jAT'L INST. OF STAND & TECH

AlllDt. 3 M 7 A NBSIR 85-3108

NBS

Reference PUBL!CAT| 0ns

A Preliminary Study of the

Vertical Stack to Horizontal Drain

Entry Condition As An Extension

to the Modeling of Unsteady

Partially Filled Pipe Flow

Dr. J.A. Swaffield, Reader

Mechanical Engineering Department

Building Technology

Brunei University

Uxbridge, England

Dr. Lawrence S. Galowin

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Gaithersburg, MD 20899

December 1984

Issued February 1985

" S. DEPARTMENT OF COMMERCE

-QC

100

.1156

85-3108

1984

iTIONAL BUREAU OF STANDARDS

NATIONAL BUREAU OF STANDARDS

LIBRARY

NBSIR 85-3108

A PRELIMINARY STUDY OF THE

VERTICAL STACK TO HORIZONTAL DRAIN

ENTRY CONDITION AS AN EXTENSION

TO THE MODELING OF UNSTEADY

PARTIALLY FILLED PIPE FLOW

Dr. J.A. Swaffield, Reader

Mechanical Engineering Department

Building Technology

Brunei University

Uxbridge, England

Dr. Lawrence S. Galowin

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Gaithersburg, MD 20899

December 1984

Issued February 1985

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
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SUMMARY

The finite difference based method of characteristics model for unsteady

partially filled pipe flow was extended to include the model for the stack

to horizontal drain entry boundary condition. The conditions at drain entry

are defined in terms of the energy of the terminal annular flow velocity in

the stack, together with an appropriate loss coefficient function at the entry.

The hydraulic solutions link the branch drains, fittings, vertical soil stack,

and building drain. The analysis permits any combination of drainage load

patterns from simultaneous, overlapping, or sequence of discharge events.

Preliminary simulations utilizing this model indicate that the modeling technique extends the existing horizontal network analysis program for determination

of multistory building drainage systems. The sizing procedure provides the

hydraulic capacity of drains for specified pipe sizes, pipe pitch, and wall

roughness factors.

Keywords: Drain stack to horizontal drain flows; drain system design;

plumbing drainage; unsteady flow in drains.

iii
NOTATION

A

C

D

Ee b

e

fg

h

ho

i*n K

k

Lm

n

Q

Qs

Re

SSo

T

tV

V 0

Vt

Zo

flow area (m^)

Chezy coefficient (m

pipe diameter (m)

flow specific energy (m)

bulk annular flow specific energy (m)

incremental annular flow specific energy (m)

friction factor, resistance coefficient

acceleration due to gravity (m/s^)

depth (m)

entry depth (m)

normal depth (m)

entry loss coefficient

pipe roughness coefficient (usually ratio of roughness height

to pipe diameter)

pipe length (m)

hydraulic radius (m)

Manning's loss coefficient (m ”1/3 S ) flow rate (£/s)

stack flow rate ( £/ s)

Reynolds number

slope energy grade line

pipe slope

water surface width (m)

thickness annular layer (m) , or time mean velocity (m/s)

entry flow velocity (m/s)

stack annular flow terminal velocity (m/s)

pipe elevation (m)

0 angular displacement (radians)

A Colebrook White loss coefficient

v kinematic viscosity (m^/s)

p density (kg/m^)

tqwall shear stress (kg/ro^)

AL pipe length increment (m)

iv
SUMMARY

The finite difference based method of characteristics model for unsteady

partially filled pipe flow was extended to include the model for the stack

to horizontal drain entry boundary condition. The conditions at drain entry

are defined in terms of the energy of the terminal annular flow velocity in

the stack, together with an appropriate loss coefficient function at the entry.

The hydraulic solutions link the branch drains, fittings, vertical soil stack,

and building drain. The analysis permits any combination of drainage load

patterns from simultaneous, overlapping, or sequence of discharge events.

Preliminary simulations utilizing this model indicate that the modeling technique extends the existing horizontal network analysis program for determination

of multistory building drainage systems. The sizing procedure provides the

hydraulic capacity of drains for specified pipe sizes, pipe pitch, and wall

roughness factors.

Keywords: Drain stack to horizontal drain flows; drain system design;

plumbing drainage; unsteady flow in drains.

iii
NOTATION

A

C

D

Ee b

e

fg

h

ho

K

k

Lm

n

Q

Qs

Re

SSo

T

tV

Vo

v t Zo

flow area (m^)

Chezy coefficient (m

pipe diameter (m)

flow specific energy (m)

bulk annular flow specific energy (m)

incremental annular flow specific energy (m)

friction factor, resistance coefficient

acceleration due to gravity (m/s^)

depth (m)

entry depth (m)

normal depth (m)

entry loss coefficient

pipe roughness coefficient (usually ratio of roughness height

to pipe diameter)

pipe length (m)

hydraulic radius (m)

Manning's loss coefficient (m ~1/3 S ) flow rate (&/s)

stack flow rate (&/ s)

Reynolds number

slope energy grade line

pipe slope

water surface width (m)

thickness annular layer (m) , or time mean velocity (m/s)

entry flow velocity (m/s)

stack annular flow terminal velocity (m/s)

pipe elevation (m)

