ME 563: Turbomachines

Estimated study time: 11 minutes

Table of contents

Sources and References

Dixon and Hall, Fluid Mechanics and Thermodynamics of Turbomachinery, 7th ed., Butterworth-Heinemann.
Cumpsty, Compressor Aerodynamics, 2nd ed., Krieger.
Japikse and Baines, Introduction to Turbomachinery, Concepts ETI.
Saravanamuttoo, Rogers, and Cohen, Gas Turbine Theory, 7th ed., Pearson.
Denton, “Loss mechanisms in turbomachines,” ASME Journal of Turbomachinery, 1993.

Chapter 1: Classification and Fundamental Equations

Turbomachines exchange energy between a fluid and a rotor through an organized, continuous flow. They range from desk-fan scale to the largest steam turbines; they compress or expand gas, pump or extract power from liquid, and provide thrust for aircraft. Despite the diversity, a compact set of equations and a handful of geometric ideas describe them all.

1.1 Classification

Turbomachines are classified by operation (producing or consuming power), by fluid (compressible or incompressible), and by the direction of through-flow relative to the axis of rotation (axial, radial, or mixed). Power-producing machines — turbines — convert fluid energy to shaft work; power-consuming machines — compressors, fans, pumps — do the reverse. Axial machines have flow parallel to the axis throughout; radial (centrifugal for compressors and pumps, centripetal for turbines) redirect flow from axial inlet to radial outlet or vice versa.

1.2 Euler’s Turbomachine Equation

Applying angular-momentum conservation to the rotor control volume gives Euler’s equation,

\[ w = U_2 c_{\theta 2} - U_1 c_{\theta 1}, \]

where \( U = \omega r \) is blade speed, \( c_\theta \) is the tangential component of absolute velocity, and \( w \) is specific work delivered to the fluid (positive for compressors, negative for turbines). Each row of blades thus exchanges angular momentum with the flow; the summed product \( U c_\theta \) across all rows equals the total specific work.

1.3 Velocity Triangles

At any radius of a rotor row the flow is described by three velocity vectors: absolute \( \mathbf{c} \), relative \( \mathbf{w} \), and blade \( \mathbf{U} \), with \( \mathbf{c} = \mathbf{w} + \mathbf{U} \). The triangle at inlet and the triangle at exit together encode the aerodynamic loading. Stage reaction

\[ R = \frac{h_2 - h_1}{h_{03} - h_{01}} \]

measures the fraction of total enthalpy rise (for a compressor) delivered in the rotor; 50 percent reaction designs give symmetric rotor–stator loading and good efficiency.

Chapter 2: Performance Parameters and Modelling

2.1 Dimensionless Groups

Similarity analysis produces flow coefficient \( \phi = c_x/U \), work coefficient \( \psi = \Delta h_0/U^2 \), pressure coefficient, Reynolds number, and — for compressible flow — corrected mass flow

\[ \dot{m}\frac{\sqrt{\theta}}{\delta} \]

and corrected speed \( N/\sqrt{\theta} \), with \( \theta \) and \( \delta \) normalized temperature and pressure. These groups collapse performance maps across operating conditions and compose the maps one sees in textbooks: compressor map lines of corrected speed, surge line, choke limit, and efficiency contours; turbine map similar with choked flow and limiting speed.

2.2 Efficiency Definitions

Isentropic efficiency compares the actual enthalpy change to that of an isentropic process at the same pressure ratio. For a compressor,

\[ \eta_c = \frac{h_{02s} - h_{01}}{h_{02} - h_{01}}, \]

and for a turbine,

\[ \eta_t = \frac{h_{01} - h_{02}}{h_{01} - h_{02s}}. \]

Polytropic (small-stage) efficiency corrects for the fact that a compressor of multiple stages has each stage dissipating into the next, and it provides a more useful figure of merit when comparing machines of different pressure ratio.

2.3 Compressible-Flow Laws of Modelling

Corrected conditions preserve similarity across inlet states. A compressor tested at altitude but matched in corrected parameters to the design point produces the same nondimensional performance as one tested at sea level. This is the basis for cross-facility comparison and for off-design prediction.

Cordier diagram. Plotting specific speed against specific diameter for a large population of efficient machines produces a narrow band (Cordier line). Preliminary design starts by reading off this band: a required duty (flow, head, speed) defines a point; the corresponding machine type — axial, mixed, radial — is read from its location.

