# Chapter 5: Mass, Bernoulli, and Energy Equations

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1 Chapter 5: Mass, Bernoulli, and Energy Equations

2 Introduction This chapter deals with 3 equations commonly used in fluid mechanics The mass equation is an expression of the conservation of mass principle. The Bernoulli equation is concerned with the conservation of kinetic, potential, and flow energies of a fluid stream and their conversion to each other. The energy equation is a statement of the conservation of energy principle. Meccanica dei Fluidi I (ME) 2

3 Objectives After completing this chapter, you should be able to Apply the mass equation to balance the incoming and outgoing flow rates in a flow system. Recognize various forms of mechanical energy, and work with energy conversion efficiencies. Understand the use and limitations of the Bernoulli equation, and apply it to solve a variety of fluid flow problems. Work with the energy equation expressed in terms of heads, and use it to determine turbine power output and pumping power requirements. Meccanica dei Fluidi I (ME) 3

4 Conservation of Mass Conservation of mass principle is one of the most fundamental principles in nature. Mass, like energy, is a conserved property, and it cannot be created or destroyed during a process. For closed systems mass conservation is implicit since the mass of the system remains constant during a process. For control volumes, mass can cross the boundaries which means that we must keep track of the amount of mass entering and leaving the control volume. Meccanica dei Fluidi I (ME) 4

5 Mass and Volume Flow Rates The amount of mass flowing through a control surface per unit time is called the mass flow rate and is denoted m& The dot over a symbol is used to indicate time rate of change. Flow rate across the entire crosssectional area of a pipe or duct is obtained by integration m& = δm= ρvnda A c While this expression for is exact, it is not always convenient for engineering analyses. A c m& c Meccanica dei Fluidi I (ME) 5

6 Average Velocity and Volume Flow Rate m& Integral in can be replaced with average values of ρ and V n 1 Vavg = VndAc A c A c For many flows, variation of ρ is very small: m& = ρv A avg Volume flow rate is given by V& = V da = V A = VA A c n c avg c c Note: many textbooks use Q to denote volume flow rate. Mass and volume flow rates are related by m& = ρv& c (use V to denote avg velocity) Meccanica dei Fluidi I (ME) 6

7 Conservation of Mass Principle The conservation of mass principle can be expressed as m& in m& = out m& dm dt CV m& Where in and out are the total rates of mass flow into and out of the CV, and dm CV /dt is the rate of change of mass within the CV. (kg/s) Meccanica dei Fluidi I (ME) 7

8 Conservation of Mass Principle For CV of arbitrary shape rate of change of mass within the CV dmcv d = dt dt ρ dv CV net mass flow rate r r m& = δm& = ρv da= ρ V. n da net. m net > 0 n CS CS CS net outflow ( ) Therefore, general conservation of mass for a fixed CV is: d r ( r ρ dv + ρ V. n) da = 0 dt CV CS (see also RTT with B = m and b = 1) Meccanica dei Fluidi I (ME) 8

9 Steady Flow Processes For steady flow, the total amount of mass contained in CV is constant: m CV = constant. Total amount of mass entering must be equal to total amount of mass leaving m& = m& in out For incompressible flows, in VA = VA n n n n out Meccanica dei Fluidi I (ME) 9

10 Mechanical Energy Mechanical energy can be defined as the form of energy that can be converted to mechanical work completely and directly by an ideal mechanical device such as an ideal turbine. Flow P/ρ, kinetic V 2 /2, and potential gz energy are forms of mech. energy (per unit mass): e mech = P/ρ + V 2 /2 + gz Mechanical energy per unit mass change of a fluid during incompressible flow becomes 2 2 P2 P1 V2 V1 emech = + + g z z ρ 2 ( ) 2 1 In the absence of losses, e mech represents the work supplied to the fluid ( e mech > 0) or extracted from the fluid ( e mech < 0). Meccanica dei Fluidi I (ME) 10

11 Mechanical Energy 2 2 P2 P1 V2 V1 emech = + + g z z ρ 2 ( ) 2 1 If no changes in flow velocity or elevation, the power produced by an ideal hydraulic turbine is proportional to the pressure drop of water across the turbine. Meccanica dei Fluidi I (ME) 11 An ideal hydraulic turbine produces the same work per unit mass w turbine = gh whether it receives water from the top or from the bottom of the container (in the absence of irreversible losses).

