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Chapter 4: Problem 26

A rice farmer needs to fill her \(150 \mathrm{m} \times 400 \mathrm{m}\) field with water to a depth of \(7.5 \mathrm{cm}\) in 1 hr. How many \(37.5-\mathrm{cm}\) diameter supply pipes are needed if the average velocity in each must be less than \(2.5 \mathrm{m} / \mathrm{s} ?\)

Short Answer

Expert verified
The farmer will need 1 pipe.

Step by step solution

01

- Calculate the volume of water needed

The volume of water needed is given by the field area multiplied by the required water depth, both of which need to be in the same units for the calculation. As 1 m equals 100 cm, convert the water depth from cm to m: \(7.5 cm = 0.075 m\). Hence, the volume of water needed \(V\) is given by \(V = 150m * 400m * 0.075m = 4500 m^3\)
02

- Calculate flow rate required

The flow rate required is the volume of water needed per unit of time, in this case, per hour. To convert volume in cubic meters to liters multiply by 1000 (since 1 cubic meter is 1000 liters) and to convert hours to seconds multiply by 3600 (since 1 hour is 3600 seconds). Therefore, the required flow rate \(Q_{req}\) in liters per second is given by \(Q_{req} = 4500m^3 * 1000L/m^3 / 3600s = 1.25 L/s\).
03

- Calculate the flow rate through one pipe

The flow rate through one pipe is given by the product of the cross-sectional area of the pipe and the velocity of the water. Firstly, convert the diameter of the pipe from cm to m: \(37.5cm = 0.375m\). The cross-sectional area \(A\) of the pipe is given by the equation for the area of a circle \(A = 蟺D^2/4 = 蟺*(0.375m)^2/4 = 0.1104 m^2\). The maximum velocity \(V_{max}\) is given as \(2.5 m/s\). Therefore, the flow rate \(Q_{pipe}\) through one pipe in liters per second is given by \(Q_{pipe} = A * V_{max} = 0.1104 m^2 * 2.5 m/s = 0.276 m^3/s = 276 L/s\).
04

- Calculate the number of pipes needed

The number of pipes needed is the required flow rate divided by the flow rate through one pipe. Therefore, \(N = Q_{req} / Q_{pipe} = 1.25 L/s / 276 L/s = 0.0045\). As we cannot have a fraction of a pipe, we will need to round up to the nearest whole number, which gives 1 pipe.

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