REFERENCE

PUBLICATIONS p, NBSIR 82-2614

NAT'L INST. OF STAND & TECH

Ainot sbissa Prediction of Floating Solid

Velocities in Unsteady Partially

Filled Pipe Flow

Drainage Research Group

Department of Building Technology

Brunei University

Uxbridge

Middx , U K.

July 1983

100 DEPARTMENT OF COMMERCE

-U56 ONAL BUREAU OF STANDARDS

82-2614

1983

•Research li •National •National biureau ot Standards

Gaithersburg, Maryland 20899

information Center

NBSIR 82-2614 Ho.

THE PREDICTION OF FLOATING SOLID

VELOCITIES IN UNSTEADY PARTIALLY

FILLED PIPE FLOW

Dr. J. A. Swaffield

Drainage Research Group

Department of Building Technology

Brunei University

Uxbridge

Middx., U K.

July 1983

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

NATIONAL BUREAU OF STANDARDS. Emest Ambler. Director

SUMMARY

The method of characteristics is applied to solve the unsteady partially filled

pipe flow equations and to predict the velocity of floating solids assumed to

travel at a fixed percentage of the local flow velocity.

Experimental verification for the technique is provided for cylindrical solids

in a 100 mm diameter drainage pipe at a range of gradients from 1/40 to 1/150.

The system upstream boundary conditions are shown to be capable of

representation in terms of the infow energy at the pipe entry section.

Steady flow floating solid to flow velocity ratios are presented at 1/150 pipe

gradient and further areas of experimental work to determine the variation of

these ratios with pipe gradient and flow depth are identified.

iii
PREFACE

This report is one of a group documenting National Bureau of Standards (NBS)

research and analysis efforts in developing water conservation test methods,

analysis, economics, and strategies for implementation and acceptance. This

work is sponsored by the Department of Housing and Urban Development/Office of

Policy Development and Research, Division of Energy Building Technology and

Standards, under HUD Interagency Agreement H-48-78.

Report prepared by Dr. J. A. Swaf field. Senior Lecturer, Drainage Research

Group, Department of Building Technology, Brunei University, Uxbridge, U.K., as

a result of a study leave period as a guest research worker at NBS/Stevens

Institute of Technology.

Experimental results included in this report are drawn from work undertaken by

the Drainage Research Group at Brunei University on behalf of NBS in June 1980.

iv
TABLE OF CONTENTS

Page

SUMMARY iii

PREFACE iv

LIST OF FIGURES AND TABLES vi

NOTATION vii

1. INTRODUCTION 1

2. BASIS FOR THE TRANSPORT MODEL 2

2.1 VELOCITY PROFILES IN OPEN CHANNEL FLOW 2 2.2 FLOATING SOLID VELOCITIES 2 2.3 PREDICTION OF FLOW ATTENUATION IN PARTIALLY FILLED PIPE FLOW ... 2

