Flow through Mouthpieces

Different types of mouthpieces, their classifications, and the concept related equations and formulas

1. Mouthpieces Submitted by :- Dheeraj Kumar Soni SID-15109008

2. 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. 

3. Classification:- 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. Flow through an External Cylindrical Mouthpiece :- Mouthpiece’s X-sectional area, a Area of jet at VC, a c Specific weight of liquid, ꝩ Height of liquid, h A bsolute pressure head at section b-b and at water surface(atmospheric) , h a Absolute pressure head at c-c, h c Head loss through the mouthpiece, h L Velocity of jet at VC, V c Velocity of jet at b-b, V

5. Flow through an External Cylindrical Mouthpiece :- Applying Bernoulli's equation between the free surface of the liquid in the tank and section b-b   , we get , h a + h = h a + (V 2 /2g) + h L (1) Similarly between the free surface of liquid and section c-c, h a + h = h c + (V c 2 /2g) (2) The expression of head loss can be written as:- h L = (V 2 /2g)[(1/C c ) – 1] 2 where C c { = a c /a } is the contraction coeff.

6. Flow through an External Cylindrical Mouthpiece :- Hence, Eqn.(1) can be written as, h = V 2 /2g[ 1 + {(1/C c ) – 1} 2 ] (3) As we know, the value of- V th = √2gh and also, V = C v . V th = C v √2gh (4) Substituting Eqn. (4) in Eqn. (3), we get C v = [1/{1 + (1/C c – 1) 2 }] 1/2 Also, the coeff. of discharge, C d = C c x C v Hence, discharge through the mouthpiece can be written as Q = C d .a.√2gh

7. 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 . However, the pressure at vena- contracta cannot be reduced below absolute zero that limits the maximum divergence to be provided for the mouthpiece.

8. Flow through a Convergent-Divergent Mouthpiece :- Applying Bernoulli's equation between free liquid surface in the reservoir, the vena- contracta ( section-cc) and exit of the mouthpiece ( section-bb), Hence, If  and  are the cross-sectional areas at vena- contracta and the outlet end of the mouthpiece, then by continuity equation,

9. 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 . a). Running full b). Running free

10. Flow through an Internal Mouthpiece :- 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 Static thrust on the fluid for the area  a = ꝩah Rate of change of momentum of the jet by this static thrust is = [ꝩ. a c .V c /g]V c Equating the above two, by applying Newton’s Law of motion, we get a c /a = gh /V c 2 (5)

11. Flow through an Internal Mouthpiece :- 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 , (6) By combining Eqn. (5) with (6), i.e. the coefficient of contraction for a Borda's mouthpiece is 0.5 .

12. Borda’s mouthpiece 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. Hence, Applying Bernoulli's equation, (7) where is the head loss due to sudden enlargement,

13. Borda’s mouthpiece running full :- Then solving for velocity V from the Eqn , we get Coefficient of velocity becomes,