# CLASS Fourth Units (Second part)

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1 CLASS Fourth Units (Second part)

2 Energy analysis of closed systems Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

3 MOVING BOUNDARY WORK Moving boundary work (P dv work): The expansion and compression work in a piston-cylinder device. Quasi-equilibrium process: A process during which the system remains nearly in equilibrium at all times. W b is positive for expansion W b is negative for compression The work associated with a moving boundary is called boundary work. A gas does a differential amount of work W b as it forces the piston to move by a differential amount ds. 3

4 The boundary work done during a process depends on the path followed as well as the end states. The area under the process curve on a P-V diagram represents the boundary work. The net work done during a cycle is the difference between the work done by the system and the work done on the system. 4

5 Boundary work for a constant-volume process

6 Boundary work for a constant-pressure process

7 Polytropic, Isothermal, and Isobaric processes Polytropic process: C, n (polytropic exponent) constants Polytropic process Polytropic and for ideal gas When n = 1 (isothermal process) Constant pressure process What is the boundary work for a constantvolume process? Schematic and P-V diagram for a polytropic process. 7

8 Boundary work for a constant-pressure process

9 ENERGY BALANCE FOR CLOSED SYSTEMS Energy balance for any system undergoing any process Energy balance in the rate form The total quantities are related to the quantities per unit time is Energy balance per unit mass basis Energy balance for a cycle Energy balance in differential form 9

10 Energy balance when sign convention is used (i.e., heat input and work output are positive; heat output and work input are negative). For a cycle E = 0, thus Q = W. Various forms of the first-law relation for closed systems when sign convention is used. The first law cannot be proven mathematically, but no process in nature is known to have violated the first law, and this should be taken as sufficient proof. 10

11 Energy balance for a constant-pressure expansion or compression process General analysis for a closed system undergoing a quasi-equilibrium constant-pressure process. Q is to the system and W is from the system. For a constant-pressure expansion or compression process: U W H b An example of constant-pressure process 11

12 SPECIFIC HEATS Specific heat at constant volume, c v : The energy required to raise the temperature of the unit mass of a substance by one degree as the volume is maintained constant. Specific heat at constant pressure, c p : The energy required to raise the temperature of the unit mass of a substance by one degree as the pressure is maintained constant. Specific heat is the energy required to raise the temperature of a unit mass of a substance by one degree in a specified way. Constantvolume and constantpressure specific heats c v and c p (values are for helium gas). 12

13 The equations in the figure are valid for any substance undergoing any process. c v and c p are properties. c v is related to the changes in internal energy and c p to the changes in enthalpy. A common unit for specific heats is kj/kg C or kj/kg K. Are these units identical? Formal definitions of c v and c p. The specific heat of a substance changes with temperature. True or False? c p is always greater than c v. 13

14 INTERNAL ENERGY, ENTHALPY, AND SPECIFIC HEATS OF IDEAL GASES Joule showed using this experimental apparatus that u=u(t) For ideal gases, u, h, c v, and c p vary with temperature only. Internal energy and enthalpy change of an ideal gas 14

15 At low pressures, all real gases approach ideal-gas behavior, and therefore their specific heats depend on temperature only. The specific heats of real gases at low pressures are called ideal-gas specific heats, or zero-pressure specific heats, and are often denoted c p0 and c v0. u and h data for a number of gases have been tabulated. These tables are obtained by choosing an arbitrary reference point and performing the integrations by treating state 1 as the reference state. Ideal-gas constantpressure specific heats for some gases (see Table A 2c for c p equations). In the preparation of ideal-gas tables, 0 K is chosen as the reference temperature. 15

16 Internal energy and enthalpy change when specific heat is taken constant at an average value (kj/kg) For small temperature intervals, the specific heats may be assumed to vary linearly with temperature. The relation u = c v T is valid for any kind of process, constantvolume or not. 16

