505
13.6 Exercises
13.6.1 Average Density with Compression Verify that the average density is
well-suited to approximate the isentropic enthalpy change with a compression process. Calculate for air as an ideal gas ( γ = 1.40) for pressure ratio values 1.2, 1.4, 1.7,
2 and infinitesimal efficiency values 1, 0.95, 0.90: Δh s , Δh r , Δp/ρ m . Establish that
Δp/ρ m is a very good approximation of Δh s (sometimes overestimation, sometimes
underestimation) and a somewhat inferior approximation of Δh r (always underestimation).
13.6.2 Average Density with Expansion The average density concept is applicable with expansion processes as well. The term Δp/ρ m constitutes an approximation of the isentropic enthalpy change, but is much less accurate than with a
compression. For an expansion the following applies:
Calculate for air as an ideal gas ( γ = 1.40) for pressure ratio values 1.2, 1.4, 1.7,
2 and infinitesimal efficiency values 1, 0.95, 0.90: Δh s , Δh r , Δp/ρ m . Establish that
Δp/ρ m results in a slight underestimation of Δh s , and a somewhat stronger underestimation of Δh r .
13.6.3 Loss Coefficient Verify numerically that the enthalpy loss coefficient ξ 1
(13.6) and the pressure loss coefficient ω 1 (13.7) are nearly equal for low inlet Mach
number. Assume a compressor cascade with air and infinitesimal efficiency 0.9.
Consider inlet Mach numbers 0.3, 0.7 and 1.2. Assume, by way of simplification,
zero outlet velocity. Observe that the energy loss coefficient increases weakly with
increasing inlet Mach number and that the pressure loss coefficient increases quite
strongly. Observe that ω 1 is always larger than ξ 1 . Remark that the Mach number
influence on ω 1 may be neutralised by taking the logarithm of pressure instead of
pressure. Such a loss coefficient is 1 minus the infinitesimal efficiency (similarly as
with expansions; see Exercise 6.10.1).
13.6.4 Lift Coefficient and Diffusion Factor Formulate the relationship between the lift coefficient and the diffusion factor. Establish that, for a given diffusion factor value, e.g., DF = 0.60, a great variation of the lift coefficient is possible,
strongly dependent on the general deceleration value w 2 /w 1 . This demonstrates that
lift coefficient is an unworkable concept with compressor cascades.
13.6.5 Radial Variation with a Fan Stage The radius ratio chosen in Sect. 13.3
is 1.60 (1.20/0.75). This ratio is smaller with many compressor stages, but it is much
greater with a fan stage, where it may be up to four. Study a fan stage with free vortex flow (same work on all radii) and radius ratio 4. Take as an example the fan of
the Tent800 (Fig. 12.9). Tip and hub radii are 1.40 m and 0.35 m. Take radius 1 m as
approximation of the equal area average radius. Consider the fan on the ground with
T 00 = 288 K and p 00 = 100 kPa. Take tip blade speed 420 m/s and inflow velocity (absolute system) in axial direction, constant over the radius, equal to 200 m/s (Mach
s
r
m
p
h
h
h .
<
≈
<
D
D
D
D
r
13.6 Exercises
13.6 Exercises
13.6.1 Average Density with Compression Verify that the average density is
well-suited to approximate the isentropic enthalpy change with a compression process. Calculate for air as an ideal gas ( γ = 1.40) for pressure ratio values 1.2, 1.4, 1.7,
2 and infinitesimal efficiency values 1, 0.95, 0.90: Δh s , Δh r , Δp/ρ m . Establish that
Δp/ρ m is a very good approximation of Δh s (sometimes overestimation, sometimes
underestimation) and a somewhat inferior approximation of Δh r (always underestimation).
13.6.2 Average Density with Expansion The average density concept is applicable with expansion processes as well. The term Δp/ρ m constitutes an approximation of the isentropic enthalpy change, but is much less accurate than with a
compression. For an expansion the following applies:
Calculate for air as an ideal gas ( γ = 1.40) for pressure ratio values 1.2, 1.4, 1.7,
2 and infinitesimal efficiency values 1, 0.95, 0.90: Δh s , Δh r , Δp/ρ m . Establish that
Δp/ρ m results in a slight underestimation of Δh s , and a somewhat stronger underestimation of Δh r .
13.6.3 Loss Coefficient Verify numerically that the enthalpy loss coefficient ξ 1
(13.6) and the pressure loss coefficient ω 1 (13.7) are nearly equal for low inlet Mach
number. Assume a compressor cascade with air and infinitesimal efficiency 0.9.
Consider inlet Mach numbers 0.3, 0.7 and 1.2. Assume, by way of simplification,
zero outlet velocity. Observe that the energy loss coefficient increases weakly with
increasing inlet Mach number and that the pressure loss coefficient increases quite
strongly. Observe that ω 1 is always larger than ξ 1 . Remark that the Mach number
influence on ω 1 may be neutralised by taking the logarithm of pressure instead of
pressure. Such a loss coefficient is 1 minus the infinitesimal efficiency (similarly as
with expansions; see Exercise 6.10.1).
13.6.4 Lift Coefficient and Diffusion Factor Formulate the relationship between the lift coefficient and the diffusion factor. Establish that, for a given diffusion factor value, e.g., DF = 0.60, a great variation of the lift coefficient is possible,
strongly dependent on the general deceleration value w 2 /w 1 . This demonstrates that
lift coefficient is an unworkable concept with compressor cascades.
13.6.5 Radial Variation with a Fan Stage The radius ratio chosen in Sect. 13.3
is 1.60 (1.20/0.75). This ratio is smaller with many compressor stages, but it is much
greater with a fan stage, where it may be up to four. Study a fan stage with free vortex flow (same work on all radii) and radius ratio 4. Take as an example the fan of
the Tent800 (Fig. 12.9). Tip and hub radii are 1.40 m and 0.35 m. Take radius 1 m as
approximation of the equal area average radius. Consider the fan on the ground with
T 00 = 288 K and p 00 = 100 kPa. Take tip blade speed 420 m/s and inflow velocity (absolute system) in axial direction, constant over the radius, equal to 200 m/s (Mach
s
r
m
p
h
h
h .
<
≈
<
D
D
D
D
r
13.6 Exercises