Capacities of

Plumbing Stacks in Buildin

United States Department of Commerce

National Bureau of Standards

Building Materials and Structures Report 132
BUILDING MATERIALS AND STRUCTURES REPORTS

On request, the Superintendent of Documents, U. S. Government Printing Office, Washington

25, D. C., will place your name on a special mailing list to receive notices of new reports in this

series as soon as they are issued. There will be no charge for receiving such notices. An alternative method is to deposit with the Superintendent of Documents the sum of $5, with

the request that the reports be sent to you as soon as issued, and that the cost thereof be

charged against your deposit. This will provide for the mailing of the publications without

delay. You will be notified when the amount of your deposit has become exhausted.

If 100 copies or more of any report are ordered at one time, a discount of 25 percent is allowed. Send all orders and remittances to the Superintendent of Documents, U. S. Government Printing

Office, Washington 26, D. C.

The following publications in this series are available by purchase from the Superintendent of Documents at the prices indicated:

BMS2 Methods of Determining the Structural Properties of Low-Cost House Constructions 10)5 BMS3 Suitability of Fiber Insulating Lath as a Plaster Base 15^ BMS4 Accelerated Aging of Fiber Building Boards 10^ BMS5 Structural Properties of Six Masonry Wall Constructions 20^

BMS6 Survey of Roofing Materials in the Southeastern States 16^ BMS8 Methods of Investigation of Surface Treatment for Corrosion Protection of Steel 15^ BMS9 Structural Properties of the Insulated Steel Construction Co.'s "Frameless-Steel" Constructions for Walls, Partitions, Floors, and Roofs 10(5 BMSIO Structural Properties of One of the "Keystone Beam Steel Floor" Constructions

Sponsored by the H. H. Robertson Co. 10^ 6MS11 Structural Properties of the Curren Fabrihome Corporation's "Fabrihome" Constructions for Walls and Partitions . 10^ BMS12 Structural Properties of "Steelox" Constructions for Walls, PartitionSf Floors, and

Roofs Sponsored by Steel Buildings, Inc 16^ BMS14 Indentation and Recovery of Low-Cost Floor Coverings 10^ BMS15 Structural Properties of "Wheeling Long-Span Steel Floor" Construction Sponsored by

the Wheeling Corrugating Co 10^ BMS16 Structural Properties of a "Tilecrete" Floor Construction Sponsored by Tilecrete

Floors, Inc - _ 10^ BMS17 Sound Insulation of Wall and Floor Constructions 20^

Supplement to BMS17, Sound Insulation of Wall and Floor Constructions 6^ Supplement No. 2 to BMS17, Sound Insulation of Wall and Floor Constructions 15^ BMS18 Structural Properties of "Pre-fab" Constructions for Walls, Partitions, and Floors Sponsored by the Harnischfeger Corporation . . 10^ BMS20 Structural Properties of "Twachtman" Constructions for Walls and Floors Sponsored

by Connecticut Pre-Cast Buildings Corporation 10^ BMS21 Structural Properties of a Concrete-Block Cavity-Wall Construction Sponsored by the National Concrete Masonry Association 10^ BMS22 Structural Properties of "Dun-Ti-Stone" Wall Construction Sponsored by the Brick

Dunn Manufacturing Co 10^ BMS23 Structural Properties of a Brick Cavity-Wall Construction Sponsored by the W. E. Manufacturers Association of New York Inc 10^ BMS24 Structural Properties of a Reinforced-Brick Wall Construction and a Brick-Tile CavityWall Construction Sponsored by the Structural Clay Products Institute 15#

BMS25 Structural Properties of Conventional Wood-Frame Constructions for Walls, Partitions, Floors, and Roofs 20#

BMS26 Structural Properties of "Nelson Pre-Cast Concrete Foundation" Wall Construction

Sponsored by the Nelson Cement Stone Co., Inc ; 100 BMS27 Structural Properties of "Bender Steel Home" Wall Construction Sponsored by the Bender Body Co 100 BMS28 Backflow Prevention in Over-Rim Water Supplies 150 BMS29 Survey of Roofing Materials in the Northeastern States 200

BMS30 Structural Properties of a Wood-Frame Wall Construction Sponsored by the Douglas

Fir Plywood Association 150 BMS32 Structural Properties of Two Brick-Concrete-Block Wall Constructions and a Concrete-Block Wall Construction Sponsored by the National Concrete Masonry

Association - 150 BMS33 Plastic Calking Materials 150 BMS34 Performance Test of Floor Coverings for Use in Low-Cost Housing: Part 1 150

{List continued on cover page ni]
UNITED STATES DEPARTMENT OF COMMERCE • Charles Sawyer, Secreiary NATIONAL BUREAU OF STANDARDS • A. V. Astin, Acting Dirccwr

Capacities of

Plumbing Stacks in Buildings

Robert S. Wyly and Herbert N. Eaton

Building Materials and Structures Report 132

Issued May 28, 1952

For sale by the Superintendent of Documents, U. S. Government Printing Office, Washington 25, D. C.

Price 20 cents
Foreword

The problem of what constitutes the capacity of a vertical plumbing

drainage stack in a building has never been solved satisfactorily. The lack of

knowledge in this respect is reflected in the glaring differences in stack-loading

tables found in different plumbing codes. Obviously no building drainage

system can be designed economically unless the minimum sizes of pipes that can be used in its various parts can be predicted, and this is particularly true

of the drainage stack and the building drain as these are the largest pipes used

in the system. Attempts have been made for many years to solve the problem,

but, owing to the complexity of the flow phenomena in the drainage system,

progress has been made only with the conditions so simplified that the practical

utility of the results is very restricted.

The investigation reported here was an attempt to solve one of the more

complicated problems of stack capacities—the maximum load that can be

brought into the stack from a horizontal branch when water is being discharged

down the stack from fixtures on higher floors. A theoretical approach to the

problem was used, and the resulting formula was tested by experiment. As a

result, a distinct step forward has been made in the solution of the problem.

The research forming the basis of this report was undertaken for the

Housing and Home Finance Agency as part of the research program of that

Agency under its statutory authority. The report presents only the results of the first part of the investigation.

A. V. AsTiN, Acting Director.

Contents

Page

Foreword ii

1. Introduction 1

2. Previous research on staclc capacities 2

2.1. National Bureau of Standards 2

2.2. State University of Iowa 4

2.3. University of Illinois 5

3. Summary of attempts to define stack capacity , 5

3.1. National Bureau of Standards 5

3.2. State University of Iowa 5

3.3. University of Illinois 5

4. The problem of stack capacity 6

4.1. Basic factors to be considered 6

a. Noise and vibration 6

b. Excessive pressures in horizontal branches 7

4.2. Tentative definition of stack capacity 8

4.3. Terminal velocity and terminal length 8

5. Terminal velocities in stacks 9

5.1. Tests at the National Bureau of Standards 9

5.2. Tests at the State University of Iowa 9

5.3. Derivation of equation for terminal velocities 11

6. Terminal lengths in stacks 13

7. Tests on stacks and drains 14

7.1. Nature of the tests 14

7.2. Simplified setup for simulating conditions at the junction of one or two horizontal branches and a vertical stack 14

8. Analysis of conditions at the stack fitting where a horizontal drain empties into a stack at an intermediate floor of a multistory building 16

8.1. Statement of the problem 16

8.2. Analysis of the problem ^ 16-

9. Summary 20

10. Application of the results of the investigation 24

11. Conclusions 26

12. Further investigation 27

13. References 27

14. Appendix. Evaluation of the sand roughness coefficient, 28
Capacities of Plumbing Stacks in Buildings

Robert S. Wyly and Herbert N. Eaton

This report gives the results of the first pliase of a theoretical and experimental investi- gation of the capacities of plumbing drainage stacks in buildings. Because of the complexity

of the problem and the wide variety of conditions that are encountered in providing sanitary

drainage for buildings, no satisfactory means of determining the sizes of pipes required in the drainage system of a particular building has ever been developed. The present investi- gation was intended to solve one of the problems involved in determining the capacity of a

vertical drainage stack, namely, the pressures that are produced in a horizontal drain by

the interference of the flow down the stack from higher levels with the flow from a horizontal drain into the stack at some intermediate level. This was accomplished by developing an

equation for the relation between rate of flow down the stack, rate of flow from the branch,

pneumatic pressures in the stack and in the branch at the branch level under consideration, the diameters of the stack and the branch, and the backpressure produced in the branch. With the aid of experiment, the unknown quantities in the equation were determined, and

the equation can be used to predict this backpressure for any given conditions when a

sanitary tee stack fitting is used to connect the branch to the stack.

1. Introduction

The soil, or waste, stack of a building is the

vertical drainage pipe into which the horizontal

branches carrying the discharge of the fixtures

empty and which carries the waste water down to the building drain. Above the highest fixture

connection on the stack, the stack continues

upward through the roof, sometimes with, sometimes without change in diameter, to serve as a vent for the stack. This portion of the stack is called the stack vent, and the portion above the

roof is frequently referred to as the roof vent. The hydraulic and pneumatic conditions exist- ing in such a stack down which water is flowing- are extremely complicated. The stack flow^s only

partly full, the water tending to form an annular

sheet around the wall of the stack. A considerable quantity of air is carried along with the water,

this air entering the system through the various

vent pipes, including the stack vent. When the

sheet of water reaches the bottom of the stack, it flows around the bend, frequently still adhering

to the wall around its entire circumference, falling

away from the upper part of the cross section only

after it has traveled a short distance. In the

lower portion of the stack there is a tendency

for positive pneumatic pressures to exist; hence

considerable quantities of air are removed from

this portion of the stack through any vent pipes

that are connected to the stack in this region of

positive pressure, that is, pressures greater than

atmospheric pressure. The water entering the building drain from the

stack is traveling at a higher velocity than can be

maintained in a , sloping building drain by the

force of gravity. The result is that the stream

slows up gradually, until finally, if the drain is long enough, it suddenlj^ increases in depth and

decreases in velocity, forming what is called a

hydraulic jump [1],^ which may or may not entirely

fill the cross section of the drain at the section at which it occurs. If the hydraulic jump fills the

entire cross section of the pipe, this forms a

barrier to the further travel of the air that is moving with the water, and the pneumatic pres- sure in the drain upstream from this section and

in the lower part of the stack increases, gradually

returning to its normal value, somewhere near atmospheric pressure, in about one story height

or thereabouts, in the usual multistory installation.

If the water is flowing down the stack in the

form of an annular sheet at a particular floor level,

coming from fixtures on higher floors, and, if water is also flowing out of a horizontal branch at that floor level, the stream from the branch interferes with the flow down the stack, with the result

that a backpressure is created in the branch, and,

if the system is not properly designed, this backpressure may be enough to flood shower stalls or

to produce sluggish flow from fixtures emptying

into the branch, or to affect the proper functioning

of vents connecting into the branch.

Where the flow from a horizontal branch enters the stack, it may fill the entire cross section of the

stack if the flow is large enough. As the velocity

of the water thus entering the stack increases

downward under the influence of gravity, the pres- sure in the stack in this region decreases, sometimes

very greatly, gradually returning to its normal

value, which is approximately atmospheric pressure, usually within a distance equal to one to three stories' height of the building.

When the building dram includes a house trap,

and when the discharge from a water closet or some

other fixture closes off the cross section of the

stack at some point, air is thus trapped between

1 Numbers in brackets indicate the references at the end ( f this report.

1
the two water seals, and, owing to the fact that

water is flowing into the space containing the

trapped air for several seconds before an equal

flow passes out through the house trap, some of the

trapped air must be displaced. This frequently

results in blowing the fixture trap seals in lines connecting into the stack between the two water

seals referred to. This phenomenon has been observed in tests. The large amount of air that is carried down a stack with the water is not generally recognized.

As a very rough approximation, we may say that

about as much air as water by volume, is carried

down the stack, although m some instances the

air flow will greatly exceed the water flow. This

air must come from somewhere, usually from the

stack vent, but if the cross section is closed off at some point by the discharge from a horizontal

branch, then it must come in part from the horizontal branches, and, if the venting system is inadequate, this results in pulling the seals of

fixture traps. In regions of positive pressure, inadequate venting results in the blowing of trap

seals. Because of the extreme complexity of the

hydraulic and pneumatic conditions existing in the

sanitary drainage systems of buildings, little progress has been made toward the establishment

of any sound basis for determining how great a load can be imiposed on a given system without

impairino; the adequacy of its operation. Eather we have learned from experience what a system of a given size will stand, and loading tables for plumbing drainage stacks have been established,

mainly on the basis of this experience. The

great variation in permissible loadings on stacks can be seen from table 1.

Table 1. Maximum fixture-unit loadings on multistory

stacks permitted by different authorities

Stack diameter

Iowa and

Wisconsin State codes

Cleveland, Ohio

St. Louis, MissouriDetroit, Michi- gan BMS66

National Plumbing Code

1941 1942 1945 1946 1940 1951

in.4

56

8

10

256

680

1,380 3,600

96

192 336

840

1,600

2, 840

300

720 1,080

1,920

265

500 1,200

3, 000

600 1,500 2,800

5, 400 8,000

14, 000

500

1,100

1,900

3, 600 5,600

12 8, 400

From table 1 it is obvious that there has been

little or no agreement as to what constitutes max- imum permissible loadings on multistory stacks.

Similar disagreement exists regarding loads on

short stacks, horizontal branches, building drains,

and building sewers, storm and combined drains

and sewers, and vents. In terms of actual fix- tures, or groups of fixtures, the variation in permissible loadings on stacks is not always as great

as indicated by table 1, owing to the fact that

fixture-unit ratings for some fixtures are assigned

different values by different codes. In this connection, table 2 shows the fixture-unit values for

water closets employed in the codes listed in

table 1.

Table 2. Fixture-unit values for water closets according to different authorities (as applied to drainage loads)

Type of supply or class of usage Authority

Public Private Flush Flush

use use valve tank

Iowa and Wisconsin State Codes (1941) 6 6 6 6 Cleveland, Ohio (1942) 4 4 4 4

St. Louis, Mo. (1945) 6 6 6 6 Detroit. Mich. n94fi1 6 4 BMS66 n9401_ 10 6 National Plumbing Code

(19511 8 4

2. Previous Research on Stack

Capacities

Several attempts have been made in the past

to obtain some knowledge of the flow and pressure

conditions in stacks and thus to offer, at least to some extent, a solid basis for establishing tables

of stack loadings. This has been very difficult, for not enough has been known about the conditions in stacks to permit a satisfactory definition

of stack capacities to be proposed. Ihe late Koy B. Hunter at the National Bureau of Standards, F. M. Dawson and A. A. Kalinske at the

State University of Iowa, and Harold E. Babbitt at the University of Illinois, have all made

contributions to the subject.

2.1 National Bureau of Standards

Hunter first reported tests on stack capacities

in 192.3 [2]. In his report are to be found some

of the first clear statements regarding the nature

of flow in partly filled stacks and the first defini- tion of stack capacity that the authors of this

paper have found. The following is quoted from

Hunter's report:

The character of the flow of water in partially filled vertical pipes varies with the extent to which the pipe

is filled. * * * For small volumes of flow, amounting to little more than a trickle, the flow is entirely on

the inner wall of the stack. With increase in volume,

this adherence of the flow to the wall continues up to the point where the frictional resistance of the air causes

it to diaphragm across the pipe, temporarily forming a

short slug of water which descends as a slug filling the

.stack until increased air pressure breaks through, the water forming the slug either being thrown against the

wall or falling a short distance as separate streamlets

in the center of the pipe. This diaphragming and forming slugs probably first appears in a 3-in. stack when

the stack is from' one-fourth to one-third full. As the volume of flow increases, the slugs thus formed are more frequent and more persistent, and in a stack open

at the lower end may not be broken through. This

intermittent rate partially accounts for the rapid erratic

oscillations of pressure in a plumbing system.

With regard to stack capacity andfitting capacity.

