An asymptotic model for the primary drying stage of vial lyophilization

Abstract

Asymptotic methods are employed to analyse a commonly used one-dimensional transient model for coupled heat and mass transfer in the primary drying stage of freeze-drying (lyophilization) in a vial. Mathematically, the problem constitutes a two-phase moving boundary problem, in which one of the phases is a frozen porous matrix that undergoes sublimation, and the other is a low-pressure binary gaseous mixture. Nondimensionalization yields a model with 19 dimensionless parameters, but a systematic separation of timescales leads to a reduced model consisting of just a second-order differential equation with two initial conditions for the location of a sublimation front; the temperature and gas partial pressures can be found a posteriori. The results of this asymptotic model are compared with those of earlier experimental and theoretical work. Most significantly, the current model would be a computationally efficient tool for predicting the onset of secondary drying.

Appendices

Appendix A: Further considerations relating to boundary condition (2.24)

Note that if $p_{\mathrm{w}}^{H}>p_{\mathrm{w}}^\mathrm{sat}\left( T_{\mathrm{init} }\right) ,$ where $p_{\mathrm{w}}^\mathrm{sat}$ is the temperature-dependent water vapour saturation pressure, then sublimation cannot begin instantaneously at $t=0.$ As $T$ at $z=H$ increases, eventually sublimation can begin when

$$\begin{aligned} p_{\mathrm{w}}^\mathrm{sat}\left( T\left( H,t\right) \right) =p_{\mathrm{w}}^{H}. \end{aligned}$$

(A.1)

On the other hand, if $p_{\mathrm{w}}^{H}<p_{\mathrm{w}}^\mathrm{sat}\left( T_{\mathrm{init}}\right) ,$ sublimation would begin instantaneously at $t=0.$ However, this would seem to be an unrealistic scenario, given that the upper surface of the frozen matrix is continuously in contact with the water vapour above it during freezing. Thus, most realistic and expedient seems to be

$$\begin{aligned} p_{\mathrm{w}}^{H}=p_{\mathrm{w}}^\mathrm{sat}\left( T_{\mathrm{init}}\right) , \end{aligned}$$

(A.2)

as this avoids a post-freezing pre-sublimation stage$;$ in turn, this means that we must have

$$\begin{aligned} p_{\mathrm{w}}^\mathrm{sat}\left( T_{\mathrm{init}}\right) =p_{\mathrm{w}} ^{H}<p_{\mathrm{c}}. \end{aligned}$$

(A.3)

In summary, it seems that setting $p_{\mathrm{c}}$ and $T_{\mathrm{init}}$ would suffice in order to determine $p_{\mathrm{a}}^{H}$ and $p_{\mathrm{w} }^{H}.$

However, there are two further points of confusion. First of all, different authors use different expressions for $p_{\mathrm{w}}^\mathrm{sat}\left( T\right) .$ Velardi and Barresi [12] use

$$\begin{aligned} p_{\mathrm{w}}^\mathrm{sat}\left( T\right) =\exp \left( -20.9470\left( \frac{273.15}{T}-1\right) -3.56654\log \left( \frac{273.15}{T}\right) +\, 2.0189\left( 1-\frac{273.15}{T}\right) -5.0983\right) , \end{aligned}$$

(A.4)

whereas

$$\begin{aligned} p_{\mathrm{w}}^\mathrm{sat}\left( T\right) =133.3224\exp \bigg (-\frac{2445.5646}{T}+8.23121\log _{10}T -0.0167006T+1.20514\times 10^{-5}T^{2}-6.757169\bigg ) \end{aligned}$$

