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Keep Your Momentum Going

Age 16 to 18 Challenge Level:

You may have learned Newton's 2nd law of motion, "force is equal to the rate of change of momentum". In fluids, the rate of change of mass, $dm/dt$, often abbreviated $\dot{m}$, is important. Use the product rule to find the form of Newton 2 that includes the possibility of mass flow.

If the flow is "steady", i.e. the mass flow in to a certain volume equals the mass flow out, the formula you derived simplifies to $F = \dot{m}v$.

When we resolve this formula in any particular direction, we call it the "steady flow momentum equation".

A tank (pictured) has a chemical (density $800kg/m^3$) flow of 1kg/s going through from left to right. The inlet pipe has an area $100cm^2$, and the outlet pipe has an area $50cm^2$. If the inlet pressure is 1MPa, what is the output pressure? Hint: mass is conserved, and the mass flow in a pipe of area A with fluid velocity V is just $\dot{m} = \rho AV$.

A supported fluid tank with an inlet and an outlet

Can you see any structural problems that might arise with this tank?

Would it make a significant difference if the tank were aligned vertically or horizontally?