Hunter wrote as follows:

2
In vertical stacks a static head can develop only

after the stack has become filled at some point. With

the water introduced at a given rate by volume, this

will first occur at the point of lowest velocity, evidently

at the point of entrance. Since the vertical component

of the entrance velocity is less than the maximum

velocity attained in the stack, the velocity increases from the point of entrance until the maximum is reached. The vertical component of the entrance

velocity depends on the rate of flow by volume, the

cross-sectional area of the inlet, and the angle of the

entrance. This points to the capacity of the fitting as a measure of the "practical capacity of the stack." We will define the capacity of the fitting as the rate of flow in gallons per minute at which the water just

begins to build up in the vertical part of the fitting

above the inlet branch of the fitting. With any rate

of flow not exceeding this value then the stack, barring

stoppage or marked retardation, will not be completely

filled at any point.

Hunter is speaking here of the conditions under

which water is flowing from the horizontal branch

into the stack, but no water is flowing down the

stack from a higher level. Continuing with

Hunter's discussion:

Determinations were made with different fittings on both 2- and 3-in. stacks as follows: The fittings were

set in the top of the stack, and water was introduced at known rates through a trap and the inlet fitting. So

long as the water flowed freely with no appreciable

retard in the fitting, the rate was considered below that

fitting capacity. The rate at which the first tendency

to back vertically into the fitting above the level of the inlet appeared, was taken as the fitting capacity.

When the water stood above the level of the inlet, even

temporarily during the sustained rate of flow, it was

considered that the capacity had been exceeded. This

last condition in the case of only one inlet to the stack would do no particular harm, and it should be noted

that the rate of flow that causes the water to stand in the fitting is greatly in excess of the rate at which the tendency to build up first appears, double the rate in some cases. The flow was maintained for from 15 to 30 seconds by means of tanks, thus giving ample time

for observation and for any effect from the lower

portions of the stack to be transmitted. Various

lengths of stacks from 30 to 45 ft., both with and with- out house drains were employed, but as no modifying

effect from the part of the system below the fltting was

observed in any case, only the fitting and tank connections are illustrated in figure 39, showing a double Y fitting with water introduced simultaneously at both

inlets. The preceding description applies to capacities with

entrance at one level with either one or two inlets. For entrance at two levels, two fittings of the kind

under test were set at two points in the stack approximately 11 ft apart, and a 1-in. glass tube bent in L shape was set in a side inlet in the lower of the two

fittings and water introduced simultaneously at known

rates through the main inlets of the two fittings as

before. Any retard or interference in the lower fitting sufficient to cause a slight backflow, neglecting

splashes, into the glass L tube was taken as exceeding

the capacity of the fitting under these conditions. As

might be expected, the tendency is toward an increase

in capacity with the water introduced at two levels. This is essentially different from the effects in pipes

flowing full. It therefore seems safe to assume that

the capacity of a given stack with entrance points at more than one level is not less than the capacity of the same stack with entrance points at one level only. In fact, there is very strong evidence that the capacity

increases with the increase in number of entrance

points at different levels, tending toward a maximum

capacity measured by the stack flowing full with the

maximum velocity attained by the water flowing in a vertical stack of that diarmiter. What w(; have

really determined is therefore, the maximum cajjacit y

of the stack for the methods of construction that give a relatively small maximum capacity compared to what might be secured by other methods of design

or construction. From these tests, which were made only on 2- and .3-in. diam.eter stacks, Hunter concluded that

the stack capacity could be represented approxim.ately by an equation of the type

Q=kd\ (1)

where

Q=t\\e perm-issible volume rate of flow into

the stack in gallons per minute

fZ=the diam.eter of the stack in inches

A: =an empirical constant. Himter gives the values ^ =22.5 for 45° Y inlets ^=11.25 for sanitary T inlets. On the basis of this reasoning and the experimental results with 2- and 3-in. stacks, Hunter

gave the following table of stack capacities in BH13 [3].

Table 3. Carrying capacity of stacks constructed with

single or double fittings

Water introduced at one level only. (Hunter)

Flow into stack throughDiameter of stack

Sanitary T 45° Y

in. gpm gpm

2 (experimental data). - 45 90

3 (exiHTiniciilul data) - . 100 200

4 (extralulali'iU 180 360

5 (extralJolatid) . _ _ . ._ 230 o60

6 (exlrapolalrd) 405 810

8 (extrapolated) 720 1,440

No tests were made with water introduced

through single fittings, and no quantitative test results were reported for the case in wliich the

water is introduced into the stack at more than

one level. An illustration of a case in which the fitting

capacity has been exceeded considerably is shown

in figure 1. Tliis photograph was taken in the

course of the investigation reported here. It shows a 3-in. double sanitary tee with 3-in. side

inlets made of transparent plastic material with

its inside dim.ensions closely simulating those of the

corresponding metal fitting. Flows of 100 gpm

are entering the fitting from each horizontal

branch, but there is no flow do^vn the stack from

higher levels. It will be observed that the water

is boiling up in the stack som.ewhat above the level of the branches. According to Hunter's formula

(eq 1, using the value 11.25 for k), the capacity of the fitting is approximately 100 gpm. Thus Hunter's criterion indicates that a flow of 200 gpm

3
Figure 1. Flow into stack from two horizontal branches at same level, showing case in wh ich the capacity of the fitting

is exceeded.

Total flow from branches is 202 gallons per minute. No flow down stack from higher levels.

would overload the fitting, and figure 1 shows that

tliis is actually the case. It \vill be observed also that the meniscus in the

manometer shown in figure 1 is slightly below the

top of the horizontal branch, showing that no backpressure in the branch is created by this condition.

This manometer is connected into the branch at a

distance of 4 branch diameters from the stack, or approximately 1 ft. In BH13 [3] Hunter makes the following further statements: With a system composed of vertical and horizontal

sections of the same diameter throughout, it is evident

that on the assumption that no static head may

develop, the limit of carrying capacity will first be reached in the horizontal sections of the system. This

at once suggests the separate consideration of the

horizontal and vertical sections. . . . Apparently, in view of the varying degrees to which

the system or parts of the systeiii may be filled, a very rough approximation of the maximum carrying capacity is all that may be hoped for. The fact that an

appreciable head of water developed in any portion of a plumbing system into which branches are discharging tends to impair the efficiency of the drainage indicates a limiting condition on which to base determination of capacities. While the development of a head of water

in certain portions of a given system—namely, in unvented small waste branches—may produce no

detrimental effects, the assumption of a condition under which no head can develop in any portion of the

stack or house drain is certainly a safe procedure and

apparently the only one that will permit a general

application of results. Hunter also prepared the stack-loading tables in

BMS66 [4]. Unfortimately, however, the data

and the reasoning on which these tables were based

are no longer available.

2.2 State University of Iowa

Dawson and Kalinske have made several contributions to the problem of stack capacities, the

most important being reported in Bulletin 10 oif the State LI niversify of Iowa[5]. In this investigation, stacks 3, 4, and 6 in. in diameter and

approximately 30 ft. high were tested. Various

flows were introduced through a horizontal branch

near the top of the stack, and pressures and velocities were measured at various points down the

stack. The results will be discussed later in this

paper. In Technical Bulletin 2 of the National

Association of Master Plumbers of the United

States[6], they make the following statements

bearing on the problem of stack capacities:

Since plumbing stacks cannot be permitted to flow completely full, one important determination is, naturally, how full they can flow without undesirable effects. In other words, what is the carrying capacity

of any given size of stack? The maximum allowable

rate of flow in a stack, assuming that proper venting

exists, will be determined ordinarily by the conditions

at the bottom of the stack and in the house drain and

sewer. The velocity of water in the building drain and

sewer must obviously be considerably less than that in the vertical stack. Therefore, if the building drain and sewer are not to be completely filled, the stack must be considerably less filled, because the ratio of the velocity of the water in the stack and in a horizontal drain laid at a J-^-inch per foot slope is of the order

of 4 to ]

.

Also, a stack which is flowing too nearly full fre- quently causes considerable noise and vibration. Tests have very definitely shown that too high a rate of flow

in a stacic produces undesirable noises due to the terrific pneumatic disturbances that are set up. A consideration of the various test data and of different practical items indicates that the stack should not flow over about Yi full when the water attains its maximum

velocity .... Dawson and Kalinske give the following table

of maximum carrying capacity of stacks:

Table 4. Recommended maximum carrying capacity of

stacks [Dawson and Kalinske]

Diameter of stack Maximum flow

in. gpm

3 90

4 180

6 350

6 560

8 1,200

4
The authors then go on to say:

Another important item that must be determined

when the capacity of stacks is under consideration is the amount of water that can enter the stack at any

one level. Since the vertical velocity of the water is least at the point of entrance, this part of the stack

will be filled up the most. If it becomes filled up too much, however, water will start to head up in the

stack, thus retarding the flow from the horizontal

drains connected to that level, and also creating undesirable pneumatic effects and considerable noise. Therefore, the allowable rate of flow into the stack at any one floor level must be limited. Our tests and

observations have shown that the allowable rate of flow into a stack at any one floor level should not be over one-third of the allowable maximum carrying;

capacity of the stacks, if the horizontal drains are con- nected to the stack with sanitary tees and not over

one-half of the maximum carrying capacity if 45° wye

connections are used. Although the procediu-e followed by Dawson

and Kalinske represents probably the most detailed experimental and analytical approach to the

problem that has yet been published, the detailed

experimental data reported consisted entirely of

those from tests in which all the water was introduced at one level on the stack, rather than at

several levels, which will, of course, be the service

condition for all buildings greater than one story

in height.

2.3. University of Illinois

Harold E. Babbitt of the Universit_y of Illinois has published the results of stack tests[7], but the

most important part of his work related to variations of pressure in the stacks with flow. He

gives a discussion of "The useful capacity of 2-inch,

3-inch, and 4-inch soil stacks", in which he treats the capacity of a soil stack to receive the discharge

from plumbing fixtures. Here he cites the results of tests on stacks of the three sizes mentioned

above, on which he gives the pressures measiu-ed

in a horizontal branch 42 ft lower on the stack

than the level at which water was introduced in

varying amounts. Plotting these results on logarithmic paper, the pressures in the horizontal

branch being plotted as ordinates and the rate of discharge down the stacks as abscissas, he shows

that these curves for the low flows are straight

lines with the slope 5/2, but that the curve to the right for higher flows in the direction of lower

powers.

Babbitt assumes that the points at which the cm-ves begin to deviate from the straight lines mark the points at which water can no longer

enter the stack freely, and hence that these points

represent the capacity of the stack to receive

water through the fitting used. The points at which these curves begin to deviate from straight

lines are not clearly markei, and it is easy to

differ hy as much as 100 percent in reading these

points. Neither does the pressure read appear to be a very good criterion for the purpose. Again

the results apply only to flow into the stack at the

highest horizontal branch and with the condition

that no more water enters the stack for at least a distance of 42 ft below this point of entry.

3. Summary of Attempts to Define

Stack Capacity

3.1 National Bureau of Standards

Hunter suggested two dift'crent criteria. The

first is based on the rate of flow entering through

two horizontal branches connecting into the stack

at the same level and 180° apart. When the

flow has reached a value that causes the water to begin to back up in the vertical part of the fitting

above the inlet branches he considers that this flow represents the capacity of the stack.

According to Hunter, eq 1, with the values of k

following it, gives the capacities of stack fittings and

hence the practical capacity of stacks. Hunter

makes the following important statement:

The fact that an appreciable head of water devel- oped in any portion of a plumbing system into which

branches are discharging tends to impair the efficiency

of the drainage indicates a limiting condition on which

to base determination of capacities . . . the assumption of a condition under which no head can develop

in any portion of the stack or house drain is certainly a safe procedure and apparently the only one that will permit a general application of results. He also states that an upper limit to stack capacity can be set on the basis that when the stack is flowing from one-fom*th to one-third full (that is, when the wetted cross section is from one-fourth

to one-tliird of the total cross section of the stack),

slugs of water form in the pipe and fall for short

distances before thev break, and, as these slugs

produce large pressure fluctuations in the stack,

their formation should be avoided. He does not

say whether this criterion refers to the sheet of

water flowing down the stack wall when it has

reached its maximum or terminal velocit}^, but it w^ould seem that this must have been his intention.

The use of this criterion makes it necessary to consider the problem of flow down the wall of a partially fllled vertical pipe, and this will be done in

section 5.

3.2. State University of Iowa

Dawson and Kalinske point out the importance

of not allowing undesirable pressure conditions to occur in smy part of the system. They state that

the stack should not flow more than about one- fourth full (pi'esiunably this refers to the cross- sectional area) when the water attains its maximum

velocity. They also suggested that the allowable

rate of flow into a stack at any one floor level should not be over one-third of the allowable maximiun carrying capacity of the stack if the horizontal branches are comiected to the stack with

sanitary tees and not over one-half of the allowable

maximimi carrying capacity if 45° Y connections

are used.

3.3. University of Illinois

Babbitt's work on stack capacities related

mostly to the positive pressures created in the lower

part of the stack b}- water flowmg from higher

5
levels. These positive pressures are probably due

to two causes. First, it appears obvious that,

with a sheet of water flowing down the inside

surface of the vertical stack, the 90° bend at the base of the stack may cause the hollow

core in the stack through which air is flowing from

the upper part of the stack to become closed off in or iust beyond the bend, thus causing an in- crease in pneumatic pressure as the water flowing

down the stack approaches the bend.

Second, such flows very possibly set up fluctuations of pressure in the lower horizontal branches

because of the disturbance of the sheet of water

flowing down the wall of the stack as it passes

the inlets from the horizontal branches, even when no discharge is taking place from these

branches. This type of disturbance has been

observed in tests with transparent plastic systems

at the National Bureau of Standards. Babbitt

makes the following general statement as to stack capacities:

. . . the indications are that a 4-in. soil stack will take all the water that can be delivered to it in a

5-story building; that a 3-in. soil stack woiild take

all the water that would be delivered to it in a 3- story residence; and that a 2-in. pipe is unsuitable

for use as a soil stack.

4. The Problem of Stack Capacity

4,1. Basic Factors To Be Considered

The basic factors that have to be considered in

connection with the problem of stack capacities are (1) the stack must not flow so full that vibration and undesirable noises will occur, (2) positive

pressures that cannot be relieved by venting and

which are sufficient to produce sluggish drainage

or the flooding of fixtures must not be permitted

to occur in any horizontal branch. In the dis- cussions to follow, it will be assumed that the

venting system is adequate to relieve partial vacumn conditions within the system, provided

that hydrostatic heads in the horizontal branches

are not excessive.

a. Noise and Vibration

Noise and vibration mav possibly be caused

by the following:

1. Formation and breaking of slugs of water

in an overloaded stack. These slugs travel with

high velocity and produce large pneumaticpressure fluctuations. Hunter says that these

slugs may form if the stack is flowing from one- fourth to one-third full. Dawson and Kalinske

state that excessive noise and vibration may

occur if the stack is flowing more than one-fourth

fufl.2. It is also possible that these undesirable effects may be clue partly to interference with the sheet

of water flowing down the stack by the opening

or the lower edge of the stack fitting into which

a horizontal branch carrying no flow may be

P'iGURE 2. Flow down stark punt a sanitary tee, showing

deflection of sheet of water toward the center of the cross

section and the rollers formed in the branches. No flow from branches. Flow down stack 124 gallons per minute. Pneumatic pressure in stack is atmospheric.

connected. This interference has been observed

visually in a transparent system in which the

sheet of water flowing down the stack passed

the side inlets of a double sanitary tee fitting

(see fig. 2.) The sheet of water was deflected

toward the center of the cross section of the

stack by its impact with the bottom of the inlets,

and rollers were formed in the inlets of the stack

fitting. The pneumatic pressure in the stack was

atmospheric.

Figure 3 shows the conditions at the junction

of the horizontal branches when there was a

slight pressure reduction in the middle of the

stack, the conditions of flow being the same

as those represented by figure 2. (No flow from

branches, 124 gpm down the stack). It will be

observed that the rollers are absent when a re- duced pressure exists in the stack owing to the

tendency for the pressure difference between the

branches and the stack to pull the sheet of water

inward toward the center of the stack. The deflection of the sheet of water inwardly by the

bottom elements of the branches where they in- tersect the stack is more pronounced than in the

former case.

