Question 11.1: The design point of a centrifugal-compressor stage (shown in......

Part a: Let us begin by computing the impeller-inlet static density:

\rho_{1}=\rho_{t\ 1}{\biggl[}1-{\biggl(}{\frac{\gamma-1}{\gamma+1}}{\biggr)}M_{c r\ 1}{}^{2}{\biggr]}^{\frac{1}{\gamma-1}}

=\left(\frac{p_{t\ 1}}{R T_{t\ 1}}\right)\biggl[1-\biggl(\frac{\gamma-1}{\gamma+1}\biggr)M_{c r\ 1}{}^{2}\biggr]^{\frac{1}{\gamma-1}}=1.23\,\mathrm{kg/m}^{3}

Next, we apply the continuity equation at the impeller inlet station, knowing that V_{1} is totally axial:

V_{z1}=V_{1}=M_{c r\ 1}V_{c r\ 1}=M_{c r\ 1}\sqrt{\left(\frac{2\gamma}{\gamma+1}\right)R T_{t\ 1}}=95.4\,\mathrm{m/s}

The impeller-inlet tip radius can now be calculated:

r_{1t}=\sqrt{r_{1h}+\left(\frac{\dot{m}}{\pi\rho_{1}V_{z1}}\right)}=0.126\,\mathrm{m}

Part b: In order to calculate the difference in the hub-to-tip blade inlet (metal) angle (\Delta\beta_{1}{}^{\prime}), we recall that the flow is fully guided (at inlet) by the impeller blades. The procedure to calculate this variable is as follows:

U_{1m}=\omega r_{1m}=106.2\,\mathrm{m/s}

W_{1m}=\sqrt{U_{1m}{}^{2}+V_{z1}{}^{2}}=142.7\,\mathrm{m/s}

U_{1h}=\omega r_{1h}=54.0\,\mathrm{m/s}

U_{1t}=\omega r_{1t}=158.3\,\mathrm{m/s}

\beta_{1h}=\tan^{-1}\left(\frac{U_{1h}}{V_{z1}}\right)=29.5^{\circ}

\beta_{1t}=\tan^{-1}\left(\frac{U_{1t}}{V_{z1}}\right)=58.9^{\circ}

\Delta\beta_{1}{}^{\prime}=\beta_{1t}-\beta_{1h}=29.4^{\circ}

Part c: Let us calculate the impeller-exit relative velocity (W_{2}):

U_{2}=\omega r_{2}=408.4\,\mathrm{m/s}

h_{2}-h_{1}=84,100\,\mathrm{J/kg}=\frac{\left(W_{1m}{}^{2}-W_{2}{}^{2}\right)}{2}+\frac{\left(U_{2}{}^{2}-U_{1m}{}^{2}\right)}{2}

Upon substitution, we get

W_{2}=87.6\,\mathrm{m/s}

As for the impeller-exit absolute velocity (V_{2}), we proceed as follows:

V_{r\,2}=W_{r\,2}=W_{2}\cos\beta_{2}=81.8\,\mathrm{m/s}

V_{\theta\ 2}=W_{\theta\ 2}+U_{2}=W_{2}\sin[-21^{\circ}]+U_{2}=377.0\mathrm{\,m/s}

V_{2}=\sqrt{V_{\theta\ 2}{}^{2}+V_{r\,2}{}^{2}}=385.8\,\mathrm{m/s}

Now, the impeller dynamic head can be calculated as follows:

(\Delta h)_{d y n.}=\frac{\left(V_{2}{}^{2}-V_{1m}{}^{2}\right)}{2}=69,860\,\mathrm{J/kg}

Part d: The stage reaction can easily be calculated as follows:

\mathrm{Stage\;Reaction}\,(R)=\frac{(\Delta h)_{s t a t i c}}{(\Delta h)_{s t a t i c}+(\Delta h)_{d y n a m i c}}=54.6%

Part e: In order to calculate the impeller-exit sidewall spacing (b_{2}), we first calculate the static-density magnitude as follows:

T_{t\ 2}=T_{t\ 1}+\frac{\Delta h_{t}}{c_{p}}=T_{t\ 1}+\frac{[(\Delta h)_{s t a t i c}+(\Delta h)_{d y n a m i c}]}{c_{p}}=455.3\,\mathrm{K}

p_{t\ {2}}=p_{t\ {1}}\left\{1+\eta_{ i m p.}\left[\left({\frac{T_{t\ {2}}}{T_{t\ \ 1}}}\right)-1\right]\right\}^{\frac{\gamma}{\gamma-1}}= 3.47\ \mathrm{bars}

\rho_{2}=\left(\frac{p_{t\ 2}}{R T_{t\ 2}}\right)\left\{1-\left(\frac{\gamma-1}{\gamma+1}\right)\left[\frac{V_{2}{}^{2}}{\left(\frac{2\gamma}{\gamma+1}\right)R T_{t\ 2}}\right]\right\}^{\frac{1}{\gamma-1}}=1.69\,\mathrm{kg/m}^{3}

Now we calculate the impeller-exit endwall spacing as follows:

b_{2}={\frac{\dot{m}}{(\rho_{2}V_{r\ 2}2\pi r_{2})}}=0.42\,{\mathrm{cm}}

Part f: In order to calculate the impeller-wise change in total relative pressure \left[(\Delta p_{t\ r})_{i m p}\right], we proceed as follows:

(T_{t\ r})_{1m}=T_{t\ 1}+\left({\frac{W_{1m}{}^{2}-V_{1m}{}^{2}}{2c_{p}}}\right)=307.6\,\mathrm{K}

