Measurement of Flow Rate Through Pipe | Fluid Mechanics for Mechanical Engineering PDF Download

Measurement Of Flow Rate Through Pipe
Flow rate through a pipe is usually measured by providing a coaxial area contraction within the pipe and by recording the pressure drop across the contraction. Therefore the determination of the flow rate from the measurement of pressure drop depends on the straight forward application of Bernoulli’s equation.
Three different flow meters operate on this principle.

• Venturimeter
• Orificemeter 
• Flow nozzle.

    
   


Venturimeter
Construction: A venturimeter is essentially a short pipe (Fig. 15.1) consisting of two conical parts with a short portion of uniform cross-section in between. This short portion has the minimum area and is known as the throat. The two conical portions have the same base diameter, but one is having a shorter length with a larger cone angle while the other is having a larger length with a smaller cone angle.

                   Fig 15.1  A Venturimeter


 Working:

• The venturimeter is always used in a way that the upstream part of the flow takes place through the short conical portion while the downstream part of the flow through the long one. 
   
• This ensures a rapid converging passage and a gradual diverging passage in the direction of flow to avoid the loss of energy due to separation. In course of a flow through the converging part, the velocity increases in the direction of flow according to the principle of continuity, while the pressure decreases according to Bernoulli’s theorem. 
   
• The velocity reaches its maximum value and pressure reaches its minimum value at the throat. Subsequently, a decrease in the velocity and an increase in the pressure takes place in course of flow through the divergent part. This typical variation of fluid velocity and pressure by allowing it to flow through such a constricted convergent-divergent passage was first demonstrated by an Italian scientist Giovanni Battista Venturi in 1797.
  
  Fig 15.2  Measurement of Flow by a Venturimeter
   
• Figure 15.2 shows that a venturimeter is inserted in an inclined pipe line in a vertical plane to measure the flow rate through the pipe. Let us consider a steady, ideal and one dimensional (along the axis of the venturi meter) flow of fluid. Under this situation, the velocity and pressure at any section will be uniform.
• Let the velocity and pressure at the inlet (Sec. 1) are V₁ and p₁ respectively, while those at the throat (Sec. 2) are V₂ and p₂. Now, applying Bernoulli’s equation between Secs 1 and 2, we get

           (15.1)
             (15.2)

where ρ is the density of fluid flowing through the venturimeter.

• From continuity,
       (15.3)
  
  where A₁ and A₂ are the cross-sectional areas of the venturi meter at its throat and inlet respectively.
   
• With the help of Eq. (15.3), Eq. (15.2) can be written as
     (15.4)

           where   are the piezometric pressure heads at sec. 1 and sec. 2 respectively, and are defined as

          

             Hence, the volume flow rate through the pipe is given by
                 (15.6)


Venturimeter... contd from previous slide

• If the pressure difference between Sections 1 and 2 is measured by a manometer as shown in Fig. 15.2, we can write
     (15.7)
  
  where ρ_m is the density of the manometric liquid
• Equation (15.7) shows that a manometer always registers a direct reading of the difference in piezometric pressures. Now, substitution of    from Eq. (15.7) in Eq. (15.6) gives
      (15.8)
   
• If the pipe along with the venturimeter is horizontal, then z₁ = z₂; and hence  becomes h₁ − h₂, where h₁ and h₂ are the static pressure heads 
   
• The manometric equation Eq. (15.7) then becomes
  
• Therefore, it is interesting to note that the final expression of flow rate, given by Eq. (15.8), in terms of manometer deflection ∆h, remains the same irrespective of whether the pipe-line along with the venturimeter connection is horizontal or not.
   
• Measured values of ∆h, the difference in piezometric pressures between Secs I and 2, for a real fluid will always be greater than that assumed in case of an ideal fluid because of frictional losses in addition to the change in momentum. 
   
