Engineering Bernoulli Equation

Transcription

1 Engineering Bernoulli Equation R. Shankar Subramanian Department of Chemical and Biomolecular Engineering Clarkon Univerity The Engineering Bernoulli equation can be derived from the principle of conervation of energy. Several book provide uch a derivation in detail. The intereted tudent i encouraged to conult White () or Denn (). Here, I have merely ummarized the important form of thi equation for your ue in olving problem. Whenever you ue thi equation, be ure to draw a ketch and clearly mark the datum from which height are meaured. When etting a term to zero, indicate the reaon for doing o. For example, when the free urface of the liquid in a tank i expoed to the atmophere, or when it i iuing a a free jet into the atmophere, the preure at that location i et equal to zero gage. When liquid i taken out of a veel through a pipe of croectional area that i mall compared with that of the veel, the velocity of the free urface will be relatively mall, and the kinetic energy term aociated with that velocity can be et equal to zero without much error. When the Engineering Bernoulli Equation i applied to fluid contained in a control volume fixed in pace, typically the control volume ha impenetrable boundarie, with the exception of one or more inlet and one or more outlet through which fluid enter and leave the control volume. During paage of fluid through the control volume, mechanical work i irreveribly tranformed by fluid friction into heat, leading to loe. Alo, the fluid may operate a turbine, performing work on the blade of the machine, or work may be performed on the fluid by a pump. Thee lead to haft work, aumed by convention to be poitive when performed by the fluid, and negative when performed on the fluid. Both loe and haft work are included in the energy form of the Engineering Bernoulli Equation on the bai of unit ma of fluid flowing through. The two mot common form of the reulting equation, auming a ingle inlet and a ingle exit, are preented next. Energy Form Here i the energy form of the Engineering Bernoulli Equation. Each term ha dimenion of energy per unit ma of fluid. p V p V ρ ρ out out in in + + gzout = + + gzin lo w In the above equation, p i preure, which can be either abolute or gage, but hould be in the ame bai on both ide, ρ repreent the denity of the fluid, aumed contant, V i the velocity of the fluid at the inlet/outlet, and z i the elevation about a datum that i pecified. Note that it i only difference in elevation that matter, o that the choice of the datum for z i arbitrary. The ymbol g tand for the magnitude of the acceleration due to gravity.

2 The term lo tand for loe per unit ma flowing through, while w repreent the haft work done by the fluid per unit ma flowing through. The form given above aume flat velocity profile acro the inlet and exit, which i a reaonable approximation in turbulent flow. In laminar flow, the velocity ditribution acro the cro-ection mut be accommodated in the kinetic energy calculation. In that cae, we ue the average velocitie at the inlet and exit, but multiply the kinetic energy term on each ide of the Engineering Bernoulli Equation by a correction factor α that account for the variation of the kinetic energy of the fluid acro the cro-ection. You can conult reference () or () to learn how to calculate thi correction factor. We expre lo uually a a certain number ( N ) of velocity head. In thi cae, lo = N( V /). At other time, lo i expreed a a certain ( M ) head of fluid. In thi cae, lo = gm. Head Form The head form of the Engineering Bernoulli Equation i obtained by dividing the energy form throughout by the magnitude of the acceleration due to gravity, g. p V p V lo w γ g γ g g g out out in in + + zout = + + zin In thi equation, the ymbol γ repreent the pecific weight of fluid. γ = ρg We define the head developed by a pump a hp = w / g. Becaue the work term w i negative for a pump, the head developed by a pump h p i alway poitive. Lo i alway poitive. We define a head lo term a lo hfriction h f g = =. Therefore, we can rewrite the head form of the Engineering Bernoulli Equation a p V p V γ g γ g out + out + in in z = + + out z in h + f hp Now, two example are preented that will help you learn how to ue the Engineering Bernoulli Equation in olving problem. In a third example, another ue of the Engineering Bernoulli equation i illutrated.

3 Example : Sizing a Pump ID Pipe pump ID Pipe 5 ft Aborber 5 ft z torage tank 5 ft The pump hown here i ued to lift a proce liquid of denity.9 lug / ft from a torage tank, and dicharge it at a rate of 0.75 cubic feet per econd into the top of an aborber. The inlet to the aborber i located 5 feet above the free urface of the liquid in the torage tank, and the pump inlet i located at an elevation of 5 feet above that of the free urface. You can aume that the aborber operate at atmopheric preure. A ID pipe lead from the torage tank to the pump, while the pipe from the pump to the top of the aborber i of ID. You can aume the loe in the ID pipe to be 4 velocity head, and the loe in the ID pipe to be 5 velocity head. Auming the pump i 85% efficient, calculate the BHP (Brake Hore Power) of the pump.

