3. Theoretical background

It is thought that the desiccation process of IJs happened by slow absorption of the aqueous suspension containing them for the carrier material, starting a reduction process of moisture, which developed at an appropriate rate, and diminished the metabolism of the IJs [5]. In fact, 52% of the variation in its survival rate is explained by the behaviour of the moisture content of the DE pellet, whereas 84% of the variation in infectivity on G. mellonella is explained by the survival of S. glaseri IJs in diatomaceous earth pellets [3]. The hypothesis is that the sudden death of the IJs formulated under these conditions is due to the diffusional migration of water molecules surrounding the IJs to the DE pellet surface followed by the contact of the DE particles with the nematode cuticle which absorbs moisture faster from the IJ's cells [3]. Also as it can be observed in Figure 4, the behaviours of drying kinetics of diatomaceous earth

Recent inspection of cross-sectional area of diatomaceous earth pellets using scanning electron microscopy showed particles as plate-form and non-uniform pore distribution that form a complex and disordered microstructure [6], probably dominated by a double porosity due to two distinct distributions, one for the region of macroscopic porosity between particles, and another for the region of microscopic porosity within particles [16]. Due to the above-mentioned facts, in next sections, we will be dealing with theories applied to understand the moisture evaporation from porous media to understand how the evaporation from pellets happened, which can be

useful to set design criteria to elaborate GB reservoirs for the optimum storage of EPNs.

Figure 4. Drying kinetics of diatomaceous earth pellets without EPNs with several initial moisture contents, stored in

pellets are different with or without S. glaseri IJs.

90 Current Perspective to Predict Actual Evapotranspiration

quiescent surrounding at room temperature (23 3C).

#### 3.1. The evaporation process in porous media

In the evaporation process, liquid water is transformed to water vapour and transferred from the surface of evaporation to the surrounding atmosphere, while a porous medium is a material with a skeletal solid structure with interconnected void spaces that allow fluid to pass through the medium and is mainly characterized by its porosity, i.e. the ratio of the void space to the total volume of the medium [17–19]. The permeability is a measure of the flow conductivity in the porous body and the tortuosity represents the hindrance to flow diffusion imposed by local boundaries or local viscosity; both these are important characteristics for the combination of the fluid and structure of the porous medium, respectively [17].

In porous media, evaporation involves mass and energy transport including phase change, vapour diffusion and liquid flow, resulting in complex displacement patterns affecting drying rates [20] and is the only one mechanism by which moisture can leave the GBs. Moreover, at the pore scale several mechanisms influence the macroscopic behaviour of the drying process. Among others, the phase change at the liquid-gas interface, diffusion and convection of mass and heat, type of flow and the effect of the combined viscous, capillary and buoyancy forces on the receding of the liquid-gas interfaces are also the influencing factors in drying process [21]. Most evaporation processes are assumed to be viewed as transport at the pore space where water is displaced by air; this moving zone between wet and partially dry zones is named the drying front in [22].

According to the theory of drying in two stages, drying rates from an initially saturated porous medium often exhibit distinct transitions whereby initially high drying rate (denoted as stage I or constant rate period) abruptly drops to a lower rate (stage II or falling rate period) governed by vapour diffusion through the porous medium [20, 23]. The drying stage 1 occurs by convection on the material surface at an evaporation rate relatively high and nearly constant. The factors involved in its development are the external conditions of mass and heat exchange in the system related to the properties of the surrounding air (temperature, pressure, humidity, convective airflow velocity and area of the exposed surface) and supplied by capillary liquid flow connecting a receding drying front with an evaporating surface [24].

The drying stage 2 starts when the moisture content of the porous media is less than its critical moisture content. At a certain drying front depth, continuous liquid pathways are interrupted when gravity and viscous effects overcome capillary forces in porous media and simultaneously the moisture content is less than its critical moisture concentration. These events marks the onset of a new evaporation regime with lower evaporation rate limited by transport properties of porous media, known as the drying stage 2 [22]. In this period the process is influenced by internal factors related to the properties of the material such as water activity, internal structure (porosity and tortuosity), chemical composition and transport properties (thermal and hydraulic conductivities, moisture diffusivity and vapour diffusion) [22–24]. The rate of drying in the two different periods has mainly been modelled numerically from a set of equations taking into account vapour and liquid transfers along with boundary conditions.

In [25], the convective drying is considered as a process of three successive steps. The first step is liquid movement in porous media from the wet interior to the gas-solid interface across internal pore, particle surface, etc. This step is slower for larger solids and/or low moisture concentration in the porous material. The second is evaporation due to heat by energy supplied to change liquid into vapour. The third step is vapour movement to the surrounding gas by diffusion and convection. In general, different stages are often used for diagnostics and classification of evaporative drying of porous media, which depend on the focus of the studies and the interpretation of the rate limiting processes. Consequently, complex and highly dynamic interactions between medium properties, transport processes and boundary conditions result in a wide range of evaporation behaviours [20] and internal and external factors are common influences on most of the evaporation behaviours.