17 Three ways of calculating u and h 1. By using the tabulated u and h data. This is the easiest and most accurate way when tables are readily available. 2. By using the c v or c p relations (Table A-2c) as a function of temperature and performing the integrations. This is very inconvenient for hand calculations but quite desirable for computerized calculations. The results obtained are very accurate. 3. By using average specific heats. This is very simple and certainly very convenient when property tables are not available. The results obtained are reasonably accurate if the temperature interval is not very large. Three ways of calculating u. 17

18 Specific Heat Relations of Ideal Gases The relationship between c p, c v and R dh = c p dt and du = c v dt On a molar basis Specific heat ratio The c p of an ideal gas can be determined from a knowledge of c v and R. The specific ratio varies with temperature, but this variation is very mild. For monatomic gases (helium, argon, etc.), its value is essentially constant at Many diatomic gases, including air, have a specific heat ratio of about 1.4 at room temperature. 18

19 INTERNAL ENERGY, ENTHALPY, AND SPECIFIC HEATS OF SOLIDS AND LIQUIDS Incompressible substance: A substance whose specific volume (or density) is constant. Solids and liquids are incompressible substances. The specific volumes of incompressible substances remain constant during a process. The c v and c p values of incompressible substances are identical and are denoted by c. 19

20 Internal Energy Changes Enthalpy Changes A more accurate relation than The enthalpy of a compressed liquid 20

21 Energy analysis of open systems Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

22 FLOW WORK AND THE ENERGY OF A FLOWING FLUID Flow work, or flow energy: The work (or energy) required to push the mass into or out of the control volume. This work is necessary for maintaining a continuous flow through a control volume. Schematic for flow work. In the absence of acceleration, the force applied on a fluid by a piston is equal to the force applied on the piston by the fluid. 22

23 Total Energy of a Flowing Fluid h = u + Pv The flow energy is automatically taken care of by enthalpy. In fact, this is the main reason for defining the property enthalpy. The total energy consists of three parts for a nonflowing fluid and four parts for a flowing fluid. 23

24 Energy Transport by Mass When the kinetic and potential energies of a fluid stream are negligible When the properties of the mass at each inlet or exit change with time as well as over the cross section The product m i i is the energy transported into control volume by mass per unit time. 24

25 ENERGY ANALYSIS OF STEADY-FLOW SYSTEMS 2 2 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 transfers during steady flow is equal to the difference between the rates of outgoing and incoming energy flows with mass.

26 ENERGY ANALYSIS OF STEADY-FLOW SYSTEMS Many engineering systems such as power plants operate under steady conditions. Under steady-flow conditions, the mass and energy contents of a control volume remain constant. Under steady-flow conditions, the fluid properties at an inlet or exit remain constant (do not change with time). 26

27 Mass and Energy balances for a steady-flow process Mass balance A water heater in steady operation. Energy balance 27

28 Energy balance relations with sign conventions (i.e., heat input and work output are positive) when kinetic and potential energy changes are negligible Some energy unit equivalents Under steady operation, shaft work and electrical work are the only forms of work a simple compressible system may involve. 28

29 SOME STEADY-FLOW ENGINEERING DEVICES Many engineering devices operate essentially under the same conditions for long periods of time. The components of a steam power plant (turbines, compressors, heat exchangers, and pumps), for example, operate nonstop for months before the system is shut down for maintenance. Therefore, these devices can be conveniently analyzed as steady-flow devices. A modern land-based gas turbine used for electric power production. This is a General Electric LM5000 turbine. It has a length of 6.2 m, it weighs 12.5 tons, and produces 55.2 MW at 3600 rpm with steam injection. At very high velocities, even small changes in velocities can cause significant changes in the kinetic energy of the fluid. 29

30 Nozzles and Diffusers Nozzles and diffusers are commonly utilized in jet engines, rockets, spacecraft, and even garden hoses. A nozzle is a device that increases the velocity of a fluid at the expense of pressure. A diffuser is a device that increases the pressure of a fluid by slowing it down. The cross-sectional area of a nozzle decreases in the flow direction for subsonic flows and increases for supersonic flows. The reverse is true for diffusers. Nozzles and diffusers are shaped so that they cause large changes in fluid velocities and thus kinetic energies. Energy balance for a nozzle or diffuser: 30

