Application of Orifices and Its Characteristics | Fluid Mechanics

An orifice may be defined as an opening provided in the side or bottom of a tank, for the purpose of discharging the liquid contained in the tank. It should be noted that the opening will be considered as an orifice only when the liquid surface in the tank is above the upper edge of the opening. Orifices may be classified based on their size, shape, sharpness and discharge conditions.

Based on their size orifices are classified into small and large orifices. In a small orifice, the size of the orifice is so small compared with the head over it, the velocity at the level of the centre of the orifice may be taken as the mean velocity through the orifice. In a large orifice however, this is not correct.

An orifice may be circular, rectangular or square though often, circular orifices are adopted.

An orifice may be sharp edged or bell mouthed depending on the shape of the entrance edge. In the case of a sharp edged orifice the inner edge (i.e., at entrance) is made sharp and is tapered to a slightly larger diameter at the outer edge. The liquid discharged through the orifice will touch only the sharp edge at entrance.

In the case of a bell mouthed orifice, a rounded passage is provided in the orifice and the discharge liquid will be in contact with the entire inner surface of the orifice. Due to decreased friction a bell mouthed orifice has a greater discharging capacity.

The orifices mentioned above may discharge a liquid either from a tank into the atmosphere or from one tank into another. If the liquid surfaces on the two sides of an orifice are above the upper edge of the orifice, then the orifice is called a submerged or drowned orifice. If an orifice discharges a liquid to the atmosphere then the discharge is said to be free.

Fig. 8.1 shows an orifice of area a provided in the side of a tank. Let H be the head of the liquid above the centre of the orifice. The liquid stream discharged by the orifice is called a jet. The liquid particles forming the jet approach the orifice from all directions and after passing through the orifice, the jet contracts and reaches a minimum sectional area a_c, at a certain section C-C called vena contracta.

The distance of vena contracta from the orifice is approximately equal to one-half the depth or diameter of the orifice. The stream lines of flow are converging upto vena contracta and beyond this section the stream lines are parallel.

Coefficient of Contraction (C_c):

The ratio of the area of the jet at vena contracta to the area of the orifice is called coefficient of contraction.

Let C_c = Coefficient of contraction

Hence, C_c = area of jet at vena contracta/area of orifice = a_c/a

The value of C_c depends on a number of factors like the size and shape of the orifice, the head on the orifice, the viscosity of the liquid etc. For sharp-edged orifice the value of C_c is generally taken equal to 0.62.

Coefficient of Velocity (C_v):

The ratio of the actual velocity of the jet at vena contracta to the theoretical velocity of the jet is called coefficient of velocity.

Coefficient of Discharge (C_d):

The ratio of the actual discharge to the theoretical discharge of the orifice is called the coefficient of discharge.

Relation between the Coefficients of an Orifice:

Determination of the Coefficients of an Orifice:

(i) Determination of the Coefficient of Discharge:

Maintaining a constant head of water in the tank over the orifice, the jet of water discharged by the orifice is collected in a measuring tank. The rise in the level of water in the measuring tank is measured in a known interval of time.

The experiment can be repeated a number of times and the values of C_d obtained may be averaged.

(ii) Determination of the Coefficient of Velocity:

First Method (Trajectory Method):

A constant head of water is maintained over the orifice.

Let v = Velocity of the jet at vena contracta.

Consider any point P on the centre-line of the jet. Let the horizontal and vertical coordinates of P be x and y with respect to the centre of the jet at vena contracta as origin.

Let the time taken by a particle of water to move from vena contracta to be P be t.

The experiment may be repeated taking other points on the centre-line of the jet and the values of C_v obtained may be averaged.

Second Method (Momentum Method):

A tank integral with a triangular beam is provided with an orifice. The tank is supported so that the triangular beam rests on knife edge supports. A horizontal lever is fixed to the tank wall opposite to the wall containing the orifice.

When the tank contains water and the orifice is closed, the tank is levelled by placing or removing weights on the lever.

When the orifice is open and the water is discharged in a jet, the liquid will exert a horizontal force P on the wall of the tank. By placing additional weight W on the lever the tank is balanced.

(iii) Determination of the Coefficient of Contraction:

First Method:

In this method the area of the jet at vena contracta is measured by using an instrument called the micrometer contraction gauge (Fig. 8.4). This instrument consists of a ring provided with four radial screw gauges, equally spaced. The ring is held at the vena contracta section so that the jet can pass through its centre.

The screws of the screw gauges are now adjusted so that their sharp points just touch the surface of the jet. Now the instrument is removed and the spacing between the opposite screw points are measured accurately.

In actual practice this method is not found to be satisfactory, due to the following reasons:

(a) The section of the jet is not absolutely circular.

(b) It is practically impossible to adjust all the four screw points in contact with the jet simultaneously.