#### 3.2. External mass transfer

#### 3.2.1. Classic evaporation theory

A brief history of the theories of evaporation is presented in reference [26]. The Dalton's assay was one of the major events in the development of the evaporation theory and states that the rate of water evaporation is proportional on the saturation deficit of the air, which is given by the difference between the saturation vapour pressure at the water temperature, ew and the actual vapour pressure of the air ea [27, 28]:

$$E\_L = \mathbb{C}(\mathbf{e}\_w - \mathbf{e}\_a) \tag{1}$$

where EL = rate of evaporation (mm/day) and C = constant; ew and ea are in mm of mercury. Eq. (1) is known as Dalton´s law of evaporation. Also, this proportionality is affected by the wind velocity, the RH and the moisture concentration in the solid surface; therefore, the moisture will evaporate from the surface until the surrounding air saturates [28]. The similarity theory is a standard basis for predicting evaporation rate from a free water surface and states that convective heat and mass transfer are completely analogous phenomena if the mass flow from the surface is caused by diffusion, which requires that the content of the diffusing species be low and the diffusional mass flux is low enough that it does not affect the imposed velocity field. However, these two analogies are not valid if applied to a capillary porous media containing a liquid [28].

#### 3.2.2. The constant rate period

According to the theory of two stages of drying of porous media, during the 1-stage, the rate of water loss per unit surface area remains nearly constant and close to evaporation rate from free water surface and is attributed to the persistence of continuous hydraulic pathways between the receding drying front and surface of porous media where liquid flow is sustained by hydraulic gradient towards the evaporation surface [29]. Numerous experimental and theoretical studies have established the existence of such a constant phase during drying of porous media, typically under mild atmospheric demand [20, 30].

The intrinsic characteristic length LC, a measure of the extent of hydraulic continuity and the strength of capillary driving force deduced from pore size distribution of a medium that control the transition from liquid-flow–supported stage 1 to diffusion-controlled stage 2 during evaporation from porous media is proposed in [20]. The LC is used for predicting the end of stage 1 evaporation and considering balance between gravitational, capillary and viscous forces. The theory assumes that in the stage-1, air first enters the largest pores in a complex porous medium while menisci in smaller pores at the evaporating surface may remain in place (albeit with decreasing radii) of curvature. Thereby, liquid moves into the pore space (large size pores with lower air-entry value to smaller size water-filled pores) and then towards the surface of the body where the moisture evaporation happens; according to [20], such mass flow may be sustained as long as capillary driving forces are higher than gravitation and viscous forces. In [20], characteristic lengths for evaporation from porous media are theoretically developed for the following three conditions:

1. pore size distribution and gravity characteristic length LG,

$$L\_G = \frac{1}{a(n-1)} \left(\frac{2n-1}{n}\right)^{(2n-1)/n} \left(\frac{n-1}{n}\right)^{(1-n)/n} \tag{2}$$

2. pore size distribution effects on the viscous characteristic length LV,

$$L\_V = \frac{K(\theta)}{\mathfrak{e}\_0} \Delta h\_{\text{cap}} \tag{3}$$

3. combined gravity and viscous length LC,

a set of equations taking into account vapour and liquid transfers along with boundary

In [25], the convective drying is considered as a process of three successive steps. The first step is liquid movement in porous media from the wet interior to the gas-solid interface across internal pore, particle surface, etc. This step is slower for larger solids and/or low moisture concentration in the porous material. The second is evaporation due to heat by energy supplied to change liquid into vapour. The third step is vapour movement to the surrounding gas by diffusion and convection. In general, different stages are often used for diagnostics and classification of evaporative drying of porous media, which depend on the focus of the studies and the interpretation of the rate limiting processes. Consequently, complex and highly dynamic interactions between medium properties, transport processes and boundary conditions result in a wide range of evaporation behaviours [20] and internal and external factors are

A brief history of the theories of evaporation is presented in reference [26]. The Dalton's assay was one of the major events in the development of the evaporation theory and states that the rate of water evaporation is proportional on the saturation deficit of the air, which is given by the difference between the saturation vapour pressure at the water temperature, ew and the

where EL = rate of evaporation (mm/day) and C = constant; ew and ea are in mm of mercury. Eq. (1) is known as Dalton´s law of evaporation. Also, this proportionality is affected by the wind velocity, the RH and the moisture concentration in the solid surface; therefore, the moisture will evaporate from the surface until the surrounding air saturates [28]. The similarity theory is a standard basis for predicting evaporation rate from a free water surface and states that convective heat and mass transfer are completely analogous phenomena if the mass flow from the surface is caused by diffusion, which requires that the content of the diffusing species be low and the diffusional mass flux is low enough that it does not affect the imposed velocity field. However, these two analogies are not valid if applied to a capillary porous media

