Channel Desing to Minimize Lining Material Costs

8/17/2019 Channel Desing to Minimize Lining Material Costs

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E C E M B E R 1 9 82

R 4

CHAN NEL D ESIGN TO MINIMIZE LINING

MA TERIAL COSTS

B y Thomas 3. Trout,

M. A S C E

ABSTRACT:

A

direct algebraic technique is developed to determine ope n chan-

nel cross-sectional designs which m inim ize l ining m aterial costs when base

and side wall unit costs are different. Solution graphs, which indicate both

the optimal parameter com bination and the costs of deviating from the optimal

design, are presented. The graphs or analytical technique are also effective

in designing any trapezoidal channe ls. Use of the technique show s that mod -

erate deviations from op timal designs are not costly.

inrrnoDuc-noN

The m aterial cost of channel linings depen ds upon the volum e of l ining m a-

terial used. On a per unit length basis, the material costs are a function of the

lining thickness and pe rimeter length, which, in turn, dep ends up on the cha nne l

cross-sectional shape. The capacity of a channel constructed on a given slope

will also be

a func tion of the cross-sec tiona l shape. Using this interrelationship,

cross sections that m inimize lining material costs can be de termined .

If the channel is lined with one material of uniform thickness, the problem

reduces to one of determ ining the best hydraulic section (1 ). How ever, if m aterial

or l ining thickness is changed along the perim eter, the problem becom es m uch

m ore com plex. This paper will present

a

solution to the cost m inim ization prob-

lem w hen the m aterial cost per unit surface area of the base o f trapezoidal (or

rectang ular) channe ls is different from the m aterial cost of the sides.

This analysis does not deal w ith placem ent or other construction c osts. If they

can be evaluated in terms of surface area, they can be com bined with ma terial

costs . Otherwise, they m ust be c onsidered separately. The technique w ill thus

be m ost useful when long chan nel sections are to b e constructed, al lowing con-

struction proced ures to be oriented toward m inimizing m aterial costs or when

labor costs are low relative to m aterial costs , such as is the case in developing

countries.

'Agricultural Engr., U.S. Dept. of Agr., Agricultural Research Service, Snake River

Conservation Research Center, Kimberly, Idaho 83341; form erly Research Asst. Prof. ,

De pt. of Ag ricultural and C hem ical Engrg., Colorad o State Univ., Fort Collins, Colo.

Note.—D iscussion open unti l May 1 , 1983. To extend the closing date one m onth, a

written request mu st be filed with the Man ager of Tech nical and Professional Publications,

ASC E. Manuscript was subm itted for review for possible publication on Janu ary 4, 1982.

This pape r is part of the Journal of the Irrigation and D rainage D ivision, Proceed ings of

the Am erican Society of Civil Engineers, ©A SCE, Vol. 108, No. IR4, Decem ber, 1982.

ISSN 0044-797 8/82/0004-0242/$01 .00.

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5/5

1 R 4

HANNEL DESIGN

49

chan nels to minimize m aterial costs when be d and side lining costs are different.

The m ethod can also be used to evaluate added costs of deviating from optimal

designs. Use of the technique show s that mode rate deviations from optimal de-

signs do not usually cause a significant increase in costs. Althou gh the ben efits

gained by minimizing costs may not be large, in the light of the simplicity of

the technique, the know ledge gained from the exercise is worthwhile.

APPENDIX I . — R E F E R E N C E S

1. Chow, V. T.,

Open Channel Hydraulics,

McGraw-Hill Book Co., Inc., New York,

N.Y., 1959, pp. 128 and 160.

2.

Nicholson, W .,

M icroecono m ic Theory: B asic Principles and Ex tensions, The Dreydon

Press, Inc., Hinsdale, Ill., 1972, pp. 27-33 and 58-61.

3. Streeter, V. L.,

Fluid M echanics,

5th ed., McGraw-Hill Book Co., Inc., New York,

N.Y., 1971, p. 599.

APPENDIX IL-- NOTAT ION

The following symbols are used in this paper:

A

B

B,,

B

b

C

Ca

C S

D „

E

E'

F

k

k

, k

,

and

k

3

n

p

Q

b

t ,

2

l

ea

= cross-sectional flow area;

= channel bottom width;

•

bottom w idth value at the graph intercept (D);

•

bottom corner w idth;

channel bottom m aterial cost per unit area;

•

channel lining m aterial cost per unit length;

= base material cost per unit length;

= side material cost per unit length;

constant in Mann ing's Equation for conversion to U.S. Cus-

tomary units (1.486);

• normal flow depth;

= depth value at the graph intercept

(B -

0);

= channel side material cost per unit area;

•

channel wetted side length;

= freeboard side length allowance;

= freeboard allowance;

•

corner m aterial cost per unit length;

= constants defined by Eqs. 12, 13, and

1 4;

= Manning

roughness coefficient;

wetted perimeter length;

= flow rate;

•

hydraulic radius;

•

channel slope;

= thickness of channel bottom lining;

= thickness of channel side lining;

•

channe l side slope (horizontal-to-vertical);

= cost of base lining material per unit volume; and

= cost of side lining material per unit volume.

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Channel Desing to Minimize Lining Material Costs