6
FiGUBE 3. Flow down stack past double sanitarrj tee, showing deflection of water toward the center of the cross section. Pneumatic pressure in stack a little less than atmospheric.

3. It is also possible that some disturbance

might be caused at the level of a stack fitting

through which so much water is discharged into

the stack that the cross section of the stack is completely filled at that level. In summary, it may be stated that iu all probability most of the noise and vibration is caused

by slugs forming and breaking. The best judgment available at the moment is that, if these

undesirable effects are to be prevented, the stack

should not be allowed to flow with niore than

one-fourth of its cross section full when the sheet

of water is traveling at its maximum or terminal

velocity. This criterion enables us to set an upper

limit to the permissible wetted cross section of a stack of any given size, and tlris can be translated

into maximum permissible flows on the basis of information given later in this paper.

b. Excessive pressures in horizontal branches

1. The question arises as to how much positive

hydrostatic pressure is permissible in a branch.

The pressure involved may be excessive for the reason that it produces sluggish flow from fixtm-es on the branch, that it causes fixtures on the branch

to be flooded, or that the function of certain vents may be affected adversely. The question of

sluggish fk)W will be ignored for the time; being.

A bathtub, shower stall, or floor drain would

be most subject to the effet't of tliis pressure in the horizontal branch as these are the lowest fix- tures installed on such hi'anches. On any horizontal branch to which a number of fixtures are connected, it may be possible to have a flow sufficient to result in the creation of an excessive

backpressure in the drain. This may cause the

water in the branch to rise through fixture drains

into the fixtures not being discharged, such as the

bathtub or shower stall, or onto the floor through

the floor drain. It will also, under certain conditions, cause sluggish drainage from any of the

fixtures mentioned that are discharging.

In adilition to the undesirability of sluggish

drainage of these fixtures, it is perfectly possible

that, where two horizontal branches connect into

the stack at the same level through a double sanitary tee fitting, one branch may be flowing under

a head sufficient to cause cross flow into the opposite branch, resulting in the creation of a hydrostatic head in the branch not flowing. This could

conceivably result in the flooding of the lower

fixtures on the branch that is not discharging,

although this is mentioned merely as a possibility,

not as a probability.

A condition similar to the one just mentioned

could occur when the stack fitting used is a single

sanitary tee fitting and little or no flow is coming

down the stack from above. In this case, an

excessive flow in the branch could result in backpressure caused by the stack becoming filled at the branch level. This, of course, represents a condition similar to that used in determining the

"practical capacity" of the fitting, on which

Hunter based some of his work on stack capacities

mentioned in section 3.1 of this paper.

Of course, the two latter conditions are likely to happen only if the load on the horizontal branch is too heavy, but it gives one more reason for limiting

the load on horizontal branches.

It is obvious that if a vent pipe connects to a

horizontal branch that is flowing under a slight

pressure at that point, the vent may be rendered

ineffective at that time. In view of the points considered above, it would seem that the maximum hydrostatic head in the

horizontal branch should be limited to an amount

that will just fill the branch near its junction with

the stack, at least in long horizontal branches to which a number of fixtures are connected.

The factors that tend to produce excessive pres- sures in horizontal branches include the foUowing:

(a) pneumatic pressures in the stack, (b) excessive

rates of flow in the branch, (c) interference with

flow from the branch by flow down the stack coming from higher levels, and (d) obstructions in the

branch or stack.

The pressure existing at any level in the stack is transmitted through the sheet of water flowing

985861'—52 2 T
down the wall of the st ack to the horizontal branch,

and, if the drain is filled with water at or near its outlet, exerts a backpressure on the branch. This

is a complicated problem, and, as far as the authors

know, has never been anal^^zed. It will be treated

later in this paper.

The question of obstructions in the stack or branch will be ignored, as the existence of such

obstructions constitutes an abnormality that will rarely, and shoidd never, occur. The other factors listed—excessive rates of flow

in the branch and the interference of flows—must

be considered together, assuming that the branch

is not so overloaded that an excessive head will be

built up even when no water is flowing down the

stack from higher levels. The principal cause of

hydrodynamic backpressure on the branch is the

interference of the sheet of water flowing down the

stack with the flow from the branch. The physics

of this phenomenon will be explained more fully later in this paper, but the cause of this backpressure will be pointed out here

The sheet of water flowing down the wall of the

stack at high velocity has considerable momentum.

In order that the water may flow out of the

branch, it is necessary that it deflect the sheet of

water flowing down the wall past the branch inlet to the stack. But, by a well-known law of physics

the deflection of the sheet, with its consequent

change in momentum, can be produced only by

means of a force exerted on it to produce the deflection, and this force is supplied through the pressure

in the stream from the branch. The greater the

the momentum of the flow clown the stack, the

greater will be the backpressure required to pex-- mit a given flow from the branch. Also, for a given rate and velocity of flow down the stack, the

greater the flow from the branch, the greater will

be the pressure required in the branch. Consequently, assuming the pneumatic pressure in the

stack to be the same at all levels for simplicity,

it is obvious that horizontal drains at the higher

floors of a multistory- building can be loaded more

heavily than can the lower branches as less water

will be passing the outlets of the higher branches

than the lower ones, and hence there will be less back-pressure caused for a given flow from a branch

at a high level than one having the same rate of outflow at a lower level. This fact, as far as the

authors know, has never been pointed out in the

literature, and yet it has an important bearing on

the design of building drainage sy^stems.

4.2. Tentative Definition of Stack Capacity

From what has been said, it is obvious that we

cannot speak of the capacity of a stack without

considering the distribution of load vertically on

the stack. Hence the capacity of a stack will vary, depending on the vertical load distribution,

and" our definition must take account of this fact. It also seems from the preceding discussion that

any definition of stack capacity must be based on

the question of pressures in the horizontal

branches. Even though it is fully realized that

the developm.ent of a sm.all head in the branches

may not necessarily be a serious objection, nevertheless, coDsideration of the hydrodynamic pres- sures in the branches offers a necessary and possibly sufficient condition on which to base the

definition. The following definition of stack capacity is offered, therefore, as a tentative definition,

with the full realization that later and m^ore detailed investigation m.ay require that it be changed.

It is believed that, if the branch pressures are kept down to reasonable values, there will be no

noise or objectionable vibration, but this m.atter is open to further investigation.

The capacity of a drainage stack at the level of any

horizontal branch is defined as the total flow at that

level which causes a preassigned positive pressure in

the horizontal branch which cannot be relieved by

venting or, in the case of horizontal branches low on

the stack, by increasing the size of the building drain.

Several com.m.ents are required to clarify the

above definition. First, by total flow we m.ean the

sum of the flows from, the horizontal branch in

cjuestion and from, fixtures higher up on the system.

This total flow m.ay be divided in any way between

the branch and the part of the stack immediately

above the branch.

Second, as it is specified that the preassigned

positive pressure cannot be relieved by venting,

this m.eans that it must be a hydrostatic pressure

due to a head of water. Were it a pneum.atic

pressure, it could be relieved by venting.

Third, as it is specified that the preassigned

positive pressure cannot be relieved in som_e in- stances by increasing the size of the building drain,

this means that the size of the building drain is adequate. Otherwise, for horizontal branches on

the lower part of the stack, the hydrostatic pres- sure under consideration could be relieved, at least

partially, by increasing the diameter of the building drain.

Fourth, if we require that this preassigned positive pressure shall never be exceeded, this will lead

to oversizing the pipes. Rather, we mean that it shall probably not be exceeded m_ore than a certain

specified portion of the time during which the fix- tures on the system, are being used with their m.ax- m.um assumed frequency, as during the m.orning

rush hours in a hotel. For a discussion of this

question of m.axim.um- frequencies, see [12].

4.3. Terminal Velocity and Terminal Length

In any adequate study of stack capacities, we

must have information as to the terminal velocities and terminal lengths of fall of the sheet of water in

the stack.

By terminal velocity we mean the m^axim.um, velocity of fall attainecl by the water column in the

pipe, whether the cross section of the pipe is filled or whether the water is flowing in a sheet down the

wall, when the w^ater is allowed to fall from, a given

8
level under the force of gravity and is retarded by

the wall friction through a long enough distance to attain this velocity. When these two forces bal- ance, the water is no longer accelerated, so that

the velocity from tliis point down is constant. By terminal length we mean the distance through

which the water must fall before it attains the

terminal velocity. Theoretically, the water would

have to fall an infinite distance before this condition would exist, but, for all practical purposes, a

finite lim-it can be established.

5. Terminal Velocities in Stacks

5.1. Tests at the National Bureau of Standards

Hunter [8] made tests on the fall of water

columns in vertical stacks up to 100 ft in height

built of 1-, 2-, and 3-in. galvanized steel pipe.

His experimental procedure was as follows: Gage

holes were tapped in the vertical stack at intervals

of 10 ft in height. A quick-opening valve was

installed at the bottom of the stack. With the

valve closed, the pipe was filled with water to the

level of the first gage hole. The valve was then

opened quickly, and the time required for the

water to flow out of the pipe was observed. This

process was repeated, the pipe being filled to the

height of each gage hole successively. To obtain

the terminal velocity for each diameter of pipe,

he plotted the lengths of the falling column as ordinates and the times of descent as abscissas.

This gave a straight line in the region where the

water was falling at terminal velocity, and the

slope of this line gave the terminal velocity.

In order to establish limits within which the

measured terminal velocities shovdd lie, Hunter

computed terminal velocities for smooth and for very rough pipe over a range of diameters of from

1 to 8 in. These curves, together with the ex- perimentally determined velocities, are shown in

figure 4. The three experimental points lie between the two curves, as they should, and they are

closer to the curve for smooth pipe than the curve

for very rough pipe, as would be expected. Because

of this good agreement of the experimental values

with the values given by the smooth-pipe curve. Hunter concluded that the smooth-pipe curve

given in figure 4 might be used to set the upper

limit of terminal velocities for flow out of plumbing stacks complete!}^ filled with water. Table 5 gives these velocities.

Table 5. Terminal velocities in vertical pipes flowing full.

Diameter of pipe Terminal velocity Nominal Actual

in. in. fps

1 1,05 18. 45

2 2. 07 28.5

3 3. 07 37.3

4 4.03 44.3

6 6. 07 57.2

10 I

I I I I I I I I I I I I I

1 1.5 2 3 4 5 6 7 8 9 10 DIAMETER OF PIPE , INCHES Figure 4. Terminal velocities for flow in vertical pipes

flowing full under the force of gravitation.

Hunter [2] also conducted tests on terminal

velocities in 2- and 3-in. cast-iron stacks flowing

partly full. He used a 3-in. stack 45 ft in height,

open top and bottom, and introduced the water

through a 45° double Y fitting at the top of the

stack. The velocities were measured with a pitot

tube at the bottom of the stack. Various heights

of fall from 5 to 45 ft were used. From these

tests he found the terminal velocity of flow in the

3-in. stack to be about 16.8 fps, attained in a

height of about 15 ft, for a flow of 100 gpm; and

32.4 fps, attained in a height of about 75 ft for a flow of 200 gpm.

He also made tests on a 2-in. stack, obtaining

the following results: for a flow of 45 gpm, a

terminal velocity of 18.5 fps, attained in a fall of

about 20 ft; and for a flow of 90 gpm, a terminal

velocity of 24 fps, attained in a fall of about 20 ft. Hunter remarks that, with flows of 90 gpm in

the 2-in. stack and 200 gpm in the 3-in. stack,

slugs of watej- completely filling a short length of

the pipe occasionally formed, and the maximimi

velocity of these slugs approached the maximum

velocity for a completely filled stack.

5.2. Tests at the State University of Iowa

Dawson and Kalinske conducted experiments on

flow in partly filled vertical pipes in order to determine terminal velocities for dift'erent conditions [5]. They tested 3-, 4-, and 6-in. installations with the water entering the stacks through

a single stack fitting at the top. They do not

state of what material the stacks were made or what the building drain lengths were. The stacks were provided with vent pipes at two levels and

discharged through a 90° bend into a short horizontal drain. However, it is not known whether

the vents were open or closed while taking velocity

measurements. The height of the stacks was

about 30 ft in all cases. Using a pitot tube,

they measured the average velocities correspond9
ing to different heights of fall and rates of flow.

Table 6 gives the values of the terminal velocities

which they obtained.

Table 6. Terminal velocities offlow in vertical pipes flowing

partly full (Dawson and Kalinske)

Diameter

of stack Flow Flow Terminal

velocity

Thickness

of sheet (computed)

in.33

3

3

npm

45

90

135 ISO

c/s

0. 1003 .2005

.3008

.4010

fps

11. 5 15.4

17.6 19.5

in. 0.135

.215

.290

..355

4

4

4

4

4

90

135

190

240 300

.2005

.3008

.4235

.5345

.6684

16.8

19.0 20.4

21. 2 22.0

.138

.185 .250

.315

.385

6

fi6

6

6

6

6

115

165 220 345

450

560 675

.256

.367 .490

.754

1.003

1.248

1. 505

12.3

14.4

17.4

21.7 25.0

28. 5

27. 5

.160 .200

.220

.275

..320 .380

.445

From the terminal velocities given in the table,

the cross-sectional areas, Oi, of the sheets of water were computed from the known values of the flows. The thickness, T, of the sheet in each case was then

computed from the equation

ai = T{di—T)T, (2)

in which di is the diameter of the stack, and the

values thus obtained are also given in the table. The same computations were made from

Hunter's data, and the results are given in table 7.

Table 7. Terminal velocities offlow in vertical pipes flowing

partly full

(Hunter)

Diameter of stack Flow Flow Terminal

velocity

Thickness of sheet computed

in. spm cfs fps in. 3 100 0. 2225 16.8 0. 223

3 200 .450 32.4 .235

2 45 .1003 18. 5 . 130 2 90 .2005 24.0 .220

; A tentative plot was made of these data with

terminal velocities plotted against thiclmess of the

sheet of water, but the results were rather disappointing. For each diameter of stack, there is a steady increase in terminal velocity as the thick- ness of the sheet increases, and the data for different diameters of stack yield separate curves, but the curves do not fall in order of increasing

diameters. Hunter's observation for the 3-in. stack with a flow 100 gpm agrees well with Dawson

and Kalinske's observation for the 3-in. stack, but

Hunter's observation for the 3-in. stack with a flow

of 200 gpm lies far above the terminal velocities

T

1 1 1 1 1 1 1 1 1 1 1 r

I I I I I I I I I I 1 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Q| (gpm) / d| (INCHES) Figure 5. Terminal velocities for flow in partially filled vertical pipes.

o Dawson. _ _ 3-inch stack. V do 4-inch stack. A do 6-inch stack. Hunter. _._ 2-ineh stack. • do.- 3-inch stack.

obtained in any of the other tests. It was observed

in connection with the last two measurements

mentioned that the thickness of the sheet is nearly

the same in the two cases, although the flow is double in one case what it is in the other. This

fact throws doubt on these two measurements.

The data given in tables 6 and 7 are plotted in

figure 5. Dawson and Kalinske [5] developed equations

for terminal velocity, starting with the equation

ma=mg~kirdi ( Ai) wv^, (3)

wheremag--

V-

^L--

wk--

-the mass of water =^{AL) Aw/g

- the acceleration of the water at any point

: the acceleration of gravity

^the diameter of the stack

- the average velocity of the water at any

cross section

= the length of the slug of water under

consideration

= the cross-sectional area of the slug of water

:the specific weight of the water

^a dimensional constant.

Although their expression for the friction term

involves a dimensional constant, whereas the con10
slant could have been made dimensionless by

writing the density, p, of the water in place of the

specific weight, w, their resulting expressions for terminal velocity and terminal length appear to be correctly derived under the simplifying as- sumptions made.

Dawson and Kalinske derive the following ex- pression for terminal velocity, which is expressed

here in the notation of the present paper for simplicity of comparison:

v,= iQ,g/M^y'' (4)

5.3. Derivation of Equation for Terminal

Velocities

The following derivation of the equation for terminal velocities is an approximate one. In it the sheet of water is treated as if it were a rigid

body instead of a fluid sheet with a radial velocity

gradient. The more accurate solution would in- volve a computation of the turbulent boundary

layer down the pipe, but, in view of the extremely

complicated equations that would be involved,

this would not be warranted in connection with our

present problem.