Chapter 3: Axial Turbomachines

3.1 Axial Compressors

An axial compressor stage comprises a rotor row that raises total enthalpy and a stator row that diffuses part of the dynamic head back to static pressure and redirects the flow for the next rotor. Diffusion factor

\[ DF = 1 - \frac{W_2}{W_1} + \frac{\Delta W_\theta}{2 \sigma W_1}, \]

with solidity \( \sigma \), quantifies the loading of a row; values above about 0.5 mark the threshold at which blade-row losses rise sharply and the rotor approaches stall. Stall and surge are instability phenomena: rotating stall is a localized cell of low-throughflow that travels around the annulus; surge is an axial oscillation of the whole compressor against downstream capacitance. Design margin and surge-control schedules keep the machine away from these limits.

3.2 Axial Turbines

Axial turbines accept high-temperature gas from the combustor or boiler and extract work. Stator (nozzle) rows accelerate the flow and turn it into the rotor; rotor rows turn and decelerate the flow relative to the blade. The Smith chart relates stage loading and flow coefficient to efficiency and guides preliminary selection. High-pressure turbines operate at gas temperatures exceeding metal melting points and rely on film cooling, internal convective cooling, and thermal-barrier coatings.

3.3 Three-Dimensional Effects

Secondary flows (tip-clearance vortex, horseshoe vortex, passage vortex, corner separation) depart from the two-dimensional cascade idealization and redistribute loss and work along the span. Compound lean, sweep, and end-bend are three-dimensional blade shaping strategies that reduce these losses. Radial equilibrium,

\[ \frac{1}{\rho} \frac{d p}{d r} = \frac{c_\theta^2}{r}, \]

fixes the radial distribution of static pressure under axisymmetric, inviscid assumptions and underlies free-vortex and forced-vortex design conventions.

Chapter 4: Radial Turbomachines

4.1 Centrifugal Compressors and Pumps

A centrifugal impeller raises the tangential velocity of the fluid from nearly zero at the eye to \( U_2 \beta_2 \) at the tip, where \( \beta_2 \) reflects blade backsweep and slip. The slip factor

\[ \sigma_s = 1 - \frac{c_{\theta 2,\mathrm{slip}}}{U_2} \]

corrects Euler work for the finite number of blades (Stanitz, Wiesner, Stodola correlations). Downstream diffusers — vaneless, vaned, or pipe — recover static pressure from the high kinetic energy leaving the impeller. Surge margin is set largely by the diffuser’s stable operating range.

4.2 Radial Turbines

Radial-inflow turbines (ninety-degree IFR) invert the centrifugal flow path: gas enters through a scroll and nozzles, flows radially inward through the rotor, and exits axially. They suit small-power applications such as turbochargers and cryogenic expanders. Optimal tip-speed ratio \( U_2/C_0 \) is around 0.7, and the blade loading is markedly different from an axial counterpart.

Chapter 5: Losses, Efficiencies, and Design Process

5.1 Loss Mechanisms

Denton’s classification separates loss into profile (boundary-layer and trailing-edge), endwall (secondary and corner), tip clearance, and shock contributions. Each scales with different design variables: profile loss with Reynolds number and blade surface finish; endwall loss with aspect ratio and loading; tip-clearance loss with gap-to-span ratio and tip geometry; shock loss with Mach number and blade thickness distribution.

Quantitatively, a row loss coefficient

\[ \omega = \frac{p_{01} - p_{02}}{p_{01} - p_{1}} \]

measures total-pressure loss and is predicted by correlations (Ainley–Mathieson, Craig–Cox, Dunham–Came) or by CFD.

5.2 Design Process

Preliminary design begins with meanline analysis: given duty and speed, choose stage count and radius distribution to hit flow coefficient and loading targets while controlling loss. Through-flow analysis (axisymmetric streamline curvature or finite-volume solvers) distributes work and loss along the span. Blade-to-blade CFD and fully three-dimensional CFD refine airfoil shapes and minimize endwall and tip-clearance losses. Structural, thermal, and aeroelastic analyses run alongside and constrain the aerodynamic choices.

Single-stage steam turbine. With inlet conditions 50 bar, 550 °C and outlet 5 bar, an ideal expansion releases roughly 750 kJ/kg of specific work. A 50-percent-reaction stage at U = 350 m/s requires ψ ≈ 1.1, consistent with well-designed modern practice. Loss analysis accounts for ~10 percent of ideal work, giving roughly 675 kJ/kg delivered.

Turbomachine design is a continuous balance of aerodynamics, materials, structural integrity, and manufacturing cost. The fundamentals above — Euler’s equation, similarity, velocity triangles, radial equilibrium — scale from hair-dryer fans to power-plant turbines.