CHAPTER 5: Mass and Energy Analysis of Control Volumes
5.1: Conservation of Mass
• Mass and energy can be converted into one another as described by the formula:
E = mc
o (Where c= speed of light in a vacuum (2.9979 x 10 m/s) 8
• This equation suggests that there is equivalence between mass and energy. All physical
and chemical systems have energy interactions with their surroundings but the amount
of energy involved is equivalent to an extremely small mass compared to the total m ass.
• In most engineering analyses, we consider both mass and energy to be conserved
• Closed systems: the conservation of mass principle is used by requiring that the mass of
the system remain constant during a process.
• Control volumes: Mass can cross the boundaries and so the amount of mass entering
and leaving the system must be calculated.
• Mass flow rate: (𝑚) the amount of mass flowing through a cross section per unit time.
• The dot over a symbol is used to indicate time rate of change
• The differential mass flow rate of fluid flowing across a small area element dA in a cross
section of a pipe is proportional to dA ic elf, the fluid density ρ, and the component of
the flow velocity normal to dA (wcich is denoted, V ) is nxpressed as:
δ𝑚 = ρ V nA c
• Both δ and d are used to indicate differential quantities but δ is typically used for
quantities such as heat, work, mass transfer which are path functions and have inexact
differentials. While d is used for properties that are point function s and have exact
• In most applications we assume density is uniform across a cross section of a pipe.
• Velocity is never uniform over a cross section of a pipe because of the no -slip condition
at the walls. Velocity varies from zero at the walls to some maximum value at or near
the centerline of the pipe.
• Average velocity: (V avg is the average value of V ncross the entire cross section of the
V avg 𝑉 𝑑𝐴
▯▯ ▯ ▯ ▯ ▯
o (Where A ic the area of the cross section normal to the flow direction)
• For incompressible flow or compressible flow where ρ is uniform across A : c
𝑚 = ρ V avgAc (kg/s)
• For compressible flow we can think of ρ as the bulk average density over the cross
• Volume flow rate: the volume of fluid flowing through a cross section per unit time
𝑉 = ▯ 𝑉▯𝑑𝐴 =▯V avg Ac V A c (m /s)
• Mass and volume flow rates are related by:
𝑚 = ρ 𝑉= ▯ (Where v is specific volume)
• Conservation of mass principle : the net mass transfer to or from a control volume
during a time interval Δ t is equal to the net change (increase or decrease) of the total
mass within the control volume during Δ t.
• The following two equations are mass balance equations and can be applie d to any
control volume undergoing any kind of process. m inm =out cvg)
• It can also be expressed in rate form:
𝑚 in𝑚 = out/dt cv
• Change in the mass of the control volume during the process:
Δm = m – m
cv final initial
• Steady-flow process: the total amount of mass contained within a control volume does
not change with time (m = constant).
• The conservation of mass principle requires that the total amount of mass entering a
control volume equals the total amount of mass leaving it .
• In steady flow devices we are considering the mass flow rate, or the amount of mass
flowing per unit time.
• The conservation of mass principle for a general steady-flow system with multiple inlets
and outlets is expressed as:
Σ in= Σ out𝑚 (kg/s)
• The total rate of mass entering a control volume is equal to the total rate of mass
• Single-stream steady flow devices: involve only one inlet and one outlet such as a
nozzle, diffuser, turbine, compressor and pump:
𝑚 1 𝑚 2à ρ V 1 1 ρ1V A 2 2 2
• If a fluid is incompressible, which is usually the case for liquids, general steady flow can
be simplified into steady, incompressible flow:
Σ in= Σ out𝑉 (m /s)
• Steady, incompressible flow (single stream) :
𝑉 = 𝑉 à V A = V A
1 2 1 1 2 2
• The volume flow rates into and out of a steady -flow device may be different.
5.2: Flow Work and the Energy of a Flowing Fluid
• Control volumes involve mass flow across their boundaries and some work is required to
push the mass into or out of the control volume.
