Thermodynamic process control based on pseudo-density root for equation of state

A system for thermodynamic modeling is provided. The system comprises a computer having a processor, a thermodynamic process simulation application, and a thermodynamic equation of state application. When executed by the processor, the thermodynamic equation of state application determines a density root based on a first and second point of departure from an equation of state and based on a first and a second extrapolation equation. The first departure point satisfies the equationThe second departure point satisfies the equationThe density root is determined as a pseudo-density in a phase two when the specified pressure is greater than the second departure point pressure and in a phase one when the specified pressure is less than the first departure point pressure. When executed by the processor, the thermodynamic process simulation application invokes the thermodynamic equation of state application to determine a result based on the density root.

CROSS-REFERENCE TO RELATED APPLICATIONS

Not applicable.

REFERENCE TO A MICROFICHE APPENDIX

Not applicable.

BACKGROUND

In the drive for ongoing improvements in operating efficiency, industrial plants such as chemical plants, refineries, food processing plants, pharmaceutical plants, breweries, and other batch and continuous plant systems may employ computer-based modeling and simulation to optimize plant operations. These modeling systems are typically used to simulate plant processes by defining components and equipment of plants in computer models and then using mathematical computations to project and/or to reveal the behavior of these systems as relevant parameters vary.

This type of modeling may be used to aid in the design and operation of such plants, as well as to provide computer-based training of operators by simulating plant and process responses to variations that can arise in real-world situations without the hazards or costs associated with subjecting plants to these events. In addition, predictions can be made about plant behavior in order to devise tactics for handling such events, should they occur. This type of modeling can also be used to assist in controlling plant operations by predicting system changes and responding accordingly by tying the information produced by the models into control loops of plant equipment.

Modeling of these systems typically involves iterative calculations of complex thermodynamic equations in order to accurately describe static views of dynamic situations. Given the rapidly changing state of these systems and the limitation of only being able to calculate discrete moments in time, this form of modeling can place heavy demands on a computer's central processing unit (CPU) as constant updating is needed to keep the model updated. This heavy processing load challenges the ability to provide accurate data with sufficient speed to obtain predictive models in time to proactively forestall critical situations, thereby rendering plant control in a real world application difficult or impossible.

SUMMARY

In an embodiment, a system is disclosed. The system comprises a computer system, a thermodynamic process simulation application, and a thermodynamic equation of state application. The computer system comprises at least one processor. The thermodynamic equation of state application, when executed by the at least one processor of the computer system, determines a density root based on at least a specified pressure, a specified temperature, and a first point of departure from an equation of state. The first point of departure is determined based on a proportional relationship between the ratio of pressure to density of the equation of state at the first point of departure and the rate of change of pressure with respect to density of the equation of state at the first point of departure, and wherein the density root is determined as a pseudo-density when the specified pressure is less than the pressure at the first point of departure. The thermodynamic process simulation application executes on the at least one processor of the computer system and invokes the thermodynamic equation of state application iteratively to determine a result based on the density root determined by the thermodynamic equation of state application.

In an embodiment, a system comprising a computer system, a thermodynamic process simulation application, and a thermodynamic equation of state application is disclosed. The computer system comprises at least one processor. The thermodynamic equation of state application, when executed by the at least one processor of the computer system, determines a density root based on at least a specified pressure, a specified temperature, a specified composition, and a second point of departure from an equation of state. The second point of departure is determined based at least in part on a proportional relationship between the rate of change of pressure with respect to density of the equation of state and the universal gas constant. The density root is determined as a pseudo-density when the specified pressure is greater than the pressure at the second point of departure. The thermodynamic process simulation application executes on the at least one processor of the computer system and invokes the thermodynamic equation of state application iteratively to determine a result based on the density root determined by the thermodynamic equation of state application.

