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Keep Your Momentum Going

Age 16 to 18 Challenge Level:

Resolving forces and the resultant mass flows in the "x" direction, $\sum{P_x A_x} = \sum{\dot{m}V_x}$.

So $P_{in}A_{in} - P_{out}A_{out} = \rho A_{in}V_{in}(V_{out} - V_{in})$.

$\therefore P_{out}A_{out} = P_{in}A_{in} - \rho A_{in}V_{in}(V_{out} - V_{in})$.

but $A_{out}V_{out} = A_{in}V_{in}$ (conserve mass), so

$P_{out}A_{out} = P_{in}A_{in} - \rho A_{in}V_{in}^2(A_{in}/A_{out} - 1)$.

But, $\dot{m} = \rho A_{in}V_{in} \therefore V_{in} = \dot{m}/\rho A_{in}$

$\therefore P_{out} = (P_{in}A_{in} - \rho A_{in}(\dot{m}/\rho/A_{in})^2)(A_{in}/A_{out} - 1)/A_{out}$

$= (P_{in}A_{in} - (\dot{m})^2/\rho/A_{in})(A_{in}/A_{out} - 1)/A_{out}$

Entering the data we have, $P_{out} = 1999975Pa$.

There might be a structural problem with the tank because of the offset distance between the pipes, such that the pressure force in the pipes would generate a twisting moment. This was a problem in the Flixborough disaster.