0

Av

P

T oAL

angular displacement (radians)

Colebrook White loss coefficient

kinematic viscosity (m^/s)

density (kg/m^)

wall shear stress (kg/ro^)

pipe length increment (m)

iv
TABLE OF CONTENTS

Page

1. INTRODUCTION 1

2. DEVELOPMENT OF THE VERTICAL STACK TO HORIZONTAL DRAIN NUMERCIAL

MODEL 2

2.1 Entry Energy Based on Fully Developed Vertical Stack Flow ...... 2 2.2 Initial Flow Profiles 3

2.3 Initial Consideration of Loss Coefficient at Entry to

Horizontal Drain 5

3. FLOW SIMULATION 7

3.1 Initial Flow Depth Profiles Along the Horizontal Drain 7 3.2 Simulation of Vertical Stack Annular Thickness and Terminal

Velocity 9

3.3 Surge Attenuation Following Stack Discharge to a Horizontal

Drain 9

3.4 Effect of Baseflow on Surge Attenuation 10

3.5 Comparison of Predicted and Observed Depth Profiles for a Horizontal Drain Supplied by a Stack 10

4. CONCLUSIONS 13

5. REFERENCES 14

v
LIST OF FIGURES

Page

Figure 1 . Summary of the methods of characteristics solution of the

unsteady partially filled pipe flow equations 15

Figure 2. Application of the method of characteristics to unsteady

supercritical partially filled pipe flow 16

Figure 3. Fully developed annular flow in vertical stacks 17

Figure 4. Flow rate vs: time inflow to a vertical stack translated

into annular thickness and terminal energy vs. time

profiles 18

Figure 5. Calculation of the energy grade line for gradually varied

flow 19

Figure 6. Development of flow depth profiles in gradually varied pipe

flow 20

Figure 7. Upstream boundary formed by an entry energy equation 21

Figure 8. Depth profiles for initial steady flow along drain at 1/50

gradient, no energy loss at entry 22

Figure 9. Depth profiles for initial steady flow along drain at a 1/100

gradient, no energy loss at entry 23

Figure 10. Influence of the entry energy loss coefficient on development

on initial steady flow depth profiles ( 1 / 100 ) 24

Figure 11. Influence of the entry energy loss coefficient on development

on initial steady flow depth profiles (1/50) 24

Figure 12. Annular stack flow thickness as a function of stack steady

flow rate 25

Figure 13. Stack annular flow terminal velocities compared to existing

experimental data 26

Figure 14. Flow depth at pipe entry compared to annular stack flow thickness for a range of steady flow rates 27

Figure 15. Peak depths predicted along the horizontal drain for a range

of inflow profiles (1/50) 28

Figure 16. Peak depths predicted along the horizontal drain for a range

of inflow profiles (1/100) 29

vi
LJST OF FJCURES ( CONTINUED)

Page

Figure 17. Peak flow rates predicted locally along the horizontal drain

for a range of inflow profiles (1/50) 30

Figure 18. Peak flow rates predicted locally along the horizontal drain

for a range of inflow profiles (1/100) 31

Figure 19. Location of predicted peak flow depth and local flow rate for

a range of inflow profiles (1/50) 32

Figure 20. Location of predicted peak flow depth and local flow rate for

a range of inflow profiles (1/100) 33

Figure 21. Peak flow depth and local flow rate envelopes along the length

of the horizontal drain (1/50) 34

Figure 22. Peak flow depth and local flow rate envelopes along the length

of the horizontal drain (1/100) 35

Figure 23. Depth vs. time profiles at a number of stations along the horizontal drain (1/50) 36

Figure 24. Depth vs. time profiles at a number of stations along the horizontal drain (1/100) 37

Figure 25. Summary of baseflow effect on the peak depth and local flow

rate prediction (1/50) 38

Figpre 26. Summary of baseflow effect on the peak depth and local flow

rate prediction (1/100) 39

Figure 27. Comparison of observed and predicted surge attenuation downstream of a vertical stack to horizontal drain entry 40

Figure 28. Comparison of observed and predicted surge attenuation downsteam of a vertical stack to horizontal drain entry 41

Figure 29. Comparison of predicted surge depths downstream of a vertical

stack to a horizontal drain entry for two entry energy loss

factors 42

Figure 30. Comparison of predicted surge profile depths in a horizontal

drain downstream of a vertical stack for composite loading

profiles 43

vii
LIST OF TABLE

Page

Table 1. Normal depths in a 100 mm smooth partially filled pipe at 1/100

and 1/50 gradients 8

viii
1 . INTRODUCTION

The unsteady flow equations defining partially filled pipe flow may be solved

by means of the finite difference method of characteristics. Programs to

utilize this technique have been developed and fully described in previous

progress reports and published papers [1].

The development of the basic equations and their application in finite

difference form to describe unsteady partially filled pipe flow are illustrated

by figures 1 and 2. In common with other finite difference applications the

methods require a knowledge of both initial conditions and the system boundary

equations to allow the solution to proceed. For a single pipe as shown in

figure 2, under supercritical flow conditions, the exit boundary is supplied

by the C+C~ characteristics from the preceding upstream section. In addition

to the boundary conditions the finite difference solution also requires knowledge of the initial flow depth and velocity along the pipe length at the

start of the simulation. Normally a small steady flow is assumed along the

whole pipe length prior to the introduction of the drain load profile.

