Impact of Factors on Flow Rate in a Pipe

What is the Hazen-Williams hydraulics formula for volume rate of flow through a pipe?

a) Q = (1/C) * (π * D² / 4) * sqrt(2gh / L)
b) Q = (C/π) * (D² / 4) * sqrt(2gh / L)
c) Q = (1/C) * (π * D² / 4) * sqrt(gh / (2L)
d) Q = (C/π) * (D² / 4) * sqrt(gh / (2L)

Answer:

The question requires the application of fluid dynamics principles to assess the impact of various factors on the flow rate in a tube. However, the Hazen-Williams formula provided is not represented in the options, and the question seems to blend concepts from the continuity equation and Bernoulli's equation.

Explanation: The question is concerned with the application of the Hazen-Williams formula which is used to calculate the volume rate of flow (Q) through a pipe under specific conditions. None of the given answer choices directly represents the Hazen-Williams formula; they appear to be variations of the Bernoulli equation or other hydraulic flow equations. In practice, the Hazen-Williams formula is given by the equation:
Q = 0.2785 * C * R^0.63 * S^0.54 where Q is the flow rate (L/s), C is the dimensionless Hazen-Williams roughness coefficient, R is the hydraulic radius (m), and S is the hydraulic gradient (m/m).

If we use the provided information to illustrate the sensitivity of flow rate to various factors in a tube: the pressure difference, viscosity, length, and diameter all influence the flow rate. Using the continuity equation and Bernoulli's equation, we can derive relationships that would explain how the flow rate changes with different situations. The continuity equation relates flow rate (Q), cross-sectional area (A), and flow velocity (v) according to Q = Av. By manipulating this equation and considering changes in pressure, viscosity, or dimensions of the tube, one can predict the effect on the flow rate.