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The Bernoulli Equation • It is an approximate relation between pressure, velocity and elevation • It is valid in regions of steady, incompressible flow where net frictional forces are negligible • Viscous effects are negligible compared to inertial, gravitational and pressure effects. • Applicable to inviscid regions of flow (flow regions outside of boundary layers) • Steady flow (no change with time at a specified location) Steady flow • The value of a quantity may change from one location to another. In the case of a garden hose nozzle, the velocity of water remains constant at a specified location but it changes from the inlet to the exit (water accelerates along the nozzle). Acceleration of a Fluid Particle • Motion of a particle in terms of distance “s” along a streamline • Velocity of the particle, V = ds/dt, which may vary along the streamline • In 2-D flow, the acceleration is decomposed into two components, streamwise acceleration as, and normal acceleration, an. V2 an R • For particles that move along a straight path, an =0 Fluid Particle Acceleration • Velocity of a particle, V (s, t) = function of s, t V V dV ds dt s t • Total differential dV V ds V or dt s dt t • In steady flow, V 0;andV V ( s ) t • Acceleration, dV V ds V dV as V V dt s dt s ds Derivation of the Bernoulli Equation (1) • Applying Newton’s second law of conservation of linear momentum relation in the flow field dV PdA ( P dP)dA W sin mV ds m V dAdsisthemass W=mg= gdAdsistheweight of the fluid sin =dz/ds Substituting, dz dV -dpdA - gdAds dAdsV ds ds dp gdz VdV , Canceling dA from each term and simplifying, 1 Note V dV= d (V 2 ), and divding by 2 dp 1 d (V 2 ) gdz 0 2 Derivation of the Bernoulli Equation (2) • Integrating For steady flow dp V 2 2 gz constant (along a streamline) For steady incompressible flow, p V2 gz constant (along a streamline) 2 Bernoulli Equation • Bernoulli Equation states that the sum of kinetic, potential and flow (pressure) energies of a fluid particle is constant along a streamline during steady flow. • Between two points: p1 V12 p2 V2 2 gz1 gz2 or, 2 2 p1 V12 p2 V2 2 z1 z2 2g 2g p V2 pressure head; velocity head, z=elevation head 2g Example 1 Figure E3.4 (p. 105) Flow of water from a syringe Example 2 • Water is flowing from a hose attached to a water main at 400 kPa (g). If the hose is held upward, what is the maximum height that the jet could achieve? Example 3 • Water discharge from a large tank. Determine the water velocity at the outlet. Limitations on the use of Bernoulli Equation • change in flow conditions • Frictional effects can not be neglected in long and narrow flow passage, diverging flow sections, flow separations • No shaft work