5 4 THE BERNOULLI EQUATION

Transcription

1 185 CHATER 5 the suounding ai). The fictional wok tem w fiction is often expessed as e loss to epesent the loss (convesion) of mechanical into themal. Fo the idealied case of fictionless motion, the last elation educes to 1 g 1 g o whee the value of the constant is C gh. That is, when the fictional effects ae negligible, the sum of the kinetic and potential enegies of the ball emains constant. Discussion This is cetainly a moe intuitive and convenient fom of the consevation of equation fo this and othe simila pocesses such as the swinging motion of the pendulum of a wall clock. The elation obtained is analogous to the Benoulli equation deived in Section 5 4. Most pocesses encounteed in pactice involve only cetain foms of, and in such cases it is moe convenient to wok with the simplified vesions of the balance. Fo systems that involve only mechanical foms of and its tansfe as shaft wok, the consevation of pinciple can be expessed conveniently as E mech, in E mech, out E mech, system E mech, loss (5 3) whee E mech, loss epesents the convesion of mechanical to themal due to ievesibilities such as fiction. Fo a system in steady opeation, the mechanical balance becomes E. mech, in E. mech, out E. mech, loss (Fig. 5 0). 5 4 THE BERNOULLI EQUATION g C constant The Benoulli equation is an appoximate elation between pessue, velocity, and elevation, and is valid in egions of steady, incompessible flow whee net fictional foces ae negligible (Fig. 5 1). Despite its simplicity, it has poven to be a vey poweful tool in fluid mechanics. In this section, we deive the Benoulli equation by applying the consevation of linea momentum pinciple, and we demonstate both its usefulness and its limitations. The key appoximation in the deivation of the Benoulli equation is that viscous effects ae negligibly small compaed to inetial, gavitational, and pessue effects. Since all flui have viscosity (thee is no such thing as an inviscid fluid ), this appoximation cannot be valid fo an entie flow field of pactical inteest. In othe wo, we cannot apply the Benoulli equation eveywhee in a flow, no matte how small the fluid s viscosity. Howeve, it tuns out that the appoximation is easonable in cetain egions of many pactical flows. We efe to such egions as inviscid egions of flow, and we stess that they ae not egions whee the fluid itself is inviscid o fictionless, but athe they ae egions whee net viscous o fictional foces ae negligibly small compaed to othe foces acting on fluid paticles. Cae must be execised when applying the Benoulli equation since it is an appoximation that applies only to inviscid egions of flow. In geneal, fictional effects ae always impotant vey close to solid walls (bounday layes) and diectly downsteam of bodies (wakes). Thus, the Benoulli W pump Steady flow 1 = = 1 + h 1 = = atm E mech, in = E mech, out + E mech, loss W pump + mg 1 = mg + E mech, loss W pump = mgh + E mech, loss FIGURE 5 0 Most fluid flow poblems involve mechanical foms of only, and such poblems ae conveniently solved by using a mechanical balance. 1 Benoulli equation valid Benoulli equation not valid FIGURE 5 1 The Benoulli equation is an appoximate equation that is valid only in inviscid egions of flow whee net viscous foces ae negligibly small compaed to inetial, gavitational, o pessue foces. Such egions occu outside of bounday layes and wakes. h