2.3.1 Initial Conditions 6

2.3.2 Boundary Conditions 6

2.4 CALCULATION OF SOLID VELOCITY 9

3. DISCUSSION OF THE FLOATING SOLID TRANSPORT MODEL 12

3.1 STEADY FLOW SOLID TRANSPORT VELOCITY MEASUREMENT 12

3.2 EXPERIMENTAL VERIFICATION 12

3.3 ENTRY BOUNDARY CONDITIONS AND UNSTEADY FLOW SOLID TRANSPORT .... 15

3.4 FLOATING SOLID TO FLOW VELOCITY RATIO AND ITS EFFECT ON

PREDICTED SOLID TRANSPORT 21

4. CONCLUSIONS AND FURTHER WORK 26

REFERENCES 27
LIST OF FIGURES AND TABLES

Page

Figure 1. Typical velocity profile in open channel or partially

filled pipe flow 3

Figure 2. Floating solid velocity in open channel flow 4

Figure 3. Summary of method of characteristics solution to the

unsteady partially filled pipe flow equations 5

Figure 4. Development of the solution in the x-t plane 7

Figure 5. Alternative entry boundary equations 8

Figure 6. Solid migration through an inflow profile 10

Figure 7. Calculation of floating solid velocity and position in

the pipe 11

Figure 8. Steady flow solid velocity results at a 1/150 pipe

gradient 13

Figure 9. Schematic layout of solid transport test rig 14

Figure 10. Typical solid transport data as produced by the Apple II

system 16

Figure 11. Comparison of observed and predicted solid velocities

along the test pipe at a gradient of 1/40 17

Figure 12. Comparison of observed and predicted solid velocities

along the test pipe at a gradient of 1/80 18

Figure 13. Comparison of observed and predicted solid velocities

along the test pipe at a gradient of 1/150 19

Figure 14. Comparison of flow velocity and depth predictions at pipe

entry for the normal depth and inflow energy boundary

conditions 20

Figure 15. Maximum predicted depths along the test pipe and both

normal flow depth and entry energy upstream boundary

conditions 22

Figure 16. Illustration of the effect of solid insertion time on predicted transport along the test pipe 25

Table 1 . Predicted Total Solid Transport Time for 14m Pipe Gradients

and Velocity Ratios 26

vi
NOTATION

A

C+, C“

C

c

D

E

F r

G

g

h

L

m

N ,n

P

Q

So

S

T

t

V

Xl-4

x

a

At

Pipe flow cross section area

Notation referring to the positive and negative characteristics

Chezy coefficient

Wave speed

Pipe diameter

Specific energy = h + V^/2g

Froude N° = V//gh

Pipe slope

Acceleration due to gravity

Flow depth

Pipe length

Hydraulic mean depth

N° of pipe length sections employed

Wetted channel perimeter

Flow rate

Pipe slope

Slope of energy grade line, defined by Manning's Equation

Surface width of flow within partially filled channel

Time

Local mean velocity

Functions of h, V, c and s calculated at each base point at each

time step

Distance, positive in initial flow direction

Pipe slope, S Q = sina

Time step

vii

1
Ax Pipe section length

9 At/Ax

p Fluid density

t 0 Wall to fluid shear stress

Subscripts

A, B, C Calculated points in an x-t grid at time t

c Critical flow conditions

n Normal flow conditions

P Calculated points in an x-t grid at time t + At

R, S, S' Interpolated points in an x-t grid at time t

1 Upstream boundary section

N+l Downstream boundary section
1 . INTRODUCTION

The transport of waste solids under unsteady flow conditions in partially

filled drainage pipes may be characterized by the specific gravity of the

solid. In the case where the specific gravity of the solid, including its

saturated state, exeeds unity, it is necessary to include solid to pipe wall

sliding friction in the model, together with a representation of the significant flow depth change that can occur across the solid. This case was the

subject of previous reports in the series (1,2).

However, if the solid specific gravity is less than unity then the solid may

be considered to "float'' in the flow if sufficient depth is provided by the

transporting fluid. Quite obviously the attenuation effects previously

described (3,4) will eventually reduce the transporting flow depth to the

point where solid to pipe wall sliding friction is again established; however,

this will indicate the imminence of solid deposition and, as such, may be used

as a cut-off indicator for the proposed floating solid transport model.

This paper presents a mathematical model, based on the application of the

method of characteristics to solve the unsteady open channel flow equations,

including boundary conditions chosen to represent the flow entry energy. The

model predictions are compared with experimental results based on NBS cylindrical model solid transport tests undertaken at Brunei University in 1980.

1
2.1 2. BASIS FOR THE TRANSPORT MODEL

VELOCITY PROFILES IN OPEN CHANNEL FLOW

Figure 1 illustrates typical fluid velocity profiles in open channel or partially filled pipe flow. These profiles are influenced by the presence of

the channel walls as well as the free surface, and, in unsteady flow will vary

with flow depth in circular cross-section channels.

