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Bernoulli2002b - CE 321 INTRODUCTION TO FLUID MECHANICS...

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1 CE 321 INTRODUCTION TO FLUID MECHANICS 2002 THE BERNOULLI EQUATION Objective: To investigate the validity of Bernoulli's Equation as applied to the flow of water in a tapering horizontal tube. Specifically, to determine if the total pressure head remains constant along the length of the tube as the equation predicts. And to determine if the variations in static pressure head along the length of the tube can be predicted with Bernoulli’s Equation. Approach: Establish a constant flow rate Q through the tube and measure it. Use a pitot probe and static probe to measure the total pressure head h Tm and static pressure head h Sm at six locations along the length of the tube. The values of h Tm will show whether total pressure head remains constant along the length of the tube as required by the Bernoulli Equation. Using the flow rate and cross sectional area of the tube, calculate the velocity head h Vc at each location. Use Bernoulli’s Equation, h Tm and h Vc to predict the variations in static pressure head h St expected along the tube. Compare the calculated and measured values of static pressure head to determine if the variations in fluid pressure along the length of the tube can be predicted with Bernoulli’s Equation. . Equipment: Hydraulic bench with Bernoulli apparatus, stop watch. Theory: Considering flow at any two positions on the central streamline of the tube (Fig. 1), Bernoulli's equation may be written as Bernoulli’s equation indicates that the sum of the velocity head (V 2 /2g), pressure head (p/ γ ), and elevation (z) are constant along the central streamline. As explained in the Appendix, Eq. 1 can be simplified for this apparatus. The tube is horizontal so z 1 =z 2 , and the pressure heads h 1 = p 1 / γ and h 2 = p 2 / γ can be measured from a common arbitrary datum so that Bernoulli’s Equation simplifies to Note that the sum on either side of the equal sign is the total pressure head h T . If Bernoulli’s Eq. is correct the total pressure head h T has the same value at all locations on the central streamline. This is not obvious since the narrowing tube causes the velocity to vary along the length of the tube. Furthermore if it is true then the pressure head must decrease as much as the velocity head increases in moving from point 1 to point 2. The pitot probe you will use measures the total pressure head h T of the fluid a short distance upstream of the probes tip. Ideally the probe’s tip must be positioned about V g p z V g p z 1 2 1 1 2 2 2 2 2 2 + + = + + γ γ (1) V g h V g h 1 2 1 2 2 2 2 2 + = + (2)
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2 3mm downstream from point where the measurement is desired and oriented so that it opens directly into the flow (Fig.1). The piezometers installed along the side of the tube measure static pressure head h Sm .
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