as the fixture unit

ratings assigned to the different kinds of fix- tures in Recommended Minimum Requirements for Plumbing [1]. This unit in the weight

scale is not a definite unit of flow but is simply

a load factor to be applied through a relation

between m and n such as developed in figures 2 and 3. However, since the term fixture unit

in this significance has become fairly well es- tablished by usage during the past 15 years, the

tei'm will be retained here to designate load

weights of fixtures. A further comparison and an illustration of the results of applying the average weights of

difl'erent kinds of fixtures to a single basic

probability curve may be made by reducing the

number of fixtures to fixture units (n times

weight) and replotting, as in figure 4. It will be observed that the curves for flush tanks and

bathtubs (total supply) lie very close together,

indicating that the relative weights, 5 to 4, are approximately correct throughout the range

illustrated for the individual volumes and rates allowed. It will also be observed that the two

curves both intersect the flush-valve cm've between 193 gpm 880 fixture units and 210 gpm

1,040 fixture units, indicating again that the

relative weights 10:5:4 are approximately cor- rect in that range. For 300 flush tanks, 1,500

[13]
500

400

300

a.

200

100

I. FLUSH VALVES

2. FLUSH TANKS

3. BATH TUBS

1000 1500 2000

FIXTURE UNITS

Figure 4.

Relation of demand to fixture units.

2500 3000

fixture units, referred separately to the flush- valve curve for an estimate, the demand esti- mate would be about f 5 percent lower than the

corresponding estimate made directly from the

flush-tank curve. The corresponding- error in the estinuite for 300 l)athtubs made in the same manner would be about 3)^ percent. These errors are immateivial. for the only result, in case the

design load was exceeded in service by that

amoiuit, would be an increase in the time re- quired to refill the fixtures by 15 percent and

3^2 percent, respectively, or, in case the same

time is occupied in refilling, a reduction by these

percentages in the volume of water used. Below the point of intersection, referring flush tanks and bathtubs separately to the flush- valve ciu-ve would result in overestimating the

demand by amounts varying from very small

percentages for 880 to 1,040 fixture units to

about 94 and 23 percent, respectively, for 100

fixture units. However, the error in an esti- mate made from curve 1 for the total demand

load for flush valves for water closets and for bathtubs will be less than the error indicated

by an estimate made separately for the bathtubs from the same curve in all cases in which

the flush valves predominate, on the basis of

total fixture units of the two kinds of fixtures. In cases where flush tanks for water closets are used exclusively or predominaritly in the sys- tem, a closer estimate could be made on the

basis of total weights, by using curve 2 and the

total fixture units for all kinds of fixtures in- volved. Obviously the error made by using curve 2 for both flush tanks and bathtubs for any number of either up to 300 would be small. Also, the demand load relative to the number

of fixture units may be approximately repre-

[14]
500

400

sented in this range by a smooth curve drawn

above the two probabihty curves and merged

with cm*ve 1 as shown by the broken hne in

figure 4, thus giving estimates shghtly in excess

of the peak demands indicated by the separate

curves for flush tanks and bathtubs. The broken hne in figure 4 is reproduced in figure 5, to- gether with curve 1. The ciu-ves in this figure

are proposed for estimating design loads for water-supply lines in general, curve 1 to be

used when flush valves predominate in the sys- tem and curve 2 to be used when flush tanks

predominate, the common curve above the

branch to be used for all weighted fixtm*es.

Of the fixtures commonly installed in the

greatest numbers, these three produce the major

part of the peak demand. Lavatories are in- stalled in most systems in as great numbers, and

frequently in greater numbers, but obviously

have a much smaller load-producing weight, and

are frequently ignored in estimating demand

and sewage loads. Because, as pointed out in discussing supply demands for bathtubs, it is impossible to estimate the values of t and T

reliably for faucet-supplied fixtures, and because

of the relatively small effect on the total

demand, it is suggested that satisfactory

fixture-unit ratings may be assigned to irregularly used faucet-supplied fixtures independently of the probability function from a consideration of the sizes of the supply outlets

and the relative quantities of water used. On

the basis of tliis reasoning, the relative weights

suggested for the kinds of fixtures most com- monly subjected to congested service are: 10:5:4:2 for flush valves for water closets.

[15]
flush tanks for -svater closets, bath tubs, and

la V a tories . respeo t i vely

.