According to the theory of two stages of drying of porous media, during the 1-stage, the rate of water loss per unit surface area remains nearly constant and close to evaporation rate from free water surface and is attributed to the persistence of continuous hydraulic pathways between the receding drying front and surface of porous media where liquid flow is sustained by hydraulic gradient towards the evaporation surface [29]. Numerous experimental and

EL ¼ Cðew � eaÞ ð1Þ

common influences on most of the evaporation behaviours.

conditions.

3.2. External mass transfer

containing a liquid [28].

3.2.2. The constant rate period

3.2.1. Classic evaporation theory

actual vapour pressure of the air ea [27, 28]:

92 Current Perspective to Predict Actual Evapotranspiration

$$L\_{\mathbb{C}} = \frac{L\_{\mathbb{C}}}{1 - \frac{\varepsilon\_{0}}{K(\theta)}}\tag{4}$$

where K(θ) is the unsaturated hydraulic conductivity, α is the inverse of a characteristic pressure head, n is the pore size distribution, e<sup>0</sup> is the water flow supporting evaporation rate and Δhcap is the maximum capillary driving force. This theoretical approach is based on experimental data of evaporative drying of two quartz sand media with particle sizes ranging from 0.1 to 0.5 mm (denoted as "fine sand") and from 0.3 to 0.9 mm ("coarse sand") to quantify the evaporation rates from sand-filled cylindrical Plexiglas columns of 54 mm in diameter and 50–350 mm in length and rectangular Hele-Shaw cells 260 mm in length, 10 mm in thickness and 75 mm in width with a top boundary open to the atmosphere. As expected, the highest initial evaporation rate corresponds to high temperature and low humidity (28�C and 31% RH), whereas the lowest evaporation occurred for cool and humid conditions (21�C and 58% RH).

#### 3.2.3. Natural convection theory

Natural convection is defined as air movements brought about by density differences in hot and cool air, whereas forced convection is the movement of air brought about by an external force. In natural convection, fluid motion is due to gravity that creates a buoyant force within the fluid, which lifts the heated fluid upward. Since the fluid velocity associated with natural convection is relatively lower than those associated with forced convection, the corresponding convection transfer rates are also smaller [31]. The Rayleigh number that measures the intensity of natural convection, based on the macroscopic length scale L, is defined in reference [19] as

$$Ra\_L = \frac{g\beta\Delta TL^3}{\nu\_f \alpha\_f} \tag{5}$$

where g is the gravity constant, af is the thermal diffusivity of fluid, β is the volumetric temperature expansion coefficient, L is the macroscopic scale length scale, ΔT is the temperature scale and vf is the kinematic viscosity of fluid. Based on natural convection theory, heat and mass transfer from porous media in quiescent fluid environments have been extensively studied by its importance in the design or performance of the systems when it is desirable to minimize heat transfer rates or to minimize operating costs [32].

#### 3.3. Internal mass flow in porous media

The internal process of mass transfer during drying is usually described using a convective model (known as a capillary-porous model) based on fluid pressure gradients (Darcy's law) and techniques of scale change as a representative elementary volume (REV) in order to express the transition from a microscopic level to a macroscopic one in the conservation equations and a diffusive model based on the gradients of moisture concentration, a phenomenon described by Fick's law. This model is a simplification of the capillary porous model if the drying is supposed isothermal with no gravity effects in the solid-liquid water system [33–35].

#### 3.3.1. Fickean diffusion

The diffusion is known as the preponderant internal mass transfer mechanism during drying of porous media. According to Fick's first law, the matter flows erratically from moist regions towards dry regions inside bodies, assuming that the moisture gradient is the unique driving force of the flow [37]. However, although several empirical equations have been proposed to predict mass transfer on this basis, much more must be explained [37]. For instance, this laws make several assumptions and simplifications that are often unrealistic to model water diffusion during drying as materials are non-heterogeneous and isotropic media and diffusion coefficients are not correlated to moisture content; samples are in most cases considered as having regular shapes; heat transfer during drying is disregarded and collapse of vegetable tissues by water loss is also neglected [36, 37].