Second Method:

Coefficient of discharge and coefficient of velocity of the orifice are first determined. Now we can find the coefficient of contraction by dividing the coefficient of discharge by the coefficient of velocity.

Time of Emptying a Tank:

Consider a tank of uniform sectional area A provided with an orifice of area a at the bottom. Let the head of water over the orifice fall from a value H₁ to a value, H₂ in an interval of time T seconds.

Let at any instant the head of water over the orifice be h. Let the water level fall by dh in an interval of dt seconds.

The numerator is twice the volume of water initially present and the denominator is the rate of discharge at the initial moment.

Volume of Fluid Discharged Under a Falling Head:

Consider a tank of uniform sectional area A with an orifice of area a at the bottom. Let the head of water over the orifice fall from a value H₁ to a value H₂ in an interval of time T.

Time of Emptying a Hemispherical Tank:

Fig. 8.14 shows a hemispherical tank of radius R and provided with an orifice of area a at the bottom.

Let the head of water over the orifice fall from a value H₁ to a value H₂ in an interval of time T seconds.

Let at any instant the head of water over the orifice be h.

Let x be the radius of the water surface at this level.

∴ Interval of time required to change the head of water from H₁ to H₂ is obtained by integrating the above quantity from the lower limit of h to upper limit of h.

Discharge from a Tank with Inflow:

Consider a tank of area A provided with an orifice of area a at its bottom. Let the tank be supplied with a uniform inflow of water at the rate Q per second. Let at any instant, the head of water over the orifice be h.

Let the change in level of the water be dh in a small interval of time dt.

Time of Flow from One Vessel to Another:

Fig 8.27. shows two tanks with a communicating orifice in the common wall. Let the area of the two tanks be A₁ and A₂ Let a be the area of the orifice.

Let at any instant, the water level in the tank of area A₁ be h units above the water level of the other tank.

i.e., h = head causing the flow.

Let in small interval of time dt the fall in water level of the first tank be dh₁ and the corresponding rise in water level of the other tank be dh₂.

Submerged Orifice:

Consider an orifice discharging a liquid from one tank into the other. The orifice is provided in a partition wall between the tanks A and B. See Fig. 8.28. The liquid levels in both the tanks are above the orifice. An orifice in such a case is called a submerged orifice.

The space in the tanks above the liquid levels may be at atmospheric pressure or may in any particular case at pressure intensities p_a and p_b. Let the heads of the liquid levels above the centre of the orifice in the tanks A and B be h₁ and h₂.

Large Vertical Orifices:

Consider a large rectangular orifice provided in the side of a tank containing a liquid. In this case, the depth of the orifice is so large that the velocity is different at different levels through the orifice.

Let L be the breadth of the orifice. Let the height of the liquid level be H₁ above the upper edge of the orifice and H₂ above the lower edge of the orifice. Consider an elemental horizontal strip of the orifice of thickness dh at depth h.

Loss of Head due to a Sudden Enlargement:

Fig. 8.30 shows a liquid of specific weight w flowing in a pipe whose sectional area suddenly increases from a₁ to a₂. As the liquid enters, the large section the area of flow of the liquid stream increases from a₁ to a₂ within a small distance.

Hence, an annular space is created for a small distance.

Gradually, this annular space also gets filled with the liquid. The liquid collected in this space has a whirling motion due to the formation of the eddies.

Consider two sections 1-1 and 2-2 before and after the enlargement. Let p₁ and p₂ be the pressure intensities at the sections 1-1 and 2-2. Let v₁ and v₂ be the velocities at these sections. Let p_o be the pressure intensity of the liquid collected in the annular space.

Loss of Head due to a Sudden Contraction:

Fig. 8.31 shows a liquid flowing in a pipe whose sectional area suddenly decreases from A to a.

As the liquid enters the smaller pipe, the area of flow goes on decreasing and reaches a minimum area a_c at a certain section C-C. Beyond this section C-C, a sudden enlargement of the area of flow takes place and the liquid fills the smaller pipe and flows with the area a.

Observations show that practically no loss of head takes place as the liquid approaches the section C-C. Hence the loss of head which takes place in this case is due to the sudden enlargement which takes place beyond the section C-C.

Loss of Head due to Obstruction:

Consider a pipe of area A and let an obstacle of area a be placed in the pipe. As the liquid flows past the obstacle the area of flow reaches a minimum value a_c = C_c (A – a) at a certain section C-C. Beyond the section C-C the area of flow suddenly, increases to the area A of the pipe. This cause a loss of head.

Loss of Head at Entrance to a Pipe:

This is similar to the case of a sudden contraction.

Loss of Head at Entrance to a Reservoir:

This is similar to the case of a sudden enlargement.

This loss of head = h_l = 0.5(v²/2g)

where v = velocity in the pipe.

Loss of Head due to Change in Direction: