Chapter CHAPTER 6 4 The Energy Equation And Its Applications
FLUID MECHANICSGazaCHAPTER 6Chapter5Chapter4The Energy Equation and its ApplicationsDr. Khalil Mahmoud ALASTAL
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGObjectives of this Chapter: Derive the Bernoulli (energy) equation. Demonstrate practical uses of the Bernoulli andcontinuity equation in the analysis of flow. Understand the use of hydraulic and energy gradelines. Apply Bernoulli Equation to solve fluid mechanicsproblems (e.g. flow measurement).K. ALASTAL2
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUG6.1 Mechanical Energy of Flowing Fluid : Bernoulli’s equation is one of the mostimportant/useful equations in fluid mechanics. The Bernoulli equation is a statement of theprinciple of conservation of energy along astreamline. It can be written:p V2 z H constant g 2 g These terms represent:Pressureenergy perunit weightK. ALASTAL KineticPotentialenergy per energy perunit weightunit weightDaniel Bernoulli(1700-1782)Total energy perunit weight3
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUG2p1 V1 z1 H constant g 2 g These term all have units of length. They are often referred to as the following: p1pressure head g2 velocity head V12g potential head z1 total head HBy the principle of conservation of energy the total energy in thesystem does not change, Thus the total head does not change.K. ALASTAL4
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGRestrictions in application of Bernoulli’s eq.:Bernoulli’s equation has some restrictions in its applicability: Flow is steady; Density is constant (which also means the fluid isincompressible); Friction losses are negligible. The equation relates the states at two points along a singlestreamline, (not conditions on two different streamlines).All these conditions are impossible to satisfy at any instant in time!Fortunately for many real situations where the conditions areapproximately satisfied, the equation gives very good results.K. ALASTAL5
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUG As stated above, the Bernoulli equation applies toconditions along a streamline. We can apply itbetween two points, 1 and 2, on the streamline:total energy per unit weight at 1 total energy per unit weight at 2orTotal head at 1 Total head at 2or22p1 V1pV z1 2 2 z2 g 2 g g 2 gK. ALASTAL6
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUG22p1 V1p2 V2 z1 z2 g 2 g g 2 g This equation assumes no energy losses (e.g. fromfriction) or energy gains (e.g. from a pump) along thestreamline. It can be expanded to include thesesimply, by adding the appropriate energy terms:Totalenergy per unit weightat 1Totalenergy perunitweight at 22 WorkEnergyLossper unit done- suppliedper unitper unitweightweightweight2p1 V1p2 V2 z1 z2 h w q g 2 g g 2 gK. ALASTAL7
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGP1 V12P2 V22 z1 hpump z2 hturbine hL 1 g 2 g 2 g 2 gK. ALASTAL8
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGExample: (Ex 6.1, page 172 Textbook)K. ALASTAL9
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGExample: (Ex 6.1, page 172 Textbook)Calculate:a) the velocity of the jet issuing from the nozzle at C.b) the pressure in the suction pipe at the inlet to the pump.K. ALASTAL10
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGExample: The indicated crosssectional areas are A0 12 cm2 and A 0.35 cm2.The two levels areseparated by a verticaldistance h 45 mm. What is the volume flowrate from the tap ?K. ALASTAL11
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUG6.5 Representation of Energy Changes ina Fluid System (HGL and EGL): It is often convenient to plot mechanical energygraphically using heights. Hydraulic Grade LinePHGL z g Energy Grade Line (ortotal energy)P V2EGL z g 2gK. ALASTAL12
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGHydraulic Gradient Line (H.G.L.): It is the line that joins all the points to which water wouldrise if piezometric tubes were inserted. or it is the line that connects the piezometric heads at allpoints ( p/g z )Energy Gradient Line (E.G.L.): K. ALASTALIt is the line that joins all the pointsthat represent the sum of kinetichead and piezometric head (V2/2gabove the HGL).13