In an embodiment, a system comprising a computer system, a thermodynamic simulation application, and a thermodynamic equation of state application is disclosed. The computer system comprises at least one processor. The thermodynamic equation of state application, when executed by the at least one processor of the computer system, determines a density root ρ based on at least a specified pressure P, a specified temperature, a specified composition, and a first point of departure from an equation of state (ρdp1,Pdp1), wherein when the specified pressure is less than Pdp1the density root ρ is determined as a pseudo-density based on an extrapolation equation that comprises a density squared term. The thermodynamic process simulation application, when executed by the at least one processor of the computer system, invokes the thermodynamic equation of state application iteratively to determine a result based on the density root determined by the thermodynamic equation of state application. The system processes the result determined by the thermodynamic process simulation application to execute at least one action from the following group of actions: controlling a thermodynamic process control component, training an operator of the thermodynamic process control component, predicting a failure time of the thermodynamic process control component, and validating a design for the thermodynamic process control component.

In an embodiment, a system comprising a computer system, a thermodynamic process simulation application, and a thermodynamic equation of state application is disclosed. The computer system comprises at least one processor. The thermodynamic equation of state application, when executed by the at least one processor of the computer system, determines a density root ρ based on at least a specified pressure P, a specified temperature, a specified composition, and a second point of departure from an equation of state (ρdp2,Pdp2), wherein when the specified pressure is greater than Pdp2the density root ρ is determined as a pseudo-density based on an extrapolation equation

P=Pdp⁢⁢2+d⁡(1-g1-k⁢⁢ρ)⁢(ρ-ρdp⁢⁢2)+m⁡(1-g1-k⁢⁢ρ)⁢(ρ-ρdp⁢⁢2)2+Ω
where d, g, k, and m are constants and where Ω is an optional offset. The thermodynamic process simulation application, when executed on the at least one processor of the computer system, invokes the thermodynamic equation of state application iteratively to determine a result based on the density root determined by the thermodynamic equation of state application. The system processes the result determined by the thermodynamic process simulation application to execute at least one action from the following group of actions: controlling a thermodynamic process control component, training an operator of the thermodynamic process control component, predicting a failure time of the thermodynamic process control component, and validating a design for the thermodynamic process control component.

In another embodiment, a system comprising a computer system, a thermodynamic process simulation application, and a thermodynamic equation of state application is disclosed. The computer system comprises at least one processor. The thermodynamic equation of state application, when executed by the at least one processor of the computer system, determines a density root based on at least a specified pressure, a specified temperature T, a specified composition, a first point of departure from an equation of state, and a second point of departure from the equation of state. The first point of departure is determined as the point (ρdp1,Pdp1) on the isothermic curve of pressure P versus density ρ at the specified temperature derived from the equation of state where

∂P∂ρ=β⁢∂P∂ρ+Ω
where β is a constant selected subject to the constraint β≧0.5 and where Ω is an optional offset. The second point of departure is determined as the point (ρdp2,Pdp2) on the isothermic curve of pressure P versus density ρ at the specified temperature derived from the equation of state where

α⁡(∂P∂ρ-R)+(1-α)⁢∂P∂ρ⁢|dp⁢⁢1=0
where a=f(T), where a is a non-negative number less than or equal to 1.0, where R is the universal gas constant, and where

∂P∂ρ⁢|dp⁢⁢1
is a constant equal to the value of the partial derivative of pressure P with respect to density ρ of the equation of state at the first point of departure for the specified temperature. Further, ρdp2is less than ρdp1. The density root is determined as a pseudo-density when the specified phase is a phase two and the specified pressure is greater than the pressure at the point (ρdp2,Pdp2). The density root is determined as a pseudo-density when the specified phase is a phase one and the specified pressure is less than the pressure at the point (ρdp1,Pdp1). The thermodynamic process simulation application executes on the at least one processor of the computer system and invokes the thermodynamic equation of state application iteratively to determine a result based on the density root determined by the thermodynamic equation of state application. The system one of controls a thermodynamic process control component, trains an operator of the thermodynamic process control component, and predicts a failure time of the thermodynamic process control component based on the result determined by the thermodynamic process simulation application.

In an embodiment, a computer program product for a thermodynamic modeling system is disclosed. The computer program product comprises a computer readable storage medium having computer usable program code embodied therein. The computer usable program code determines a density root based on at least a specified pressure, a specified temperature, a specified state, and a first point of departure from an equation of state. The first point of departure is determined based on a proportional relationship between the ratio of pressure to density of the equation of state at the first point of departure and the rate of change of pressure with respect to density of the equation of state at the first point of departure. The density root is determined as a pseudo-density when the specified state is a first state and when the specified pressure is less than the pressure at the first point of departure.