4 Solution Firt, we mut identify location and for applying the Engineering Bernoulli Equation. Recall that the preure, velocity, and elevation at each of thee location appear in the Engineering Bernoulli Equation. Thu, we elect thee location in uch a way a to be able to pecify the maximum amount of information poible at each. With thi in mind, location i elected to be at the free urface of the liquid in the torage tank, and location at the entrance to the aborber. Further, we elect the datum for meauring height to be at the free urface of the liquid in the torage tank, a hown in the ketch. Note that the height of the pump above that free urface i given, but it i not a good idea to chooe our location at either the inlet or the exit of the pump, becaue it would unnecearily add to the calculational burden. Now lit all the known information at the two location. p = 0 gage (Open to atmophere) V = 0 (Large cro-ectional area) z = (By choice of datum) 0 From the given dicharge rate and the diameter of the pipe at the aborber inlet, we can calculate ft π V. Q = 0.75 = VA = V ( 0.5 ft) yielding V = 5. ft /. 4 p = 0 gage (Aborber i at atmopheric preure) V = 5. ft / (from pecified data) z = 5 ft (pecified) Let u write the Engineering Bernoulli Equation. We ue location for in and location for out. p V p V + + gz = + + gz lo w ρ ρ Subtituting ome of the known information into the above equation, we obtain V gz = gz lo w V or w = lo + + g( z z ) The implication of the above equation i a follow. The haft work delivered by the pump to the fluid account for the loe in the flow through the pipe and any fitting, the exit velocity 4

5 head that mut be delivered, and the lifting of the fluid from elevation z to elevation z. If the fluid enter the control volume with ome kinetic energy (that i, if V 0 ), then that kinetic energy would help reduce the haft work needed. Likewie, we can ee from the Engineering Bernoulli Equation that the haft work alo mut upply any needed preurizing of the fluid ( p p ) / ρ. In thi problem, p = p. We have choen each to be zero by uing gage preure, but even if we had ued abolute preure, the difference would till be zero and no haft work would be needed for increaing the preure. Let u firt calculate the lo. It i pecified a 4 velocity head in the ID pipe, and 5 velocity head in the ID pipe. We already calculated the velocity in the ID pipe to be V = 5. ft /. Becaue Q = VA, and the cro ectional area are proportional to the quare of the diameter, we can write inche ft 9 ft V" ID pipe = V" ID pipe = 5. = 4.4 inche 4 Therefore, we can find the lo a ft ft V V ft lo = = =.95 0 " ID pipe " ID pipe Subtituting in the reult for the haft work, ft 5. V ft ft w = lo + + g z z = ft + + =.87 0 ft ( ) ( ) The unit of the haft work appear to be trange, but they are not. Let u invetigate thi further. Recall that each term in thi verion of the Engineering Bernoulli Equation mut have the ame unit a the lo or haft work, which are in energy per unit ma flowing through the control volume. Let u work out the unit of energy per unit ma in the Britih ytem. Energy ha the ft lb ame unit a work, which i force time ditance. Therefore, the unit of haft work are f lug ft. One lb f i the force required to accelerate a ma of one lug by one, i.e., lug ft ft lbf ft lug ft ft lbf =. Therefore, the unit of haft work are = =. So, lug lug 5

6 we ee that the unit of ft lb f lug are equivalent to ft. In the metric ytem, the correponding unit would be J kg and m, which you hould verify are equivalent to each other. Therefore, the work to be delivered by the pump to the fluid i multiply thi by the ma flow rate, which i given by.87 0 ft lb f lug. We mut now m = ρq = =.45 ft lug ft lug The power upplied to the fluid by the pump i given by lug ft lbf ft lb Power to Fluid = m ( w ) = = lug Power i uually reported in Hore Power abbreviated a HP or hp. One Hore Power i ft lb equivalent to 550 f or 746 W in the metric ytem. Therefore, ft lbf hp Power to Fluid = = 0. hp ft lbf 550 There are loe that occur inide a pump. Therefore, the power upplied to a pump, called the Brake Hore Power or BHP i larger than the power delivered to the fluid. The ratio of the power delivered to the fluid to the Brake Hore Power i termed the efficiency of the pump repreented by the ymbol η. Therefore, f BHP Power to Fluid 0. hp = = = η hp 6