31 Turbines and Compressors Energy balance for the compressor in this figure: Turbine drives the electric generator In steam, gas, or hydroelectric power plants. As the fluid passes through the turbine, work is done against the blades, which are attached to the shaft. As a result, the shaft rotates, and the turbine produces work. Compressors, as well as pumps and fans, are devices used to increase the pressure of a fluid. Work is supplied to these devices from an external source through a rotating shaft. A fan increases the pressure of a gas slightly and is mainly used to mobilize a gas. A compressor is capable of compressing the gas to very high pressures. Pumps work very much like compressors except that they handle liquids instead of gases. 31

32 Throttling valves Throttling valves are any kind of flow-restricting devices that cause a significant pressure drop in the fluid. What is the difference between a turbine and a throttling valve? The pressure drop in the fluid is often accompanied by a large drop in temperature, and for that reason throttling devices are commonly used in refrigeration and airconditioning applications. Energy balance The temperature of an ideal gas does not change during a throttling (h = constant) process since h = h(t). During a throttling process, the enthalpy of a fluid remains constant. But internal and flow energies may be converted to each other. 32

33 Mixing chambers 60 C In engineering applications, the section where the mixing process takes place is commonly referred to as a mixing chamber. 140 kpa 10 C 43 C Energy balance for the adiabatic mixing chamber in the figure is: The T-elbow of an ordinary shower serves as the mixing chamber for the hot- and the cold-water streams. 33

34 Heat exchangers Heat exchangers are devices where two moving fluid streams exchange heat without mixing. Heat exchangers are widely used in various industries, and they come in various designs. The heat transfer associated with a heat exchanger may be zero or nonzero depending on how the control volume is selected. Mass and energy balances for the adiabatic heat exchanger in the figure is: A heat exchanger can be as simple as two concentric pipes. 34

35 Pipe and duct fow The transport of liquids or gases in pipes and ducts is of great importance in many engineering applications. Flow through a pipe or a duct usually satisfies the steady-flow conditions. Pipe or duct flow may involve more than one form of work at the same time. Heat losses from a hot fluid flowing through an uninsulated pipe or duct to the cooler environment may be very significant. Energy balance for the pipe flow shown in the figure is 35

36 ENERGY ANALYSIS OF STEADY-FLOW SYSTEMS For single-stream devices, mass flow rate is constant. 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 net, in P V P V gz1 wpump gz2 wturbine emech, loss

37 ENERGY ANALYSIS OF STEADY-FLOW SYSTEMS Divide by g to get each term in units of length P V P V z1 hpump z2 hturbine hl 1g 2g 2g 2g Magnitude of each term is now expressed as an equivalent column height of fluid, i.e., Head

38 The Bernoulli Equation If we neglect piping losses, and have a system without pumps or turbines P V P V z1 z2 1g 2g 2g 2g This is the Bernoulli equation It can also be derived using Newton's second law of motion (see text, p. 187). 3 terms correspond to: Static, dynamic, and hydrostatic head (or pressure).

39 HGL and EGL It is often convenient to plot mechanical energy graphically using heights. Hydraulic Grade Line HGL P g z Energy Grade Line (or total energy) 2 P V EGL z g 2g

40 The Bernoulli Equation 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.

41 The Bernoulli Equation Limitations on the use of the Bernoulli Equation Steady flow: d/dt = 0 Frictionless flow No shaft work: w pump =w turbine =0 Incompressible flow: = constant No heat transfer: q net,in =0 Applied along a streamline (except for irrotational flow)

42 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/, kinetic V 2 /g, and potential gz energy are the forms of mechanical energy e mech = P/ V 2 /g + gz Mechanical energy change of a fluid during incompressible flow becomes 2 2 P2 P1 V2 V1 e mech g z z In the absence of loses, e mech represents the work supplied to the fluid ( e mech >0) or extracted from the fluid ( e mech <0).