The effect of microscopic structure on mass transfer has been completely discarded, but a disordered internal geometry caused by the percolation phenomena is very common in the structure of most porous media, sometimes described by the fractal geometry. The complex

microstructure affects the water diffusion phenomena, resulting in anomalous diffusion and involving complex parameters such as fractal dimension and spectral dimension [37]. The inclusion of a porous medium affects the forms of Fick's first and second laws for diffusion of chemical species in aqueous solution in two general ways: first, the existence of the solid particles comprising the porous medium results in diffusion pathways that are more tortuous. This increased tortuosity reduces the macroscopic concentration gradient and, therefore, reduces the diffusive mass flux relative to that which would exist in the absence of the porous medium as in single droplets. Second, there may be interactions between the diffusing species and the solid porous media that either directly affect the mass of the diffusing species in aqueous solution (e.g. sorption) and/or result in physicochemical interactions that affect the tortuosity [18]. All forms of Fick's first and second laws for governing macroscopic diffusion through porous media include an effective porosity, εff and a mass diffusion coefficient D [18].

#### 3.3.2. Capillarity

ð5Þ

3.2.3. Natural convection theory

94 Current Perspective to Predict Actual Evapotranspiration

Natural convection is defined as air movements brought about by density differences in hot and cool air, whereas forced convection is the movement of air brought about by an external force. In natural convection, fluid motion is due to gravity that creates a buoyant force within the fluid, which lifts the heated fluid upward. Since the fluid velocity associated with natural convection is relatively lower than those associated with forced convection, the corresponding convection transfer rates are also smaller [31]. The Rayleigh number that measures the intensity of natural

> RaL <sup>¼</sup> <sup>g</sup>βΔTL<sup>3</sup> ν<sup>f</sup> α<sup>f</sup>

where g is the gravity constant, af is the thermal diffusivity of fluid, β is the volumetric temperature expansion coefficient, L is the macroscopic scale length scale, ΔT is the temperature scale and vf is the kinematic viscosity of fluid. Based on natural convection theory, heat and mass transfer from porous media in quiescent fluid environments have been extensively studied by its importance in the design or performance of the systems when it is desirable to

The internal process of mass transfer during drying is usually described using a convective model (known as a capillary-porous model) based on fluid pressure gradients (Darcy's law) and techniques of scale change as a representative elementary volume (REV) in order to express the transition from a microscopic level to a macroscopic one in the conservation equations and a diffusive model based on the gradients of moisture concentration, a phenomenon described by Fick's law. This model is a simplification of the capillary porous model if the drying is supposed isothermal with no gravity effects in the solid-liquid water system [33–35].

The diffusion is known as the preponderant internal mass transfer mechanism during drying of porous media. According to Fick's first law, the matter flows erratically from moist regions towards dry regions inside bodies, assuming that the moisture gradient is the unique driving force of the flow [37]. However, although several empirical equations have been proposed to predict mass transfer on this basis, much more must be explained [37]. For instance, this laws make several assumptions and simplifications that are often unrealistic to model water diffusion during drying as materials are non-heterogeneous and isotropic media and diffusion coefficients are not correlated to moisture content; samples are in most cases considered as having regular shapes; heat transfer during drying is disregarded and collapse of vegetable

The effect of microscopic structure on mass transfer has been completely discarded, but a disordered internal geometry caused by the percolation phenomena is very common in the structure of most porous media, sometimes described by the fractal geometry. The complex

convection, based on the macroscopic length scale L, is defined in reference [19] as

minimize heat transfer rates or to minimize operating costs [32].

3.3. Internal mass flow in porous media

tissues by water loss is also neglected [36, 37].

3.3.1. Fickean diffusion

The migration of the liquid phase in a deformable porous matrix, by convective transport, is managed by the generalized Darcy's law [33–35]. Darcy's law in its simplest form expresses the proportionality between the average velocity v of a fluid flow and the flow potential, comprised by the pressure gradient Δp existent through porous media and the gravitational contribution and is applicable to multiphase mixtures as opposed to Fick's law, which requires the assumption of a homogeneous mixture [19]. The proposed relationship is as follows:

$$\mathbf{v} = k \frac{\Delta p}{\Delta \mathbf{x}} \tag{6}$$

in this expression, k is the hydraulic conductivity and describes the ease with which a fluid can flow through the pore spaces. Darcy's equation is valid for incompressible and isothermal creeping flows. In a complex form of Darcy's law, the rate of flow is related to the pressure gradient in the liquid, ∇〈Vl〉 <sup>l</sup> [33]. Eq. (7) expresses the relationship between the liquid-phase velocity 〈Vl〉 l and solid one 〈Vs〉 <sup>s</sup> as follows:

$$
\langle V\_l \rangle^l = \langle V\_s \rangle^s - \frac{\overline{\overline{K}}}{\varepsilon\_l \mu\_l} \cdot \left( \nabla \langle P\_l \rangle^l \right) \tag{7}
$$

where K is the permeability of the porous medium, ε<sup>l</sup> is the liquid fraction, μ<sup>l</sup> is the liquid dynamic viscosity and Δ〈Pl〉 l is the gradient of fluid pressure.

