thermo
7.2 Duct Flow of Compressible Fluids
Such problems as the sizing of pipes and the shaping of nozzles require application of the momentum principle of fluid mechanic,
and therefore do not lie within the province of thermodynamics. However, thermodynamics does provide equations that interrelate
the changes occurring in pressure, velocity, cross-sectional area, enthalpy, entropy, and specific volume of a flowing stream. We
consider here the adiabatic, steady-state, one-dimensional flow of a compressible fluid in the absence of shaft work and of
changes in potential energy. The pertinent thermodynamic equations are first derived; they are then applied to flow in pipes and
nozzles.
The appropriate energy balance is (7.17). With Q, W, and Z all set, equal to zero, it reduces to
Which in differential form becomes (7.22)
The continuity equation, (7.11), is also applicable. Since m is constant, its differential form is
Or (7.23)
The appropriate fundamental property relation applicable to a unit mass of fluid is (6.8)
In addition, we may consider the specific volume of the fluid a function of its entropy and pressure: V=V(S,P). Then
This equation is put into more convenient form as follows. First, we write the mathematical identity
Substituting for the two partial derivatives on the right by (3.2) and (6.17) gives
Where B is the volume expansivity. The equation derived in physics for the speed of sound C in a fluid is
We may therefore write
Substituting for the two partial derivatives in the equation for dV now yields (7.24)
Equations (7.22), (7.23), (6.8), and (7.24) are four expressions relating the six differentials-dH, du, dV, dA, dS, and dP. Thus we
may treat dS and dA as the two independent variables and develop equations that express the remaining variables as functions of
these two.
First, (7.22) and (6.8) are combined: (7.25)
Or
Substituting this equation and (7.24) into (7.23) gives after rearrangement (7.26)
Where M is the Mach number, defined as the ratio of the speed of the fluid in the duct to the speed of sound in the fluid, u/c.
equation (7.26) relates dP to dS and dA.
Equations (7.25) and (7.26) may be combined to eliminate V dP, giving after rearrangement (7.27)
This equation relates du to dS and dA. Combined with (7.22) it relates dH to dS and dA, and combined with (7.23) it relates dV to
these same independent variables.
The differentials in the preceding equations represent changes in the fluid as it traverses a differential length of its path. If this
length is dx, then each of the equations of flow may be divided through by dx. Equations (7.26) and (7.27) then become (7.28)
And (7.29)
For adiabatic flow it follows from the second law that the irreversibilities due to fluid friction cause the entropy to increase in the
direction of flow, with the limiting value of the rate of increase equal to zero when the flow approaches reversibility. In general,
then, we have
Pipe Flow
Consider a compressible fluid in steady-state adiabatic flow in a horizontal pipe of constant cross-sectional area. For this case,
dA/dx=0 and (7.28) and (7.29) reduce to
And
For subsonic flow, M
Such problems as the sizing of pipes and the shaping of nozzles require application of the momentum principle of fluid mechanic,
and therefore do not lie within the province of thermodynamics. However, thermodynamics does provide equations that interrelate
the changes occurring in pressure, velocity, cross-sectional area, enthalpy, entropy, and specific volume of a flowing stream. We
consider here the adiabatic, steady-state, one-dimensional flow of a compressible fluid in the absence of shaft work and of
changes in potential energy. The pertinent thermodynamic equations are first derived; they are then applied to flow in pipes and
nozzles.
The appropriate energy balance is (7.17). With Q, W, and Z all set, equal to zero, it reduces to
Which in differential form becomes (7.22)
The continuity equation, (7.11), is also applicable. Since m is constant, its differential form is
Or (7.23)
The appropriate fundamental property relation applicable to a unit mass of fluid is (6.8)
In addition, we may consider the specific volume of the fluid a function of its entropy and pressure: V=V(S,P). Then
This equation is put into more convenient form as follows. First, we write the mathematical identity
Substituting for the two partial derivatives on the right by (3.2) and (6.17) gives
Where B is the volume expansivity. The equation derived in physics for the speed of sound C in a fluid is
We may therefore write
Substituting for the two partial derivatives in the equation for dV now yields (7.24)
Equations (7.22), (7.23), (6.8), and (7.24) are four expressions relating the six differentials-dH, du, dV, dA, dS, and dP. Thus we
may treat dS and dA as the two independent variables and develop equations that express the remaining variables as functions of
these two.
First, (7.22) and (6.8) are combined: (7.25)
Or
Substituting this equation and (7.24) into (7.23) gives after rearrangement (7.26)
Where M is the Mach number, defined as the ratio of the speed of the fluid in the duct to the speed of sound in the fluid, u/c.
equation (7.26) relates dP to dS and dA.
Equations (7.25) and (7.26) may be combined to eliminate V dP, giving after rearrangement (7.27)
This equation relates du to dS and dA. Combined with (7.22) it relates dH to dS and dA, and combined with (7.23) it relates dV to
these same independent variables.
The differentials in the preceding equations represent changes in the fluid as it traverses a differential length of its path. If this
length is dx, then each of the equations of flow may be divided through by dx. Equations (7.26) and (7.27) then become (7.28)
And (7.29)
For adiabatic flow it follows from the second law that the irreversibilities due to fluid friction cause the entropy to increase in the
direction of flow, with the limiting value of the rate of increase equal to zero when the flow approaches reversibility. In general,
then, we have
Pipe Flow
Consider a compressible fluid in steady-state adiabatic flow in a horizontal pipe of constant cross-sectional area. For this case,
dA/dx=0 and (7.28) and (7.29) reduce to
And
For subsonic flow, M