A transition is placed in a rectangular air duct as shown in Fig. 5–14. If 2 kg/s of air flows steadily through the duct, determine the pressure change that occurs between the ends of the transition. Take ρa = 1.23 kg/m³.

Question

A transition is placed in a rectangular air duct as shown in Fig. 5–14. If 2 kg/s of air flows steadily through the duct, determine the pressure change that occurs between the ends of the transition. Take [latex] \rho_a=1.23 kg / m ^3 [/latex].

Accepted Answer

Fluid Description. We have steady flow. At slow velocities the air passing through the duct will be considered an ideal fluid, that is, incompressible and inviscid.

Analysis. To solve this problem, we will first use the continuity equation to obtain the average velocity of flow at A and at B. Then we will use the Bernoulli equation to determine the pressure difference between A and B.

Continuity Equation. We will consider a fixed control volume that contains the air within the duct, Fig. 5–14. Thus, for steady flow,

[latex] \begin{gathered} \frac{\partial}{\partial t} \int_{ cv } ho d \sout{V}+\int_{ cs } ho V _{f / cs } \cdot d A =0 \ 0-V_A A_A+V_B A_B=0 \ Q=V_A A_A=V_B A_B \end{gathered} [/latex]

But

[latex] Q=\frac{\dot{m}}{ ho}=\frac{2 kg / s }{1.23 kg / m ^3}=1.626 m ^3 / s [/latex]

so that

[latex] V_A=\frac{Q}{A_A}=\frac{1.626 m ^3 / s }{(0.45 m )(0.3 m )}=12.04 m / s [/latex]

and

[latex] V_B=\frac{Q}{A_B}=\frac{1.626 m ^3 / s }{(0.45 m )(0.15 m )}=24.09 m / s [/latex]

Bernoulli Equation. Selecting points at A and B on the horizontal streamline, we have

[latex] \begin{gathered} \frac{p_A}{\gamma_a}+\frac{V_A^2}{2 g}+z_A=\frac{p_B}{\gamma_a}+\frac{V_B^2}{2 g}+z_B \ \frac{p_A}{\left(1.23 kg / m ^3 ight)\left(9.81 m / s ^2 ight)}+\frac{(12.04 m / s )^2}{2\left(9.81 m / s ^2 ight)}+0=\frac{p_B}{\left(1.23 kg / m ^3 ight)\left(9.81 m / s ^2 ight)}+\frac{(24.09 m / s )^2}{2\left(9.81 m / s ^2 ight)}+0 \end{gathered} [/latex]

[latex] p_A-p_B=267.66 Pa =0.268 kPa [/latex]

This small drop in pressure, or the low velocities, will not significantly change the density of the air, and so here it is reasonable to have assumed the air to be incompressible.

Question 5.4: A transition is placed in a rectangular air duct as shown in...

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