# Solid Phase Modeling in the Symmetry Platform

##### Introduction

The prediction of solubility of a solid in a liquid is important in different types of applications; for instance, separation processes that use crystallization as a route to purification of a species, design of heat exchangers for fluids where a dissolved solid can change the freezing point, or in flow assurance where cold temperatures can induce the formation of solids resulting in expansion and eventual bursting of pipes.

The objective of this communication is to present the implementation of the solid phase modeling in the Symmetry platform. The addition of the thermodynamic solid phase model is accompanied by the development of a liquid-solid flash and calculations to predict solid formation curves.

*The Solid phase model is available in the Symmetry platform’s Advanced Peng Robinson - based property packages: Advanced Peng Robinson (APR), Advanced Peng Robinson for Natural Gas
(APRNG) and Advanced Peng Robinson for Natural Gas 2 (APRNG2).*

##### Solid Phase Modeling

This section is dedicated to the consideration of thermodynamic relations that allow computation of the solubility of solids in liquids as a function of temperature.

The variation of the fugacity of the solid (*f ^{s}*) and its subcooled liquid (

*f*) can be evaluated in terms of the fundamental equation 1 [1]:

^{l}, (1)

where the second term of the right-hand side is known as the Poynting pressure correction. When equation 1 is applied to each phase, the conclusion is:

. (2)

For practical purposes the difference of specific volumes of condensed phases may be taken independent of pressure, but the enthalpy of fusion (*H _{l}* –

*H*) vary appreciably with the temperature and this behavior is described by [1]:

_{s}(3)

Where the subscript *tm* designated the melting temperature. When equation 3 is substituted in equation 2, the result is:

(4)

where the solid properties *T ^{tm}* and

*ΔH*represent the fusion temperature and the enthalpy of fusion, respectively. These pure component thermophysical properties can be obtained from correlations presented by Riazi [2].

^{tm}##### Solid Phase Equilibrium

To apply the solid thermodynamic model, it is necessary to compute the equilibrium between solid and fluid phases. By using the equations above coupled with a cubic equation of state (Advanced Peng Robinson for instance) for the fluid phases, it is possible to compute the Gibbs energy of any phase. Equilibrium occurs for the phase distribution for which the Gibbs energy is at a minimum.

To solve this problem, a solid PT flash algorithm was added to VMGThermo (the Symmetry platform’s Thermodynamic Engine), the algorithm uses a phase -stability test and a phase split procedure to compute the final material balance. The stability test can determine if it is possible to split a new phase off from an existing phase. The phase split procedure finds the equilibrium point for a given set of coexisting phases by adjusting the phase fractions and compositions until the Gibbs energy is at a minimum.

##### Solubility of Solids in Liquids

The prediction of the correct amount of solid solute present in the liquid phase is important in cryogenic processes; although the presented solid model has shown good accuracy results respect to freezing temperatures, some tuning is necessary to match experimental solubility data.

As an illustration, the solid- liquid equilibrium data obtained from the work of Kuebler and McKinley [3] was used to tune the interaction parameters of the cubic equation of state coupled with the solid model. The following plots show the comparison of the solid solute composition in the liquid phase after tuning using the APR property package for the following binary mixtures:

Methane – n-Hexane

Methane – n-Heptane

Methane – Benzene

Methane – Toluene

**Figure 1** Liquid composition at Solid – Liquid Equilibrium (SLE) of n-Hexane, n-Heptane, Benzene and Toluene with Methane after fine tuning.

The average absolute error between calculated and experimental data after tuning was less than 10% for all the studied binaries.

To avoid affecting any vapor-liquid equilibrium calculations, these new interaction parameters are part of a new set of matrices that are specifically used when solid flash calculations are
performed. *This feature can be exploited by users that want to run their own solid-liquid equilibrium regressions within the Symmetry platform*.

The APR matrix for vapor liquid solid equilibrium can be found in the list of Kij matrices as *Advanced Peng Robinson Z Factor (VLS)* and the APRNG and APRNG2 matrix is listed as
*APR for Natural Gas Interaction Parameter (VLS)*.

##### Pure Components Freezing Temperature

The freezing temperatures of most components is well predicted across different pressures, special efforts have been done for components of interest like Water and CO_{2}.

##### Solid Calculations in the Symmetry Platform

The Solids calculations are located in the *Material Stream*, *Envelope* and *Pipe Segment* unit operations from the Symmetry platform.

##### Material Stream Unit Operation

The Solid formation calculations can be found in the **Solids Formation** tab from the Material Stream, this tab needs to be activated from the **More Properties**
tab. The solid formation properties are under the **Solids** frame.

This frame shows different information about the solid phase; for instance; the solid formed and temperature of formation can be inspected here. A list to select components that can be frozen is also available.

A nice feature is the addition of Solid Formation Warning messages that can alert the user when the temperature of the stream is close to the solid temperature. This is very useful when monitoring freezing conditions across a flowsheet.

The solid phase properties can be enabled in the Equilibrium results tab by checking the Show Solid Phase in Eq. Results box.

##### Envelope Unit Operation

The solid formers curves can be seen in this unit operation; to activate their calculation, the **Solid** box at the bottom of the unit operation must be activated.

The Solid Formers list as well as the options for the solid curves calculations can be found in the **Settings** tab when the Solid box is activated.

##### Pipe Segment Unit Operation

Solid calculations are also available in the Pipe Segment trough the **Solids Formation** Tab from the **Profiles** Tab. If the **Pipeline Path** option
is selected, the freezing temperature of the first solid that can be formed is plotted across the pipe segment. Warnings regarding the appearance solids at the pipe conditions can be also be
triggered from here.

##### Example

Liquified Natural Gas (LNG) freezes at about -300 F (about 90 K). A typical LNG composition is given in the table below [4].

If we specify this composition in the Symmetry platform using the APRNG2 property package, the calculated freezing temperature is 90.1 K which is virtually the same as the expected value of 90 K. The lightest and heaviest components were chosen as solid formers and, as one can anticipate, the first compound that crystallizes at the specified pressure (6400 kPa) is the heaviest one (n-Pentane).

##### References

[1] **Walas, S. M.**, Phase Equilibria in Chemical Engineering, Stoneham, MA: Butterworth Publishers, 1985

[2] **Riazi, M. R.** Characterization and Properties of Petroleum Fractions, West Conshohocken, PA: ASTM International, 2005

[3] **Kuebler, G.P., McKinley, C.**, Solubility of Solid Benzene, Toluene, n-Hexane, and n-Heptane in Liquid Methane" *Advancements in Cryogenic Enginnering,* vol. 19,
1995, 320-326

[4] **Liquid Gas Carrier**,Safety and Operational Matters – LNG Handling, 2018

Herbert Loria, Ph.D, P.Eng, VMG Calgary

Please contact your local VMG office for more information.