We consider the sheet of water as if it were a

rigid body moving down a plane vertical wall,

acted on by the oppositely directed forces of gravitation and friction. We assume that the sheet

starts with an initial vertically directed velocity

downward having the magnitude Vq, where t'o is less than the terminal velocity, Vt, which the sheet

will attain ultimately if the length of fall is great

enough. This is a further simplification as the

water actually flows down the interior surface of a pipe. As the velocity of fall increases with z, where z denotes the distance measured down the

stack from the point of entry, the thickness of the

sheet decreases with z, becoming constant when z attains the value of Z(, the terminal length. Other

symbols that will be required are: di=tiie diameter of the stack

(?i =the volume rate of flow down the wall

y=the velocity of fall corresponding to any

given distance of fall, z p=the density of the water

^=the acceleration of gravity

AZ=the length of an elementary volume of

water

T=the thickness of the sheet corresponding

to any given value of v f=time

X=a dimensionless constant

We start with the fact that the accelerating

force is equal to the gravitational force less the

frictional resistance.

m ^=mg— ToTrdiAL, (5)

where tq is the wall shear per unit area, and the

elementary volume of water is assumed to be -irdi

units wide. But

m = p QiAt, so pQi-^ M= pQi gAt — T „7rr/

,

AZ .

We now define tq by the relation,

ro=^ V\ (6)

and replace AZ by its value, vM. Equation 5 then becomes

pQ, ^

M=pQ,GM-'^ d.v'M,

or dv irX , o -dt = ^J-2Q,'^^'-

We now obtain the expression for the terminal

velocity, Vt, by placing dv/dt equal to zero:

It can be shown easily that X is equal to //4, where / is the dimensionless friction coefficient in the Darcy-Weisbach formula for pipe flow. It is a function of the Reynolds number, Re=TvJv,

approximately, and of the roughness ratio, ks/R

or ks/ T, where v is the kinematic viscosity of water,

and ks is the average height of a roughness element.

The above expression for the terminal velocity

contains the friction coefficient X, and it will be

convenient to modify the expression to eliminate

this. To do this we use the following equation

given by Keulegan for flow in open channels, the

equation being based on the Manning formula [9]:

where v is the mean velocity in the cross section, R

is the hydraulic radius, and v-t: is the shear velocity,

defined by the relation,

t^*-=VWp, (10)

and ks is the sand roughness, which may be'looked on as representing the distances that the sand

grains would project from a surface having the same frictional resistance as the sm-face being

considered.

We note that the U in eq 9 is the same as the v in the other equations, and we eliminate tq

between eq 6 and 9, and then between eq 9

and 10, obtaining finally

/k \^'^ X = 0.0303 (^^ V , (11)

11
if we replace R by T, as we can do approximately.

Substituting this value of X in eq 8, we have

Tt being written for T to indicate that the value

of Tis that corresponding to the terminal velocity.

One further change will be made in the expression for terminal velocit}^ by eliminating the

thickness Ti of the sheet of water b}'' means of

the relation between the thickness of the sheet,

the cross-sectional area of the sheet, and the rate

of flow. This relation is given by the approximate expression

Q, =Trd,T^V„ (13)

or by the more accurate expression

Q, =ir{d,~T,)T,v,. (14)

If we use the approximate expression in eq 13,

there results:

or

The phenomena of flow down a vertical stack

is independent of the Reynolds number for two

reasons. First, because of the fact that the water

is falling rapidly under the influence of gravitation, the velocity is nowhere small enough, except

possibly for an extremely short distance at the

point of entrance, for the fluid friction to play an

important part. Second, as the Reynolds number

for the flow can be written as Tvjv, and by the

approximate equation, eq 13, we see that for a given flow, T is inversely proportional to v, then

the Reynolds number for the given flow down the

stack will be constant as long as there are no

additions to the flow as it passes down the stack.

Thus the flow phenomena can be considered a

function of the relative roughness, but the Reynolds number can be neglected.

An appropriate value of ks has been determined

from data on friction losses for flow through new

cast-iron soil pipe tested by Hunter [10] and

yields the approximate value, 0.00083 ft. (See Appendix I for the derivation of this value.)

Rouse [11] gives the value, 0.00085 ft for new cast-iron pipe, so the agreement is good.

We can express eq 12 in more convenient form

for computation. We replace Tt by its approximate value from eq 13, take the value of as 0.00083 ft, and reduce the resulting equation,

obtaining (for cast-iron soil pipe)

Vt = l2M(^'\ (16a)

where Vi is expressed in feet per second, is in cubic feet per second, and di is in feet. If we wish to express Qi in gallons per minute

and di in inches, we write (for cast-iron soil pipe)

eq 16a as

Equation 16b is plotted in figure 5, together

with the experimental data obtained by Hunter

and by Dawson and Kalinske. An average cm-ve having the equation

.,=4.3 (ly, (17)

is drawn through the experimental points.

As would be expected, the experimental points

scatter considerably, for the terminal velocity is a quantity that is very difficult to measure. The

fact that stands out strikingly is that all the

measurements of the experimenters referred to above lie above the curve that has been established

from resistance measurements on cast-iron soil pipe. There are certain reasons why this tendency

might exist, the most probable one relating to the

velocity distribution in the sheet of water. A

very steep velocity gradient exists in the sheet

of water, increasing from zero at the wall to a maximum at the inner surface of the sheet. The

experimenters made their observations with a

pitot tube with its dynamic opening pointing

vertically upward through the vertical sheet of water. It was impossible to get the pitot tube

close to the wall where the lowest velocities existed,

and hence, in averaging their measured velocities,

they ma}^ not have taken into account adequately

the lower velocities near the wall. A conversation with Professor Dawson bore

out the above supposition. He stated that his

measurements were made in the outer layers of

the sheet and hence did not represent a true aver- age of the velocities in the sheet. The difficulties involved in measuring the true terminal velocity

are very great, and too great accuracy must not

be expected.

One other point should be mentioned here.

Experiments made by one of the authors since

the earlier work reported in this paper have shown

that there is considerable variation in air content

radially in the sheet of water. At the wall the

sheet is made up entirely of water. As we pass

inward toward the water surface in the pipe, the

air content in the sheet increases, until near its inner edge it is roughly 50 percent air by volume.

This would have a considerable efl^ect on pitot

tube readings made in the sheet, but the tendency

wordd be to decrease the measured vertical velocities because of the reduction in the effective

density of the mixture.

It seems probable that the computed curve

expressed by eq 16b is more reliable than are the

12
measurements of terminal velocity. Fxirtlier evidence of this was obtained by one of the authors

in a series of careful measurements of terminal

velocity. With the above undei'standing of the

conditions in the sheet of water, he was able to get measurements approximating the computed

curve in figure 5.

We can determine tlie variation in thickness of

the sheet of water as it flows down tlie wall for any rate of flow, diameter of stack, entrance

velocity, and heiglit of fall by integrating eq 20

for values of v ranging from zero to v,,. Practically,

it is easier and sufficiently accurate to integrate

the equation graphically. To do this, we write

6. Terminal Lengths in Stacks

To derive the equation for terminal lengths in

sheet flow down the walls of vertical stacks, we

return to eq 7, make the necessary transformations, and integrate the result to obtain the ex- pression for the terminal length. We first note

that

dv dv dz

dz dt

dv

dz (18)

Making this substitution in eq 7 and solving for

dz, we have

dz =—

g

vdv

^ TT \ di „

1 11 (19)

and assuming that X is not a function of v, we

have from eq 8

^^_v^t {vlv,)d{vlvt) _v^, Odd

1 i-{viv,y -Ji-e^' ^^^^

where 6=v/vt.

We can integrate eq 20 directly, obtaining

Li = z 0 3 g ^log.

1 +

tan (21)

We note that if we substitute the upper limit,

v/vt—1, we obtain an infinite result as the denominator of the first term in the parenthesis becomes

zero. We avoid this difficulty by assuming that,

when the velocity has reached a value equal to 0.99 times the true terminal velocity, the terminal

length has been attained for all practical purposes.

Making this substitution for v/vt in eq 21, and

inserting the limits, we obtain

X,=||[ilog, 29,700-tan-^ 2.98-^ log, 1 +

tan-' ij, or

Z,=0.052;>/. (22)

(23)

where 6 represents the ratio v/vt, and we plot

values of 6/(1-6^) against 6 over the range,

^=0 to 6=1. By measuring the areas under the curve from ^=0 to any other desired value, we

obtain the corresponding values of

J* I'^'gs' wliich is equal to J^^>

and by multiplying these values by vf/'g, we

obtain the distance through which the water had

to fall to acquire this mean velocity, v. However, our principal interest is to obtain a knowledge of how far a sheet of water must fall before it attains a velocity appreciably equal to the terminal velocity. Tliis can be com.puted

from eq 21, placing v equal to 0.99i'/, or it can be

read from figure 6, which is a plot of eq 22 for a new cast-iron stack. The necessity of having

a knowledge of terminal lengths will appear later in tliis paper when a m.ethod of computing

backpressures on horizontal branches at inter0 5 10 15 20 25 30 TERMINAL VELOCITY, fps

Figure 6. Terminal lengths for flow in partiaUy filled vertical pipes.

Computf d from the equatioQ £(=0.052(5-

13
mediate floor levels will be given. Our knowledge

of terminal lengths indicates that in practically

all cases the sheet of water in falling through the

height of one story, or a little more, attains

practically its term.inal velocity, wliich simplifies

the application of the theory presented. Consequently, the exact value of the numerical

coefficient in eq 22 is not of great importance, and

the equation will be used onh^ to give approximate

values of the terminal lengths.

If we com.pare eq 22 with the equation of free

fall under the influence of gravity, we find that

the sheet of water in the pipe m^ust fall approximately three tim.es as far to attain a given velocity

of fall as a body falling freely under the influence

of gravity.

7. Tests on Stacks and Drains

7.1. Nature of the Tests

Because the suggested definition of stack

capacity in section 4.2 is based on hydrostatic

pressures in the horizontal branches caused by

conditions in the stack, it appeared advisable to study first the conditions existing at the junction of one or two horizontal branches connecting into the

stack at a given level with water flowing down

the stack from, higher floors. It was planned

to try to relate the conditions of flow in the

stack with the flow conditions in the drain, pres- sures in the stack, and pressures in the drain.

For this purpose a 3-in. stack and drains of Iji-, 2-, and 3-in. diameter were selected for the first tests. Double sanitary tee stack fittings were

used in the first phase of the investigation.

Drawings of these fittings are shown in figures

7, 8, and 9. Id order to clarify the terms applied to the

drainage fittings used at the junction of the stack

and the horizontal branches, a word of explanation

is in order. The sanitary tee fittings shown in

figures 7, 8, and 9 are also known as short-turn

T-Y and 90° short Y -branch fittings. A sanitary

wye fitting is also known as a long-turn T-Y or a 90° long Y-branch fitting, and if the fitting is a caulked soil-pipe fitting, it may be known as a combination Y and bend.

Figure 8. Drainage fittings used in stack tests.

3 X 3 X 2 X 2-inch doubli' sanitaiy tee.

Figure 7. Drainage fittings used in stack tests.

3 X 3 X Hi X 1 }2-iiich double sanitary tee.

Figure 9. Drainage fittings used in stack tests.

3 X 3 X 3 X 3-inch double sanitary tee.

7.2. Simplified Setup for Simulating Conditions at the Junction of One or Two

Horizontal Branches and a Vertical Stack

In planning the tests the idea was conceived

of eliminating a high stack and attaining a higher

degree of accuracy for the first tests by substituting the equipment shown in figures 10, 11,

and 12. Figure 12 shows the equipment with a

transparent-plastic stack fitting and stack substituted for the conventional m.etal fitting and

stack to permit the visual observation of the

flow conditions. The test results, however, are based entirely on the results obtained with the

usual metal stack and fittings.

In the equipment described, the sheet of water

fiowing dowii the stack is produced at the level of the stack fitting by installing a second piece

of pipe smaller in diameter than the stack in the

stack just above the stack fitting through wliich

the horizontal branches discharge into the stack.

As can be seen from figures 11 and 12, the stack

14
terminated in a closed chamber or drum supplied

with water around its circumference through six pipes 1 in. in diameter. This drum then dis- charged water through the annular space between

the stack wall and the inserted pipe, thus forming

the sheet of water representing the sheet that

would under service conditions be flowing down

the stack from_ higher floors. By using difl'erent diameters for the inserted pipe, the thickness of the sheet of water could be varied, and by changing the pressure in the drum, the velocity of flow

in the sheet could be varied. Thus the equipment

afforded a means of obtaining much m^ore flex- ibility in the experim.ents than could be obtained

with an actual stack.

There is, however, one definite difference

between the conditions in the sheet produced

artificially and those in the sheet flowing down

an actual stack. As has already been pointed

out, in an actual stack there will be a very marked

velocity gradient at any horizontal section of the

sheet, the velocity increasing from zero at the

wall to a maximum^ value at the inner surface of

the sheet. This is not true in the artificially

produced sheet of water, in which the velocity

distribution is probably nearly uniform. This

fact will very possibly cause a slight difference

in the results obtained in the two cases. Hence

certain conclusions based on results obtained

with the special setup, were later checked by

making tests on an actual stack. Nevertheless,

it was much simpler to experiment with the

special setup and the range of conditions that

could be tested was much greater than with an

actual stack. Furthermore, the difference in

velocity distributions in the two cases was believed

to be a minor consideration.

WATER INLET

ANNULAR SPACE

STACK FITTING 'A'

l-\ li PLASTIC

/ TUBING

MANOMETER

CONNECTIONS

Figure 11. Details of apparatus for simulating flow conditions in stack.

AUXILIARY

TO ORIFICE METER

Figure 10. Outline drawing of apparatus for simulating

flow conditions in stack. Figure 12. Apparatus for simulating floio conditions -in stack.

15
8. Analysis of Conditions at the Stack

Fitting Where a Horizontal Drain

Empties Into a Stack at an Intermediate Floor of a Multistory Building

8.1. Statement of the Problem

Hunter defined the capacity oj a stack fitting as the rate of flow in gallons per minute at which

water flowing out of the horizontal drain into the

stack just begins to build up in the vertical part

of the fitting above the inlet branch of the fitting.

This definition seems to apply only to the highest

horizontal branch on the stack where no water

would be coming down the stack from a higher

level. Obviously, such a building-up of water

would create a backpressure on the drain and

would retard the flow from the drain into the stack.

However, the conditions might be quite dift'er- ent at an intermediate floor of a multistory building. Here the flow from the drain may have to mix with or deflect a high-velocity sheet of water

flowing down the stack from the higher floors in

order to get into the stack. The act of mixing

with the vertically flowing sheet of water or deflecting it will necessarily create a backpressure on

the drain whether or not the water builds up in

the stack above the level of the drain. This backpressure will be very considerable if the flow down

the stack at the level of the horizontal drain under

consideration is large and if it has, as it usually

would, a relatively high velocity. This would restrict the rate of flow out of the drain and would

tend to back up the flow in the fixture drain

unduly, unless this fact is taken into consideration

in the design of the stack and drain.

The tests carried out in the basic investigation

with the special equipment described earlier in

this paper constitute the first step in the attempt

to furnish the necessary data to make an adequate

investigation of this problem and hence to furnish

the necessary means for designing a stack and

horizontal branches. As has been pointed out al- ready, the tests reported in this paper were made

only with sanitary tee stack fittings for a 3-in. stack,

and it appeared desirable to carry out a similar

investigation with long-turn T-Y stack fittings, as

these are frequently used in the construction of plumbing systems. The tests with these latter

fittings have already been made. Although the

tests with the long-turn T-Y fittings gave some- what different backpressures than did those with

the sanitary tee fittings, the results with the two

types of fittings are consistent with each other.

8.2. Analysis of the Problem

The problem will be investigated for the some- what idealized conditions shown in figure 13. The

drain is assumed to connect straight into the stack

instead of through a sanitary tee, as it did in the

Figure 13. Diagrammatic representation of flow from

horizontal branch deflecting sheet flow down stack.

tests. The analvsis can be made equall}- well for the actual condition, but it leads to a somewhat

more complicated equation, and it was found that

any effect of simplifying the conditions merely

influenced the empirical coefficient in the equation.