5. Relative Load Weights foe Different

CoxDiTioNS OF Service

I'p to this point the discussion has been confined to four kuuls of tixtin-es under congested

conditions of service. There are several considerations that slunild be taken into account in determinmg load ratings for other fixtures and

for other conditions of serA'ice, amoui:- which

the following are especially important

{!) Fixtures that are relatively few in number

and unlikely to be used when the predominant

kinds are being used most frequently will add

very little to the demand or to the peak sewage

load and hence may be ignored, except in regard

to the branch supply lines and branch drain

pipes of these fixtures. Slop or service sinks in

ofhce building's, which are in use to any considerable extent only before or after office hours,

add a negligible load to the peak loads of the

day. Kitchen sinks and laundry trays in dwelling's and residential apartments may also be placed in this category.

{2) Fixtures so installed that they cannot in

general be subject to congested conditions of service in the same sense as fixtures installed in

public comfort stations, general toilet rooms in

office building's, and other buildings in which

each fixture is open and accessible for use at all times, should be given a rating in accordance

with the possible extent or frequency of their

use. Bathi'ooms in private dwellings, resi- dential apartments, and private bathrooms in

hotels may be considered m this class, and can

be rated advantageously as a group.

(3) Water services that demand a continuous

flow, such as la'Rm sprmklei'S. air-conditioning

equipment, gang showers m factories, and

athletic dressing rooms, present no element of

chance in regard to overlapping and are not

susceptible of a logical weighting in relation to water closets and other fixtures that use water

at high rates for comparatively short periods of

time. Hence the demand for this type of

supply should be considered separately and esti- mated separately. If the use of these continu- ous water supplies is such that they overlap the

rush period of the day for the weighted fixtm*es

in the system, the separate estimates for the

two classes of supply should be added to obtain

the estimate for total demand on any supply ; pipe common to both services. If the two types ; of demand do not come at the same time of day,

^ the greater demand of the two may be taken as : the peak demand, or design estimate.

' A bathroom group of fixtures can probably be ! more closely weighted to the selected scale of 1 to 10 by taking the three common bathroom

fixtures—water closet, lavatory, and bathtub or shower—as a unit thait by weighting each ' fixture separately. In hotels and apartment

houses the bathrooms are not all in use at one I time, and the loading weight of any fixture of the group—for example, a water closet—will , be much lower than for the same fixtiu'c in | congested service, for which it has been assumed ; that all fixtures will be in continuous use during ' the period of congestion. One peak period

occiu's m hotels and apartments in the morning ' durmg the rismg hom- and another in the after- noon durmg the preparations for dinner.

Probably the best mdication of the distribution ) of the use of the bathrooms in hotels and apartment buildings obtainable will be given by i records of the acti-s-ity dtu'ing the hours when

the peak use of the plumbing fixttu'es occui-s,

Table 4 is a stunmary of recorded morning i (rising) calls from the switchboard for two large

hotels in New York City for 7 consecutive days

m one case and 9 days in the other. Hotel 1 is t a large hotel in which the guests were prmcipally

transient. Hotel 2 is a large apartment hotel i in which the greater part of the guests were ' resident but which also received and accom- ' modated transients.

Table 4.—Distribution of morning calls in hotels

\

Time of call

Place Before

6:S0

6:31) 7:00 7:15 7:30 7:45 S:0l) After

'«:00

Total

calls

Hotel 1 Hotel 2

Total calls

174

19S

372

22S

212 9S

133 4S6ci99 146

176

700

7S0 1S5

176 463 442

382

341 2.S62 3.057

440 231 1,085 322 1.4ai 361 905 r23 .T,919

It will be observed that the time of rising I in the two hotels is distributed m approximately < the same manner, the greatest number of calls i in any 15-minute period being for that follow-- ; ing 7:30 a. m., and that slightly more than t

[16]
one-half of all calls fall within the hour from

7:00 to 8:00 a. m. This is significant, and

it may be assumed that the time of rising of guests leaving no morning call will be similarlydistributed in time.

Table 5.

