Population balance modeling: application in nanoparticle formation through rapid expansion of supercritical solution

Abstract

The rapid expansion of supercritical solution (RESS) process is a novel method to produce free-solvent particles with narrow particle size distribution (PSD), and it is offered to replace conventional particle formation methods. The present study is aimed to investigate the numerical simulation of the RESS process comprising hydrodynamic and particle formation and solving population balance equation to compute PSD of solute-supercritical CO₂ (SC-CO₂) systems. Mass, momentum and energy balances in addition to an accurate equation of state are used to calculate hydrodynamic behavior of SC-CO₂ at various conditions through a nozzle and supersonic free jet with considering heat exchange in the nozzle. The solubility of TBTPP (5, 10, 15, 20-tetrakis (3, 5-bis (trifluoromethyl) phenyl) porphyrin) in SC-CO₂ is calculated using the Soave–Redlich–Kwong equation of state with van der Waals mixing rule. After that, supersaturation and nucleation of TBTPP and three other drugs, i.e., aspirin, salicylic acid and ibuprofen, are investigated. Eventually, the method of moments is used to solve the population balance equations to determine PSD of the solutes. The results are presented with and without coagulation. Furthermore, the effect of nozzle length and pre-expansion temperature on PSD is studied. Furthermore, to improve the results of coagulation model on PSD, collision frequency and lognormal function are modified. The modifications improve the coagulation results. The average absolute percent deviation of TBTPP solubility is 1.86. The CO₂ hydrodynamic behavior shows the same trend as reported in the literature. Moreover, results of PSD calculation with coagulation indicated there is a good agreement with the experimental ones.

Appendix

Soave wrote SRK EoS in the following form [26]:

$$ P = \frac{RT}{v - b} - \frac{{a_{c} \alpha }}{{v\left( {v + b} \right)}} $$

(36)

$$ a_{c} = 0.42747\frac{{\left( {RT_{c} } \right)^{2} }}{{P_{c} }} $$

(37)

$$ b = 0.08664\frac{{RT_{c} }}{{P_{c} }} $$

(38)

$$ \alpha = \left[ {1 + m\left( {1 - \sqrt {\frac{T}{{T_{c} }}} } \right)} \right]^{2} $$

(39)

$$ m = 0.48 + 1.574\omega - 0.176\omega^{2} $$

(40)

where $P_{c}$, $T_{c}$ and $\omega$ are critical pressure, temperature and acentric factor, respectively. Also, Eq. (36) in terms of compressibility factor would be [26]:

$$ Z^{3} - Z^{2} + \left( {A - B - B^{2} } \right)Z - AB = 0 $$

(41)

$$ A = \frac{aP}{{\left( {RT} \right)^{2} }} $$

(42)

$$ B = \frac{bP}{RT} $$

(43)

Equations of state are applied to multicomponent systems by using mixing rules to determine their parameters for mixtures. In this study, vdW mixing rule is used for a and b parameters [27]:

$$ a = \mathop \sum \limits_{i} \mathop \sum \limits_{j} y_{i} y_{j} \sqrt {a_{i} a_{j} } \left( {1 - k_{ij} } \right) $$

(44)

$$ b = \mathop \sum \limits_{i} y_{i} b_{i} $$

(45)

where k_ij is binary interaction parameters. The fugacity coefficient of a solute in a SCF is derived [26]:

$$ \ln \hat{\varphi }_{i} = \frac{{b_{i} }}{b}\left( {Z - 1} \right) - \ln \left( {Z - B} \right) + \frac{A}{B}\left( {\frac{{2\mathop \sum \nolimits_{j} y_{j} \sqrt {a_{i} a_{j} } \left( {1 - k_{ij} } \right)}}{a} - \frac{{b_{i} }}{b}} \right)\ln \left( {\frac{Z}{Z + B}} \right) $$

(46)