12 Francis Hydraulic Turbine Distributor, Francis runner and shaft The choice of Francis, Kaplan or Pelton runners is based on the water mass flow rate and on the available head. Meccanica dei Fluidi I (ME) 12

13 Efficiency Transfer of e mech is usually accomplished by a rotating shaft: shaft work Pump, fan, compressor: receives shaft work (e.g., from an electric motor) and transfers it to the fluid as mechanical energy (less frictional losses) fluid pressure is raised Turbine: converts e mech of a fluid to shaft work. In the absence of irreversibilities (e.g. friction), the mechanical efficiency of a device or process can be defined as E E η mech = = 1 E E mech, out mech, loss mech, in mech, in If η mech < 100% losses have occurred during conversion. Meccanica dei Fluidi I (ME) 13

14 Pump and Turbine Efficiencies In fluid systems, we are usually interested in increasing/decreasing the pressure, velocity, and/or elevation of a fluid: pumps or turbines. In these cases, efficiency is better defined as the ratio of supplied or extracted work vs. rate of increase in mechanical energy η η pump turbine = E& W& W& = E& mech, fluid shaft, in shaft, out mech, fluid Overall efficiency must include motor or generator efficiency, ratio of power output to input (shaft power and electric power). = rate of increase in the mechanical energy of the fluid mechanical energy input Meccanica dei Fluidi I (ME) 14

15 Turbine and Generator Efficiencies Example: Determine: (a) overall efficiency of the turbine-generator, (b) mechanical efficiency of the turbine, (c) shaft power supplied by the turbine to the generator, and (d) the irreversible losses through each component. Meccanica dei Fluidi I (ME) 15

16 General Energy Equation One of the most fundamental laws in nature is the 1st law of thermodynamics, which is also known as the conservation of energy principle: energy can be neither created nor destroyed during a process; it can only change forms Falling rock, picks up speed as PE is converted to KE. If air resistance is neglected, PE + KE = constant Meccanica dei Fluidi I (ME) 16

17 General Energy Equation The energy content of a fixed quantity of mass (closed system) can be changed by two mechanisms: heat transfer Q and work transfer W. Conservation of energy for a closed system can be expressed in rate form as Q& + W& = net, in net, in de dt sys Net rate of heat transfer to the system (> 0 if the system is heated): Q& = Q& Q& net, in in out Net power input to the system (> 0 if work is done on the system): W& = W& W& net, in in out Meccanica dei Fluidi I (ME) 17

18 General Energy Equation Recall general RTT db dt sys d r ( r = ρbdv b + ρb V. ) CV CS r n da dt Derive energy equation using B=E and b=e de dt sys d r Q& ( ) net, in W& r net, in ρedv ρe V. CV CS r n da dt = + = + Break power into rate of shaft work and pressure work (neglect viscous work, and that done by other forces such as electric, magnetic and surface tension) r W& ( ) net, in = W& shaft, net, in + W& pressure, net, in = W& r shaft, net, in P V. n da A Meccanica dei Fluidi I (ME) 18

19 General Energy Equation Where does expression for pressure work come from? When piston moves down by ds under the influence of F=PA, the work done on the system is δw boundary =PAds. If we divide both sides by dt, we have ds δw& pressure = δw& boundary = PA = PAVpiston dt For generalized system deforming under the influence of pressure: r δ W& r = PdAV = PdA V n pressure ( ) Note sign conventions: n r is outward pointing normal negative sign ensures that work done is positive when is done on the system. n Meccanica dei Fluidi I (ME) 19

20 General Energy Equation Moving integral for rate of pressure work to RHS of energy equation results in:.. d P r Q W ρedv e e( r + = & + + ρ V n) da net, in shaft, net, in r dt ρ CV CS Recall that P/ρ is the flow work, which is the work associated with pushing a fluid into or out of a CV per unit mass. For fixed CV pressure work can exist only where the fluid enters and leaves the CV. Meccanica dei Fluidi I (ME) 20

21 General Energy Equation As with the mass equation, practical analysis is often facilitated when P/ρ + e is averaged across inlets and outlets of a fixed CV: d P P Qnet, in + Wshaft, net, in = edv m e m e dt ρe ρ & + & + + CV out ρ & in ρ. r r m= ρ V n da.. A C ( ) c Since e=u + ke + pe = u + V 2 /2 + gz d P V P V Qnet, in + Wshaft, net, in = edv m u gz m u gz dt ρ e & + & & out ρ 2 in ρ 2 CV Meccanica dei Fluidi I (ME) 21

22 Energy Analysis of Steady Flows V V Qnet, in + Wshaft, net, in = m& h+ + gz m& h+ + gz out 2 in 2 For steady flow, time rate of change of the energy content of the CV is zero. This equation states: the net rate of energy transfer to a CV by heat and work during steady flow is equal to the difference between the rates of outgoing and incoming energy flows with mass. Meccanica dei Fluidi I (ME) 22