Water is assumed to be flowing down the stack

from higher floors in a sheet which, as it approaches

the level of the drain under consideration, has the

thickness T and has a mean velocity of flow, Vi. The pneumatic pressure in the core of air at the

center of the stack is 2)i, and in most of the tests

it was fairly constant and slightly less than at- mospheric pressure at a point just above the level of the horizontal branch. Water is assumed to be

flowing out of the branch at the volume rate Q2,

with the mean velocity V2, it being assumed

throughout the analysis that the branch is flowing

full near its entrance into the stack, as it did in the tests. At a distance L along the branch from

the stack the head /12 corresponding to the pressure

P2 is measured.

Although it is assumed that the velocitj^ in a

horizontal section of the vertical sheet is uniform,

this is far from being the case, as has already

been pointed out in this paper. Actually, a very

steep velocity gradient exists, the velocitj^ being

zero at the wall and very high in the outer layer.

However, this simplifying assumption will be

taken into accoimt automatically in the determination of the empirical constants in the equation,

and hence the actual velocity distribution will be

neglected. The actual conditions in the vertical

sheet before it comes in contact with the flow

from the branch can be evaluated by investigating the formation of the turbulent boundary layer

in the sheet as it flows downward. The equations

16
are very complicated, however, and their solution

would be ver}^ time consuming, so that the increased accuracy resulting from their use would probably not be worth the effort of solving them.

The theoretical analysis of the problem will be

based on the equation of momentum, considering

the horizontal component of the momentum of the

water flowing out of the branch. The simplest

approach is to consider the slug of water between

a vertical section through the point at which the

pressure is measured and the vertical section

coinciding with the end of the branch where it enters the stack. The change in momentum of

the water in flowing from the upstream section to the downstream section is equal to the sum of

the external forces acting axially on the slug of water. This will be put in mathematical form as

follows:

The flux of momentum across the upstream

section is equal to PQ2V2, if we assume uniform

velocity distribution. As the cross section of the

flow is constant between these two sections (we assume that the drain is flowing full), and there

is no change in the quantity of water flowing as it passes between the two sections, the flux of momentum across the downstream section is also

PQ2V2, again assuming uniform velocity distribution.

The external forces acting on the slug of water

are (fig. 14):

1. The force due to 1)2, tending to produce flow

out of the branch.

2. The force due to pi, tending to oppose flow

out of the branch.

3. The force due to pa, the backpressure caused

by the deflection of the sheet of water flowing

down the stack by the flow out of the branch, and

4. The frictional resistance in the branch

between the two sections, tending to oppose flow

out of the branch.

It is assumed that the branch is horizontal,

instead of slightly sloping as it would be actually, so that gravitational forces do not have to be

considered. Furthermore, if the velocity distribution changes between the two sections considered,

there would be a slight change in the flux of momentmn between the sections, but this will be

neglected here.

Summarizing, we have the force 2>202, tending

to force the water into the stack. Opposing this

we have the pressure forces, pia2 and paC-i, together

with the friction force, F=Trd2LTo, where tq is the

frictional force per imit area exerted by the walls

of the pipe on the slug of water opposing the flow

into the stack. Hence

F

F

L Figure 14. External forces acting on slug of water in branch.

PQ2V2 —P<'M = (pi+Pci)a2 + W^Lto— =0, or P2a2={pi+Pa)a2 + Trd2LTo. (24)

The friction force can be represented as a backpressure, Pf, opposing the flow of water out

of the branch. Thus

Pfa2=pghfa2, (25)

and, using the Darcy-Weisbach equation for flow

in pipes, we obtain

L

p^^=fa^pg — -l, (26)

where / is the Darcy-Weisbach dunensionless

friction coefficient. Keplacing also the other

pressures by their equivalent expression, density

times acceleration of gravity times height of water

column corresponding to the pressure concerned,

we obtain

fe=A,+A.+/||f, (27)

which could equally well have been developed from

the energy equation, assuming uniform velocity

distribution.

All the quantities in eq 27 are known or can be

m_easured except the dynamjc head, ha. The in- teraction between the vertical sheet of water flow

ing in the stack with the flow from the branch is too com.plicated to analyze, except in a very general and approximate way. In m.aking this analj'--

sis, we assum.e that the flow lines in the vertical

sheet of water are changed from straight lines to arcs of circles of radius r by the stream from the

branch (flg. 15). Obviously tlris is a sim.phfnng

assumption that probably does not fit the facts exactly, yet it yields results that bring order out of

FiGUBE 15. Evaluation of backpressure on branch.

17
Figure 16. Diagram for evaluating centrifugal force acting on an elementary mass of vjater moving in a circular path.

the experimental data and hence may be considered justified by that fact. Under this assumption, we can write the centripetal force, Fa, which the flow from the branch

exerts on the vertical sheet as (fig. 16):

pTAsAbvl

(28)

where m is the mass of the elementary volume

A,^A6^. Then

Pd-- (29)

if we assum_e an average value of r for all elementary volumes. It follows then that

rg

(30)

and, substituting this result in eq 27, we have

Dividing by hi —hi, and rearranging:

d2 2g{h,-h,y

1- 2T

r 2g{h2 —hi)

(31)

(32)

The quantities in eq 32 are all capable of determination by measurement or otherwise, except

the radius of curvature of the flow lines. We proceed to consider the quantity r in more detail. Other tilings being equal, it may be assumed that

r is directly proportional to the momentum, pQiVi,

of the sheet of w^ater flowing clown the stack.

Likemse, other tilings being equal, it may be as- sumed that r is inversely proportional to the m.o- meiitum, pQ-nk, of the flow out of the branch. The

validity of these and other assumptions made in

evaluating r will be determined by the success that we have in fitting the final equation to the experimental data. In view of the above assumptions, we can write

=functionC^^\

But r m.ust be placed equal to a length tim_es this

function if our equation is to be dim.ensionally correct, and the dim.ension that seem.s to be "most

closely involved with r is the diam_eter, c^s, of the

branch. Hence we can write

^ =^^^(4v)=^^^('-^> (33)

where Oi is the cross-sectional area of the sheet of

water in the stack at the level of the branch, and

Oo is the cross-sectional area of the flow in the

branch for the branch flowing full. Hence

the constant 4 being absorbed in the coefficient C. We go one step further and assume that the

function <^ can be taken as a sim_ple power. This was found to be approxim^ately true for the ratio

T/c?2, and hence it should be true for the other

two ratios. Thus

r=Cd.

--Cd2

2/ \d2) \vl

dX / TV

d2) \dj

2g (hj—hi)

^1 ^2g{h2-hi)^

(35)

We now substitute the value of r given by eq 35

in eq 32, waiting for sim_plicity:

2g{h2-hi) =X, and 2g{h2-K

2T

Y:

d2 ^ ^ 'Cd2 /dX/TX /X

or

d

(11(0(1:

Here there are tw^o quantities, d and a, to be

determined from the experimental data. It is in- teresting to note that, before the relation between

the exponents of X and Y in the right member of

eq 36 had been demonstrated by analysis, it had

been found from a study of the experimental data

that the exponent of Y was very close to 3/8, and

the exponent of X was very close to 5/8, thus

substantiating the relation between these ex- ponents given by eq 36.

18
Before proceeding to the evaluation of C, and

a, a further discussion regarding details that have

not been taken into account in deriving eq 36 will be given. Some of these have already been

pointed out, but we are now in a position to consider them m.ore definitely. It has been assumed that the horizontal branch

enters the stack wdthout tiu'ning downward

slightly, as a sanitary tee fitting actually does.

In practice, either a sanitary tee or a long-turn

T-Y is used as the stack fitting (see fig. 7, 8, and

9 for drawings of sanitary tee fittings). Either

t,ype of fitting turns the water downward some- what at the entrance to the stack, the long-turn

fitting somewhat m^ore than the sanitar}" tee. Furthermore, horizontal branches are actually

laid on a slight slope instead of being level, as was assum.ed in developing eq 36. If these facts are taken into account, tlu-ee m.ore term.s Avill have

to be added to eq 36. In the first place, the

horizontal component of the momentum, will not

cancel out for the two sections, as the inward flux

of m.omentum at the upstream section will be

p 02?2, whereas at the downstream, section the

outward flux of mom.entum ^\dll be p 0-2^2 cos /3, (3 being the angle through wliich the stack fitting deflects the water downward. In addition, there

"will be a gravitational force tending to produce

flow out of the branch, this force being due to the

additional head between the center line of the

branch and the center of the section of the branch

opening into the stack. Like\vise, if the branch were laid on a slope instead of being horizontal as assum.ed, the effect of this slope would be taken

into account through the addition of a gravitational force just as has been described for the

stack fitting. A further loss due to the defiection of the stream, of water by the stack fitting is also involved.

Although all these factors would have to be

considered in a complete analysis of the problem,

it is believed that for our present purpose their

effect will be taken care of adequately in the values

determined for the empirical constants that will be found from the experimental data. Hence they

will not be considered further at this time.

As a preliminary step in determining the average

value of the coefficient Ci and the exponent a in eq 36, the value of the Darcy-Weisbach / was

taken as 0.03. Inserting this value of / in eq 36,

setting Ljdo equal to 4, as it was in the tests, l-0.12r

{Tld,)X

"

The experimental data afforded the necessary

information for computing X, Y, and T/d2, and

the value of a was then computed for each test run from the first member of the equation. The

next step was to determine the value of the ex- ponent a of the quantity, Y/X. Values of a were plotted against YjX on logaritlmiic paper,

and the resulting series of straight lines had so

nearly a slope of 3/8 that this value was assumed

for convenience.

The next step was to ch'termine the value of the

exponent a of the quantity, (f/2AA)(''4/7')- Although theory indicated that the three ratios in the right member of the above equation had the

common exponent a, it was decided to check this

from the data. It was not feasible to separate

the ratios, d^/d^ and dilT in this procedure, so it was assumed that both had the same exponent.

Wlien values of a were plotted against (f^AA)

(di/T), another series of straight lines with a

slope very closely equal to 3/8 again resulted.

Hence it was concluded that the three ratios all had the same exponent, 3/8.

Values of Ci were then computed for each run from

aseparately for the runs with double-branch flow

and for the runs with single-branch flow. The

average value of Ci for double-branch flow was

a little less than 1.78, and for single-branch flow

it was about 1.80, showing that, except for two

or thi'ee runs with large flows from both branches,

in which the capacity of the fitting seems to have

been exceeded, the same value of Ci applies

approximately, whether the flow is from one or from both branches. The value 1.78 was adopted

for Ci. The average deviation of d from the average

value of 1.78 was found to be 0.24 for the doublebranch flow and 0.25 for the single-branch flow.

The probable error in Ci was computed from the

formula:

Probable error =0.84

average deviation

. ? Vnumber of observations

and was found to be 0.03. Ci can be given as Thus the value of

Ci = 1.78±0.03.

The formula for the probable error assumes

that the deviations are wholly due to accidental

errors, and hence it does not apply strictly to the

data in question. Nevertheless, it seemed desu-- able to get at least a rough idea of the closeness

with which the average value of 1.78 represents the

results obtained from tests made under a widely

varying set of conditions.

Using the values of Ci and a thus determined, we write eq 36 in the form

'TV /doV^'

or

0.12r=l-1.78 (jj' ' ^3/s^Y=/^ (3 7)

r= 8.33 -14.8 (j^" (jf^'"

Y'/'X^'^ (38)

19
Equation 38 can be solved for corresponding

values of X and Y for any selected values of

{Tld.yi^{d2ld,yi\ This has "been done, and the

resulting curves are plotted in figure 17, together

with the experimental data, showing how well the

data fit the curves resulting from the analysis

of the problem. The experimental points fit quite well except for two of the intermediate

curves, the points generally lying too low for one

curve and too high for the other. In the runs

represented by these points, it was noted several

times that peculiar pressure conditions existed

in the stack in the vicinity of the branch opening.

It must be pointed out here that, theoretically

there are certain restrictions on the applicability

of eq 38. In the first place, it applies only to stacks in which it can be assumed that the water

flows in an annular sheet on the wall of the stack

at the level of the horizontal branch under consideration, without an appreciable content of entrained air. Although it is believed that these

idealized conditions are not entirely realized in an

actual stack, our information regarding the actual

conditions is limited, and consequently we shall assume that eq 38 applies to the actual conditions.

Second, eq 38 applies only when sanitary tee

fittings are used to connect the horizontal branches

to the stack and only when the head, /i2, exists at a distance of four pipe diameters upstream from

the branch outlet. The horizontal branches are assumed to have a roughness expressed by a value

of the Darcy-Weisbach / of 0.03. However, as

in eq 38 the friction head loss is generally small in comparison to the dynamic backpressure, it is unhkely that under most conditions small variations in L and / would have much efi^ect on the

results. Finally, the data on which eq 38 is based were

obtained for a 3-in. stack only. However, the

equation should be applicable to stacks of other

diameters through the ratio dildi.

9. Summary

The special equipment described in section 7.2 and shown in figures 11 and 12 was used in the

tests described earlier, using double sanitary tee drainage fittings. The tests were made with 1}^-, 2-, and 3-in. -diameter horizontal branches, and

with thiclmesses of the annular sheet of water in the stack of 0.16, 0.34, and 0.58 in. These

represent nine dift'erent conditions for which the

pressures and velocities (or rates of flow) were measured in the stack and the branches.

Preliminary cm-ves were prepared for each rate

of flow and thickness of water sheet in the stack,

showing the relationship between pressure head

and rate of flow in the branch. In order to obtain

from these curves the data shown in tables 8 and

9, several values of the ratio of the pressure drop

tln-ough the drain to the drain diameter, Qh—hi)!

dz, were selected. Corresponding flow rates (or

velocities) and pressure heads in the branches for

each selected value of {h2~hi)/d2 were obtained

by interpolation from each of the preliminary

cm-ves. The data for tables 8 and 9 could also have been obtained by making the computationF

for each separate run instead of interpolating fron.

the curves, but this would have been more

laborious and would have made only an insignificant difference in the results. The data from tests with the approximately

equal rates of flow from both branches are shown

in table 8. Flows from one branch only are given

in table 9. Figure 17 shows a comparison of the

experimental data in tables 8 and 9 with the series of curves representing eq 38. It is obvious that the data from the experiments

with single-branch flow appear to fit the computed

curves as well as do the data from the doublebranch flow for all except one of the curves. The

reason for this exception has not been determined.

As eq 38 does not take into account variations in

the pipe-friction coefficient with variations in the

Reynolds number, and as other minor factors are neglected for the sake of simplicity in the development of this relation, these things may possibly

account for some of the inconsistencies of the plot.

In addition, certain difficulties in test procedure

and measurements of the fiuctuating pressures may

have caused some irregularities. But, on the other

hand, it will be noted that the fit is good for the

two extreme curves and does not seem to deviate

in an}" consistent manner from the intermediate

curves. Generally the agreement of the data with

the computed curves is fairly good if we consider

Table 8. Data from tests with double sanitary tee drainage

fittings, 3-in. stack, flow froin both branches

Flow

in each branch

Stack

flow T hi—hi Vl V2 Symbol

2g{hi-h) 2gQii-hi) in fig. 17

gpm gpm in. in. ft- fps fps

5.0 98 0. 58 1.62 0.094 6. 92 0. 78 7. 85 0. 100

1.8 158 . 58 1. 62 .094 11. 2 .28 21.0 .013 7.4 98 . 58 1. 62 .135 6. 92 1. 15 5, 5 . 162 3.4 158 . 58 1. 62 . 135 11.2 0. 53 14,6 .032

1.8 203 . 58 1. 62 . 136 14.3 .28 23, 7 .010

12. 2 98 . 58 1. 62 .203 6. 92 1. 89 .3. 67 .273

6.0 158 . 58 1. 62 .203 11. 2 0.936 9.65 .067

.3.8 203 . .58 1. 62 .203 14,3 .59 15.8 .027

11.2 158 . 58 1. 62 .3.37 11.2 1.74 6.80 . 139 7.6 203 . 58 1. 62 .337 14.3 1. 18 9.95 .064

5.

