# Objectives. Conservation of mass principle: Mass Equation The Bernoulli equation Conservation of energy principle: Energy equation

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2 Objectives Conservation of mass principle: Mass Equation The Bernoulli equation Conservation of energy principle: Energy equation

3 Conservation of Mass Conservation of Mass Mass, like energy, is a conserved property, and it cannot be created or destroyed during a process. For closed systems, the conservation of mass principle is implicitly used by requiring that the mass of the system remain constant during a process. For control volumes, however, mass can cross the boundaries, and so we must keep track of the amount of mass entering and leaving the control volume.

4 Conservation of Mass The amount of mass flowing through a cross section per unit time: mass flow rate ( )

5 Conservation of Mass for incompressible flow: Ac: cross-sectional area normal to the flow direction

6 Conservation of Mass The volume of the fluid flowing through a cross section per unit time: volume flow rate ( )

7 Objectives Conservation of Mass Principle change in the mass of the control volume during the process: It can also be expressed in rate form: often referred to as the mass balance and are applicable to any control volume undergoing any kind of process

8 Conservation of Mass It can also be expressed in rate form Consider a control volume of arbitrary shape: Mass of a differential volume dv within the control volume is:

9 Conservation of Mass Total mass within the control volume at any instant in time t: Rate of change of mass within the CV:

10 Conservation of Mass considered.

11 Conservation of Mass The time rate of change of mass within the control volume plus the net mass flow rate through the control surface is equal to zero. Other forms of mass conservation principle:

12 Conservation of Mass A simple rule in selecting a control volume is to make the control surface normal to flow at all locations where it crosses fluid flow, whenever possible.

13 Conservation of Mass Moving Control Volumes valid for moving or deforming control volumes provided that the absolute velocity is replaced by the relative velocity, which is the fluid velocity relative to the control surface. In the case of a nondeforming control volume, relative velocity is the fluid velocity observed by a person moving with the control volume

14 Conservation of Mass Deforming Control Volumes Some practical problems (such as the injection of medication through the needle of a syringe by the forced motion of the plunger) involve deforming control volumes.

15 Conservation of Mass The general conservation of mass relation for a control volume can also be derived using the Reynolds transport theorem (RTT) by taking the property B to be the mass m. Mass of a system is constant.

16 Conservation of Mass Mass Balance for Steady-Flow Processes During a steady-flow process, the total amount of mass contained within a control volume does not change with time. the total amount of mass entering a control volume is equal the total amount of mass leaving it PER unit time. Conservation of mass principle for a general steady-flow system with multiple inlets and outlets:

17 Conservation of Mass Many engineering devices such as nozzles, diffusers, turbines, compressors, and pumps involve a single stream (only one inlet and one outlet) single-stream steady-flow systems

18 Conservation of Mass Incompressible Flow The conservation of mass relations can be simplified even further when the fluid is incompressible, which is usually the case for liquids. Steady incompressible flow (single-stream) Conservation of Volume Principle

19 Conservation of Mass

20 Conservation of Mass

21 Conservation of Mass

22 Conservation of Mass

23 Conservation of Mass

24 Mechanical Energy Mechanical Energy A pump transfers mechanical energy to a fluid by raising its pressure, & a turbine extracts mechanical energy from a fluid by dropping its pressure. mechanical energy of a flowing fluid PER unit-mass:

25 Mechanical Energy mechanical energy change of a fluid during incompressible flow: mechanical energy of a fluid does not change during flow if its pressure, density, velocity, and elevation remain constant.

26 Mechanical Energy An ideal hydraulic turbine would produce the same work per unit mass whether it receives water (or any other fluid with constant density) from the top or from the bottom of the container. ideal flow (no irreversible losses)

27 Bernoulli Equation Note that we are also assuming through the pipe leading from the tank to the turbine. Therefore, the total mechanical energy of water at the bottom is equivalent to that at the top. 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. Derive by applying: conservation of linear momentum principle

28 Bernoulli Equation Bernoulli Equation key approximation in the derivation of the Bernoulli equation: Viscous effects are negligibly small compared to inertial, gravitational, and pressure effects: Inviscid Fluid Pressure Velocity Elevation

29 Bernoulli Equation

30 Bernoulli Equation Derivation of the Bernoulli Equation Derive Bernoulli equation? Steady Flow Consider the motion of a fluid particle in a flow field in steady flow:

31 Bernoulli Equation

32 Bernoulli Equation

33 Bernoulli Equation Bernoulli equation, which is commonly used in fluid mechanics for steady, incompressible flow along a streamline in inviscid flow: between any two points on the same streamline as

34 Bernoulli Equation Bernoulli equation: mechanical energy balance The sum of the kinetic, potential, and flow energies of a fluid particle is constant along a streamline during steady flow when the compressibility and frictional effects are negligible.

35 Bernoulli Equation Static, Dynamic, and Stagnation Pressures P: Static pressure/ Thermodynamic pressure of the fluid : Dynamic pressure; it represents the pressure rise when the fluid in motion is brought to a stop isentropically. : hydrostatic pressure, which is not pressure in a real sense since its value depends on the reference level selected; it accounts for the elevation effects, i.e., of fluid weight on pressure.

36 Bernoulli Equation

37 Bernoulli Equation The sum of the static and dynamic pressures is called the stagnation pressure, The stagnation pressure represents the pressure at a point where the fluid is brought to a complete stop isentropically. Fluid velocity at stagnation point:

38 Bernoulli Equation The flow streamline that extends from far upstream to the stagnation point is called the stagnation streamline.