The flow profiles entering the drain may be assumed for initial case studies.

For example, a high energy steady inflow will become progressively deeper

along the pipe's length as its energy is dissipated. The flow depth will

eventually reach the normal steady uniform flow depth, however the distance

required for this will depend on the entry energy, the flow rate and pipe

parameters such as diameter, roughness, slope, etc. The entry condition at

the head of the drain depends on the inflow conditions resulting from the flow

upstream of the entry section. In this report the upstream flow conditions

are derived to represent fully developed vertical stack flow entering the

"horizontal" drain. slope etc. Determination of this initial flow depth

profile as a function of time at the horizontal drain entry will be presented

based upon gradually varied flow theory.

1
2. DEVELOPMENT OF THE VERTICAL STACK TO HORIZONTAL DRAIN NUMERICAL MODEL

2.i ENTRY ENERGY BASED ON FULLY DEVELOPED VERTICAL STACK FLOW

Figure 3 illustrates the annular nature of vertical stack flow under fully

developed flow conditions [2].

For a section AL of the stack a force balance equation for the falling water

may be expressed under steady developed flow conditions, where stress at the

air water interface is neglected, as

tiD t 0 (AL) = pir Dt(AL)g (1)

where D = pipe diameter

t = annular thickness

x 0 = wall shear stress resisting motion

It follows that:

= P t g ( 2 )

Following the work of Colebrook White it may be shown that the shear stress may

be expressed, for full bore pipe flow by the equation

/p~y2

V S T0

2 log 10 [

k

3.7 D +

2.51

Re

(3)

where V is flow mean velocity

k is surface roughness coefficient

and Re is Reynolds Number, based on pipe diameter.

Ackers (3) has shown that equation (3) may be applied to partially filled pipe

flow if the characteristic length D is replaced by the hydraulic radius:

m Area

Wetted Perimeter

which for a full bore pipe flow is

m = D/4

Replacing Re by VD/ v and substituting for D in terms of m yields

(4)

P V^

8 x. = - 2 log 10

k +

2.51v /p V

14.8 m 4 Vm J 8 t o

( 5 )

For the annular flow condition in the vertical stack, it has been shown that

= P t g

2
and, it also follows that

( 6 )

if t << D, as is the case here [2,4]

Substituting from (2) and (6) into (5) yields an expression for annular thickness, t, where the volumetric flow Q is taken as

This expression may be solved to yield t for a given Q and hence the annular

velocity V.

The specific kinetic energy in the annular downflow then becomes

Naturally, the expressions above only apply strictly to fully developed flow.

Previous work [4] has shown that the necessary vertical stack length to reach

this condition is less than 3m in many design cases, i.e., less than a story

height. In those cases the derivation above may therefore be utilized. The

influence of vertical stack pipe wall roughness for given flow rate Q, upon

x 0 follows from (8) and (2).

The flow rate in the stack may vary with time due to the imposed load from the

branch drain, as sketched, figure 4. However, it may be assumed that the flow

rate profile is made up of an infinite number of fully developed flow sections,

thus allowing the annular thickness and energy vs. time profiles to be developed as shown. There profiles may then be taken as the entry energy profiles

at the drain upstream boundary. With sufficient height of fall in the vertical

stack the equilibrium terminal velocity, Vt , is attained.

2.2 INITIAL FLOW PROFILES

The development of flow depth along the length of the horizontal drain pipe

under the initial steady flow conditions may be generated by reference to gradually varied flow theory. The basic assumption is that the local head at any

section is given by the equivalent steady flow loss equation, either Manning's

or Darcy's equation, utilizing the Colebrook White friction factor expression.

Depth profile predictions by numerical integration are based on this assumption,

expressed in figure 5 by

Q

= it Dt V ( 7 )

where V is the mean annular velocity

( 8 )

2 g

3
d_

dL

^

+ (Z Q - S oL > + h = - S (9)

where S is the slope of the energy grade line determined from Mannings equation

or from the general Chezy expression:

v/T = n Q

S = _v

C^m

A m^/3

2

(Manning, metric)

(Chezy)

( 10 )

( 11 )

Hence - 1 dv + ^ - dh = S

g dL ° dL

Then, from, Q

= VA

dV A + V dA = o dL DL

Also, as dA = Tdh, where T is the water surface width (which does not change

with L, or changes slowly) it follows that

dV - _ V dA - _ VT dh = _ QT dh

dL A dL A dL A2 dL

and substituting yields

/ Q

2 T\ dh + Sq

V gAVdL

- dh = s dL

4L i

- s) ,< -v

]

dh

or the distance along the flow profile, between two depths hQ and hj , becomes

h l r~ - I

1 ~ Q

2 T/gA3

(SQ-S) ]

dh ( 12 )

Figure 6 illustrates this integration.

In order to numerically evaluate (12) it is necessary to define a boundary

depth at the pipe entry.