In view of the complexity of velocity profiles across a pipe flowing partially

full, and the lack of both detailed experimental data and theoretical velocity

distribution expressions, similar to those available for full bore pipe flow,

it is necessary to generalize the flow velocity profiles. For open channel

flow in channels having circular cross-sections, it is generally accepted (5)

that the maximum velocity occurs at about 15-25 percent depth below the free

surface and that this free surface velocity is approximately 80-85 percent of

the mean velocity, which occurs at about 60 percent depth below the free

surface. These figures will change with depth-diameter ratio, roughness, etc.

2.2

FLOATING SOLID VELOCITIES

If a model solid has a specific gravity less than unity and is sufficiently

small to cause little flow obstruction, it may be assumed to travel at the

local flow velocity appropriate to its position on the fluid velocity profile

described above. Thus, a small solid floating on, or just below, the flow free

surface would be expected to move at 80 percent, or more, or the flow mean

velocity, as shown in Figure 2.

In this case, a calculation technique that predicts unsteady flow local mean

velocities may be simply utilized to yield both the solid velocity and a measure

of its progress along any open channel.

2.3

PREDICTION OF FLOW ATTENUATION IN PARTIALLY-FILLED PIPE FLOW

The application of the method of characteristics to the prediction of flow

depth, mean velocity and associated attenuating flow rate in partially filled

pipe flow has been detailed previously (3) and the basic theory will not be

restated here. Figure 3 illustrates the transformation of the unsteady flow

equations of continuity and motion:

gili + g (S-S ) + viZ+_av=o

3x ° 3x 3t

VT 3h 4. T 3h + A av = n 3x 3t 3x

into a pair of total differential equations solvable along two characteristic

lines, C*, C“ drawn in the x-t plane. In terms of Figure 3, these equations

may be expressed as

dV +£dh + g ( S _ S ) =0

dt " c dt 0

2
Figure 1. Typical velocity profile in open channel or partially filled

pipe flow

3
Figure 2. Floating solid velocity in open channel flow

4
Figure 3. Summary of method of characteristics solution of the

unsteady partially filled pipe flow equations

5
provided that j** = V + c where c iz^

dt / T

The slope of the characteristic lines in Figure 3 depends on both flow mean

velocity, V, and local wave speed, c, and thus depend on the type of flow

being considered. If V < c then the flow is termed subcritical and disturbances traveling at wave speed may propogate both upstream and downstream.

Conversely, if V > c, the flow is termed supercritical and disturbances can only propogate downstream. In the case of partially filled pipe flow in

drainage sized pipes, normally of 100 mm diameter, and rarely set at slopes

less than 1/100, it is supercritical flow that is the predominant design condition. The analysis presented in this paper, particularly with reference to

pipe boundary conditions, is therefore limited to this flow condition.

2.3.1 Initial Conditions

The solution technique requires that flow depth and velocity are known at all

nodes along the pipe at time zero. This requires the establishment of a small

initial steady flow, (3) where the relevant depths, mean velocities and wave

speed are calculated from the Manning steady flow equation:

a 2/3 1/2

Q - 1

- s °

and c _ gA

T

where n is the surface roughness; a schematic of the solution technique is

depicted in Figure 4.

2.3.2 Boundary Conditions

In addition to the initial conditions, mentioned above, it is necessary to

define the boundary equations that determine flow depth or velocity at pipe

inlet and exit. As the current analysis deals only with supercritical flow,

the exit conditions are solely governed by the upstream flow conditions, i.e.,

the presence of the pipe exit cannot be communicated upstream as the

approaching flow velocity exceeds the necessary wave speed.

The entry boundary condition requires more consideration. Figure 5 illustrates

two boundary equations considered in this analysis, namely the simple assumption that the entry depth conforms to the "normal" flow depth appropriate to

that flow rate, and calculated via the Manning equation, and a more realistic

boundary condition incorporating the entry energy of the flow.

Referring to Figure 5, the entry energy of the flow may be determined in terms

of the water jet velocity; u,

2 § 2 g a^

where a, is the area of the delivering tube.