DETAILED DESCRIPTION

The present disclosure teaches a system and method for modeling and controlling thermodynamic systems. The method can be executed on a computer to calculate and thereby simulate and/or model the characteristics of thermodynamic systems. The method comprises determining pseudo-properties over a dynamically determined portion of the range of an independent variable of a thermodynamic equation of state. Some equations of state may have a form P=EOS(T,x,ρ), where EOS( ) represents the subject equation of state, where P represents pressure, T represents temperature,x={x1, x2, . . . , xn} represents a mole fraction of an n-component material mixture and/or composition that is the subject of the thermodynamic analysis, for example a mixture of ethane, butane, methane, and other hydrocarbons, and ρ is the density of the material mixture. In an embodiment, the method comprises identifying a first departure point from a curve of pressure versus density at a constant temperature and for a given material composition determined from the equation of state, a second departure point from the curve of pressure versus density. The first departure point is associated with a first phase of the material and the second departure point is associated with a second phase of the material, for example a liquid phase and a vapor phase.

The method also comprises identifying a first extrapolation equation associated with the first phase of the material and a second extrapolation equation associated with the second phase of the material. When the method is invoked for a material in the first phase at a specified pressure lower than the pressure at the first departure point, the first extrapolation equation is used to determine a pseudo-density property. When the method is invoked for a material in the second phase at a specified pressure higher than the pressure at the second departure point, the second extrapolation equation is used to determine the pseudo-density property. In an embodiment, the first departure point (ρdp1,Pdp1) is determined based on the equation

∂P∂ρ∝Pρ.
In an embodiment, the second departure point (ρdp2,Pdp2) is determined based on the equation

∂P∂ρ=R,
where R is the universal gas constant. In an embodiment, the first extrapolation equation has the form P=Pdp1+b(ρ−ρdp1)+c(ρ−ρdp1)2, where b and c are constants. In an embodiment, the second extrapolation equation has the form P=f(ρ), where f(ρ) is quadratic in ρ and where f(ρ) asymptotically approaches the equation of state as P increases and/or at high values of the specified pressure P.

In some known thermodynamic modeling and/or simulation systems the algorithms may too frequently fail to converge to a consistent thermodynamic state solution and the algorithm fails. In other circumstances the known algorithms may converge to a consistent thermodynamic state solution, but too slowly to permit use of the thermodynamic state solution in real-time applications. The known equation of state algorithms are, at least in part, the cause of these failures to converge timely to a consistent thermodynamic state solution. As known to those skilled in the art, computer solutions for quadratic functions are generally more efficient than computer solutions for logarithmic functions, hence the two extrapolation equations identified above may promote improved computational efficiency when determining thermodynamic properties versus other known extrapolation equations, hence enabling the use of the thermodynamic simulation and/or modeling system in real-time applications. Further, the methods for choosing the first and second departure points taught by the present disclosure may, at least in some circumstances, promote more consistent and reliable convergence on a consistent thermodynamic state solution.

FIG. 1illustrates a system100suitable for generating models that simulate and control the physical characteristics of a thermodynamic system according to the embodiments of the disclosure. A computer110includes a memory that stores and a processor that invokes a thermodynamic process simulation application120and a thermodynamic equation of state application130. The thermodynamic process simulation application120and the thermodynamic equation of state application130together implement a thermodynamic model that can be used, for example, to control a thermodynamic process in a plant, train an operator of the thermodynamic process or the plant, to predict a future behavior of the thermodynamic process, and to validate a design of a thermodynamic process component, a thermodynamic process control component, and/or a thermodynamic process.

