Thermodynamics

832 | ThermodynamicsFluidFluidThroatConverging nozzleThroatConverging–diverging nozzleFIGURE 17–14The cross section of a nozzle at thesmallest flow area is called the throat.designed by a Swedish engineer, Carl G. B. de Laval (1845–1913), andtherefore converging–diverging nozzles are often called Laval nozzles.Variation of Fluid Velocity with Flow AreaIt is clear from Example 17–3 that the couplings among the velocity, density,and flow areas for isentropic duct flow are rather complex. In theremainder of this section we investigate these couplings more thoroughly,and we develop relations for the variation of static-to-stagnation propertyratios with the Mach number for pressure, temperature, and density.We begin our investigation by seeking relationships among the pressure,temperature, density, velocity, flow area, and Mach number for onedimensionalisentropic flow. Consider the mass balance for a steady-flowprocess:m # rAV constantDifferentiating and dividing the resultant equation by the mass flow rate, weobtaindrr dA A dV V 0(17–13)CONSERVATION OF ENERGY(steady flow, w = 0, q = 0, ∆pe = 0)h 1 + V 1 2= h2 2 + V 222orh + V 22 = constantDifferentiate,dh + V dV = 0Also,0 (isentropic)T ds = dh – v dPdh = v dP = r1 dPSubstitute,dPr + V dV = 0FIGURE 17–15Derivation of the differential form ofthe energy equation for steadyisentropic flow.Neglecting the potential energy, the energy balance for an isentropic flow withno work interactions can be expressed in the differential form as (Fig. 17–15)dPr V dV 0(17–14)This relation is also the differential form of Bernoulli’s equation whenchanges in potential energy are negligible, which is a form of the conservationof momentum principle for steady-flow control volumes. CombiningEqs. 17–13 and 17–14 givesdAA dP r a 1 V 2 drdP b(17–15)Rearranging Eq. 17–9 as (∂r/∂P) s 1/c 2 and substituting into Eq. 17–15 yielddAA dPrV 2 11 Ma2 2(17–16)This is an important relation for isentropic flow in ducts since it describesthe variation of pressure with flow area. We note that A, r, and V are positivequantities. For subsonic flow (Ma 1), the term 1 Ma 2 is positive; andthus dA and dP must have the same sign. That is, the pressure of the fluidmust increase as the flow area of the duct increases and must decrease as theflow area of the duct decreases. Thus, at subsonic velocities, the pressuredecreases in converging ducts (subsonic nozzles) and increases in divergingducts (subsonic diffusers).In supersonic flow (Ma 1), the term 1 Ma 2 is negative, and thus dAand dP must have opposite signs. That is, the pressure of the fluid must