6
N ! concAit »or\5 <K at ± = a

at a\\ r\ocA « s 1 — at \ .

Figure 4. Schematic of the development of the solution in the x-t plane
V* > 'V

Figure 5. Alternative entry boundary conditions

8
This entry energy is equated to the flow specific energy at entry to the

partially filled pipe flow,

where h and V are the flow depth and mean velocity at pipe entry as shown. By

continuity it follows that:

2 g A2

where A is the flow area, and a function of depth h. The inflow Q is known as

a function of time and the flow depth may then be calculated by the expression:

h = -Si. (1/a2 - 1/A2

) 2 g

An iterative procedure is used, employing trial values to calculate A and

satisfy the equivalence between entry and initial flow at entry specific

energies

.

2.4 CALCULATION OF SOLID VELOCITY

Figure 6 illustrates the effect on the solid if its floating velocity, Vs , is

less than the mean flow velocity, Vj. Relative to the carrying flow, the solid

is seen to move back along the inflow profile, initially accelerating, but,

depending on the velocity decrement, eventually passing behind the peak of the

inflow profile and hence decelerating towards possible deposition in the "tail"

of the inflow profile. It must be remembered that this description is based on solid velocity relative to the surrounding water, and while this migration is

in process, the solid is being swept downstream at absolute velocities that

depend on pipe gradient, diameter, pipe slope and the magnitude and shape of

the inflow profile.

Figure 7 illustrates the calculation technique employed to determine the solid

absolute velocity and its position along the pipe at any given time. At any

time the solid velocity is assumed to be given by:

V s = DV -Vi,

where DV is <1, the velocity ratio between the solid and the surrounding water

defined as:

DV = ratio of the solid velocity, Vs , to the local mean water velocity,

Note that V^ is calculated by interpolation between nodes

bracketing the solid position at any time.

Once the V s value is known, the solid position at the end of the next time

step, X t + dt = Xt + V s *At

and the process repeated at each subsequent time step.

9
Figure 6. Solid migration through the inflow profile

10
hrvoc^Q.. L u\

Figure 7. Calculation of floating solid velocity and position in the pipe

11
3. DISCUSSION OF THE FLOATING SOLID TRANSPORT MODEL

3.1 STEADY FLOW SOLID TRANSPORT VELOCITY MEASUREMENT

Figure 8 illustrates the results of a primary study of transport of a typical

NBS cylindrical model solid along a 1/150 gradient, 100 mm diameter test pipe

at two steady water flow rates of 2.0 and 1.0 1/s. In each case the mean water

flow velocity has been calculated from Manning's equation.

In the case of the empty solid is may be seen that the velocity ratio was roughly constant for both the steady flow rates. In these cases the solid

floated at about its centre line. Some variations in the velocity ratio would

be expected as the velocity profiles at the two flow depths would vary. However,

the values are sufficiently close to support the use of a constant value in the

simulation.

This would not appear to be the case for the solid when filled with water as a means of producing a specific gravity slightly in excess of unity. At 2 1/s

the solid diameter and flow depth coincided almost exactly and the solid was observed to move steadily along the pipe with no disruption to the local water

surface. However, at the 1 1/s flow rate the solid was uncovered in the flow,

resulting in the flow disruption shown and a much lower transport velocity. It

may be concluded from these results that the assumption of a constant ratio

between floating solid and accompanying water velocity is only acceptable above

certain limiting flow rates, sufficient to provide adequate depths of flow. In

the case of the empty solid flow depths less than 18 mm would result in the same effect, i.e. flow rates, at 1/150 gradient, of approximately 0.4 1/s. In

the full solid case the limiting flow rate would be in the region of 2 1/s. In

both cases the flow disruption at flow rates below the values mentioned would

be expected to result in deviations between measured solid velocities and the

output of a simulation based on constant velocity ratios.

Measurement of similar velocity ratios at various gradients and flow rates is

currently part of the NBS research programme in conjunction with work at Brunei.