7 Example : A Turbine Problem 5 m z turbine V In the above example problem, a hydroelectric turbine at the bae of a dam i hown chematically. The height of the water above the turbine tation given a 5 m. Thi turbine produce 4.6 MW of electricity, and you may aume the loe in the ytem to be equivalent to 0 m of head of water. The diameter of the pipe at the turbine exit i 0.75 m, and the velocity of water exiting from the pipe i.5 m/. Calculate the efficiency of the turbine. Solution A in the earlier example, we firt identify location and for applying the Engineering Bernoulli Equation. Becaue the preure, velocity, and elevation at each of thee location appear in the Engineering Bernoulli Equation, we again elect thee location in uch a way a to be able to pecify the maximum amount of information poible at them. With thi in mind, we chooe location at the free urface of the liquid in the reervoir, and location at the exit from the turbine. The datum for meauring height i choen logically at the bae of the reervoir where the turbine i located, a hown in the ketch. We proceed to lit all the known information at the two location. p = 0 gage (Open to atmophere) V = 0 (Large cro-ectional area) z = 5 m (Specified) 7

8 p = 0 gage (Free jet into the atmophere) V =.5 m/ (given) z = 0 (By choice of datum) Uing location for in and location for out, the Engineering Bernoulli Equation i written a follow. p V p V + + gz = + + gz lo w ρ ρ A in Example, we ubtitute ome of the known information into the above equation, yielding V w = g( z z) lo Thi equation tell u that the haft work we can obtain from the turbine i proportional to the head of water available minu the lo in the ytem minu the velocity head in the exit tream. If the inlet tream had a ignificant amount of velocity head, that would help increae the haft work, but becaue of the large cro-ectional area of the reervoir, the velocity at the free urface can be aumed to be negligible. We are given the lo a 0 m of head of water, o it i eaiet to cat it a lo = g 0 ( m). Subtituting the value of g, z, z, V, and the reult for the lo in the equation for the haft work, we obtain ( m ) m m.5 / m w = 9.8 ( 5 0)( m) ( m ) = 95 Thi i the work performed on the turbine by the water per unit ma flowing through. Therefore, we mut multiply thi by the ma flow rate m to obtain the power delivered by the water to the turbine blade, which can be written a m = ρ Q= ρva. To etimate the denity of the water, we need the temperature. The temperature i not pecified, but we can aume it to lie in the ordinary range of temperature for the environment. Given the variation of the temperature depending on the time of the year, it i afe to ue a denity of water ρ = 0 kg / m, recognizing that thi value i ubject to an error of a couple of kg / m. The diameter of the pipe at the turbine exit i given a D = 0.75 m, o that it cro-ectional area i A π π ( ) = D = 0.75 m = 0.44 m 4 4 8

9 Now, we can proceed to calculate the ma flow rate of water out of the turbine a kg m kg m = ρ VA = ( m) = m Therefore, the power delivered to the turbine blade i given by m kg kg m Power to turbine blade = w m = = Let u work out the unit of thi power. 6 kg m kg m m N m J = = = = W Thu, the power i in Watt. The power output of the turbine i given a 4.6 MW or. Therefore, the efficiency of the turbine can be calculated a follow W 6 Power output from the turbine W Efficiencyη = = = Power to turbine blade W The efficiency i alo reported commonly a a per cent. In that cae, we would tate it a 87.5%. 9