The available flow energy at the horizontal drain entry, assuming no losses, is

based on the vertical stack annular specific flow energy,

E = h

2la

2g + Xa (13)

4
where hQ and VQ are the flow depth and velocity at entry to the horizontal

drain.

1 V

t 2

Substituting E = — —— , yields

2 g

1

2 h

o + h

o +

1_

2g

= F(h0

) (13a)

and solving for h0 , with VQ = Qq/Aq and A0 = f(hQ ) yields the upstream boundary depth.

The downstream boundary may be taken as the flow normal depth, hn

at the

distance downstream to be determined from the integration of (12). As the

flow is supercritical, the depth may rise gradually to this value with no discontinuity, such as a hydraulic jump.

The choice of dh values in the numerical integration is based on the difference

between the control depth, h Q and the downstream "target" depth, h n. A reasonable integration step size is given by

dh = (hn

- ho )/30 (14)

Once the depth reaches hn

the remaining downstream profile is assumed to be at

that constant depth.

For short pipe lengths, or high energy steady initial flows, it may be that

the normal depth is not achieved within the available pipe length. For this case

the integration is terminated when SAL exceeds the pipe length, i.e., ZAL>L.

A valid flow profile is however achieved in this case.

2.3 INITIAL CONSIDERATION OF LOSS COEFFICIENT AT ENTRY TO HORIZONTAL DRAIN

Figure 7 illustrates the conditions for an initial derivation of the energy

loss at the entry elbow bend to the horizontal drain. It is assumed that

within the annular flow the specific energy, e, along "streamlines" of the

downflow is represented by the kinetic energy at the terminal velocity condition. Degradation of the energy is assumed to result from mixing at the bend

due to the varying orientation of flow directions from the vertical to

horizontal. The equation may be assumed as:

V

2 e = ~ (sin2 I o) (14a)

where Vt is the mean annular (terminal) velocity. The equation (14) indicates

the vertically falling flow at 0=0 loses its entire energy content and only

adds mass at the drain inlet and at 0=tt the streamline total energy content is

5
preserved. Then a relation for the specific energy content of the annular

mass flow* may be obtained as:

/ e(|- - t) t d 0 = E

fi JL £d2 - (D-2t) (15)

Where Eg is the annular energy. Substituting for e yields

£d 2 - (D-2t) 2

J

= 2 1 (D-t) t — f sin 2 JLVQ

0 2 2g \ 2/

= (D - t ) 1 V

2

2 g

C 7

0

^sin2 ©JdO

D t V’- 2 = ( I

"

g

' (tt/2)

v

2 „ (D/2-t) V

t (D-t) 2g

For t « D, Eg +

2 2g

(16)

The model does not apply for t -* D/2 (full bore pipe flow) since the turning

losses at an elbow for full bore flow do not have comparable mixing. Flow

separation from full bore downflow into a partially filled horizontal drain

requires different physical modeling.

* The term (D/2-t) includes other losses for annual flow not in contact with

the pipe wall. The use of (D-t) 1/2 (in average cross section area formulation for equation (15)) leads to a constant energy loss of 50 percent.

6
r\ Values of the factor modifying the kinetic energy term, V

t z /2g, are indicated

below:

Pipe Diameter

(mm)

Annular Film

Thickness

Ratio

Factor

(mm) (D/2-t)/(D-t)

100 25.

12.5

2.5

33

43

49

Measured values from experiments for the annular thicknesses are difficult to

obtain [2, 4]; however, indicated ranges are much less than 20 percent of the

diameter and are a function of flow rate in the pipe, the pipe diameter, and

wall roughness. For D>t, (D/2-t)/(D-t) -» 0.5 hence an energy "loss" term of

50 percent may be a reasonable assumption for preliminary studies. Conditions

other than for terminal velocity should retain the relationship (16) with

replaced by the falling velocity as a function of length along the stack.

A study of entry loss effects could naturally include K values greater than

unity. That condition would imply that the entry flow energy was higher than

that appropriate to gravity driven stack flow as might be the case if the

drain were supplied from a pressurized source. Surge flows generated in tests

reported in [5] appear to correspond to pumped head conditions; there, other

elbow losses may also be significant and required for inclusion of the analysis

of the experimental results.

3. FLOW SIMULATION

3.1 INITIAL FLOW DEPTH PROFILES ALONG THE HORIZONTAL DRAIN

Figures 8 and 9 illustrates the depth profiles along a drain set at slopes of

0.02 and 0.01 and the other identified parameters for a range of initial flow

rates. The profiles attain the normal depth at some distance along the pipe,

thereafter the flow is assumed to continue at that depth until discharge at

the pipe exit. The pipe length necessary to attain the steady normal flow

depth increases for larger values of the steady flows rates, or as the pipe

slope is increased.

Figures 10 and 11 illustrate the changes in profile for the 8 i/s case in pipes

pitched at 1/100 and 1/50 for both a decreasing and increasing inlet energy

factors. The reduction in entry energy indicated by the introduction of a loss coefficient will steepen the depth profile resulting in normal depths

being achieved earlier in the pipe. The determination of the correct flow

depth profile for the start of experiments or predictive simulation studies Is important since it effectively provides the flow depth condition immediately

ahead of the time-dependent surge wave.

