Note that if the elevations z 2 and z 1 are the same

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Note that if the elevations Z 2 and Z 1 are the same, and the fluid is incompressible (constant ϒ ) the term ϒ z can be eliminated from Euler’s equation. Example: The object shown is accelerated to the right at acceleration = a Bernoulli’s Equation: Integrating Euler’s Equation along a pathline “s” instead of an axis “l”, and replacing the acceleration “a l ” by convective, wields a general equation for moving fluids called: Bernoulli’s Equation Another form of the equation: A B D C ( ) ( ) ( ) ( ) Apply Euler’s Equation (after integration) between points A and B: Points A and B are on the same elevation. Acceleration is substituted positive, since the direction of integration (A to B) coincides with direction of acceleration. L ( ) ( ) ( ) ( ) ( ) Apply Euler’s Equation (after integration) between points B and C: Since there is no acceleration in the vertical direction, Euler’s integrated equation reduces to hydrostatic equation
Last update: Spring2014 Society of Civil & Arch. Engineering Students Assistance Committee *This summary is done by students, it may contain some errors. Note: p + ϒ z is called Piezometric pressure (P z ), and p/ ϒ + z is called Piezometric head (h) Applying Bernoulli’s Equation: Like Euler’s equation, this equation can be applied between any two locations in a flowing fluid (provided that it is the same fluid (constant ϒ )). Important assumptions: o Fluids exposed to atmosphere have pressure = 0 gage. o In cases like the figure below, V1 can be assumed = 0 since V2 >> V1 ( Note: Euler’s Equation reduces to hydrostatic differential equation when acceleration = 0 Bernoulli’s Equation reduces to hydrostatic equation when velocities = 0 1 2 Fluid in this container flows out from the outlet denoted by “2”. Pressure at both locations is equal to 0 gage Velocity at the outlet is much more than the velocity of decrease of the fluid’s surface, therefore V1 is neglected
Last update: Spring2014 Society of Civil & Arch. Engineering Students Assistance Committee *This summary is done by students, it may contain some errors. Chapter 5: Control Volume Approach and Continuity Equation Two approaches for finding fluid properties: 1. Lagrangian Approach: Calculating properties by considering particle Pathlines, and applying Euler’s and Bernoulli’s Equations. 2. Eulerian Approach: Calculating properties by finding general equations and observing the field’s general properties.. Analysis of the Eulerian Approach: This approach is also called control volume approach. Important definitions: 1. Control volume : An imaginary volume that is specified in a fluid field to simplify the analysis. All properties in the Eulerian approach are measured inside a control volume. Control volumes can rotate, move, and deform. 2. Control surface : The surface which encloses a control volume. All matter passes in and out of the control volume through the control surface. Usually the control surface is shown by a dashed line. Example: Flow rate (or discharge): A property of a flowing fluid, which measures the amount of volume that passes a certain cross section in a fluid field per second.

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