Thermodynamics

The net flow rate into or out of the control volume through the entire controlsurface is obtained by integrating dṁ over the entire control surface,Chapter 5 | 223Net mass flow rate: m # net dm # (5–14)CS rV n dA CS r 1V S # S n 2 dACSNote that V → · n → V cos u is positive for u 90° (outflow) and negative foru 90° (inflow). Therefore, the direction of flow is automaticallyaccounted for, and the surface integral in Eq. 5–14 directly gives the netmass flow rate. A positive value for ṁ net indicates net outflow, and a negativevalue indicates a net inflow of mass.Rearranging Eq. 5–9 as dm CV /dt ṁ out ṁ in 0, the conservation ofmass relation for a fixed control volume can then be expressed asdGeneral conservation of mass: r dV (5–15)dtCV r 1V S # S n 2 dA 0CSIt states that the time rate of change of mass within the control volume plusthe net mass flow rate through the control surface is equal to zero.Splitting the surface integral in Eq. 5–15 into two parts—one for the outgoingflow streams (positive) and one for the incoming streams (negative)—the general conservation of mass relation can also be expressed asdr dV (5–16)dt aoutrV n dA ainrV n dA 0CVAAwhere A represents the area for an inlet or outlet, and the summation signsare used to emphasize that all the inlets and outlets are to be considered.Using the definition of mass flow rate, Eq. 5–16 can also be expressed asddtCVr dV ainm # aoutm # or dm CVdt ainm # aoutm #(5–17)Equations 5–15 and 5–16 are also valid for moving or deforming control volumesprovided that the absolute velocity V → is replaced by the relative velocityV → r , which is the fluid velocity relative to the control surface.Mass Balance for Steady-Flow ProcessesDuring a steady-flow process, the total amount of mass contained within acontrol volume does not change with time (m CV constant). Then the conservationof mass principle requires that the total amount of mass entering acontrol volume equal the total amount of mass leaving it. For a gardenhose nozzle in steady operation, for example, the amount of water enteringthe nozzle per unit time is equal to the amount of water leaving it perunit time.When dealing with steady-flow processes, we are not interested in theamount of mass that flows in or out of a device over time; instead, we areinterested in the amount of mass flowing per unit time, that is, the massflow rate ṁ. The conservation of mass principle for a general steady-flowsystem with multiple inlets and outlets can be expressed in rate form as(Fig. 5–7)Steady flow: a m # aoutm # 1kg>s2(5–18)inm˙ 1 = 2 kg/sCVm˙ 3 = m˙ 1 + ˙m 2 = 5 kg/sṁ 2 = 3 kg/sFIGURE 5–7Conservation of mass principle for atwo-inlet–one-outlet steady-flowsystem.