For the simulation presented in this paper the velocity ratios are assumed to

apply at all pipe gradients, however, as flow depth is dependent on gradient,

the limiting flow rates mentioned will also be dependent on gradient. For

example, the 18 mm depth limit for the empty solid will be reached at a flow

rate of 0.75 1/s at a 1/80 gradient and 1.1 1/s at 1/40. In view of these

limits, it might be expected that the model, as presented, will perform best at

the flatter pipe gradients.

3.2 EXPERIMENTAL VERIFICATION

Figure 9 illustrates the test pipe configuration employed to record the

transport performance of NBS cylindrical solids in June 1980. These results

have been utilized in the current paper to provide experimental data as a comparison to the output of the simulation described.

The test pipe consisted of 14 m of 100 mm diameter clear UPVC pipe, Manning

coefficient 0.008, capable of gradient adjustment from 1/40 to 1/300. Solid

12
Figure 8. Steady flow solid velocity results at a 1/150 pipe gradient

13

velocity was recorded by six pairs of photoelectric cells at 0.3 m separation

along the first 12 m of pipe. Output from the photocells was fed directly to a microcomputer used produce the data illustrated in Figure 10.

The volume of water leaving the pipe ahead of the solid was recorded by means

of level sensing in the terminal collection tank.

The solids tested were initially placed in the test pipe and inflow to the pipe

was provided by a Geberit drop valve cistern, set to 10.5 liter capacity, and

supplying water to the test pipe via a 43 mm diameter downpipe as shown.

Discharge rates from the cistern were monitored by means of a pressure transducer and a high speed pen recorder and provided the input data for the current

simulation. The solid used in all tests considered in this report was an NBS

hollow cylindrical model solid, length 80 mm, 37 mm diameter, initially

stationary in the pipe 1 m from entry.

3.3 ENTRY BOUNDARY CONDITIONS AND UNSTEADY FLOW SOLID TRANSPORT

Figures 11, 12 and 13 illustrate the predicted solid velocities along 14 m of

100 mm diameter pipe following a 10.5 1 discharge into the system, the peak

inflow being 6 1/s and the inflow duration being less than three seconds. Two

model boundary conditions are illustrated: namely normal depth at entry and

the dependence of the entry depth on flow energy. A velocity ratio of 0.9 was used to illustrate the following effects of the boundary condition choice on

the predicted solid velocity.

(i) Normal depth at entry dictates that the maximum flow depth occurs at this

point as the inflow profile peaks. Thereafter the depth decreases along

the whole pipe length. This condition also limits the velocity at entry

to that calculated from Manning's equation.

(ii) Normal depth is dependent on pipe gradient so that the entry depth and

velocity depend on pipe gradient. However, as the cases chosen are restricted to supercritical flow, and as an air gap exists between the

cistern discharge pipe and the test drainage pipework, the entry conditions would be expected to remain constant and independent of gradient.

Figure 14 illustrates the predicted flow depths and velocities at the

pipe entry for both the entry energy and normal depth models. It will

be seen that the energy model predictions are unaffected by pipe

gradient. Further, it will be seen that entry depth decreases with

increasing flow rate in the energy model, as the kinetic energy term

predominates, the opposite to the normal depth model where flow depth

and velocity vary together through Manning's equation.

Comparison of the predicted solid velocity results indicate that the normal

depth assumptions are incapable of providing solid velocities of the same magnitude as those observed on the test rig and included in Figures 11 to 13.

The assumption that the entry conditions are dependent on the inflow energy,

however, give solid velocities close to those recorded; i.e. agreement better

than 10 percent at pipe entry.

