An Enthalpy-Balance Model for Timewise Evolution of Temperature during Wet Stirred Media Milling of Drug Suspensions

Abstract

Purpose

Nanosuspensions have been used for enhancing the bioavailability of poorly soluble drugs. This study explores the temperature evolution during their preparation in a wet stirred media mill using a coupled experimental–enthalpy balance approach.

Methods

Milling was performed at three levels of stirrer speed, bead loading, and bead sizes. Temperatures were recorded over time, then simulated using an enthalpy balance model by fitting the fraction of power converted to heat ξ. Moreover, initial and final power, ξ, and temperature profiles at 5 different test runs were predicted by power-law (PL) and machine learning (ML) approaches.

Results

Heat generation was higher at the higher stirrer speed and bead loading/size, which was explained by the higher power consumption. Despite its simplicity with a single fitting parameter ξ, the enthalpy balance model fitted the temperature evolution well with root mean squared error (RMSE) of 0.40–2.34°C. PL and ML approaches provided decent predictions of the temperature profiles in the test runs, with RMSE of 0.93–4.17 and 1.00–2.17°C, respectively.

Conclusions

We established the impact of milling parameters on heat generation–power and demonstrated the simulation–prediction capability of an enthalpy balance model when coupled to the PL–ML approaches.

Graphical abstract

Appendices

Appendix A

The integration of a differential enthalpy balance for the coolant (chiller liquid) passing through the jacket [51] of the milling chamber leads to following expression for the heat removal rate from the milling chamber and overall enthalpy balance for the chiller liquid:

$${Q}_{\mathrm{ch}}=U{A}_{\mathrm{m}}\frac{T_{\mathrm{ch},\mathrm{in}}-{T}_{\mathrm{ch},\mathrm{out}}}{\ln \left(\left({T}_{\mathrm{ch},\mathrm{in}}-{T}_{\mathrm{s},\mathrm{m}}\right)/\left({T}_{\mathrm{ch},\mathrm{out}}-{T}_{\mathrm{s},\mathrm{m}}\right)\right)}$$

(14)

$${m}_{\mathrm{ch}}{C}_{\mathrm{p},\mathrm{ch}}\left({T}_{\mathrm{ch},\mathrm{out}}-{T}_{\mathrm{ch},\mathrm{in}}\right)=-{Q}_{\mathrm{ch}}=U{A}_{\mathrm{m}}\frac{T_{\mathrm{ch},\mathrm{out}}-{T}_{\mathrm{ch},\mathrm{in}}}{\ln \left(\left({T}_{\mathrm{ch},\mathrm{in}}-{T}_{\mathrm{s},\mathrm{m}}\right)/\left({T}_{\mathrm{ch},\mathrm{out}}-{T}_{\mathrm{s},\mathrm{m}}\right)\right)}$$

(15)

In the derivation of the above expressions, one assumes plug flow of the chiller liquid inside the jacket [51] and makes a pseudo steady-state approximation due to time-dependence of the variables along with the well-mixedness assumption for the suspension in the mill and the holding tank (spatial invariance of T_s,m and T_s.ht). The chiller liquid temperature entering the milling chamber T_ch,in was measured and recorded for each sampling time. Rearrangement of Eq. 15 yields

$$\ln \left(\left({T}_{\mathrm{ch},\mathrm{in}}-{T}_{\mathrm{s},\mathrm{m}}\right)/\left({T}_{\mathrm{ch},\mathrm{out}}-{T}_{\mathrm{s},\mathrm{m}}\right)\right)=\frac{U{A}_{\mathrm{m}}}{m_{\mathrm{ch}}{C}_{\mathrm{p},\mathrm{ch}}}$$

(16)

This equation can be simplified by defining number of transfer units (NTU) as follows:

$$\frac{U{A}_{\mathrm{m}}}{m_{\mathrm{ch}}{C}_{\mathrm{p},\mathrm{ch}}}=\mathrm{NTU}\kern1.75em \ln \left(\frac{T_{\mathrm{ch},\mathrm{in}}-{T}_{\mathrm{s},\mathrm{m}}}{T_{\mathrm{ch},\mathrm{out}}-{T}_{\mathrm{s},\mathrm{m}}}\right)=\mathrm{NTU}\kern0.75em$$

(17)

$${T}_{\mathrm{ch},\mathrm{out}}={T}_{\mathrm{s},\mathrm{m}}+\left({T}_{\mathrm{ch},\mathrm{in}}-{T}_{\mathrm{s},\mathrm{m}}\right)\exp \left(-\mathrm{NTU}\right)$$