It will be seen that if the inlet energy is artificially increased (e.g. pumped

inlet condition [5] then the normal flow depth development is delayed. Th<

corollary to this is that a fixed length pipe may be seen to carry rather ;i >r.

7
flow than suggested by the steady flow calculations of normal depth. For

example, with K > 1.0 with pipe lengths less than 16ra, the depths are less

than those which occur with gravity drain conditions, from which it would be

implied that higher flow rates can exist in the pipe without violating permissible half-filled depths. Care should be taken in any experimental work in

this area to ensure that the entry energy is appropriate. Failure to do this

could suggest enhanced carrying capacities that will not be found in practice.

The presence of pipe fittings, such as a junction will result in substantial

energy dissipation and the flow will therefore rapidly increase in depth,

could fill the pipe cross section and be improperly interpreted as the hydraulic

jump.

Table 1 presents normal depth values for 100 mm diameter pipes at gradients of

1/100 and 1/50.

Table 1. Normal Depths in a 100 mm Diameter Smooth Drain,

k = 0.0, Calculated via the Colebrook White

Friction Loss Expression

Pipe Gradient Flow Rate

SL/s

Normal Depth

mm

1/100

1.0 23.6

2.0 33.6

3.0 41.8

4.0 49.2

5.0 56.2

6.0 63.2

7.0 70.8

8.0 79.4

1.0 19.5

2.0 27.5

3.0 33.9

4.0 39.5

5.0 44.8

6.0 49.7

7.0 54.5

8.0 59.3

9.0 64.1

10.0 69.2

11.0 74.7

12.0 81.1

13.0 91.7

8
3.2 SIMULATION OF VERTICAL STACK ANNULAR THICKNESS AND TERMINAL VELOCITY

Equation 8 may be utilized to calculate the annular flow thickness in the

vertical stack for a range of steady flow rates. The terminal velocity

corresponding to this thickness and flow rate may then be calculated from

equation 7.

Figures 12 and 13 illustrates the variation in terminal velocity and annular

thickness with flow rate and also include previous experimental work reported

elsewhere by Wise and Wyly [2,4]. The experimental values of terminal velocity

probably exceed those predicted due to the difficutly in measuring the thickness of the annular layer (the air core interface with the water edge is at the same velocity due to the shear interaction) and as some of the flow will be

’’lost" as spray falling down the center of the pipe. Thus the value of terminal

velocity calculated from equation 7 will be excessive as t is underestimated.

Wise's results also indicate this tendency.

With no energy loss in the 4'ntry bend the entry pipe flow at velocity is close

to the stack terminal velocity. The results presented in figures 13 and 14

conform to the energy boundary equation (13a).

3.3 SURGE ATTENUATION FOLLOWING STACK DISCHARGE TO A HORIZONTAL DRAIN

The necessary equations to simulate wave attenuation along the length of the

pipe have been developed [6] . The transition from the stack to the horizontal

drain may be characterized by the entry energy boundary condition described

previously, based on the annular terminal flow in the stack. Initial conditions along the horizontal drain may be set up utilizing the gradually varied

flow theory outlined.

The wave formation in the horizontal drain may be viewed as being made up of

two separate but superimposed mechanisms. As the surge progresses down the

drain its depth increases as energy is dissipated. At the same time the wave attenuates due to the diffusion so that the superposition of these two effects

results in a peak flow depth downstream of the entry along the pipe length.

Figures 15 and 16 indicate the peak flow depths predicted for a range of inflow

profiles. Figures 17 and 18 present the calculated peak flow rates, calculated at each pipe section as the product of (VxA). It will be seen that

the peak flow rates exceed the maximum inflow rates. This apparent anomaly is

explained as the rising inflow surge generates waves moving with increasing

wave speed greater than the mean velocity. The waves "overtake" the preceding

flow and hence generate both depths and flow rates in excess of those predicted

by a steady flow analysis.

It will also be seen that any surge alleviation provided by the reduction in

surge flow is only manifested for the triangular inflow profiles. Profiles

displaying a plateau at peak inflow rate tend to produce nearly similar maxi- mum depths. This would suggest that the increased flow capacity often referred

to as a result of attenuating surge flows is applicable only to relatively

"sharp", triangular inflow rate vs. time profiles.

9
Figures 19 and 20 indicate the location of the peak flow and peak depth along

the horizontal drain for a range of inflow profiles (the calculation nodes are approximately 0.5m apart). It will be noted that the peak flow rate invariably preceeds the location of peak depth. The same effect is also noticeable

in figures 21 and 22 that indicate the peak depth and flow rate "envelopes"

along the horizontal drain at gradients of 1/50 and 1/100. Figures 23 and 24

illustrate the flow depths at three stations along the drain over the duration

of the simulation.

3.4 EFFECT OF BASEFLOW ON SURGE ATTENUATION

Figures 25 and 26 present peak flow depths and rates, together with their

location for two computational test cases, for baseflows 0.2 £/s and 2 Si/s

with triangular flow profiles rising to 12 1/ s.

