Question Video: Using the Continuity Equation to Model The Flow of a Compressible Fluid Physics • 9th Grade

A gas flows smoothly through a pipe. The pipe’s cross-sectional area contracts from 0.075 m² to 0.025 m². The gas enters the pipe moving at 1.8 m/s and leaves the pipe moving at 2.0 m/s. The density of the gas as it enters the pipe is 1.4 kg/m³. What is the ratio of the density of the gas where it enters the pipe to its density where it exits the pipe?

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Video Transcript

A gas flows smoothly through a pipe. The pipe’s cross-sectional area contracts from 0.075 meters squared to 0.025 meters squared. The gas enters the pipe moving at 1.8 meters per second and leaves the pipe moving at 2.0 meters per second. The density of the gas as it enters the pipe is 1.4 kilograms per meter cubed. What is the ratio of the density of the gas where it enters the pipe to its density where it exits the pipe?

We’ll start by drawing a diagram of this scenario. We have a pipe that contracts from one cross-sectional area that we will call 𝐴 one to a second cross-sectional area that we will call 𝐴 two. Gas flows smoothly through this pipe and enters the pipe with a velocity that we will call 𝑣 one. The gas leaves the pipe at a velocity that we will call 𝑣 two. Before the contraction, the gas has a density that we will call 𝜌 one. After the contraction, the gas has a density that we will call 𝜌 two.

The question asks us to calculate the ratio of the density of the gas where it enters the pipe to its density where it exits the pipe. So the question wants us to calculate 𝜌 one divided by 𝜌 two. In the question, we are told that the pipe’s cross-sectional area contracts from 0.075 meters squared to 0.025 meters squared. This means that 𝐴 one is equal to 0.075 meters squared and 𝐴 two is equal to 0.025 meters squared. The question also tells us that the gas enters the pipe moving at 1.8 meters per second and leaves the pipe moving at 2.0 meters per second. So 𝑣 one is equal to 1.8 meters per second and 𝑣 two is equal to 2.0 meters per second. Finally, the question tells us that the density of the gas as it enters the pipe is 1.4 kilograms per meter cubed. So 𝜌 one is equal to 1.4 kilograms per meter cubed.

Before we go any further, it’s important to notice that all the values given to us are in SI units. This means that we don’t have to convert any of them before we substitute them into any equations or use them for any calculations.

To answer this question, we’re going to use the continuity equation for fluids, which states that a fluid’s density multiplied by the cross-sectional area of the pipe it is flowing in multiplied by the fluid’s velocity is constant. This means that we can write density multiplied by cross-sectional area multiplied by velocity before contraction is equal to density multiplied by cross-sectional area multiplied by velocity after contraction. And we would like to rearrange this to get an expression for 𝜌 one divided by 𝜌 two.

We will start by dividing both sides of the equation by 𝜌 two. And we see that these 𝜌 twos on the right cancel. Next, we will divide both sides of the equation by 𝐴 one multiplied by 𝑣 one. And we see that these 𝐴 ones cancel. And also these 𝑣 ones cancel on the left. And this gives us our expression for 𝜌 one divided by 𝜌 two, which is equal to 𝐴 two multiplied by 𝑣 two divided by 𝐴 one multiplied by 𝑣 one. And we actually see that we don’t need the value of 𝜌 one to calculate 𝜌 one divided by 𝜌 two.

We can now continue and substitute our values for 𝐴 one, 𝑣 one, 𝐴 two, and 𝑣 two into this equation, which gives us 𝜌 one divided by 𝜌 two is equal to 0.025 meters squared multiplied by 2.0 meters per second divided by 0.075 meters squared multiplied by 1.8 meters per second. Evaluating this expression gives us 𝜌 one divided by 𝜌 two is equal to 0.37 to two decimal places.

The ratio of the density of the gas where it enters the pipe to its density where it exits the pipe is 0.37.