Distrihidion of morning calls in apartments

for 9 days

Time of call

Period Total Before 6:30 6:45 7:00 7:15 7:30 7:45 8:00 After

8:00

calls

6:30

1st day.__ ... 3 2 7 11 3 6 3 1 1 37 2d day 8 4 8 12 2 9 3 1 2 49 3d day__ 6 2 9 12 3 9 3 2 2 48

4th day 4 3 9 12 3 8 3 1 1 44 5th day 3 2 9 12 2 8 3 1 1 41

lith day 5 4 7 11 2 8 2 1 1 41

7th day __- 8 4 7 13 3 11 4 2 1 53 8th day 4 4 9 11 4 11 2 2 3 50 9th day 4 3 10 12 4 10 3 1 1 48

Total calls 45 28 75 106 26 80 26 12 13 411

Table 6.

Distribution of morning exits in apartments

for 2 days

Period

Interval of time

Total

6:45 exits to 7:007:00

to 7:15

7:15

to 7:30

7:30

to 7:45

7:45

to 8:00

8:00

to 8:15

8:15

to 8:30

8:30

to 8:45

8:45

to 9:00

1st day

2d day

Total exits

11

6

17

4

8

12

18

15

31274749

96

70

59

41

54

31

23 26

21 279262

33 58 129 95 54 47 541

Table 5 gives the summarized record of morning calls for a large apartment house in Washington, D. C, for 9 different days other

than Sundays during a period of 22 days. Table

6 gives the summarized record of morning

exits from the same apartment house. Comparing tables 4 and 5, it will be observed that

the number of morning calls in 15-minute

periods in hotels and apartments fluctuates in

about the same manner, the principal difference

being that the peak number of calls in any 15- minute period is about 10 to 15 percent of the

total number higher for the apartment house

than for either the transient or apartment

hotel. Also comparing tables 5 and 6, it will be observed that the peak of exits from the

apartment house, indicating the beginning of the business or social activity of the day, occurs approximately 1 hour later than the peak of the rising activity. This indicates that extensive use of phMnl)ing fixtures in any single

apartment or in any gi'oup of apartments in which the occupants rise at the same hour will

last for about an hour. The maximum number

of calls within any single hour ranges from 69

to 75 percent for different days. The corre- sponding percentages for the hotels range from

47 to 62. It therefore seems reasonable to assume that not more than 75 percent of the

bathrooms in an apartment house or in any

hotel will be occupied and in use at one time.

There may be exceptional apartment houses in

which the occupants are all engaged in the same occupation that would show a higher concentration in rising time. However, it will be

the exception rather than the rule and

should be treated as an exception in estimating

water-supply demands and sewage loads.

There is another consideration that is important in estimates of loads for bathrooms. In a

leisurely use of a bathroom, the fixtures will ordinarily be used one at a time. In cases of

hurry or congestion, two fixtures may be in use at the same time in a bathroom, in the sense that water is being drawn for or is being dis- charged from both at the same time. Hence

it seems reasonable to assume that, on the

average, not more than two-thirds of the total number of fixtures in all occupied bathrooms

will be in use at the same time. The total number in use at one time on this basis, will not

exceed two-thirds of 75 percent, or 50 percent,

of the total number of fixtures in bathrooms in

the building. Accordingly, it is suggested that a bathroom group in a residence or apartment,

or a private bathroom in a hotel be given onehalf the total weight for the same fixtures in

public or congested service in estimating supply

demands.

The term "bathroom group" is here defined

as consisting of a water closet, a lavatory, and

a bathtub with or without a shower head, or of a water closet, a lavatory, and a shower stall. These suggested weights become (10+4 +2)h2 =8. and (5-f 4 +2)^2 = 5.5, respectively, for installations using flush valves and flush tanks.

Again, in accordance with the convention pre- viously used in selecting fixture weights, the

nearest higher integral value will be assigned,

giving for a bathroom group using a flush tank

the weight of 6.

[17]
In suggesting- this rating for a bathroom

group, it is recognized that the condition of service for which the rating is made does not

completely block out of use one-half of each

kind of fixture, as would be required for the

result obtained to be exact. What actually occurs in the installations to which the proposed

rating would apply is that one-fourth or more

of the fixtures of each kind are not in service

during the time the other three-fourths or less are in actual use, and that the ratio of tjT for each kind of fixture is greatly decreased below

the ratio applying to congested conditions of

service. For example, if each of the watercloset flush valves is operated twice dming the

hour in which it was assumed that two-thirds

of the bathrooms were in use, ^/J'=9/1800 =

1/200; if used four times, f/r=9/900= 1/100, as compared to ^/T=9/300 =3/100 for congested

service. A similar relation will exist between

the ratios of tjT for flush tanks and bathtubs

under the conditions described and under

congested conditions. It may be noted in this connection that assigning a bathroom group

one-half the rating that is given the same fixtures

in congested service would give an estimate of

the demand greater than one-half of that for congested service. For example, 10 flush valves for water closets, 10 bathtubs, and 10

lavatories in congested service total 160 fixture

units, which by curve 1 in figure 5 gives an

estimated demand of 81 gpm. The proposed

rating for 10 bathroom groups with flush valves,

80 fixture units, gives an estimated demand of 64 gpm.

