Determination of Coefficient of Discharge of a Rectangular & Triangular sharp crested weir

Objective:

Objective of the experiment is to determine the (i) Coefficient of discharge of a Venturi meter (ii) Calibration curve of meter between manometric head difference & actual flow rate & (iii) Variation of Cd with Reynolds number. 

Introduction:

Flow meters are used in the industry to measure the volumetric flow rate of fluids. Differential pressure type flow meters (Head flow meters) measure flow rate by introducing a constriction in the flow. The pressure difference caused by the constriction is correlated to the flow rate using Bernoulli’s theorem.If a constriction is placed in a pipe carrying a stream of fluid, there will be an increase in velocity, and hence an increase in kinetic energy, at the point of constriction. From energy balance as given by Bernoulli’s theorem, there must be a corresponding reduction in pressure. Rate of discharge from the constriction can be calculated by knowing this pressure reduction, the area available for flow at the constriction, the density of the fluid and the coefficient of discharge Cd. Coefficient of discharge is the ratio of actual flow to the theoretical flow and makes allowances for stream contraction and frictional effects. Venturi meter, orifice meter, and Pitot tube are widely used head flow meters in the industry. The Pitot-static is often used for measuring the local velocity in pipes or ducts. For measuring flow in enclosed ducts, the Venturi meter and orifice meters are more convenient and more frequently used. The Venturi is widely used particularly for large volume liquid and gas flows since it exhibits little pressure loss. However, for smaller pipes orifice meter is a suitable choice. In order to use any of these devices for measurement it is necessary to empirically calibrate them. That is, pass a known volume through the meter and note the reading in order to provide a standard for measuring other quantities. 

Principle:

Consider in-compressible steady flow of a fluid in a horizontal pipe of diameter D that is constricted to a flow area of diameter d, as shown in Fig.

The mass balance and the Bernoulli equations between a location before the constriction (point 1) and the location where constriction occurs (point 2) can be written as    

  • Continuity Equation:

Q=A1v1=A2v2=V1=V2A2/A1=V2(d/D)2  

  • Bernoulli’s Equation (z1=z2):

P1/ρg+(v21)/2g=P2/ρg+(v22)/2g

Combining the above equations

For the obstruction with no loss   V2=(2(p1p2)/ρ(1β4))

Where β= d/D is the diameter ratio. Once Vis known, the flow rate can be determined from  Q=A2v2=(π2)/4xV2

This simple analysis shows that the flow rate through a pipe can be determined by constricting the flow and measuring the decrease in pressure due to the increase in velocity at the construction site. Noting that the pressure drop between two points along the flow can be measured easily by a differential pressure transducer or manometer, it appears that a simple flow rate measurement device can be built by obstructing the flow. Flow meters based on this principle are called obstruction flow meters and are widely used to measure flow rates of gases and liquids.

The velocity is obtained by assuming no loss, and thus it is the maximum velocity that can occur at the constriction site. In reality, some pressure losses due to frictional effects are inevitable, and thus the velocity will be less. Losses can be accounted for by incorporating a correction factor called the discharge coefficient Cd whose value (which is less than 1) is determined experimentally. Then the flow rate for obstruction flow meters can be expressed as

Q=CdAt(2(P1p2)/ρ(1β4))

where At=is the cross-sectional area of the throat and β= d/D is the ratio of constriction throat diameter to pipe diameter.

The value of Cd depends on both β and the Reynolds number (Re), and charts and curve-fit correlations for Cd are available for various types of obstruction meters.

However, the most precise and most expensive of the three obstruction-type flow meters(orifice, nozzle & venturi) is the Venturi meter shown in Fig (in the name of G. B. Venturi (1746–1822)). Although the operating principle for this device is the same as for the orifice or nozzle meters, the geometry of the Venturi meter is designed to reduce head losses to a minimum. This is accomplished by providing a relatively streamlined contraction (which eliminates separation ahead of the throat) and a very gradual expansion downstream of the throat (which eliminates separation in this decelerating portion of the device). Most of the head loss that occurs in a well-designed Venturi meter is due to friction losses along the walls rather than losses associated with separated flows and the inefficient mixing motion that accompanies such flow. The range of values of the Venturi discharge coefficient, is given in Fig.The throat-to-pipe diameter ratio β, the Reynolds number Re, and the shape of the converging and diverging sections of the meter are among the parameters that affect the value of Cd.

Now, Pressure difference P1-P2 can be measured using a differential manometer connected between the two pressure tapings. If the head difference of the manometric liquid in two columns is  & density of the manometric liquid & working liquid in the pipe are respectively  &  then it can be easily shown from hydrostatic equation of incompressible fluid

p1p2=(ρmρ)gΔh

So, Flow rate

Q=CdAt(2(P1p2)/ρ(1β4))
where Q is actual flow rate.

Q can be measured using volumetric tank & stop watch.

Therefore,

Cd=QxA(t1)x(2(ρm/ρ1)gΔh/((1β4)))(1/2)

Apparatus Used:

  • A venturi meter fitted across a pipeline leading to a collecting tank
  • Stop watch & volumetric tank
  • U-Tube manometer

Procedure:

  • Note the diameter at the inlet of pipe (D1= D) and the diameter of the venturi tube at throat (Dt= d).
  • Note the density of manometric liquid i.e. mercury (ρm) and that of fluid flowing through pipeline i.e. water (ρ).
  • Connect the U-tube manometer to the pressure tapings of venturi meter as shown in figure.
  • Start the flow and adjust the control valve in pipeline to get the required discharge.
  • Note down the reading of two mercury columns of manometer and hence measure.
  • Measure flow rate i.e. actual discharge (Q) through venturi meter by means collecting the water in collecting tank for a specified period of time.
  • Q=(Volumeofwatercollectedintank())/(time(t))=(baseareaofthetank(A)XHeadriseinthetank(ΔH))/(time(t))
  • Change the flow rate by adjusting the control valve and repeat the process or at least five times.
  • Calculate Cd corresponding to each flow rate.
  • Plot a calibration curve between Q(actual discharge) & (manometric liquid difference)
  • Draw the variation of Cd with Reynolds number in a semi-log graph paper.

Diameter of pipe, D1= D = ______ m

Diameter of throat, Dt =d= ______ m

β=d/D=_______________

Base area of collecting tank, A = ______×______ = ________ m2

Area of pipe at entry, A1 = ________ m2.

Area of throat, At = ________ m2.

Density of mercury, ρm =13600 kg / m3.

Density of water, ρ =1000 kg / m3

Safety:

  • At the time of starting the flow through the pipe manometer ports should be closed. Manometer ports are opened when flow through the pipe is almost steady.
  • Readings are to be taken carefully so that chances of errors (like parallax error, human error during stop watch reading etc.) are minimized.

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