It would appear from these results that the baseflow only becomes an important

parameter when the inflow surge time is sufficiently long so that the beneficial attenuation of the peak does not occur. Reference to the earlier figures

15 and 16 indicate that only the 2 second base triangle surge suffered any

appreciable attenuation below the steady flow normal depth associated with

its peak.

It is apparent that further data is required to be more specific as regards

the effects of baseflow.

3.5 COMPARISON OF PREDICTED AND OBSERVED DEPTH PROFILES FOR A HORIZONTAL DRAIN

SUPPLIED BY A VERTICAL STACK

Figures 27 and 28 present comparisons between experimentally measured depth vs. time profiles in a 75 mm pipe set at a gradient of 0.06. Surges enter the pipe

via a vertical stack supplied by a constant baseflow plus discharges from water

closets at higher levels. The results presented were recorded in a dynamics

of plumbing system test program [7,8].

Unfortunately no data is available on the soil stack or drain inflow entry

flow rate profile which is governed by the mixing at the horizontal drain

branch elbows into the soil stack downflow. The flow rate profiles presented

in figures 27 and 28, for the two baseflow cases, were best estimates of the

likely water closet discharge profiles determined from the measured depth vs. time records in the horizontal branch drains at the second and third (story)

level leading to the vertical stack. A single water closet on the third level

and two back-to-back water closets on the second level were simultaneously discharged into constant downflows in the vertical stack.

The estimated discharge profiles were summed (as shown in the figures) and

used to establish the inlet conditions to the drain model based upon the terminal velocity kinetic energy equation. Comparison between observed and

predicted depth variation at a downstream depth tapping are shown. The time

scale of the observed results have been adjusted so that the observed and

predicted profiles move from the steady base flow condition at the same time.

10
This approximate adjustment may account for some of the time scale deviations

shown. The predicted normal depths with the steady baseflow (after the surge)

are within 2 percent of measured values.

The results presented in figures 27 and 28 support the proposed model for

stack to horizontal drain entry model. The baseflow does not affect the

predicted peaks appreciably; however that was expected from the results shown

in figures 25 and 26. Any improvement on the agreement requires an improved

knowledge of the inflow flow rate profile, or at least a better estimate of

the water closet flow rate discharge curves as modified by the junction fittings at the upper levels from the horizontal branch drains. Further research

is necessary to fully account for mixing at entry fittings into the soil stack

downflow. There have been no measurements of the distribution of annular

flow properties in the vertical soil stack which are required to provide

understanding of such gravity driven flow regimes. Investigations of annular

flow phenomena indicate that surface waves are observed on vertical falling

films. That suggests that superposition of surges on steady soil stack flows

may take place, wave velocities greater than the mean falling flow velocities

can occur, and flow redistribution may result, in addition to mass mixing at

junctions or fittings.

For the same surge profile conditions of figure 27, the predicted effects of

the entry loss factor for K=0.5 and 1.0 are shown in figure 29 at three stations along the length of the drain. The higher inlet energy condition

(K=1.0) result in smaller peak depths compared with the lower inlet energy

condition (K=0.5). A steepened surge wave develops with higher energy content,

due to the higher wave velocity (/gA/T) overtaking the upstream flow which

reinforces the surge surge wave front. Also it is noted that the surge depth

attenuation for the higher energy case has not yet occurred up to the 10 m station. At that station the 1/ D ratio (length of pipe to pipe diameter) is

about 130; the prediction of large values of 2,/D were reported in earlier

experiments, reported in [5] with pumped inlet conditions. The sharp falloff

from peak depth for the high energy computations at the 10.0 m station indicates that higher wave velocities have overtaken the surge. Those comparisons

indicate that a need exists to improve the simplified drain inlet energy profile

derivation. Observations made in experiments [6, 7] suggest that a transition

entry length in the horizontal drain, downstream of the elbow, may be required

to account for a flow mixing transition in the modeling.

11
In figure 30, the results for the drain depths for three inlet profiles are

shown. The inlet profiles considered are indicated below.

Flow

The assumed conditions represent waste discharge loading patterns due to a short time interval surge followed, after an interval, by a rise to a steady

discharge flow. In all cases, the initial surge rapidly attenuates (as has

been noted for figures 15 and 16). With increasing times the transient depths

occur greater than the steady-state conditions before trending towards the

normal depths (compare with values of table 1). For case A, the peak depths

are 90 percent of full bore flow, whereas the normal depth was shown to be

about 80 percent of full bore. Currently accepted plumbing code criteria for

allowable partially filled drain pipe depths permit only half-filled pipes

(although the report [9] on which the codes are based, assumed 75 percent for

surge flows). Neither the transient measurement methods or calculation methods

were available to the investigators or code writing panels at that time,

consequently the effects discussed herein were only available by observations and

greatly simplified measurement methods. Here, the larger wave speeds at the

greater rate of increase to the higher steady flow rates cause a steepening of

the wave and occurrence at somewhat earlier times for the same station. The

peak depths occur at pipe length to diameter (A/D) ratios of about 100 for case

A, and 75 for cases B and C. The predicted results are consistent with the

data from the pumped experiments [5] that showed the limit of hydraulic drain

capacity, based upon a prescribed depth for a fixed length, is limited by lowest

steady flow rate surge component and similar ranges of (A/D) values.

