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(a) Derive the given equation. (b) If the density increases by $1.50 \%$ from point 1 to point 2, what happens to the volume flow rate?

Solution

Verified$\textbf{A.}$

Deriving continuity equation for compressible fluids from incompressible fluids:

$\begin{align*} A_1v_1 = A_2 v_2 \end{align*}$

We know that:

$\begin{align*} m &= \rho V \\ \end{align*}$

and the total mass flow rate for steady flow condition is:

$\begin{align*} \mathrm{\frac{dM}{dt}} &= 0 \end{align*}$

Using the relation of mass flow rate to volume flow rate:

$\begin{align} \mathrm{\frac{dM}{dt}} &= \rho \mathrm{\frac{dV}{dt}} \end{align}$

Solving for the mass in and out:

$\begin{align*} \mathrm{\frac{dm_1}{dt}} &= \rho A_1 v_1\\ \mathrm{\frac{dm_2}{dt}} &= \rho A_2 v_2\\ \Rightarrow & \rho A_1 v_1 - \rho A_2 v_2 = 0\\ \Rightarrow & \boxed{\rho A_1 v_1 = \rho A_2 v_2} \end{align*}$

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