209
8
Variable Area Flow
Introduction
The steady ow of a gas through a duct (or a streamtube) that has a varying cross-sectional
area will be considered in this chapter. Such ows, i.e., compressible gas ows through a
duct whose cross-sectional area is varying, occur in many engineering devices, e.g., in the
nozzle of a rocket engine and in the blade passages in turbo machines.
It will be assumed throughout this chapter that the ow can be adequately modeled by
assuming it to be one-dimensional at all sections of the duct, i.e., quasi-one-dimensional
ow will be assumed in this chapter. This means, by virtue of the discussion given in
Chapter 2, that the rate of change of cross-sectional area with distance along the duct is
not very large. It will also be assumed in this chapter in studying the effects of changes in
area on compressible gas ow that the ow is isentropic everywhere except through any
shock waves that may occur in the ow. This assumption is usually quite adequate since
the effects of friction and heat transfer are usually restricted to a thin boundary layer adja-
cent to the walls in the types of ows here being considered and their effects therefore can
often be ignored or be adequately accounted for by introducing empirical constants. The
presence of shock waves will have to be accounted for in the work of this chapter and the
ow through these waves is, as discussed before, not isentropic.
Effects of Area Changes on Flow
Consider, rst, the general effects of a change in area on isentropic ow through a chan-
nel. The situation considered is shown in Figure 8.1, i.e., the effects of a differentially small
change in area, dA, on the other variables, i.e., V, p, T, and ρ, are considered. The effects of
dA on the changes in pressure, density, velocity, etc., i.e., on dp, dρ, dV, will be derived using
the governing equations discussed earlier. The analysis presented here is an extension of
some of the analyses given earlier and there is some overlap with earlier work.
First, it is recalled that the continuity equation gives
ρAV = mass ow rate = constant (8.1)
where A is the cross-sectional area of the duct at any point. Applying this to the ow being
considered gives
ρAV = (ρ + dρ)(A + dA)(V + dV)
210 Introduction to Compressible Fluid Flow
Since dp, dρ, dV, and dA are, by assumption, all small, this equation becomes, to rst-order
accuracy (i.e., if terms involving the products and squares of the differentially small quan-
tities such as dρ × dA are ignored)
ρAV = AVdρ + ρVdA + ρAdV
i.e., dividing through by ρAV
ddA
A
dV
V
ρ
ρ
++= 0 (8.2)
Next, it is recalled that the energy equation gives
cT
V
p
+=
2
2
constant
which gives for the situation being considered
cT
V
cT dT
VdV
pp
+= ++
+
22
22
()
()
i.e., to rst-order accuracy
c
p
dT + V dV = 0 (8.3)
Further, the equation of state gives
p = ρRT and p + dp = (ρ + dρ) R(T + dT)
A
V
p
T
A + dA
V + dV
p + dp
T + dT
Control
volume
FIGURE 8.1
Flow changes considered in variable area channel.
211Variable Area Flow
Subtracting these two equations and dividing the result by the rst of the two equations
give, to rst-order accuracy,
dp
p
dd
T
T
=+
ρ
ρ
(8.4)
Lastly, since the ow being considered is, by assumption, isentropic, it follows that
pp
dp
d
ρρ
ρ
γγ
=
+
+
=constant an
dc
onstant
()
(8.5)
Because dp/p and dρ/ρ are by assumption small, the second of the above two equations
gives
p
dp
p
d
ρ
ρ
ρ
γγ
1
1
+
+
= constant
i.e., to rst-order accuracy
p
dp
p
d
ρ
γ
ρ
ρ
γ
1
1
+
+
= constant
i.e.,
pdp
p
d
ρ
γ
ρ
ρ
γ
1+−
= constant
Combining this with the rst equation then gives to rst-order accuracy
dp
p
d
=γ
ρ
ρ
(8.6)
Equations 8.2, 8.3, 8.4, and 8.6 together are sufcient to determine the required results,
i.e., to determine the relationship between the four variables dp/p, dV/V, dT/T, and dρ/ρ,
and the fractional area change dA/A. As discussed before, because isentropic ow is being
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