• Flow work: (or flow energy) is the work required to push mass into/out of the control
volume and is necessary for maintaining a continuous flow through a control volume.
• The force applied on the fluid element by the imaginary piston is:
F = PA (where P = fluid pressure and A = cross -sectional area of the fluid
• To push the entire fluid element into the control volume, this force must act through a
distance L. The work done in pushing the fluid element across the boundary (i.e. flow
W flow= FL = PAL = PV (kJ)
• The flow work per unit mass is obtained by dividing both sides of this equation by the
mass of the fluid element:
w flow = Pv (kJ/kg)
• the flow work relation is the same whether the fluid is pushed into or out of the control
• Flow work is expressed in terms of properties because it is the product of two fluid
properties and should be treated as work.
• A simple compressible system co nsists of three parts: internal, kinetic and potential
energies and can be expressed as: ▯ ▯
e = u + ke + pe = u + ▯ + gz (kJ/kg)
(where V is velocity and z is the elevation of the system relative to some external
• The fluid entering or leaving a control volume possesses an additional form of energy,
flow energy (Pv). The total energy of a flowing fluid on a unit-mass basis is:
θ = Pv + e = Pv + (u + ke + pe)
• The combination Pv + u has been previously defined as enthalpy, h.
θ = h + ke + pe = h + ▯ + gz (kJ/kg)
• By using enthalpy instead of the internal energy to represent the energy of a flowing
fluid, one does not need to be concerned about the flow work. Thus the energy
associated with pushing the fluid into/out of the contr ol volume is automatically taken
care of by enthalpy.
• Total energy per unit mass θ combined with the total energy of a flowing fluid of mass m
is simply θm (provided the properties of mass are uniform)
• When a fluid stream with uniform properties is flowing at a mass flow rate 𝑚, the rate
of energy flow with that stream is 𝑚θ.
• Amount of energy transport :
Emass = mθ = m(h + ▯ + gz) (kJ)
• Rate of energy transport :
𝐸 mass= 𝑚θ = 𝑚(h + + gz) (kW)
• When kinetic and potential energies of a fluid stream are negligible (often the case)
these relations can simplify to E mass= mh
• The total energy transported by mass into/out of the control volume is not easy to
determine since the properties of mas s at each inlet or exit may be changing with time
as well as over the cross section.
• The way to determine energy transport is to consider small differential masses dm that
have uniform properties and add their total energies during flow.
• The total energy of a flowing fluid of mass δ m is θδm and the total energy transported
by mass through an inlet or exit is obtained by integration
E in, mass ▯ ▯θδ𝑚 = ▯▯(ℎ + ▯ + 𝑔𝑧) δm i
5.3: Energy Analysis of Steady -Flow Systems
• Steady-flow devices: operate for a long period of time under the same conditions once
the transient start-up period is completed and steady operation is established.
• Steady-slow process: an idealized process involving steady-flow devices defined as a
process during which a fluid flow s through a control volume steadily.
• the fluid properties can change from point to point within the control volume, but at
any point, they remain constant during the entire process.
• Steady = no change with time
• During a steady-flow process, no intensive or extensive properties within the control
volume change with time. Thus, the volume, mass and total energy content of the
control volume remain constant.
• The boundary work is zero for a steady -flow process (since V = cocvtant) • The total mass or energy entering the control volume must be equal to the total mass or
energy leaving it (since m = cvnstant and E = conscvnt).
• The fluid properties at an inlet or exit remain constant during a steady -flow process, the
properties may be different at differ ent inlets/exits.
• All properties including velocity and elevation must remain constant with time at a fixed
point at an inlet or exit.
• The mass flow rate of the fluid at an opening must remain constant.
• Heat and work interactions between a steady -flow system and its surroundings do not
change with time. Thus the power delivered by a system and the rate of heat transfer to
or from a system remain constant.
• The mass balance for a ge