1

252 .58 1.62 .3.37 17. 7 0.79 14.7 .029

7.8 56 .34 1.62 .094 6.2 1. 22 6, 30 .245 11.4 56 .34 1.62 . 135 6.2 1. 78 4, 42 ..366 5.4 98 .34 1. 62 .135 10.9 0. 84 13.6 .081

18. 1 56 . 34 1. 62 . 203 6.2 2. 82 2.94 .600 8.4 98 . 34 1. 62 .203 10.9 1,31 9. 05 .131 14.8 98 .34 1.62 .337 10.9 2. 30 5.40 .242

9.3 137 .34 1.62 .3.37 15.1 1,45 10.6 .096

7.8 40 .16 1. 62 .094 8. 76 1.22 12.6 .245

12.4 40 . 16' 1. 62 . 135 8. 75 1.93 8. 85 .428

12.6 80 . 16' 1. 62 .3.37 17.6 1,97 14.

1

.178

8.6 98 .16 1.62 .3.37 21.5 1,34 21.2 .083! 10.4 98 . 68 2. 06 . 120 6. 92 1,01 6.40 . 130 4.6 158 . .58 2. 06 , 120 11. 2 0, 446 16.3 .026

17.6 98 . 58 2. 06 . 172 6. 92 1.69 4. 32 .269

7.7 158 . 58 2. 06 . 172 11. 2 0, 744 11.4 .060

31.8 98 . 58 2. 06 .268 6. 92 3.31 2.88 .560 ' 13. 1 158 . 58 2. 06 .258 11.2 1,27 7.68 .096

7.8 203 .58 2. 06 .258 14.3 0. 754 12.4 .034

24.5 158 .58 2. 06 " .429 11.2 2. 37 4.56 .203

16.

1

203 .5812.06 .429 14.3 1,66 7. 46 . 145 10.6 252 .68 2. 06 .429 17.7 1.02 11.5 . 037

20
Table 8. Data from tests with double sanitary tee drainage

fittings, 3-in. stack, flow from both branches—Continued

Flow

in each branchStack

flow T d2 Vl Vl

1)2* Symbol

9/7f/)o —/ill in fig. 17 ^yv^i "'1/

gpm gpm in. in. ft. fps fps 32.4 56 -.34 2. 06 . 172 6. 20 3. 12 3. 47 .882

10. 2 98 . 34 2. 06 . 172 10. 9 0. 983 10. 7 . 087

5. 5 137 . 34 2. 06 . 172 15. 1 .533 20. 7 . 050

17.8 98 . 34 2. 06 . 258 10. 9 1. 72 7. 10 . 177 i 38.3 98 . 34 2. 06 . 429 10. 9 3. 69 4. 27 .493

19.

1

137 . 34 2. 06 . 429 15.

1

1.84 8. 30 . 123 12.3 180 .34 2. 06 .429 19.9 1.19 14.4 .051

15. 2 40 . 16 2. 06 . 120 8. 75 1.47 9.93 . 288

10. 2 54 . 16 2. 06 . 120 11.8 0. 982 18. 2 . 125 26.4 40 . 16 2. 06 . 172 8. 75 2. 55 6.91 .585

13.8 54 . 16 2. 06 . 172 11.8 1. 33 12. 7 . 160 24,8 54 . 16 2. 06 . 258 11. 8 2. 39 8. 93 .344

24. 4 80 . 16 2. 06 .429 17. 6 2. 36 11. 2 .201

16. 3 98 . 16 2. 06 . 429 21. 5 1.57 16.8 .090

61. 0 98 . 58 3. 06 . 179 6. 92 2. 71 4. 15 .635

26.9 158 . 58 3. 06 . 179 11. 2 1. 19 10.9 . 123

16.

1

203 . 58 3. 06 . 179 14.3 0.714 17.8 .032

7.

8

252 . 58 3. 06 . 179 17. 7 . 346 27. 9 . Oil 77.0 98 . 58 3. 06 . 255 6. 92 3. 42 2. 96 . 713 43.8 158 . 58 3. 06 . 255 11. 2 1.94 7. 66 .229 25.6 203 . 58,3. 06 . 255 14. 3 1. 14 12. 5 .078

15.5 252 . 58 3. 06 . 255 17. 7 0. 687 19.4 .029 65.4 158 . 58 3. 06 .383 11. 2 2. 91 6. 17 .343 44.6 203 . 68 3. 06 .383 14.3 1. 98 8. 37 . 159 28.3 252 . 58 3. 06 .383 17.7 1. 25 12.9 .064

, 44.8 98

j

. 34 3. 06 . 179 10. 9 1. 98 10. 2 . 352

27.0 137 . 34 3. 06 . 179 15. 1 1. 19 19. 8 . 124 69.2 98 . 34 3. 06 . 255 10. 9 3. 06 7. 18 . 670 44.6 137 . 34 3. 06 . 255 15.

1

1.97 13.9 . 238

25.8 180 . 34 3. 06 . 255 19. 9 1. 14 24.

1

.079

73. 8 1.37 . 34 3. 06 . 383 15.

1

3. 28 9. 31 . 437 45.0 180 . 34 3. 06 .383 19.9 1.99 16.1 .161

83. 0 180 . 34 3. 06 . 638 19.

9

3. 68 9. 66 . 331

24. 7 80 . 16 3. 06 . 179 17. 6 1.09 26.8 . 103 38.0 80 . 16 3. 06 . 255 17.6 1. 68 18.8 . 172 30.0 98 . 16 3. 06 .255 21. 5 1.33 28.

1

. 108

74. 1 % 0 80 . 16 3. 06 . 383 17. 6 3. 27 12. 5 . 435 43.5 98 .16,3. 06 . 383 21. 5 1. 92 18.8 . 150 90.0 98 .16 3. 06 . 638 21. 5 3. 96 11.3 .385

3.6 98 .34 1.62 .0945 10. 9 0. 662 19. 4 . 052 1 3. 1 137 ..34 1.62 . 135 15.

1

.482 26.3 .027

5. 1 137 .34 1.62 .203 15.

1

.799 17.6 .049

6. 1 180 .34 1.62 . 338 19.9 . 951 18. 2 .041

5. 8 54 .16 1.62 . 094.') 11.

8

. 902 23. 0 . 134

9. 2 54 .16 1.62 . 135 11.8 1. 43 16.

1

. 234 20.9 40 .16 1.62 . 203 8. 76 3. 26 6. 86 . 810

15.0 54 .16 1.62 . 203 11.8 2. 34 10 8 .419 <

7.0 80 .16 1.62 .203 17.6 1.09 23.7 .091

29. 4 54 . 16 1. 62 .338 11.8 4. 58 6. 40 .960

5.5 203 .58 2.06 . 172 14.3 0. 347 18. 6 .011

11.0 56 .34 2 06 . 120 6.20 1.06 4.98 . 146

1 5.7 98 .34 2. 06 . 120 10.9 0. 550 15.7 .039

9.9 137 .34 2. 06 .258 16.

1

6.1 180 .34 2. 06 .258 19.9 .956

.589 23.8

13.8 .056* .021 h 6.3 80 .16 2.06 . 172 17.6 .610 27.8 .034'

11. 2 80 .16 2.06 .258 17.6 1.08 18.6 .071'

1 -

7.3 98 . 16 2. 06 .258 21.5 0.705 27.8 .030

Table 9. Data from tests with double sanitary tee drainage

fittings, 3-in. stack, one-branch flow

Flow

in Stack T /l2 —Al Vl V2 Symbol

each in flg. 17 branchflow 2g{h2-hi) 2g(h2-hi)

gpm gpm in. in. ft- fps fps

5. 40 98 0. 58 1. 62 0. 094 6. 92 0. 841 7. 85 0.116 2.70 168 .58 1.62 .094 11.2 .419 21.

1

.029

8.40 98 .58 1.62 . 135 6. 92 1.32 5.50 . 197

4. 40 158 .58 1. 62 . 135 11.2 0.686 14.6 .054

2.60 203 .58 1.62 . 135 14.3 .405 23.7 .019

13.6 98 .58 1.62 .203 6. 92 2.31 3. 67 .409

7. 10 158 .68 L 62 .203 n. 2 1. 11 9. 65 .094 0

4.40 203 .58 1.62 .203 14.3 0. 685 16.8 .036

29.0 98 .58 L 62 .337 6. 92 4. 52 2.20 .936

12.9 158 .58 1.62 .337 11.2 2.01 5. 80 . 136 8.4 203 .68 1.62 .337 14.3 1.31 9. 46 .078 5.7 252 .68! 1.62 .337 17.7 0.890 14. 66 .036

Table 9. Data from tests with double sanitary tee drainage

fittings, 3-in. stack, one-branch flow—Continued

r low

in each branchotacK

flow T d2 hi—hi Vl Vl

Vl^ bymDot

2g(.hi—hi) 2g(.hi-hi) in fig. 17

gpm gpm in. in. fl. fps fps

8.

1

56 ..34 1. 62 .094 6. 20 1. 26 6, 3( . 261 3.2 98 .34 1.62 .094 10.9 0, 500 19.4 .041 10.8 56 ..34 1. 62 . 1.36 6.20 1.08 4. 42 .324 4.8 98 ..34 1.62 . 135 10.9 0. 749 13.6 . 065 \ 17.8 56 ..34 1.62 .203 6.2' 2. 77 2.94 ..588 ( ^

7.8 98 ..34 1.62 .203 10.9 1. 16 9. 05 . 1 13 14.2 98 .34 1.62 .337 10.9 2.21 6.4 . 224 8.4 1,37 ..34 1.62 .337 15.

1

1.31 10.5 .079

7.80 40 .16 1.62 .094 8. 75 1. 22 12.6 .234

4.60 64 . 16 1.62 .094 11.8 0.734 23. 1 .089

12.5 40 . 16 1.62 . 135 8. 76 1.94 8. 85 .436

8. 10 54 . 16 L 62 . 135 1L8 1.26 16. 1 .183 ' <1 6.80 80 . 16 L 62 .203 17.6 1.06 23.6 .091 12 3 80 .16 1.62 ..337 17.6 1.87 14.2 . 160 8.50 98 . 16 1.62 .337 2L5 L 32 21.2 .080

0. oU OQyis . 00 2. Of] 120 0. yz 0. 849 A ACi 0. *±U . UtfO 4.20 158 .58 2. 06 .' 120 11.2 .405 16.3 .021

14. 8 98 . 68 2. 06 . 172 6. 92 1. 43 4. 32 . 184

7.

1

158 .68 2 06 . 172 11.2 0.686 n.4 .043 26.4 98 .58 2. 06 .258 6. 92 2. 55 2. 88 ..391 1 A

12. 1 168 .58 2 06 .258 11. 2 1. 17 7. 57 .082

7. 60 203 .,^8 2.06 .258 14. 3. 0.698 12.4 .029

24.0 158 .58 2. 06 .429 U. 2 2. 32 4. 56 .196 15.7 203 .58 2. 06 .429 14.3 1.47 7. 46 .079

10.4 262 .58 2. 06 .429 17.7 I. 01 11.5 .037

7. 30 98 .34 2. 06 . 120 10.9 0. 708 15. 7 . 065 25.6 56 ..34 2. 06 . 172 6. 20 2. 47 3. 47 . 651 11.4 98 .34 2. 06 . 172 10.9 1.09 15. 6 . 109

5. 50 137 .34 2. 06 .258 15.

1

0. 529 20.7 .025

18.4 98 .34 2. 06 .429 10.9 1.78 7. 10 .190 ' V

35. 2 98 .34 2. 06 .429 10.9 3. 42 4. 27 .421 20.2 137 .34 2 06 .429 15.

1

1.95 8.30 .138 14.0 180 .34 2. 06 .429 19.9 1.35 14.4 .066

9.0 54 .16 2. 06 . 120 11. 8 0. 866 18. 2 .097 21.0 40 . 16 2. 06 .172 8. 75 2.03 6.91 .371 13.6 54 . 16 2. 06 . 172 11.8 1.31 12.8 . 156 41.8 40 .16 2. 06 .258 8. 76 4. 04 4. 61 .978 23.6 54 . 16 2. 06 .258 11.8 2.28 8. 42 .312

11.2 80 . 16 2. 06 . 258 17.6 L 09 18.6 .071 23.0 80 .16 2. 06 .429 17.6 2.20 11.2 . 179 16.3 98 .16 2. 06 .429 21.6 1.57 16.8 .090

lOo ..68 3. 06 . 179 11. 2 1. 10 A

10. 9 . 104 16.0 203 .58 3. 06 . 179 14.3 0.708 17.9 .043 8.2 252 .58 3. 06 . 179 17.7 .364 27.7 .012

35 8 100 .58 3. 06 . 255 11. 2 1. 58 7. 66 . 153 25 7 203 .58 3. 06 . 255 14. 3 1. 14 12. 5 . 079

16 0 252 .58 3. 06 . 255 17. 7 0. 707 19. 4 . 030

' 42 0 . 58 3. 06 . 383 14. 3 1. 86 8. 37 . 140

-So. 0 252 .58,3. 06 . 383 17. 7 1. 27 13. 0 . 066

00. / 203 . 58 3. 06 . 638 14. 3 3. 80 5. 02 . 333

54 7 252 .58 3. 06 . 638 17. 7 2. 42 7. 77 . 143

35.6 98 .34 3. 06 .179 10.9 1.58 10.2 .216

24 3 137 . 34'3. 06 . 179 15. 1 1. 08 19. 9 . 101 59 2 QQ .34 3. 06 . 255 10. 9

CO

7. 18 . 419

34 3 io / . 34 3. 06 . 255 15. 1 1. 52 13. 9 . 141 22 3

1 QA ioU . 34 3. 06 . 255 19. 9 0. 988 24. 1 . 059

53 2 137 .34 3.06 . 383 15.

1

2. 36 9. 31 . 228 37^7 180 .34 3. 06 .383 19.9 1. 67 16.1 .113 69.4 180 . 34 3. 06 .638 19.9 3. 06 9. 66 .229

27 8 .16 3. 06 . 179 1 / . b 1. 16 26. 7 . 1.31 4L5 80 . 16 3. 06 .265 17.6 1. 84 18.8 . 206 31.5 98 .16 3. 06 .256 2L6 1.40 28.

1

. 119 70.6 80 .16'3. 06 ..383 17.6 3.12 12,6 .396 49.8 98 . 16 3. 06 .383 2L5 2. 20 18,8 . 198

106. 0 98 . 16 3. 06 . 638 21. 6 4. 69 11, 2 . 536

3.2 137 .34 1. 62 . 135 15.

1

0. 500 26,4 .029

4. 7 137 .34 I. 62 . 203 15. 1 . 734 17, 6 . 041

5. 9 180 .34 I. 62 . 337 19. 9 . 916 18, 2 . 038

27.9 64 .16 1. 62 .337 11.8 4.36 6, 40 .872 <

0.9 203 .58 2. 06 .120 14,3 0.087 26,6 .001 3.4 203 .58 2. 06 . 172 14.3 ..328 18,6 .010

14.2 56 .34 2, 06 . 120 6,20 1.37 4, 97 .243

2.7 137 . 34 2. 06 . 120 15,

1

0. 26 29,7 .009 10.2 137 .34 2. 06 .258 15. 1 .985 13,8 .068

6.2 180 .34^2. 06 .258 19.9 .599 23.8 .022

7.4 98 . 16 2. 06 .258 21,5 .715 27. 9. .031

21
1.0

0.9

0.8

0.7

0.6

0,5 -

0.4-

0.3-

0,2 -

0.1 -

: 0,156

= 0.175

= 0.185

0,253

= 0.280

= 0.297

= 0.353'

= 0.390'

= 0.414'

I L J L J L

4

14 16 X

20 22 24 26 28 30

Figure 17. Results of tests to determine backpressures in horizontal branches due to interference of flow down stack and

flow from branches {3-inch stack).

Curves are plotted from ths equation F=8.33-14.8{r/rf2)5/H<i2/<<i)3"F3/sX5/8.

