Flow Through Orifices And Mouthpieces and Flow from an Orifice at the Side of a Tank under a Constant Head

Flow Through Orifices And Mouthpieces

• An orifice is a small aperture through which the fluid passes. The thickness of an orifice in the direction of flow is very small in comparison to its other dimensions.
  
  
• If a tank containing a liquid has a hole made on the side or base through which liquid flows, then such a hole may be termed as an orifice.The rate of flow of the liquid through such an orifice at a given time will depend partly on the shape, size and form of the orifice.
  
• An orifice usually has a sharp edge so that there is minimum contact with the fluid and consequently minimum frictional resistance at the sides of the orifice. If a sharp edge is not provided, the flow depends on the thickness of the orifice and the roughness of its boundary surface too.

Flow from an Orifice at the Side of a Tank under a Constant Head

• Let us consider a tank containing a liquid and with an orifice at its side wall as shown in Fig. 16.5. The orifice has a sharp edge with the bevelled side facing downstream. Let the height of the free surface of liquid above the centre line of the orifice be kept fixed by some adjustable arrangements of inflow to the tank.
  
• The liquid issues from the orifice as a free jet under the influence of gravity only. The streamlines approaching the orifice converges towards it. Since an instantaneous change of direction is not possible, the streamlines continue to converge beyond the orifice until they become parallel at the Sec. c-c (Fig. 16.5).
  
• For an ideal fluid, streamlines will strictly be parallel at an infinite distance, but however fluid friction in practice produce parallel flow at only a short distance from the orifice. The area of the jet at the Sec. c-c is lower than the area of the orifice. The Sec. c-c is known as the vena contracta.

Fig 16.5  Flow from a Sharp edged Orifice

• The contraction of the jet can be attributed to the action of a lateral force on the jet due to a change in the direction of flow velocity when the fluid approaches the orifice. Since the streamlines become parallel at vena contracta, the pressure at this section is assumed to be uniform.
  
• If the pressure difference due to surface tension is neglected, the pressure in the jet at vena contracta becomes equal to that of the ambience surrounding the jet.
  
• Considering the flow to be steady and frictional effects to be negligible, we can write by the application of Bernoulli’s equation between two points 1 and 2 on a particular stream-line with point 2 being at vena contracta (Fig 16.5).

(16.10)

• The horizontal plane through the centre of the orifice has been taken as datum level for determining the potential head.
  
• If the area of the tank is large enough as compared to that of the orifice, the velocity at point 1 becomes negligibly small and pressure p1 equals to the hydrostatic pressure p₁ equals to the hydrostatic pressure at that point as p₁=p_atm +ρg(h-z₁).

• Therefore, Eq. (16.10) becomes

	(16.11)
or,	(16.12)

• If the orifice is small in comparison to h, the velocity of the jet is constant across the vena contracta. The Eq. (16.12) states that the velocity with which a jet of liquid escapes from a small orifice is proportional to the square root of the head above the orifice, and is known as Torricelli’s formula.
  
• The velocity V₂ in Eq. (16.12) represents the ideal velocity since the frictional effects were neglected in the derivation. Therefore, a multiplying factor C_v known ascoefficient of velocity is introduced to determine the actual velocity as

• Since the role of friction is to reduce the velocity, C_v is always less than unity. The rate of discharge through the orifice can then be written as,
  

  (16.13)

where a_c is the cross-sectional area of the jet at vena contracta.

• Defining a coefficient of contraction C_c as the ratio of the area of vena contracta to the area of orifiice, Eq. (16.8) can be written as                               

(16.14)

where, a₀ is the cross-sectional area of the orifice. The product of C_c and C_v is written as C_d and is termed as coefficient of discharge. Therefore,