Modern Engineering Thermodynamics

16.5 Converging-Diverging Flows 661
or
∂h s = −V ∂V s /g c
Now, we combine this with Gibbs Eq. (7.21) for an isentropic process to get
or
then,
T ∂s s = ∂h s − v ∂p s = 0
∂h s = v ∂p s = ∂p s /ρ = −V ∂V s /g c
∂V s /V = −g c ∂p s /ρV 2 (16.16)
Substituting Eq. (16.16) into (16.15) and using Eqs. (16.5) and (16.9) gives the desired result
∂A s /A = −∂V s /V −∂ρ s /ρ = g c /V 2 − ∂ρ
ð∂p s /ρÞ
∂p
= ½g c /V 2 − g c /c 2 Šð∂p s /ρÞ = 1 − M 2 ðg c /V 2 Þð∂p s /ρÞ
s
or
∂A
= ð1 − M 2 Ag c
Þ
∂p
s
ρV 2
(16.17a)
Using Eq. (16.16), we can rewrite Eq. (16.17a) as
∂A s
A
= ð1 − M2 Þ g c∂p s
ρV 2
= ðM2 − 1Þ ∂V s
V
(16.17b)
Equations (16.17a) and (16.17b) lead to slightly different but related conclusions about the nature of convergingdiverging
flows.
Converging subsonic flow: ∂A s < 0 and M < 1
■ Equation (16.17a). Converging flows are characterized by the fact that the area
becomes smaller in the direction of flow, or ∂A s < 0. Subsonic flows are
characterized by the fact that M < 1. Since A, g c , ρ, andV are all > 0, Eq. (16.17a)
shows that (∂A/∂p) s must be > 0. Then, ∂p s must be < 0since∂A s < 0for
converging flows. Consequently, the pressure must decrease in the direction of a
subsonic converging flow.
■ Equation (16.17b). Since M < 1 and ∂A s < 0 for subsonic converging flows
(Figure 16.7), then Eq. (16.17b) shows that ∂V s must be > 0 for these flows.
That is, the flow velocity increases in subsonic converging flow. In Chapter 6,
we define a nozzle as a flow geometry that converts pressure into kinetic
energy, or ∂p s < 0 and ∂V s > 0 in the direction of flow. Consequently,
converging subsonic flow corresponds to what we traditionally call nozzle flow.
Therefore, a converging passage carrying subsonic flow is called a subsonic
nozzle.
Converging supersonic flow: ∂A s < 0 and M >1
■ Equation (16.17a). Here, ∂A s is still < 0, but now M >1. Then, Eq (16.17a)
tells us that ∂p s must be > 0 and the pressure increases in the direction
of flow.
■ Equation (16.17b). If∂A s < 0 and M >1, then Eq. (16.17b) tells us that ∂V s
must be < 0, or the flow velocity must decrease in the direction of flow. In
Chapter 6, we define a diffuser as a flow geometry that converts kinetic energy
into pressure, or ∂V s < 0 and ∂p s > 0 in the direction of flow. Consequently,
a converging passage carrying a supersonic flow is called a supersonic diffuser
(Figure 16.8).
Diverging subsonic flow: ∂A s > 0 and M < 1
■ Equation (16.17a). Diverging flows are characterized by the fact that the
area becomes larger in the direction of flow, or ∂A s > 0. If the flow is
M > 1
Δp > 0
ΔV < 0
FIGURE 16.7
Converging subsonic nozzle.
M >1
Δp > 0
ΔV < 0 t
Decreasing pressure
Throat
Increasing pressure
FIGURE 16.8
Converging supersonic diffuser.
Throat