2 186 FLUID MECHANICS FIGURE 5 Duing steady flow, a fluid may not acceleate in time at a fixed point, but it may acceleate in space. appoximation is typically useful in flow egions outside of bounday layes and wakes, whee the fluid motion is govened by the combined effects of pessue and gavity foces. The motion of a paticle and the path it follows ae descibed by the velocity vecto as a function of time and space coodinates and the initial position of the paticle. When the flow is steady (no change with time at a specified location), all paticles that pass though the same point follow the same path (which is the steamline), and the velocity vectos emain tangent to the path at evey point. Acceleation of a Fluid aticle Often it is convenient to descibe the motion of a paticle in tems of its distance s along a steamline togethe with the adius of cuvatue along the steamline. The velocity of the paticle is elated to the distance by /, which may vay along the steamline. In two-dimensional flow,the acceleation can be decomposed into two components: steamwise acceleation a s along the steamline and nomal acceleation a n in the diection nomal to the steamline, which is given as a n /R. Note that steamwise acceleation is due to a change in speed along a steamline, and nomal acceleation is due to a change in diection. Fo paticles that move along a staight path, a n 0 since the adius of cuvatue is infinity and thus thee is no change in diection. The Benoulli equation esults fom a foce balance along a steamline. One may be tempted to think that acceleation is eo in steady flow since acceleation is the ate of change of velocity with time, and in steady flow thee is no change with time. Well, a gaden hose nole tells us that this undestanding is not coect. Even in steady flow and thus constant mass flow ate, wate acceleates though the nole (Fig. 5 as discussed in Chap. 4). Steady simply means no change with time at a specified location, but the value of a quantity may change fom one location to anothe. In the case of a nole, the velocity of wate emains constant at a specified point, but it changes fom the inlet to the exit (wate acceleates along the nole). Mathematically, this can be expessed as follows: We take the velocity of a fluid paticle to be a function of s and t. Taking the total diffeential of (s, t) and dividing both sides by give d (5 33) In steady flow / t 0 and thus (s), and the acceleation in the s- diection becomes a s d t and d d t (5 34) whee / if we ae following a fluid paticle as it moves along a steamline. Theefoe, acceleation in steady flow is due to the change of velocity with position. Deivation of the Benoulli Equation Conside the motion of a fluid paticle in a flow field in steady flow descibed in detail in Chap. 4. Applying Newton s second law (which is

3 187 CHATER 5 Steady flow along a steamline ( + d) da n s da W u u dx d x FIGURE 5 3 The foces acting on a fluid paticle along a steamline. efeed to as the consevation of linea momentum elation in fluid mechanics) in the s-diection on a paticle moving along a steamline gives (5 35) a F s ma s In egions of flow whee net fictional foces ae negligible, the significant foces acting in the s-diection ae the pessue (acting on both sides) and the component of the weight of the paticle in the s-diection (Fig. 5 3). Theefoe, Eq becomes (5 36) whee u is the angle between the nomal of the steamline and the vetical - axis at that point, m da is the mass, W mg g da is the weight of the fluid paticle, and sin u d/. Substituting, Canceling da fom each tem and simplifying, Noting that d da ( d) da W sin u m d d da g da d d da 1 d g d d d( ) and dividing each tem by gives d 1 d( ) g d 0 (5 37) (5 38) (5 39) Integating (Fig. 5 4), Steady flow: d (5 40) g constant (along a steamline) since the last two tems ae exact diffeentials. In the case of incompessible flow, the fist tem also becomes an exact diffeential, and its integation gives Steady, incompessible flow: (5 41) g constant (along a steamline) This is the famous Benoulli equation, which is commonly used in fluid mechanics fo steady, incompessible flow along a steamline in inviscid (Steady flow along a steamline) Geneal: d + Incompessible flow ( = constant): + FIGURE 5 4 The Benoulli equation is deived assuming incompessible flow, and thus it should not be used fo flows with significant compessibility effects.

