# Finite elements model of heat transfer predicts the edge vial effect in different freeze-drying configurations

Bernadette Scutella, Stéphanie Passot, Erwan Bourlés, Fernanda Fonseca, Artemio Plana-Fattori, Denis Flick, Ioan-Cristian Trelea

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Bernadette Scutella, Stéphanie Passot, Erwan Bourlés, Fernanda Fonseca, Artemio Plana-Fattori, et al.. Finite elements model of heat transfer predicts the edge vial effect in different freeze-drying configurations. 2016 CPPR Conference Freeze-Drying of Pharmaceuticals and Biologicals, Jul 2016, Breckenridge, United States. hal-01567200

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Submitted on 5 Jun 2020

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<figure><img src="image_1.png" /></figure>
# Finite elements model of heat transfer predicts the edge vial effect in different freeze-drying configurations

B. Scutella<sup>1,2</sup>, S. Passot<sup>1</sup>, E. Bourlès<sup>2</sup>, F. Fonseca<sup>1</sup>, A. Plana-Fattori<sup>3</sup>, D. Flick<sup>3</sup>, I.C. Trelea<sup>1</sup>

E-mail: bernadette.scutella@grignon.inra.fr

Dam A. Bourlès
CPPR Center for
Pharmaceutical
Processing Research

AgroParisTech

Agronext

www.agroparistech.com

## What is the “edge vial effect”?

<figure><img src="image_2.png" /></figure>

### What is the “edge vial effect”?

During freeze-drying process, products in vials located at the periphery of the shelf are warmer and have an higher sublimation rate of about 15% compared to vials in the centre [1]. This phenomenon results from the combined effects of radiation from external sources (walls, rail) and conduction due to direct contact of the vial with the rail.

### Which are the consequences?

The “edge vial effect” results in different thermal histories among the vials and thus in a non-homogeneous final quality of the product.

## Mathematical model of heat transfer in freeze-drying

### Main hypotheses

* Pseudo-stationary state: all the heat received from the vial is used for ice sublimation;

* Constant ice thickness in the vials and flat ice-vapour interface;

* Pure water vapour in the drying chamber;

* Heat transfer by convection is negligible.

### Relevant parameters

<table><thead><tr><td>Parameter</td><td>Symbol</td><td>Value</td></tr></thead><tbody><tr><td>"Bulk" vapour conductivity</td><td>λ<sub>w</sub></td><td>0.025 [W m<sup>-1</sup> K<sup>-1</sup>]</td></tr><tr><td>Knudsen layer conductivity</td><td>λ<sub>kn</sub></td><td>2.10<sup>3</sup> Pr [W m<sup>-1</sup> K<sup>-1</sup>]</td></tr><tr><td>Depth of vial bottom curvature</td><td>l<sub>w</sub></td><td>0.1 [mm]</td></tr><tr><td>Knudsen layer thickness</td><td>l<sub>kn</sub></td><td>0.1 [mm]</td></tr></tbody></table>

### From the vial...

### ...to the system

<figure><img src="image_3.png" /></figure>

### Knudsen layer

* Limit layer to simulate the heat transfer through the gas at low pressure in Knudsen regime;

* Covers all the solid surfaces;

* The thermal conductivity λ<sub>th</sub> of the Knudsen layer is a function of the pressure and can be defined as:

$$ \lambda_{K_n} = 2\alpha\Lambda_{\alpha}P_c l_{K_n} $$

where:

* l<sub>kn</sub> is the thickness of the Knudsen layer;

* P<sub>c</sub> is the chamber pressure;

* αλ<sub>α</sub> is the dimensionless accommodation coefficient (a) times the molecular thermal conductivity of the gas at 0°C [A<sub>0</sub>]. This product was experimentally evaluated (0.7 W m<sup>-1</sup> K<sup>-1</sup> Pa<sup>-1</sup>).

### Heat transfer in solid and gaseous bodies

<figure><img src="image_4.png" /></figure>

$$ \bar{q}_c = -\lambda_1 \nabla T $$

* $\bar{q}_c$: heat flux by conduction in the solid or gas [W m<sup>-2</sup>];

* λ<sub>th</sub>: thermal conductivity of the body material [W m<sup>-1</sup> K<sup>-1</sup>];

* $\nabla T$: gradient of temperature [K].

### Heat transfer by contact conduction

<figure><img src="image_5.png" /></figure>

$$ q_{cc} = K_{cc} (T_1 - T_2) $$

* $Q_{cc}$: heat flux by contact conduction [W m<sup>-2</sup>];

* $K_{cc}$: heat transfer coefficient of contact conduction [W m<sup>-1</sup> K<sup>-1</sup>];

* $T_{1,2}$: temperatures of the bodies [K].

