The inside diameters of the larger portions of the...

The inside diameters of the larger portions of the horizontal pipe depicted in Figure $\mathrm{P} 9.55$ are $2.50 \mathrm{~cm} .$ Water flows to the right at a rate of $1.80 \times 10^{-4} \mathrm{~m}^{3} / \mathrm{s}$. Determine the inside diameter of the constriction.

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Key Concepts

Cross-sectional Area Calculation

Calculating the cross-sectional area of a pipe is key when working with fluid flow in conduits. The area of a circular pipe is determined using the formula A = (?/4)·d², where d is the inside diameter. This calculation is critical in converting between flow rate and velocity, since even a small change in diameter leads to a significant change in area and therefore the flow characteristics.

Volumetric Flow Rate

The volumetric flow rate is a measure of the volume of fluid that passes through a cross-section of the pipe per unit time. It is defined by the product of the cross-sectional area and the velocity of the fluid and is conserved in an incompressible, steady flow system. Understanding this concept helps in relating the known flow rate in one section of a pipe to the unknown parameters in another section, such as a constricted area.

Continuity Equation

The continuity equation is a statement of mass conservation for incompressible fluids, which asserts that the mass flow rate must remain constant along a streamline. In the context of flow through pipes, this means that the product of the cross-sectional area and the fluid velocity at any point remains constant. This principle is essential when analyzing how a change in the pipe’s diameter affects the velocity of the fluid.

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