11. Realistic Equations of State 1
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PVT Behavior of Fluids and the Theorem of Corresponding States Compressibility Factor •The volumetric PVT behavior of a material is often characterized by the compressibility factor Z •Typical values: ideal gas, Z = 1.0; liquids, Z ~ 0.01 to 0.2; critical point, Z c ~ 0.27 to 0.29 Theorem of Corresponding States •This theorem suggests that values of thermodynamic properties of different fluids can be compared when properties are normalized by their values at the critical point RT PV NRT V P Z r r c P T Z f Z , ,
description
Equations of State for Physical Fluids
Transcript of 11. Realistic Equations of State 1
No Slide TitlePVT Behavior of Fluids and the Theorem of Corresponding States
Compressibility Factor
The volumetric PVT behavior of a material is often characterized by the compressibility factor Z
Typical values: ideal gas, Z = 1.0; liquids, Z ~ 0.01 to 0.2;
critical point, Zc ~ 0.27 to 0.29
Theorem of Corresponding States
This theorem suggests that values of thermodynamic properties of different fluids can be compared when properties are normalized by their values at the critical point
PVT Behavior of Fluids and the Theorem of Corresponding States
Theorem of Corresponding States
The reduced temperature and
constant for all materials
support this approach
The relation above indicates that all fluids have the same reduced volume Vr (as well as other properties) at a specified Tr and Pr
In Example 7.2 the principle of corresponding states was used to relate the a and b constants of the van der Waals equation of state to a fluids critical properties, specifically Tc and Pc
and
PVT Behavior of Fluids and the Theorem of Corresponding States
Generalized Coexistence Curve
The theorem of corresponding states can be used to obtain a generalized phase diagram
“Law of rectilinear diameters” – states that the average density in the L-V region varies linearly over a wide temperature range below the critical point
Zeno line – the Z = 1.0 line on
the phase diagram
repulsive forces dominate and
conversely, for states where
are dominant
PVT Behavior of Fluids and the Theorem of Corresponding States
Acentric Factor
To improve the accuracy of property predictions, Pitzer and coworkers introduced the acentric factor w as a third correlating parameter
The acentric factor was developed
as a measure of the difference in
structure between the material of
interest and a spherically symmetric
gas such as argon
to a molecular property
reduced vapor pressure
Overview
Although the corresponding states correlations are useful, the data is displayed in charts and graphs are therefore cumbersome to use
A more useful means to express a fluids PVT behavior is through an analytical expression (PVT equation of state)
Two general approaches
Empirical expressions – models formulated by determining parameters in a mathematical expression through a fit to experimental data (preferred by chemical engineers)
PVTN Equations of State for Fluids
Relating EOS Parameters to Experimental Data
Empirical models have parameters that must be determined to calculate PVT data
A common way to determine these parameters is to relate the model parameters to a fluid’s critical properties
By adopting this approach we are utilizing the theorem of corresponding states
The critical point criteria for a single-component system require
Recall Example 7.2, where these criteria were used to determine the parameters for the van der Waals EOS
and
Cubic Equations of State
Since the pioneering work of van der Waals, numerous modifications of the original vdW EOS have been published
Most are pressure-explicit cubic forms, i.e.,
van der Waals EOS
Redlich-Kwong
with
with
Redlich-Kwong-Soave
The attractive term is made temperature dependent and the acentric factor is introduced
with
Peng-Robinson
with
Hard-Sphere Equation of State
A class of EOSs that incorporate knowledge of the specific molecular interactions with additional fitted parameters
The repulsive term is theoretically based
The term bo below accounts for the excluded volume of a molecule
The attractive term is empirical
Here is one example of a hard-sphere EOS that uses the Carnahan-Starling repulsive term and van der Waals attractive term
s → molecular diameter
Virial-Type Equation of State
An important EOS class that has a rigorous basis in molecular theory (may be derived from statistical mechanics)
The compressibility is expressed as a polynomial in volume
Or as a polynomial expansion in pressure
Where
The coefficients B, C, D, … are termed the second, third, fourth, … virial coefficients
PVTN Equations of State for Fluids
Method of Abbott
The virial equation is truncated after the second virial coefficient term and the pressure is obtained
The second virial coefficient is expressed in terms of two functions of temperature B(0) and B(1) as well as the acentric factor
Where
and
Benedict-Webb-Rubin EOS
Has eight parameters
Frequently used for modeling the PVT behavior of light hydrocarbons and gases in petroleum and natural gas applications
Evaluating Changes in Properties Using Departure Functions