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGNote: The E.G.L. (total energy) falls due to friction losses(hL). This loss can be, also, caused by any variations in thecross-section of the pipe such as enlargement,contraction, or because the presence of entrances orvalves and so on .K. ALASTAL14
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGExample: (page 180 Textbook)K. ALASTAL15
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGExample: (page 181 Textbook)K. ALASTAL16
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGApplications of Bernoulli’sEquationK. ALASTAL17
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUG6.6 The Pitot Tube: The Pitot tube is used to measure thevelocity of a stream. It consists of a simple L-shaped tube facinginto the incoming flow.Simple Pitot Tube If the velocity of the stream at A is u,a particle moving from A to themouth of the tube B will be boughtto rest so that u0 at B is zero.A point in a fluid stream where the velocity isreduced to zero is known as a stagnation point.( Points B and 2 )K. ALASTAL18
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUG Apply Bernoulli’s equation between points A and B :Total head at A Total head at B2p0 u0pu2 g 2 g g 2 gp0pu2 g g 2 gp z gandThus, p0 will be greater than pp z h gp0p0 pu2p h2 g g g gVelocity at AK. ALASTALu 2 gh19
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGHow we can use the Pitot tube in the pipe?:Method 1 Two piezometers, one as normal and one as a Pitot tubewithin the pipe can be used in an arrangement shown belowto measure velocity of flow in pipes.22p1 u1pu z1 2 2 z2 g 2 g g 2 g gh1 u12 gh2 g2g gV u1 2 g h2 h1 K. ALASTAL20
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGHow we can use the Pitot tube in the pipe?:Method 2 Using a static pressure taping in the pipe wall with adifferential pressure gauge to measure the difference betweenthe static pressure and the pressure at the impact holeV ? (HW)K. ALASTAL21
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGHow we can use the Pitot tube in the pipe?:Method 3 Using combined Pitot static tube. In which the inner tube isused to measure the impact pressure while the outer sheathhas holes in its surface to measure the static pressureV ? (HW)K. ALASTAL22
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGNote: Theoretically:u 2 gh However, Pitot tubes may require calibration The true velocity is given by:V C 2 gh Where C is the coefficient of the instrumentFor example: C 1 for Pitot static tube (when Re 3000)K. ALASTAL23
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGDisadvantages : The Pitot/Pitot-static tubes give velocities at pointsin the flow. It does not give the overall discharge ofthe stream, which is often what is wanted. It also has the drawback that it is liable to blockeasily, particularly if there is significant debris in theflow.K. ALASTAL24
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUG6.9 Changes of Pressure in a tapering pipe: Changes of velocity in a tapering pipe were determinedby using the continuity equation. Changes of velocity will accompanied by a changed inpressure, modified by any changed in elevation orenergy loss, which can be determined by the use ofBernoulli’s equation.K. ALASTAL25
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGExample: (Ex 6.2, page 185 Textbook)Find: the pressuredifference across the2m length ignoringany losses of energy. the difference in levelthat would be shownon a mercurymanometerconnected across thislength.K. ALASTAL26
IUGFLUID MECHANICS, IUGSolution: From continuity equation : V2 8m/s Applying Bernoulli’s equation between section 1 and 2:(Ignoring losses)22p1 V1pV z1 2 2 z2 g 2 g g 2 g p1 V12 p2 V2 2 g z1 g z2 g 2 g g 2 g 11p1 V12 gz1 p2 V22 gz2221p1 p2 oil V22 V12 g z2 z1 2 Substituting with V1, V2, and observing that z2-z1 2sin45 1.41mp1 p2 39484 N/m2K. ALASTAL27
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUG For the manometer: The pressure at level XX is the same ineach limbp1 oil gz1 p2 oil g ( z2 h) man gh oil p1 p2 h z1 z2 man oil oil g Substituting with p1, p2, andobserving that z2-z1 2sin45 1.41mh 0.217mK. ALASTAL28
