Daniel Bernoulli, born February 8th, 1700, is the most acclaimed member of the Swiss family of mathematicians known as the Bernoulli family. Not only was he a mathematician (among many other things), but also a physicist who made monumental discoveries in hydrodynamics. Bernoulli is best known for his published book, the “Hydrodynamica”, in which he discussed the relationship and properties of basic fluid flow we which know today as “Bernoulli’s Principle” (The Doc). Though many are informed on the basis of Bernoulli’s Principle, how many of us can say we are familiar with the math behind this theory? To refresh your memory, Bernoulli’s Principle states that the pressure of a fluid decreases in areas where the flow velocity increases. Imagine
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“P1” stands for pressure energy (pressure in the fluid), “ρ” is the density, “v” is the velocity of the fluid, “g” stands for the acceleration due to gravity (9.8 or 10, whichever you prefer), and “h” equals the height of fluid off the ground (“Bernoulli’s Equation”). Let us do an example problem to help put this into perspective:
“Say that an amount water flows through an ‘s’ shaped pipe. At one end of the pipe, the water has a pressure of 160,000 Pascal (Pa), a velocity of 7.0 m/s, and has a height of 1.2 m. At the other end of the pipe, the velocity of the water increased to 11 m/s, and the height has raised to 4.0 m. The density of the water is 2000 kg/m^3” (Friedl). Now, all you have to do is the plug the information into the equation: ρ = 2000 kg/m^3, g = 9.8 m/s^2, P1 = 160,000 Pa, v1 = 7.0 m/s, h1 = 1 m, v2 = 11 m/s, h2 = 4.0 m, and P2 is…?
Wait! You are missing the pressure at the second point in the pipe. To solve for a missing variable, all you have to do is simply rearrange the …show more content…
Volume flow rate is how much space a fluid occupies while moving through an object at a given time. The equation is written as Q = AV, which states that volume flow rate is equal to the area times the flow velocity (Khan). Mass flow rate is the amount of mass of the fluid that is moving through an object at a given time. Its equation declares that the mass flow rate is equal to density times the area times the velocity, written as Mdot = ρAV. Furthermore, the Equation of Continuity (ρ1A1V1 = ρ2A2V2) could also be written as Mdot1 = Mdot2, saying that the mass flow rate of a fluid at one point is equivalent to the mass flow rate of the fluid at another point. Or, you could also write it as ρ1Q1 = ρ2Q2, saying that the density of the fluid times the volume of the flow rate in one spot is equal to the density of the fluid times the volume of the flow rate at a different spot. You can then simplify this version of the formula because the density of the fluid is frequently the same on both sides of the equation, so you are left with Q1 = Q2, also written as A1V1 = A2V2, which is the simpler version of Equation of Continuity (Drew). Though Bernoulli’s Equation is associated with many different formulas and can be written various ways, every formula has the same principle. Bernoulli’s Principle! No matter the volume, area, or type of watercourse, the amount of fluid in equals the amount of fluid out.
“P1” stands for pressure energy (pressure in the fluid), “ρ” is the density, “v” is the velocity of the fluid, “g” stands for the acceleration due to gravity (9.8 or 10, whichever you prefer), and “h” equals the height of fluid off the ground (“Bernoulli’s Equation”). Let us do an example problem to help put this into perspective:
“Say that an amount water flows through an ‘s’ shaped pipe. At one end of the pipe, the water has a pressure of 160,000 Pascal (Pa), a velocity of 7.0 m/s, and has a height of 1.2 m. At the other end of the pipe, the velocity of the water increased to 11 m/s, and the height has raised to 4.0 m. The density of the water is 2000 kg/m^3” (Friedl). Now, all you have to do is the plug the information into the equation: ρ = 2000 kg/m^3, g = 9.8 m/s^2, P1 = 160,000 Pa, v1 = 7.0 m/s, h1 = 1 m, v2 = 11 m/s, h2 = 4.0 m, and P2 is…?
Wait! You are missing the pressure at the second point in the pipe. To solve for a missing variable, all you have to do is simply rearrange the …show more content…
Volume flow rate is how much space a fluid occupies while moving through an object at a given time. The equation is written as Q = AV, which states that volume flow rate is equal to the area times the flow velocity (Khan). Mass flow rate is the amount of mass of the fluid that is moving through an object at a given time. Its equation declares that the mass flow rate is equal to density times the area times the velocity, written as Mdot = ρAV. Furthermore, the Equation of Continuity (ρ1A1V1 = ρ2A2V2) could also be written as Mdot1 = Mdot2, saying that the mass flow rate of a fluid at one point is equivalent to the mass flow rate of the fluid at another point. Or, you could also write it as ρ1Q1 = ρ2Q2, saying that the density of the fluid times the volume of the flow rate in one spot is equal to the density of the fluid times the volume of the flow rate at a different spot. You can then simplify this version of the formula because the density of the fluid is frequently the same on both sides of the equation, so you are left with Q1 = Q2, also written as A1V1 = A2V2, which is the simpler version of Equation of Continuity (Drew). Though Bernoulli’s Equation is associated with many different formulas and can be written various ways, every formula has the same principle. Bernoulli’s Principle! No matter the volume, area, or type of watercourse, the amount of fluid in equals the amount of fluid out.