6-2canal Design Viscous Flow

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OPTIMAL OPEN CHANNEL

SECTIONS FOR VISCOUS FLOW

Article in ISH Journal of Hydraulic Engineering · June 2012

DOI: 10.1080/09715010.2000.10514676

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Prabhata K. Swamee

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40

THE

INDIAN

SOCIETY

FO R

HYDRAULICS

JOURNAL

OF

HYDRAULIC ENGINEERING

OPTIMAL O P EN C HA NN EL S E CT IO N S FOR

VISCOUS

FLOW

by

Prabhata

K. Swamee , M.ISH

ABSTRACT

VOL. 6. 2)

Th e

minimum

area

open channel section is generally adopted in chemical plants for

transferring viscous fluids. Such a section is e co no mi ca l as it in vol ves the least amount of

ma te ri al and the l east s ur fa ce area for its construction. In this paper explicit equation for the

optimal dimensions of various open channel sections carrying viscous fluids have been obtained

and their salient features have been pointed out. It is hoped that these equations will be useful

to the design engineer.

KEY W O R D S : Channel design, Explicit equations, Optimal sections, Optimization, Uniform

flow and Viscous flow.

INTRODUCTION

Design

of

open channels carrying viscous fluids involves optimization

o.f

cost

by

reducing

the flow area and flow perimeter to a minimum. A review of literature reveals that although the

e qu at io ns for t he o pt im al c ha nne l d im en si on s are a va il abl e f or t ur bu le nt flow e.g. S tr ee te r

1945;

Swamee and

B ha ti a 1972; Tr out 1982; S ak hu ja et al. 1984;

Gu o

and Hu gh es 1984;

Flynn and Mariamno 1987; Loganathan 1991; Froehlich 1994; Monadjemi 1994; and

Swamee

1995; Swamee et al. 2000 a-b , no such equations are available for viscous flow.

ANALYTICAL CONSIDERATIONS

For steady and unifonn viscous flow the Navier-Stokes equations of motion may

be

reduced

to the following two dimensional

fonn of

Poission s equation:

, 2

a v

a v

g

- -2

I)

y z v

w he re y

and

z

=

horizontal and vertical distances, respectively, from the

center ofthe

channel

boltom;

=

velocity at t he p oint y, z); g

=

gravitational acceleration; So

=

bed slope; and v

=

kinematic viscosity

of

the fluid. The solution

of

Eq. I for discharge Q in an open rectangular

channel as derived by Boussinesq in 1868 and later modified by Cornish 1928), and Woo and

Brater

1 961) is as shown in Eq. 2).

1. Professor

of

Civil Engineering, University

of

Roorkee, Roorkee V.P. .

Note: Written discussion of this paper will be

open

until 31 st December, 2000.

ISH JOURNAL OF HYDRAULIC ENGINEERING, VOL.

6

2000. NO.2


3/6

VOL. 6, Z

OPTIMAL OPEN CHANNEL

SECTIONS FOR

VISCOUS FLOW

41

2)

4a)

4b)

g b Y ~ [

8 Y n ~

-5 2n+l7


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42)

OPTIM LOP N H NNELSEcnONS FOR VIS OUS FLOW VOL. 6 2

• Jl m

m

b = 3.561 0.375 L

2 J I + m

2

m

=1.781 2J1+m

2

m f 3 7 5 L

p =3.561 2J1+m

2

m

f 6 2 5

L

r 0.25

A · = 3. 1712V 1+ m

2

m L

2

SALIENT FEATURES

7a )

7b

7c

7d

Equation 5 reveals that v has a strong influence on the optimal open channel geometry.

On

the other hand, for a smooth turbulent flow, as depicted by the length scale Swamee

1995 . L is given by

vO.0

4

Q0375

L = g

S

O)O.208 8

Equation 8 shows the kinematic viscosity has little role to play in determining the optimal

channel dimensions in turbulent flow

A perusal

of

Table I shows that the semicircular section

of

diameter 3L has the least flow

arca and thc flow perimeter. The other optimal sections, following the constraints

of

the shape,

are closest to this semicircular section

DESIGN EXAMPLE

The objective

is

to

design a rectangular channel section

for

carrying a discharge

0.025

m ls on a slope

of

0.005. The kinematic viscosity

of

fluid is 4 x 10 m /s Adopting g = 9.8 mis

and using Eq. 5 , L = 0.213 m. Further, using Eqs. 4 a and 4 b , b = 0.585 m; and =

0.293 m. The corresponding flow area

A

= b Y = 0.1714

m

yielding Y = 0.146 m/s

jTRIANGULAR

SECTION hI RECTANGULAR SECTION

c

TRAPEZOIDAL SECTION d

CIRCULAR

SECTION

FIG CHANNEL SECTIONS:

ISH

JOURNAL OF HYDR UliC ENGINEERING VOL 6, 2000. NO.2


5/6

VOL. 6, 2)