In an embodiment, by modeling thermodynamic processes to determine a result, such as a flash condition, the computer110may control the thermodynamic process in a plant170. The flash condition may be considered in association with a thermodynamic component, for example, but not by way of limitation, a stripper column, a distillation column, an extractor column, an absorber column, and a compressor. In an embodiment, the result also may comprise a flash condition in a flash evaporator, a distillation condition in a distillation column, an absorption condition in an absorber column, and/or a stripping condition in a stripper column. In another embodiment, by modeling thermodynamic processes, possibly faster than in real-time, to predict a future state of the thermodynamic process, the computer110may anticipate an undesirable and/or hazardous operating condition before it occurs and take corrective action automatically and/or notify an operator to take corrective action. Corrective actions may comprise adjusting one or more operating parameters, for example a material mixture input stream flow rate associated with the thermodynamic process. Corrective actions may comprise shutting down one or more motors and/or heaters coupled to the thermodynamic process. By modeling thermodynamic processes, the computer110may forecast operating conditions of the thermodynamic process one minute, five minute, ten minutes ahead, based on current process parameters and control inputs. In an embodiment, the thermodynamic process modeling and/or simulation, executed by the computer110, may forecast the future failure of a thermodynamic process component and/or a thermodynamic process control component, allowing a replacement to be ordered and replacement activities to be scheduled, for example at a time which does not interrupt an in-progress batch process, thereby averting costs associated with wasted materials.

In an embodiment, the computer110may receive measurements of thermodynamic variables from the plant170via a network150, for example from sensors coupled to thermodynamic components in the plant170such as chambers of a fractionation tower and/or a distillation tower. Sensors of thermodynamic variables may include temperature sensors, pressure sensors, and the like.

The network150may be provided by any of a local area network, a public switched telephone network (PSTN), a public data network (PDN), and a combination thereof. Portions of the network150may be provided by wired connections while other portions of the network150may be provided by wireless connections. Based on the values of the thermodynamic variables, the computer110may invoke the thermodynamic process simulation application120to determine control and/or command values. The computer110may then transmit the control and/or command values via the network150to a process controller160, where the process controller160is coupled to the plant170and/or a thermodynamic process component in the plant170via network150. The process controller160may control the plant170and/or one or more thermodynamic process components in the plant170based on the control and/or command values received from the computer110.

The system100may further comprise a workstation140that may provide a user interface for an operator to interact with the computer110and/or the thermodynamic process simulation application120. In an embodiment, a trainee may use the workstation140, in association with the computer110and the thermodynamic process simulation application120, to simulate a variety of virtual events associated with the plant170, for example a motor tripping off line, and the result of the trainee's response to the virtual event in the simulated behavior of the plant170. This may permit trainees to learn valuable plant management lessons in a safe and consequence-free environment. In an embodiment, a manager of the plant170may use the workstation140to model the operation of a variety of thermodynamic process components of the plant170at different operating points, to analyze advantages and disadvantages associated with operating the plant170at these operating points. For example, an increased material throughput may be associated with higher operating costs per unit of product output, but in market conditions of elevated prices for the product, greater profit may nevertheless result from the increased throughput.

The thermodynamic process simulation application120and the thermodynamic equation of state application130may be stored in the memory of the computer110. Computers are discussed in more detail hereinafter. In an embodiment, other thermodynamics applications may be stored in the memory of the computer110and executed by the processor of the computer110. Alternatively or additionally, the thermodynamic process simulation application120and the thermodynamic equation of state application130may be stored on one or more computer readable media, for example floppy disks, compact disks, optical disks, magnetic tapes, magnetic disks, and other computer readable storage media. In an embodiment, the thermodynamic process simulation application120and/or the thermodynamic equation of state application130may be copied and/or loaded from the computer readable media to the computer110, for example to a secondary storage of the computer110, to a non-volatile memory of the computer110, or to a volatile memory of the computer110. In an embodiment, the thermodynamic process simulation application120and/or the thermodynamic equation of state application130may be executed by a processor of the computer110reading the instructions implementing the thermodynamic process simulation application120and/or the thermodynamic equation of state application130from the computer readable media, from secondary storage, from non-volatile memory, and/or from volatile memory. In an embodiment, the instructions or a portion of the instructions implementing the thermodynamic process simulation application120and/or the thermodynamic equation of state application130may be transmitted to the computer110from the network150, via either a wired and/or a wireless communication link.

