NPTEL Course Developer for Fluid Mechanics Dr. Niranjan Sahoo
Module 04; Lecture 30 IIT-Guwahati
DYMAMICS OF FLUID FLOW
FLOW THROUGH MOUTHPIECE
A mouthpiece is a short tube of length not more than two to three times its diameter, which is fitted to a tank for measuring discharge of the flow from the tank. By fitting the mouthpiece, the discharge through an orifice of the tank can be increased. Mouthpieces are classified on the basis of their shape, position and discharge conditions.
· According to the shape, they may be classified as, cylindrical, convergent, divergent and convergent-divergent.
· Based on the positions, they may be external or internal mouthpieces with respect to reservoir/tank to which it is connected. An external mouthpiece projects outside the tank whereas the internal mouthpiece projects inside the tank.
· On the basis of discharge conditions, they may be classified as running full and running free mouthpieces.
Flow through an External Cylindrical Mouthpiece
Consider a cylindrical mouthpiece of cross-sectional area , which is attached externally to the tank as shown in Fig. 1. The tank is filled with a liquid of specific weight up to a constant height above the center of the mouthpiece. The discharge from the tank through the mouthpiece, compared to that of orifice, can be increased by
§ running the mouthpiece full so that the jets of liquid emerging from the mouthpiece will be of same diameter as that of mouthpiece.
§ maintaining a sufficient pressure-head in the tank so as to achieve the mouthpiece running full.
A mouthpiece will be running full, if its length is equal to about two to three times its diameter and the pressure head in the tank is maintained at some critical level. Any deviation in full running condition of mouthpiece will result in the formation of vena-contracta as in case of orifice. Referring to the Fig. 1, if and represents the absolute pressure head at section b-b (atmospheric) and at vena-contracta, is the head loss through the mouthpiece, is the velocity of the jet at vena-contracta, then applying Bernoulli’s equation between the free surface of the liquid in the tank and section b-b, we get,
Fig. 1: Flow through an external cylindrical mouthpiece.
(1)
Similarly, between the free surface of the liquid and section c-c
(2)
The expression for head loss can be written as,
where is the contraction coefficient.
Hence, Eq. (1) can be written as,
(3)
Substituting Eq. (3) in Eq. (2) and using the definition of contraction coefficient, we get,
(4)
The minimum possible value up to which the pressure at vena-contracta may be reduced is the absolute zero pressure. The limiting value of available pressure head in the tank corresponding to zero pressure head at vena-contracta is given by,
(5)
The discharge through mouthpiece can be written as,
(6)
where, is the coefficient of discharge of the mouthpiece.
Flow through a Convergent -Divergent Mouthpiece
The formation of vena-contracta and subsequent enlargement of the jet causes the loss of energy, which results in reducing the coefficient of discharge of the mouthpiece. The energy loss of the jet can be minimized by designing the shape of the mouthpiece similar to that of the flow pattern of the jet at the entry, vena-contracta and exit of the mouthpiece. So, the mouthpiece is made to conform the shape of the jet up to vena-contracta, then making it gradually diverging. Such a mouthpiece is known as convergent-divergent mouthpiece (Fig. 2). In such a mouthpiece, the velocity of the jet increases in the convergent portion at the expense of pressure up to vena-contracta where the jet experiences minimum area. Beyond the point of vena-contracta, the pressure increases, as a result velocity of the jet drops down. However, the pressure at vena-contracta cannot be reduced below absolute zero that limits the maximum divergence to be provided for the mouthpiece.
Referring to Fig. 2 and applying Bernoulli’s equation between free liquid surface in the reservoir, the vena-contracta (section-CC) and exit of the mouthpiece (section-BB),
(7)
where is the atmospheric pressure head in terms of flowing liquid, is the free liquid surface above the center of the mouthpiece, is the absolute pressure head at vena-contracta, and are the velocities of flow at vena-contracta and exit of the mouthpiece respectively.
Fig. 2: Convergent-divergent mouthpiece.