Table 7 gives the fixture weights suggested in

accordance with the use to which the fixtures are subjected and the manner in which they

are installed. The term "public" refers to fix- tures which are individually open for use at all times when the building is open, as in public

toilets or general toilets in office buildings.

"Private" refers to fixtures installed in groups

in such a manner that the entire group may be

and generally is confined to the use of one per- son at a time, as in residences or private baths

of hotels. "Total" refers to hot and cold

supply combined. "Hot or cold" refers to hot

or cold water supply only.

Table 7.Demand weights of plumbing fixtures

Fixture or group

Water closet Do

Pedestal urinal

Stall or wall urinal _ Do

Lavatory Do

Bathtubs Do

Shower head Do

Bathroom ?roup_ Do.-- Do

Do

Do- - Bathroom group with

separate shower

.

Occu- pancy

Public-

..-do. .-do .- _-do--.

.-do--,

.-do...

_-do---

. do..- -do... .do--.

_-do..-

Private ...do... .--do...

..-do .. ---do...

...do...

Type of supply

Flush valve Flush tank

Flush valve do

Flush tank

Total Hot or cold Total Hot or cold Total Hot or cold Flush valve (total) Flush valve (cold only) Flush tank (total) B'lush tank (cold only)

.

Hot water only - . Add to corresponding group above tor total,

2; for cold or hot

Weight

per fix- ture or group in

fixture units

10

5 10

5

321.5

43

43

31.5

6. Application of Load Chart and Weight

Table

In estimating the demand load for a supply

pipe in any building, the total number of each

kind of weighted fixtures or weighted groups of fixtures will be multiplied by the weight of that

fixture or group (table 7) and the products added

to obtain the total number of fixture units.

The demand load is given by the corresponding

ordinate of the appropriate curve in figure 5. For example, assume that an apartment house

or a hotel has 100 bathrooms with flush-tank

supply for all water closets and that any other

flxtures in the building are negligible * in rela- tion to peak demand. The total number of fixture units wiU be 100X6, or 600. The ordinate on curve 2 of figure 5 corresponding to this abscissa is 147 gpm. This estimate, of course, applies only to the main supply or service pipe

for the building and does not include any con- tinuous demand, such as that for lawn sprinklers or air conditioning, etc. Again, assume that an office building has 100 water closets with

flush-valve supply, 25 stall urinals with flush- valve supply, and 100 lavatories, and that the

demand of other plumbing flxtures is negligible

in relation to these, but that the building has an air-cooling system which demands a maximum rate of 225 gpm. By referring to table

' See discussion of negligible flxtures on page 16.

[18]
5 it is found that the total number of fixture

units is 100X10+25X5+100X2, or 1,325.

The corresponding estimated demand load is 247

gpm, to which 225 gpm must be added, giving

472 gpm for the total estimated demand on the

sei'vice pipe of the building. The method of estimating for branch risers and other dis- tributing pipes is similar, the estimate being

based on the number of fixtures or groups

supplied through the branch, and the weights

being selected according to whether the branch

supply pipe carried the total supply, the cold

water only, or the hot water only.

X. ADEQUACY OF THE METHOD

The proof of the adequacy of the proposed

method of estimating the demand loads to be

expected in building-water-supply systems will, in the end, depend on its success in actual trial over a period of years. Fortunately, in this case we have a means of comparison with a

similar method that has been given a trial. Table 8 is an abridgement of a table from the

construction manual [5] used by the mechanical

engineering section of one of the Federal departments, based on the method of estimate by

means of the probability function as originally

proposed [1] in 1923. This table has been used

for a number of years by that office with reason- ably successful results. The only complaint

against the results of using this table and method

that has come to the author's attention is that

it tends to give larger estimates than have

been found necessary for satisfactory service.