The computer110invokes the thermodynamic process simulation application120, and the thermodynamic process simulation application120may iteratively invoke the thermodynamic equation of state application130to determine a thermodynamic result. As known to those skilled in the art, the thermodynamic process simulation application120may invoke the thermodynamic equation of state application130with specified values that deviate from feasible thermodynamic state values, for example while the thermodynamic process simulation application120is in the process of converging on a consistent solution of thermodynamic state for a thermodynamic system, volume, and/or process. In an embodiment, the thermodynamic equation of state application130may return pseudo-properties when invoked with infeasible values. In an embodiment, it may be desirable that the pseudo-properties returned by the thermodynamic equation of state application130promote convergence of the solution sought by the thermodynamic process simulation application120.

In an embodiment, the thermodynamic equation of state application130determines a density ρ from an equation of state based on at least a specified temperature T and a specified pressure P. Temperature T may be represented in units of degrees Kelvin. Pressure P may be represented in units of Pascals. Density ρ may be represented in mole/liter. In an embodiment, a compositionx={x1, x2, . . . , xn} that represents a mole fraction of an n-component material mixture and/or composition that is the subject of the thermodynamic analysis, also may be specified. The compositionxhaving n components may comprise n variables {x1, x2, . . . , xn}, where x comprises a mole of the subject material and each xirepresents the mole fraction of the associated component. For example, a mole quantity of a material mixture comprised of 0.2 mole of ethane, 0.3 mole of butane, and 0.5 mole of methane may be represented asx={x1=0.2,x2=0.3,x3=0.5}. While the quantity of the subject material may be analyzed based on a reference mole quantity, the actual quantity of the subject material within the thermodynamic process may be a different quantity. While the above units are consistent with the International System of Units (SI), any consistent system of units may be employed. In an embodiment, a phase or an assumed phase of the material may also be specified.

The equation of state solved by the thermodynamic equation of state application130may be represented as:
P=EOS(T,x,ρ)  (1)
A number of different equations of state are known to those skilled in the art. In an embodiment, the processing of the thermodynamic equation of state application130may be based on a Soave-Redlich-Kwong (SRK) equation of state or another equation of state derived from the SRK equation of state, but in other embodiments the thermodynamic equation of state application130may be based on a different equation of state. Different equations of state may be preferred for modeling and analyzing different thermodynamic systems, and it is contemplated that the teachings of the present disclosure may be applied to these different equations of state.

FIG. 2illustrates a family of pressure versus density curves200plotted from an equation of state having the general form of equation 1 set forth above. A first curve230represents pressure versus density at a constant temperature T1for the specified materialx. A second curve220represents pressure versus density at a constant temperature T2for the specified materialx. A third curve210represents pressure versus density at a constant temperature T3for the specified materialx. In the case of the exemplary curves210,220,230, temperature T1is less than temperature T2, and temperature T2is less than temperature T3. Note that at low temperature T1, the curve230plotted based on the equation of state may exhibit generally cubic structure while at high temperature T3, the curve210plotted based on the equation of state may exhibit generally linear structure. Further, note that over a middle portion of curve230, the pressure P decreases with increasing density ρ (the equation of state is a decreasing function having negative slope over this middle portion of curve230) while over an initial portion and over a later portion of the curve230, the pressure P increases with increasing density ρ (the equation of state is an increasing function having positive slope over these initial and later portions of the curve230).

In some thermodynamic analysis models it may be preferred to employ one or more extrapolation equations in the place of the equation of state to determine thermodynamic properties at least over a portion of the range of the density ρ, for example over a middle portion of the range of the density ρ. The properties determined based on the extrapolation equations may be referred to as pseudo-properties. Generally, it may also be preferred to minimize the range of density ρ over which pseudo-properties are determined.

Turning now toFIG. 3, a first set of extrapolation curves associated with the temperature T1is described. In an embodiment, a first departure point242associated with a first phase of the material is located at point (ρdp1,Pdp1) and a second departure point244associated with a second phase of the material is located at point (ρdp2,Pdp2). In some contexts, the first phase of the material may be referred to as phase one and the second phase of the material may be referred to as phase two. In an embodiment, the phase one may be a liquid phase of the material and the phase two may be a vapor phase of the material. In other embodiments, however, the phase one and the phase two may be other phases of the material. When the material is specified to be in phase one and the pressure is greater than Pdp1, the thermodynamic properties of the material may be determined using the equation of state. When the material is specified to be in phase one and the pressure is less than Pdp1, the thermodynamic properties of the material are determined as pseudo-properties based on using a first extrapolation equation and the dotted line curve246is produced. For example, when the material is specified to be in phase one and the pressure is less than Pdp1, a density root may be determined as a pseudo-density by the thermodynamic equation of state application130using the first extrapolation equation and returned to the thermodynamic process simulation application120.