Overview
We now develop methods for determining changes in derived thermodynamic properties of a pure substance
Because properties are state functions, we can choose any convenient path to calculate the change in the property of interest
(e.g., H, S, U, A, or G)
To perform the calculation, we must generally have access to two constitutive relationships
A PVT or PVTN equation of state (EOS)
An ideal-gas state heat capacity equation
Evaluating Changes in Properties Using Departure Functions
Convenient Path
To calculate property changes between two well-defined states (T1,P1,V1) and (T2,P2,V2) we use the following three-step process
Isothermal expansion at T1 from P = P1 to 0 (or V = V1 to ∞
Isobaric (or isochoric) heating from T1 to T2 in an attenuated ideal gas state [P = 0 (or V = ∞)]
Isothermal compression at T2 from P = 0 to P2
(or from V = ∞ to V2)
Evaluating Changes in Properties Using Departure Functions
Convenient Path
The total property change between states 1 and 2, DB (where B is one of H, S, U, A, or G), is the sum over these three steps
Step (2) occurs entirely within the ideal-gas state
Recall that only 2 of the 3 intensive variables T, P, and V are independent
We select T as one independent variable and choose P or V based on the PVT EOS we have
If we have a pressure-explicit EOS (P = f (T,V)), we choose V
If we have a volume-explicit EOS (V = f (T,P)), we choose P
step (1)
step (2)
step (3)
Example 8.3
Develop expressions for DS between states 1 and 2 assuming you have analytic forms for (1) a volume-explicit EOS and (2) a pressure-explicit EOS and
Summary
Departure (Residual) Functions
A departure function is the difference between the property of interest in its real state at a specified T, P, and V and in an ideal-gas state at the same temperature T and pressure P
There are two equivalent forms of the departure function
B is any derived property (H, S, U, A, or G) and Vo = RT / P(T,V)
Evaluating Changes in Properties Using Departure Functions
Residual Entropy
Based on Example 8.3, the residual entropy is defined as
Evaluating Changes in Properties Using Departure Functions
Residual Helmholtz Energy
When working with pressure-explicit EOSs, the departure function for the Hemlholtz energy is particularly useful
The differential expression for A is
Integrating at constant T between A(T,V) and A(T,∞) gives
To obtain the departure function, we must also account for the change from V → ∞ to V = Vo, which occurs in the ideal-gas state
To avoid a singularity we also add and subtract
Evaluating Changes in Properties Using Departure Functions
Residual Helmholtz Energy
Residual Entropy
The entropy is obtained by differentiating A with respect to T
Evaluating Changes in Properties Using Departure Functions
Other Departure Functions
Now that A and S are known, other departure functions can be obtained through simple algebra
General Result
To determine the change in a thermodynamic property between two states, we evaluate two isothermal departure functions (at T1 and T2) and a temperature change contribution in the ideal gas state
departure function at T1
departure function at T2
Departure Functions for Volume-Explicit PVT EOSs
When volume-explicit PVT EOSs are available the following isothermal departure functions should be used
Evaluating Changes in Properties Using Departure Functions
Useful Ideal-Gas Property Relationships
Evaluating Changes in Properties Using Departure Functions
Heat Capacity Departure Functions (Pressure-Explicit PVT EOS)
The heat capacities are related to differentials of the entropy with temperature, for example Cv is defined as
Evaluating the derivative yields
The constant pressure heat capacity departure function requires a bit more work to obtain
Often times it is more convenient to obtain Cp – Cpo by first determining Cv – Cvo and then using an expression that relates Cv and Cp that will be introduced below
Evaluating Changes in Properties Using Departure Functions
Heat Capacity Departure Functions (Volume-Explicit PVT EOS)
With a volume-explicit PVT EOS the constant pressure heat capacity departure function is relatively straightforward to obtain
Evaluating the derivative yields
General Relationships Between Cv and Cp
Using the formalism introduced in Chapter 5 one can obtain the following relationship between Cv and Cp
The above expression is completely general and therefore can be used to relate heat capacities for real or ideal gases as well as for liquids and solids
Recalling the definitions of kT and ap, the expression above can equivalently be written as
Compressibility and Heat Capacity of Solids
Incompressible Substances
Most solids and many liquids far below their critical temperatures have very low compressibilities
In these situations one can assume perfectly or nearly incompressible behavior as a first-order approximation
An incompressible substance is defined as having a constant volume (V = constant) independent of T and P
From this definition, it follows that ap = kT = 0 for an incompressible substance
For a nearly incompressible substance the following expressions are appropriate (for a perfectly incompressible substance, the approximation becomes a true equality)