OPTIM L OPEN CH NNEL

SECTIONS FOR VISCOUS

R OW

TABLE-}

PROPERTIES

OF

OPTIMAL

CHANNEL SECTIONS

43)

Section Shape Side Slope Section Shape Coefficients

k

b

or

m

k

D

k

y

k

p

k

A

k

v

1 2) 3) 4) 5) 6) 7)

Triangular 1.000 0.000 1.942 5.492 3.771 0.265

Rectangular

0.000 2.746 1.373 5.492

3.771 0.265

Trapezoidal 0.577 1.673 1.449 5.020 3.638

0.275

Circular

3.004 1.502

4.720 3.535 0.282

• Not applicable,

CONCLUSIONS

has been possible to obt in optim l dimensions open ch nnel sections for viscous

flow These dimensions are strongly influenced by the kinematic viscosity of the fluid. Subject

to the geometrical restriction placed by the way of defining the shape, the optimal channel

section becomes closest to the corresponding semicircular section

which

is

the best

hydraulic

section for open ch nnel flow

REFERENCES

Cornish, R. J. 1928). Flow in a Pipe ofRectangular Cross Section. Proc.

Roy

Soc, of London,

Series A, 120, pp. 691-700.

Davis, S. J. and White, C

M

1928), An Experimental Study

of

Flow of Water in Pipes of

Rectangular Section. Proc,

Roy

Soc. ofLondon, Series A, 119, pp. 92-107.

Flynn, and Marimno, M. A. 1987). Canal Design; Optimal Sections. 1 Irrig. and Drain.

Engg., ASCE, 113 3), pp. 335-355.

Froehlich, 0, C. 1994). Width and Depth-Constrained Best Trapezoidal Section. J. Irrig. and

Drain. Engg., ASCE, 120 4), pp. 828-834.

Guo, C.

Y

and Hughes,

W

C. 1984). Optimal Channel Sections with Free Board. J Irrig. and

Drain. Engg., ASCE,

110 3),

pp.

304-313,

Loganathan, G V 1991). Optimal Design of Parabolic Canals. 1 Irrig. and Drain. Engg.,

ASCE, II 7 5), pp. 716-735.

Monadjemi, P

1994).

General Formulation

of

Best Hydraulic Channel Section. J. Irrig. and

Drain. Engg., ASCE, 120 1), pp. 27-35.

Sakhuja, V S., Singh, S.

and

Paul, T C.

1984).

Discussion

of

Channel Design to Minimize

Lining Material

Costs

by

Thomas J.

Trout

J. Irrig. and

Drain.

Engg., ASCE, II

0 2), 253-254.

Streeter, V 194 . Frooomic CanaJ Cross Sections. Trans. ASCE, 110, pp. 421-430.

Straub, G., Sir - H.C 195 ). Open Channel Flow at SmaIl Reynolds

Number. Trans. ASCE 1 - -Po


6/6

44)

OPTIMAL

OPEN CHANNEL

SEmONS

FOR VISCOUS FLOW

VOL. 6. 2)

Swamee, P

K

1995). Optimal Irrigation Canal Sections. 1 Irrig. and Drain. Engg., ASCE,

121 6), pp. 467-469.

Swamee, P K and Bhatia, K G 1972). Economic Open-Channel Sections. 1 Irrig. and

Power, CBIP, New Delhi, April, pp. 169-176.

Swamee,

P

K., Mishra, G. C. and Chahar,

B

R. 2oooa). Design

of

Minimum Seepageloss

Canal Sections. 1 Irrig. and Drain. Engg., ASCE, 126 1), pp. 28-32.

Swamee, K., Mishra, G C. and Chahar, B R. 2ooob). Minimum Cost Design of Lined

Canal Sections. 1 Wat Resour. Mangt., Kluwer Academic Publishers, 14

I

pp. 1-12.

Trout, T 1 1982). Channel Design to Minimize Lining Material Costs. 1 rrig. and Drain.

Engg., ASCE, 108 4),242-249.

Woener, 1 L., lones, B. A Ir. and Frenzl, R N 1968). Laminar Flow in Finitely Wide

Rectangular Channels. 1 Hydr. Div., ASCE, 94 3), pp. 691-704.

Woo, D C. and Brater, E. F 1961). Laminar Flow in Rough Rectangular Channels. Geoph.

Res., AGU, 66 12), 4207-4217.

NOT TIONS

A Flow area,

b

Bed width

iameter

g

Gravitational acceleration

k

A

Area coefficient

k

b

Bed width coefficient

k

D

iameter coefficient

k

p

Perimeter coefficient

k

v

verage velocity coefficient

k

y

Normal depth coefficient

L

Length scale

m

Side slope

P

Flow perimeter

Q

Discharge

Bed slope

V

verage velocity

v

Point velocity

y

ertic l

distance

Y

n

Normal depth

z Horizontal distance

V

Kinematic viscosity

Superscript

•

Optimal

ISH JOURNAL OF

HYDRAULIC

ENGINEERING VOL 2000

NO

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6-2canal Design Viscous Flow