12
4. CONCLUSIONS

This report has introduced the concept of an energy entry condition to simulate

vertical stack flow entering a horizontal drain. In general the model has been

satisfactory in that no numerical instability problems have been encountered.

The predictions of flow depth and flow rate through the stack and drain match

the physical mechanisms known to occur for these flow situations. The predictive model provides for the first time, a comprehensive theoretical basis for

explanation of the earlier experiments at NBS to establish the hydraulic capacity of drains.

It may be concluded that the simplified annular stack flow and entry energy

model provides the first theoretical basis as a suitable method of linking

networks of a number of horizontal drains with vertical stack discharging to

a lower level pipe. Network horizontal pipe junctions solutions have

progressed, in a parallel study [10], so that complex interconnected drainage

system analysis is feasible now through combination of the predictive models.

The models will be developed further and validated during the remainder of the

projected program.

13
5. REFERENCES

1. Swaf field, J.A. and Bridge, S., Quarterly Progress Reports to NBS under

Contact No. DA 2004.

2. Wise, AFE, "Drainage Services in Buildings," BRE Paper CP 35/73.

3. Ackers, P., "Resistance of Fluids Flowing in Channels and Pipes," HRP No.

1, HMSO 1958.

4. Wyly, R.S. and Eaton, H.N., "Capacities of Stacks in Sanitary Drainage

Systems for Buildings," NBS Monograph 31, 1961.

5. Wyly, R. S., Investigation of the Hydraulics of Horizontal Drains in

Plumbing Systems, NBS Monograph 1, 86, 1964.

6. Swaffield, J. A., "Application of Method of Characteristics to Model

the Transport of Discrete Solids in Partially-Filled Pipe Flow

NBS BSS 139, February 1982.

7. Galowin, L. S., Dynamics of Drain-Waste-Vent Plumbing Systems - A Preliminary Study, CIB W62 Water Supply and Drainage Seminar, Zurich, Switzerland,

1982.

8. Galowin, L. S., Dynamics of Plumbing System Drainage and Vents, CIB W62

Water Supply and Drainage Seminar, Tel Aviv, Israel, 1983.

9. Hunter, R. B. , Methods of Estimating Loads in Plumbing Systems, NBS

Building Materials and Structures Report, BMS 65, 1940.

10.

Swaffield, J. A., Bridge, S., Galowin, L. S., "Development of Junction

Model Suitable for Inclusion in an Unsteady Partially Filled Pipe Flow

Network Analysis, CIB W62 Water Supply and Drainage Seminar, Tel Aviv,

Israel, 1983.

14
yaoU

f FrouJL-* < l e,uV>cr.t ve^ • >\ Su^«rcr^iea\ fou).

Hot* : 0*0^ - V ± C

eU

t.K«o 4^ -4-^4^ -4* q CS-So^ ~ O

5T

* 3 ** *

Figure 1. Summary of method of characteristics solution of the unsteady

partially filled pipe flow equations

15
Figure 2. Application of the method of characteristics to unsteady supercritical partially filled pipeflow

16
0 cl'\ a^C< o oV «. r »r\ cl

r * m * ^

Cm\Jl 5

/H"J> -Tc d\- -= Q^TT^-fc kL

a0 _ n*

Figure 3. Fully developed annular flow in vertical stacks

17
Figure 4. low rate vs. time Inflow to a vertical stack t inular thickness and terminal energy vs.

ranslated

profiles

into

18
si

GodooW^ *«<•;•.<* “-’'•'V'

1 t *'' el otv V'** J N °' s Q<; °‘l

J.ajt.-.A t>--J ^U-.N A tX. *<,-•— J\-w \..S -t

Cko-t ^"W rati <x~<A f<v*.0'«'t«f *

Figure 5. Calculation of the energy grade line for gradually varied flow

19
(0 ine.r«o. S< S a 3 L*

Of) j-\o — a

I *xma c W-« 5 rv c?i « V\ f V> ^ # /)0«to^VNj o»\\.^ \*^ S«p*f t^tt- »‘ca) ^Iouj

0»v) R , T Vdo^iVn ^

CV^

(j'v} S C «N c JN fll "V -€ cl -Jy

o(V\ C&\<V /’oe-i^.

Figure 6. Development of flow depth profiles in gradually vared pip flow

20
Figure 7c Upstream boundary formed by an entry energy equation

21
Figure 8. Depth profiles for an initial steady flow along the drain at a 1/50 gradient, no energy loss at entry

22
Figure 9. Depth profiles for an initial steady flow along the drain at a 1/100 gradient, no energy loss at entry

23
Figure 10. Influence of the entry energy loss coefficient on initial steady

flow depth profiles (1/100)

Figure 11. Influence of the entry energy loss coefficient on initial steady

flow depth profiles (1/50)

24
s*

in4o 3

6C

25

Annular

stack

flow

thickness

asa

function

of

stack

steady

flow

rate
Figure 13. Stack annular flow terminal velocities compared to existing

experimental data

26
Figure 14. Flow depth at pipe entry compared to annular stack flow tM

for a range of steady flow rates

27
Figure 15. Peak depths predicted along the horizontal drain for a range of

inflow profiles (1/50)