#### 3.3.3. The falling rate period

The stage 2 is governed by vapour diffusion through the porous medium. This period is divided in a first decreasing-rate period (FDR) characterized by breaks in the uniformity of water content close to the surface and slight augmentation of the heat surface; and a second decreasing-rate period (SDR), correlated with a discontinuous liquid network into the porous medium and with the development of a dry receding front from the free surface of the porous sample [38].

The falling rate period is expected to be short for two reasons: (1) the liquid mass corresponding to the films is small compared to the mass of liquid initially present in the medium and (2) the external mass transfer length scale (the mass external boundary layer typically) is typically greater than the thickness of the medium, which implies that the mass transfer resistance due to the receding of film tips within the medium is weak compared to the external mass transfer resistance [39].

#### 3.4. Mathematical models of moisture evaporation from porous media

The drying behaviour of porous material can be described with a model and the porous media is described in multiple length scales [40, 41]. The macroscopic length scale is defined by the overall physical domain indicated by the length scale L. The microscopic length scale captures the detailed morphology and is indicated by d. A REV is defined as a volume whose size LREV lies between length scale d and L, i.e. d ≪ LREV ≪ L [19]. Thus, the modelling of drying process of porous media can be developed at different scales in a process that should be initiated typically to the pore-scale and then model it at larger and larger scales up to dryer scale or product scale in the case of designs of dryers.

The macroscopic variables of the drying process are commonly defined by the volume average of the microscopic variables over the REV and the nature of the product is the result of a diversity of multiple factors and their relationships among themselves that increases the complexity of the evaporation process [19, 41]. In the formulation of EPNs, the interesting scale is the product scale, being one single granular biopesticide. The continuum approach and pore network models are approaches for the modelling of the drying process of the porous media [41].

#### 3.4.1. The continuum approach

In the continuum approach, variables (e.g. temperature) are averaged over the volume, the REV. Equations for the conservation of liquid, air and energy are supplemented with boundary and initial conditions. The continuum approach can be solved by efficient numerical techniques at a large scale in comparison to the pore scale, which is a great advantage. The effective parameters, such as vapour diffusivity, permeability, thermal conductivity and capillary pressure have to be determined by dedicated experiments. The continuum approach fails when the pores are large compared with the system and is not able to easily take structural features of the medium into account. Moreover, the computation of the effective properties at the scale of a REV is necessary in the continuum approach [41]. The parameters of the continuum model can be assessed for a certain pore structure using a pore network model.

#### 3.4.2. The pore network approach

Porosity is a primary property of the granules, which dominates a wide range of secondary properties as water flow or air entrapment, both strongly linked to the microstructure [42, 43]. Pore network models are based on a porous structure represented as a network of pores and throats and these models can be used to simulate the drying process at the pore level because they can take into account important features of the microstructure as the role of large pores and their distribution on motion of the gas-liquid menisci in the pores, diffusion, viscous flow, capillarity and liquid flow [40, 41, 44]. The developing of models that permit to analyse the influence of the porous microstructure has at least two motivations. One is the computation of the effective parameters at the scale of a representative elementary volume REV of the microstructure. A second one is to analyse drying at the scale of the product without assuming a priori existence of the REV that is associated with the continuum approach. Pore network models have been used in both cases and have been described in two and three dimensions [44].

#### 3.4.3. Pore network models

medium and with the development of a dry receding front from the free surface of the porous

The falling rate period is expected to be short for two reasons: (1) the liquid mass corresponding to the films is small compared to the mass of liquid initially present in the medium and (2) the external mass transfer length scale (the mass external boundary layer typically) is typically greater than the thickness of the medium, which implies that the mass transfer resistance due to the receding of film tips within the medium is weak compared to the external

The drying behaviour of porous material can be described with a model and the porous media is described in multiple length scales [40, 41]. The macroscopic length scale is defined by the overall physical domain indicated by the length scale L. The microscopic length scale captures the detailed morphology and is indicated by d. A REV is defined as a volume whose size LREV lies between length scale d and L, i.e. d ≪ LREV ≪ L [19]. Thus, the modelling of drying process of porous media can be developed at different scales in a process that should be initiated typically to the pore-scale and then model it at larger and larger scales up to dryer scale or

The macroscopic variables of the drying process are commonly defined by the volume average of the microscopic variables over the REV and the nature of the product is the result of a diversity of multiple factors and their relationships among themselves that increases the complexity of the evaporation process [19, 41]. In the formulation of EPNs, the interesting scale is the product scale, being one single granular biopesticide. The continuum approach and pore network models are approaches for the modelling of the drying process of the porous media [41].

