Summary
Highlights
This problem demonstrates how to compute the discharge of water through a pipe given its diameter and the mean velocity. The discharge is calculated using the formula Q = A * V, resulting in 0.022 m³/s.
This example shows how to calculate the mean velocity of air in a pipe when the discharge and pipe diameter are known. The velocity is found by rearranging the formula V = Q/A, yielding 15.591 m/s.
Flow rate, or discharge (Q), is defined as the amount of fluid passing through a section of a stream in a unit of time. It is measured in cubic meters per second (m³/s). Q can also be calculated as the product of the cross-sectional area (A) of the pipe and the average velocity (V) of the fluid (Q = A * V).
The Reynolds number (Re) is a dimensionless quantity used to predict fluid flow patterns. It helps determine whether the flow is laminar or turbulent. The formula for the Reynolds number is Re = (ρVD)/μ or Re = (VD)/ν, where ρ is mass density, V is flow velocity, D is the inside diameter, μ is dynamic viscosity, and ν is kinematic viscosity.
Laminar flow occurs when fluid particles move in smooth, parallel paths without crossing. For circular cross-section pipes, flow is considered laminar if the computed Reynolds number is less than 2100.
Turbulent flow is characterized by irregular and crossing path lines of fluid particles. The flow is turbulent when the Reynolds number is greater than 2100.
For continuous flow in a pipe with varying cross-sections, the principle of conservation of mass dictates that the discharge (Q) remains constant at every section. This means that Q1 = Q2 = Q3, regardless of changes in pipe diameter.
This example calculates the Reynolds number for glycerin flowing in a pipe with a given diameter, velocity, mass density, and dynamic viscosity. The calculated Reynolds number (708.1875) is less than 2100, indicating a laminar flow.
This problem explores a pipeline with successive sections of different diameters. It demonstrates how to calculate the mean velocity in each section, given a continuous flow rate. The results show that as the pipe diameter decreases, the fluid velocity increases, illustrating the inverse relationship between diameter and velocity due to the conservation of mass.