10 Example : Etablihing the Flow Direction in a Pipe B 0 ft A pa = 8 pig pb = 5 pig What i the direction of flow of water in thi pipe? You can aume that the flow i teady, and that the denity of water i contant. Solution We can determine the direction of flow of water in thi pipe in a couple of way. One way i to determine the hydrotatic preure difference between A and B. If the water i tationary in the pipe, the preure at A would be larger than that at B by γ h, where γ i the pecific weight of the fluid and h i the elevation difference between A and B. The pecific weight of water i lbf γ water = 6.5. Therefore, we can write ft lbf lbf ( pa pb ) = γ water h = ( ft) pi hydrotatic = = ft ft But the actual preure difference pa pb = 8 pig 5pig = pi (Note that preure difference are abolute they are not meaured relative to any other preure) So, we ee that the preure at A i not large enough for a hydrotatic balance, and flow mut occur from point B to point A. The econd way ue the idea that loe in a flow mut alway be poitive, becaue they arie from irreverible converion of mechanical work into heat. To ue thi idea, we apply the Engineering Bernoulli equation to the ytem. Let u begin by auming that the flow occur from A to B. Therefore, location A i the in location, and location B i the out location for the control volume and we can write 0

11 p B B A A + V + gzb = p + V + gza lo w ρ ρ where we have croed out the haft work term, becaue there i none in the control volume. Becaue the diameter of the pipe i contant and the denity of water can be aumed to be contant, VA = VB. Therefore, we can write lbf 44 in ( 8 5) lo pa p in ft B = ( zb za) = 0 ft = ( 6.9 0) ft =.09 ft g γ lbf 6.5 ft o that the lo i 6 ft / and i negative. Thi i not poible. Therefore, our aumption that flow occur from A to B mut be wrong, and the actual flow direction i from B to A. We can confirm the correctne of thi concluion by recating the flow a occurring from B to A in the Engineering Bernoulli equation. In that cae, the reult for the lo i lo pa pb = ( zb za) = ( 0 6.9) ft =.09 ft g γ Therefore, the lo i poitive, and equal to + 6 ft /. Even though the two approache we ued appear to be different uperficially, they are baically the ame. To ee thi, examine the lo calculation when the flow i aumed to be from A to B. The firt term on the right ide i the amount by which the preure at A i larger than the preure at B, expreed in head of water. Thi i only 6.9 feet, wherea the econd term i the actual elevation difference between B and A, which i 0 feet. Thu, the preure difference i maller than what we d ee if the fluid were tationary. The lo work out to be the preure difference between A and B expreed a head of water minu the elevation difference between B and A. Thi i a negative quantity, and it i impoible to have uch a flow. If we aume the flow direction to be from B to A, which i the correct direction, then the preure difference till i equivalent to 6.9 feet of water, but the lo now i the elevation difference between B and A minu the preure difference between A and B in head of water, and become poitive, telling u that thi i the correct direction of flow.

12 Bernoulli Equation In ome application, there i no machinery uch a a pump or turbine in the control volume. In uch cae, the haft work term in the Engineering Bernoulli Equation i zero. Furthermore, it i convenient to neglect loe in hort ection of piping. In other cae, where there are loe they are accommodated eparately uing a o-called lo coefficient. To handle thee type of application, we ue a impler verion of the Bernoulli Equation that contain neither haft work nor a lo term. It i imply called the Bernoulli Equation. Bernoulli Equation The energy form i written a follow. Each term ha dimenion of energy per unit ma of fluid. We ue the convention that location i the inlet and location i the outlet. p V p V ρ ρ + + gz = + + gz The correponding Head form i p V p V γ g γ g + + z = + + z While it i true that the Bernoulli Equation can be derived rigorouly along a treamline for a pecial cla of flow called potential flow, thi i not particularly important when we apply it to practical problem. Some uer try to draw contrived treamline for thi purpoe, which are unneceary. The main idea i that haft work i zero, and loe in the control volume are treated a being negligible in the actual ituation. A few ituation where the Bernoulli Equation prove ueful are lited below.. Etimating the flow rate out of a tank, given the diameter of the exit pipe. Evaluating flow rate and preure in a iphon. Evaluating the volumetric flow rate through a pipe uing a flow meaurement device uch a a venturimeter 4. Determining a local velocity uing the Pitot-Static Tube Each of thee example i illutrated next.

