22-Mec-B6 Advanced Fluid Mechanics · December 2019
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format: National Examinations, December 2019 — 16-Mec-B6 Fluid Machinery. Closed book, three hours, 22 pages. Section A is calculative (Questions 1–5) and Section B descriptive (Questions 6–8); a candidate answers four questions from Section A and two from Section B — six questions of ten marks each, 60 marks in all. Reference data for individual questions are supplied in the Attachments on pages 12–17, and the nomenclature, general constants and reference equations on pages 18–22. All eight questions are solved below, because this set is a study resource rather than a three-hour sitting.
Check — angle conventions are taken from the paper's own attachments, and on this sitting they are uniform. The gas-turbine velocity diagram on page 13 strikes $\theta$, $\phi$, $\gamma$ and $\delta$ off the tangential direction (the plane of rotation); the pump diagram on page 14 strikes $\alpha_1$, $\alpha_2$, $\beta_1$ and $\beta_2$ off the tangential direction as well; and the Francis diagram on page 15 defines $\alpha_1$ between $V_1$ and $u_1$, again tangential. Every angle below is therefore measured from the direction of blade motion. Every constant used is the paper's own page-19 value: $g = 9.81\ \text{m}\,\text{s}^{-2}$, $\rho_{\text{water}} = 1000\ \text{kg}\,\text{m}^{-3}$, $p_{\text{atm}} = 100\ \text{kPa}$ and $p_{\text{vapour}} = 1.71\ \text{kPa}$ at $15^{\circ}\text{C}$.
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
The picture shows the familiar asymmetry of every axial gas turbine engine ever built: a long stack of ten or a dozen compressor rows feeding a combustor, and behind it only two or three turbine rows driving them all. Since the turbine and the compressor sit on the same shaft and pass very nearly the same mass flow, the turbine must produce every joule the compressor absorbs, plus whatever the propelling nozzle or the load takes. The two components therefore do the same total work, and the difference in stage count is entirely a statement about how much work one row of each type can do before it stops working properly. The answer, in a sentence, is that a turbine stage can be loaded three or four times as heavily as a compressor stage, because the flow through a turbine accelerates while the flow through a compressor decelerates.
Raising the pressure of a gas in a blade passage means diffusing it — slowing the flow so that kinetic energy converts to static pressure. Diffusion is intrinsically fragile. The boundary layer on the suction surface of a compressor blade is being pushed against a rising pressure, so it thickens rapidly, and if the demanded pressure rise is too large it separates and the row stalls. The practical measure of this limit is the de Haller number, the ratio of the relative velocity leaving a rotor row to that entering it, which must not fall much below about 0.72; equivalently, the diffusion factor must stay under roughly 0.6. Both criteria cap the turning a blade can impose, and hence cap the whirl change, and hence — through the Euler equation $w = U\,\Delta C_y$ — the work per stage. In a modern engine that ceiling corresponds to a stage temperature rise of perhaps 25 to 45 K and a stage pressure ratio between about 1.15 and 1.4.
A second constraint bites at the front of the machine. The relative Mach number at the rotor tip of the first stage is set by the flight speed, the axial velocity and the blade speed; push it much past unity and shock losses and shock–boundary-layer interaction erode the very pressure rise the stage is trying to produce. This caps blade speed, which caps $U\Delta C_y$ from the other direction. A third constraint is off-design behaviour: a compressor with a high overall pressure ratio has a rear-stage matching problem at part speed, because the density in the back stages is far from its design value and the rear blades choke while the front blades stall. That is why high-pressure-ratio compressors are split into two or three separately-shafted spools, and why variable inlet guide vanes and interstage bleed valves exist at all.
Together these fix the minimum number of compressor stages. If an engine needs an overall pressure ratio of, say, 30, and the safe stage pressure ratio is 1.3, then the required number of stages is $\ln 30/\ln 1.3 = 13$. Fewer stages than that cannot be built without exceeding the stall or Mach limit somewhere; more stages than that are possible but add weight, length and cost, so the designer sits just above the minimum. Note that the total temperature rise is not the number of stages times a stage rise computed from the inlet temperature: each stage compresses a hotter gas than the one before it, so the correct calculation is always $\Delta T_{\text{total}} = T_1\left(r_c^{(k-1)/k} - 1\right)$ taken in one step.
In a turbine the gas expands, so the pressure falls in the direction of flow. A boundary layer running downhill in pressure stays firmly attached, and the blade can be turned through 80° or 100° without any risk of the kind of separation that limits a compressor. The flow leaving each row is faster than the flow entering it, so the passage acts as a nozzle rather than a diffuser. The consequence is that a turbine stage can take a temperature drop of 150 to 250 K and a pressure ratio of two or three, which is roughly four times the work of a compressor stage. That alone accounts for the observed ratio of stage counts: if the compressor needs twelve rows and the turbine stage is four times as capable, three turbine rows will do.
Three quite different limits stop a designer from putting the whole expansion into a single turbine row. The first is metallurgical. Turbine entry temperature is the single most valuable parameter in the whole engine, and it is set by what the first-row blade material and its cooling can survive. Loading a single stage with the whole expansion means the largest possible gas velocities and the largest possible heat transfer coefficients at exactly the hottest point in the machine, which is precisely where designers least want them. Splitting the drop lets the first stage be modest and cooled hard, and the later stages run cooler and can be uncooled.
The second is mechanical. Stage work is $U\,\Delta C_y$; a single-stage machine absorbing the whole drop would need either an enormous blade speed, which the disc rim stress cannot carry at turbine temperatures, or an enormous whirl change, which means very high absolute velocities, high friction losses and a large residual swirl leaving the machine. Rim stress scales with the square of blade speed and falls off sharply in allowable value as metal temperature rises, so the two demands fight each other.
The third is aerodynamic and often decisive: exit kinetic energy. Whatever velocity the gas still has when it leaves the last row is energy the shaft never receives. In the four-stage carbon-dioxide turbine of Questions 1 and 2 this showed up quantitatively — the velocity-diagram power came out at 242.3 MW against the thermodynamic 243.6 MW, and the 1.3 MW difference is exactly the leaving loss. A single-stage machine taking four times the drop would have roughly twice the absolute exit velocity and four times that loss. Multi-staging also keeps the flow coefficient and stage loading of every row in the range where blade efficiency is highest, and it allows the annulus to open out gradually as the gas expands, which keeps the axial velocity nearly constant and the incidence angles well behaved from hub to tip.
The design sequence in both components is the same, and it is exactly the one used in Question 1. Fix the rotational speed from the load or the fan; fix the mean blade speed from stress and Mach limits; fix the axial velocity from annulus size and choking margin; then compute the work one stage can do, $w = U\Delta C_y$, and divide the required total work by it. For the compressor the answer is rounded up and generally a stage or two is added for surge margin and off-design matching; for the turbine the answer is rounded up as well but the number is small, so the choice between two and three stages is usually decided by exit swirl and by whether the last row can be left uncooled. The result is the picture in the question: many small steps up, a few big steps down.