Branches

flowing T

/ y\ 5/8 /rfjXS/S

One Both

in. in. D 9 3. 06 0. 16 0.158 Q 2. 06 .16 .175 < 1.62 .16 .185

t> 3.06 .34 .235 V T 2. 06 .34 .280 o 1.62 .34 .297

3. 06 .58 .353 A 2. 06 .58 .390 O • 1.62 .58 .414

the difRculty of measuring fluctuating pressures, and the fact that eq 38 is necessarily approximate

in its nature and contains experimentally determined constants. A few tests were made in which the rates of flow

from the branches were evidently sufficient to fill the stack at the point of entrance, after which a

slight increase in flow from the branch caused a

large increase in backpressure. It cannot be

expected that eq 38 will apply when this condition

is attained.

These data from 3-in. branches are shown in

figure 18.- It is noteworthy that the abovementioned phenomena occurred to a noticeable extent only when 3-in. branches were used with the

3-in. stack. The reason for this is simply that,

for a given velocity of flow in a horizontal branch,

a large branch delivers more water than a smaller

one, and, if the velocity is great enough, the large

branch will cause the stack to be completely fllled at the point of water entrance, whereas a smaller

branch will not cause the stack to be filled. Equation 38 indicates that for a given thickness of the

sheet of water and velocity of fiow in the stack,

and for a given magnitude of the backpressure,

large branches will deliver more water to the stack

22
TOTAL FLOW IN BRANCHES , gpm Figure 18. Effect of backpressures in horizontal branches when the capacity of the stack is reached.

Diameter of stack, 3 inches. Stack fitting used was a 3 x 3 x 3 x 3-inch double

sanitary t(e. Pneumatic pressure reduction in stack was 0.3 inch of water column below atmospheiic piessure. Flow rates in the two branches approximately equal.

Flow in stack Thickness, T,

of water sheet Symbol

gpm hi. 98 0.34

137 .34 o

98 .58 A

158 .58 o

303 .58 V

than smaller ones. This was found to be true in

tests made in this investigation. It also suggests

that, for a given condition of flow in the stack and

a given backpressure in the branch, large branches

are more likely to fill the stack than smaller ones.. Figure 19 shows large flows from both horizontal

branches (123 gpm total) interfering with a still larger flow (145 gpm) down the stack and shows the

large backpressure caused by this interference.

The total flow from the two branches in this illustra- tion was approximately equally divided between

the two branches, although it appears that the flow

from the right-hand branch was a little greater

than the flow from the left-hand branch. The

data were not taken with a high degree of accuracy

as the photograph was intended merely as an

illustration of the phenomenon, not as a means of obtaining data.

The shadows in figure 19 indicate that the entire

section of the stack was filled just below the level

of the branches and that a mass of water was

thrown upward by the impact of the streams of

water meeting in the center of the cross section of the stack.

It is likely that this eft'ect of an additional backpressure on the branch due to the stack becoming

almost or completely filled at the branch level would have to be considered in an accurate analysis of the problem. It has been neglected here

for the sake of simplicity, and, as it occurs only in

the largest branch size and then onh^ to a limited

Figure 19. Backpressure produced in a branch when the stack fitting is filled at the branch level by flows from the two

branches meeting the flow down the stack.

Total flow in both branches is 123 gallons per minute. Flow in stack is 145 gallons per minute.

extent for any reasonable velocity of flow, it probably did not affect very much of the data. On the

other hand, for two of the three curves in figure 17

that do not fit the experimental data very well,

this additional backpressure due to the tendency

of the stack to be filled existed to a limited extent,

hence may account for some of the scatter. Another possible reason why two of the computed curves in figure 17 applying to the 3-in. branches do not agree well with the experimental

data is that in these two cases there existed a

considerable negative pneumatic pressure in the

stack a few inches below the branch opening,

although just above the branch openings the

pneumatic pressure was nearly atmospheric. This

again was probably due to the stack being filled or nearly filled at the branch level, resulting in the

air flow past the fitting being seriously restricted.

The reduced pressure below the fitting may have

been transmitted to some extent to the cross section

of the branch opening, so that the actual pneumatic

pressure acting on the branch outlet would have

been somewhat less than that indicated by the

readings just above the fitting. If this latter factor were taken into account, the eft'ect would be

23
to move the experimental data in the direction of the computed curves in figure 17. However, it appears extremely difficult to actually apply this

correction numerically as it is not known just what

the mean value of the pneumatic pressure acting on the branch opening was. Possibly the answer

is that no combination of flows in the stack and

from the branch should be permitted that will allow the stack to fill sufficiently to produce these

extreme changes in pressure near the branch

opening. This points to the necessity of giving more attention to what Hunter called the fitting

capacity.

10. Application of the Results of the

Investigation

Now that a relation has been developed involving the pressures, velocities of flow, and the

diameters of the stack and of the horizontal

branches em.ptjdng into the stack, we m.ay well

ask what practical application of tliis relation can be made in designing plum.bing systems and

in preparing loading tables. What follows should

be considered m^erely as some preliminary suggestions as to how the results can be utilized. Later

inform.ation and further study of the resiflts may

conceivably m.odify the views expressed here as there has not been an opportunity as yet to test fully the applications of the results of the paper.

There are two main steps involved in applying

the results of tliis investigation to the design of

vertical stacks and horizontal branches for any

given system. The first step is the determination

of the maximum rates of flow that can be per- mitted to occur simultaneously in the stack and

in a horizontal branch at any given point on the

stack, from the standpoint of back pressure

produced thereby in the horizontal branch. In- formation given in this paper allows this to be

done for a stack in which sanitary tee fittings are used to connect the horizontal branches to the

stack, if we assume what maximum, backpressure

is to be permitted in the horizontal branches.

The second step in the design process involves

the application of the theory of probabilit}^ in

arriving at the num.ber of fixture units that can

be permitted on the stack and on any horizontal

branch without probably exceeding the permissible

simultaneous rates of flow from the branch and

in the stack at the given branch level more than

a selected portion of the tim.e. Or, to put the

matter in another way, we wish to determine

what diameters of stack and branch m.ust be used

in order to take care adequately of simultaneous

flows that probably will not be exceeded m.ore than a given portion of the tim.e from the stack

and its branches. For a multistory biulding tliis is a complicated problem in com.pound probabilities, and no analysis appears to have been m.ade

of it as j^et. Important as is Hunter's application of the

theory of probabihty to the determination of

probable loads [12, 13], it had to do with over-all

loads on the entire system, or som.e given part of

the system, and does not apply to the probable

flow distribution between difi'erent parts of the

sj-stem.. We stiU have the problem of determining

what com.binations of flow probably ^vill occur not m.ore than a given portion of the tim.e at a given branch level in a m.ultistory building when

this flow is considered in two parts, the first com.prising the flow coming down the stack from

higher levels and com.prising the flow from a number of horizontal branches, the second being the

flow from, the particular horizontal branctx that

is being considered.

The assumption made at the outset in developing the relation under discussion—that the drain

or branch is flowing full at the outlet—requires

som.e consideration as, in general, a fixture drain

that flows full at the outlet m.ay subject the fixture

trap on it to serious self-siphonage effects if it is only stack-vented. As the present discussion

relates to large buildings, that is, buildings with

large numbers of fixtures, we assum.e that the

horizontal branches entering the stack have a

considerable num.ber of fixtures connected to them and that they are back-vented, loop-vented,

or otherwise adequately vented. Tliis being the

case, the fact that the horizontal branch to which

the fixtures are attached may flow full at its outlet

need not necessarily have any serious effects as

far as self-siphonage is concerned.

In order to make eq 38 m.ore convenient to use, we replace ih in term.s of Q2 and c^2, Vi in term.s of

Qx and d^, and eliminate T. Using the relation

replacing Tt by its value from eq 13, and Vi from

eq 15b, we obtain

or

(39b)

All of the quantities in eq 39b are expressed in

foot-second-units, and the equation is applicable

only when the sheet of water is traveling at ter- minal velocity, when the pressure head, Ao, is measured at a distance of 4 diameters from, the

stack, and when a sanitary tee stack fitting is used. This means that the assigned value of

h2—hi used in preparing a working diagram., such

as that shown in figure 20, would be for a point

4 diameters away from the stack. A slight effect due to friction and slope would have to be taken

into account if we consider points further from

the stack than this, and this could be com.puted

from the standard pipe-flow equation for full pipes.

24
O 20 40 60 80 100 120 140 160

Q|, FLOW IN ST4CK , gpm Figure 20. Perviissihle flow combinations at the junction

of a horizontal branch and a 3-inch stack. The curves shown for bianch diameters of 1!^, 2, and 3 inches represent eq 39b when the branches are flowing just full but under no pressure (A2— Ai = 0.5d2) at a distance of 4 drain diameters from the stack and when sanitary tee drainage fittings are used to connect the stack and the horizontal branches. The dashed lines show the computed rates of flow which would cause the hydraulic gradient in a horizontal branch to equal H inch per foot. But as the alteration of eq 39a or 39b to include

this factor would cause the equation to be more

complicated, and as this effect can be taken into

account by assigning a limiting value to the rate

of branch flow that will result in a friction loss per unit length of branch equal to or somewhat

less than the slope of the branch, we are justified

in using eq 39a and 39b as they are derived, at

least for our present purposes.

Equation 39 b can be solved for Q2 in terms of

Qi for a chosen value of hz —hy and for selected

values of stack and branch diameters. Such a

solution for a 3-in. stack is shown in figure 20

superposed on a series of straight lines that re- present the branch flow as various fractional

parts of the flow in the stack above the branch

level under consideration.

The curved lines in figure 20 are plotted from

eq 39b and give the corresponding flows in the

stack and in a single horizontal branch for an assumed pressure clift'erence between the stack

and a point in the branch 4 branch diameters

back from the stack, h2~hi =0.5do. The straight

lines, Q2=Qi, enable one to read directly the ratio

of Q2 to Qi.

A modification of this plot, in wliich Q2 is plotted against ^?i + ^?2, would enable one to determine the limiting ratios of branch flow to flow

in the stack just below the branch connection at any branch level. Trial computations indicate

that such a procedure will not serve to furnish

the actual design but will rather furnish a series of limiting values of the ratio Q2IQ1, or $2/(^2 + ^1),

which must not be exceeded. Otlier considerations will be involved in the actual design, and

these require further study.

Another matter that requires investigation is the suitability of the assumption that the upper

limit of the head, h2—hi, should be taken as O.-or/j. This is equivalent to letting a horizontal bi'anch

flow just full near its junction with the stack,

providing the pneumatic pressure in the stack is the same as that in a vent connecting to the

horizontal branch. If the pneumatic pressure in

the stack is less than in the vent, the drain will flow less than full, and if greater the drain will flow imder a slight head. However, it seems

reasonable that under most conditions the pneumatic pressures in vented horizontal branches

will not be greatly different from the pneumatic

pressures in the stack where the horizontal

branches connect. In view of this assumption

and of the fact that eq 39b is applicable only

where a horizontal drain flows full at its connection

to the stack, and as it does not appear that such

a horizontal branch flowing just full for a small

fraction of the time could result in adverse effects,

it seems reasonable to set the limiting value of

h2~-hx equal to 0.5f/2- Furthermore, in the actual

design of a system it would be anticipated that

this limit would be reached for only a small

fraction of the time, and that the head would be

smaller for the major portion of the time owing to probability considerations.

At this time it must be pointed out that eq 39a

and 39b are subject to the same limitations as those stated previously for eq 38, as eq 39a and

39b are based on eq 38. In addition, the latter equations are, at least theoretically, applicable

only for stacks having a roughness expressed by

a value of ks of 0.00083, corresponding to new

cast-iron soil pipe. Any effect due to roughness

will depend on its effect on the terminal velocity

and terminal length. According to eqs 15b and

22, the effect of roughness on terminal velocity

and terminal length is inversely proportional to the 1/10 and 1/5 powers of the roughness factor,

ks, respectively. Hence it would appear that for

the range of roughness encountered in commonly

used stack materials, there would be only a small

difference in the computed terminal velocities and

terminal lengths due to different degrees of

roughness.

Another possible logical criterion for the maximum permissible flow in a horizontal branch under

any condition is to permit no flow in the branch

greater than that which results in a friction loss that will make the hydraulic gradient steepei" than

the slope of the branch, such a limit to be exceeded

not more than a selected fraction of the time. If this is done, the static pressure head in the branch

will nowhere be greater than at a point near the

stack. If tliis limit is exceeded, excessive pres- sures may develop in the branch at some distance

from the stack for high rates of flow. For long

horizontal branches loaded with a ninnber of

25
fixtures, this could result in sluggish drainage and

possible backwater effects or overflow of certain

low-set fixtures sucli as bathtubs or shower stalls, and would be likely to reduce the effectiveness of

vents connecting to the horizontal branch.

One other point that deserves consideration here

is the effect of hydrostatic heads in small, short

horizontal branches receiving the discharge from

only a few fixtures, such as a 2-in. horizontal

branch to which the waste pipes from a com.bination sink-and-laiuidrv tray fixture, a washbasin,

and a bathtub are connected. It can be shown

that a reasonable probability exists for the simultaneous discharge of two of these fixtures, and

that the corresponding flow rate could very well

exceed that resulting in a hydraulic gradient equal

to the slope of the branch. Yet it is generally

conceded that in a sm_all installation, such as that

described above, no harmful effects are experienced

due to this fact. Thus it might at first appear that

the above-naentioned criterion is unwarranted.

Probably, in all fairness, it should be admitted

that flows in such a horizontal branch as that dis- cussed here m.ay be allowed to exceed somewhat

those indicated by the above criterion, at least in

single family dwellings, owing to the fact that the

maximum flows and pressures occur in the horizontal branch at such a point and at such times

that it is not likely that any harm may be done.

However, in larger and longer horizontal branches

loaded with a larger number of fixtures, and where

there may be several such branches at diflerent

levels connecting to a soil stack, it is easy to think

of situations where certain fixtures might be subjected to adverse hydraulic and pneum^atic effects should the horizontal branches be flowing for an

appreciable portion of the time imder an excessive

hydraidic gradient, or an excessive hydrostatic backpressure, even if the branch were vented. Hence

for large systems it would appear that criteria of the

nature suggested above are necessary. In this connection, it would probably have been better

for illustrative purposes to have prepared figure

20 for a stack larger than 3 in. in diameter as

figure 20 indicates horizontal-branch flow limits

that, under certain conditions, are lower than are known to be satisfactory from a practical viewpoint, for the reasons stated above. However, as eq 39 was derived on the basis of tests made on a 3-in.-diam.eter stack, figure 20 was prepared for

this diam^eter of stack, and will, for the present,

suffice for illustration of the basic principles in- volved in determining limiting flow com^binations

at the point of junction between a horizontal

branch and a stack. Granting that eq 39b is applicable to larger diameters of stack through

the ratio d^jdi, figure 20 could just as well have

been prepared for a larger stack.

11. Conclusions

The terminal velocity of a sheet of water flowingdown the wall of a vertical pipe, providing the

tliickness of the sheet is small compared with the

diameter of the pipe, can be computed from, the

formula

?'i=the terminal velocity in feet per second

(?i = the rate of flow in gallons per minute

di — the diam.eter of the stack in inches.

2. The distance through wliich a sheet of water

flowing do\yii the wall of a vertical pipe must fall before attaining approxim.ately its terminal velocity can be computed from the formula i,=0.052«f,

where Lt is the terminal length in feet, and Vt is the terminal velocity in feet per second.

3. Stack capacity is closely related to horizontal

branch capacity, and the two m.ust be considered

together.

4. The pneumatic pressures in the stack and

the terminal velocity of the sheet of water flowingdown the stack influence stack capacity. Thus

the whole design of the building stack and buildingdrain and the submergence or nonsubmergence of

the building sewer affect stack capacity, particularly in installations that are inadequately vented,

or in wliich tlie building drain or building sewer is inadequate for one reason or another.

5. The principal contribution of the present

paper to the problem of capacities of stacks and

horizontal branches is in the development of what

appears to be a new and important additional

limiting criterion. In this connection, the fol- lowing criterion is proposed. The capacity of a drainage stack at the level of any horizontal

branch is defined as the total flow at that level which causes a preassigned positive pressure in the horizontal branch that cannot be relieved by

venting or, in the case of horizontal branches low on the stack, by increasing the size of the building

drain.