13 Example : Tank Draining Through a Pipe Obtain an expreion for the velocity and the volumetric and ma flow rate out of a pipe of diameter d at the bottom of a torage tank in which a liquid i maintained at a height H. The tank i open to the atmophere, and the liquid iue out of the pipe a a free jet into the atmophere. H z A hown in the ketch, we place location at the free liquid urface, and location at the exit of the pipe. Clearly, there i no pump or turbine here. If we neglect loe, which i reaonable if the pipe i hort and the torage tank i large and wide, we can ue the Bernoulli Equation, written between location and in the ketch. p V p V + + gz = + + gz ρ ρ The preure at location and are both zero gage. By choice of datum, z = H and z = 0. Becaue of the large cro-ectional area at location, the velocity V can be et equal to zero. Thu, we can make thee ubtitution into the Bernoulli Equation to obtain V = gh which yield V = gh. Uually, any mall loe are accommodated by multiplying thi reult by a lo coefficient. The volumetric flow rate ( π /4) i given a m = ρ Q= ( πd /4) ρv. Q= d V, and the ma flow rate Note that we aumed that the liquid level in the tank i maintained contant at H. The reult we obtained can be ued even when that level i changing becaue of lo of liquid through the exit, o long a the rate of decreae of H i ufficiently mall. In fact, the rate of lo of liquid can be ued to calculate the rate at which H decreae with time, uing an unteady ma balance on the content of the tank. Thi type of aumption i known a the quai-teady-tate aumption. In other word, we aume teady condition for calculating the exit velocity out of the pipe, even though the condition are truly unteady. The quai-teady aumption i commonly ued in heat and ma tranfer operation in ituation where it ue i jutified.

14 Example : Siphon A iphon i a device for removing liquid from a container uing a pipe that rie above the liquid level in the container. A ketch of a typical iphon i hown below. h H z A we can ee there are no pump or turbine here, and if we neglect the mall loe involved, we can apply the Bernoulli Equation to obtain an etimate of the velocity out of the pipe, V. All that matter i the height H of the free urface of the liquid in the container above the location of the iphon outlet. We have therefore elected the location a datum, and location a the free urface of the liquid in the container. The Bernoulli Equation i p V p V ρ ρ + + gz = + + gz Becaue location and are both open to the atmophere, the gage preure i zero at both location. The velocity at location can be et equal to zero becaue of the large cro-ectional area in the container. By choice of datum, z = H and z = 0. Subtituting all of thi information into the Bernoulli Equation yield V = gh or V = gh, the ame reult that we obtained in Example for the velocity out of the pipe at the bottom of a tank when the liquid level wa located at a height H above the pipe. Of coure, there would be ome lo aociated with the flow through the iphon pipe, which will lead to a maller value of V than gh. 4

15 Now, let u find the gage preure at location. We can write the Bernoulli Equation between location and. p V p V ρ ρ + + gz = + + gz Becaue the pipe ha the ame diameter throughout, the velocity V = V. Subtituting all the known information, we obtain 0 + V p V + 0 = + + g( H + h) ρ o that p = ρg( H + h) Thi mean that the preure at location i le than atmopheric preure. Clearly there i a limit to how low we can go below atmopheric preure. At firt glance, you might think that we can increae the height difference between the lowet and highet point in the iphon pipe up to the value where the abolute preure at location will be zero. Thi would be an incorrect concluion. Actually, when the preure i gradually lowered in a liquid, it will firt reach a value where it equal the vapor preure at room temperature. When it goe lightly below thi value, vapor bubble will begin to form, typically at location on the pipe wall that contain crevice with trapped air, a proce known a heterogeneou nucleation. When vapor bubble are nucleated, we ay that cavitation i occurring in the liquid. The preence of uch vapor bubble will caue problem with operating the iphon, interfering with the flow. Cavitation can be a eriou problem in machinery in which the preure drop below atmopheric preure, perhap becaue of a high velocity being reached at certain location. The vapor bubble that form will collape when the liquid move to a different location where the preure i higher. Such collaping of vapor bubble can be violent and can caue pitting and eroion of material on the blade of centrifugal pump, and on propeller blade ued on hip. 5

