Fluid Mechanics Flow Rate and Continuity Equation If you have a pipe that is flowing a liquid you will have a flow rate. The flow rate is the volume of fluid that passes any particular point per unit of time. f = Av f= flow rate (m 3 /s) A = cross-sectional area of the pipe (m 2 ) v= flow speed (m/s) If the pipe is carrying a liquid (which is considered incompressible - unlike a gas) then the flow rate must be the same anywhere in the pipe. This is called the Continuity Equation. Continuity Equation The flow rate at any point is the same as any other point (assuming no other sources or sinks in the system) f 1 = f 2 A 1 v 1 = A 2 v 2 Example F8 : A pipe of non-uniform diameter is carrying water. At one point the diameter of the pipe is 2cm. If the flow speed is 6m/s a) What is the flow rate? b) What would be the flow speed if the pipe constricts (gets smaller) to a diameter of 1cm? F1
Example F9: What would happen to the flow speed in a pipe that expands from 4cm to 16cm? Bernoulli's Equation This is the statement of the conservation of energy per unit volume for ideal fluid flow. In order to have ideal fluid flow the following conditions must exist: the fluid is incompressible (works best for liquids but can also apply to gases if pressure change stays small) the fluid's viscosity* is negligible the water is streamline/laminar ** *viscosity is force of cohesion in a fluid. If a liquid has high viscosity and is stickier it has a higher resistance to flow and therefore will usually have poor results when Bernoulli's equation is applied. **streamline/laminar means the individual lines of water don't form little vortices (areas of curling). If you put a dye in streamline waster you would see it in nice streak. The opposite is turbulent flow which has curling whirlpools. This flow is unpredictable and therefore won't follow the equation. Understanding fluid flow is important in piping systems for homes, building and large pipelines as well as internal engine design. Turbulent flow in an engine cylinder head can cause power loss. Bernouilli's Equation states that the total pressure from all sources of pressure is constant at all points assuming all conditions are met (which they should be for the exam). Pressure can be from a source such as atmospheric pressure or a pump (P o ). Pressure due to height above a reference height (ρgh or ρgy) and pressure from fluid flow (1/2ρv 2 ). F2
Bernouilli's Equation P 1 + ρgy 1 + 1/2ρv 1 2 = P 2 + ρgy 2 + 1/2ρv 2 2 Sometimes written as: P 1 + ρgy 1 + 1/2ρv 1 2 = constant P = pressure from atmosphere or a pipe (Pa or N/m 3 )- they are the same ρ = density of fluid (kg/m 3 ) y = height above reference point (m) v = flow speed of fluid (m/s) *Note the similarity to the conservation of energy equation: 1/2kx 2 + mgh + 1/2mv 2 = mgh + 1/2mv 2 + 1/2kx 2 Example F10: A pump is used to pump water!! The pump is connected to a pipe that decreases in diameter at the end to 1/3 of its original size at the pump. (Think water hose) The water exits at a height of 60cm above its original height at the pump. If the flow speed at the pump is 1m/s what is the gauge pressure at the point 1 (the initial pressure at the pump)? Example F.11 A container full of salt water is open to the atmosphere. a) What is the pressure at a depth of 2m. (Solve using Bernoulli's equation) b) Someone punctures a hole in the side of the container at a position 3m below the surface. What is the velocity of the water s it leaves the container? If the container is 5m tall to the water level determine where the stream of water will hit the ground with reference to the side of the tank. F3
The Bernoulli Effect The Bernoulli Effect says basically that a fast moving fluid creates lower pressure than a a slow moving fluid. It is sometimes called the Venture effect also. This is based on Bernoulli's Equation. In the diagram below you can see that at the two points chosen that the height is the same so the pressure due to change in height between the two points is zero. Applying Bernoulli's equation you would get P 1 + ½ ρv 1 2 = P 2 + ½ ρv 2 2 A 1 v 1 = A 2 v 2 we can also apply the continuity equation Since the area is smaller the velocity must be greater to maintain equality. Therefore when applied to the first equation you would find that P2 would decrease. The Pressure is lower when the speed is greater. Look at the diagram above. You can see as the fluid is forced through the middle section it must speed up. At the top of each section is a tube that is open to the atmosphere. In the middle section the pressure is lower as evidenced by the height difference. The Bernoulli effect accounts for many of the phenomena that you experience everyday. - in the summer when the breeze blows by a window the curtain may blow out the window - allow airplanes to fly - it makes a motorcycle driver's jacket puff out when they are travelling through the air. The air inside the jacket is higher pressure than the air outside - a carburetor on a car or motorcycle creates a low pressure area that draws fuel into the intake - in some cases when winds from a tornado or hurricane pass over a house the outside pressure can be raised to such a level that a house with closed windows experiences such a huge pressure difference that the roof can blow off! You can read more in the handout notes. F4
Example F.12 The pump in your house has a gauge pressure of 40psi and is located in the basement. The hose has diameter of 1.5cm at the pump. The velocity out of the pump into the hose is 0.5m/s. You are outside with a garden hose. The nozzle has a diameter of 5mm. If the house is connected 2m above the and you hold it 1m above that height determine the velocity of the water when it leaves the nozzle. b) If you hold the nozzle at an angle of 60o how far away will the water land if the hose connection is 0.5m off the ground. Example F.13 - A garden hose has an inner radius of 1.0cm. The hose has a nozzle with an inside radius of 1.5mm. Water flows through the hose with an average speed of 25.0 cm/s. a) What is the speed of the water as it emerges from the nozzle? b) How long does it take to spray 4.0x10-3 m 3 of water from the hose? Example F.14 A liquid (ρ = 1.65 g/cm 3 ) flows through two horizontal sections of tubing joined end to end. In the first section the cross-sectional area is 10cm 2, the flow velocity is 275m/s and the pressure is 1.2x10 5 Pa. The liquid then enters the second section, where the cross-sectional area is 2.5cm 2. Calculate a) the flow velocity b) the pressure in the section of smaller diameter. F5
Example F. 15 In the diagram below there are two different fluids. If the blue liquid is water determine the density of the other liquid given that h 1 is 20cm and h 2 is 4m. Example F.16 The figure below is of a pipe that is fitted with a Venturi U-tube. A fluid with a density ρ A flows through the pipe at a constant flow rate and negligible viscosity. The pipe constricts from cross-sectional area A 1 to A 2. The Venturi tube contains a different fluid of density ρ L. Fluid in the Venturi is stationary. The pressure at point 1 is P 1 and the velocity is v 1 and the pressure at point 2 is P 2 and the velocity is v 2. a) What is P x, the hydrostatic pressure at Point X. Your answer should be in terms of P 1, ρ A, h 1 and g b) What is the P y, the hydostatic pressure at Point Y. Your answer should be in terms of P 2, ρ A, ρ L, h 2, d and g c) Apply Bernoulli's equation at points 1 and 2 in the pipe to solve for P 1 - P 2. d) Given that P x = P y, set the equations in a) and b) equal and solve for P 1 - P 2. e) Derive and equation for v 2 and the flow rate f, in terms of A 1,A 2, d, ρ A, ρ L, and g. Show that v 2 and f are proportional to d F6
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