CHE 204: TRANSPORT PHENOMENA ITERM 102ALSHAMI

Chapter 2: Bernoullis equation (simplified)In the previous lecture we derived the generalized Bernoulli equation, Eqn. (1) by performing an overall energy balance around the defined system. (1)Eqn. (1) is further simplified for most application if the following assumptions hold valid:1. The flow is steady (no accumulation/disappearance of mass and energy with time)2. The fluid is incompressible (constant density)3. The flow is inviscid (Frictionless)4. There are no work effects: no work applied or producedUnder these circumstances, eqn. (1) is reduced to: (2)Eqn. (2) is the famous Bernoullis equation: one of the most important and used relation in fluid mechanics. This equation simply says that there is NO net change in energy between the inlet and outlet of the system (ie, location 1 and 2):

Head of fluidA quantity often used to describe overall pressure magnitude is pressure head H. We already introduced the terminology in the previous lectures when analyzing hydrostatics. Extending the analysis to a moving fluid, we can see if we divide through eqn. (2) by g, we result with: (3)Examining eqn. (3) shows that each of its 3 terms has a dimension of length (L), and consequently called pressure head, velocity (dynamic) head, and static head.

Applications of Bernoullis equationI. Tank drainingDetermine the velocity and mass flow rate of water efflux from the circular orifice (0.1 m ID.) at the bottom of the water tank (at this instant). The tank is open to the atmosphere and H = 4 m.

H

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Solution1. Conservation of mass:

- No incoming mass into the system ==> min = 0- we are interested in find the instantaneous flow rate ==> dM/dt = constant

(- sign indicates that mass is leaving the system)

2. Conservation of energy:i. No work effectsii. No friction losses (no long length of conduits)

Simplified Bernoullis equation is applicable:

Also, P1 - P2 = 0 open to the atmosphere u1=0 (fluid movement at location 1 is negligible compared to movement in location 2) z = H = 4 m Solving for u2 gives,

Orifices correction factorFluids flow through orifices is generally accompanied by disruptions and must be accounted for, especially at high velocities. A correction factor is commonly used to correct for the sudden area contraction, and is defined as: II. Orifice Plate meter

Applying Bernoullis between inlet and outlet, which are at the same elevation (z=0), yields:

From the continuity equation, we get:

Solving for u1 yields:

And the volumetric flow rate Q is:

A more commonly used equation is:

CD is known as the discharge coefficient that is generally determined experimentally and most often provided by the vendor of the meter (usually is around 0.63).

III. Pitot tubeThe Pitot tube (named after the French Scientist and engineer Henry Pitot, 1695-1771) is perhaps the simplest and most useful fluid flow instrument ever devised.

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Applying Bernoullis equation between locations 1 and 2, yields:

Pressure at location 2 is atmospheric, which is the same as on water surface (water-air interface).

The pressure at location 1 from hydrostatics is:

Substituting for P1 and rearranging yields:

Alternatively, if we apply Bernoullis equation between location 1 and the stagnation point; ie., tip of the tube where the fluid is no moving (location 3), then the following can be derived:

Solving for u1 yields:

Also,

And

Subtracting P3 from P1 results with:

Yielding same results for the fluid (or the craft) velocity obtained above.

Similar analysis can be carried out for another type of Pitot tube, commonly called Pitot-Static tube, sketched in your textbook in Fig. 2.12. This tube involves 2 tubes where the differential fluid level (h) is used in the equation above to calculate the fluid or the craft moving speed. IV. Venturi Tube meter

Venturi tube is simply a tube that is made up of 3 distinct sections: 1) convergent region, 2) throat, and 3) divergent region. The design of these tubes aims at minimizing, as much as possible, losses due to friction and flow separation. In addition to using these tubes for pressure (velocity) measurements, they are also commonly used as atomizers. Applying Bernoullis equation between the inlet and throat (ie., 1 and 2), which are at same elevation (ie., z = 0) gives:

From the continuity equation,

Substituting for u1 yields,

Solving for u2 ( velocity at throat) yields,

And the volumetric flow rate Q is:

A more commonly used equation is:

Cv is known as the discharge coefficient that is generally determined experimentally and most often provided by the vendor of the meter (generally it is around 0.98).

V. Siphon

Siphonh2Dischargeh1z123

A siphon can be used to drain liquid from an open tank. Siphon lefts the liquid to an elevation higher than its free surface and discharges it at a lower elevation than the level of the liquid. The top point (2) of the siphon is called summit and it is where low pressure occurs, sometimes to levels close to the liquid vapor pressure. When vapor pressure is reached at the summit, liquid begins to boil (ie., evaporates) resulting with the so-called vapor lock, and flow stops.

Applying the Bernoulli equation between locations (1) and (3):

Where,P = 0 (atmospheric pressure)u1 = 0z3 = 0z1 = h1 Substituting yields,

Solving for discharging velocity (u3), gives:

Similarly, applying Bernoullis equation between locations (1) and (2) results with:

Where,P1 = 0 (atmospheric pressure)u1 = 0z1 = h1z2 = (h1+ h2) Substituting yields,

Since the Siphon (tube) cross-sectional area does not change, then velocity remains constant, ie.:

Substituting for u2 in above equation yields:

Rearranging gives:

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