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUG6.10 Principle of the Venturi Meter: The Venturi meter is a device formeasuring discharge in a pipe. It consists of a rapidly convergingsection, which increases the velocity offlow and hence reduces the pressure. It then returns to the originaldimensions of the pipe by a gentlydiverging ‘diffuser’ section. By measuring the pressure differencesthe discharge can be calculated. This is a particularly accurate methodof flow measurement as energy lossesare very small.K. ALASTAL29
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONS Applying Bernoulli equationbetween sections 1 and 2, wehave: (assuming no losses)FLUID MECHANICS, IUG22p1 V1pV z1 2 2 z2 g 2 g g 2 g p p2 V22 V12 2 g 1 ( z1 z 2 ) g From continuity equation:A1V1 A2V2 A 2 p p2 V12 1 1 2 g 1 ( z1 z 2 ) A2 g V1 A1 V2 V1 A2 p1 p2 2g ( z1 z 2 ) g A2A12 A22 Volume flow rate (Q):Q A1V1 K. ALASTALA1 A2A12 A22 p p2 2g 1 ( z1 z 2 ) g 30
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSQ A1V1 A1A1 A2A12 A22orQ Where:p p2H 1 ( z1 z 2 ) gm 122 gHFLUID MECHANICS, IUG p p2 2g 1 ( z1 z 2 ) g This is the theoretical dischargeA1and m A2 The value of H can also be expressed in terms of themanometer readingsp1 g ( z1 z) p2 g ( z2 z h) man gh man p1 p2H ( z1 z2 ) h 1 g Q K. ALASTAL man 2 gh 1 2m 1 A1This is also the theoreticaldischarge in terms ofmanometer readings31
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUG In practice, some losses of energy between section 1 and 2occurs. Therefore, we include a coefficient of discharge to get theactual dischargeQactual Cd QtheoriticalK. ALASTAL32
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGExample: (Ex 6.3, page 189 Textbook)A venturi meter having a throat diameter d2 of 100mm isfitted into a pipeline which has a diameter d1 of 250mmthrough which oil of specific gravity 0.9 is flowing.The pressure differencebetween the entry and throattapings is measured by U-tubemanometer, containingmercury of specific gravity13.6.If the difference of level ofmanometer is 0.63m, calculatethe theoretical dischargeK. ALASTAL33
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUG6.11 Pipe Orifices: A similar effect as the venturimeter can be achieved byinserting an orifice plate The orifice plate has an openingin it smaller than the internalpipe diameterQ A1m 12Where:p p2H 1 ( z1 z 2 ) gandQactual Cd Qtheoritical2 gHor man H h 1 For Sharp-edged orifice Cd 0.65K. ALASTAL34
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUG6.11 Theory of Small Orifices Discharging toAtmosphere An orifice is an opening in theside or base of a tank orreservoir through which fluid isdischarge in the form of a jet. The discharge will depend uponthe head of the fluid (H) abovethe level of the orifice.H The term small orifice means that the diameter of the orifice issmall compared with the head producing flow (it can beassumed that the head does not vary appreciably from point topoint across the orifice).K. ALASTAL35
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUG Applying Bernoulli equationbetween sections A and B, wehave: (assuming no losses)22pA vApv z A B B zB g 2 g g 2 gVelocity of jet v 2 gHThis result is known asTorricelli's Theorem. Theoretically, if A is the cross sectional area of the orifice,then:Discharge QTheoritical A 2 gH The actual discharge, is given by:QActual Cd QTheoritical Cd A 2 gH Where: Cd is the coefficient of dischargeK. ALASTAL36
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUG Two reasons for the difference between theoretical andactual discharges. FIRST: the velocity of jet is less than the velocity calculatedbecause there is losses of energy between A and B.Actual velocity at B Cv v Cv 2 gH Where Cv is the coefficient of velocity SECOND: The streamlines at the orifice contract reducing thearea of flow. (This contraction is called the vena contracta.)Actual area of jet at B Cc A Where Cc is the coefficient of contractionK. ALASTAL37
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGActual discharge Actual area at B Actual velocity at B Cc A Cv 2 gH Cc Cv A 2 gHNote that: Cd Cc CvThese values are determined experimentally, where:Actual measured dischargeCd Theoretical dischargeArea of jet at vena contractaCc Area of orificeCv K. ALASTALVelocity at vena contractaTheoretical velocity38