From the Eq. (7),
If and are the cross-sectional areas at vena-contracta and the outlet end of the mouthpiece, then by continuity equation,
(8)
Flow through an Internal Mouthpiece
An internal mouthpiece fitted in to a tank or reservoir projects in to the tank and is generally of cylindrical shape only. It is also called re-entrant or Borda’s mouthpiece and can be operated in running free or running full.
§ When the internal mouthpiece runs full, the flow pattern is same as in the case of external cylindrical mouthpiece because the mouthpiece is sufficiently long enough to expand the jet of liquid completely (Fig. 3-a).
(a) (b)
Fig. 3: Internal mouthpiece; (a) running full; (b) running free.
In the case, if the internal mouthpiece runs free, the length of the mouthpiece is small enough to ensure full expansion of the jet. As a result, a vena-contracta is formed as shown in Fig. 3-b.
(a) Free running Borda’s mouthpiece
Let us consider an internal mouthpiece as shown in Fig. 3-b. It has a cross-sectional area , discharging the liquid under a constant head above the mouthpiece. Since the mouthpiece is running free, a vena-contracta will be formed. Let the cross-sectional area and velocity of the emerging jet at vena-contracta be and respectively. Now,
Static thrust on the fluid for the area
Rate of change of momentum of the jet by this static thrust
Equating the above two, by applying Newton’s law of motion, we get,
(9)
Since, the jet is not expanding, so the loss of the energy can be neglected. Then, applying the energy equation between the free surface of the liquid and outside the mouthpiece,
(10)
By combining Eqs. (9) and (10),
i.e. the coefficient of contraction for a Borda’s mouthpiece is 0.5.
(b) Borda’s mouthpiece running full
As discussed earlier, the flow pattern in this mouthpiece will be same as that of an external mouthpiece (Fig. 3-a). In this case,
Applying Bernoulli’s equation,
(11)
where is the head loss due to sudden enlargement. Then, solving for velocity from the Eq.(11), we get
(12)
Coefficient of velocity becomes,
FLOW OVER NOTCHES AND WEIRS
A notch may be defined as an obstruction over which the flow of liquid occurs. As the depth of flow above the base of the notch is related to the discharge, the notch forms a useful measuring device. In case of measuring tank or reservoir, the opening is provided at the side of the tank such that the liquid surface in the tank is below the top edge of the opening. In fact, this is a large orifice, which has no upper edge, so that it has a variable area depending upon the level of the free surface.
A weir is a notch on a large scale used for measuring the flow of a river. It is a concrete or masonry structure of substantial breadth built across the river in the direction of flow. This allows the excess water to flow over its entire length to the downstream side. Thus a weir is similar to a small dam constructed across the river, with a difference that the excess water flows downstream only through a small portion called spillway and incase of weir, the excess water flows over its entire length.
The sheet of water flowing through a notch or over a weir is known as nappe or vein. The bottom edge of the notch or the top of a weir over which water flows is known as sill or crest. The height above the bottom of the tank or channel is known as crest height.
Notches and weirs can be classified based on the followings;
(a) According to the shape of the openings, notches/weirs may be classified
§ Rectangular type
§ Triangular or V-type
§ Trapezoidal type
§ Stepped type
(b) Based on the shape of the crest, weirs are classified as,
§ Sharp-crested weir
§ Broad-crested weir
§ Submerged weir
§ Ogee-shaped weir
Elementary Theory of Notches and Weirs
There exists a considerable similarity between the patterns of flow over a notch/weir of same shape. As such, the same expressions will hold good for notch/weir of same shape. Moreover, due to this similarity, a notch is often termed as sharp-crested weir.
The method of determining the theoretical flow through a notch is the same as that adopted for the large orifice. For a notch of any shape (Fig. 4) with height , consider a horizontal strip of width at a depth below the free surface.
Fig. 4: Discharge through a notch/weir of any shape.
Area of the elementary strip
Velocity through the strip
Discharge through strip,
Integrating from at the free surface to at the bottom of the notch,
Total theoretical discharge,
(13)
Before the integration of the Eq. (13) to be carried out, must be expressed in terms of . As in the case of orifices, the actual discharge through a notch/weir can be found by multiplying the theoretical discharge by the discharge coefficient . Then Eq. (13) can be written for actual discharge as,
(14)
Special cases
§ For a rectangular notch/weir (Fig. 5-a), (i.e. the width of the crest of the rectangular notch/weir), Eq. (14) can be written as,
(15)
(a) (b)
(c)
(d)
(e)
Fig. 5: (a) Rectangular type; (b) Triangular/V- type; (c) Trapezoidal type;
(d) Cipolletti type; (e) Stepped type.