16 Example : Venturimeter p p V V throat diffuer A venturimeter i a device that i placed in a pipe to permit the meaurement of the flow rate of the fluid through the pipe. The meaurement device i named in honor of Giovanni Venturi. The principle of the venturimeter i imple. The cro-ectional area of the flow i reduced uddenly to form a throat region. A the fluid accelerate to a higher velocity through the throat to accommodate the contancy of the volumetric flow rate, the kinetic energy mut increae, and thi mut be accompanied by a drop in preure. Thi preure change i meaured uing preure tap placed on the wall of the tube uptream of the contriction and in the throat. There are ome loe aociated with the contraction, but uually the larget loe accompany a udden expanion in cro-ectional area that create large recirculating eddie right after the expanion. Thee loe are avoided in the venturimeter by having the cro-ection gradually expand through a diffuer ection to the original pipe diameter. A well-deigned venturimeter ha minimal loe, and mot of the preure-drop i recovered downtream. Given the fact that location and in the chematic are at the ame elevation, regardle of our choice of datum, z = z. In the abence of any haft work, and neglecting loe, the Bernoulli Equation reduce to p V p V + = + ρ ρ Therefore, we can write V V = p, where p= p p. From conervation of ma, ρ VA = VA. We know that the area are proportional to the quare of the two diameter at location and. Therefore, we may write V diameter. Uing thi information, we can obtain = β V, where β = D / D, the ratio of the V p = ρ( β ) 4 / 6

17 π The volumetric flow rate through the pipe i then obtained from Q= D V, and the ma flow 4 rate, if deired, a ρ Q. In pite of the careful deign to minimize the loe, there till are ome, and thi fact i accommodated by multiplying the reult for V by a dicharge coefficient C d which i a function of the diameter ratio β. The dicharge coefficient i le than unity, but typically lie between 0.9 and.0. The venturimeter i relatively expenive, but minimize operating cot by keeping the loe mall. At the other end of the pectrum i the inexpenive orifice-meter, which involve placing an orifice-plate in the path of the flow. Thi plate contain a mall orifice. The fluid accelerate through the orifice, and expand to fill the pipe again a it flow downtream. A noted earlier, in the udden expanion of the cro-ection pat the orifice, recirculating eddie are formed, and the energy to maintain thee eddie come from the preure-head. Therefore, while the principle of operation of both the venturimeter and the orifice meter are the ame, the loe are much larger in the orifice meter. Modern orifice meter are relatively inexpenive and eay to intall, but add to the pumping cot becaue of the loe. The loe are uually reported a Non-recoverable head lo, meaured in term of throat velocity head. The non-recoverable head lo for a venturimeter i typically of the order of a quarter throat velocity head, wherea that for orifice meter can range from 0.5 to.5 throat velocity head, depending on the diameter ratio β. 7

18 Example 4: Pitot-Static Tube The Pitot-Static tube or imply Pitot tube, named after Henri Pitot, i a device that i employed to determine a local velocity within a flow field. It i baed on the idea that when fluid moving at a certain velocity i brought to a top at a tagnation point, it preure rie. From the increae in preure, we can infer the value of the velocity. Conider point and in the ketch, and aume that point i a tagnation point; that i, the velocity at point, V = 0. Becaue both point are at the ame elevation, regardle of datum, z = z. With no haft work and negligible loe, the Bernoulli Equation reduce to V p ρ + p V = + ρ or p ρv = p + the preure p i called the tagnation preure for thi treamline. We can olve the above equation for the velocity V. V = ( ) p p ρ Thi i the principle on which the Pitot tube i baed. A typical Pitot tube conit of two hollow tube of mall diameter, one inide the other, a hown in the ketch adapted from White (). flow hole preure tranducer 8

19 The annulu formed by the two tube i cloed off at the end facing the flow, and a few hole on the cylindrical urface of the outer tube permit the fluid to enter a the fluid flow pat the Pitot tube in order to ample the preure in the fluid flowing pat the annulu. Therefore, the preure of the fluid entering the annulu i the local preure in the flow. The end of the inner hollow tube i connected to one ide of a differential preure tranducer, o that the fluid ha to come to a top within that tube, raiing the preure to the tagnation preure. The end of the annulu i connected to the other ide of the ame preure tranducer. Thu the preure difference p = p p meaured by the tranducer can be ued in the equation we obtained earlier to calculate the velocity. The intrument i quite mall in diameter compared with the length cale on which the velocity varie in the flow, o that it can be aumed to meaure the local velocity at the point where the ampling of the flow i performed. The Pitot tube i employed on airplane in flight to meaure the peed of the plane, but a correction ha to be applied for compreibility effect, becaue of the relatively large velocitie involved. It alo can be ued to meaure the local velocity of a liquid, uch a that of blood in arterie and vein (). Reference. F.M. White, Fluid Mechanic, Seventh Edition, McGraw-Hill, New York (0).. M.M. Denn, Proce Fluid Mechanic, Prentice-Hall, Englewood Cliff (980). 9