When the material is specified to be in phase two and the pressure is less than Pdp2, the thermodynamic properties of the material may be determined using the equation of state. When the material is specified to be in phase two and the pressure is greater than Pdp2, the thermodynamic properties of the material are determined as pseudo-properties based on using a second extrapolation equation and the dotted line curve248is produced. For example, when the material is specified to be in phase two and the pressure is greater than Pdp2, a density root may be determined as a pseudo-density by the thermodynamic equation of state application130using the second extrapolation equation and returned to the thermodynamic process simulation application120.

In some contexts, the first departure point242may be referred to as a phase one departure point and the second departure point244may be referred to as a phase two departure point. When phase one corresponds to a liquid phase of the material, the first departure point242may be referred to as a liquid departure point. When phase two corresponds to a vapor phase of the material, the second departure point may be referred to as a vapor departure point.

In an embodiment, the first departure point242may be determined, for a specified temperature T, based on a proportional relationship between the ratio of pressure to density of the equation of state at the first point of departure and the rate of change of pressure with respect to density of the equation of state at the first point of departure. In an embodiment, the first departure point242may be determined as the point (ρdp1,Pdp1) that satisfies the equation

∂P∂ρ=Pρ(2)
In words, the first departure point242may be determined as the point (ρdp1,Pdp1) where the tangent line to the curve of pressure P versus density ρ coincides with a line through the point and the origin of the pressure P versus density ρ axes. In another embodiment, however, the criteria of equation 2 may be relaxed somewhat, and the first departure point242may be determined as the point (ρdp1,Pdp1) that satisfies the equation

∂P∂ρ=β⁢Pρ+Ω(3)
where β is a constant chosen based on the constraint β>0.5 and where Ω is an optional offset. Finding the departure point as the point that satisfies equation 3 still may be said to be based on a proportional relationship between the ratio of pressure to density of the equation of state at the first point of departure and the rate of change of pressure with respect to density of the equation of state at the first point of departure, notwithstanding the inclusion of the Ω optional offset. The value of Ω may be zero (0), may be a non-zero constant value, or may be a function of one or more thermodynamic parameters of the plant170and/or the process controller160. In an embodiment, the value of Ω may be a function of temperature T. In this case, for a specific value of temperature T, the value of Ω may be considered to be a constant. In another embodiment, β is a constant chosen based on the constraint 3.0≧β≧0.7. In another embodiment, β=1.0, in which case equation 3 is substantially identical to equation 2. In another embodiment, β is a constant chosen so that the tangent line to the curve of pressure P versus density ρ at the point (ρdp1,Pdp1) makes an acute angle of less than 20 degrees with the line through the point (ρ=0,P=0) and the point (ρdp1,Pdp1). In another embodiment, β is a function of temperature T. In this case, for a specific value of temperature T, β may be considered to be a constant.

In an embodiment, the first extrapolation equation comprises a density squared term and may be said to be quadratic in density ρ. In an embodiment, the first extrapolation equation may be defined as
P=Pdp1+b(ρ−ρdp1)+c(ρ−ρdp1)2+Γ  (4)
where b and c are constants and where F is an optional offset. The value of Γ may be zero (0), may be a non-zero constant value, or may be a function of one or more thermodynamic parameters of the plant170and/or the process controller160. In an embodiment, the constants b and c may be defined by

∂P∂ρ⁢|dp⁢⁢1
is the partial derivative of pressure P with respect to density ρ at (ρdp1,Pdp1), where ρspis the density ρ at the spinoidal point of the equation of state and where zdp1is the compressibility of the material at the first departure point. In another embodiment, however, the constants b and c may be defined differently and have different values. In an embodiment, the first extrapolation equation promotes convergence of the thermodynamic state solution processing in the thermodynamic process simulation application120.