(A.5)

is used in [2, 7, 25, 26]; these are plotted in Fig. 9, along with tabulated values [24], and suggest that the units for (A.4) are bars, rather than Pa as given in [12]. Moreover, it is far from clear that all earlier models are thermodynamically consistent: Velardi and Barresi [12] took $p_{\mathrm{c}}=26$ Pa, $T_{\mathrm{init} }=228$ K, yet $p_{\mathrm{w}}^\mathrm{sat}\approx 28.7$ Pa; others [2, 7, 25, 26] took $p_{\mathrm{c}}\approx 5\hbox { Pa, }T_{\mathrm{init}}=233$ K, yet $p_{\mathrm{w}}^\mathrm{sat}\approx 48.9\hbox { Pa};$ Mascarenhas et al. [6] took $p_{\mathrm{w}}^{H}=5.3\hbox { Pa, }T_{\mathrm{init}}=242$ K, but $p_{\mathrm{c}}$ was not specified at all. For the computations, we therefore use the data for $p_{\mathrm{w}}^\mathrm{sat}$ given in [24] and the value for $p_{\mathrm{c} }$ used in [12] and as given in Table 1.

Fig. 9

Water vapour saturation pressure $p_{\mathrm{w}}^\mathrm{sat}$ versus temperature $T$

Appendix B: Forms for $Q_{\mathrm{v}}$

For $Q_{\mathrm{v}},$ some authors [13, 26, 27] simply use

$$\begin{aligned} Q_{\mathrm{v}}=1.5358p_{\mathrm{c}}, \end{aligned}$$

(B.1)

whereas others [2, 7, 12] take the expression given in (2.26); this expression takes into heat conduction through the glass of the vial and heat transfer in the air gap between the vial and the heating surface, which may be expected to consist of both conduction and radiation. For ${\mathfrak {h}}_{\mathrm{v}}$ in (2.26), some [2, 7] take $30\hbox { Wm}^{-2} \hbox {K}^{-1},$ whereas others [12] take the linear sum

$$\begin{aligned} {\mathfrak {h}}_{\mathrm{v}}=\frac{k_{\mathrm{a}}a^{*}\Lambda _{0} p_{\mathrm{c}}\left( 273.15/T_{\mathrm{g}}\right) ^{1/2}}{k_{\mathrm{a} }+\left[ h_{\mathrm{a}}\right] a^{*}\Lambda _{0}p_{\mathrm{c}}\left( 273.15/T_{\mathrm{g}}\right) ^{1/2}}+\frac{\sigma \left( T_{\mathrm{sh}} ^{2}+T_{\mathrm{gl}}^{2}\right) \left( T_{\mathrm{sh}}+T_{\mathrm{gl} }\right) }{2+\varepsilon _{\mathrm{sh}}^{-1}+\varepsilon _{\mathrm{gl}}^{-1} }, \end{aligned}$$

(B.2)

where $k_{\mathrm{a}}$ is the thermal conductivity of the gas in the air gap, $\Lambda _{0}$ is the free molecular conductivity of the gas at 273.15 K, $T_{\mathrm{g}}$ is equal to the mean temperature value between the bottom surface of the vial and the shelf surface, $a^{*}$ is an accommodation coefficient which takes into account the efficiency of energy transfer between gas molecules striking the surface of the glass and of the shelf, $\left[ h_{\mathrm{a}}\right]$ is the characteristic air-gap width, which can typically vary between 0.05 and 0.6 mm [9], $T_{\mathrm{gl}}$ is the temperature of the bottom surface of the vial, and $\varepsilon _{\mathrm{sh}}$ and $\varepsilon _{\mathrm{gl}}$ denote the radiative emissivities of shelf and the glass vial, respectively. Further, we have [4, 28]

$$\begin{aligned} \varLambda _{0}=\left\{ \begin{array} [c]{ll} 3.5\text { Wm}^{-2}\,\text {K}^{-1}\,\text {Pa}^{-1} &{}\quad \text {H}_{2}\text {O},\\ 1.3\text { Wm}^{-2}\,\text {K}^{-1}\,\text {Pa}^{-1} &{}\quad \text {N}_{2}, \end{array} \right. \end{aligned}$$

