Question Video: Calculating Density Change Using the Continuity Equation for Fluids Physics • 9th Grade

A gas flows smoothly through a pipe that expands from having a cross-sectional area of 0.10 m² to having a cross-sectional area of 0.15 m². The gas enters the pipe moving at 1.4 m/s and leaves it moving at 1.2 m/s. The density of the gas as it enters the pipe is 0.98 kg/m³. How much does the gas decrease in density between entering the pipe and exiting it?

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Video Transcript

A gas flows smoothly through a pipe that expands from having a cross-sectional area of 0.10 meters squared to having a cross-sectional area of 0.15 meters squared. The gas enters the pipe moving at 1.4 meters per second and leaves it moving at 1.2 meters per second. The density of the gas as it enters the pipe is 0.98 kilograms per meter cubed. How much does the gas decrease in density between entering the pipe and exiting it?

Let’s start by drawing a diagram of the pipe. As we can see, the pipe expands, starting with a cross-sectional area that we will call 𝐴 one and finishing with a cross-sectional area that we will call 𝐴 two. Gas enters the pipe moving at a speed that we will call 𝑣 one and leaves the pipe traveling at a speed that we will call 𝑣 two. When the gas enters the pipe, it has a density that we will call 𝜌 one. And when it leaves the pipe, it has a density that we will call 𝜌 two.

The question asks us how much does the gas decrease in density between entering the pipe and exiting it. So we must calculate 𝜌 one minus 𝜌 two. The question tells us the pipe expands from having a cross-sectional area of 0.10 meters squared to having a cross-sectional area of 0.15 meters squared. So 𝐴 one is equal to 0.10 meters squared and 𝐴 two is equal to 0.15 meters squared. We are also told that the gas enters the pipe moving at 1.4 meters per second and leaves it moving at 1.2 meters per second. So 𝑣 one is equal to 1.4 meters per second, and 𝑣 two is equal to 1.2 meters per second. Finally, we are told that the density of the gas as it enters the pipe is 0.98 kilograms per meter cubed. So 𝜌 one is equal to 0.98 kilograms per meter cubed. And we don’t know the value of the density of the gas when it exits the pipe. So 𝜌 two is unknown.

So we must first calculate 𝜌 two. We will use the continuity equation for fluids to calculate 𝜌 two, 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 speed is constant. This means that we can write the gas’s density multiplied by the cross-sectional area of the pipe multiplied by the gas’s speed before the expansion is equal to the gas’s density multiplied by the cross-sectional area of the pipe multiplied by the gas’s speed after the expansion. And we will rearrange this to get an expression for 𝜌 two.

We will start by dividing both sides of the equation by 𝑣 two. And we see that the 𝑣 twos on the right cancel. Next, we will divide both sides of the equation by 𝐴 two. And again, we see that the 𝐴 twos on the right cancel. And this gives us our expression for 𝜌 two. Writing this a bit more neatly, 𝜌 two is equal to 𝐴 one 𝑣 one divided by 𝐴 two 𝑣 two multiplied by 𝜌 one.

Before substituting anything into this equation, we should note that all of our values for 𝐴 one, 𝑣 one, 𝐴 two, 𝑣 two, and 𝜌 one are in SI units. So we do not need to convert any of them before we continue. Continuing, 𝜌 two is equal to 0.10 meters squared multiplied by 1.4 meters per second divided by 0.15 meters squared multiplied by 1.2 meters per second multiplied by 0.98 kilograms per meter cubed. Evaluating this gives 𝜌 two is equal to 0.76 kilograms per meter cubed to two decimal places. And we’ll keep a note of this over here on the left.

Our final step is to calculate the difference between the density of the gas when it enters the pipe and when it exits the pipe, which is equal to 𝜌 one minus 𝜌 two. Using our known value of 𝜌 one and our calculated value of 𝜌 two, this is equal to 0.98 kilograms per meter cubed minus 0.76 kilograms per meter cubed, which is equal to 0.22 kilograms per meter cubed to two decimal places. The answer to “How much does the gas decrease in density between entering the pipe and exiting it?” is 0.22 kilograms per meter cubed to two decimal places.

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