Pressure loss equations

Theory and Concepts
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list the important equations for flow in a pipe

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PK Bot’s answer:

  1. ENGINEERING CONTEXT: Design and analysis of pump‐pipe systems require relating pressure drop, flow rate, and viscous losses in a conduit.

  2. PHYSICS: The interplay between kinetic energy (inertia) and viscous forces, which dissipate energy as heat, governs the flow behavior in a pipe.

  3. ASSUMPTIONS: Assume fully developed, steady, one-dimensional, incompressible flow in a circular pipe with uniform properties so that a modified Bernoulli analysis with viscous loss terms appropriately describes the system.

  4. SOLUTION TECHNIQUE:
    • Continuity:

Q=UA

• Modified Bernoulli (averaged, with losses):

\frac{\Delta\overline{B}_{loss}}{L}= f_{\mathrm{D}}\,\frac{\frac{1}{2}\rho\overline{U}^{2}}{d}

• Viscous dissipation factor:

f_{\mathrm{D}}=\frac{d\Delta\overline{B}_{loss}}{\frac{1}{2}\rho\overline{U}^{2}L}

• Reynolds number definition:

Re_d=\frac{\rho\, U\, d}{\eta}

• For laminar flow (Hagen–Poiseuille relation in viscous-dissipation-factor form):

f_{\mathrm{D}}=\frac{64}{Re_d}

• Loss coefficient in terms of viscous dissipation factor:

K=f_{\mathrm{D}}\frac{L}{d}

• Shear stress at the wall:

\tau=\frac{1}{8}\,\rho\,U^{2}\,f_{\mathrm{D}}
  1. REFLECTION: These equations interconnect pressure drop, flow velocity, and wall friction, enabling iterative or empirical methods (e.g., Colebrook–White or Haaland equation) to resolve turbulent flow; the accuracy of these approximations under steady, fully developed conditions supports practical pump and pipe selection in engineering applications.

To formulate this response, I consulted 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 6.4: Bernoulli constant drop in pipes Part 3 (HIGHLY RELEVANT)
  2. Appendix C.5: Viscous Dissipation Factor (HIGHLY RELEVANT)
  3. Chapter 6.4: Bernoulli constant drop in pipes Part 2 (HIGHLY RELEVANT)
  4. Chapter 6.2: Modified Bernoulli equation Part 2 (HIGHLY RELEVANT)
  5. Appendix C.2: Dimensional Analysis to Generalize Experiments (HIGHLY RELEVANT)

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