Technology & Engineering
Isochoric Process
An isochoric process is a thermodynamic process in which the volume of the system remains constant while the pressure and temperature change. This process is also known as an isometric process or a constant volume process. In an isochoric process, no work is done by or on the system.
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No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
When R and M are taken as constant, then pressure P can stay constant as the density-temperature quadrant (ρ, T ) undergoes a squeeze mapping. Isochoric Process An Isochoric Process , also called a constant-volume process , an isovolumetric process , or an isometric process , is a thermodynamic process during which the volume of the closed system undergoing such a process remains constant. In nontechnical terms, an Isochoric Process is exemplified by the heating or the cooling of the contents of a sealed non-deformable container: The thermodynamic process is the addition or removal of heat; the isolation of the contents of the container establishes the closed system; and the inability of the container to deform imposes the constant-volume condition. Formalism An isochoric thermodynamic process is characterized by constant volume, i.e. Δ V = 0. The process does no pressure-volume work, since such work is defined by ____________________ WORLD TECHNOLOGIES ____________________ Δ W = P Δ V , where P is pressure. The sign convention is such that positive work is performed by the system on the environment. For a reversible process, the first law of thermodynamics gives the change in the system's internal energy: dU = dQ − dW Replacing work with a change in volume gives dU = dQ − PdV Since the process is isochoric, dV = 0, the previous equation now gives dU = dQ Using the definition of specific heat capacity at constant volume, C v = dU / dT , dQ = nC v dT Integrating both sides yields Where C is the specific heat capacity at constant volume, a is initial temperature and b is final temperature. We conclude with: ____________________ WORLD TECHNOLOGIES ____________________ Isochoric Process in the Pressure volume diagram. In this diagram, pressure increases, but volume remains constant. On a pressure volume diagram, an Isochoric Process appears as a straight vertical line. Its thermodynamic conjugate, an isobaric process would appear as a straight horizontal line.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
When R and M are taken as constant, then pressure P can stay constant as the density-temperature quadrant (ρ, T ) undergoes a squeeze mapping. Isochoric Process An Isochoric Process , also called a constant-volume process , an isovolumetric pro-cess , or an isometric process , is a thermodynamic process during which the volume of the closed system undergoing such a process remains constant. In nontechnical terms, an Isochoric Process is exemplified by the heating or the cooling of the contents of a sealed non-deformable container: The thermodynamic process is the addition or removal of heat; the isolation of the contents of the container establishes the closed system; and the inability of the container to deform imposes the constant-volume condition. Formalism An isochoric thermodynamic process is characterized by constant volume, i.e. Δ V = 0. The process does no pressure-volume work, since such work is defined by ________________________ WORLD TECHNOLOGIES ________________________ Δ W = P Δ V , where P is pressure. The sign convention is such that positive work is performed by the system on the environment. For a reversible process, the first law of thermodynamics gives the change in the system's internal energy: dU = dQ − dW Replacing work with a change in volume gives dU = dQ − PdV Since the process is isochoric, dV = 0, the previous equation now gives dU = dQ Using the definition of specific heat capacity at constant volume, C v = dU / dT , dQ = nC v dT Integrating both sides yields Where C is the specific heat capacity at constant volume, a is initial temperature and b is final temperature. We conclude with: ________________________ WORLD TECHNOLOGIES ________________________ Isochoric Process in the Pressure volume diagram. In this diagram, pressure increases, but volume remains constant. On a pressure volume diagram, an Isochoric Process appears as a straight vertical line.