The principal reason for this tendency to oversize supply pipes does not lie in any inherent

fault in the probability function, but in the

application of the method by means of a table

which does not provide for the probability,

or rather the improbability, of overlapping between or among two or more groups of different

kinds.

Table 8.

Demand estimates in gallons per minute

from federal table

Number of fixtures Kind of fixture 2 4 8 12 16 24 40 50 80 100 150 200 300

Water-closet flush tanks 8 12 16 20 30 38 50 58 80 95 140 21."; 290 Water-closet flnsh valves 36 46 ni 72 83 100 128 143 180 200 2.50 295 375 Urinal flush valves . 18 23 30 3R 42 50 64 72 W 100 125 148 188 Lavatories . _ _ 6 10 1,5 IS 20 25 30 36 51 60 75 85 115

The inherent fault of applying the probability function by means of a table is illus- trated in the comparisons which follow. Table

9 gives estimated demand loads for flush valves

for water closets, flush valves for urinals, flush tanks, and lavatory faucets, taken from the curves of figure 4 over the range in number of fixtures included in table 8. The values for flush valves for urinals were taken from the curve for flush valves for water closets on the

basis of five fixture units per fixture, and the

values for lavatories from the curve for bathtubs on the basis of two fixture units per fixture

(table 7). Table 9.

Demand estimates in gallons per minute from

figure 4 Number of fixtures Kmd of fixture

2 4 8 12 16 24 40 50 80 100 150 200 300

Water-closet flush tanks 8 12 18 24 30 39 56 66 96 117 167 216 310 Water-closet flush valves 36 47 64 70 82 98 125 140 182 205 269 327 435 Urinal flush valves _ . 27 36 47 54 61 71 89 98 125 140 175 205 269 Lavatories _ 8 12 16 20 23 29 39 43 58 65 84 102 1.37

There are obvious irregularities in the esti- mates given by table 8, especially in the esti- mates for flush tanks and lavatory faucets.

This may be seen by plotting these values or by comparing them by steps as is done in

table 10. Table 10.

Demand estimates in gallons per minute per

fixture by table 8

Number of fixtures Kind of fixture First 50 Second

50 Third 50 Fourth

50 Last 100

Flush tanks 1. 16

0. 72

0. 74 .480. 90

.30 1.50 0.20

0. 75 Lavatories - .60

This makes the irregularities in table 7 obvious. To be consistent, the increment in the

estimate for a given increment in number of

fixtures should gradually approach a constant

minimum as the total number increases.

Table 11 shows the regular decrease in demand per fixture as given in table 9. Table 11.

Demand estimates in gallons per minute per

fixture by table 9

Kind of fixture Number of fixture?

first 50 second 50 third 50 fourth 50 last 100

Flush tanks 1.32 0.86 L02

0.54

1.00 0.380.98

.36

0. 94

I^avatories.-- ... . .35

[19]
The regularly decreasing allowances per

fixture in this table are proper and permissible.

However, although the demand per fixture decreases with increasing number of fixtures,

tending toward the average demands for entu-e systems, the estimate per fixture for satis- factory service of the entire group of fixtures

cannot fall to that average, approximately 0.8 gpm per water closet and 0.16 gpm per lavatory,

if the supply pipes are to be capable of supplying

the peak demands.

There are two particular requirements that

any practical method of estimating demand

loads should meet. First, the method should

give estimates greater than the average demand

for all fixtures in the system during the periods

of heaviest use; otherwise complete interruption of service will occur. Second, the method

should give satisfactory and economical esti- mates for groups of fixtures all of the same kind

and for groups composed of any number of different kinds of fixtures. The inherent defects of a table in the form of tables 8 or 9 in meeting the second requirement will become

apparent from the comparisons given in table 12.

Table 12.

Comparison of demand estimates for plumbing systems

Number and kind of fixtures

Estimates by

Table 7 and

figure 5 Table9 Table

8

fixture units gpm gpm gpm

12 flush valves (W. C.) » 12040

24

70

4720

72

30

18 8 flush valves (XIr.) ^ 12 lavatories

Total 184 86- 137 120

50 flush valves (W. C.) 500

120 100

140

7143

143

5036

24 flush valves (Ur.)

50 lavatories .. _

Total 720 171 264 229

200 flush valves (W. C.)

100 flush valves (Ur.)