(B.3)

with

$$\begin{aligned} k_{\mathrm{a}}\approx \displaystyle \left\{ \begin{array} [c]{ll} \displaystyle 0.004T_{\mathrm{sg}}^{1/2}\,\left( \frac{1-1.7\times 10^{-4}T_{\mathrm{sg}} }{1+659/T_{\mathrm{sg}}}\right) &{} \quad \text {H}_{2}\text {O},\\ \displaystyle 0.001T_{\mathrm{sg}}^{1/2}\,\left( \frac{1+1.7\times 10^{-5}T_{\mathrm{sg}} }{1+52/T_{\mathrm{sg}}}\right) &{} \quad \text {N}_{2}, \end{array} \right. \end{aligned}$$

(B.4)

where $T_{\mathrm{sg}}=(T_{\mathrm{sh}}+T_{\mathrm{gl}})/2$ [9], and

$$\begin{aligned} \frac{a^{*}}{2-a^{*}}=\frac{a_{\mathrm{gl}}^{*}a_{\mathrm{sh}}^{*}}{a_{\mathrm{gl}}^{*}+a_{\mathrm{sh}}^{*}-a_{\mathrm{gl}}^{*}a_{\mathrm{sh}}^{*}}, \end{aligned}$$

(B.5)

where $a_{\mathrm{gl}}^{*}$ is the accommodation coefficient for the gas and the glass and $a_{\mathrm{sh}}^{*}$ is the accommodation coefficient for the gas and the shelf [29]. In turn,

$$\begin{aligned} a_{\mathrm{j}}^{*}=\varphi \left( \frac{M_{\mathrm{g}}}{6.8+M_{\mathrm{g}} }\right) +\left( 1-\varphi \right) \left( \frac{2.4{\mathfrak {M}}}{\left( 1+{\mathfrak {M}}\right) ^{2}}\right) ,\quad {{j}}={\mathrm{gl}} ,{\mathrm{sh}}, \end{aligned}$$

(B.6)

Fig. 10

Heat transfer coefficient ${\mathfrak {h}}_{\mathrm{v}}$ versus temperature $T$ as given by Eq. (B.2) for two values of vial–shelf gap width $\left[ h_{a}\right]$

where $M_{\mathrm{g}}$ is the molecular weight of the gas, ${\mathfrak {M}}$ is the ratio of molecular weights of the gas and the surface, i.e. the bottom of the glass vial and the shelf, and $\varphi$ is the fractional coverage of the adsorption layer, which depends on the nature of the surface; one form for this is [30]

$$\begin{aligned} \varphi =\mathrm{e}^{\Phi \left( T_{\mathrm{r}}-273\right) /273}, \end{aligned}$$

(B.7)

where $T_{\mathrm{r}}$ is the temperature of the gas leaving the surface, which can be taken to be the mean of $T_{\mathrm{sh}}$ and $T_{\mathrm{gl}},$ i.e. $T_{\mathrm{sg}}$ as defined earlier$,$ and $\Phi$ is a parameter related to the nature of the surface, which was taken to be 2 in [9]. Although Eqs. (B.2)–(B.7) were not implemented in the final model, it is interesting to note that they lead to ${\mathfrak {h}}_{\mathrm{v}}\sim$ 15–25 $\hbox {Wm}^{-2}~\hbox {K}^{-1},$ as shown in Fig. 10, which is of the same order of magnitude as that given by [2, 7]; furthermore, for $p_{\mathrm{c} }\approx 20$ Pa, all approaches lead to $Q_{\mathrm{v}}\sim 30\hbox { Wm}^{-2}\,\hbox { K}^{-1}.$ Consequently, as a balance between oversimplicity and overcomplexity, and so as to be able to allow for the explicit effect of the properties of the vial, i.e. $w$ and $k_{\mathrm{gl}},$ (2.26) has been used, in tandem with ${\mathfrak {h}}_{\mathrm{v}}=30\hbox { Wm}^{-2}\,\hbox {K}^{-1} .$