§ In case of V-notch with included angle (Fig. 5-b), . So, Eq. (14) gives,
or,
(16)
§ Referring to the Fig. 5-c, a trapezoidal weir/notch is a combination of a rectangular and a V-type weir/notch. The discharge over such notch/weir can be determined by adding the individual discharges over the two different types i.e.,
(17)
where are the discharge coefficients of rectangular and triangular weir or notch respectively.
§ A particular type of trapezoidal weir with included angle of 140 is called “Cipolletti weir”, invented by an Italian engineer Cipolletti in 1887. For this weir, the decrease in the discharge over rectangular weir due to end contraction is compensated by the increase in discharge through the two triangular portions. Hence, the total discharge over a Cipolletti weir may be computed using the Eq. (15) as used for rectangular weir (Fig. 5-d).
§ A stepped notch is a combination of rectangular notches shown in Fig. 5-e. The discharge through a stepped notch is equal to sum of the discharges through different rectangular notches. Referring to Fig. 5-e, the total discharge for a stepped notch can be calculated as,
(18)
Example-1
Determine the discharge through a 100mm diameter internal mouthpiece under a head of 2m when it flows freely. What will be the discharge and pressure at vena-contracta if it runs full? Take and atmospheric pressure as 10.33 m of water
Solution
Internal mouthpieces running free and full are shown in Fig. 3-a and 3-b respectively.
(a) For free flow conditions, a vena-contracta is formed as shown.
Applying Bernoulli’s equation, between section “1” and at the vena-contracta (section c-c)
So,
Then,
(b) For free flow conditions, a vena-contracta is formed but the pressure difference is sufficient to expand the jet at exit.
Applying Bernoulli’s equation, between section “1” and “2”
Further, continuity equation yields,
Hence,
(c) Applying Bernoulli’s equation, between section “1” and “section c-c”
where is the atmospheric pressure head.
or,
EXERCISES
1. A tank carrying water is fitted with a cylindrical mouthpiece fitted externally. When water flows in the mouthpiece, a vena-contracta is formed at some section of the mouthpiece. For a given height of water in the tank and diameter of the vena-contracta, determine the diameter of the mouthpiece for which the flow rate is maximum.
2. A convergent-divergent mouthpiece is fitted to the side of a tank. At constant head of 1.7m, the discharge is 200lits/min. The head loss in the divergent portion is 0.1 times the kinetic head at outlet. Find the throat and exit diameters if the separation pressure is 2.5m and the atmospheric pressure head is 10.33m of water.
3. An external cylindrical mouthpiece of diameter 12cm is discharging water under a head of 6m. Find
(i) Discharge through the mouthpiece
(ii) Absolute pressure head of water at vena-contracta
Take ,and atmospheric head is 10.33m of water.
4. An external mouthpiece converges from inlet up to the vena-contracta to the shape of the jet and than diverges gradually. The diameter of the vena-contracta is 2cm and the head of the water over the center of the mouthpiece is 1.5m. The head loss in the contraction is 1% and that in divergent portion is 5% of total energy before the inlet. The pressure in the system falls up to 8m below atmosphere. Find the maximum discharge that can be drawn through outlet and the corresponding diameter at the outlet.
5. A streamlined nozzle of diameter is supplied at constant water head whose magnitude is larger compared to . The nozzle discharges water directly into the atmosphere and is so shaped that the issuing jet is parallel at the nozzle exit. In order to increase the flow rate a shroud of the diameter is firmly secured to the nozzle as shown in the figure.
The jet expands to fill the shroud and the shroud is long enough to ensure that the flow leaving is steady and parallel. Determine,
(i) the diameter of the shroud in order to achieve maximum flow rate.
(ii) percentage increase in discharge.
Neglect shear stresses at the walls of the shroud.
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