39 Bernoulli Equation Steady Flow Frictionless flow No heat transfer (density variation) Incompressible flow Flow along a streamline

40 HGL & EGL Hydraulic Grade Line (HGL) & Energy Grade Line (EGL) : pressure head; it represents the height of a fluid column that produces the static pressure P : velocity head : elevation head H is the total head for the flow

41 HGL & EGL If a piezometer (measures static pressure) is tapped into a pipe, the liquid would rise to a height of above the pipe center. The hydraulic grade line (HGL) is obtained by doing this at several locations along the pipe and drawing a line through the liquid levels in the piezometers.

42 HGL & EGL if a Pitot tube (measures static+dynamic pressure) is tapped into a pipe, the liquid would rise to a height of distance of above the HGL. + above the pipe center, or a The energy grade line (EGL) is obtained by doing this at several locations along the pipe and drawing a line through the liquid levels in the Pitot tubes.

43 HGL & EGL

44 HGL & EGL Noting that the fluid also has elevation head z (unless the reference level is taken to be the centerline of the pipe), the HGL and EGL can be defined as follows: The line that represents the sum of the static pressure and the elevation heads, +, is called the HGL. The line that represents the total head of the fluid, the energy grade line. + +,is called The difference between the heights of EGL and HGL is equal to the dynamic head,.

45 HGL & EGL For stationary bodies such as reservoirs or lakes, the EGL and HGL coincide with the free surface of the liquid. The EGL is always a distance above the HGL. These two lines approach each other as the velocity decreases, and they diverge as the velocity increases. In an idealized Bernoulli-type flow, EGL is horizontal and its height remains constant. This would also be the case for HGL when the flow velocity is constant.

46 Bernoulli Equation

47 Bernoulli Equation

48 Bernoulli Equation

49 Bernoulli Equation

50 Bernoulli Equation

51 Bernoulli Equation

52 Bernoulli Equation

53 Bernoulli Equation

54 Bernoulli Equation

55 General Energy Equation General Energy Equation first law of thermodynamics, also known as the conservation of energy principle. total energy consists of internal, kinetic, and potential energies

56 General Energy Equation Energy Transfer by Heat, Q Transfer of thermal energy from one system to another as a result of a temperature difference is called heat transfer. A process during which there is no heat transfer is called an adiabatic process. There are two ways a process can be adiabatic: Either the system is well insulated so that only a negligible amount of heat can pass through the system boundary, or both the system and the surroundings are at the same temperature and therefore there is no driving force (temperature difference) for heat transfer.

57 General Energy Equation Energy Transfer by Work, W An energy interaction is work if it is associated with a force acting through a distance. The time rate of doing work is called power: A system may involve numerous forms of work, and the total work can be expressed as:

58 General Energy Equation Shaft Work Many flow systems involve a machine such as a pump, a turbine, a fan, or a compressor whose shaft protrudes through the control surface, and the work transfer associated with all such devices is simply referred to as shaft work. The power transmitted via a rotating shaft:

59 General Energy Equation Work Done by Pressure Forces: Flow Work piston-cylinder devices:

60 General Energy Equation Pressure always acts inward and normal to the surface;

61 General Energy Equation For a closed system:

62 General Energy Equation

63 General Energy Equation

64 General Energy Equation In the case of a deforming control volume:

65 General Energy Equation For a fixed control volume (no motion or deformation of control volume) This equation is not in a convenient form for solving practical engineering problems because of the integrals, and thus it is desirable to rewrite it in terms of average velocities and mass flow rates through inlets and outlets. Approximated by:

66 General Energy Equation Or: Used definition of enthalpy: general expressions of conservation of energy, but their use is still limited to fixed control volumes, uniform flow at inlets and outlets, and negligible work due to viscous forces and other effects. Also, the subscript net in stands for net input, and thus any heat or work transfer is positive if to the system and negative if from the system.

67 General Energy Equation Energy Analysis of Steady Flows For steady flows, the time rate of change of the energy content of the control volume is zero, for such single-stream devices:

68 General Energy Equation

69 General Energy Equation unit-mass basis: mechanical energy input mechanical energy output If the flow is ideal with no irreversibilities such as friction, the total mechanical energy must be conserved,

70 General Energy Equation For single-phase fluids (a gas or a liquid):

71 General Energy Equation

72 General Energy Equation Multiplying by the mass flow rate: frictional losses in the piping network)

73 General Energy Equation Thus the energy equation can be expressed in its most common form in terms of heads as: useful head delivered to the fluid by the pump

74 General Energy Equation extracted head removed from the fluid by the turbine. due to all components of the piping system other than the pump or turbine. represents the frictional losses associated with fluid flow in piping, and it does not include the losses that occur within the pump or turbine due to the inefficiencies of these devices.

75 General Energy Equation Special Case: Incompressible Flow with No Mechanical Work Devices and Negligible Friction which is the Bernoulli equation derived earlier using Newton s second law of motion.

76 General Energy Equation Kinetic Energy Correction Factor The kinetic energy correction factors are often ignored (i.e., a is set equal to 1) in an elementary analysis since (1) most flows encountered in practice are turbulent, for which the correction factor is near unity, and

77 General Energy Equation (2) the kinetic energy terms are often small relative to the other terms in the energy equation, and multiplying them by a factor less than 2.0 does not make much difference.

78 General Energy Equation

79 General Energy Equation

80 General Energy Equation

81 General Energy Equation

82 General Energy Equation

83 General Energy Equation

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