4.1 Thermodynamic Analysis of Control Volumes A large number of engineering problems involve mass flow in and out of a system and, therefore, are modeled as control volumes (e.g. water heater, radiator, turbine, compressor, etc...) In general, any arbitrary region may be selected as a control volume, but making a proper choice simplifies the solution process. The boundaries of a control volume are called the control surface, and they can be real or imaginary (e.g. see nozzle below). Real Boundary
Imaginary Boundary
Control Volume
(a nozzle)
A control volume can be fixed in size and shape or possess moving boundaries (e.g. shock absorber).
A Few Definitions: ‚ steady: implies no change with time; the opposite of transient ‚ uniform: implies no change with location It is very important that you learn the significance of these definitions!
Conservation of Mass Principle: The conservation of mass is one of the fundamental principles in nature. Simply stated, it asserts that mass is a conserved property and can not be created or destroyed. The conservation of mass principle for a control volume (CV) undergoing a process can be expressed as:
mi
−
me
=
m CV
(4.1.1)
Total mass entering Total mass leaving Net change in mass control volume control volume within control volume where the subscripts i, e, and CV stand for inlet, exit, and control volume, respectively. The conservation of mass equation may also be expressed on a time rate basis by expressing all quantities per unit time. The conservation of mass principle is also often referred to the continuity equation in fluid mechanics.
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Mass and Volume Flow Rates: The amount of mass flowing through a cross-section per unit time is called the mass flow rate . and is denoted by m . In most practical applications, the mass flow rate in a pipe or duct can be evaluated using the following expression ... (4.1.2) where Vav 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 given by:
. V = V av A
(4.1.3)
.
.
Thus, the mass flow and volume flow rates are related by: m = V
(4.1.4)
Conservation of Energy Principle: The first law of thermodynamics attributes the changes in total energy of a closed system to heat and work interactions. For control volumes, however, an additional mechanism can change the energy of a system: mass flow in and out of the control volume! When mass enters a control volume, the energy of the control volume increases because the entering mass carries energy with it. Likewise, when some mass leaves the control volume, the energy contained within the control volume decreases because the leaving mass takes out some energy with it. Then the conservation of energy equation for a control volume undergoing a process can be expressed as: (4.1.5)
Q−W
Total energy crossing boundary as heat and work
+
E in Total energy of mass entering CV
−
E out
=
Total energy of mass leaving CV
E CV Net change in in energy of CV
If no mass enters or leaves the control volume, the second and third terms drop out and the above equation becomes the first law for closed systems.
Flow Work: Unlike closed systems, control volumes involve mass flow across their boundaries, and some work is required to push the mass into or out of the control volume. This is known as flow work, or flow energy. 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)
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The fluid entering or leaving a control volume possesses an additional form of energy--the flow energy 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 reduces to: (4.1.9)
Note! By using the enthalpy instead of the internal energy to represent the energy of a flowing fluid, you do not need to be concerned about the flow work!
4.2 The Steady-Flow Process Processes involving steady-flow devices (turbines, compressors, nozzles, etc...) can be represented reasonably well by a somewhat idealized process, called the steady-flow process. A steady-flow process can be defined as a process during which a fluid flows through a control volume steadily. That is, the fluid properties can change from point to point within the control volume, but at any fixed point they remain the same during the entire process. A steady-flow process is characterized by the 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 not change with time.
Conservation of Mass: During a steady-flow process, the total amount of mass contained within a control volume does not change with time. The conservation of mass principle for steady-flow systems requires that the total amount of mass entering a control volume equal the total amount leaving it (e.g. see figure below).
m
m
1
CV m
2
m m 3 = 1+ 2
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The conservation of mass principle for a general steady-flow system with multiple inlets and exits can be expressed in the rate form as:
. mi
. me
=
Total mass entering CV per unit time
(4.2.1)
Total mass leaving CV per unit time
where the subscripts i stands for inlet and e for exit.
Question to Think About! If the conservation of mass principle exists, what about conservation of volume? (Hint: think about the definition of density (or specific volume) and compressibility!)
Conservation of Energy: It was pointed out earlier that system properties remain constant for the duration of a steady-state process. In order for the total energy of an open system undergoing a steady-state process to remain constant, the amount of energy entering a control volume in all forms (heat, work, mass transfer) must be equal to the amount of energy leaving it. By this line of reasoning, the conservation of energy principle for a general steady-flow system with multiple inlets and exits can be mathematically stated as:
. . Q − W
. me
=
Total energy crossing boundary as heat and work per unit time
e
−
Total energy transported out of CV with mass per unit time
. mi
i
(4.2.2) Total energy transported into CV with mass per unit time
where θ is the tot al energy of the flowing fluid, including the flow work, per unit mass. Eq. (4.2.2) can also be expressed as: (4.2.3) . Dividing Eq. (4.2.3) by m gives the first law relation for control volumes on a unit-mass basis, or ...
q − w = h e +
V 2e 2g c
+
gz e gc
− hi +
V 2i 2g c
+
gz i gc
(4.2.4)
4.3 Some Steady-Flow Engineering Devices Many engineering devices operate essentially under the same conditions for long periods of time (e.g. components of a steam power plant). Therefore, these devices can be conveniently analyzed as steady-flow devices.