(18)

which is identical to Eq. 7 in the main text. Continuing derivation as:

$${T}_{\mathrm{ch},\mathrm{in}}-{T}_{\mathrm{ch},\mathrm{out}}={T}_{\mathrm{ch},\mathrm{in}}\left(1-\exp \left(-\mathrm{NTU}\right)\right)-{T}_{\mathrm{s,m}}\left(1-\exp \left(-\mathrm{NTU}\right)\right)=\left({T}_{\mathrm{ch},\mathrm{in}}-{T}_{\mathrm{s},\mathrm{m}}\right)\left(1-\exp \left(-\mathrm{NTU}\right)\right)$$

(19)

The r.h.s. of Eq. 15 could be rewritten with NTU as follows:

$${Q}_{\mathrm{ch}}=U{A}_{\mathrm{m}}\left({T}_{\mathrm{ch},\mathrm{in}}-{T}_{\mathrm{ch},\mathrm{out}}\right)/\mathrm{NTU}$$

(20)

By inserting Eq. 19 and NTU definition into Eq. 20, Eq. 6 was obtained. Eq. 8 was obtained in a similar fashion.

Appendix B

Overall heat transfer coefficient U for the mill chamber and holding tank was calculated from [51]:

$$\frac{1}{U}=\frac{1}{h_{\mathrm{B}}}+\frac{\left({R}_{\mathrm{o}}-{R}_{\mathrm{i}}\right){A}_i}{k_{\mathrm{wall}}{A}_{\mathrm{lm}}}+\frac{A_{\mathrm{i}}}{A_{\mathrm{o}}{h}_{\mathrm{j}}}$$

(21)

where h_B is the heat transfer coefficient of the liquid (product batch) inside the mill chamber or the holding tank, h_j is the heat transfer coefficient of the jacket side chiller liquid, R and A are the radius and surface area of the respective chambers, where i and o indices stand for inside and outside. A_lm is the logarithmic mean of inside and outside areas. k_wall is the thermal conductivity of the wall, which is zirconia for mill chamber with 2.5 W/m°C [52] and stainless steel for holding tank with 15 W/m°C [53]. When U is written in the form as in Eq. 21, the surface area in UA is taken as A_i. h_B was calculated using:

$${h}_{\mathrm{B}}=\frac{k{A}_2}{D}{\operatorname{Re}}^{2/3}\ {\Pr}^{1/3},\kern0.5em \operatorname{Re}=\frac{D^2 N\rho}{\mu },\kern0.5em \Pr =\frac{C_{\mathrm{p}}\mu }{k}$$

(22)

in which k is the thermal conductivity of the liquid, A₂ is a constant that depends on agitator type, which was taken as 0.54 for mill chamber (disk agitator) and 0.36 for holding tank (paddle agitator). D is the diameter of the chamber, Re is the Reynolds number, and Pr is the Prandtl number. N is the stirrer speed (1/s), μ is viscosity and ρ is density. For the mill chamber, k, μ and ρ were found for the bead–suspension mixture as follows [54,55]:

$${k}_{\mathrm{mix}}=\frac{k_{\mathrm{b}}+\left(n-1\right){k}_{\mathrm{s}}+\left(n-1\right)\left({k}_{\mathrm{b}}-{k}_{\mathrm{s}}\right)c}{k_b+\left(n-1\right){k}_s-\left({k}_b-{k}_s\right)c}{k}_{\mathrm{b}}$$

(23)

$${\mu}_{\mathrm{mix}}={\upmu}_{\mathrm{L}}\ \left[1+2.5c+10{c}^2+0.0019\exp (20c)\right],\kern0.5em {\rho}_{\mathrm{mix}}={\rho}_{\mathrm{b}}\ c+{\rho}_{\mathrm{L}}\ \left(1-c\right)$$

(24)

where k_b and k_s are thermal conductivities of the beads and the suspension, respectively. k_b is 1.8 W/m°C [56] and k_s is assumed to be equal to that of water, i.e., 0.607 W/m°C [57]. n was taken as 3 for spherical beads [54]. c is the bead loading. μ_L and ρ_L are the viscosity and density of the drug pre-suspension, which were measured as 198 mPa.s and 1030 kg/m³. For h_j, the following correlation [51] was used:

$${h}_{\mathrm{j}}=\frac{k_{\mathrm{j}}{A}_1}{D_{\mathrm{j}}}{\operatorname{Re}}_{\mathrm{j}}^{2/3}\ {\Pr}_{\mathrm{j}}^b,\kern0.5em {\operatorname{Re}}_{\mathrm{j}}=\frac{D_{\mathrm{j}}v{\rho}_{\mathrm{j}}}{\mu_{\mathrm{j}}}$$

(25)

Here, A₁ and b are constants that are recommended to be 0.0265 and 0.3, respectively, for a cooling system (jacket). v is the jacket liquid velocity. ρ_j, μ_j, and Pr_j were taken as 1.13 g/cm³, 1.448 mPa.s, and 15 respectively, for the glycol–water mixture [58] used in our chiller.