28
Figure 16. Peak depths predicted along the horizontal drain for a range of

inflow profiles (1/100)

29
Figure 17. Peak Flow rates predicted locally along the horizontal drain for a range of inflow profiles (1/50)

30
Figure 18. Peak flow rates predicted locally along the horizontal drain for .1 range of inflow profiles (1/100)

31
Figure 19. Location of predicted peak flow depth and local flow rate for a range of inflow profiles ( 1 / 50

)

32
Figure 20. Location of predicted peak flow depth and local flow rate for a range of inflow profiles (1/100)

33
90

*V

df

0 0?

**% t \ <* iT—

o

i -. 4% •*

0

v0-

*

o vtV

t>l 6

oo

o

c<

V

0

r-o 00 OD u:N

y J f< >

j- S 0

*/C

~V

0 o o

V0-

vr> 0 VP

0

CD

9

34

Figure

21.

Peak

flow

depth

and

local

flow

rate

envelopes

along

the

length

of

the

horizontal

drain

(

1/

SO

)
35

Figure

??.

Peak

flow

depth

and

local

flow

rate

envelopes

along

the

length

of

the

horizontal

drain

(1/1

00

)
36

Figure

23.

Depth

vs.

time

profiles

at

a

number

of

stations

along

the

horizontal

drain

(1/50)
I©

(p

mm

«oou

c.oE

c. k

CM

t-3

oc

37

number

of

stations

along

the

horizontal

drain

(1/100)
38

Figure

25.

Summary

of

baseflow

effect

on

peak

depth

and

local

flow

rate

predictions

(1/50)
9vou<^V«i

«tl

^»00

(

N«

No»s«\

Jfi. V-

^

0 2 S'

1 CD J®

0 - c \

6 i t< *

«*

" * # C o * J

(5 O -P (3

* j*_ •* H 9

6

*

v m

ii tl ^ **0 o 6

'

.

. * ®

*> .1) <3 <3

© G

39

Fleur*

2

A.

Summary

of

haseflow

effect

on

peak

depth

end

local

flow

rate

predictions

(I

/inn)
Figure 27. Comparison of observed and predicted surge attenuation downstream

of a vertical stack to horizontal drain entry

40
Figure 28. Comparison of observed and predicted surge attenuation downstream

of a vertical stack to horizontal drain entry

41
Pe'rrri

/ im*" (sec)

Oerrt*

f'/r m)

Pept),

(trt m)

Figure 29. Comparison of predicted surge depths downstream of a vertical stack

to a horizontal drain entry for two entry energy loss factors

42
100 —

80 -

60

£ 40

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. * * X XX X X

*«

> a

©O 0 0 oo 0 • • ° n ’• *•» » • # |T* • • Station

a (m)

x^> **** o 0°

0 o°° .0

a ooo° o ® ^ ^

. 1 1 1

2.5 • 5 O

7.5 x

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12.5 0

$ 0 C 0$

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14 16 18 20

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Q

« O • • • o • • *X X X K

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* • ® *° e o°o o *

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Figure 30. Comparison of predicted surge profile depths in a horizontal Ir.iln

downstrem of a vertical stack for composite loading profiles

43
NBS-114A (rev. 2-80

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

. PUBLICATION OR

REPORT NO. NBSIR 85-3108

2. Performing Organ. Report No. 3. Publication Date February 1985

4. TITLE AND SUBTITLE

A PRELIMINARY STUDY OF THE VERTICAL STACK TO HORIZONTAL DRAIN ENTRY CONDITION AS AN

EXTENSION TO THE MODELING OF UNSTEADY PARTIALLY FILLED PIPE FLOW

5. AUTHOR(S)

Dr. J. A. Swaf field, Lawrence S. Galowin

6. PERFORMING ORGANIZATION (If joint or other than N BS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

10.

SUPPLEMENTARY NOTES

Q2] Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bl iography or literature survey, mention it here)

The finite difference based method of characteristics model for unsteady partially

filled pipe flow was extended to include the stack to horizontal drain entry boundary

condition. The conditions at drain entry are defined in terms of the energy of the

terminal annular flow velocity in the stack, together with an appropriate loss

coefficient as the entry function. The hydraulic solutions link the branch drains,

fittings, vertical soil stack and building drain. The analysis permits any combination of drainage load patterns from simultaneous, overlapping or sequence of

discharge events. Preliminary simulations utilizing this model indicates that the

modeling technique extends the existing horizontal network analysis program to a simulation of multistory building drainage systems. The sizing procedure determines

the hydraulic capacity of drains for specified pipe sizes, pipe pitch and well

roughness factors.

12. KEY WORDS (Six to twelve entries ; alphabetical order; capitalize only proper names; and separate key words by semicolons)

drain stack to horizontal drain flows; drain system design; plumbing drainage;

unsteady flow in drainage

13. AVAILABILITY 14. NO. OF

[X] Uni imited

| | For Official Distribution. Do Not Release to NTIS

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53

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