Thermodynamics

Now we begin to reduce the back pressure and observe the resultingeffects on the pressure distribution along the length of the nozzle, as shownin Fig. 17–20. If the back pressure P b is equal to P 1 , which is equal to P r ,there is no flow and the pressure distribution is uniform along the nozzle.When the back pressure is reduced to P 2 , the exit plane pressure P e alsodrops to P 2 . This causes the pressure along the nozzle to decrease in theflow direction.When the back pressure is reduced to P 3 ( P*, which is the pressurerequired to increase the fluid velocity to the speed of sound at the exit planeor throat), the mass flow reaches a maximum value and the flow is said tobe choked. Further reduction of the back pressure to level P 4 or below doesnot result in additional changes in the pressure distribution, or anything elsealong the nozzle length.Under steady-flow conditions, the mass flow rate through the nozzle isconstant and can be expressed asChapter 17 | 837m # rAV a P RT b A 1Ma2kRT2 PAMa kB RTSolving for T from Eq. 17–18 and for P from Eq. 17–19 and substituting,m # AMaP 0 2k>1RT 0 231 1k 12Ma 2 >24 1k12>321k124(17–24)Thus the mass flow rate of a particular fluid through a nozzle is a functionof the stagnation properties of the fluid, the flow area, and the Mach number.Equation 17–24 is valid at any cross section, and thus ṁ can be evaluatedat any location along the length of the nozzle.For a specified flow area A and stagnation properties T 0 and P 0 , the maximummass flow rate can be determined by differentiating Eq. 17–24 withrespect to Ma and setting the result equal to zero. It yields Ma 1. Sincethe only location in a nozzle where the Mach number can be unity is thelocation of minimum flow area (the throat), the mass flow rate through anozzle is a maximum when Ma 1 at the throat. Denoting this area by A*,we obtain an expression for the maximum mass flow rate by substitutingMa 1 in Eq. 17–24:⋅⋅mm max/P 01.05 4 3P e5 4 3P*P 0211.01P bP 0m # 1k12> 321k124k 2max A*P 0 aB RT 0 k 1 b(17–25)Thus, for a particular ideal gas, the maximum mass flow rate through anozzle with a given throat area is fixed by the stagnation pressure and temperatureof the inlet flow. The flow rate can be controlled by changingthe stagnation pressure or temperature, and thus a converging nozzle can beused as a flowmeter. The flow rate can also be controlled, of course, byvarying the throat area. This principle is vitally important for chemicalprocesses, medical devices, flowmeters, and anywhere the mass flux of agas must be known and controlled.A plot of ṁ versus P b /P 0 for a converging nozzle is shown in Fig. 17–21.Notice that the mass flow rate increases with decreasing P b /P 0 , reaches amaximum at P b P*, and remains constant for P b /P 0 values less than thisP*P 00P*P 021.0P bP 0FIGURE 17–21The effect of back pressure P b on themass flow rate m # and the exit pressureP e of a converging nozzle.