In an embodiment, the second departure point244may be determined based, at least in part, on a proportional relationship between the rate of change of pressure with respect to density of the equation of state and the universal gas constant. In an embodiment, the second departure point244may be determined as the point (ρdp2,Pdp2) that satisfies the equation

∂P∂ρ=R(8)
where R is the universal gas constant. In one system of units, the universal gas constant R may be approximated as

R≈8.314472⁢Pascals×Meters3mole×°⁢⁢K.(9)
The universal gas constant R in some embodiments may be extended to additional significant figures, shortened to fewer significant figures, expressed as a different value, and/or expressed according to a different system of units. In another embodiment, the criteria of equation 8 may be relaxed somewhat, and the second departure point244may be determined as the point (ρdp2,Pdp2) that satisfies the equation

∂P∂ρ=δ⁢⁢R(10)
where R is the universal gas constant and where δ is a constant chosen based on the constraint 0≦δ≦10.

In another embodiment, the second departure point244may be defined as the point (ρdp2,Pdp2) that satisfies the equation

α⁡(∂P∂ρ-R)+(1-α)⁢∂P∂ρ⁢|dp⁢⁢1=Δ(11)
where a=f(T), with the constraint that 0<a≦1 and

ⅆαⅆT<0,
and where Ω is an optional offset. The value of Δ may be zero (0), may be a non-zero constant value, or may be a function of one or more thermodynamic parameters of the plant170and/or the process controller160. In an embodiment, the value of Δ may be a function of temperature T. In this case, for a specific value of temperature T, the value of Δ may be considered to be a constant. The general effect of equation 11 is that for low temperatures and for Δ=0, the second departure point244is determined substantially according to equation 8 above while for high temperatures, the second departure point244approaches the first departure point242from below (e.g., where ρdp2<ρdp1and Pdp2<Pdp1). In an embodiment, f(T)<0.2 for T>2000° K. In another embodiment, f(T)=1.0 for all non-negative values of T, under which condition equation 11 is substantially the same as equation 8. In some thermodynamic simulation and/or modeling situations, equation 11 may provide advantages that overcome the disadvantages of the added complexity.

In an embodiment, the second extrapolation equation comprises a density squared term and may be said to be quadratic in density ρ. In an embodiment, the second extrapolation equation may be defined as

P=Pdp⁢⁢2+d⁡(1-g1-k⁢⁢ρ)⁢(ρ-ρdp⁢⁢2)+m⁡(1-g1-k⁢⁢ρ)⁢(ρ-ρdp⁢⁢2)2+Λ(12)
where d, g, k, and m are constants, and where Λ is an optional offset. The value of Λ may be zero (0), may be a non-zero constant value, or may be a function of one or more thermodynamic parameters of the plant170and/or the process controller160. In an embodiment, the value of Λ may be a function of temperature T. In this case, for a specific value of temperature T, the value of Λ may be considered to be a constant. In an embodiment, the constants d, g, and m may be defined as

∂P∂ρ⁢|dp⁢⁢2
is the partial derivative of pressure P with respect to density ρ of the equation of state at the second departure point (ρdp2,Pdp2). In an embodiment, k is chosen such that 1/k is equal to the asymptote of the equation of state as pressure P increases. Alternatively, in an embodiment, k is chosen such that second extrapolation equation asymptotically approaches the equation of state at high specified pressure. In an embodiment, the second extrapolation equation promotes convergence of the thermodynamic state solution processing in the thermodynamic process simulation application120.

The equations for determining the departure points242,244and the first and second extrapolation equations may be employed to determine thermodynamic properties at any temperature. Turning now toFIG. 4, a second set of extrapolation curves associated with the temperature T2is described. A third departure point252and a fourth departure point254determined based on the equations above for the exemplary curve220are illustrated. Note that the third departure point252and the fourth departure point254are closer to each other than the first departure point242is to the second departure point244, a general result of the approach to determining departure points taught by the present disclosure. The first extrapolation equation may be used to generate the dotted line curve256and the second extrapolation equation may be used to generate the dotted line curve258, each of which represent pseudo-properties.

