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Chapter 4: Control Volume Analysis Using Energy

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Chapter 4: Control Volume Analysis Using Energy Chapter 4: Control Volume Analysis Using Energy Conservation of Mass and Conservation of Energy ENGINEERING CONTEXT The objective of this chapter is to develop and illustrate the use of the control volume forms of the conservation of mass and conservation of energy principles. Mass and energy balances for control volumes are introduced in Secs. 4.1 and 4.2, respectively. These balances are applied in Sec. 4.3 to control volumes at steady state and in Sec. 4.4 for transient applications. Although devices such as turbines, pumps, and compressors through which mass flows can be analyzed in principle by studying a particular quantity of matter (a closed system) as it passes through the device, it is normally preferable to think of a region of space through which mass flows (a control volume). As in the case of a closed system, energy transfer across the boundary of a control volume can occur by means of work and heat. In addition, another type of energy transfer must be accounted for- the energy accompanying mass as it enters or exits. Examples of Open Systems How do we get from the fixed mass (closed) systems that we know, to the open systems we are interested in? Mass crosses boundary- W ?? What about energy? W Q T P Consequences of mass and energy crossing boundary? Where does mass cross boundary? Yes No Yes Yes Does total mass in the system change over time? Where does energy cross boundary? ?? Yes ?? Yes What is the purpose of this device? What performance questions might we ask? Which energies are most significant? What is the relationship between these energies? Does the total mass and/or energy of system change over time? What forms of energy cross boundary? Chemical Thermal (In) Mechanical Work (Out) Rotational, Torque Non-reacted Gas, Hot Gas (Out) Chemical Energy (In) (Stored in Chemical Bonds) One Inlet, One Exit Control Volume Boundary Dashed line defines control volume boundary Inlet, i Exit, e Mass is shown in green Developing Control Volume Mass Balance Start with fixed mass (closed system) and allow an interaction with mass crossing a boundary. Transform to a form that accounts for flow across boundary. Total mass is: mmmttotal=+ i cv ( ) At time t mi mcv (t) Inlet, i Exit, e Push mass mi into CV during interval ∆t Developing Control Volume Mass Balance At time t + ∆t m (t + ∆t) cv m Inlet, i e Exit, e mtotal=+ mmt i cv( ) = mt cv( +∆+ t) m e mtcv( + ∆− t) mt cv( ) = mm i − e Developing Control Volume Mass Balance Mass balance Express on rate basis over time interval ∆t, as ∆t goes to zero: mtcv( + ∆− t) mt cv( ) = mm i − e mt( +∆ t) − mt( ) mm cv cv =−ie ∆ttt∆∆ mtcv( +∆ t) − mt cv ( ) mm lim()∆→t 0 =ie − ∆ttt∆∆ dm cv = mm− dt ie Multiple Inlets and Exits Control Volume Boundary Dashed line defines control volume boundary Inlets, i Exits, e dm cv =−∑∑mmie dt ie Conservation of Mass for Control Volume time rate of change of time rate of flow time rate of flow mass contained within=− of mass in across of mass out across the control volume at time tinlet i at time texit e at time t • • dmcv =−∑∑mmie dt ie Suppose ∑∑mmie> ie dm What happens to cv dt In = Stored + Out Conservation of Mass for Control Volume time rate of change of time rate of flow time rate of flow mass contained within=− of mass in across of mass out across the control volume at time tinlet i at time texit e at time t • • dmcv =−∑∑mmie dt ie dm What happens to cv if: dt dm cv ∑∑mmie< < 0 ie dt mm= dm ∑∑ie cv = 0 Steady State ie dt Evaluating Mass Flow Rate at Ports amount of mass crossing dA during =∆ρ VtdA ()n the time interval ∆t mL2 3 tL∼ m Lt instantaneous rate of mass flow = ρVdA n across dA mL2 m 3 L ∼ L tt Integrate over port area: mVdA = ρ ∫ n A Forms of Mass Rate Balance dm cv =−∑ mmie∑ dt ie Alternative forms convenient for particular cases: One-dimensional Flow Steady State Integral One Dimensional Flow Model How to handle non-uniform velocity/properties Actual Two-Dimensional Flow: V = V(r) mVdArVrrdr ==∫∫ρρ( ) ( )[2 π] AA One-Dimensional Flow Model or Approximation: mVdAAV ==∫ ρρ A V = constant Note simplifying 2-D to 1-D One Dimensional Flow Model mAV = ρ 1 v = ρ AV m = v AV is the Volumetric Flow Rate (e.g. m3/kg) dmcv AV i i A e V e =−∑∑ = ∑ρρiii()AVAV − ∑ eee() dtie vie v i e Convenient form when Volume Flow Rates known Other forms of Conservation of Mass In many cases, the mass in balances the mass out Steady-State Flow: No changes in time All derivatives with respect to time are zero 0 dm cv =−∑ mmie∑ dt ie •• What goes in, goes out! ∑∑mmie= ie Other forms of the Conservation of Mass Fluid density may vary within the control volume Fluid velocity and density may vary across flow area dm cv =−∑ mmie∑ dt ie mt= ρ dV cv () ∫ mVdA = ρ n V ∫ A Integral Form: d ρρdV=− Vnn dA ρ V dA ∫∫VA∑∑( )(ie ∫ A ) dt ie Text Examples Feedwater Heater at Steady State Filling a barrel with water Energy Conservation in Open Systems • Mass crosses the system boundary, transporting – Internal thermal energy per unit mass, (u) – Kinetic energy per unit mass – Potential energy per unit mass – Chemical energy in chemical bonds • Perform Transformation of Control Mass to Control Volume Conservation of Energy for Control Volume Approach similar to Mass Conservation derivation: Closed system to open system Mass balance to rate basis Below: Stored = HeatIn –WorkOut + Net “Mass-Energy” In time rate of changenet rate at which net rate at which net rate of energy of the energy energy is being energy is being transfer into the contained within=− transferred in transferred out + control volume the control volume at by heat transfer by work accompanying time t at time t at time t mass flow Conservation of Energy for Control Volume Single Inlet, Single Outlet time rate of changenet rate at which net rate at which net rate of energy of the energy energy is being energy is being transfer into the contained within=− transferred in transferred out + control volume the control volume at by heat transfer by work accompanying time t at time t at time t mass flow •• •22 • dEcv V i V e =−+Q W mii u + + gz i − m ee u + + gz e dt 22 Conservation of Energy for Control Volume Single Inlet, Single Outlet •• •22 • dEcv V i V e =−+Q W mii u + + gz i − m ee u + + gz e dt 22 Note: Internal energy, Kinetic energy, Potential energy associated with fluid flow at inlet & exit Conservation of Energy for Control Volume Work term includes flow across boundary and all other forms (Work?) Wshaft (Work?) •• •22 • dEcv V i V e =−+QW mii u + + gz i − m ee u + + gz e dt 22 Concept of Flow Work F = P A i i i mi mcv (t) Inlet, i Exit, e Push mass mi into Control Volume Force through a displacement is Work Work IN, is done ON system at inlets Concept of Flow Work mcv (t + ∆t) Fe = PeAe m Inlet, i e Exit, e Push mass me out of Control Volume Force through a displacement is Work Work OUT, is done BY system at outlets Concept of Flow Work Rate mcv (t + ∆t) Fe = PeAe m Exit, e Inlet, i e δWdx Work Rate: ==WF = FV δtdt Work Rate in terms of pressure & velocity WFVPAV ==( ) time rate of energy transfer by work from the control = PA V ()ee e volume at exit, e Evaluating Rate of Flow Work Out: WW=+cv( PAV e e)() e − PAV i i i A V mAV = ρ or m = v (PA) V= m ( Pv) WW=+cv mPvmPv e( e e) − i( i i ) v is specific volume Rate of Flow Work & Rate of All other forms of work included (rotating shafts, displacement of boundary, electrical work, etc.) Other forms of Conservation of Energy Single Inlet, Single Outlet •• •• • WW=+cvFlow Work = W cv + mPvmPv e() e e − i () i i Recall: huPv= +⋅ •• •22 • dEcv V i V e =−+QW mii u + + gz i − m ee u + + gz e dt 22 Energy Rate Balance: •••22 • dEcv V i V e =−+Qcv W cv m i h i ++ gz i − m e h e ++ gz e dt 22 Multiple Ports Energy Rate Balance: •• •22 • dEcv V i V e =−+Qcv W cv∑∑ m i h i ++ gz i − m e h e ++ gz e dt ie22 Variations within CV and at boundaries: V 2 E t==++ρρ edV u gz dV cv () ∫∫ VV2 Integral Form: dVV•• 22 ρρρedV=−+ Qcv W cv h ++ gz V n dA − h ++ gz V n dA dt ∫∫VA∑∑22 ∫ A ie ie Analyzing Control Volumes at Steady State Transient start up and shutdown periods not included here Steady-State Forms of Mass and Energy Rate Balances Control volume at Steady-State (SS): • Conditions (states/properties) of mass within CV and at boundaries do not vary with time • No change of mass (+ or -) within CV • Mass flow rates are constant • Energy transfer rates by heat & work are constant Steady-State Forms of Mass and Energy Rate Balances Control volume at Steady-State (SS): • Conditions (states/properties) of mass within CV & at boundaries do not vary with time ρ (x, y, z) P1 time time T (x, y, z) h2 time time Steady-State Forms of Mass and Energy Rate Balances Control volume at Steady-State (SS): • No change of mass (+ or -) within CV mcv Yes: time No: mcv time Steady-State Forms of Mass and Energy Rate Balances Control volume at Steady-State (SS): • Mass flow rates are constant m e Yes: time No: m i time Steady-State Forms of Mass and Energy Rate Balances Control volume at Steady-State (SS): • Energy transfer rates by heat & work are constant Qin Qout time Wnet time Steady State Special case when no changes occur in time 0 • • dmcv mmie= =−∑∑mmie ∑ ∑ dt ie ie Steady State Flow: •• •22 • VVie 0 =−+Qcv W cv∑∑ m i h i ++ gz i − m e h e ++ gz e ie22 Net Energy Flow In = Net Energy Flow Out VV22 ie QmhgzWmhgzcv+++=+++∑∑ i i i cv e e e ie22 One Inlet, One Outlet Steady State Special case when no changes occur in time mmmie= = 22 (VVie− ) 0 =−+QWmhh () −+ + gzz() − cv cv i e2
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