15
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CORRELATION

COEFFICIENT <

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17

Figure

11:

Comparison

of

Observed

and

Predicted

Solid

Velocities

Along

the

Test

Pipe

at

a

Gradient

of

1/40.
18

Figure

12:

Comparison

of

the

Observed

and

Predicted

Solid

Velocities

Along

the

Test

Pipe

at

Gradient

of

1/80.
19

Figure

13:

Comparison

of

the

Observed

and

Predicted

Solid

Velocities

Along

the

Test

Pipe

at

Gradient

of

1/150.
I

20

Figure

14:

Comparison

of

the

Flow

Velocity

and

Depth

Predictions

at

Pipe

Entry

for

the

Normal

Depth

and

Inflow

Energy

Boundary

Conditions.
Figure 15 illustrates the effect on flow depth along the pipe for the two types

of boundary conditions above. As mentioned, the normal depth condition results

in a gradually reducing maximum depth along the whole pipe, indicating a progressive attenuation along the system. The energy model however, indicates

a maximum depth point reached between 5-7 m from pipe entry, the exact location

moving downstream with increasing flowrate. This is effectively the transition

zone from the entry energy to the normal flow depth modified slightly by an overlying attenuation effect.

For each of the 3 gradients shown (1/40 1/80, 1/150) maximum depth is slightly

less than the normal depth appropriate to the maximum inflow of 6 1/s, the

divergence being greatest, as expected, in the 1/150 case and least in the 1/40 case.

These results are considered sufficient to show that the energy entry condition

is the more realistic model for the flow conditions that existed in the experiments. It also indicates that the method of characteristics solution is capable of dealing with transition phenomena under unsteady flow conditions. This

observation is limited to supercritical flow at present due to the possibility

of hydraulic jump formation in subcritical flows set up under the condiitons

described, however, this is not considered a major limitation as pipe gradients

flatter than 1/150 are rare in most practical applications. The subcritical

case will be the subject of further investigations as Brunei experimental data

is available for the NBS cylindrical solid at gradients down to 1/300.

3.4 FLOATING SOLID TO FLOW VELOCITY RATIO AND ITS EFFECT ON PREDICTED SOLID

TRANSPORT

The steady flow test results at 1/150 gradient discussed in section 3.1 indicated

that the assumption of a constant velocity ratio between solid and accompanying

flow, even for a nominally floating solid, broke down in shallow flow conditions.

Reference to Figure 13, for 1/150 gradient, indicates that in this case, close

agreement was achieved between the measured and predicted solid velocities over

the majority of the pipe length, however a study of the depth versus time

profiles for this simulation indicates that the flow depths were sufficient to

comply with the restrictions identified earlier.

As might be expected, with steepening gradient, this is not the case at the

1/80 and 1/40 gradients, thus the predicted solid velocities in these two cases would be expected to exceed those recorded during the tests. Due to the

reduced depth characteristic of the steeper pipe gradients, the effective blockage of the solid within the pipe flow cross section increases which produces a secondary "blocking" effect, identifiable in Figures 11 and 12. The increased

blockage tends to 'store' more water behind the solid in practice than predicted by the simulation, which assumes zero disruption to the flow around the

solid. Hence in the later pipe length stage, the solid transport velocity will

tend to be higher in practice than the value predicted by the simulation. This

effect is clearly seen for both the 1/40 and 1/80 cases, of Figures 11 or 12

particularly for the 'full' solid, which has the larger flow disruption effect

as seen in Figure 8.

21
Figure 15: Maximum Predicted Depths Along the Test Pipe for both

Normal Flow Depth and Entry Energy Upstream Boundary

Conditions

.

22
While Figures 11 to 13 illustrate the variations in solid velocity along the

waste pipe, it will be noted that the total transport time increases with

reductions in both pipe gradient, from 1/40 to 1/150, and solid velocity ratio,

from 1.0 to 0.8. Table 1 contains predicted results for a 14 m pipelength and

reinforces the 'backward' movement of a solid through the inflow profile

illustrated in Figure 6. Table 1 was compiled for the constant inflow profile

illustrated in Figure 11. This relative motion of the solid through the inflow

profile is also dependent on the initial position of the solid in the profile.

Figure 16 illustrates, for a velocity ratio of 0.8, the effect of introducing

the solid at various times during the inflow duration. These results indicate

that solid transport efficiency is severely reduced by late insertion. The

results are significant for WC design, particularly in water conservation and

reduced flush volume designs.