Similarly,

T_{t\ r\ 2}=385.0\,\mathrm{K}

(p_{t\ r})_{1m}=p_{t\ 1}\biggl(\frac{T_{t\ r\ 1m}}{T_{t\ 1}}\biggr)^{\frac{\gamma}{\gamma-1}}=1.15\,\mathrm{bars}

(p_{t\ r})_{2}=p_{t\ 2}\biggl(\frac{T_{t\ r\ 2}}{T_{t\ 2}}\biggr)^{\frac{\gamma}{\gamma-1}}=1.93\,\mathrm{bars}

(\Delta p_{t\ r})_{i m p.}=p_{t\ r\ 2}-p_{t\ r\,1m}=0.78\,\mathrm{bars}

In reference to the preceding results, note the following:

1) Although (\Delta p_{t\ r})_{i m p.} encompasses the profile (or skin friction) losses, it also reflects the fact that the total relative pressure will rise as a result of the radius change along the impeller (master) streamline.
2) In referring to the inlet station, it was necessary to use the subscript “1m,” signifying the mean radius, since the total relative temperature is a function of velocity-diagram variables, where the solid-body rotational velocity U is radius-dependent. The total relative pressure, by reference to the last three computational steps (above), is (in turn) a function of the total-relative temperature.

Part g: Let us now perform the critical computational step of verifying that the impeller-exit (absolute) critical Mach number is not greater than unity. Choking, if present, will now take place immediately outside the impeller exit station. Note that the impeller-exit magnitude of relative velocity (W_{2}) will always be too small to warrant verification of the exit relative critical Mach number, a compressor-rotor feature that is quite the opposite when it comes to turbine aerodynamics.

M_{c r\ 2}=\frac{V_{2}}{\sqrt{\left(\frac{2\gamma}{\gamma+1}\right)R T_{t\ 2}}}=0.988\left(\mathrm{acceptably\;subsonic}\right)

The impeller-exit static pressure can now be calculated:

p_{2}=p_{t\ 2}{\biggl[}1-{\left({\frac{\gamma-1}{\gamma+1}}\right)}M_{c r\ 2}{}^{2}{\biggr]}^{\frac{\gamma}{\gamma-1}}=1.86\,\mathrm{bars}

Under the free-vortex flow-structure assumption across the vaneless diffuser, we can calculate the flow variables at the vaned-diffuser inlet station (station 3) as follows:

V_{r\ 3}=\left(\frac{r_{2}}{r_{3}}\right)V_{r\ 2}=66.7\,\mathrm{m/s}

V_{\theta\ 3}=\left(\frac{r_{2}}{r_{3}}\right)V_{\theta\ 2}=307.5\,\mathrm{m/s}

V_{3}=\sqrt{V_{r\ 3}{}^{2}+V_{\theta\ 3}{}^{2}}=314.6\,\mathrm{m/s}

V_{c r\ 3}=\sqrt{\left(\frac{2\gamma}{\gamma+1}\right)R T_{t\ 3}}=390.4\,\mathrm{m/s}

p_{3}=p_{t\ 3}\bigg[1-\bigg(\frac{\gamma-1}{\gamma+1}\bigg)M_{c r,3}{}^{2}\bigg]^{\frac{\gamma}{\gamma-1}}=2.32\,\mathrm{bars}

Proceeding to the vaned-stator exit station, we have

p_{t\ 4}=(1-0.063)p_{t\ 3}=3.25\,\mathrm{bars}

T_{t\ 4}=T_{t\ 3}=\,T_{t\ 2}=455.3\,\mathrm{K\,(adialatic\;flow)}

In this part of the problem, we are required to calculate the exit value of the vaned-diffuser static pressure (p_{4}). Unfortunately, there is no direct way of achieving this.
With the vaned-stator exit value of total pressure now known, we first have to compute the exit magnitude of the critical Mach number (M_{cr\ 4}) or, equivalently, the exit velocity (V_{4}), which we know to be totally radial (from the problem statement).
Naturally, in this case, we would think of applying the continuity equation at the vaned-stator exit station. However, such a step requires knowledge of the exit static density magnitude (ρ_{4}), which itself is a function of the critical Mach number.
The procedure, under these circumstances, has to be iterative, whereby the following computational procedure is repeated towards convergence:

• Assume the M_{cr\ 4} magnitude.
• Calculate the corresponding magnitude of static density (ρ_{4}).
• Apply the continuity equation at station 4, and find V_{r\ 4} (=V_{4}).
• Now calculate the exit critical Mach no. (M_{cr\ 4}).
• Compare the computed M_{cr\ 4} to the assumed value (above).

The foregoing computational procedure should be repeated until the point is reached where the assumed and computed critical Mach numbers are sufficiently close to one another.
The iterative procedure (above) was executed and the final results obtained:

M_{c r\ 4}=0.303

\rho_{4}=2.399\,\mathrm{kg/m^{3}}

V_{4}=V_{r\ 4}=118.5\,\mathrm{m/s}

The exit magnitude of static pressure (p_{4}) can now be calculated as

p_{4}=p_{t\ 4}\biggl[1-\biggl({\frac{\gamma-1}{\gamma+1}}\biggr)M_{c r\ 4}{}^{2}\biggr]^{\frac{\gamma}{\gamma-1}}=3.08\,\mathrm{bars}

The static-pressure rise across the vaned stator can now be determined:

\Delta p_{s t a t o r}=p_{4}-p_{3}=0.76\,\mathrm{bars}

Part h: Having computed the stage-exit static pressure (p_{4}), determination of the stage total-to-static efficiency is straightforward:

\eta_{t-s}={\frac{\left({\frac{p_{4}}{p_{t\ 1}}}\right)^{\frac{\gamma-1}{\gamma}}-1}{\left({\frac{T_{t\ 4}}{T_{t\ 1}}}\right)-1}}=68.6%