6. Based on information contained in this paper,

it appears that the ideal solution to the problem

of maximum loadings on stacks and horizontal

branches would have to satisfy three requirements,

at least for the major portion of the time during

which the fixtures on the S3"stem are in congested

or frequent use. The three conditions that should

be satisfied for the selected fraction of the time

are:

(a) The total load on a stack must not be so great as to cause the stack to flow more than

one-fourth to one-third full wherever terminal

velocity exists. According to both Hunter and

Dawson, this limit is necessary in order to prevent

undue noise, vibration, and pressure fluctuations,

and in order to allow the satisfactory functioning

of vents connecting to the lower portions of the

stack.

(b) No horizontal branch shall be loaded so heavily as to cause the hA^draulic gradient to exceed the slope of the drain, or to cause the

stack to be complete^ filled where the branch

discharges into it. 26
(c) At any point of connection of a horizontal

branch to a stack no simultaneous combination

of flows in the stack and in the horizontal branch

shall occur that cause the horizontal branch to flow under an excessive hydrostatic head, or backpressure. A logical limit would appear to be that

of allowing a horizontal branch to be loaded no more heavily than to result in its flowing just full near its junction with the stack. Equation 39b

may be used to determine various flow combinations that give this result. The reasons for this requirement, of course, are similar to those given

under (b) above.

In connection with the preceding discussion,

Hunter's experimental studies and his application

of the theory of probability to the problem afford an approximate method of determining the fixtureunit loadings that would satisfy points (a) and

(b). However, in order to determine maximum

fixture-unit loadings that would satisfy point (c),

it will first be necessary to develop an application

of the theory of probability dealing with the probable combinations of flow resulting from the

random discharges of two separate groups of

plumbing fixtures, one group being those on a

particular horizontal branch, and the other being

those draining into the stack from all higher levels. Until this problem is solved it will be unpossible

to realize the full utility of the investigation

reported in this paper.

7. In some cases the limiting load on the stack

at a given point may very well be determined by

noise and vibration considerations, or by pressure

fluctuations, rather than by backpressure on the

branches. In other cases, the size or nature of

the venting system or the diameter of the house

drain may be the limiting factor in determining

the practical capacity of the stack.

12. Further Investigation

The results obtained in this investigation repre- sent but a single step toward a solution of the

problem of stack capacities. The tests involved

the use of a sanitary tee stack fitting. Similar

tests made with long-turn T-Y stack fittings, which

are also cjuite commonly used, have been completed since this paper was started, and these

results are consistent with those presented in this paper. Similarly, as the tests made with the

apparatus shown in figures 10, 11, and 12 do not

simulate in all details the conditions of flow in actual drainage systems, the tests have been re- peated with an actual stack. The results obtained

with this stack (which are, in general, consistent

with the results reported in this paper) will be

published at a later date.

Other data obtained include measurements of terminal velocities, variation in air content radially through the sheet of water, and pncumaUf;

pressures in the stack just below the f)()inl of entry of water from tlic branch.

Acknowledgment is made to tlie ffousing and

Home Finance Agency for its support of the investigation reported here. John L. French contributed a number of useful ideas, including the idea

of using the special apparatus for sim.ulating the

sheet flow down the stack. Vincent T. Manas

gave the authors much useful advice in cormection

with planning the investigation. Otto Hintz carried out much of the laborious computation in- volved. Lewis A. Lembeck was responsible for the conduct of the tests with the assistance of

Edward J. Norcom.e, and the results are due in no

sm.all m.easure to their careful and thorough work.

13. References

[1] Sherman M. Woodward and Chesley J. Posey, The

hydraulics of steady flow in open channels (.John Wiley & Sons, Inc., "New York, N. Y., 1941).

[2] Recommended minimum requirements for plumbing

in dwellings and similar buildings; Final Report of the Subcommittee on Plumbing of the Building

Code Committee, U. S. Department of Commerce,

Ehmination ofWaste Series, BH2 (1924).

[3] Recommended minimum recjuirements for plumbing,

revision of the above, BH13 (1932).

[4] Plumbing manual, NBS Building Materials and

Structures Report BMS66, Report of the Subcommittee on Plumbing of the Central Housing Committee on Research, Design, and Construction

(1940).

[5] F. M. Dawson and A. A. Kalinske, Report on hydraulics and pneumatics of plumbing drainage systems,

I., University of Iowa Studies in Engineering, Bulletin 10 (1937).

[6] F. M. Dawson and A. A. Kalinske, Report on the hydraulics and pneumatics of the plumbing drainage

system, Technical Bulletin 2 of the National Association of Master Plumbers of the United States, Washington, D. C. (1939).

[7] Harold E. Babbitt, Tests on the hydraulics and pneumatics of house plumbing, Engineering Experiment

Station Bulletin No. 143, University of Illinois, XXI, No. 47 (Julv 21, 1924).

[8] Roy B. Hunter, Gene E. Golden, and Herbert N.

Eaton; Cross-connections in plumbing systems,

NBS .J. Research 20, 479 (1938) RP1086.

[9] Garbis H. Keulegan, Laws of turbulent flow in open

channels, NBS J. Research 21, 707 (1938) RP1151.

[10] Roy B. Hunter, unpublished manuscript. National Bureau of Standards.

[11] Hunter Rouse, Elementary mechanics of fluids, p. 211 (John Wiley & Sons, Inc., New York, N. Y., 1946).

[12] Roy B. Hunter, Methods of estimating loads in plumbing systems, NBS Building Materials and Structures Report BMS65 (1940).

[13] Herbert N. Eaton and John L. French, Fixture unit

ratings as used in plumbing system design, Housing

Research Paper 15, Housing and Home Finance

Agency (1951).

[14] Garbis H. Keulegan, Friction losses in short pipes and

culverts; unpublished manuscript. National Bureau

of Standards (1948).

27
14. Appendix. Evaluation of the Sand

Roughness Coefficient,

Table 10. Computation of the sand roughness coefficient, ka from Hunter's data [10] on friction losses for new cast-iron

soil pipe flowing full

Nominal

diameter Acrualdiameter / SIf

in. in. 2 2.06 0. 0301 267 16. 35 0. 0128

3 2. 90 .0258 310 17. 60 .176

4 3. 93 .0243 329 18. 15 .301

5 4.85 .0250 320 17. 90 .398

In table 10, / is the Darcy-Weisbach friction coefficient, and R is the radius of the pipe. From an unpublished paper by Keulegan [14], we have

o

^f8^f= 4.75- 2.5 + 5.75(1 + A:./i?)nogio(l + R/K)

.

As ks/R is very small

V8Zf= 4. 75-5.75 \og,o{R/k,).

Now if we plot the data in the last two columns of table

10, fair a curve through the points, and extend the curve back to the point, log R= 0, corresponding to R=l in., we

can solve the last equation for ks by substituting the value

read from the plot for ^/S|k,:

16.24=4.75-5.75 logiofc., whence

log,ofc,= -2.00,

and

k, =0.010 in., or 0.00083 ft. Washington, September 4, 1951.

28


BUILDING MATERIALS AND STRUCTURES REPORTS

[Continued from cover page ii] BMS36 Structural Properties of Wood-Frame Wall, Partition, Floor, and Roof Conatruotions With "Red Strjpe" Lath Sponsored by The Weston Paper and Manufacturing Co 10^ BMS38 Structural Properties of Two "Dunstone" Wall Constructions Sponsored by the

W. E. Dunn Manufacturing Co 10^ BMS39 Structural Properties of a Wall Construction of "Pfeifer Units" Sponsored by the Wisconsin Units Co 10^ BMS42 Structural Properties of Wood-Frame Wall and Partition Construction with "Celotex"

Insulating Boards Sponsored by The Celotex Corporation 15^ BMS43 Performance Test of Floor Coverings for Use in Low-Cost Housing: Part 2 15^ BMS44 Surface Treatment of Steel Prior to Painting 10^ BMS47 Structural Properties of Prefabricated Wood-Frame Constructions for Walls, Partitions, and Floors Sponsored by American Houses, Inc 20^ BMS48 Structural Properties of "Precision-Built" Frame Wall and Partition Constructions

Sponsored by the Homasote Co 15^ BMS49 Metallic Roofing for Low-Cost House Construction 20(i BMS50 Stability of Fiber Building Boards as Determined by Accelerated Aging 10^ BMS51 Structural Properties of "Tilecrete Type A" Floor Construction Sponsored by the

Tilecrete Co 10$

BMS52 Effect of Ceiling Insulation Upon Summer Comfort 15^ BMS53 Structural Properties of a Masonry Wall Construction of "Munlock Dry Wall Brick"

Sponsored by the Munlock Engineering Co 10^ BMS54 Effect of Soot on the Rating of an Oil-Fired Heating Boiler. 10^ BMS55 Effects of Wetting and Drying on the Permeability of Masonry Walls 10j5 BMS58 Strength of Soft-Soldered Joints in Copper Tubing 10^ BMS59 Properties of Adhesives for Floor Coverings 15^ BMS60 Strength, Absorption, and Resistance to Laboratory Freezing and Thawing of Building

Bricks Produced in the United States 30<5 BMS61 Structural Properties of Two Nonreinforced Monolithic Concrete Wall Constructions. _ 10^ BMS62 Structural Properties of a Precast Joist Concrete Floor Construction Sponsored by

the Portland Cement Association 15^ BMS63 Moisture Condensation in Building Walls 15<5 BMS64 Solar Heating of Various Surfaces 10^ BMS65 Methods of Estimating Loads in Plumbing Systems 15^ BMS66 Plumbing Manual 35^ BMS67 Structural Properties of "Mu-Steei" Prefabricated Sheet-Steel Constructions for Walls,

Partitions, Floors, and Roofs, Sponsored by Herman A. Mugler—. 15^ BMS68 Performance Test for Floor Coverings for Use in Low-Cost Housing: Part 3 20^ BMS69 Stability of Fiber Sheathing Boards as Determined by Accelerated Aging 10^ BMS70 Asphalt-Prepared Roll Roofings and Shingles 20^5 BMS71 Fire Tests of Wood- and Metal-Framed Partitions 200 BMS72 Structural Properties of "Precision-Built, Jr." Prefabricated Wood-Frame Wall Construction Sponsored by the Homasote Co 100 BMS73 Indentation Characteristics of Floor Coverings 100 BMS74 Structural and Heat-Transfer Properties of '^U. S. S. Panelbilt" Prefabricated SheetSteel Constructions for Walls, Partitions, and Roofs Sponsored by the Tennessee

Coal, Iron & Railroad Co 200 BMS75 Survey of Roofing Materials in the North Central States. ^ 150 BMS76 Effect of Outdoor Exposure on the Water Permeability of Masonry Walls 150 BMS77 Properties and Performance of Fiber Tile Boards 100 BMS78 Structural, Heat-Transfer, and Water-Permeability Properties of Five Earth-Wall

Constructions 250 BMS79 Water-Distributing Systems for Buildings 200 BMS80 Performance Test of Floor Coverings for Use in Low-Cost Housing: Part 4 150 BMS81 Field Inspectors' Check List for Building Constructions (cloth cover, 5 n 7H inches).. 300 BMS82 Water Permeability of Walls Built of Masonry Units. 250

BMS83 Strength of Sleeve Joints in Copper Tubing Made With Various Lead-Base Solders 150 BMS84 Survey of Roofing Materials in the South Central States 150 BMS85 Dimensional Changes of Floor Coverings With Changes in Relative Humidity and

Temperature 100 BMS86 Structural, Heat-Transfer, and Water-Permeability Properties of "Speedbrik" Wall

Construction Sponsored by the General Shale Products Corporation 150 BMS87 A Method for Developing Specifications for Building Construction—Report of Subcommittee on Specifications of the Central Housing Committee on Research,

Design, and Construction 200

BMS89 Structural Properties of "Precision-Built, Jr." (Second Construction) Prefabricated

Wood-Frame Wall Construction Sponsored by the Homasote Co 150 BMS90 Structural Properties of "PHC" Prefabricated Wood-Frame Construction for Walls,

Floors, and Roofs Sponsored by the PHC Housing Corporation 150 BMS92 Fire-Resistance Classifications of Building Constructions 300 BMS93 Accumulation of moisture in Walls of Frame Construction During Winter Exposure.. 100

[List continued on cover page iv]
BMS94

BMS95

BMS96

BMS97

BMS98

BMS99

BMSlOO

BMSlOl

BMS102

BMS103

BMS104

BMS105

BMS106

BSM108

BMS109

BSMUO

BMSIU

BMS112

BMS113

BMS114

BMS115

BMS116

BMS117

BMS118

BMS119

BMS120

BMS121

BMS122

BMS123

BMS124

BMS125

BMS126

BMS127

BMS128

BMS129

BMS130

BMS131

BMS132

BUILDING MATERIALS AND STRUCTURES REPORTS ^

(Continued from cover page iiij Water Permeability and Weathering Resistance of Stucco-Faced, Gunite-Faoed and

"Kn«ro Concrete-Unit" Walls 55^ Tests of Cement-Water Paints and Other Waterproofings for XJnit-Wasonry'Walls 251

Properties of a Porous Concrete of Cement and Uniform-Sized Gravel IQi Experimental Dry-Wall Construction With Fiber Insulating Board loi Physical Properties of Terrazzo Aggregates I5S

Structural and Heat-Transfer Properties of "Multiple Box-Gir¥e'r"nywoVd"Pan^^^^

Walls, Floors, and Roofs , 15^ Relative Slipperiness of Floor and Deck Surfaces 10< Strength and Resistance to Corrosion of Ties for Cavity Walls iQi Painting Steel jq5

Measurements of Heat Losses From Slab Floors-IIIIIIIIIIIIIIIIIIII 16*

Structural Properties of Prefabricated Plywood Lightweight Constructi'o'nVfor "Wails" Partitions, Floors, and Roofs Sponsored by the Douglas Fir Plywood Association ' SOi

Paint Manual With Particular Reference to Federal Specifications... $L 25

Laboratory Observations of Condensation in Wall Specimens ' 10^ Temperature Distribution in a Test Bungalow With Various Heating Devices'..'. 10j5 Strength of Houses: Application of Engineering Principles to Structural Design. $1. 50 Paints for Exterior Masonry Walls ' 151 Performance of a Coal-Fired Boiler Converted to Oil'.. "_'."."_".'.""_"_' is*

Properties of Some Lightweight-Aggregate Concretes With 'and"'without 'an "Air- entraining Admixture 10<

Fire Resistance of Structural Clay Tile Partitions 'III" " I5i Temperature in a Test Bungalow With Some Radiant and jacketed Space Heaters""'" 25* A Study of a Baseboard Convector Heating System in a Test Bungalow 15< Preparation and Revision of Building Codes Zl 15*

Fire Resistance of WaJls of Lightweight Aggregate Concrete Masonry "UnitsI 20i

Stack Venting of Plumbing Fixtures 15^ Wet Venting of Plumbing Fixtures I._...I_ri'I'I" 20<5 Fire Resistance of Walls of Gravel-Aggregate Concrete Masonry Ij'nits "

15i Investigation of Failures of White-Coat Plasters "

" 25?

Physical Properties of Some Samples of Asbestos-Cement Siding "" ib*

Fire Tests of Wood-Framed Walls and Partitions With Asbestos-Cement Facings ' 15«S Fire Tests of Steel Columns Protected With Siliceous Aggregate Concrete 15* Stone Exposure Test Wall... " 3ol The Self-Siphonage of Fixture Traps .IIIIIIII'I " 20*

Effect of Aging on the Soundness of Regularly Hydrated Dolomitic Li'me'Putties 15* Atm(^pheric Exposure Tests of Nailed Sheet Metal Building Materials 20^

Fire Endurance of Shutters for Moving-Stairway Openings _ 10* Methods and Equipment for Testing Printed-Enamel Felt-Base Floor "Covering"

"' 15*

Fire Tests of Gunite Slabs and Partitions 15* Capacities of Plumbing Stacks in Buildings 20(S