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGExample: (Ex 6.4, page 192 Textbook) A jet of water discharge horizontally into the atmosphere froman orifice as shown. Drive an expression for the actual velocity vof a jet at the vena contracta if the jet falls a distance y verticallyin a horizontal distance x, measured from the vena contracta. Ifthe head of water above the orifice is H, determine thecoefficient of velocity. If the orifice has an area of 650 mm2and the jet falls a distance y 0.5min a horizontal distance x 1.5m. Calculate Cc , Cv ,Cd. Given that thevolume flow rate of flow is0.117m3/min and the head H abovethe orifice is 1.2mK. ALASTAL39
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUG6.14 Theory of Large Orifices : It is an orifice with large vertical height. So that the head producing flow is substantially less at the topof the opening than at the bottom (and so do the velocity) The method adopted is to calculate the flow through a thinhorizontal strip and then integrate from top to bottom toobtain the theoretical dischargeK. ALASTAL40
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGExample: (Ex 6.5, page 194 Textbook) 1.2.3.K. ALASTALA reservoir discharges through a rectangular sluice gate ofwidth B and height D. the top and bottom of the opening areat depths H1 and H2 below the free surface.Derive an expression for the theoretical discharge through theopening.If H1 0.4m and B 0.7m and D 1.5m, find Qtheoretical.What would be the percentage of error if the opening treatedas a small orifice41
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGDerivation : Consider a horizontal strip of height dh at a depth hbelow the free surfaceArea of strip B dhVelocity through the strip 2 ghDischarge through the strip dQ V dA B 2 gh dhQ B 2g H h1/ 2 dhH21Q K. ALASTAL 2B 2g H23 / 2 H13 / 2 342
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUG6.15 Elementary Theory of Notches & Weirs: A notch is an opening in the side of a tank or reservoir whichextends above the surface of the liquid. (Large orifice with noupper edge) It is usually a device for measuring discharge. A weir is a notch on a larger scale - usually found in rivers. It is used as both a flow measuring device and a device toraise water levels.K. ALASTAL43
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGA General Weir Equation (As in large orifice) To determine an expression for the theoretical flow through anotch we will consider a horizontal strip of width b and depthh below the free surface, as shown:Area of strip b dhVelocity t hrough the strip 2 ghDischarge through the strip dQ AV B 2 gh dhHQtheoritical 2 g bh1/ 2 dh0 Before the integration of the above equation, b must beexpressed in terms of hK. ALASTAL44
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGRectangular Notch For a rectangular notch the width does not change withdepth so there is no relationship between b and depth h.We have the equation Put b constant BHQtheoritical 2 g bh1/ 2 dh0HQtheoritical B 2 g h1/ 2 dh0Qtheoritical 2B 2g H 3/ 23QActual Cd QTheoritica lK. ALASTAL45
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGVee Notch For the “V” notch the relationship between width anddepth is dependent on the angle of the V notch (q ). Put b 2 (H-h) tan(q/2)HQtheoritical 2 g bh1/ 2 dh0Qtheoritical q H 2 2 g tan ( H h)h1/ 2 dh 2 0Qtheoritical 8 q 2 g tan H 5 / 215 2 QActual Cd QTheoriticalK. ALASTAL46
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGExample: (Ex 6.6, page 196 Textbook) It is proposed to use a notch to measure the flow of water froma reservoir and it is estimated that the error in measuring thehead above the bottom of the notch could be 1.5mm. For a discharge of 0.28m3/s, determine the percentage errorwhich may occur, using right triangular notch with a coefficientof discharge of 0.6K. ALASTAL47
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUG6.16 The Power of a Stream of Fluid : A stream of fluid can do work as a result of its pressure p,velocity v and elevation z. Remember that the total energy per unit weight H of the fluidis given by:p V2Energy per unit weigh t z g 2 g The power of the stream can be calculated as:Power Energy per unit TimeWeightEnergyPower Unit time unit weigh t p V2 Power gQH gQ z g 2 g K. ALASTAL48