- S. Bobby Rauf(Author)
- 2021(Publication Date)
- River Publishers(Publisher)
The specifc en-thalpy of the fuid inside the pressure vessel is the same as the specifc Figure 8-3. Thermodynamic Processes in a Typical Refrigeration Cycle 154 Thermodynamics Made Simple for Energy Engineers enthalpy of the same fuid immediately after it escapes the vessel. In such a scenario, the temperature and velocity of the escaping fuid can be calculated if the enthalpy is known. In Figures 8-1 and 8-2, an isenthalpic process follows the isotherm line at a specifc temperature, and along the isotherm the following re-lationship between enthalpy, temperature and specifc heat holds true: dh = c p dT = 0 Additional examples of isenthalpic process are referenced later in this chapter, under the heat engine cycle discussion. Constant Pressure or Isobaric Process An isobaric process is a thermodynamic process in which the pressure remains constant. See Figure 8-4, where the curve represents an isobar. Even though the temperature varies as a function of the en-tropy in this graph, the pressure stays constant. Isobaric Process Example I: Evaporation Stage of a Refrigeration Cycle Evaporation stage of a refrigeration cycle represents an isobaric process in that the pressure remains constant as the low pressure liq-uid system evaporates or changes phases from liquid to gaseous by ab-sorbing the heat energy from the air passing through the heat exchang-er. This absorption of heat by the system—refrigerant or the working fuid— from the surroundings (ambient air) is shown in Figure 8-4 as the shaded area under the isobar, between entropies s 1 and s 2 . In an isobaric process: Δp = 0 and, Q = ΔH The latter mathematical statement, Q = ∆H, implies that in this isobaric process, the heat absorbed by the refrigerant, during the evap-oration phase, results in a net increase in the enthalpy of the refriger-ant. Some of the equations with practical applications in closed sys-tem isobaric processes are listed below:
eBook - PDF- Donald Olander(Author)
- 2007(Publication Date)
- CRC Press(Publisher)
w pdv v v = ∫ 1 2 84 General Thermodynamics Numerical examples of each of the four “iso” processes are presented. The exam-ples will be based on the properties of ideal gases, water, and simple solids. An example of a process that does not follow any of the four “iso” restraints is also analyzed. 3.2 THE Isochoric Process An Isochoric Process in p -v coordinates is shown in Figure 3.1. Because the system boundary is rigid, no work is performed ( w = 0). Passage of the system from initial state 1 to final state 2 is induced by adding a quantity q of heat per mole of substance. The initial state is specified by a property pair such as p 1 and v , where v is the constant specific volume maintained during the process. The first law for the Isochoric Process is: u 2 – u 1 = q (3.3) The final state is determined by v and u 2 . To translate this information to T 2 and p 2 , both p -v -T and thermal equations of state for the substance involved are needed. The diagram in Figure 3.1 shows a rigid vessel with an internal electrical resistor inside to heat the contents. The first law (or energy conservation) for this system is: E el = I 2 R t = Δ u = q where I is the electric current, R is the resistance and t is time. Example 1. 1 mole of air (an ideal gas) initially at p 1 = 5 MPa and T 1 = 60 ° C heated in a rigid vessel until p 2 = 15 MPa. How much heat is required? The vessel volume is obtained from the ideal-gas law: v = RT 1 / p 1 : The final temperature is T 2 = p 2 v / R = 15 × 10 6 × 5.5 × 10 –4 /8.314 = 999 K = 726°C. FIGURE 3.1 An Isochoric Process. The contents of the rigid vessel are heated respectively. Pressure Specific Volume V p 2 p 1 1 2 v Pa Pa = − × × = × − 8 314 333 5 10 5 5 10 6 4 . . m /mole-K K m 3 3 Application of the First and Second Laws to Processes in Closed Systems 85 For an ideal gas, u is independent of pressure.
- Keith Stowe(Author)
- 2007(Publication Date)
- Cambridge University Press(Publisher)
Special processes 227 Figure 12.1 (a) Processes throughout vast regions of space are isobaric and/or isothermal. The gravitational collapse of interstellar clouds, which forms stars, is nearly adiabatic. (Courtesy of NASA) (b) A thunderhead is the result of the rapid adiabatic cooling of air as it rises and expands. (Plymouth State University photograph, courtesy of Bill Schmitz) (c) The physical and chemical processes carried out through marine plankton are isobaric and isothermal. (Courtesy of Mark Moline, California Polytechnic State University) (b) 228 Introduction to thermodynamics and statistical mechanics (c) Figure 12.1 ( cont .) We choose our variables to be N , p , T because N and p are constant and we are measuring changes in T . The change in internal energy is given by the first law, 1 d E = T d S − p d V = T ∂ S ∂ T p d T − p ∂ V ∂ T p d T . With the help of Table 11.1 we can convert these partial derivatives to give d E = ( C p − pV β )d T (isobaric processes) . (12.1) This is what we wanted. If we know how the properties C p , p , V , and β depend on T then we can integrate equation 12.1 to find the relationship for finite changes. B Isothermal processes Isothermal nondiffusive processes also operate under two constraints, d N = d T = 0. So again, there is just one remaining independent variable, which we are free to choose. 1 Note the convention that we usually display only two variables for nondiffusive (d N = 0) processes. Special processes 229 Example 12.2 How does a system’s internal energy vary with volume for isother-mal, nondiffusive, processes (d T = d N = 0)? We choose our variables to be T , N , V because T and N are constant and we are measuring changes in V . The change in internal energy is given by the first law: d E = T d S − p d V = T ∂ S ∂ V T d V − p d V . Again, we use Table 11.1 to convert the partial derivative to easily measured properties: d E = T β κ − p d V (isothermal processes) .
