4.1 Thermodynamic Analysis of Control Volumes
4.1 Thermodynamic Analysis of Control Volumes 4.1 Thermodynamic Analysis of Control Volumes
Mass and Volume Flow Rates:The amount of mass flowing through a cross-section per unit time is called the mass flow rateand is denoted by ṁ . In most practical applications, the mass flow rate in a pipe or duct can beevaluated using the following expression ...(4.1.2)where V av is the average bulk fluid velocity and A is the cross-sectional area normal to the flow direction.The volume flow rate V . is the volume of fluid flowing through a cross section per unit time and is givenby:V . = V av A(4.1.3)Thus, the mass flow and volume flow rates are related by: ṁ = V .(4.1.4)Conservation of Energy Principle:The first law of thermodynamics attributes the changes in total energy of a closed system toheat and work interactions. For control volumes, however, an additional mechanism can change theenergy of a system: mass flow in and out of the control volume! When mass enters a control volume, theenergy of the control volume increases because the entering mass carries energy with it. Likewise, whensome mass leaves the control volume, the energy contained within the control volume decreasesbecause the leaving mass takes out some energy with it. Then the conservation of energy equation for acontrol volume undergoing a process can be expressed as:(4.1.5)Q − W + E in − E out = E CVTotal energy cross- Total energy of Total energy of Net change ining boundary as mass entering CV mass leaving CV in energy of CVheat and workIf no mass enters or leaves the control volume, the second and third terms drop out and theabove equation becomes the first law for closed systems.Flow Work:Unlike closed systems, control volumes involve mass flow across their boundaries, and somework is required to push the mass into or out of the control volume. This is known as flow work, or flowenergy. The work done in pushing the fluid across the boundary (i.e. flow work) is:W flow = PV... and on a mass basis ... w flow = Pv(4.1.6)Total Energy of a Flowing Fluid:The total energy of closed system (non-flowing fluid) is expressed as:(4.1.7)ENGS205--Introductory Thermodynamics page 39
The fluid entering or leaving a control volume possesses an additional form of energy--the flowenergy Pv ! Then the total energy of a flowing fluid on a unit-mass basis (denoted θ) becomes:= Pv + e = Pv + ⎛ ⎝u + ke + pe⎞ ⎠(4.1.8)But the combination Pv+u has been previously defined as the enthalpy h. So the above relation reducesto:Note!(4.1.9)By using the enthalpy instead of the internal energy to represent the energy of aflowing fluid, you do not need to be concerned about the flow work!4.2 The Steady-Flow ProcessProcesses involving steady-flow devices (turbines, compressors, nozzles, etc...) can berepresented reasonably well by a somewhat idealized process, called the steady-flow process. Asteady-flow process can be defined as a process during which a fluid flows through a control volumesteadily. That is, the fluid properties can change from point to point within the control volume, but at anyfixed point they remain the same during the entire process. A steady-flow process is characterized bythe following:‚ No properties (intensive or extensive) within the control volume change with time. As a result,boundary work is zero for steady-flow systems.‚ No properties change at the boundaries (control surface) of the control volume with time.‚ The heat and work interactions between a steady-flow system and its surroundings do notchange with time.Conservation of Mass:During a steady-flow process, the total amount of mass contained within a control volume doesnot change with time. The conservation of mass principle for steady-flow systems requires that thetotal amount of mass entering a control volume equal the total amount leaving it (e.g. see figure below).m m 1 2CVm 3= m 1 + m 2ENGS205--Introductory Thermodynamics page 39
- Page 4 and 5: The conservation of mass principle
- Page 6 and 7: CompressorShaftWork (-)TurbineShaft
- Page 8 and 9: ‚ Ẇ = 0: Heat exchangers typica
- Page 10: The total energy of a flowing fluid
Mass and Volume Flow Rates:The amount <strong>of</strong> mass flowing through a cross-section per unit time is called the mass flow rateand is denoted by ṁ . In most practical applications, the mass flow rate in a pipe or duct can beevaluated using the following expression ...(<strong>4.1</strong>.2)where V av is the average bulk fluid velocity and A is the cross-sectional area normal to the flow direction.The volume flow rate V . is the volume <strong>of</strong> fluid flowing through a cross section per unit time and is givenby:V . = V av A(<strong>4.1</strong>.3)Thus, the mass flow and volume flow rates are related by: ṁ = V .(<strong>4.1</strong>.4)Conservation <strong>of</strong> Energy Principle:The first law <strong>of</strong> thermodynamics attributes the changes in total energy <strong>of</strong> a closed system toheat and work interactions. For control volumes, however, an additional mechanism can change theenergy <strong>of</strong> a system: mass flow in and out <strong>of</strong> the control volume! When mass enters a control volume, theenergy <strong>of</strong> the control volume increases because the entering mass carries energy with it. Likewise, whensome mass leaves the control volume, the energy contained within the control volume decreasesbecause the leaving mass takes out some energy with it. Then the conservation <strong>of</strong> energy equation for acontrol volume undergoing a process can be expressed as:(<strong>4.1</strong>.5)Q − W + E in − E out = E CVTotal energy cross- Total energy <strong>of</strong> Total energy <strong>of</strong> Net change ining boundary as mass entering CV mass leaving CV in energy <strong>of</strong> CVheat and workIf no mass enters or leaves the control volume, the second and third terms drop out and theabove equation becomes the first law for closed systems.Flow Work:Unlike closed systems, control volumes involve mass flow across their boundaries, and somework is required to push the mass into or out <strong>of</strong> the control volume. This is known as flow work, or flowenergy. The work done in pushing the fluid across the boundary (i.e. flow work) is:W flow = PV... and on a mass basis ... w flow = Pv(<strong>4.1</strong>.6)Total Energy <strong>of</strong> a Flowing Fluid:The total energy <strong>of</strong> closed system (non-flowing fluid) is expressed as:(<strong>4.1</strong>.7)ENGS205--Introductory <strong>Thermodynamic</strong>s page 39
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