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGExample: (Ex 6.8, page 199 Textbook)Water is drawn from a reservoir, in which the water level is240m above datum, at rate of 0.13m3/s the outlet of thepipeline is at datum level and is fitted with a nozzle to producea high speed jet to drive a turbine of Pelton wheel type. If thevelocity of jet is 66m/s, calculate:1. The power of the jet.2. The power supplied from the reservoir3. The head used to overcome losses.4. The efficiency of the pipeline and nozzle in transmittedpower.K. ALASTAL49
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGExample:Point 1 Given: Velocity in outlet pipe fromreservoir is 6 m/s and h 15 m.Find: Pressure at A.Solution: Bernoulli equation V12 p AV A2 z1 zA g2gg2gp10pAV A2 h 0 g2g g2g0Point AV A218p A g ( h ) 9810(15 )2g9.81p A 129.2 kPaK. ALASTAL50
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGExample: Given: D 30 in, d 1 in, h 4 ftFind: VA Solution: Bernoulli equationV12 p AV A2 z1 zA g2gg2gp1Point 1Point A0 0V A2 h 0 g2g g2g0V A 2 gh 16 ft / sK. ALASTAL51
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGExample:K. ALASTAL52
CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONSFLUID MECHANICS, IUGExample: Given: Velocity in circular duct 100 ft/s, air density 0.075lbm/ft3.Find: Pressure changebetween circular and squaresection.Solution: Continuity equationVc Ac Vs As 100( D 2 ) Vs D 24Air conditioning ( 60 oF) Vs 100
Derive the Bernoulli (energy) equation. Demonstrate practical uses of the Bernoulli and continuity equation in the analysis of flow. Understand the use of hydraulic and energy grade lines. Apply Bernoulli Equation to solve fluid mechanics problems (e.g. flow measurement). K. ALASTAL 2 CHAPTER 6: ENERGY EQUATION AND ITS APPLICATIONS FLUID MECHANICS, IUG
May 02, 2018 · D. Program Evaluation ͟The organization has provided a description of the framework for how each program will be evaluated. The framework should include all the elements below: ͟The evaluation methods are cost-effective for the organization ͟Quantitative and qualitative data is being collected (at Basics tier, data collection must have begun)
Silat is a combative art of self-defense and survival rooted from Matay archipelago. It was traced at thé early of Langkasuka Kingdom (2nd century CE) till thé reign of Melaka (Malaysia) Sultanate era (13th century). Silat has now evolved to become part of social culture and tradition with thé appearance of a fine physical and spiritual .
On an exceptional basis, Member States may request UNESCO to provide thé candidates with access to thé platform so they can complète thé form by themselves. Thèse requests must be addressed to esd rize unesco. or by 15 A ril 2021 UNESCO will provide thé nomineewith accessto thé platform via their émail address.
̶The leading indicator of employee engagement is based on the quality of the relationship between employee and supervisor Empower your managers! ̶Help them understand the impact on the organization ̶Share important changes, plan options, tasks, and deadlines ̶Provide key messages and talking points ̶Prepare them to answer employee questions
Dr. Sunita Bharatwal** Dr. Pawan Garga*** Abstract Customer satisfaction is derived from thè functionalities and values, a product or Service can provide. The current study aims to segregate thè dimensions of ordine Service quality and gather insights on its impact on web shopping. The trends of purchases have
Chính Văn.- Còn đức Thế tôn thì tuệ giác cực kỳ trong sạch 8: hiện hành bất nhị 9, đạt đến vô tướng 10, đứng vào chỗ đứng của các đức Thế tôn 11, thể hiện tính bình đẳng của các Ngài, đến chỗ không còn chướng ngại 12, giáo pháp không thể khuynh đảo, tâm thức không bị cản trở, cái được
Le genou de Lucy. Odile Jacob. 1999. Coppens Y. Pré-textes. L’homme préhistorique en morceaux. Eds Odile Jacob. 2011. Costentin J., Delaveau P. Café, thé, chocolat, les bons effets sur le cerveau et pour le corps. Editions Odile Jacob. 2010. Crawford M., Marsh D. The driving force : food in human evolution and the future.
Le genou de Lucy. Odile Jacob. 1999. Coppens Y. Pré-textes. L’homme préhistorique en morceaux. Eds Odile Jacob. 2011. Costentin J., Delaveau P. Café, thé, chocolat, les bons effets sur le cerveau et pour le corps. Editions Odile Jacob. 2010. 3 Crawford M., Marsh D. The driving force : food in human evolution and the future.