In the continuum approach, variables (e.g. temperature) are averaged over the volume, the REV. Equations for the conservation of liquid, air and energy are supplemented with boundary and initial conditions. The continuum approach can be solved by efficient numerical techniques at a large scale in comparison to the pore scale, which is a great advantage. The effective parameters, such as vapour diffusivity, permeability, thermal conductivity and capillary pressure have to be determined by dedicated experiments. The continuum approach fails when the pores are large compared with the system and is not able to easily take structural features of the medium into account. Moreover, the computation of the effective properties at the scale of a REV is necessary in the continuum approach [41]. The parameters of the continuum model

Porosity is a primary property of the granules, which dominates a wide range of secondary properties as water flow or air entrapment, both strongly linked to the microstructure [42, 43]. Pore network models are based on a porous structure represented as a network of pores and

can be assessed for a certain pore structure using a pore network model.

3.4. Mathematical models of moisture evaporation from porous media

sample [38].

mass transfer resistance [39].

96 Current Perspective to Predict Actual Evapotranspiration

3.4.1. The continuum approach

3.4.2. The pore network approach

product scale in the case of designs of dryers.

A pore network model for the evaporative drying of macroporous media was presented in [30]. The model takes into account the heterogeneity of the pore size distribution and the pore wall microstructure, expressed through the degree of pore wall roundness for viscous flow through liquid films, gravity, and for mass transfer, both within the dry medium and also through a mass boundary layer over the external surface of the medium. The model is used to study capillary, gravity and external mass transfer effects through the variation of the three dimensionless numbers: a film-based capillary number Ca<sup>0</sup> <sup>f</sup>, that expresses the ratio of viscous forces to capillary forces in the films;

$$\mathbf{C}d'\_f = \frac{4\mu\_1 D\_\mathbf{M} \mathbf{C}\_\mathbf{c}}{\rho\_1 \chi \overline{r}\_t} \tag{8}$$

the Bond number, Bo, that expresses the ratio of viscous forces to capillary forces in the films;

$$B\sigma = \frac{\mathcal{g}\_x \rho\_1 \overline{r}\_t^2}{\mathcal{Y}},\tag{9}$$

and the Sherwood number, Sh, that describes mass transfer conditions within the mass boundary layer over the product surface.

$$\text{Sh} = \frac{\lambda}{\delta} = \frac{\lambda \overline{r}\_t}{\Delta},\tag{10}$$

where <sup>λ</sup> <sup>¼</sup> Def f ,s<sup>þ</sup> Def f ,s� <sup>&</sup>gt; 1 is the ratio of external to internal effective (volume-averaged) diffusivities, δ ¼ Δ=rt is the dimensionless value of the mass boundary layer and Δ is its corresponding thickness. gx is the gravity acceleration component in the flow direction, μ<sup>1</sup> and ρ<sup>1</sup> are the liquid-phase viscosity and density, γ is the interfacial tension, rt is the average throat radius within the porous medium, DM is the molecular diffusivity and Ce is the equilibrium (at vapor pressure) concentration of the volatile species. The film and dry pore regions are coupled through mass conservation at the front evaporation in a single scalar variable, Φ, mass transport through both the film and dry regions:

$$\Phi = \frac{J(\rho) - \text{Bolx}(\xi) + \text{Ca}\_f^\prime \zeta}{J\_p + \text{Ca}\_f^\prime} \tag{11}$$

where the variable Φ is subject to the following boundary conditions; at the percolation front P, where the films emanate as ζ = 1 and ρ = 1, at the evaporation front I as ζ = 1 and ρ = p and at the top of the mass boundary layer as ζ = 0 and ρ = 0. The effect of gravity is analysed for two cases, when it is opposing and when it is improving drying. For the second case, a two-constant rate period evaporation curve was found when viscous forces are strong and mass transfer in the dry region is fast enough compared to gravity forces. Especially in this regime, water flow is driven primarily by gravity to compensate for evaporation occurring at the film tips [30].

The incorporation of gravity can be done by considering a well-chosen invasion throat potential dependent on variables such as its width of the throat, the relative position in the gravity field and the Bond number [30]. If thermal effects are not included, the pore network model requires to be coupled with mass transfer at the open surfaces and under isothermal conditions. But certainly, temperature has an effect on viscosity and surface tension of the liquid and vapour diffusion coefficient, among others. Moreover, the temperature gradient affects the drying process and distribution of moisture during its development. The effect of heat and mass flow on the drying process in simulation has been reviewed recently in [45].

