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Aula Teórica 11 Integral Budgets: Momentum

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General Principle & Mass The rate of accumulation inside a Control Volume balances the fluxes plus production minus consumption: Fluid Mass has no source/sink and no diffusion and consequently the accumulation balances the advective flux. If the fluid is incompressible there is no accumulation and thus mass flowing in balances the mass flowing out and so does the volume in incompressible flows:

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Momentum In case of momentum In lecture 4 we have seen that the Sources/Sinks of momentum are the pressure and gravity forces. Momentum diffusive flux is in fact the Shear stress that we can compute explicitly from the velocity derivative only if we know the velocity profile. For that reason we will call it shear stress in the integral budget approach.

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Integral momentum budget Let us consider: incompressible and stationary flow. If we assume that the velocity is uniform at the inlet and the outlet of the volume:

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If p is uniform along inlet and outlet: And finally we get:

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Integral Momentum Budget This is an algebraic equation applicable if: – Stationary and incompressible flow, – Velocity is uniform at each inlet and outlet, – Pressure is uniform along surfaces (e.g. inlets and outlets). Can inlets be located in zones where streamlines have curvature? Being a budget, this equation permits the calculation of a term knowing all the others. Where is the summation of pressure forces other than those acting at inlet and outlet and is the summation of the friction forces.

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Example 1 Calculate the force exerted by the fluid over the deflector neglecting friction V=2 m/s and jet radius is 2 cm and theta is 45º. We have a flow with an inlet and an outlet. Velocity has a component at the inlet and two at the outlet Pressure is atmospheric at inlet and outlets and thus the velocity modulus remains constant. We have to compute budgets along both directions x and y. D=2cmA=0.000315m2 V=2 m/s Q=0.000629m3/s Theta451.26N 0.88971N N Fx=-0.37N Fy=0.88971N F=0.963015N

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Example 2 The Jet is hitting the surface perpendicularly (Vj=3m/s), but the surface is moving (Vc=1m/s). D=10 cm. Calculate the force and power supplied. If the control volume is moving, fluxes depend on the flow velocity relative to the control volume:

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Relative or absolute reference Discharge must be computed using relative velocity. Transported velocity can be the relative or the absolute velocity. Usually the relative velocity is more intuitive.

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Example 2Vc=1m/s D=10cm Vj=3m/sA=0.007864m2 VRj=2Q=0.015728m3/s Theta9031.46N 0N 0Symmetrical Fx=-31.46N Fy=0N F=31.456N Symmetrical

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Summary The integral momentum equation states describes the momentum conservation principle (Newton law) assuming simplified solutions for momentum flux calculations. It is useful when flow is stationary and incompressible. Becomes more useful when associated to the Bernoulli Energy Conservation Equation.

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