Question 13.10: The compressor in a single-spool turbojet engine is composed......

(a) The temperature difference across these compressors is given by

(T_{0\text{e}}-T_{0\text{i}})_{\text{axial}}=N \Delta T_{0\text{s}} \\ (T_{0\text{e}}-T_{0\text{i}})_{\text{centrifugal}}=\frac{\Psi \sigma U^2_{\text{tip}}}{Cp}

Both are constants for the two arrangements.
Arrangement 1
The pressure and temperature at the inlets and outlets of compressor in arrangement (1) is shown in Figure 13.58b.

\Delta T_{0\text{axial}}=N \Delta T_{0\text{s}}=4 \times 25=100 \text{ K} \\ T_{01}=298 \text{ K}, \quad T_{02}=398 \text{ K} \\ \frac{n}{n-1} =\eta_{\text{PC}}\frac{\gamma}{\gamma-1} =0.9\frac{1.4}{0.4} =3.15 \\ \pi_{\text{axial}}=\left(1+\frac{\Delta T_0}{T_{01}} \right) ^{n/(n-1)} \\ \pi_{\text{axial}}=\left(1+\frac{100}{298}\right)^{n/(n-1)} =(1.3356)^{3.15}=2.488

For the centrifugal compressor

U_{\text{tip}}=\pi d_{\text{t}}N=\pi \times 0.5 \times 290=455.5 \text{ m/s} \\ (\Delta T_0)_{\text{centrifugal}}=\frac{\Psi \sigma U^2_{\text{tip}}}{Cp} \\ (\Delta T_0)_{\text{centrifugal}}=\frac{1.04 \times 0.9 \times (455.5)^2}{1005} =193 \text{ K} \\ T_{03}=T_{02}+(\Delta T_0)_{\text{centrifugal}}=591 \text{ K} \\ \pi_{\text{centrifugal}}=\left(1+\eta_{\text{c}}\frac{\Delta T_0}{T_{02}} \right) ^{\gamma/(\gamma-1)}=\left(1+0.9\frac{193}{398} \right) ^{3.5}=3.5522 \\ \pi_{\text{total}}=\pi_{\text{axial}} \times \pi_{\text{centrifugal}}=8.838

The overall temperature rise across the two compressors is constant in both arrangements and equal to 293 K.

Arrangement 2
(a) The pressure and temperature at the inlets and outlets of compressor in arrangement (2) is shown in Figure 13.58c.

\pi_{\text{centrifugal}}=\left(1+0.9\frac{193}{298} \right) ^{3.5}=(1.5828859)^{3.5}=4.98969 \\ \pi_{\text{axial}}=\left(1+\frac{100}{491} \right) ^{3.15}=1.793 \\ \pi_{\text{total}}=8.9465

(b) The results for both arrangements are summarized in Tables 13.11 and 13.12 for axial and centrifugal compressors, respectively. The compressor map for axial compressor is shown in Figure 13.58d, while Figure 13.58e illustrates the compressor map for centrifugal compressor.
(c) The overall efficiency (\eta_{\text{a}}=\eta_{\text{c}}=0.9)

\eta=\frac{T_{02^\prime}-T_{01}}{T_{02}-T_{01}} =\frac{T_{01}\left\{(p_{02}/p_{01})^{(\gamma-1)/\gamma}-1\right\} }{T_{02}  –  T_{01}} \\ \eta_{\text{arrangement(1)}}=\frac{298\left\{(8.839)^{0.4/1.4}-1\right\} }{293} =\frac{298(1.8638-1)}{293} =0.8785 \\ \eta_{\text{arrangement(2)}}=\frac{298\left\{(8.9465)^{0.4/1.4}-1\right\} }{293} =\frac{298(1.8702-1)}{293} =0.8851

(d) The efficiency of the axial compressor is unchanged while the efficiency of the centrifugal compressor drops, or \eta_{\text{a}}=0.9,\eta_{\text{c}}=0.8.
Arrangement 1

T_{01}=298 \text{ K}

The compression ratio for the axial compressor will not change.

\therefore \pi_{\text{axial}}=2.488, \quad T_{02}=398 \text{ K} \\ \pi_{\text{centrifugal }} =\left(1+0.8\frac{193}{398} \right) ^{3.5}=(1.3879396)^{3.5}=3.1496 \\ T_{03}=591 \text{ K} \\ \pi_{\text{total}}=7.836

Arrangement 2

T_{01}=298 \text{ K} \\ \pi_{\text{centrifugal}}=\left(1+0.8\frac{193}{298} \right) ^{3.5}=4.3099 \\ T_{02}=491 \text{ K} \\ \pi_{\text{axial}}=\left(1+\frac{100}{491} \right) ^{3.15}=1.793 \\ \pi_{\text{total}}=7.7276

The overall efficiency (Table 13.13)

\eta_{\text{arrangement(1)}}=\frac{298\left\{(7.836)^{0.4/1.4}-1\right\} }{293} =\frac{298(1.8007-1)}{293} =0.81436 \\ \eta_{\text{arrangement(2)}}=\frac{298\left[(7.7276)^{0.4/1.4}-1\right] }{293} =\frac{298(1.7936-1)}{293} =0.8071

(e) From the above calculations, which is summarized in Table 13.13, it can be deduced that
1. The overall pressure ratio is slightly higher for arrangement (2) if both compressors have the same efficiency, but if the efficiency of the centrifugal compressor is dropped then arrangement (1) gives a higher pressure ratio.
2. When the efficiency of centrifugal compressor is less than that of the axial one, then the overall efficiency of the compressor in arrangement (1) is larger than the overall efficiency in arrangement (2).
3. Arrangement (1) is appropriate for the installation of combustion chamber while arrangement (2) needs some piping system between both compressors, which adds some mechanical complications to the compressor section. This piping increases the skin friction and thus reduces the total pressure upstream of the axial compressor.
The external drag force in arrangement (2) is larger than that in arrangement (1) as shown in Figure 13.58f.