in the book it says that energy is dissipated proportional to 1/2 rho U2, how do we know that? why is it?
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:
The statement that energy dissipation is proportional to \frac{1}{2} \rho \overline{U}^2 arises from the analysis of viscous losses in fluid flow, particularly in the context of pipe flow. This is a key concept in understanding how energy is lost due to viscous dissipation as fluid moves through a conduit.
Explanation:
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Viscous Dissipation: In fluid mechanics, viscous dissipation refers to the conversion of mechanical energy into thermal energy due to the action of viscous forces. As fluid flows through a pipe, the interaction between the fluid and the pipe wall generates shear stresses, which dissipate energy as heat.
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Kinetic Energy: The term \frac{1}{2} \rho \overline{U}^2 represents the kinetic energy per unit volume of the fluid, where \rho is the fluid density and \overline{U} is the mean flow velocity. This expression is derived from the kinetic energy formula for a moving mass, adapted for a fluid element.
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Proportionality: The proportionality of energy dissipation to \frac{1}{2} \rho \overline{U}^2 is based on the observation that the rate of energy loss due to viscous effects is related to the kinetic energy of the fluid. In pipe flow, the viscous dissipation factor f_{\mathrm{D}} is used to quantify these losses, and the energy loss per unit length of pipe is given by:
\frac{\partial \overline{B}}{\partial s} = -f_{\mathrm{D}} \frac{\frac{1}{2} \rho \overline{U}^2}{d}Here, \overline{B} is the Bernoulli constant, s is the distance along the pipe, and d is the pipe diameter. The term \frac{1}{2} \rho \overline{U}^2 appears in the expression for energy loss because it represents the dynamic pressure, which is a measure of the fluid’s kinetic energy.
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Engineering Context: In practical engineering applications, this relationship helps in designing piping systems by allowing engineers to estimate the pressure drop and energy loss due to viscous effects. The proportionality to \frac{1}{2} \rho \overline{U}^2 provides a convenient way to relate the energy dissipation to the flow velocity and density, which are typically known or can be measured.
In summary, the proportionality of energy dissipation to \frac{1}{2} \rho \overline{U}^2 is a result of the relationship between viscous forces and the kinetic energy of the fluid, and it is a fundamental concept in the analysis of fluid flow in pipes.
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:
- Chapter 6.4: Bernoulli constant drop in pipesIn practice, the hydraulic engineering design of pumpin… (HIGHLY RELEVANT)
- Appendix D Fundamentals: Why … (HIGHLY RELEVANT)
- Chapter 6.3: Modified Bernoulli equation Bird Watching Haiku the cats watch TV but their favorite ch… (HIGHLY RELEVANT)
- Chapter 6 Fundamentals: Area-averaged speed (HIGHLY RELEVANT)
- Chapter 6.1: Pumps With a few exceptions, for example gravity-fed systems, we transport fluid throug… (HIGHLY RELEVANT)
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I feel like PK Bot added so many words but it isn’t clear to me that it said very much.
energy drop per length along a pipe is proportional to the dynamic pressure at high Re. It is proportional to \mu U/\ell at low Re. dimensional analysis says it has to be proportional to one or a combination of those factors.
We know that it is proportional to dynamic pressure at high Re from experimental measurements.
pressure drop in a pipe, for any turbulent flow, scales approximately parabolically with flowrate. for a laminar flow, it scales linearly with flowrate.