Manipulate impeller inlet/outlet conditions (peripheral velocity U, axial velocity Cm, inlet angle α₁, relative exit angle β₂) to draw velocity triangles and calculate specific work and degree of reaction via the Euler equation in real time.
I've heard the term "velocity triangle," but I can't picture what's happening inside the impeller of a turbomachine...
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It's like walking inside a train. If the train is moving at 100 km/h and you walk inside at 5 km/h, your speed seen from outside (absolute velocity) is 105 km/h, and your speed seen from the train (relative velocity) is 5 km/h. Exactly the same thing happens to the fluid inside the impeller. The velocity $C$ seen from the ground (stationary frame) is the absolute velocity, the velocity $W$ seen from the rotating blades is the relative velocity, and the blade's own rotational speed is $U$. The vector relationship $C = U + W$ connects these three, forming a triangle. Check it out in the "Velocity Triangle" tab of the simulator.
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I see! So why do we draw two triangles, one for "inlet" and one for "outlet"?
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Because the direction and magnitude of the velocity change when the fluid enters and leaves the impeller. That "change" corresponds to the energy given to the fluid, i.e., the "work." Try changing the outlet angle β₂ in the simulator. The shape of the right-side (outlet) velocity triangle changes, and at the same time, the specific work w value also changes, right? The more negative β₂ becomes (bending the fluid more at the outlet), the more work is done on the fluid.
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Indeed, when I changed β₂, the "specific work" changed! Looking at the "β₂ Sensitivity Analysis" tab, the specific work is minimal around β₂=0.
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That's right. If β₂ is zero, the outlet relative velocity is purely axial, and the pump does no work (theoretically zero). As β₂ becomes larger, the fluid is accelerated in the circumferential direction, increasing the work. In actual centrifugal pumps, β₂ = 20–35° (backward-curved blades) is common for a balance with efficiency. Also, the value called "reaction degree" is important. When it is 0.5 (50% reaction), the inlet and outlet velocity triangles become symmetric, which is the standard design for axial-flow turbomachinery. Check it out with the "50% Reaction Stage" preset.
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When I switched presets, the triangle shapes changed drastically between the centrifugal pump and the axial turbine! The velocity scales are different.
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Exactly! Centrifugal pumps have large peripheral velocity U and large absolute velocity C. Axial machines have relatively small U but large Cm. Each has an optimal shape according to its application (high head vs. large flow rate). The ability to "read" velocity triangles is a fundamental literacy for turbomachinery engineers.
Euler's Work Equation and Velocity Triangle Formulas
The specific work transferred by the impeller to a unit mass of fluid, corresponding to total-pressure rise, is given by Euler's work equation:
$$w = U_2 C_{2u} - U_1 C_{1u}$$
$U_1, U_2$: inlet and outlet peripheral velocity [m/s]; $C_{1u}, C_{2u}$: tangential components of absolute velocity [m/s]. Compute them from $C_{1u}=C_{m1}\tan\alpha_1$ and $C_{2u}=C_{m2}\tan\beta_2 + U_2$.
Degree of reaction and the nondimensional loading coefficient ψ are important design indicators:
ψ is a nondimensional work coefficient. Values around 0.2 to 0.5 are commonly efficient design ranges.
Application in Actual Machinery
Aircraft engine compressors: Multistage axial compressors optimize velocity triangles at each stage and stack stage pressure ratios of roughly 1.1 to 1.3 to achieve high overall compression. Inlet guide vanes (α₁ ≠ 0) add pre-swirl to keep blade loading appropriate.
Steam turbines: Impulse stages (R≈0) and reaction stages (R≈0.5) are combined in multistage layouts to convert steam thermal energy into mechanical work efficiently. The outlet velocity triangle is shaped to optimize inlet conditions for the next stage.
Centrifugal pumps and compressors: Backward-curved blades (β₂ < 0) improve efficiency and help maintain stable operation across a wide flow range. Outlet width and angle optimization are key to preventing surge.