• Therefore, Eq. (15.8) always overestimates the actual flow rate. In order to take this into account, a multiplying factor C_d, called the coefficient of discharge, is incorporated in the Eq. (15.8) as
  
   
• The coefficient of discharge C_d is always less than unity and is defined as
  
  
  where, the theoretical discharge rate is predicted by the Eq. (15.8) with the measured value of ∆h, and the actual rate of discharge is the discharge rate measured in practice. Value of C_d for a venturimeter usually lies between 0.95 to 0.98
   

Orificemeter
Construction: An orificemeter provides a simpler and cheaper arrangement for the measurement of fow through a pipe. An orificemeter is essentially a thin circular plate with a sharp edged concentric circular hole in it.

Working:

• The orifice plate, being fixed at a section of the pipe, (Fig. 15.3) creates an obstruction to the flow by providing an opening in the form of an orifice to the flow passage.
   
  Fig 15.3   Flow through an Orificemeter
   
• The area A₀ of the orifice is much smaller than the cross-sectional area of the pipe. The flow from an upstream section, where it is uniform, adjusts itself in such a way that it contracts until a section downstream the orifice plate is reached, where the vena contracta is formed, and then expands to fill the passage of the pipe.
   
• One of the pressure tapings is usually provided at a distance of one diameter upstream the orifice plate where the flow is almost uniform (Sec. 1-1) and the other at a distance of half a diameter downstream the orifice plate. 
   
• Considering the fluid to be ideal and the downstream pressure taping to be at the vena contracta (Sec. c-c), we can write, by applying Bernoulli’s theorem between Sec. 1-1 and Sec. c-c,
      (15.10)
  where    are the piezometric pressures at Sec.1-1 and c-c respectively.
   
• From the equation of continuity,
      (15.11)
  
  where A_c is the area of the vena contracta.
   
• With the help of Eq. (15.11), Eq. (15.10) can be written as
    (15.12)

Correction in Velocity

• Recalling the fact that the measured value of the piezometric pressure drop for a real fluid is always more due to friction than that assumed in case of an inviscid flow, a coefficient of velocity C_v (always less than 1) has to be introduced to determine the actual velocity V_c when the pressure drop    in Eq. (15.12) is substituted by its measured value in terms of the manometer deflection 
  '∆h

Hence, 
                (15.13)
where '∆h' is the difference in liquid levels in the manometer and ρ_m is the density of the manometric liquid.
Volumetric flow rate 
       (15.14)

• If a coefficient of contraction C_c is defined as, C_c = A_c /A₀, where A₀ is the area of the orifice, then Eq.(15.14) can be written, with the help of Eq. (15.13),
    (15.15)
   

  The value of C depends upon the ratio of orifice to duct area, and the Reynolds number of flow.

• The main job in measuring the flow rate with the help of an orificemeter, is to find out accurately the value of C at the operating condition. 
   
• The downstream manometer connection should strictly be made to the section where the vena contracta occurs, but this is not feasible as the vena contracta is somewhat variable in position and is difficult to realize. 
   
• In practice, various positions are used for the manometer connections and C is thereby affected. Determination of accurate values of C of an orificemeter at different operating conditions is known as calibration of the orifice meter.

Flow Nozzle
 

• The flow nozzle as shown in Fig.15.4 is essentially a venturi meter with the divergent part omitted. Therefore the basic equations for calculation of flow rate are the same as those for a venturimeter. 
   
• The dissipation of energy downstream of the throat due to flow separation is greater than that for a venturimeter. But this disadvantage is often offset by the lower cost of the nozzle. 
   
• The downstream connection of the manometer may not necessarily be at the throat of the nozzle or at a point sufficiently far from the nozzle.
   
• The deviations are taken care of in the values of C_d, The coefficient C_d depends on the shape of the nozzle, the ratio of pipe to nozzle diameter and the Reynolds number of flow.
  
  
   
• A comparative picture of the typical values of C_d, accuracy, and the cost of three flow meters (venturimeter, orificemeter and flow nozzle) is given below: