Numerical Methods For Engineers, 7 Ed
Numerical Methods For Engineers, 7 Ed
7th Edition
ISBN: 9789352602131
Author: Canale Chapra
Publisher: MCGRAW-HILL HIGHER EDUCATION
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Chapter 16, Problem 19P

The flow Q ( m 3 / s ) in an open channel can be predicted with the Manning equation

Q = 1 n A c R 2 / 3 S 1 / 2

where n = Manning roughness coefficient (a dimensionless number used to parameterize the channel friction), A c = cross-sectional area of the channel ( m 2 ) , S = channel slope (dimensionless, meters drop per meter length), and R = hydraulic radius (m), which is related to more fundamental parameters by R = A c / P ,  where  P = wetted perimeter (m). As the name implies, the wetted perimeter is the length of the channel sides and bottom that is under water. For example, for a rectangular channel, it is defined as P = B + 2 H ,  where H= depth (m). Suppose that you are using this formula to design a lined canal (note that farmers line canals to minimize leak-age losses).

(a) Given the parameters n n = 0.035 , S = 0.003 , S  and  Q = 1 m 3 / s , determine the values of B and H that minimize the wetted perimeter. Note that such a calculation would minimize cost if lining costs were much larger than excavation costs.

(b) Repeat part (a), but include the cost of excavation. To do this minimize the following cost function,

C = c 1 A c + c 2 P

where c 1 is a cost factor for excavation = $ 100 / m 2  and  c 2 is a cost factor for lining $50/m.

(c) Discuss the implications of your results.

(a)

Expert Solution
Check Mark
To determine

To calculate: The values of that will minimize the wetted perimeter P if the Manning equation of flow Q (m3/s)

can be written as Q=1nAcR2/3S1/2.

Answer to Problem 19P

Solution:

The values of BandH

that will minimize the wetted perimeter P is 1.547046mand0.77328m

respectively.

Explanation of Solution

Given Information:

The Manning equation of flow Q (m3/s)

can be written as Q=1nAcR2/3S1/2

where n is Manning roughness coefficient, Ac

is cross-sectional are of channel (m2), S is channel slope and R is hydraulic radius (m)

is given by R=Ac/P

where P is wetted perimeter and it is defined by P=B+2H

where H is depth (m)

and the parameters value are given as n=0.035,S=0.003andQ=1m3/s.

Calculation:

Consider the equation,

P=B+2H

The constrained can be written as,

Q=1nAcR2/3S1/2

This problem can be solved out by Linear programming formulation.

To minimize the wetted perimeter P function with the given constraint, the excel solver can be used.

The excel solver steps are,

Step 1. Initiate quantity BandH=1

and then write the parameter as shown below,

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  1

Step 2. Apply the formula in Ac

as shown below,

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  2

Step 3. Apply the formula in P as shown below,

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  3

Step 4. Apply the formula in R as shown below,

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  4

Step 5. Apply the formula in Qmanning as shown below,

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  5

Step 6. Go to DATA and then click on Solver. This dialog box will appear.

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  6

Step 7. Select the set objective, min, changing variable and then add,

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  7

Step 8. Click OK then this dialog box appears.

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  8

Step 9. Click on Solve and then OK.

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  9

Hence, the values of BandH

that will minimize the wetted perimeter P is 1.547046mand0.77328m

respectively.

(b)

Expert Solution
Check Mark
To determine

To calculate: The values of BandH

that will minimize the excavation cost function C=c1Ac+c2P where cost factor for excavation c1

is $100/m2

and cost factor for lining c2 is $50/m

if the Manning equation of flow Q (m3/s)

can be written as Q=1nAcR2/3S1/2.

Answer to Problem 19P

Solution:

The values of BandH

that will minimize the cost function P is 1.547046mand0.77328m

respectively.

Explanation of Solution

Given Information:

The Manning equation of flow Q (m3/s)

can be written as Q=1nAcR2/3S1/2

where n is Manning roughness coefficient, Ac

is cross-sectional are of channel (m2), S is channel slope and R is hydraulic radius (m)

is given by R=Ac/P

where P is wetted perimeter and it is defined by P=B+2H

where H is depth (m)

and the parameters value are given as n=0.035,S=0.003andQ=1m3/s.

Calculation:

Consider the function,

C=c1Ac+c2P

The constrained can be written as,

Q=1nAcR2/3S1/2

This problem can be solved out by Linear programming formulation.

To minimize the cost function with the given constraint, the excel solver can be used.

The excel solver steps are,

Step 1. Initiate quantity BandH=1

and then write the parameter as shown below,

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  10

Step 2. Apply the formula in Ac

as shown below,

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  11

Step 3. Apply the formula in P as shown below,

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  12

Step 4. Apply the formula in R as shown below,

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  13

Step 5. Apply the formula in Qmanning as shown below,

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  14

Step 6. Apply the formula in Cost as shown below,

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  15

Step 7. Go to DATA and then click on Solver. This dialog box will appear.

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  16

Step 8. Select the set objective, min, changing variable and then add,

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  17

Step 9. Click OK then this dialog box appears.

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  18

Step 10. Click on Solve and then OK.

Numerical Methods For Engineers, 7 Ed, Chapter 16, Problem 19P , additional homework tip  19

Hence, the values of BandH

that will minimize the cost function is 1.547046mand0.77328m

respectively.

(c)

Expert Solution
Check Mark
To determine

The implication of the result obtained in part (a) and part (b).

Answer to Problem 19P

Solution:

This can be interpreted that both the excavation and limiting cost can be minimized simultaneously by considering the bottom width B that is twice the length of each vertical side H.

Explanation of Solution

To interpret the result, first consider the constraint equation,

1nBH(BHB+2H)2/3S1/2=Q

And,

BH(BHB+2H)=(nQS1/2)3/2(BH)5/2(B+2H)=(nQS1/2)3/2=Constant

On further simplification,

BH=Constant×(B+2H)2/5

As AcandP

are dependent on B and H, thus both have minimized.

Thus, the excavation cost function C=c1Ac+c2P

is minimized and as excavation cost C is directly proportional to cross-sectional area.

Hence, both the excavation and limiting cost can be minimized simultaneously by considering the bottom width B that is twice the length of each vertical side H as obtained in part (a) and (b) as 1.547046mand0.77328m.

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Chapter 16 Solutions

Numerical Methods For Engineers, 7 Ed

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