Frequently Asked Questions
Since the inlet absolute velocity is purely axial (C₁ᵤ=0), the Euler equation simplifies to w=U₂C₂ᵤ. This makes design and analysis easier, and eliminates the need for inlet guide vanes (IGV), resulting in a compact design. Many simple pumps and compressors are designed under this condition. On the other hand, setting α₁≠0 (giving pre-swirl) reduces work with positive swirl (α₁>0) to improve efficiency, and increases work with reverse swirl (α₁<0).
A negative β₂ (outlet relative velocity opposite to rotation) is called a backward-curved blade, and a positive value is called a forward-curved blade. Backward-curved (β₂ < 0) blades have high efficiency and stable flow characteristics, and are widely used in centrifugal pumps and compressors. Forward-curved (β₂ > 0) blades can achieve high work at the same rotational speed, but have lower efficiency and tend to be unstable with flow changes. Comparing positive and negative β₂ values in this simulator shows the difference in specific work.
When the reaction degree R=0.5, the inlet and outlet velocity triangles are mirror images. This means that equal velocity changes occur in both the stator and rotor, avoiding excessive load concentration on one side. Also, the blade shape becomes nearly symmetric front-to-back, making design and manufacturing relatively easier. Furthermore, the pressure difference across the blade becomes uniform, tending to stabilize the flow over the blade profile and reduce secondary losses. Check the symmetry of the inlet and outlet triangles with the "50% Reaction Stage" preset.
This tool calculates an ideal 1D flow. In reality, actual machine performance is affected by: ① blade surface friction loss (profile loss), ② leakage loss at the flow path ends (tip clearance loss), ③ loss due to radial secondary flow, ④ unstable phenomena such as surge and choking, and ⑤ a decrease in actual C₂ᵤ due to the slip factor (effect of finite blade count). Detailed design considering these factors requires CFD analysis.
Generally, for axial-flow machines (compressors and turbines), ψ = 0.2–0.4 is an efficient design range. If ψ is too large, blade stall is likely to occur, and efficiency drops sharply. For centrifugal machines, due to different blade shapes, ψ ≈ 0.5–0.7 may be acceptable. Checking the ψ values of each preset in this simulator allows you to understand typical design points for different machine types.
In centrifugal (radial) machines, the impeller inlet radius r₁ and outlet radius r₂ are different, so even at the same rotational speed ω [rad/s], the peripheral velocity U=rω differs. Generally, U₂ > U₁, and the larger this difference (larger diameter ratio r₂/r₁), the larger the right-hand side of the Euler equation, resulting in higher specific work (head). This is why centrifugal pumps are suitable for high head. On the other hand, in axial-flow machines, since they rotate at the same radius, U₁ ≈ U₂.
What is Turbomachinery Velocity?
Turbomachinery Velocity is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.
By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition — making it an effective learning tool for students and a rapid-verification tool for practicing engineers.
Physical Model & Key Equations
The simulator is based on the governing equations behind Turbomachinery Velocity Triangle Analyzer. Understanding these equations is key to interpreting the results correctly.
Each parameter in the equations corresponds to a slider in the control panel. Moving a slider changes the equation's solution in real time, helping you build a direct connection between mathematical expressions and physical behavior.
Real-World Applications
Engineering Design: The concepts behind Turbomachinery Velocity Triangle Analyzer are applied across mechanical, structural, electrical, and fluid engineering disciplines. This tool provides a quick way to estimate design parameters and sensitivity before committing to full CAE analysis.
Education & Research: Widely used in engineering curricula to connect theory with numerical computation. Also serves as a first-pass validation tool in research settings.
CAE Workflow Integration: Before running finite element (FEM) or computational fluid dynamics (CFD) simulations, engineers use simplified models like this to establish physical scale, identify dominant parameters, and define realistic boundary conditions.
Common Misconceptions and Points of Caution
Model assumptions: The mathematical model used here relies on simplifying assumptions such as linearity, homogeneity, and isotropy. Always verify that your real system satisfies these assumptions before applying results directly to design decisions.
Units and scale: Many calculation errors arise from unit conversion mistakes or order-of-magnitude errors. Pay close attention to the units shown next to each parameter input.
Validating results: Always sanity-check simulator output against physical intuition or hand calculations. If a result seems unexpected, review your input parameters or verify with an independent method.