43 Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. example Consider a river flowing toward a lake at an average velocity of 3 m/s at a rate of 500 m 3 /s at a location 90 m above the lake surface. Determine the total mechanical energy of the river water per unit mass and the power generation potential of the entire river at that location. 3-6

44 Efficiency Transfer of e mech is usually accomplished by a rotating shaft: shaft work Pump, fan, propulsion: receives shaft work (e.g., from an electric motor) and transfers it to the fluid as mechanical energy Turbine: converts e mech of a fluid to shaft work. In the absence of irreversibilities (e.g., friction), mechanical efficiency of a device or process can be defined as E E hmech 1 E E mech, out mech, loss mech, in mech, in If h mech < 100%, losses have occurred during conversion.

45 Pump and Turbine Efficiencies In fluid systems, we are usually interested in increasing the pressure, velocity, and/or elevation of a fluid. In these cases, efficiency is better defined as the ratio of (supplied or extracted work) vs. rate of increase in mechanical energy h h pump turbine E W W E mech, fluid shaft, in shaft, out mech, fluid Overall efficiency must include motor or generator efficiency.

46 Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. In a hydroelectric power plant, 120 m 3 /s of water flows from an elevation of 100 m to a turbine, where electric power is generated. The overall efficiency of the turbine generator is 80 percent. Disregarding frictional losses in piping, estimate the electric power output of this plant. 3-6 Example:

47 Example: A garden hose attached with a nozzle is used to fill a 20-gal bucket. The inner diameter of the hose is 1 in and it reduces to 0.5 in at the nozzle exit. If the average velocity in the hose is 8 ft/s, determine: (a) the volume and mass flow rates of water through the hose (b) how long it will take to fill the bucket with water, and (c) the average velocity of water at the nozzle exit.

48 Example:

49 Example: Water flows through a horizontal pipe at a rate of 1 gal/s. The pipe consist of two sections of diameters 4 in and 2 in with a smooth reducing section. The pressure difference between the two pipe sections is measured by a mercury manometer. Neglecting frictional effects, determine the differential height of mercury between the two pipe sections.

50 Example: The liquid in the Figure has a s = 0,85. Estimate the flow rate from the tank for a) no losses and b) pipe losses h L = 4.5 V 2 /(2g).

51 Example: Air enters an adiabatic nozzle steadily at 300 kpa, 200 C, and 30 m/s and leaves at 100 kpa and 180 m/s. The inlet area of the nozzle is 80 cm2. Determine: (a) the mass flow rate through the nozzle, (b) the exit temperature of the air, and (c) the exit area of the nozzle

52 Example: A hot-water stream at 80 C enters a mixing chamber with a mass flow rate of 0.5 kg/s where it is mixed with a stream of cold water at 20 C. If it is desired that the mixture leave the chamber at 42 C, determine the mass flow rate of the coldwater stream. Assume all the streams are at a pressure of 250 kpa.

53 Example: An oil pump is drawing 35 kw of electric power while pumping oil with ρ=860 kg/m 3 at a rate of 0.15m 3 /s. The inlet and outlet diameters of the pipe are 8 cm and 12 cm, respectively. If the pressure rise of oil in the pump is measured to be 450kPa and the motor efficiency is 80%, determine the mechanical efficiency of the pump.

54 MOMENTUM ANALYSIS

55 NEWTON S LAWS Newton s laws: Relations between motions of bodies and the forces acting on them. Newton s first law: A body at rest remains at rest, and a body in motion remains in motion at the same velocity in a straight path when the net force acting on it is zero. Therefore, a body tends to preserve its state of inertia. Newton s second law: The acceleration of a body is proportional to the net force acting on it and is inversely proportional to its mass. Newton s third law: When a body exerts a force on a second body, the second body exerts an equal and opposite force on the first. Therefore, the direction of an exposed reaction force depends on the body taken as the system. 55

56 Linear momentum or just the momentum of the body: The product of the mass and the velocity of a body. Newton s second law is usually referred to as the linear momentum equation. Conservation of momentum principle: The momentum of a system remains constant only when the net force acting on it is zero. Linear momentum is the product of mass and velocity, and its direction is the direction of velocity. Newton s second law is also expressed as the rate of change of the momentum of a body is equal to the net force acting on it. 56

57 The conservation of angular momentum Principle: The total angular momentum of a rotating body remains constant when the net torque acting on it is zero, and thus the angular momentum of such systems is conserved. The rate of change of the angular momentum of a body is equal to the net torque acting on it. 57

58 CHOOSING A CONTROL VOLUME A control volume can be selected as any arbitrary region in space through which fluid flows, and its bounding control surface can be fixed, moving, and even deforming during flow. Many flow systems involve stationary hardware firmly fixed to a stationary surface, and such systems are best analyzed using fixed control volumes. When analyzing flow systems that are moving or deforming, it is usually more convenient to allow the control volume to move or deform. In deforming control volume, part of the control surface moves relative to other parts. 58

59 FORCES ACTING ON A CONTROL VOLUME The forces acting on a control volume consist of body forces that act throughout the entire body of the control volume (such as gravity, electric, and magnetic forces) and surface forces that act on the control surface (such as pressure and viscous forces and reaction forces at points of contact). Only external forces are considered in the analysis. The total force acting on a control volume is composed of body forces and surface forces; body force is shown on a differential volume element, and surface force is shown on a differential surface element. 59

60 The most common body force is that of gravity, which exerts a downward force on every differential element of the control volume. Surface forces are not as simple to analyze since they consist of both normal and tangential components. Normal stresses are composed of pressure (which always acts inwardly normal) and viscous stresses. Shear stresses are composed entirely of viscous stresses. The gravitational force acting on a differential volume element of fluid is equal to its weight; the axes have been rotated so that the gravity vector acts downward in the negative z-direction. 60

61 A common simplification in the application of Newton s laws of motion is to subtract the atmospheric pressure and work with gage pressures. This is because atmospheric pressure acts in all directions, and its effect cancels out in every direction. This means we can also ignore the pressure forces at outlet sections where the fluid is discharged to the atmosphere since the discharge pressure in such cases is very near atmospheric pressure at subsonic velocities. Atmospheric pressure acts in all directions, and thus it can be ignored when performing force balances since its effect cancels out in every direction. Cross section through a faucet assembly, illustrating the importance of choosing a control volume wisely; CV B is much easier to work with than CV A. 61

62 THE LINEAR MOMENTUM EQUATION Newton s second law can be stated as the sum of all external forces acting on a system is equal to the time rate of change of linear momentum of the system. This statement is valid for a coordinate system that is at rest or moves with a constant velocity, called an inertial coordinate system or inertial reference frame. 62

63 63

64 The momentum equation is commonly used to calculate the forces (usually on support systems or connectors) induced by the flow. 64

65 Steady flow Special Cases Mass flow rate across an inlet or outlet Momentum flow rate across a uniform inlet or outlet: In a typical engineering problem, the control volume may contain many inlets and outlets; at each inlet or outlet we define the mass flow rate and the average velocity. 65

66 Examples of inlets or outlets in which the uniform flow approximation is reasonable: (a) the well-rounded entrance to a pipe, (b) the entrance to a wind tunnel test section, and (c) a slice through a free water jet in air. 66

67 Momentum-Flux Correction Factor, The velocity across most inlets and outlets is not uniform. The control surface integral of Eq may be converted into algebraic form using a dimensionless correction factor, called the momentum-flux correction factor. (13-13) is always greater than or equal to 1. is close to 1 for turbulent flow and not very close to 1 for fully developed laminar flow. 67

68 Steady Flow The net force acting on the control volume during steady flow is equal to the difference between the rates of outgoing and incoming momentum flows. The net force acting on the control volume during steady flow is equal to the difference between the outgoing and the incoming momentum fluxes. 68

