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PP205_FluidFlow - Fluids Fluid Flow RHJansen Ideal Fluids...

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Unformatted text preview: Fluids Fluid Flow RHJansen Ideal Fluids Fluids are complicated because of the number of particles. An ideal fluid is a simplified model (like ideal gases learned in chemistry) to help describe fluid behavior. 1. Steady Flow: particles making up a fluid have the same velocity (direction and magnitude) when they pass the same point. 2. Irrotational Flow: No turbulence, whirlpools, etc. 3. Nonviscous Flow: Viscosity is negligible. Viscosity is thickness of a fluid and causes resistance to flow. 4. Incompressible Flow: Density is constant. RHJansen Flow Rate The velocity of a fluid is related to the size of the pipe. A1v1 = 2 2 A1 v1 v2 A2 When a pipe narrows the fluid moves faster. If the radius of the pipe is cut in half what happens to speed? = 2 2 2 1 1 2 2 1 = 4 1 2 2 2 The radius is squared, so the new area is 1/4 th the original size. The velocity has to increase by 4 times to maintain the equality. RHJansen Bernoulli's Equation 1 2 P + + = 2 This equation takes some explaining. However, you can quickly see that if velocity increases, then the pressure of the fluid decreases to maintain the equality. This is the opposite of what you would expect. When a hose shoots fast moving water at you it hurts, and you interpret this as greater pressure not less. The pressure they are talking about is the pressure in the fluid (between the molecules of the fluid). This is more like traffic on a freeway: In a traffic jam, it is very tight and slow, and the pressure between cars is high. However, when you pick up speed the flow seems smoother and there is less pressure. RHJansen This pipe below covers most changes that can occur in a pipe 1 2 It changes height, diameter, P + + = and it has chimney sections. 2 Key points 1 and 2 are located at critical points in the pipe. Bernoulli's Equation has to be true at both points. 2 1 Direction of fluid flow RHJansen Two points in the same pipe must have the same constant 1 2 P + + = 1 1 1 2 1 P2 + + 2 = 2 2 2 Since this is the same pipe and the same fluid it has to have the same constant. These equations are equal to each other. 2 1 Direction of fluid flow RHJansen This is the form of Bernoulli's equation you will be using 1 2 1 P + + = + + 2 1 1 2 2 2 1 2 2 Area A and velocity v are the same as in the flow rate equation Pressure P is different at each point, but the density is constant 2 A1 1 v1 v2 A2 RHJansen Measuring the height of each point 1 2 1 P + + = + + 2 1 1 2 2 2 1 2 2 To measure vertical distances draw an arbitrary line below the tube Distance y is measured from the arbitrary line to the tubes center 2 A1 1 v1 v2 A2 Arbitrary horizontal line RHJansen The height of the fluid columns in the chimneys 1 2 1 P + + = + + 2 1 1 2 2 2 1 2 2 The height h of the fluid columns in the chimneys does not appear in the formula yet, but it will be needed in some problems. Height h is measured from each point to the surface of the water above it. h1 A1 1 v1 h 22 v2 A2 Arbitrary horizontal line RHJansen How does height factor in ? 1 2 1 P + + = + + 2 1 1 2 2 2 1 2 2 1 2 1 2 ( P0 + ) + + = ( + 2 )2 + + 1 1 0 2 2 1 2 2 The pressure at each point is due to the fluid column above that point P = P0 + gh h1 A1 1 v1 h 22 v2 A2 Arbitrary horizontal line RHJansen The only way to simplify the formula 1 2 1 P + + = + + 2 1 1 2 2 2 1 2 2 The arbitrary line can be drawn anywhere If it is drawn thru the lowest point (1) 1 2 1 P + = + + 2 1 2 2 2 1 2 2 h 22 v1 Arbitrary horizontal line v2 h1 A1 1 A2 RHJansen Example 1 Apply Bernoulli's Equation to the scenario shown. a. Draw the arbitrary horizontal line. Label the y's, A's, and v's Draw a reference line down the middle of each section of the tube (in this case it is only one line since the middle of each section is at the same height). Put points 1 and 2 on the line in sections of the tube being investigated. Label the areas and velocities. Add an arbitrary horizontal line below the diagram. Now add the y distances measured from the line. A1 1 v1 2 v2 A2 RHJansen Example 1 Apply Bernoulli's Equation to the scenario shown. b. Write the general version of Bernoulli's Equation 1 2 1 P + + = + + 2 1 1 1 2 2 2 2 2 1 2 1 P + = + 2 1 2 2 1 2 2 A1 1 v1 2 v2 A2 c. Simplify the equation as much as possible RHJansen Example 1 Apply Bernoulli's Equation to the scenario shown. Or you could have drawn the arbitrary line thru the lower of the two points (which is both of them) and you would eliminate 1 2 1 2 P + + = + + 1 1 2 2 2 1 2 2 1 2 1 P + = + 2 1 2 2 1 2 2 A1 1 v1 2 v2 A2 RHJansen Example 1 Apply Bernoulli's Equation to the scenario shown. Once you get this far you can go further by combining Bernoulli's Equation and the Flow Rate Equation 1 2 1 P + = + 2 1 2 2 1 2 2 A1v1 = 2 2 This would probably be two steps: You are asked to solve one first and the answer is used to solve the other equation. A1 1 v1 2 v2 A2 RHJansen ...
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