An efficient numerical prediction of the crust onset of a drying colloidal drop

Highlights

• An approach based on the spectral method of collocations at Chebyshev-Gauss-Lobatto points was proposed to solve the drying of a colloidal drop.

• The locking point was used as a parameter to assess the global performance of the numerical scheme.

• It was demonstrated that the spectral collocations method provides an accurate way to solve moving boundary problems.

• Fewer grid points (200 times lesser) were required compared to the finite difference method, providing then a credible alternative for implementation in more complex models involving a large number of droplets.

• Three types of drying were identified based on the ratio of crust formation duration to the consolidation time and the Peclet number.

Abstract

An efficient numerical approach is proposed on the crust formation of an isothermal drying spherical colloidal droplet in a stagnant gas, where the heat and mass transfer were governed by the Stefan flow. A purely diffusive mechanism with a moving boundary was considered. The droplet shrinking induced an accumulation of particles, in the vicinity of the droplet interface, up to the onset of a crust formation known as the locking point. The latter was used as a control parameter of the numerical procedure global behaviour. The performances of the present numerical scheme compared to classical methods, showed a rapid convergence with a drastic reduction in the number of nodes, at least two-hundred fold less. Furthermore, the accuracy and speed of the proposed method allowed to carry out an extensive study with respect to the Péclet number, thus enabling to point out three drying regimes.

Introduction

The evaporation of colloidal drops is a subject of great interest with respect to its fundamental aspect involving diffusion with moving front and for its practical aspect in many technological fields such as spray drying [1], [2], [3], [4], [5], spray pyrolysis [6], fluidized bed drying [7], freeze-drying [8]. A colloid droplet evaporation process is generallly subdivided into three stages: crust formation, crust growth and drying of the resulting wet particle. During the first phase, which is of particular interest to the present study, the evaporation of the liquid, constituting the colloid, causes a shrinking of the drop, consequently particles accumulates in vicinity of the droplet interface. The first stage ends with the formation of a first crust layer, which occurs when the volume fraction at the interface reaches the close packing range. This critical instant is commonly known as the ”locking point”. The crust formation is the common step between all mathematical models describing this process [9], [10], [11], [12], and most of them considered its determination important in the understanding of the drying process, to predict, sometimes, the total drying time and the final size of the dried particle [13], [14], [15], [16].

The problem complexity includes a moving boundary whose location, in general, is a priori unknown, in addition of the particles concentration field whose knolewdge in particularely of interest in the vicinity of the interface. The interface movement is governed by a mass balance equation coupling the particule transport and the liquid evaporation. This equation, which contains all information with respect to the agglomeration of particles at the interface, is nonlinear and similar to the well known Stefan’s condition [17]. Consequently, analytical solutions are often difficult to implement.

The lack of exact analytical solutions for Stefan’s like problem, similar to this one, leads usually to semi-analytical or numerical methods. Semi-analytical methods are generally integral methods based on energy or mass balance [18], [19], [20], [21], [22], [23]. These inexpensive methods would provide explicit, though not very accurate, approximate analytical solutions. However, one should note such methods would work better in cartesian coordinates were several approaches exist [24]. More accurate solutions can be obtained through numerical methods associated with interface localization techniques. Two techniques, commonly used, are the variable space grid method (VSGM) and the boundary immobilization method (BIM) respectively developed by Murray and Landis [25] and Crank [26]. These two methods were used for solving Stefan’s problem [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], as well as for droplet evaporation [39]. Nevertheless, Sadoun et al. [40] showed that despite differences in the approach, both techniques lead to the same numerical scheme. The numerical procedure adopted hereafter is based on the boundary immobilization method (BIM).

Up to our knowledge, the use of finite volume methods, highly appreciated for the treatment of transfer problems due to its conservative properties, is not possible by the boundary immobilization. The methods, available for solving this type of problems, are the finite-difference [27], [28], [29], [33], [34], [41], [42] and finite element methods [35]. These methods have a low order accuracy, often linear and rarely quadratic namely through the use of a finite difference Keller Box scheme [32], [39], [43].

The spectral collocations method is a numerical method for solving a wide variety of PDEs in physics. Its high convergence rate allows achieving a great accuracy with a reduced number of nodes. Its performance makes it a valuable method to solve transfer phenomena problems including radiative heat transfer [44], [45], [46], [47], [48], [49], magnetohydrodynamics and computational fluid dynamics [50], [51]. In case of moving boundary problems, this method showed its ability to deal with Stefan’s problem in cartesian coordinates [52], [53] and radial coordinates [54], with high accuracy and convergence rate, compared to those obtained by finite difference methods.

A mathematical model, describing the drying of a colloidal drop, was proposed on the bases of heat and mass balances [11]. This model, with the objective of improving the fundamental understandings on the evaporation process of a colloidal drop, identified the governing key parameter as being the Péclet number $Pe$. The authors discussed the existence of two drying phases of the drop. The first phase includes concentration of particles, at the interface, up to a maximum value, corresponding to the formation of the first skin of the crust, while the growth and consolidation of the crust define the second phase. The durations of these two phases will be used to identify the drying regimes of the drop. Nevertheless, the numerical resolution was done by classical finite difference method whose convergence and precision constraints, mentioned before, are particularly felt for fast drying, due to strong gradients at the interface.

In order to overcome the reduced accuracy of the finite difference method [11] and to include the moving boundary, a numerical approach has been developed on the basis of a spectral collocation method, for the determination of the onset locking point of a drying colloidal drop. Similarly to the suggestion of Mitchell et al. [39], on the complete evaporation time of pure liquid droplet, the resulting error on the locking point was used as an indicator of the overall behaviour on the numerical scheme for a large range of Péclet numbers.