## Related questions with answers

An incompressible liquid with negligible viscosity and density $\rho=1250 \mathrm{kg} / \mathrm{m}^{3}$ flows steadily through a 5-m-long convergent-divergent section of pipe for which the area varies as $A(x)=A_{0}\left(1+e^{-x / a}-e^{-x / 2 a}\right)$ where $A_{0}=0.25 \mathrm{m}^{2}$ and a=1.5 m. Plot the area for the first 5 m. Develop an expression for and plot the pressure gradient and pressure versus position along the pipe, for the first 5 m, if the inlet centerline velocity is 10 m/s and inlet pressure is 300 kPa. Hint: Use relation $u \frac{\partial u}{\partial x}=\frac{1}{2} \frac{\partial}{\partial x}\left(u^{2}\right)$.

Solution

Verified$\textbf{Given:}$

$\begin{align*} A(x)&= A_0(1+e^{\dfrac{-x}{a}}-e^{\dfrac{-x}{2a}}) \end{align*}$

Where,

$\begin{align*} A&= 0.25\hspace{0.1cm} \text{m}^2\\ a&= 1.5\hspace{0.1cm} \text{m}\\ u_0&= 10 \hspace{0.1cm} \frac{\text{m}}{\text{s}}\\ p_0&= 300 \hspace{0.1cm} \text{kPa} \end{align*}$

$\textbf{Find:}$

Plot the area for the first 5 m. Develop an expression for and plot the pressure gradient and pressure versus position along the pipe, for the first 5 m.

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