Derived Property Representations
Overview
Before the infusion of computers, charts or tubular data were used widely for engineering problem solving
Although computer-generated representations of thermodynamic properties are preferred, many graphical forms of data are still used in engineering practice
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)
Compressibility Factor
The volumetric PVT behavior of a material is often characterized by the compressibility factor Z
Typical values: ideal gas, Z = 1.0; liquids, Z ~ 0.01 to 0.2;
critical point, Zc ~ 0.27 to 0.29
Theorem of Corresponding States
This theorem suggests that values of thermodynamic properties of different fluids can be compared when properties are normalized by their values at the critical point
PVT Behavior of Fluids and the Theorem of Corresponding States
Theorem of Corresponding States
The reduced temperature and
constant for all materials
support this approach
The relation above indicates that all fluids have the same reduced volume Vr (as well as other properties) at a specified Tr and Pr
In Example 7.2 the principle of corresponding states was used to relate the a and b constants of the van der Waals equation of state to a fluids critical properties, specifically Tc and Pc
and
PVT Behavior of Fluids and the Theorem of Corresponding States
Generalized Coexistence Curve
The theorem of corresponding states can be used to obtain a generalized phase diagram
“Law of rectilinear diameters” – states that the average density in the L-V region varies linearly over a wide temperature range below the critical point
Zeno line – the Z = 1.0 line on
the phase diagram
repulsive forces dominate and
conversely, for states where
are dominant
PVT Behavior of Fluids and the Theorem of Corresponding States
Acentric Factor
To improve the accuracy of property predictions, Pitzer and coworkers introduced the acentric factor w as a third correlating parameter
The acentric factor was developed
as a measure of the difference in
structure between the material of
interest and a spherically symmetric
gas such as argon
to a molecular property
reduced vapor pressure
Overview
Although the corresponding states correlations are useful, the data is displayed in charts and graphs are therefore cumbersome to use
A more useful means to express a fluids PVT behavior is through an analytical expression (PVT equation of state)
Two general approaches
Empirical expressions – models formulated by determining parameters in a mathematical expression through a fit to experimental data (preferred by chemical engineers)
PVTN Equations of State for Fluids
Relating EOS Parameters to Experimental Data
Empirical models have parameters that must be determined to calculate PVT data
A common way to determine these parameters is to relate the model parameters to a fluid’s critical properties
By adopting this approach we are utilizing the theorem of corresponding states
The critical point criteria for a single-component system require
Recall Example 7.2, where these criteria were used to determine the parameters for the van der Waals EOS
and
Cubic Equations of State
Since the pioneering work of van der Waals, numerous modifications of the original vdW EOS have been published
Most are pressure-explicit cubic forms, i.e.,
van der Waals EOS
Redlich-Kwong
with
with
Redlich-Kwong-Soave
The attractive term is made temperature dependent and the acentric factor is introduced
with
Peng-Robinson
with
Hard-Sphere Equation of State
A class of EOSs that incorporate knowledge of the specific molecular interactions with additional fitted parameters
The repulsive term is theoretically based
The term bo below accounts for the excluded volume of a molecule
The attractive term is empirical
Here is one example of a hard-sphere EOS that uses the Carnahan-Starling repulsive term and van der Waals attractive term
s → molecular diameter
Virial-Type Equation of State
An important EOS class that has a rigorous basis in molecular theory (may be derived from statistical mechanics)
The compressibility is expressed as a polynomial in volume
Or as a polynomial expansion in pressure
Where
The coefficients B, C, D, … are termed the second, third, fourth, … virial coefficients
PVTN Equations of State for Fluids
Method of Abbott
The virial equation is truncated after the second virial coefficient term and the pressure is obtained
The second virial coefficient is expressed in terms of two functions of temperature B(0) and B(1) as well as the acentric factor
Where
and
Benedict-Webb-Rubin EOS
Has eight parameters
Frequently used for modeling the PVT behavior of light hydrocarbons and gases in petroleum and natural gas applications
Evaluating Changes in Properties Using Departure Functions
Overview
We now develop methods for determining changes in derived thermodynamic properties of a pure substance
Because properties are state functions, we can choose any convenient path to calculate the change in the property of interest
(e.g., H, S, U, A, or G)
To perform the calculation, we must generally have access to two constitutive relationships
A PVT or PVTN equation of state (EOS)
