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Unformatted text preview: Lecture 19: Sound and Hearing 1 REVIEW: Equation of Continuity and Bernoulli’s Equation Equations in Fluid Dynamics For moving incompressible fluids there are two important laws of fluid dynamics: 1) The Equation of Continuity, and 2) Bernoulli’s Equation. These you have to know, and know how to use to solve problems. The Equation of Continuity The continuity equation derives directly from the incompressible nature of the fluid. Suppose you have a pipe filled with a moving fluid. If you want to compute the amount of mass moving by a point in the pipe, all you need to know is the density ρ of the fluid, the cross sectional area A of the pipe, and the velocity v of the fluid. Then the mass flow is given by ρ · A · v because ρ · A · v = ρ · A · Δ x Δ t = ρ Δ V Δ t = Δ m Δ t (the “mass flow”) If the fluid is truly incompressible, then the mass flow is the same at all points in the pipe, and the density is the same at all points in the pipe: ρA 1 v 1 = ρA 2 v 2 = ⇒ A 1 v 1 = A 2 v 2 (the equation of continuity) (14 . 10) Bernoulli’s Equation Bernoulli’s equation is very powerful equation for moving, incompressible fluids, and can be derived using the conservation of Mechanical Energy. The Bernoulli’s Equation states that if you have a fluid moving in a pipe at point 1 with pressure P 1 , speed v 1 , and height y 1 , and the fluid moves to point 2 with pressure P 2 , speed v 2 , and height y 2 , then these six quantities are related as follows P 1 + 1 2 ρv 2 1 + ρgy 1 = P 2 + 1 2 ρv 2 2 + ρgy 2 (Energy Conservation) P + 1 2 ρv 2 + ρgy = constant (Alternate form) (14 . 17) Lecture 19: Sound and Hearing 2 Using Bernoulli’s Equation: Venturi Tube and Torricelli’s Law Worked Example: The Venturi Tube A horizontal pipe with a constriction is called a Venturi Tube and is used to measure flow velocities by measuring the pressure at two different cross sectional areas of the pipe. Given two pressures P 1 and P 2 where the areas are A 2 and A 1 respectively, determine the flow velocity at point 2 in terms of these quantities and the fluid density ρ . First use Bernoulli’s law, and take the heights y 1 = y 2 = 0: P 1 + 1 2 ρv 2 1 = P 2 + 1 2 ρv 2 2 Now substitute for one of the velocities, v 1 , by using the continuity equation: A 1 v 1 = A 2 v 2 = ⇒ v 1 = A 2 A 1 v 2 = ⇒ P 1 + 1 2 ρ ˆ A 2 A 1 v 2 ! 2 = P 2 + 1 2 ρv 2 2 = ⇒ v 2 = A 1 v u u u t 2( P 1- P 2 ) ρ ( A 2 1- A 2 2 ) Example Torricelli’s Law (speed of efflux) A tank with a surface pressure P (at point 2) and a surface area A 2 has a small hole of area A 1 << A 2 at a distance of h below the surface. What is the velocity of the escaping fluid which has density ρ ?...
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- Spring '08