23
Table 1. Predicted Total Solid Transport Time for A 14 m Pipe for a Range of

Gradients and Velocity Ratios

Time to reach pipe discharge, secs

.

Gradient 1/40 1/80 1/150

Velocity 1.0 8.50 10.20 12.08

Ratio

0.9 10.48 13.40 16.80

0.8 13.78 17.80 24.68

24
SoV.cJ.

25

Figure

16:

Illustration

of

the

Effect

of

Solid

Insertion

Time

on

Predicted

Transport

Along

the

Test

Pipe.
4. CONCLUSIONS AND FURTHER WORK

The application of a method of characteristics solution to the unsteady flow

equations describing partially filled pipeflow in the special case of floating

solids with known solid to flow velocity ratios has been demonstrated. Experimental verification has indicated that, provided the flow depth is sufficient

to give flotation and provided that the disruption to the flow is slight, accurate predictions of solid velocity may be achieved. The study has also shown

that the inclusion of flow entry energy as an alternative upstream boundary

condition in supercritical channel flow is practical and is responsible for the success of the model presented. This upstream boundary condition will be utilized further in parallel work at Brunei concerned with flow attenuation prediction. The method is also shown to be capable of predicting the transition to

normal flow depth, including the overlying attenuation of the flow depth profile as it progresses downstream. The results are considered to have particular application for water conserving w.c. application with existing drainage

pipe networks.

Further work will be required to determine the solid to water mean velocity

ratios and to identify the dependence of such ratios on pipe slope and flow

depth. An extension of the model to the subcritical case, including provision

for a potential hydraulic jump will be considered.

26
REFERENCES

1. J. A. Swaf field "An Initial Study of the Application of the Method of

Characteristics to Unsteady Flow in Partially Filled Pipe Flow”. NBSIR

81-2308, July 1981.

2. J. A. Swaf field "Application of the Method of Characteristics to Model

the Transport of Discrete Solids in Partially Filled Pipe Flow”. BSS 139,

February 1982.

3. J. A. Swaf field "Application of the Method of Characteristics to Predict

Attenuation in Unsteady Partially Filled Pipe Flow." NBSIR 82-2478,

December 1981.

4. P. Burberry "Techniques for the Estimation of Flows in Foul Drains in the

Light of Present Water Conservation Proposals". Seminar on Drainage

Design, Brunei University, Uxbridge, May 1978.

5. V. T. Chow "Open Channel Hydraulics", McGraw Hill, 1970.

27

NBS-114A (rev. 2-80

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

82-2614

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

4. TITLE AND SUBTITLE

The Prediction of Floating Solid Velocities in Unsteady Partially Filled Pipe Flow

5. AUTHOR(S)

John A. Swaffield

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions) 7. Contract/Grant No. national bureau of standards

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

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

Department of Housing and Urban Development

451 7th Street, SW

Washington, D.C. 20410

10. SUPPLEMENTARY NOTES

| | 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 si gn i fi cant bibliography or literature survey, mention it here)

The method of characteristics is applied to solve the unsteady partially filled

pipe flow equations and to predict the velocity of floating solids assumed to travel at

a fixed percentage of the local flow velocity.

Experimental verification for the technique is provided for cylindrical solids in

100 mm diameter drainage pipe at a range of gradients from 1/40 to 1/150.

The system upstream boundary conditions are shown to be capable of representation

in terms of the inflow energy at the pipe entry section.

Steady flow floating solid to flow velocity ratios are presented at 1/150 pipe

gradient and further areas of experimental work to determine the variation of these

ratios with pipe gradient and flow depth are identified.

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

floating solids; partially filled pipe flows; pipe flow with solids; plumbing drains.

id. AVAILABILITY

| XI Unlimited

|

| For Official Distribution. Do Not Release to NTIS

| | Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

[}Q Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

15. Price

USCOMM-OC 6043-P80
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