Cordier diagram

Theory and Concepts
1

why does the cordier diagram have the shape it does? can i predict that with an equation? what analysis would I do to do that?

2

This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook

PK Bot’s answer:

  1. PURPOSE OF THE EXERCISE:
    Demonstrate how to use nondimensional analysis and pump affinity laws to explain the hump‐shaped efficiency curve on a Cordier diagram and to predict its key features with an equation.

  2. ENGINEERING CONTEXT:
    In pump selection and design the Cordier diagram is used to identify which operating conditions (or specific speeds) yield high thermodynamic efficiency. Mastering this analysis helps engineers select or design centrifugal pumps that perform near optimum conditions, thereby ensuring efficient energy transfer to the fluid.

  3. PHYSICAL CONTEXT:
    The hump‐shaped diagram results from the balance between impeller geometry, inertial forces, and energy conversion; too low a specific speed leads to poor flow admission and separation, and too high a value causes excessive recirculation and hydraulic losses.

  4. ASSUMPTIONS:
    Assume steady, incompressible, inviscid flow and that the pump is operating in the turbulent regime where inertial effects dominate; these assumptions are justified because dynamic pumps are generally designed for high Reynolds number water flow.

  5. DRAWINGS:
    A useful sketch would show the Cordier diagram with nondimensional specific speed on the horizontal axis and efficiency on the vertical axis, featuring a clear efficiency peak. An additional schematic of a centrifugal pump with labeled impeller, volute, and flow directions can help relate the performance curve to pump geometry.

  6. SOLUTION TECHNIQUE:
    Begin by considering the nondimensional specific speed defined as

\omega_p^* = \frac{\omega\,Q^{1/2}\,\rho^{3/4}}{\Delta p_0^{3/4}},

where \omega is the pump’s angular speed, Q is the volumetric flow rate, \rho is the fluid density, and \Delta p_0 is the total pressure change. Scaling arguments based on the pump affinity laws indicate that water horsepower is proportional to Q\,\Delta p_0 and the brake horsepower to \omega\,T. One may approximate the efficiency \eta near the optimum as controlled by the deviation of \omega_p^* from an optimal value \omega_{p,\mathrm{opt}}^*, for instance by writing a simple quadratic model

\eta = \eta_{\max} - a\,\left(\omega_p^* - \omega_{p,\mathrm{opt}}^*\right)^2,

with a>0. The peak (hump) in the Cordier diagram thus corresponds to \omega_p^* \approx \omega_{p,\mathrm{opt}}^*; deviations in either direction lead to lower efficiency due to either flow separation or excessive internal recirculation. This analysis shows that by nondimensionalizing the key operating parameters—with the specific speed as the control parameter—we capture the competing effects that set the shape of the diagram.

  1. REFLECTION:
    The analysis encapsulates how efficiency in a centrifugal pump is maximized over a narrow band of nondimensional specific speed, as demonstrated in the Cordier diagram. Although the quadratic model is a simplified approximation, it highlights that efficiency is peaked and rapidly decreases when operating away from the optimum value. Engineers use such insights to select the motor speed and impeller geometry that ensure the pump’s operating point will fall within the high-efficiency region of the diagram.

In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections according to their relevance. If they are marked as relevant, I encourage you to explore them:

Textbook sections:

  1. Chapter 2.1: Turbomachines Part 3 Part 1 (HIGHLY RELEVANT)
  2. Chapter 9.3: Why We Use Irrotational Flow Analysis Part 5 (HIGHLY RELEVANT)
  3. Chapter 6.4: Bernoulli constant drop in pipes Part 3 (HIGHLY RELEVANT)
  4. Chapter 6.3: Pump types Part 2 (HIGHLY RELEVANT)
  5. Chapter 6.2: Modified Bernoulli equation Part 1 (HIGHLY RELEVANT)

Disclaimer:

See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.

3

you can curve fit the cordier diagram but there is no clean analysis that results in the Cordier diagram.

The reason for the peak efficiency of a turbomachine comes from minimizing losses, notably separation at the leading or trailing edge of the impeller blades. So the blade geometry and spin rate has to match the flow rate and stagnation-pressure rise.

radial-flow pumps tend to have fluid that contacts the impeller longer and is turned more sharply, with larger energy change. That is easier to achieve with a radial-flow geometry and a shrouding.

axial-flow pumps have fluid that contacts the impeller for a shorter time and is turned less, with lower energy change. That works easily with axial-flow geometry and does not always need a shrouding.

the shape of the cordier diagram is a readout of the blade angles that prevent separation at leading and trailing edges. but the catch is that finding an equation to predict that from first-principles doesn’t really drive pump design or pump selection.

4

This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook

PK Bot’s answer:
Below is an analysis that complements your remarks:

  1. The quadratic model I presented is indeed a curve‐fit that captures the gross features of the Cordier diagram. A complete first–principles derivation is elusive because the Cordier shape results from many interacting effects—most notably the blade geometry, the matching of impeller spin rate with the incoming flow, and the subtle onset of leading– or trailing–edge separation. In practice, the phenomena are more complex than can be captured by a single clean equation.

  2. From a physical standpoint, pump efficiency peaks when the impeller operates in a regime where the flow remains attached to the blades without excessive recirculation. For radial–flow pumps, the longer contact time and sharper turning yield larger energy exchange but also require careful control of separation. In contrast, axial–flow pumps—by turning the fluid less and providing a shorter interaction length—tend to operate under lower energy changes. This difference in blade loading is reflected in the design limits that show up as the hump in the Cordier diagram.

  3. Nondimensional analysis, using the specific speed

\omega_p^* = \frac{\omega\,Q^{1/2}\,\rho^{3/4}}{\Delta p_0^{3/4}},

captures the trade–offs. The quadratic model

\eta = \eta_{\max} - a\,\left(\omega_p^* - \omega_{p,\mathrm{opt}}^*\right)^2,

serves as a curve–fit that quantifies the rapid efficiency drop when operating away from the optimum specific speed \omega_{p,\mathrm{opt}}^*. However, the value of a and the precise form do not emerge from a pure first–principles derivation because we must incorporate empirical inputs to account for the onset of separation and internal recirculation.

  1. Thus, although the underlying physics of blade separation and recirculation are well understood from an airfoil design and thin–airfoil analysis standpoint, those insights are used to guide pump design rather than to yield a closed–form formula for the Cordier diagram. In practice, engineers rely on performance charts and hydraulic coverage curves because they incorporate both the theoretical scaling (from nondimensional analysis) and the empirical adjustments needed to account for three–dimensional effects and losses.

Your point—that the diagram is best thought of as a readout of the blade geometry and loss mechanisms rather than something directly derivable from first principles—is absolutely correct. The interaction of the flow with the impeller blades, the onset of separation, and the resulting losses make the complete analysis prohibitively complex.

In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections according to their relevance. If they are marked as relevant, I encourage you to explore them:

Textbook sections:

  1. Chapter 2.1: Turbomachines Part 3 Part 1 (HIGHLY RELEVANT)
  2. Chapter 6.3: Pump types Part 2 (HIGHLY RELEVANT)
  3. Chapter 9.6: Centrifugal Pump Design Part 2 (HIGHLY RELEVANT)
  4. Chapter 2.2: Nondimensional Turbomachine Parameters Part 2 (HIGHLY RELEVANT)
  5. Chapter 9.6: Centrifugal Pump Design Part 3 Part 1 (HIGHLY RELEVANT)

Disclaimer:

See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.