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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 the 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.
Nozzle
Diffuser
The relative importance of the terms appearing in the energy equation for nozzles and diffusers is as follows:
.
‚ Q 0: The rate of heat transfer between the fluid flowing through a nozzle or a diffuser and the surroundings is usually very small. This is mainly due to the fluid's having high velocities and thus not spending enough time in the device for any significant heat transfer to take place. Therefore, in the absence of heat transfer data, the flow through nozzles and diffusers may be assumed to be adiabatic.
.
‚W = 0: The work term for nozzles and diffusers is zero since these devices basically are properly shaped ducts and they involve no shaft or electrical work. ‚ KE 0 : Nozzles and diffusers usually involve large changes in velocity. Therefore, kinetic energy changes must be accounted for in analyzing the flow through these devices. ‚ PE 0 : The fluid usually experiences little or no change in elevation and therefore the potential energy term can be neglected.
Turbines and Compressors: In power plants, the device that drives the electric generator is the turbine. As the fluid passes through the turbine, work is done against a blade that is attached to a shaft. As a result, the shaft rotates, and the turbine produces work. Compressors, as well as pumps, are devices used to increase the pressure of a fluid (compressors are for gases and pumps are for liquids). Work is supplied to these devices from an external source through a rotating shaft (e.g. the A/C compressor in your car is driven off a belt). Typical schematics of a turbine and compressor are shown on the next page. For turbines and compressors, the relative magnitudes of the various terms appearing in the energy equation are as follows:
.
‚ Q 0: The heat transfer in these devices is generally small relative to the shaft work, unless there is intentional cooling.
.
‚ W 0: All of these devices involve rotating shafts crossing their boundaries, and therefore the . work term is important. For turbines, W represents the power output; for pumps and compressors, it represents the power input. ENGS205--Introductory Thermodynamics
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Turbine
Compressor Shaft (-) Work
Shaft Work (+)
‚ KE 0 : The velocities involved with these devices, with the exception of turbines, are usually too low to cause any significant change in kinetic energy. In turbines, this change in kinetic energy is usually very small relative to the change in enthalpy, and is often disregarded. ‚ PE
0 : This energy term is typically small for turbines/compressors.
Throttling Valves: Throttling valves are any kind of flow-restricting devices that cause a significant pressure drop in the fluid. Unlike turbines, throttling valves produce a pressure drop without any kind of work. 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 air-conditioning applications. An adjustable throttling valve is shown below.
The relative magnitudes of the energy equation terms is discussed below ...
.
‚Q
0: Flow through these devices is usually adiabatic.
.
‚ W = 0: No work is done on or by the fluid. ‚ KE 0 : Even though the exit velocity is often considerably higher, the change in kinetic energy is insignificant. ‚ PE
0 : Potential energy changes are very small.
The conservation of energy equation for a throttling valve readily reduces to:
he
hi
(4.3.1)
... and for this reason, throttling valves are sometimes called isenthalpic devices.
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Mixing Chambers: In engineering applications, mixing two streams of fluids is not a rare occurrence. The section where the mixing process takes place is commonly referred to as a mixing chamber.
Mixing Chamber
Thermodynamic consideration of the mixing chamber reveals ...
.
‚Q
0: Mixing chambers are usually well-insulated.
.
‚ W = 0: There are no work interactions. ‚ KE
0 : Changes in kinetic energy are negligible.
‚ PE
0 : Changes in potential energy are negligible.
Heat Exchangers: Heat exchangers are devices where two moving fluid streams exchange heat without mixing. Heat exchangers are used widely in industries, and they come in numerous designs. The simplest form of a heat exchanger is a double-tube (shown below) or tube-and-shell heat exchanger. It's composed of two concentric pipes of different diameters (i.e., a pipe within a pipe). One fluid flow in the inner pipe, and the other in the annular space between the pipes. Heat is transferred through the wall separating the two fluids.
Fluid B
Fluid A
Here are the relevant energy terms ...
.
‚ Q 0: This must be absolutely true! Otherwise, the heat exchanger you're analyzing doesn't exchange heat!
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.