eBook - PDF- P.F. Kelly(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
Isothermic Processes Isothermic processes occur at constant temperature. The terms appearing in the First Law, specialised to isothermic ideal gas systems, are enumerated below. Δ U Owing to the constancy of the temperature, the internal energy of the system remains unchanged, i.e., Δ U = 0. Q Thus, according to the First Law, the flows of work and heat must cancel: 0 = Δ U = Q − W = ⇒ Q = W. W The work done by an ideal gas expanding isothermically from an initial volume, V i , to a final volume, V f , was computed in Chapter 10: W if bracketleftbig P bracketrightbig = nRT ln bracketleftbigg V f V i bracketrightbigg . The ideal gas law and the internal energy EoS remain operative and hence the tools that we shall use for our analyses of isothermic transitions of an ideal gas system are W = nRT ln bracketleftbigg V f V i bracketrightbigg , Q = W , and PV = nRT. Adiabatic Processes Adiabatic processes occur with no heat flow. The application of this constraint to an ideal gas system has several implications. FIRST LAW In the absence of heat flow, changes in the internal energy of the system are identical in magnitude and opposite in sign to the work done by the system: Δ U = − W. Any work done by the gas is accompanied by a concomitant reduction in the system’s internal energy; work input to the system is transformed into internal energy. 45–295 45–296 Properties of Materials TRAJECTORY The phase space trajectory of an ideal gas undergoing an adiabatic transi-tion [ quasi-statically ] satisfies PV γ = constant , where γ is a parameter dependent on properties of the gas, and the constant factor is system-and context-dependent. For monatomic gasses [ e.g., He, Ne, etc . ], γ ≃ 5 / 3. For air at room temperature, γ ≃ 7 / 5. ASIDE: The exponential factor governing the shape of the adiabatic phase trajectory, γ , is the ratio of the molar specific heats of the gas at constant pressure, f C P , and at constant volume, f C V , γ ≡ f C P f C V = C P C V .
eBook - PDF- Jean-Philippe Ansermet, Sylvain D. Brechet(Authors)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
This idealised cycle is defined by three reversible processes: • 1 −→ 2 isochoric compression • 2 −→ 3 adiabatic expansion • 3 −→ 1 isobaric contraction Assume that the cycle is performed on an ideal gas characterised by the coefficient c found in relation (5.62). The following values of some state variables of the gas are assumed to be known: the pressure p 1 , volumes V 1 and V 3 , temperature T 1 and the number of moles of gas N. Analyse this cycle by using the following instructions : a) Draw the (p, V) and (T, S) diagrams of the cycle. b) Determine the entropy variation ΔS 12 of the gas during the Isochoric Process 1 −→ 2. c) Express the temperature T 2 in terms of the heat exchanged Q 12 during the Isochoric Process 1 −→ 2. d) Determine the pressure p 2 in terms of the pressure p 1 , the volume V 1 and the heat exchanged Q 12 . e) Determine the pressure p 3 in terms of the pressure p 2 and volumes V 2 and V 3 . f) Determine the work W 23 performed during the adiabatic process 2 −→ 3 and the heat Q 23 exchanged during this process. g) Find the work W 31 performed during the isobaric process 3 −→ 1 and the heat Q 31 exchanged during this process. h) Find the efficiency of the cycle η L defined in conformity with relation (7.38) as, η L = − W 23 + W 31 Q 12 Express the efficiency η L in terms of the temperatures T 1 , T 2 and T 3 . 7.5 Otto Cycle The Otto cycle is a model for a spark ignition engine and represents the mode of operation of most non-diesel car engines. It consists of four processes when the 198 Heat Engines system is closed, and of two additional isobaric processes when the system is open, corresponding to air intake and exhaust.
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