Thermodynamics

860 | ThermodynamicsSolution We are to calculate the Mach number and pressure downstream ofa sudden expansion along a wall.Assumptions 1 The flow is steady. 2 The boundary layer on the wall isvery thin.Properties The fluid is air with k 1.4.Analysis Because of assumption 2, we approximate the total deflectionangle as equal to the wall expansion angle (i.e., u d 10°). With Ma 1 2.0, we solve Eq. 17–49 for the upstream Prandtl–Meyer function,n 1Ma2 Bk 1k 1 tan1 c Bk 1k 1 1Ma2 1 2 d tan 1 a 2Ma 2 1 b B1.4 11.4 1 tan1 c B1.4 11.4 1 12.02 1 2 d tan 1 a 22.0 2 1 b 26.38°Next, we use Eq. 17–48 to calculate the downstream Prandtl–Meyer function,u n 1Ma 2 2 n 1Ma 1 2 S n 1Ma 2 2 u n 1Ma 1 2 10° 26.38° 36.38°Ma 2 is found by solving Eq. 17–49, which is implicit—an equation solver ishelpful. We get Ma 2 2.385. There are also compressible flow calculatorson the Internet that solve these implicit equations, along with both normaland oblique shock equations; e.g., see www.aoe.vt.edu/~devenpor/aoe3114/calc.html.We use the isentropic relations to calculate the downstream pressure,P 2 P 2>P 0P 1 >P 0P 1 c 1 a k 1 k>1k12b Ma 2 2 d2c 1 a k 1 k>1k12b Ma 2 1 d21230 kPa2 126 kPaSince this is an expansion, Mach number increases and pressure decreases,as expected.Discussion We could also solve for downstream temperature, density, etc.,using the appropriate isentropic relations.Air inletFuel nozzles or spray barsFlame holdersFIGURE 17–50Many practical compressible flowproblems involve combustion, whichmay be modeled as heat gain throughthe duct wall.17–6 ■ DUCT FLOW WITH HEAT TRANSFER ANDNEGLIGIBLE FRICTION (RAYLEIGH FLOW)So far we have limited our consideration mostly to isentropic flow, alsocalled reversible adiabatic flow since it involves no heat transfer and noirreversibilities such as friction. Many compressible flow problems encounteredin practice involve chemical reactions such as combustion, nuclearreactions, evaporation, and condensation as well as heat gain or heat lossthrough the duct wall. Such problems are difficult to analyze exactly sincethey may involve significant changes in chemical composition during flow,and the conversion of latent, chemical, and nuclear energies to thermalenergy (Fig. 17–50).The essential features of such complex flows can still be captured by asimple analysis by modeling the generation or absorption of thermal energy