- 1 Derive Bernoullis Equation Using The Work Energy Principle See Pg 275 Av Constant P Fa P 1 Pv Pg Sy Consta 1 (17.72 KiB) Viewed 56 times
- 1 Derive Bernoullis Equation Using The Work Energy Principle See Pg 275 Av Constant P Fa P 1 Pv Pg Sy Consta 2 (78.35 KiB) Viewed 56 times
To derive Bernoulli's equation, we assume the flow is steady and laminar, the - A1 fluid is incompressible, and the viscosity is small enough to be ignored. To be general, we assume the fluid is flowing in a tube of nonuniform cross section that varies in height above some reference level, Fig. 10-22. We will consider the volume of fluid shown in color and calculate the work done to move it from the position shown in Fig. 10-22a to that shown in Fig. 10-22h. In this process, fluid entering area A flows a distance Al, and forces the fluid at area A, to move a distance al. The fluid to the left of area A, exerts a pressure P on our section (a) of fluid and does an amount of work W FAC Ρ Α, ΔΕ,. - 01 - sli (since P = F/A). At point 2, the work done on our section of fluid is W = -PA, AL, The negative sign is present because the force exerted on the fluid is opposite to the displacement. Work is also done on the fluid by the force of gravity. The net La (b) effect of the process shown in Fig. 10-22 is to move a mass m of volume A, A, FIGURE 10-22 Fluid flow: for 4. Al since the fluid is incompressible) from point to point 2, so the work derivation of Bernoulli's equation done by gravity is W, = -mg > -). where y, and y, are heights of the center of the tube above some (arbitrary) refer- ence level. In the case shown in Fig. 10-22, this term is negative since the motion is uphill against the force of gravity. The network W done on the fluid is thus W = W + W + W PAAL-BA, Al; -mgy+mgy According to the work energy principle (Section 6-3), the network done on a system is equal to its clfange in kinetic energy. Hence mo? - mo PAAL - P: 1,11-mgy+mgy The mass mt has volume 1, al= , Al, for an incompressible fluid. Thus we can substitute m-pA, A = pA,1l, and then divide through by A, AL = 4, A. to obtain po- pe P-P-P8y + PgY! which we rearrange to get w B + pus + PBY- P + ! pe + peyi (10-5) Beroun This is Bernoulli's equation. Since points 1 and 2 can be any two points along a tube of flow. Bernoulli's equation can be written as P + pv? + PEY constant at every point in the fluid, where y is the height of the center of the tube above a fixed reference level. [Note that if there is no flow (= = 0), then Eq. 10-5 reduces to the hydrostatic equation, Eq. 10-3bore P P-pg- SECTION 109 Bernoulli's Equation 275