APPLICATION OF ENERGY EQUATION: FLOW-MEASURING INSTRUMENTS A.
VENTURI METERS
A device usually insta lled across the p ipeline for the purp ose of measuring the q uantity of fluid flowin g in the line. Invented by the american engineer, Clemens Hershel, in honor of an early Italian hydraulician, Venturi.
Main parts of a Venturi Meter: (See (See Fig. 1) 1) 1) A converging section or inlet section in the shape of a frustum of a cone 2) A short section of uniform dia meter known as the throat , and 3) A diverging or outlet se ction having a shape similar to that of the inlet section. To minimize frictional losses in the meter, the angle of divergenceof the outlet section is limited to about 6O. Consider section (1) and (2) and neglecting frictional losses:
Where: V 1 and V 2 2 are the, mean velocities at sections (1) and (2), respectively. In horizontal position (Z (Z 1 = Z 2 2 ): ): (see (see fig.1) fig.1) Eq. (5.24)
Where the left side represents the increase in kinetic energy per unit weight from section (1) and (2), (2), while the right side gives the corresponding decrease in the potential (or pressure) energy per unit weight. Using continuity between two sections: Eq. (5.25)
Where: A Where: A1 and A and A2 being the cross sectional area at sections (1) and (2), respectively. Substitution of equation (5.25) into (5.24) gives the velocity of the throat, Eq. (5.26)
Eq. (5.26) gives only the mean theoretical velocity at the throat since initial considerations did not include frictional losses. The discharge computed with this velocity also represents the theoretical flow, or Eq. (5.27)
The actual discharge is Eq. (5.27a)
In case the meters is not positioned horizontally or Z or Z 1 = Z 2 2 , then the difference in elevation heads, Z 1 – Z 2 2, must be included in the parenthesis, together with the pressure-head difference, in equation (5.26), Eq. (5.26a)
Example # 1: A 30 cm by 15 cm Venturi meter is installed in a ve rtical pipe. The flo w is upward through th e meter. A differential man ometer containing a gage liquid (s = 1.50) is attached to inlet base and throat, the deflection being 80 cm. Determine the actual flow if C = 0.97. Example # 2: In a test to determine the discharge coefficient C of a 5 cm by 1.25 cm Venturi meter, the total weight of water passing through the meter in 5 min. was measured as 3420 N. A mercury-water differential gage connected to inlet base and throat showed an average deflection of 38 cm. Determine the meter coefficient.
B.
NOZZLE
Is a converging tube (shape of a frustum of a cone) connected to the end of a pipe or hose and is used in engineering practices for the creation of jets and streams for a variety of purposes as well as for metering.
Considering the mean velocities at sections (1) and (2), and assuming no frictional losses (see figure 2 ), the energy equation between the two sections is Eq. (5.28)
Where P 2 is taken as atmospheric. The mean velocity V1 may be expressed in terms of V 2 with the use of a continuity equation, or Where:
The velocity V2 which will result from equation (5.28) is the theoretical velocity of the jet and this is corrected by a correlation factor called the “coefficient of velocity”, C v , or the actual velocity is Eq. (5.29)
Where:
Equation (5.29) may now be written as Eq. (5.30)
The actual flow may be computed as Eq. (5.31)
REMARK: If
is very small relative to
, the head H =
is approximately. It is the usual practice, however, to
Express V 1 in terms of V 2 , so in the solution of the theoretical velocity V 2 or V t , only the pressure term significant term.
remains as the
Headloss in a Nozzle
If frictional losses are eliminated, the total energy per unit of weight at the base of the nozzle, that is exactly be the same as the total energy per unit weight in the issuing jet, is obtained by considering the actual position at the tip, that is (Initial Head)1 – (Actual Head)2 = Head Loss
will
. Actually, there is a loss of head between these points and this
or
Square equation (5.30), solve for H and substitute in the equation above, to get Eq. (5.32)
Equation (5.32) gives the head loss in the nozzle as a fractional part of the actual velocity head. If this loss of head is included in equation (5.28), given the value of the coefficient C V, the velocity V 2 = V a, and no correction is necessary anymore. Example # 1: A nozzle 10 cm by 4 cm is trained vertically upward. At a point 30 cm below the tip, the pressure i s maintained at 145 KPa. If the available power in the jet is 2.6 KW, find (a) the height through which the jet of water will rise, neglecting air friction, (b) the value of CV, and (c) the diameter of the jet 6 m above the tip of the nozzle.
C. PITOT TUBE A tube with circular cross section bent in the shape of an L (see Figure 3), with both ends open was first used by the French scientist, Henry Pitot, in 1732 for measuring the velocity of liquids in an open flow. If the tube is placed in a moving stream in such a way that the horizontal diverging part points against the direction of motion, the liquid enters the opening at point (2) and fills the tube until the vertical part is filled up to a height of h2 . As long as steady flow condition is maintained in the moving stream, the contained volume inside the tube will remain constant and static equilibrium is established within the tube. So, if point (2) represents a particle of liquid just inside the tip of the tube, the velocity V 2 = 0 and the static pressure at this point, called stagnation pressure, will have a value of p2 = wh2 . As a particle moves from point (1) to point (2), its velocity V 1 decreases until it becomes zero upon reaching the stagnation point , point (2). For the pathline connecting (1) and (2), the energy equation, with frictional losses eliminated, is. Eq. (5.33)
in which
and
. Considering h2 – h1 = h, the above equation is reduced to Eq. (5.33a)
or the velocity at the upstream point (1) is Eq. (5.34)
REMARKS: i. Equation (5.33a) states that the velocity head at the upstream point (1) is completely transformed into a pressure head at point (2). ii. Since frictional losses have not been considered, equation (5.34) gives only the theoretical velocity at point (1). Proper correction must therefore be introduced into equation (5.34) to obtain the actual velocity. iii. For a tube positioned with the tip of the horizontal part pointing downstream (see Figure 4a) or positioned in the moving stream with the horizontal leg normal to the direction of motion (see figure 4b), the formula made by Darcy should be used:
For the pitot tube contains a combination of tube as shown in the figure 5, the formula can be used: or Where: Consider the type of Pitot Tube shown in the figure 6. The pressure difference between the stagnation point at point (2) and the upstream point (1) is measured by means of the differential manometer attached to the tube. Energy equation written between points (1) and (2) gives Eq. (5.36)
Summation of pressures between points (2) and (1) along the manometer results into or
Eq. (5.37)
Where d is the deflection of the gage liquid in the manometer. Since the specific weight w in the equation (5.36) is that of the flowing fluid, or w = 9.81 s’ KN/m 3, substitution of equation (5.37) reduces to
Example # 1: The flowing fluid of the figure is water and the gage liquid is mercury (s = 13.6). If the Pitot tube is centered inside the pipe, what is the center velocity if the deflection of mercury is 5 cm?
