Chapter 20, Problem 50A

(a)

To determine

To explain: The effect on the speed for the given condition.

Answer to Problem 50A

The speed of gas increases for the given condition.

Explanation of Solution

Introduction:

The flow rate of a fluid is defined as the amount of liquid that will pass through an area in given time. According to equation of continuity, the rate of a fluid flowing in is equal to the rate of fluid flowing out.

  $\begin{array}{l}{Q}_1=Q_2\\ v_1\times a_1=v_2\times a_2\end{array}$

When a gas flows to a smaller diameter pipe from larger diameter pipe, then according to equation of continuity the speed of the gas will be increased because the smaller diameter pipe will carry the same volume of gas per second as the large diameter pipe.

Conclusion:

Therefore, the speed of the gas will increase.

(b)

To determine

To explain: The effect on the pressure for the given condition.

Answer to Problem 50A

The pressure decreases for a steadily flowing gas.

Explanation of Solution

Introduction:

According to Bernoulli’s principle, there will be less pressure at the points of higher fluid speed than at the points of lower fluid speed within a horizontal flow of fluid.

The Bernoulli equation is,

  $P+\rho gh+\frac{\rho v^2}{2}=\text{Constant}$

The pressure of a fluid relates to its elevation as well its speed by Bernoulli principle. When a gas flows to a smaller diameter pipe from larger diameter pipe, then speed of a fluid will increase. So, according to Bernoulli’s principle, the pressure will be less outside the pipe, when the gas is moving faster with small diameter pipe compared to the surface inside the pipe having large diameter.

Conclusion:

Therefore, the pressure of fluid will decrease.

(c)

To determine

To explain: The effect on the space between the streamlines for the given condition.

Answer to Problem 50A

The space between the streamlines will become closer for a steadily flowing gas.

Explanation of Solution

Introduction:

Streamline concept is very useful in explaining the flow field in terms of speed as well as its direction. With the help of continuity equation, it can be shown that speed of fluid and spacing between streamlines are inversely proportional to each other.

The speed of the gas present in the smaller diameter pipe will be increased, so the space between the streamlines will be close together.For example, if a river or stream flow in canyon is observed, then it is seen that the edges of the river or stream will get closer to each other.

Conclusion:

Therefore, the spacing between streamlines will decrease.