2. 000

500

400

327

140 102

295

10085

Total 2,900 432 569 480

" Water closet,

k Urinal. A comparison of these estimates shows clearly

the tendency to overestimate when the separate

estimates for the different kinds of fixtures are added directly to obtain the estimate for total demand for all fixtures. Although the manual

from which the estimate values in table 8 were

taken does not specifically state that the sum

of the separate estimates for each kind of

fixture is to be taken as the total estimate, it does not provide any other means of obtaining

the total, and presumably this is the method

that has been used in applying the table. If this is true, it accoimts to a large extent for the

overestimates complained of and previously

referred to. Furthermore, no provision is made in the manual for the difference in demand

for the same fixtures in public (congested)

service and in private (noncongested) service, an omission that would still further increase

the overestimates for the latter. In addition to the fact that the method of weighting fixtures as to their demands (table 7) and applying the weighted total to probable

use curves (fig. 5) will at least partially correct the errors inherent in the use of tables as

illustrated, once the weighted method is understood thoroughly, it will be simpler to use. Tables in an abbreviated form, as illustrated

by tables 8 and 9, require interpolation for values not appearing in the table, whereas

interpolation is automatically taken care of in using the curve.

XI. ESTIMATES OF SEWAGE LOADS

It has already been pointed out on pages

4 and 5 that the probability function has only

a limited application to the drainage systems

of buildings. Insofar as the probability function is applicable to the estimation of peak

sewage loads to be provided for, the same function as applied to the peak water demand (fig. 5) will apply also, with certain modifications,

to the drainage system. In the main, the same

table of fixture weights (fixture units) will also apply to both the supply and drainage systems.

The principal exception is the water closet,

which for the drainage system will in general

have the same weight for a given congestion of service whether the water-supply control is by

flush valve or flush tank. In certain other

cases, it may become advisable to assign dif- ferent weights to the same fixture, depending on the sizes of the outlet from the fixture and

of the fixture drain. Because the details of the

modifications indicated are closely related to the

capacities of drains, the further development

[20]
of the method of estimating sewage load w ill be treated in the paper dealing with capacities

of building drains. Also, before the method

can be successfully applied in practice, the

abbreviated table of fixture weights for demand

loads will have to be ex[)anded and modified.

These modifications will also be developed in the later paper.

XII. DISCUSSION

The principal purpose of this paper is to present the fundamental principles and data on

which the proposed method of estimating

loads to be provided for in plumbing systems

is based. It has been pointed out that the

details of application of any method in practice

must be guided to a large extent by engineering

judgment in order that it may lead to satis- factory results. The choice of values employed

in evaluating the probability function and in converting estimates in number of fixtxu'es flowing to estimates in gallons per minute represents the author's judgment.

It has also been pointed out that it is impossible to determine or to estimate closely either

the maximum demand load or the maximum

sewage load that will occur in service. It is only possible to estimate the loads having a

certain probability of being exceeded in a given

system. In this respect, it is believed that

table 7 and figure 5 will enable the engineer to obtain that estimate as closel}' as the varying

conditions encountered in phunbing .systems

permit, and that the results of applying the

data presented in that form will in general lead

to the selection of adequate sizes of sii])ply

pipes and as economical sizes as are consistent

with safety and satisfactory operation, pro- vided the proper judgment is exercised in

estimating the capacity of the pipes under the

particular conditions of installation.

XIII. REFERENCES

[1] Recommended Minimum Requirements for Plumbing, Department of Commerce Building and

Housing Series, BH13, 1924. Revised by addition of a supplement, 1932.

[2] Prohahility and Its Engineering Uses, by Thornton

C. Fry, (D. Van Nostrand and Co., Inc., 1928).

[3] The Hydraulics of Water Closet Bowls and Flushing

Devices, by Thomas R. Camp, associate professor

of Sanitary Engineering, Massachusetts Institute

of Technology. Published in mimeographed

form in 1936 b.v the Massachusetts State Association of Master Plumbers.

[4] G. E. Golden, Research Associate at the National Bureau of Standards for the Plumbing Manufacturers Research Associateship. (Data unpublished.)

[5] Manual of Procedure for the Mechanical Section, Public Works Branch, Procurement Division,

Treasury Department (Sept. 1935).

[6] Interior Water Supply Piping for Residential

Buildings, Bulletin of the University of Wisconsin, Engineering Experiment Station Series, No.

77.

Washington, August 30, 1940.

[21]

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