1. Flow measuring device
mouthpiece

2. Contents
Classification of mouthpiece
Flow through external
cylindrical mouthpiece
Convergent-divergent mouthpiece
Borda’s mouthpiece

3. Classification of 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.

4. Cylindrical mouthpiece
Flow through 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.

5. 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,

6. Similarly, between the free surface of the liquid and section c-c
The expression for head loss can be written as,
where is the contraction coefficient.
Hence, Eq. (1) can be written as,
Substituting Eq. (3) in Eq. (2) and using the definition of contraction coefficient, we get,

7. 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,
The discharge through mouthpiece can be written as,
where, is the coefficient of discharge of the mouthpiece.

8. Convergent-divergent
mouthpiece
In this type of mouthpiece, the mouthpiece is first made convergent up to the vena contracta of the jet and beyond that it is made divergent. Such a mouthpiece, which is first convergent is known as convergent-divergent mouthpiece as shown in figure

9. Convergent-divergent
Discharge through a
Convergent-divergent
the mouthpiece
The discharge through a convergent-divergent mouthpiece is same as convergent mouthpiece. In such a mouthpiece, there will be no loss of head due to sudden expansion. The coefficient of discharge Cd in the case of convergent-divergent mouthpiece is also 1.
The diameter of the mouthpiece, for the purpose of calculating the discharge, is taken at the vena-contracta i.e., at C (or in other words where the convergent and divergent pieces meet). It is also known as throat diameter of the mouthpiece.

10. Example - Discharge through a Convergent - Divergent Mouthpiece
Problem
A convergent-divergent mouthpiece having 80mm throat diameter is discharging water under a constant head of 4.5m. Find the discharge through the mouthpiece.
Workings
Given,
H = 4.5m
d = 80mm = 0.08m
The area of the mouthpiece,
=0.005*9.39
=.04695
=46 litres/sec.

11. Borda’s mouthpiece
A reentrant tube in a hydraulic reservoir, whose contraction coefficient (the ratio of the cross section of the issuing jet of liquid to that of the opening) can be calculated more simply than for other discharge openings.

12. The above plots replicate William Kahan's dramatic example of the difference between the proper and improper use (or nonuse) of signed zero on branch cuts. See his article on The Baleful Effect of Computer Benchmarks for the original plots. That article was incorporated into his 1997 S.I.A.M., John von Neumann lecture.
The contours in the plots are images F(z) of rays z = r eiθ at constant θ, where
F(z) = 1 + g(z) + log g(z)
g(z) = z2 + z sqrt (z2 + 1)
using principal branch definitions of the complex square root and natural logarithm. The red contours have parts that are right on the branch cuts, which causes trouble when signed zero is not treated correctly.

13. Prepared by:
Yesha patel
Siddhant verma
Kanku pelerin
Kapadia krunal

14. Thank you