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 t inlet i at time t exit 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 t inlet i at time t exit 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( ) ( π)[2 ] 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 change net 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 change net 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?)