69 Steady Flow with One Inlet and One Outlet One inlet and one outlet Along x- coordinate A control volume with only one inlet and one outlet. The determination by vector addition of the reaction force on the support caused by a change of direction of water. 69

70 Flow with No External Forces In the absence of external forces, the rate of change of the momentum of a control volume is equal to the difference between the rates of incoming and outgoing momentum flow rates. The thrust needed to lift the space shuttle is generated by the rocket engines as a result of momentum change of the fuel as it is accelerated from about zero to an exit speed of about 2000 m/s after combustion. 70

71 Example:

72 Example:

73 Example:

74 Example:

75 REVIEW OF ROTATIONAL MOTION AND ANGULAR MOMENTUM Rotational motion: A motion during which all points in the body move in circles about the axis of rotation. Rotational motion is described with angular quantities such as the angular distance, angular velocity, and angular acceleration. Angular velocity: The angular distance traveled per unit time. Angular acceleration: The rate of change of angular velocity. The relations between angular distance, angular velocity, and linear velocity V. 75

76 Newton s second law requires that there must be a force acting in the tangential direction to cause angular acceleration. The strength of the rotating effect, called the moment or torque, is proportional to the magnitude of the force and its distance from the axis of rotation. The perpendicular distance from the axis of rotation to the line of action of the force is called the moment arm, and the torque M acting on a point mass m at a normal distance r from the axis of rotation is expressed as Torque I is the moment of inertia of the body about the axis of rotation, which is a measure of the inertia of a body against rotation. Unlike mass, the rotational inertia of a body also depends on the distribution of the mass of the body with respect to the axis of rotation. Analogy between corresponding linear and angular quantities. 76

77 Angular momentum Angular momentum equation Angular velocity versus rpm Angular momentum of point mass m rotating at angular velocity at distance r from the axis of rotation. The relations between angular velocity, rpm, and the power transmitted through a shaft. 77

78 Rotational kinetic energy Shaft power During rotational motion, the direction of velocity changes even when its magnitude remains constant. Velocity is a vector quantity, and thus a change in direction constitutes a change in velocity with time, and thus acceleration. This is called centripetal acceleration. Centripetal acceleration is directed toward the axis of rotation (opposite direction of radial acceleration), and thus the radial acceleration is negative. Centripetal acceleration is the result of a force acting on an element of the body toward the axis of rotation, known as the centripetal force, whose magnitude is F r = mv 2 /r. Tangential and radial accelerations are perpendicular to each other, and the total linear acceleration is determined by their vector sum: 78

79 THE ANGULAR MOMENTUM EQUATION Many engineering problems involve the moment of the linear momentum of flow streams, and the rotational effects caused by them. Such problems are best analyzed by the angular momentum equation, also called the moment of momentum equation. An important class of fluid devices, called turbomachines, which include centrifugal pumps, turbines, and fans, is analyzed by the angular momentum equation. A force whose line of action passes through point O produces zero moment about point O. The determination of the direction of the moment by the right-hand rule. 79

80 Moment of momentum Moment of momentum (system) Angular momentum equation for a system Rate of change of moment of momentum 80

81 Special Cases During steady flow, the amount of angular momentum within the control volume remains constant, and thus the time rate of change of angular momentum of the contents of the control volume is zero. An approximate form of the angular momentum equation in terms of average properties at inlets and outlets: The net torque acting on the control volume during steady flow is equal to the difference between the outgoing and incoming angular momentum flow rates. scalar form of angular momentum equation 81

82 Radial-Flow Devices Radial-flow devices: Many rotary-flow devices such as centrifugal pumps and fans involve flow in the radial direction normal to the axis of rotation. Axial-flow devices are easily analyzed using the linear momentum equation. Radial-flow devices involve large changes in angular momentum of the fluid and are best analyzed with the help of the angular momentum equation. Side and frontal views of a typical centrifugal pump. 82

83 The conservation of mass equation for steady incompressible flow angular momentum equation Euler s turbine formula When An annular control volume that encloses the impeller section of a centrifugal pump. 83

84 Example:

85 Example:

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