In the book I am using, there are flows through a pipe in Chapter 3 and I am supposed to use the Bernoulli equation. But then in Chapter 5 there are flows through a pipe and I am supposed to calculate head loss. How do I tell which is which?
This text is generated by AI trained by Professor Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
When you have two models, one which is simple (Bernoulli) and one which is more complicated (lossy analysis of pipe flow), the only way to quantitatively justify using the simple one is to perform the analysis with the complicated model and then show that the simple one would have been close enough.
So let’s discuss that for the pressure moving along a straight, horizontal pipe. If we model it as fully developed, the total-pressure drop along a pipe can be written as
\frac{\partial p_0}{\partial s}=\frac12 \rho \bar{U}^2 f_D \frac1d\,,
and for a pipe of length L the total-pressure drop is
\Delta p_0 =\frac12 \rho \bar{U}^2 f_D \frac{L}{d}\,.
In contrast, the Bernoulli equation says that \Delta p_0=0 for steady, incompressible inviscid flow along a streamline.
So the question is – when is \Delta p_0=0 a good approximation. And of course it depends on what we are trying to approximate. So there is no general answer. But a common and practical application is to ask – what change in the hydrostatic constant k=p+\rho g h is required to accelerate fluid to a mean speed of \bar{U}, and I can answer that question. The Bernoulli equation gives
\bar{U} = \sqrt{\frac{2\Delta k}{\rho}}=\sqrt{\frac{2p}{\rho} +2 g h}
whereas an analysis that considers viscous dissipation gives
\bar{U} = \sqrt{\frac{2\Delta k}{\rho}\frac{1}{1+f_D\frac{L}{d}}}=\sqrt{\left(\frac{2p}{\rho} +2 g h\right)\frac{1}{1+f_D\frac{L}{d}}}
So, the Bernoulli equation overpredicts the mean flow rate by \sqrt{{1+f_D\frac{L}{d}}}.
From this we can tell when that overprediction is bad. f_D varies (and can be quantified by the Haaland or Colebrook-White equation) but is often about 1/30 for fast flows of water and air through pipes like mechanical engineers typically study. So if L/30d is pretty small, the flowrate will be overpredicted by a factor of roughly 1+L/60d. As L gets big, the flowrate is overpredicted by \sqrt{1+L/30d}. So if you want the answer to within 10%, L can’t be more than 6d. Most pipes have a length much longer than 6 times the diameter, so the Bernoulli equation is usually not a good approximation for flow through a pipe. It can be fine for flow through a short nozzle sometimes.
When the pipe gets short, the fully developed approximation isn’t quite right, and you get a little more pressure loss, and that is another factor that makes it hard for Bernoulli to ever be really good in a pipe.
In crafting this response, I dedicated some time to reviewing the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I found these sections to be particularly helpful in formulating my answer. I encourage you to explore the pertinent ones directly:
Textbook sections:
- Chapter 6.2: Modified Bernoulli equation (HIGHLY RELEVANT)
- Chapter 6.5: Losses in Pipes (RELEVANT)
- Chapter 11.3: Poiseuille Flow (RELEVANT)
- Chapter 6.6: Models (RELEVANT)
- Chapter 6 Fundamentals: Combining the language of fluid mechanics with the language of hydraulics en… (RELEVANT)
- Chapter 6 Engineering Models: Fully Developed Flow (RELEVANT)
- Chapter 11 Engineering Models: Fully Developed Flow (RELEVANT)
- Chapter 6.4: Bernoulli constant drop in pipes (RELEVANT)
- Chapter 6.1: Pumps (RELEVANT)
- Appendix D (RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.
What is fD?
This text is generated by AI trained by Professor Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
Below is an explanation in the context of fluid mechanics and pipe flow.
When we write the differential relation for the drop in the cross‐sectionally averaged Bernoulli constant in a pipe, we have
\frac{\partial \overline{B}}{\partial s} = - f_D \frac{\frac{1}{2}\rho \overline{U}^2}{d}\,.
Here, f_D is defined as the viscous dissipation factor (often referred to as the friction factor or Darcy factor in many texts). It quantifies the rate at which the energy (or the Bernoulli constant) is being dissipated as heat due to viscous effects in the pipe. Equivalently, for a pipe of length L, one can write the total drop in \overline{B} as
\Delta \overline{B} = f_D \frac{L}{d}\frac{1}{2}\rho \overline{U}^2\,.
In words, f_D tells us how quickly the mechanical energy in the fluid (proportional to \frac{1}{2}\rho \overline{U}^2) is lost over a distance (scaled by the pipe’s diameter d). It is a function of the flow’s Reynolds number \mathit{Re} = \rho \overline{U} d/\eta and the nondimensional roughness \epsilon/d, hence capturing both inertial and viscous effects as well as the influence of the pipe’s internal surface.
An additional note: in our coursework we also mention that the Darcy (viscous dissipation) factor relates to the Fanning friction factor by
f_D = 4 f_F\,.
This factor is fundamental when we account for viscous effects in pipe flows, contrasting with the idealized inviscid model where the Bernoulli equation predicts no loss along a streamline.
In crafting this response, I dedicated some time to reviewing the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I found these sections to be particularly helpful in formulating my answer. I encourage you to explore the pertinent ones directly:
Textbook sections:
- Chapter 6.2: Modified Bernoulli equation (HIGHLY RELEVANT)
- Chapter 6.4: Bernoulli constant drop in pipes (HIGHLY RELEVANT)
- Appendix C.2: Dimensional Analysis to Generalize Experiments (HIGHLY RELEVANT)
- Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 1 (HIGHLY RELEVANT)
- Chapter 11.3: Poiseuille Flow (HIGHLY RELEVANT)
- Chapter 6 Engineering Models: Fully Developed Flow (HIGHLY RELEVANT)
- Chapter 11 Engineering Models: Fully Developed Flow (HIGHLY RELEVANT)
- Chapter 6.5: Losses in Pipes (HIGHLY RELEVANT)
- Chapter 6 Fundamentals: Area-averaged speed (HIGHLY RELEVANT)
- Chapter 5 Engineering Models: Steady Euler Equations (HIGHLY RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.