Personal Technical Note
Convergent-Divergent Nozzle So, = ≡
Phenomena : Thermodynamic Derivation
from First Principle (July ’21)
Speed of Sound, = …….Eqn.8
Assumption : Compressible fluid and flow is isentropic
where
Continuity equation for steady compressible flow :
• P is the pressure;
ρ.u.A = Constant or d(ρ.u.A) = 0 …Eqn. 1
• ρ is the density and the derivative is taken
Where , ρ= Density of fluid isentropically, that is, at constant entropy s. This is
u = Velocity because a sound wave travels so fast that its
propagation can be approximated as an adiabatic
A = Flow Area
process.
Eqn. 1 can be re-written as + + = 0..Eqn.2 For an isentropic change, equation of state is
PVγ = Constant (K) …Eqn. 9
1st Law of Thermodynamics :
Or P = ργ.K
Heat Input = Change in Internal Energy + Work Done .
So, = . . = = . .
i.e. dQ = dE + pdV….Eqn. 3
where dE is the increase in internal energy per unit mass Hence, speed of sound can be re-written as
of fluid and PdV is the work of expansion on the fluid
layers ahead for a net addition of heat dQ to the system. = = √ . . …..Eqn. 10
Energy balance for unit mass of a compressible fluid = or = or = ….Eqn. 11
flowing in a pipe or tube can be expressed as
Substituting for du/u and dρlρ in equation 2 using equations 6
g dz + V dP + d(u2/2) + dF = dW….Eqn. 4
and 11 gives
where dF is the energy per unit mass required to
overcome friction. V is specific volume, 1/ρ. − . 2
+ 2 = 0…….Eqn. 12
For isentropic flow with negligible change of elevation
Mach Number, Ma = u/c
(dz =0) and no shaft work (dW=0), equation 4 reduces
Rewriting eqn. 12 in terms of Mach number,
to
2
VdP + udu = 0….Eqn. 5 = − 2 = (1 − )
. 2 . 2
Or udu = -VdP or =− =− …Eqn. 6
. 2
Or =( − 1) ……Eqn.13
Speed of sound (c) :
Consider the sound wave propagating at Observations based on eqn. 13 :
speed v through a pipe aligned with the X axis and with • For a subsonic flow in an expanding conduit (Ma <1
and dA>0), the flow is decelerating (du <0).
a cross-sectional area of A.
• For a subsonic flow in a converging conduit (Ma <1
In time interval dt it moves length dx = v dt
and dA <0), the flow is accelerating (du >0).
In steady state, the mass flow rate ṁ=ρ v A must be • For a supersonic flow in an expanding conduit (Ma
the same at the two ends of the tube, therefore the mass >1 and dA >0), the flow is accelerating (du >0).
flux ρ.v = constant i.e. vdρ = - ρdv • For a supersonic flow in a converging conduit (Ma
Per Newton's second law, the pressure-gradient >1 and dA <0), the flow is decelerating (du <0).
force provides the acceleration:
• At a throat where dA =0, either Ma =1 or du =0 (the
=− or flow could be accelerating through Ma =1, or it may
.
reach a velocity such that du =0).
= (− ) =( ) …Eqn 7
Manatosh Hajra
Convergent-Divergent Nozzle So, = ≡
Phenomena : Thermodynamic Derivation
from First Principle (July ’21)
Speed of Sound, = …….Eqn.8
Assumption : Compressible fluid and flow is isentropic
where
Continuity equation for steady compressible flow :
• P is the pressure;
ρ.u.A = Constant or d(ρ.u.A) = 0 …Eqn. 1
• ρ is the density and the derivative is taken
Where , ρ= Density of fluid isentropically, that is, at constant entropy s. This is
u = Velocity because a sound wave travels so fast that its
propagation can be approximated as an adiabatic
A = Flow Area
process.
Eqn. 1 can be re-written as + + = 0..Eqn.2 For an isentropic change, equation of state is
PVγ = Constant (K) …Eqn. 9
1st Law of Thermodynamics :
Or P = ργ.K
Heat Input = Change in Internal Energy + Work Done .
So, = . . = = . .
i.e. dQ = dE + pdV….Eqn. 3
where dE is the increase in internal energy per unit mass Hence, speed of sound can be re-written as
of fluid and PdV is the work of expansion on the fluid
layers ahead for a net addition of heat dQ to the system. = = √ . . …..Eqn. 10
Energy balance for unit mass of a compressible fluid = or = or = ….Eqn. 11
flowing in a pipe or tube can be expressed as
Substituting for du/u and dρlρ in equation 2 using equations 6
g dz + V dP + d(u2/2) + dF = dW….Eqn. 4
and 11 gives
where dF is the energy per unit mass required to
overcome friction. V is specific volume, 1/ρ. − . 2
+ 2 = 0…….Eqn. 12
For isentropic flow with negligible change of elevation
Mach Number, Ma = u/c
(dz =0) and no shaft work (dW=0), equation 4 reduces
Rewriting eqn. 12 in terms of Mach number,
to
2
VdP + udu = 0….Eqn. 5 = − 2 = (1 − )
. 2 . 2
Or udu = -VdP or =− =− …Eqn. 6
. 2
Or =( − 1) ……Eqn.13
Speed of sound (c) :
Consider the sound wave propagating at Observations based on eqn. 13 :
speed v through a pipe aligned with the X axis and with • For a subsonic flow in an expanding conduit (Ma <1
and dA>0), the flow is decelerating (du <0).
a cross-sectional area of A.
• For a subsonic flow in a converging conduit (Ma <1
In time interval dt it moves length dx = v dt
and dA <0), the flow is accelerating (du >0).
In steady state, the mass flow rate ṁ=ρ v A must be • For a supersonic flow in an expanding conduit (Ma
the same at the two ends of the tube, therefore the mass >1 and dA >0), the flow is accelerating (du >0).
flux ρ.v = constant i.e. vdρ = - ρdv • For a supersonic flow in a converging conduit (Ma
Per Newton's second law, the pressure-gradient >1 and dA <0), the flow is decelerating (du <0).
force provides the acceleration:
• At a throat where dA =0, either Ma =1 or du =0 (the
=− or flow could be accelerating through Ma =1, or it may
.
reach a velocity such that du =0).
= (− ) =( ) …Eqn 7
Manatosh Hajra
