Question
Download Solution PDFIn a steady flow through a nozzle, the flow velocity on the nozzle axis is given by \(v\; = \;u_o\left( {1 + \frac{{3{\rm{x}}}}{L}} \right){\rm{}},\) where x is the distance along the axis of the nozzle from its inlet plane and L is the length of the nozzle. The time required for a fluid particle on the axis to travel from the inlet to the exit plane of the nozzle is
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDF\(V = {u_0}\left( {1 + \frac{{3x}}{L}} \right)i\)
\(\Rightarrow V_x = \frac{{dx}}{{dt}} = {u_0}\left( {1 + \frac{{3x}}{L}} \right)\)
Separation of variable
\(\Rightarrow \frac{{dx}}{{{u_0}\left( {1 + \frac{{3x}}{L}} \right)}} = dt\)
Integrating the equation form 0 to T (Required Time)
\(\mathop \smallint \limits_0^T dt = \mathop \smallint \limits_0^L \frac{{dx}}{{{u_0}\left( {1 + \frac{{3x}}{L}} \right)}} = \frac{1}{{{u_0}}}\frac{L}{3} \cdot \left[ {\ln \left( {1 + \frac{{3x}}{L}} \right)} \right]_0^L\)
\(T = \frac{1}{{{u_0}}}\left[ {\ln \left( {1 + \frac{{3x}}{L}} \right)} \right]_0^L \cdot \left( {\frac{L}{3}} \right)\)
\(= \frac{L}{{3{u_0}}}{\rm{ln}}\left[ {1 + 3} \right]\)
\(T = \frac{L}{{3{u_0}}}\ln \left[ 4 \right]\)
Last updated on Jul 9, 2025
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