Question

The cross-section of a rough, rectangular, concrete() channel measures . The channel slope is 0.02ft/ft. Using the Darcy-Weisbach friction method, determine the maximum allowable flow rate through the channel to maintain one foot of free board(freeboard is the vertical distance form the water surface to the overtopping level of the channel). For these conditions, find the following characteristics(note that FlowMaster may not directly report all of these):

a) Flow area
b) Wetted perimeter
c) Hydraulic radius(A/P) :
d) Velocity
e) Froude number

Answer 1

The characteristics of flow through a concrete channel can be determined by calculating the flow area, wetted perimeter, hydraulic radius, velocity, and Froude number using given formulas, but specific dimensions of the channel are required to calculate these parameters.

Step-by-step explanation:

To determine the characteristics of flow through a concrete channel using the Darcy-Weisbach friction method, we must first find the flow area (A), the wetted perimeter (P), the hydraulic radius (R), the flow velocity (v), and the Froude number (Fr). However, the question seems to be missing specific dimensions for the rectangular channel. Assuming we had the width and depth of the channel, the flow area would be the product of width and depth. The wetted perimeter is the sum of the sides in contact with the water. The hydraulic radius is computed as the flow area divided by the wetted perimeter.

The flow velocity can then be derived using the relationship Q = Av, where Q is the flow rate, A is the cross-sectional area, and v is the average velocity. Finally, the Froude number, which is a dimensionless number comparing inertial and gravitational forces, can be calculated using the formula Fr = v / √(gD), where g is the acceleration due to gravity and D is the hydraulic depth.

Answer 2

The following are the answer to this question:

Step-by-step explanation:

In point a, Calculating the are of flow:

`\bold{Area =B * D_f}`

`=6* 5\\\\=30 \ ft^2`

In point b, Calculating the wetter perimeter.

`\bold{P_w =B+2* D_f}`

`= 6 +2* (5)\\\\= 6 +10 \\\\=16 \ ft`

In point c, Calculating the hydraulic radius:

`\bold{R=(A)/(P_w)}`

`=(30)/(16)\\\\= 1.875 \ ft`

In point d, Calculating the value of Reynolds's number.

`\bold{Re =(4VR)/(v)}`

`=(4V * 1.875)/(1 * 10^(-5) (ft^2)/(s))\\\\`

`=750,000 V`

Calculating the velocity:

`V= \sqrt{(8gRS)/(f)}`

`= \sqrt{(8* 32.2 * 1.875 * 0.02)/(f)}\\\\=(3.108)/(√(f))\\\\`

`√(f)=(3.108)/(V)\\\\`

calculating the Cole-brook-White value:

`(1)/(√(f))= -2 \log ((K)/(12 R) +(2.51)/(R_e √(f)))\\\\ (1)/((3.108)/(V))= -2 \log ((2 * 10^(-2))/(12 * 1.875) +(2.51)/(750,000V√(f)))\\`

`(V)/(3.108) =-2\log(8.88 * 10^(-5) + (3.346 * 10^(-6))/(750,000(3.108)))`

After calculating the value of V it will give:

`V= 25.18 \ (ft)/(s^2)\\`

In point a, Calculating the value of Froude:

`F= (V)/(√(gD))`

`= \frac{V}{\sqrt{g\frac{A}{\text{Width flow}}}}\\`

`= \frac{25.18}{\sqrt{32.2(30)/(6)}}\\\\= (25.18)/(√(32.2 * 5))\\\\= (25.18)/(√(161))\\\\= (25.18)/(12.68)\\\\= 1.98`

The flow is supercritical because the amount of Froude is greater than 1.

Calculating the channel flow rate.

`Q= AV`

`=30x 25.18\\\\= 755.4 \ (ft^3)/(s)\\`