#### 3.4.4. The diffusion model

Until now, the unique effort to study the moisture migration from GB was the application of a classical temporal surface evaporation model (Eq. (12)) of Crank [7] based in the Fick's second law to calculate the moisture content of a diatomaceous earth pellet at any given time in the drying process [3]. In this model, differences at initial and final concentration are the driving force of change of moisture in time of a sphere and to take into account all relevant physical transport mechanism in the drying process all effects are lumped on the diffusion coefficient, implying that the detailed effects of the pore microstructure are ignored.

$$\frac{M\mathbb{C}(t) - M\mathbb{C}\_1}{M\mathbb{C}\_1 - M\mathbb{C}\_0} = \sum\_{n=1}^{\circ} \left[ \frac{\theta \cdot L^2 \cdot \exp^{(-\beta\_n^2 Dt/R^2)}}{\beta\_n^2 \{\beta\_n^2 + L \cdot (L-1)\}} \right] \tag{12}$$

The use of a large number of series term in Eq. (12) makes their practical use difficult. Also, this approach model is limited to the two phases' system (solid and liquid) and the experimental determination of the effective diffusion coefficient is difficult because of its variation in space [3]. The evaluation of moisture diffusivity using numerical techniques has become a usual methodology in recent years [46] and these numerical methods seem to be a powerful tool for the researchers in formulation of EPNs in GBs.

#### 3.4.5. Drying-strain relation

Studies for optimal control of drying of porous media focused on the assessment of drying effectiveness were found in the literature review. Although for now the use of dryer technology where GBs could be properly dried is non-existent, these approaches are of interest for the formulation process of EPNs. The work of Kowalski and co-workers [47] is dedicated to numerical simulations of optimal control applied to saturated capillary-porous materials subjected to convective drying based in a thermo-hydro-mechanical model. The differential equation expressed in terms of strains, temperature and moisture content is as follows:

$$\left[2\mathcal{M}\varepsilon\_{\vec{\eta}} + (A\varepsilon + \mathcal{\boldsymbol{\gamma}}\_{T}\mathfrak{P} - \mathcal{\boldsymbol{\gamma}}\_{X}\mathfrak{G})\delta\_{\vec{\eta}}\right]\_{,\vec{\jmath}} + \rho \mathfrak{g}\_{i} = \mathbf{0} \tag{13}$$

where K and M are the elastic shear and bulk modulus, ε is the volumetric strain, ρ is the mass density of the body, g is the gravity acceleration (neglected in further considerations), T is the temperature, X is the moisture content and the index (j) denotes differentiation with respect to coordinate <sup>j</sup> <sup>¼</sup> {x, <sup>y</sup>, <sup>z</sup>}. <sup>A</sup> ¼ ð<sup>K</sup> � <sup>2</sup>MÞ<sup>⁄</sup> 3, <sup>γ</sup><sup>T</sup> <sup>¼</sup> <sup>3</sup>KkðT<sup>Þ</sup> , <sup>γ</sup><sup>X</sup> <sup>¼</sup> <sup>3</sup>KkðX<sup>Þ</sup> . Here, k (T) and k(X) are the respective coefficients of linear thermal and humid expansion. Eq. (13), after differentiating with respect to the coordinate j and using the tensor of small strains expressed by the derivative of displacement ui, becomes

$$\varepsilon\_{i\circ} = \frac{1}{2} (u\_{i,j} - u\_{j,i}) \tag{14}$$

which is the displacement differential equation, jointly with the differential equations expressing liquid concentration and temperature supplemented with appropriated initial and boundary conditions that allow to realize numerical estimations of the drying kinetics and deformations of the drying media, and by implication, of the drying-induced stresses. The whole set of differential equations were initially elaborated for the 2-D geometry of a cylindrical shape [47]. The optimization procedure is illustrated on the kaolin-clay material in the form of cylindrical samples (40 mm in radius and 40 mm in height) and the genetic algorithm method was used to simulate the optimal work of the dryer. The authors conclude that drying rates are accelerated if the drying induced stresses are small, and slowed down if the stresses tend to overcome the strength of the material. The formulation of mathematical optimization procedure based in such rigorous principles of rational control of drying and their experimental validation are useful to find optimal drying processes of porous media [47].

#### 3.4.6. Drying of droplets

<sup>Φ</sup> <sup>¼</sup> <sup>J</sup>ðρÞ � BoIxðξÞ þ Ca<sup>0</sup>

where the variable Φ is subject to the following boundary conditions; at the percolation front P, where the films emanate as ζ = 1 and ρ = 1, at the evaporation front I as ζ = 1 and ρ = p and at the top of the mass boundary layer as ζ = 0 and ρ = 0. The effect of gravity is analysed for two cases, when it is opposing and when it is improving drying. For the second case, a two-constant rate period evaporation curve was found when viscous forces are strong and mass transfer in the dry region is fast enough compared to gravity forces. Especially in this regime, water flow is driven primarily by gravity to compensate for evaporation occurring at