23 Energy Analysis of Steady Flows For single-stream devices, mass flow rate is constant. The steady flow energy equation per unit mass reads: V V q w h h g z z 2 P V P V w gz gz u u q net, in + shaft, net, in = ( ) shaft, net, in = , ρ1 2 ρ2 2 P V P V ρ ( net in ) gz1+ wpump = + + gz2 + wturbine + emech, loss 1 2 ρ2 2 since e mech,loss = u 2 u 1 q net in, and w shaft,net,in = w shaft,in w shaft,out = w pump w turbine Meccanica dei Fluidi I (ME) 23

24 Energy Analysis of Steady Flows Divide by g to get each term in units of length P V P V ρ z1+ hpump = + + z2 + hturbine + hl 1g 2g ρ2g 2g Magnitude of each term is now expressed as an equivalent column height of fluid, i.e., Head Meccanica dei Fluidi I (ME) 24

25 The Bernoulli Equation If we neglect piping losses, and have a system without pumps or turbines P V P V + + = + + = C ρ z1 z2 1g 2g ρ2g 2g This is Bernoulli s equation It can be derived in a formal way by using Newton's second law of motion (see text, p. 187). The formal derivation holds that in steady flow the sum of the kinetic, potential, and flow energies of a fluid particle is constant along a streamline, when compressibility and frictional effects can be neglected. Also: total pressure along a streamline is constant. 3 terms above correspond to static, dynamic, and hydrostatic head (pressure). Meccanica dei Fluidi I (ME) 25

26 The Pitot Tube The sum of static and dynamic pressure is called the stagnation pressure, i.e. P stag = P + ρ V 2 /2. It represents the pressure at a point where the fluid is brought to a complete stop isentropically. Meccanica dei Fluidi I (ME) 26

27 The Bernoulli Equation stagnation streamline The Bernoulli equation is an approximate relation between pressure, velocity, and elevation and is valid in regions of steady, incompressible flow where net frictional forces are negligible. Equation is useful in flow regions outside of boundary layers and wakes. Meccanica dei Fluidi I (ME) 27

28 The Bernoulli Equation Limitations on the use of the Bernoulli equation: P + ρ V 2 /2 + ρ g z = constant (along a streamline) Steady flow: d/dt = 0 Frictionless flow No shaft work: w pump = w turbine = 0 Incompressible flow: ρ = constant No heat transfer: q net,in = 0 Applied along a streamline (except when the flow is irrotational, in which case the constant C is the same throughout the flow field, Chapt. 10) Meccanica dei Fluidi I (ME) 28

29 HGL and EGL Frictional losses are present. Meccanica dei Fluidi I (ME) 29 It is often convenient to plot mechanical energy graphically using heights. Hydraulic Grade Line P HGL = + z ρg Energy Grade Line (or total energy) 2 P V EGL = + + z ρg 2g

30 HGL and EGL Steep jump (or drop) in EGL and HGL appear whenever mechanical energy is added to (or removed from) the fluid by a pump (or a turbine). Meccanica dei Fluidi I (ME) 30

31 Examples and Applications Bernoulli equation Flow irrotational and incompressible; flow quasi-steady (water drains slowly); no frictional losses nor flow disturbances. Exit velocity is given by V 2 = (2 g z 1 ) 1/2 (Torricelli equation) Meccanica dei Fluidi I (ME) 31

32 Examples and Applications Bernoulli equation Siphon diameter = 4 mm; ρ gasoline = 750 kg/m 3 ; no frictional losses. Determine the time to siphon 4L of gasoline, and the pressure at point 3 (careful of cavitation at point 3!). Meccanica dei Fluidi I (ME) 32

33 Examples and Applications Bernoulli equation When compressibility effects are important, the Bernoulli equation takes the form: d P r Q + W = ρedv& e e( r + + V n) da net, in shaft, net, in r dt ρ CV CS dp/ρ + V 2 /2 + gz = constant (along a streamline) Simplify the relation above for (a) the isothermal expansion or compression of an ideal gas, and (b) the isentropic flow of an ideal gas. Meccanica dei Fluidi I (ME) 33

34 Examples and Applications Energy analysis of steady flows 2 2 P2 P1 V2 V1 emech = + + g z z ρ 2.. ( ) P2 P1 V2 V1 emech = + + g z z ρ 2 ( ) E mech,fluid = m [ ] 2 1 Determine: (a) the mechanical efficiency of the pump, and (b) the temperature rise of water as it flows through the pump, because of mechanical inefficiency. The specific heat of water is 4.18 kj/(kg C) Meccanica dei Fluidi I (ME) 34

35 Examples and Applications V V Energy analysis of steady flows q w h h g z z 2 P V P V w gz gz u u q net, in + shaft, net, in = ( ) shaft, net, in = , ρ1 2 ρ2 2 P V P V ρ ( net in ) gz1+ wpump = + + gz2 + wturbine + emech, loss 1 2 ρ2 2 Determine the mechanical power lost (and the head loss h L ) while pumping water to the upper reservoir. Meccanica dei Fluidi I (ME) 35

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