The water flowing through a 1.9 cm (inside diameter) pipe flows out through three 1.3 cm pipes

step 1 We're told that we have a pipe with a diameter of 1 .9 centimeters and that we have water flowin

The water flowing through a 1.9 cm (inside diameter) pipe...

The water flowing through a $1.9 \mathrm{~cm}$ (inside diameter) pipe flows out through three $1.3 \mathrm{~cm}$ pipes. (a) If the flow rates in the three smaller pipes are 26,19 , and $11 \mathrm{~L} / \mathrm{min}$, what is the flow rate in the $1.9 \mathrm{~cm}$ pipe? (b) What is the ratio of the speed in the $1.9 \mathrm{~cm}$ pipe to that in the pipe carrying $26 \mathrm{~L} / \mathrm{min}$ ?

Key Concepts

Volumetric Flow Rate and Flow Speed Relationship

The volumetric flow rate in a pipe is defined as the product of the cross?sectional area and the fluid speed (Q = A × v). This relationship is central to problems where the flow splits or combines, as it helps determine how changes in pipe diameter affect the velocity of the fluid, ensuring that the flow rates balance in accordance with the continuity equation.

Cross?Sectional Area of a Pipe

Calculating the area of the circular cross section of a pipe is crucial in determining the flow speed and volumetric rate. The area is computed using the formula A = ?r², where r is the radius of the pipe. This allows one to compare flow capacities between pipes of different diameters.

Continuity Equation

The continuity equation relates the volumetric flow rate (which is the product of the pipe's cross-sectional area and the flow speed) in different parts of a system. When a fluid flows from a larger pipe into smaller ones, the total flow rate must remain constant, allowing us to set up equations that equate the flow in terms of areas and speeds.

Conservation of Mass

This principle states that mass can neither be created nor destroyed in a closed system. In the context of fluid dynamics, it means that the volume (and hence mass) of fluid entering a system must equal the volume leaving the system, assuming no accumulation. This is the fundamental idea behind the continuity equation used in analyzing fluid flow through pipes.

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