‚ W = 0: Heat exchangers typically involve no work interactions. ‚ KE
PE
0 : Changes in these forms of energy are usually negligible.
Pipe and Duct Flow: The transport of liquids or gases in pipes and ducts is of great importance in many engineering applications. When flow through pipe or ducts is analyzed, the following points should be considered:
.
‚ Q 0: Under normal operating conditions, the amount of heat gained or lost by the fluid can be very significant, particularly if the pipe or duct is long and not insulated.
.
‚ W 0: If the control volume involves a heating section (electric wires, a fan, or a pump (shaft), the work interaction should be considered. ‚ KE
0 : For constant-diameter pipes/ducts, this energy term is usually negligible.
‚ PE 0 : Potential energy consideration are important when a fluid is pump through great elevation changes.
4.4 Unsteady-Flow Processes Many processes of engineering interest involve changes within the control volume with time. Such processes are called unsteady-flow, or transient-flow processes.
Important point! The steady-flow relations developed in the previous sections are not applicable to these processes! When an unsteady-flow process is analyzed, it is important to keep track of the mass and energy contents of the control volume as well as the energy interactions across the boundary. Some familiar unsteady flow processes are the charging of rigid vessels from supply lines, discharging a fluid from a pressure vessel, inflating tires or balloons, and even cooking with an ordinary pressure cooker. Unlike steady-flow processes, unsteady-flow processes start and end over a finite time period ∆t. Another difference between steady and unsteady-flow processes is that steady-flow systems are fixed in space, in size, and in shape. Unsteady -flow systems, however, are not. They are usually stationary, but involve moving boundaries (boundary work!)
Conservation of Mass: Unlike the case of steady-flow processes, the amount of mass within the control volume does change with time during an unsteady-flow process. The degree of change depends on the amount of mass entering and leaving the control volume during the process. The conservation of mass principle for unsteady-flow processes is mathematically stated as ...
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−
mi Total mass entering CV during ∆t
me
=
m CV
Total mass leaving CV during ∆t
(4.4.1)
Net change in mass within CV during ∆t
If all the terms in Eq. (4.4.1) are divided by ∆t and taking the limit as t
0 gives ..
(4.4.2) where dmCV / dt is the time rate of change of mass contained within the control volume.
Conservation of Energy: Unlike the steady-flow process, the energy content of a control volume changes with time during an unsteady-flow process. The degree of change depends on the amount of energy transfer across the system boundaries as heat and work as well as on the amount of energy transported into and out of the control volume by mass during the process. That is, we have to keep track of the energy flowing into and out of the control volume for the duration of the unsteady process ∆t. The conservation of energy principle for a control volume undergoing an unsteady process for a time interval ∆t is
Q−W
+
Total energy crossing boundary as heat and work during ∆t
i
−
Total energy transported by mass into CV during ∆t
e
=
Total energy transported by mass out of CV during ∆t
E CV
(4.4.3) Net change in energy of CV during ∆t
If all the terms in Eq. (4.4.3) are divided by ∆t and taking the limit as t
0 gives ...
(4.4.4) . . where i and e are the rates at which energy is transported with mass into and out of the control volume respectively, and dECV / dt is the time rate of change of energy within the control volume. In Eq. (4.4.3), the heat and work terms (Q and W) can be determined by external measurements. The total energy of the control volume at the beginning and end states of the process (E1 and E 2 ) can be easily determined by measuring the relevant properties of the substance at these two states. The total energy transported into or out of the control volume (Θi , Θe), however, is not as easy to determine since the properties of the mass at each inlet or exit may be changing with time as well as over the cross-section. Thus, the only way to determine the energy transport through an opening as a result of mass flow is to consider sufficiently small differential masses δm that have uniform properties and add to their total energies.
Total mass, m divided up into numerous differential masses, δm
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The total energy of a flowing fluid of mass δm is θδm. Then the total energy transported by mass through an inlet or exit (Θi , Θe) is obtained by integrating. At an inlet, for example, it becomes ... (4.4.5) or, in the rate form ... (4.4.6) The energy equation for unsteady-processes in control volumes becomes: (4.4.7) ... and ... (4.4.8)
Uniform-Flow Processes: A uniform-flow process is a simplified unsteady-flow process involving the following idealizations: ‚ At any instant during the process, the state of the control volume is uniform (the same throughout spatially). Therefore, the state of the mass exiting the control volume at any instant is the same as the state of the mass in the control volume at that instant. ‚ The fluid properties may differ from one inlet or exit to another, but the fluid flow at an inlet or exit is uniform and steady. Under these idealizations, Eq. (4.4.7) becomes ... (4.4.9) where u1 and u2 are the initial and final specific internal energies of the system.
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