### Heat transfer by radiation

<figure><img src="image_6.png" /></figure>

$$ q_r = F_{r-2} \sigma (T_1^4 - T_2^4) $$

* $Q_r$: heat flux by radiation [W m<sup>-2</sup>];

* $F_{r-2}$: visualization factor [J];

* $T_{1,2}$: temperatures of the bodies [K];

* $Q_r$: Stephen-Boltzmann constant [W m<sup>-2</sup> K<sup>-1</sup>].

## “Edge vial effect” in different freeze-drying configurations

### A. Edge vials completely shielded by the rail

* Heat transfer by radiation from the wall to the rail;

* Heat transfer by contact conduction and radiation from the rail to the vials.

<figure><img src="image_7.png" /></figure>

### B. Edge vials partially shielded by the rail

* Heat transfer by radiation from the wall to the rail and vials;

* Heat transfer by contact conduction and radiation from the rail to the vials.

<figure><img src="image_8.png" /></figure>

### C. Edge vials completely exposed to the wall

* Heat transfer by radiation from the wall to the vials.

<figure><img src="image_9.png" /></figure>

## Conclusions and perspective

* ✓ The developed model predicts the heat flow rate in edge and central vials in different freeze-drying configurations;

* ✓ The edge effect is less important if the edge vials are shielded by the rail but not in contact with it;

* ✓ The edge effect is more important at low shelf temperature and high chamber pressure (for a range of -10 to -35°C and 4 to 9Pa);

* ✉ Next step will involve the use of the model as smart tool to predict the “edge vial effect” in freeze-dryers at commercial scale.

<figure><img src="image_10.png" /></figure>

## Why this study and which strategy?

The
“edge vial effect”
causes...

Heterogeneity
within product
batch

Issues in
process control

Problematic
scale-up

### QUESTION(S)

Is it possible to predict the “edge vial effect”? How?

Which is the effect of the freeze-drying configuration on this phenomenon?

Mathematical model of heat transfer in freeze-drying in
3D using COMSOL MULTIPHYSICS®

<figure><img src="image_11.png" /></figure>

## Experimental procedure

### Gravimetric method

Fill the vials with 1.8ml of purified water...

...weight them to evaluate the initial mass (m)...

...freeze-dry them until 30% of ice is sublimed...

...re-weight them to evaluate the final mass (m)...

$\dot{m} = m_1 - m_2$

sublimation time

<figure><img src="image_12.png" /></figure>

### Sublimation test

* Freezing step: 2h at -50°C;

* Ramp: 1°C/min;

* Sublimation:

* 9Pa and -25°C for configuration A and C (MillRock freeze-dryer);

* 6Pa and -40°C and 0°C for configuration B (Aqua-Finn GT6 freeze-dryer).

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

## Virtual experiments

<figure><img src="image_16.png" /></figure>

### Impact of the operating parameters

The heat flux difference between central and edge vials...

* ... is higher if the shelf temperature is lower (in a range of -10 to -25°C).

* ... is higher if the chamber pressure is higher (in a range of 4 to 9Pa).

<figure><img src="image_17.png" /></figure>

### Impact of the edge configuration

* Configurations A and B are equivalent (±5%) except at low shelf temperature (-35°C) and pressure (4Pa);

* The best edge configuration appears to be the configuration A, in all the possible combination of operating parameters.

<figure><img src="image_18.png" /></figure>

### References

[1] Rambhalta, S. & Pikal, M. J., 2003. Heat and mass transfer scale-up issues during freeze-drying: atypical radiation and the edge vial effect. Aaps Pharmaceutics 4, 22–31.

[2] Ganguly, A., Nai, S., Alexeevskiy, A., 2013. Experimental determination of the key heat transfer mechanisms on pharmaceutical freeze-drying. J. Pharm Sci 102(5).

### Acknowledgement

Thanks to Benoit Moreau and Yves Mayeresie (GSK Vaccines) for their contributions in reviewing this work.

Envan Bourlès is an employee of the GSK group of companies. This study was financially supported by GlaxoSmithKline Biologicals SA.

<footnote n="1"><sup>1</sup> UMR GMPA, AgroParisTech, INRA, Université Paris Saclay, 78 850 Thiverval-Grignon, France</footnote>

<footnote n="2"><sup>2</sup> Technical R&D Drug Product, GSK Vaccines, 1330 Rixensart, Belgium</footnote>

<footnote n="3"><sup>3</sup> UMR Ingénierie Procédés Aliments, AgroParisTech, INRA, Université Paris-Saclay, 91300 Massy, France</footnote>