In an embodiment, the thermodynamic equation of state application130may employ the determination of the first departure point (ρdp1,Pdp1) from an equation of state as described above in combination with previously known methods for determining thermodynamic properties and parameters from equations of state. For example, the paper “Effective Utilization of Equations of State for Thermodynamic Properties in Process Simulation” by P. M. Mathias, et al., published March 1984, in American Institute of Chemical Engineers Journal Volume 30, number 2, which is hereby incorporated by reference for all purposes, describes known techniques for determining thermodynamic properties. In an embodiment, the thermodynamic equation of state application130may employ the determination of the second departure point (ρdp2,Pdp2) from an equation of state as described above in combination with previously known methods for determining thermodynamic properties and parameters from equations of state. In an embodiment, the thermodynamic equation of state application130may employ the first extrapolation equation from an equation of state as described above in combination with previously known methods for determining thermodynamic properties and parameters from equations of state. In an embodiment, the thermodynamic equation of state application130may employ the second extrapolation equation from an equation of state as described above in combination with previously known methods for determining thermodynamic properties and parameters from equations of state.

In some situations, multiple of the determinations of departure points taught by the present disclosure may be combined with previously known extrapolation equations by the thermodynamic equation of state application130to determine thermodynamic properties and/or parameters. For example, the first departure point (ρdp1,Pdp1) and the second departure point (ρdp2,Pdp2) determined as described above may be combined with previously known extrapolation equations by the thermodynamic equation of state application130to determine thermodynamic properties and/or parameters. In some situations, multiple of the extrapolation equations taught by the present disclosure may be combined with previously known techniques for determining departure points from equations of state by the thermodynamic equation of state application130to determine thermodynamic properties and/or parameters. In a preferred embodiment, the determinations of the first and second departure points and two of the extrapolation equations taught by the present disclosure may be employed by the thermodynamic equation of state application130.

In an embodiment, portions of the system100described above may be provided as a computer program product. For example, in an embodiment, the thermodynamic process simulator application120and/or the thermodynamic equation of state application130may be provided as a computer program product. In another embodiment, the thermodynamic process simulator application120and/or the thermodynamic equation of state application130may be part of a thermodynamics modeling application (not shown) that may be provided as a computer program product. The computer program product may comprise one or more computer readable storage medium having computer usable program code embodied therein implementing the functionality of the thermodynamics modeling application, the thermodynamic simulation application120, and/or the thermodynamic equation of sate application130. The computer program product may be embodied in removable computer storage media and/or non-removable computer storage media. The computer program product may be suitable for loading, by the computer110, at least portions of the contents of the computer program product to secondary storage, non-volatile memory, and/or volatile memory of the computer110. The computer program product may include data, data structures, files, executable instructions, and other information. A portion of the computer program product may comprise instructions that promote the loading and/or copying of data, data structures, files, and/or executable instructions to the secondary storage, the non-volatile memory, and/or volatile memory of the computer110.

FIG. 5illustrates a computer system380suitable for implementing the computer110described above. The computer system380includes a processor382(which may be referred to as a central processor unit or CPU) that is in communication with memory devices including secondary storage384, read only memory (ROM)386, random access memory (RAM)388, input/output (I/O) devices390, and network connectivity devices392. The processor382may be implemented as one or more CPU chips.

The secondary storage384is typically comprised of one or more disk drives or tape drives and is used for non-volatile storage of data and as an over-flow data storage device if RAM388is not large enough to hold all working data. Secondary storage384may be used to store programs which are loaded into RAM388when such programs are selected for execution. The ROM386is used to store instructions and perhaps data which are read during program execution. ROM386is a non-volatile memory device which typically has a small memory capacity relative to the larger memory capacity of secondary storage384. The RAM388is used to store volatile data and perhaps to store instructions. Access to both ROM386and RAM388is typically faster than to secondary storage384.

The processor382executes instructions, codes, computer programs, scripts which it accesses from hard disk, floppy disk, optical disk (these various disk based systems may all be considered secondary storage384), ROM386, RAM388, or the network connectivity devices392. While only one processor382is shown, multiple processors may be present. Thus, while instructions may be discussed as executed by a processor, the instructions may be executed simultaneously, serially, or otherwise executed by one or multiple processors.