Thermodynamics

differential), but this is not the case for the integral of dṁ (thus the namespath function and inexact differential).The mass flow rate through the entire cross-sectional area of a pipe orduct is obtained by integration:Chapter 5 | 221m # A cdm # A crV ndA c 1kg>s2(5–3)While Eq. 5–3 is always valid (in fact it is exact), it is not always practicalfor engineering analyses because of the integral. We would like insteadto express mass flow rate in terms of average values over a cross section ofthe pipe. In a general compressible flow, both r and V n vary across the pipe.In many practical applications, however, the density is essentially uniformover the pipe cross section, and we can take r outside the integral of Eq.5–3. Velocity, however, is never uniform over a cross section of a pipebecause of the fluid sticking to the surface and thus having zero velocity atthe wall (the no-slip condition). Rather, the velocity varies from zero at thewalls to some maximum value at or near the centerline of the pipe. Wedefine the average velocity V avg as the average value of V n across the entirecross section (Fig. 5–3),Average velocity: V avg 1 V (5–4)A n dA ccA cwhere A c is the area of the cross section normal to the flow direction. Notethat if the velocity were V avg all through the cross section, the mass flowrate would be identical to that obtained by integrating the actual velocityprofile. Thus for incompressible flow or even for compressible flow wherer is uniform across A c , Eq. 5–3 becomesm # rV avg A c 1kg>s2(5–5)For compressible flow, we can think of r as the bulk average densityover the cross section, and then Eq. 5–5 can still be used as a reasonableapproximation.For simplicity, we drop the subscript on the average velocity. Unlessotherwise stated, V denotes the average velocity in the flow direction. Also,A c denotes the cross-sectional area normal to the flow direction.The volume of the fluid flowing through a cross section per unit time iscalled the volume flow rate V . (Fig. 5–4) and is given byV # A cV n dA c V avg A c VA c 1m 3 >s2(5–6)A cV avgFIGURE 5–3The average velocity V avg is defined asthe average speed through a crosssection.V avgV = V avg A cCross sectionFIGURE 5–4The volume flow rate is the volume offluid flowing through a cross sectionper unit time.An early form of Eq. 5–6 was published in 1628 by the Italian monkBenedetto Castelli (circa 1577–1644). Note that most fluid mechanics textbooksuse Q instead of V . for volume flow rate. We use V . to avoid confusionwith heat transfer.The mass and volume flow rates are related bym # rV # V#v(5–7)where v is the specific volume. This relation is analogous to m rV V/v, which is the relation between the mass and the volume of a fluid in acontainer.