Fox and McDonald's Introduction to Fluid Mechanics
Fox and McDonald's Introduction to Fluid Mechanics
9th Edition
ISBN: 9781118912652
Author: Philip J. Pritchard, John W. Mitchell
Publisher: WILEY
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Chapter 11, Problem 1P

Verify the equation given in Table 11.1 for the hydraulic radius of a circular channel. Evaluate and plot the ratio R/D, for liquid depths between 0 and D.

Expert Solution & Answer
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To determine

The expression for the hydraulic radius of the circular channel and plot the ratio R/D for liquid depths between 0 and D.

Explanation of Solution

Calculation:

The area of the circular channel can be molded as the summation of two triangles and the segment of the chord.

Calculate the area of the circular channel (A).

  A=AC+AT+AT=AC+2AT=(α24D2)+2(12bh)=α8D2+bh

  =α8D2+[D2sin(πα2)][D2cos(πα2)]=α8D2+[D24sin(πα2)cos(πα2)]=α8D2+D28sin(2πα)

  =D28αD28sinα=D28(αsinα)

Calculate the hydraulic radius of the circular channel (R).

  R=AP=(D28(αsinα))(Dα2)=14(1sinαα)D

Thus, the hydraulic radius of the circular channel is 14(1sinαα)D.

Calculate the ratio between the hydraulic radius and the depth of the circular channel.

  R=14(1sinαα)DRD=14(1sinαα)        (I)

Calculate the relation between the height of the channel and the angle made by the channel surface.

  y=D2+D2cos(πα2)=D2+D2cos(πα2)

  yD=12[1+cos(πα2)]        (II)

For plotting the graph between RD and yD, the term α has to be removed in Equation (I) and Equation (II) or else each value in terms of α has to be determined.

The values of RD and yD are determined for different values of α and the readings are tabulated as in Table (1).

αRDyD
0.000000.0000
0.39270.01250.0096
0.78540.04980.0381
1.17810.10790.0843
1.57080.18170.1464
1.96350.26470.2222
2.35620.34990.3087
2.74890.43040.4025
3.14160.50000.5000
3.53430.55410.5975
3.92700.59000.6913
4.31670.60690.7778
4.17240.60610.8536
5.10510.59050.9157
5.49780.56430.9619
5.89050.53250.9904
6.28320.50001.0000

Plot the ratio R/D for liquid depths between 0 and D as in Figure (1).

Fox and McDonald's Introduction to Fluid Mechanics, Chapter 11, Problem 1P

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Chapter 11 Solutions

Fox and McDonald's Introduction to Fluid Mechanics

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