What is the mathematical formulation of Prandtl’s one-seventh power law for turbulent pipe flow, where r denotes pipe radius and x is the radial distance ?
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Answer: A. V/Vmax = (x/r)1/7
Explanation
The equation models turbulent velocity by taking the ratio of radial distance to pipe radius. Raising this specific distance ratio to the one-seventh power accurately represents the turbulent profile. Prandtl established this empirical formulation specifically for fully developed turbulent flow inside circular pipes. Prandtl developed early empirical power-law velocity profiles to describe turbulent fluid mechanics behavior. The variable r represents the total inner radius of the circular pipe under consideration. The variable x denotes the radial distance measured outward from the exact centerline axis.
Engineers use this velocity distribution to calculate local friction factors in industrial piping. Understanding turbulent velocity profiles helps optimize the transport efficiency of various liquid process streams. This foundational turbulent flow model underpins many practical pipe network design calculations. Inverting the radial distance ratio fails to properly capture the zero velocity at walls. Multiplying the radial distance by the radius yields dimensionally incorrect mathematical velocity relations. Alternative formulations exist but do not use the specific one-seventh power law structure.
- Subject
- Engineering
- Category
- Chemical Engineering
- Topic
- —
- Difficulty
- medium
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