Chapter 8, Problem 19TE

(a)

To determine

The variation in the speed of gas, as it moves from lager diameter pipe to a smaller-diameter pipe.

Answer to Problem 19TE

The speed of gas increases as it moves from a lager diameter pipe to a smaller-diameter pipe.

Explanation of Solution

Write the expression for the equation of continuity.

$\begin{array}{c}{A}_i{v}_i=A_f{v}_f\\ \left(\frac{\pi }{4}{d}_i^2\right)v_i=\left(\frac{\pi }{4}{d}_f^2\right)v_f\\ d_i^2{v}_i=d_f^2{v}_f\end{array}$

Here,

d_i is the initial diameter of pipe

d_f is the final diameter of pipe

v_i is the initial speed of the fluid

v_f is the final speed of the fluid

The product of the fluid's speed at any point in the pipe and cross-sectional area of pipe remain constant. As the fluid moves from a pipe with a large diameter to the pipe with a lower diameter, its speed increases. The liquid must carry the same amount of volume, and when the diameter of the pipe gets reduced, the speed increases in order to keep the same output flow.

Conclusion:

Therefore, the speed of gas increases as it moves from a lager diameter pipe to a smaller-diameter pipe.

(b)

To determine

The variation in the pressure of the gas, as it moves from a large diameter pipe to a smaller diameter pipe.

Answer to Problem 19TE

The pressure of the gas decreases, as it moves from a large diameter pipe to a smaller diameter pipe.

Explanation of Solution

The equation that establishes the relation between the flowing fluid's physical properties is known as Bernoulli's equation. These physical properties are speed, pressure, and volume. The faster-moving fluid in the large-diameter pipe will exert more pressure on the inside walls as compared to the fluid moving with the equivalent speed in the smaller diameter.

So, a large diameter pipe required stronger walls to withstand the pressure. Thus, on decreasing the diameter of the pipe, the pressure exerted by the gas decreases.

Conclusion:

Therefore, the pressure of the gas decreases, as it moves from a large diameter pipe to a smaller diameter pipe.

(c)

To determine

The variation on the spacing between streamlines, as the gas moves from a large diameter pipe to a smaller diameter pipe.

Answer to Problem 19TE

The spacing between streamlines decreases, as the gas moves from a large diameter pipe to a smaller diameter pipe.

Explanation of Solution

The spacing between the streamlines of the fluid varies linearly with the cross-sectional area of the pipe. As the density of the fluid is constant, the speed increases as the diameter of the pipe decrease.

For a continuous density flow, when the pipe's diameter decreases, the streamlines come very close. This can be seen in rivers when it enters the places having a bottleneck section. As the cross-sectional area of the section gets reduced, the river's edges come closer to each other.

Conclusion:

Therefore, the spacing between streamlines decreases, as the gas moves from a large diameter pipe to a smaller diameter pipe.