CIV1021 Bernoulli's Theorem Date: Laboratory 2, Experiment 1 Instructions: Follow the laboratory booklet. Work in your group to undertake the experiment and share the results. Do not hand in the completed booklet. Use the booklet to write up a full laboratory report. Follow the guidance on writing a full laboratory report. Student Name: Student Number: CIV1021 - Hydraulics 1 - Hydrodynamics Experiment #1: Verification of Bernoulli's Theorem Learning Objectives: ● ● ● To experimentally determine the flow rate through a pipe system To experimentally determine the coefficient of discharge at the outlet and to estimate the time to empty the reservoir To experimentally determine the exit velocity & discharge based on the trajectory of the exit jet Description Reservoir Tank Piezometers Venturi Flow Tube School of Natural and Built Environment, QUB Discharge Discharge Basin Figure 1. Experimental setup of the Bernoulli flow experiment. In this experiment, we will characterize unsteady, gravitationally driven flow through a tube. Water flows from a reservoir tank through the Venturi tube, where the fluid pressure heads are measured, and discharges into a basin below (Figure 1). This allows us to consider the validity of Bernoulli's theorem using the pressure heads and velocity through the system, as calculated from the discharge distance. Page 1 CIV1021 - Hydraulics 1 - Hydrodynamics Experiment #1: Verification of Bernoulli's Theorem Theory Bernoulli's Theorem In steady, incompressible, inviscid flow, the sum of pressure head, velocity head and datum head is constant along a stream line; 2 P V² pg H This is known as Bernoulli's theorem, and is derived from the conservation of energy in the system. Deviations from Bernoulli's theorem may be characterized by head losses in the flow, i.e. hl. A + + Z = Const. 2g piezometers -11 B reservoir entry point H= A P₁ V² + +Z₁ pg 2g A -2 B pg 2g PB = Figure 2. Schematic illustration of experiment. Flow reference points A, B, C, and D correspond with the water surface in the basin, the first piezometer, the second piezometer, and the exit, respectively. Pr. V PB + Applying Bernoulli's theorem with head losses between the reservoir and the inlet of the tube (points A and B, respectively) we obtain; B 2g pg 2g + +h₂ +h/³. v² 2g +ZB+h₂ +h/³, L School of Natural and Built Environment, QUB (1) D 2g exit (2a) assuming the centreline through the pipe to be the datum. The equation simplifies because V₁ = 0, PA = 0, and ZĄ – ZB = H; NA DO V (2b) Page 2 CIV1021 - Hydraulics 1 - Hydrodynamics Experiment #1: Verification of Bernoulli's Theorem Similarly, applying Bernoulli's equation between the reservoir and the far side of the Venturi tube (points A to C), we obtain; or or where P₁_V²₁₂ +Z₁ = A A + A pg 2g ● or H= + V² pg 2g P_V² A + H Pc Finally, applying Bernoulli's equation between the reservoir and the discharge valve (A and D), we obtain; pg 2g - A V² hB 2g hB - +Z₁ h-entry head loss fB hB - friction head loss from the entry point to point B v² + pg 2g ·+h² +h/c. +h² +h/³₂ hic - friction head loss from the entry point to point C fC hD - friction head loss from the entry point to point D Pc PB V + +h² +h/B pg 2g V²2² B 28 - hc PD + V₁ + Z₂ +h₂ +h₂, = pg 2g Flow Rate through Venturi Tube We can combine equations (2b) and (3b) to determine the flow rate across the Venturi tube using the pressure difference data, = fc If we ignore the friction head loss between B and C, hic – hĹ³, we obtain; fB P V₂ P V² B + + pg 2g pg 2g Substituting for the definition of pressure head, V² C 2g = = +Zc+h² + hic, =hc + = V² V2² C 2g 2g P. Vč + pg 2g +h₂ +h{c. (3a) (3b) School of Natural and Built Environment, QUB (4a) (4b) (5) (6a) (6b) (6c) Page 3 CIV1021 - Hydraulics 1 - Hydrodynamics Experiment #1: Verification of Bernoulli's Theorem By continuity, the volumetric flow rate, or discharge, is conserved, so Q₁ = c ₂ and equivalently or a BVB = acV₁₂ B = ac hB-hc C' = where Q and Q are the flow rates and a and a are the pipe cross sectional areas at points B and C, respectively. Combining with equation (6c) leads to; ac a B V² 2g fB + h√³ ) = H - h₂ + From equation (9), we can specifically obtain the flow velocities, Vc or (√²/2g), and by the value of substituting V into equation (8) we can obtain V or /2g). Hence, we can determine the flow rate through the system at a given head, Q = aBVB = acVc • B B Given the velocity at B, the head difference between A and B can also be used to determine the entry loss; 2 V² B 2g == (7) (8) AR (9) Coefficient of Discharge and Drainage Time From equation (4b) we can also obtain an expression for the theoretical flow velocity (ignoring all the head losses) at the discharge valve, point D; VDL = √2gH. The actual discharge through the pipe at point D is given by; Q₁ = Cap√√2gH, a dH dt School of Natural and Built Environment, QUB (12) where C is the coefficient of discharge and an is the cross-sectional area of the pipe at point D. Substituting for the volumetric flow rate in the reservoir, we get the differential equation; Q₁ = Cap√√2gH (10) (11) (13) Page 4