18. Examples of application of bernoulli's equation to engineering problems

Bernoulli's equation is. an expression of the basic law of steady flow. It can be used in the form developed here to investigate and analyse the work of a number of devices whose design is based on this fundamental law. Let us consider some of them.

The venluri meter{Fig, 32) is used to measure pipe flow. It consists of a tube with a conical entrance and constricted throat {the nozzle) followed by a gradually diverging portion fthediffuser). In the converging portion the veloc­ity of the flow increases and the pressure decreases. The pressure drop is meas­ured by means of a pair of piezometer tubes or a U-tube differential gauge. The relation between the pressure drop and the rate of discharge is found as follows.

Let the velocity of flow at section 1-1 immediately beforef-the nozzle be v_v The pressure p, and the cfoss-seetional area S₁; at the throat 2-2 the cross-sectional area is S₁ and the velocity and pressure v and p_f respectively. The difference between the levels in the piezometer tubes at the respective sections is .

Assuming a uniform velocity distribution, write Bernoulli's equation andthe'continuity equation for the two cross-sections:

where h_t = loss of head between sections 1-1 and 2-2.

Taking into account that

and

we can use these equations to deter­mine one of the velocities, say v%. We have

whence the volume rate of dis­charge is

(4.21)

or

is a constant for the given meter;

Knowing С and observing the piezometer [readings, it is easy to determine the rate of discharge at anympnient of time from Eq. (4.21). The constant С can be computed theoretically, but greater accuracy is obtained by experiment, that is, by calibrating the meter. .

The relation between and Q is that of a parabolic function, and the square of the rate of discharge laid off on the axis of abscissas gives a straight line.

Often a differential mercury gauge is used instead of a pair of piezometer tubes to measure the pressure drop (see Fig. 32). Taking into account that the liquid above the mercury is the same in both limbs and its specific weight is y,we can write

Incidentally, a diffuser is not essential for flow measurement, and a flow nozzle alone placed inside a pipe (Fig. 33a) or between flanges (Fig. ЗЩ can be used. In this case the convergence of the flow is gradual, as in the case of a venturi meter, but the divergence beyond the nozzle is abrupt and eddies form. The resistance of a flow nozzle is greater than that of a venturi meter with a diffuser.

Another instrument for measuring fluid flow is an orifice meter (Fig. 33c). Owing to additional compression experienced by the fluid, the stream is nar­ rowest a bit downstream from the orifice and its diameter is slightly less than the orifice diameter.

Equation (4.21'} is valid for these flow meters, with correction coefficients available for standard-type flow meters in relevant- handbooks.

2. Carburettors (Fig. 34) are used in reciprocating internal-combustion en­ gines to meter, atomise and mix the fuel with air. The stream of air passes through a venturi. tube m which a fuel jet is placed: The velocity, of the air increases in the venturi and the pressure drops according to Bernoulli's equation. The reduc­ tion in pressure at the venturi throat causes fuel to flow tnrough the fuel jet into tbe.air stream;

Let us find the relation between the weight rate of discharge of the fuel (gasoline) G_g and air G_a if the di­mensions D and d and the loss coeffi­cients ξ_a of the venturi throat and ξ_a of the jet are known (neglecting the - resistance of the gasoline supply pipe).

Writing Bernoulli's equation for the air stream (sections 0-0 and 2-2) and for the gasoline stream (sections 1-1 and 2-2), we obtain (at z, = z% and a ε 1)

whence

As the weight rates of discharge are

and

we obtain

3. An air ejector (Fig. 35) consists of a nozzle A, in which the stream con­verges, and a gradually widening diffuser С placed at a distance from the nozzle inside a suction chamber B. The increased velocity of the stream leaving the nozzle causes the pressure in the stream and in the whole of the suction chamber to drop considerably. In the diffuser the velocity decreases and the pressureincreases approximately to atmospheric (if the fluid is ejected into the air). Consequently, the pressure in the suction chamber is also below atmospheric, i. e., a partial vacuum is created.

This vacuum causes the liquid in the bottom reservoir to rise along pipe D to the suction chamber where it is entrained by the stream issuing from the noz­zle. Ejectors are used in various machines, notably in liquid-propellant rocket motors.

4. A Pi tot tube (Fig. 36) is used to measure flow velocity. Let a liquid be flowing in an open channel with a velocity v. If a bent tube is immersed with the opening facing the flow as shown, the liquid will rise in the vertical limb above the free surface to a height equal to the velocity head. This is because the veloc ity of the liquid particles entering the Pitot tube at point A (the stagnation point) is

zero and, consequently, the pressure increases by the value of the veloc­ity head. By observing the height of the liquid in the tube the velocity may be calculated.

The measurement of the velocity of a flying aircraft is based on the same prin­ciple. Fig. 37 presents a schematic drawing of a Pitot tube for measuring veloc­ities comparable with the speed of sound.

Let us write Bernoulli's equation for an elementary stream tube approaching the Pitot tube along its centre line and flowing about it. Taking sections 0-0 (the undisturbed stream) and 1-1 (where v = 0), we have

As the pressure at the side openings is approximately that of the undisturbed stream, Consequently, from the foregoing we have

5. Tank pressurisation (Fig. 38) is widely employed in aircraft to increase the pressure in fuel and other tanks. At low flight velocities the gauge pressure in the tank is approximately equal to the dynamic pressure as calculated from the velocity of flight and air densitx.