4 188 FLUID MECHANICS Flow + ρ Kinetic otential FIGURE 5 5 The Benoulli equation states that the sum of the kinetic, potential, and flow enegies of a fluid paticle is constant along a steamline duing steady flow. egions of flow. The value of the constant can be evaluated at any point on the steamline whee the pessue, density, velocity, and elevation ae known. The Benoulli equation can also be witten between any two points on the same steamline as 1 Steady, incompessible flow: (5 4) 1 g 1 g The Benoulli equation is obtained fom the consevation of momentum fo a fluid paticle moving along a steamline. It can also be obtained fom the fist law of themodynamics applied to a steady-flow system, as shown in Section 5 7. The Benoulli equation was fist stated in wo by the Swiss mathematician Daniel Benoulli ( ) in a text witten in 1738 when he was woking in St. etesbug, Russia. It was late deived in equation fom by his associate Leonhad Eule in We ecognie / as kinetic, g as potential, and / as flow, all pe unit mass. Theefoe, the Benoulli equation can be viewed as an expession of mechanical balance and can be stated as follows (Fig. 5 5): The sum of the kinetic, potential, and flow enegies of a fluid paticle is constant along a steamline duing steady flow when the compessibility and fictional effects ae negligible. The kinetic, potential, and flow enegies ae the mechanical foms of, as discussed in Section 5 3, and the Benoulli equation can be viewed as the consevation of mechanical pinciple. This is equivalent to the geneal consevation of pinciple fo systems that do not involve any convesion of mechanical and themal to each othe, and thus the mechanical and themal ae conseved sepaately. The Benoulli equation states that duing steady, incompessible flow with negligible fiction, the vaious foms of mechanical ae conveted to each othe, but thei sum emains constant. In othe wo, thee is no dissipation of mechanical duing such flows since thee is no fiction that convets mechanical to sensible themal (intenal). Recall that is tansfeed to a system as wok when a foce is applied to a system though a distance. In the light of Newton s second law of motion, the Benoulli equation can also be viewed as: The wok done by the pessue and gavity foces on the fluid paticle is equal to the incease in the kinetic of the paticle. Despite the highly estictive appoximations used in its deivation, the Benoulli equation is commonly used in pactice since a vaiety of pactical fluid flow poblems can be analyed to easonable accuacy with it. This is because many flows of pactical engineeing inteest ae steady (o at least steady in the mean), compessibility effects ae elatively small, and net fictional foces ae negligible in egions of inteest in the flow. Foce Balance acoss Steamlines It is left as an execise to show that a foce balance in the diection n nomal to the steamline yiel the following elation applicable acoss the steamlines fo steady, incompessible flow: dn g constant (acoss steamlines) R (5 43)

5 Fo flow along a staight line, R and thus elation (Eq. 5 44) educes to / g constant o g constant, which is an expession fo the vaiation of hydostatic pessue with vetical distance fo a stationay fluid body. Theefoe, the vaiation of pessue with elevation in steady, incompessible flow along a staight line is the same as that in the stationay fluid (Fig. 5 6). 189 CHATER 5 A B Stationay fluid C D Flowing fluid Unsteady, Compessible Flow Similaly, using both tems in the acceleation expession (Eq. 5 33), it can be shown that the Benoulli equation fo unsteady, compessible flow is d Unsteady, compessible flow: g constant (5 44) t Static, Dynamic, and Stagnation essues The Benoulli equation states that the sum of the flow, kinetic, and potential enegies of a fluid paticle along a steamline is constant. Theefoe, the kinetic and potential enegies of the fluid can be conveted to flow (and vice vesa) duing flow, causing the pessue to change. This phenomenon can be made moe visible by multiplying the Benoulli equation by the density, g constant (along a steamline) (5 45) Each tem in this equation has pessue units, and thus each tem epesents some kind of pessue: is the static pessue (it does not incopoate any dynamic effects); it epesents the actual themodynamic pessue of the fluid. This is the same as the pessue used in themodynamics and popety tables. / is the dynamic pessue; it epesents the pessue ise when the fluid in motion is bought to a stop isentopically. g is the hydostatic pessue, which is not pessue in a eal sense since its value depen on the efeence level selected; it accounts fo the elevation effects, i.e., of fluid weight on pessue. The sum of the static, dynamic, and hydostatic pessues is called the total pessue. Theefoe, the Benoulli equation states that the total pessue along a steamline is constant. The sum of the static and dynamic pessues is called the stagnation pessue, and it is expessed as stag (ka) (5 46) The stagnation pessue epesents the pessue at a point whee the fluid is bought to a complete stop isentopically. The static, dynamic, and stagnation pessues ae shown in Fig When static and stagnation pessues ae measued at a specified location, the fluid velocity at that location can be calculated fom ( stag ) (5 47) B B A = D C FIGURE 5 6 The vaiation of pessue with elevation in steady, incompessible flow along a staight line is the same as that in the stationay fluid (but this is not the case fo a cuved flow section). Static pessue, = ieomete Stagnation point ( stag ) Dynamic pessue Stagnation pessue, stag itot tube FIGURE 5 7 The static, dynamic, and stagnation pessues.