Turbomachine equations

Theory and Concepts
1

summarize the most important turbomachine equations

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PK Bot’s answer:

  1. ENGINEERING CONTEXT: Turbomachines transfer energy between a rotating element (airfoil) and a fluid, and performance is evaluated by comparing the useful energy change (in pressure and kinetic energy) to the motor power input. My answer below focuses on system-level performance (which is more a focus in Professor Kirby’s textbook) and not equations related to impeller design e.g., the Euler turbomachine equations

  2. PHYSICS: The dominant physical effects are the conversion between kinetic energy and pressure via centrifugal acceleration and the resultant energy transfer in the fluid, with losses appearing as heat.

  3. ASSUMPTIONS: Assume low-Mach number conditions so that changes in density and internal energy are negligible and that subsystem losses (e.g., viscous dissipation) can be lumped into efficiency; these assumptions are typical for high-Reynolds number fluid flows in pumps and turbines.

  4. SOLUTION TECHNIQUE:

    • Write the total pressure definition:

      p_0 = p + \frac{1}{2}\rho U^2 + \rho g z

    • For pumps, define power relationships:

      P_\mathrm{brake}=\omega T,\quad P_\mathrm{water}=Q\Delta p_0,\quad\eta=\frac{Q\Delta p_0}{\omega T}

    • Introduce performance coefficients to nondimensionalize pump parameters:

      C_Q=\frac{Q}{\omega D^3},\quad C_H=\frac{\Delta p_0}{\rho\omega^2 D^2},\quad C_P=\frac{P}{\rho\omega^3D^5}

    • Apply pump scaling relations (for homologous designs):

      \frac{\omega_2}{\omega_1}=\left(\frac{Q_2}{Q_1}\right)^{-1/2}\left(\frac{\Delta p_{02}}{\Delta p_{01}}\right)^{3/4},\quad\frac{D_2}{D_1}=\left(\frac{Q_2}{Q_1}\right)^{1/2}\left(\frac{\Delta p_{02}}{\Delta p_{01}}\right)^{-1/4}

    • Define the specific speed and specific diameter:

\omega_p=\frac{\omega\rho^{3/4}Q^{1/2}}{\Delta p^{3/4}}\,,\quad D_p=\frac{D\Delta p^{1/4} }{\rho^{1/4}Q^{1/2}}
  1. REFLECTION: These equations quantitatively relate motor input, fluid energy change, and machine geometry; they enable system-level design and scaling by ensuring that performance coefficients remain consistent across designs, although actual performance may be affected by losses not captured in the idealized equations. They largely take as given that the detailed geometry of the pump is already known, and are used primarily to describe the pump at its best efficiency point (BEP).

To formulate this response, I reviewed the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2). Below, I have organized the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:

Textbook sections:

  1. Chapter 2.1: Turbomachines Part 2 (HIGHLY RELEVANT)
  2. Chapter 2.1: Turbomachines Part 1 (HIGHLY RELEVANT)
  3. Chapter 2.1: Turbomachines Part 3 Part 1 (HIGHLY RELEVANT)
  4. Chapter 2.2: Nondimensional Turbomachine Parameters Part 2 (HIGHLY RELEVANT)
  5. Chapter 2.5: Combining and Resizing Turbomachines Part 2 (HIGHLY RELEVANT)

Disclaimer:

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