An ideal-gas state heat capacity equation
Evaluating Changes in Properties Using Departure Functions
Convenient Path
To calculate property changes between two well-defined states (T1,P1,V1) and (T2,P2,V2) we use the following three-step process
Isothermal expansion at T1 from P = P1 to 0 (or V = V1 to ∞
Isobaric (or isochoric) heating from T1 to T2 in an attenuated ideal gas state [P = 0 (or V = ∞)]
Isothermal compression at T2 from P = 0 to P2
(or from V = ∞ to V2)
Evaluating Changes in Properties Using Departure Functions
Convenient Path
The total property change between states 1 and 2, DB (where B is one of H, S, U, A, or G), is the sum over these three steps
Step (2) occurs entirely within the ideal-gas state
Recall that only 2 of the 3 intensive variables T, P, and V are independent
We select T as one independent variable and choose P or V based on the PVT EOS we have
If we have a pressure-explicit EOS (P = f (T,V)), we choose V
If we have a volume-explicit EOS (V = f (T,P)), we choose P
step (1)
step (2)
step (3)
Example 8.3
Develop expressions for DS between states 1 and 2 assuming you have analytic forms for (1) a volume-explicit EOS and (2) a pressure-explicit EOS and
Summary
Departure (Residual) Functions
A departure function is the difference between the property of interest in its real state at a specified T, P, and V and in an ideal-gas state at the same temperature T and pressure P
There are two equivalent forms of the departure function
B is any derived property (H, S, U, A, or G) and Vo = RT / P(T,V)
Evaluating Changes in Properties Using Departure Functions
Residual Entropy
Based on Example 8.3, the residual entropy is defined as
Evaluating Changes in Properties Using Departure Functions
Residual Helmholtz Energy
When working with pressure-explicit EOSs, the departure function for the Hemlholtz energy is particularly useful
The differential expression for A is
Integrating at constant T between A(T,V) and A(T,∞) gives
To obtain the departure function, we must also account for the change from V → ∞ to V = Vo, which occurs in the ideal-gas state
To avoid a singularity we also add and subtract
Evaluating Changes in Properties Using Departure Functions
Residual Helmholtz Energy
Residual Entropy
The entropy is obtained by differentiating A with respect to T
Evaluating Changes in Properties Using Departure Functions
Other Departure Functions
Now that A and S are known, other departure functions can be obtained through simple algebra
General Result
To determine the change in a thermodynamic property between two states, we evaluate two isothermal departure functions (at T1 and T2) and a temperature change contribution in the ideal gas state
departure function at T1
departure function at T2
Departure Functions for Volume-Explicit PVT EOSs
When volume-explicit PVT EOSs are available the following isothermal departure functions should be used
Evaluating Changes in Properties Using Departure Functions
Useful Ideal-Gas Property Relationships
Evaluating Changes in Properties Using Departure Functions
Heat Capacity Departure Functions (Pressure-Explicit PVT EOS)
The heat capacities are related to differentials of the entropy with temperature, for example Cv is defined as
Evaluating the derivative yields
The constant pressure heat capacity departure function requires a bit more work to obtain
Often times it is more convenient to obtain Cp – Cpo by first determining Cv – Cvo and then using an expression that relates Cv and Cp that will be introduced below
Evaluating Changes in Properties Using Departure Functions
Heat Capacity Departure Functions (Volume-Explicit PVT EOS)
With a volume-explicit PVT EOS the constant pressure heat capacity departure function is relatively straightforward to obtain
Evaluating the derivative yields
General Relationships Between Cv and Cp
Using the formalism introduced in Chapter 5 one can obtain the following relationship between Cv and Cp
The above expression is completely general and therefore can be used to relate heat capacities for real or ideal gases as well as for liquids and solids
Recalling the definitions of kT and ap, the expression above can equivalently be written as
Compressibility and Heat Capacity of Solids
Incompressible Substances
Most solids and many liquids far below their critical temperatures have very low compressibilities
In these situations one can assume perfectly or nearly incompressible behavior as a first-order approximation
An incompressible substance is defined as having a constant volume (V = constant) independent of T and P
From this definition, it follows that ap = kT = 0 for an incompressible substance
For a nearly incompressible substance the following expressions are appropriate (for a perfectly incompressible substance, the approximation becomes a true equality)
Derived Property Representations
Overview
Before the infusion of computers, charts or tubular data were used widely for engineering problem solving
Although computer-generated representations of thermodynamic properties are preferred, many graphical forms of data are still used in engineering practice
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