Thermodynamics

226 | Thermodynamicswhere A jet pD 2 jet /4 is the cross-sectional area of the jet, which is constant.Noting that the density of water is constant, the mass of water in the tank atany time ism CV rV rA tank hwhere A tank pD 2 tank /4 is the base area of the cylindrical tank. SubstitutingEqs. 2 and 3 into the mass balance relation (Eq. 1) givesr22ghA jet d(rA tankh)→ r22gh(pD 2 jet /4) r(pD 2tank/4) dhdtdtCanceling the densities and other common terms and separating the variablesgivedt D 2tank dh2D jet 22ghIntegrating from t 0 at which h h 0 to t t at which h h 2 gives(3) t0dt D jet2D tank2 22g h 2Substituting, the time of discharge ish 0dh2h S t 2h 0 2h 22g/22bD jeta D tank224 ft 22 ft 3 12 int a b 757 s 12.6 min2232.2/2 ft/s 0.5 inTherefore, half of the tank is emptied in 12.6 min after the discharge hole isunplugged.Discussion Using the same relation with h 2 0 gives t 43.1 min for thedischarge of the entire amount of water in the tank. Therefore, emptyingthe bottom half of the tank takes much longer than emptying the top half.This is due to the decrease in the average discharge velocity of water withdecreasing h.FImaginarypistonAVPmLINTERACTIVETUTORIALSEE TUTORIAL CH. 5, SEC. 2 ON THE DVD.CVFIGURE 5–11Schematic for flow work.5–2 ■ FLOW WORK AND THE ENERGYOF A FLOWING FLUIDUnlike closed systems, control volumes involve mass flow across theirboundaries, and some work is required to push the mass into or out of thecontrol volume. This work is known as the flow work, or flow energy, andis necessary for maintaining a continuous flow through a control volume.To obtain a relation for flow work, consider a fluid element of volume Vas shown in Fig. 5–11. The fluid immediately upstream forces this fluid elementto enter the control volume; thus, it can be regarded as an imaginarypiston. The fluid element can be chosen to be sufficiently small so that ithas uniform properties throughout.If the fluid pressure is P and the cross-sectional area of the fluid elementis A (Fig. 5–12), the force applied on the fluid element by the imaginarypiston isF PA(5–22)