The incorporation of gravity can be done by considering a well-chosen invasion throat potential dependent on variables such as its width of the throat, the relative position in the gravity field and the Bond number [30]. If thermal effects are not included, the pore network model requires to be coupled with mass transfer at the open surfaces and under isothermal conditions. But certainly, temperature has an effect on viscosity and surface tension of the liquid and vapour diffusion coefficient, among others. Moreover, the temperature gradient affects the drying process and distribution of moisture during its development. The effect of heat and

Until now, the unique effort to study the moisture migration from GB was the application of a classical temporal surface evaporation model (Eq. (12)) of Crank [7] based in the Fick's second law to calculate the moisture content of a diatomaceous earth pellet at any given time in the drying process [3]. In this model, differences at initial and final concentration are the driving force of change of moisture in time of a sphere and to take into account all relevant physical transport mechanism in the drying process all effects are lumped on the diffusion coefficient,

<sup>6</sup> � <sup>L</sup><sup>2</sup> � expð�β<sup>2</sup>

" #

β2 nfβ<sup>2</sup>

The use of a large number of series term in Eq. (12) makes their practical use difficult. Also, this approach model is limited to the two phases' system (solid and liquid) and the experimental determination of the effective diffusion coefficient is difficult because of its variation in space [3]. The evaluation of moisture diffusivity using numerical techniques has become a usual methodology in recent years [46] and these numerical methods seem to be a powerful

Studies for optimal control of drying of porous media focused on the assessment of drying effectiveness were found in the literature review. Although for now the use of dryer

<sup>n</sup>Dt=R2<sup>Þ</sup>

<sup>n</sup> þ L � ðL � 1Þg

mass flow on the drying process in simulation has been reviewed recently in [45].

implying that the detailed effects of the pore microstructure are ignored.

<sup>¼</sup> <sup>X</sup><sup>∞</sup> n¼1

MCðtÞ � MC<sup>1</sup> MC<sup>1</sup> � MC<sup>0</sup>

tool for the researchers in formulation of EPNs in GBs.

the film tips [30].

98 Current Perspective to Predict Actual Evapotranspiration

3.4.4. The diffusion model

3.4.5. Drying-strain relation

Jp þ Ca<sup>0</sup> f f ζ

ð11Þ

ð12Þ

Drops are subject to theoretical and experimental analysis to determine how the moisture is lost under different conditions [41, 48–56]. Theoretical models for the drying of single droplets are of interest for applications of formulation of EPNs since GBs can be made by liquid penetration in powders or can contain soluble and insoluble adjuvants subject to evaporation. The theoretical study of evaporation of liquid droplets on solid substrate is based on the assumptions of diffusion-controlled mass transfer in the gas phase, constant temperature over the whole system (isothermal conditions) and neglects the effect of convection in the vapour phase [41]. However, when the thermal effects due to evaporative cooling in the classic model are introduced, the results show that the evaporation slows down by increase of the latent heat of evaporation and the substrate thickness as well as by a decrease of the substrate thermo-conductivity. The theoretical predictions using this model do not have good agreement if the substrate temperature deviates from the room temperature and the possible reason of this deviation is the increasing importance on thermal-buoyancy convection at higher temperatures [54].

The regular regime method is useful to determine the content-dependent diffusion coefficient for systems in which the relation among moisture diffusivity and moisture content are lineal and the last one decreasing below the critical moisture concentration or also for situations where the drying rate is dominated by mass transfer inside the drying specimen [57]. Under this theory, the drying curves show an induction period in which the drying rate is conditioned by the moisture distribution at the beginning of the drying process and a regular regime period in which the drying rate is not correlated and thereby of the moisture distribution at the beginning of the process. The establishment of the moisture range at which the regular regime occurs during the isothermal drying is the condition to apply this method, particularly for a given material and conditions, the drying curves will converge in a regular regime curve, even for different moisture contents in a single curve named the regular regime curve [49].

The objective of the work of [49] was to develop a method for quantitatively calculating the effective moisture diffusivity of isothermally dried biopolymer drops and to acquire activation energy to be used as a discriminating parameter for selecting effective wall materials against lipid oxidation. The biopolymer´s effective moisture diffusivity was dependent on moisture content and temperature. Therefore, air temperatures must be lower than 80C for an appropriate analysis of the water diffusion mechanism using the regular regime methodology. Also, the activation energy provides a quantitative measurement for selecting potential good wall materials against lipid oxidation [49].

In other approach, the interactions of droplets during its deposition on porous material for the agglomeration of particles by spray-fluidized process were studied by [55]. In this work, the penetration of liquid into the porous layer was assumed to be governed by Darcy's law. In reference [48] the molecular kinetic theory was used to model the droplet spreading, and Darcy's law to describe the one-dimensional liquid penetration into the substrate. Particularly, this approach is of interest for applications of formulation of EPNs because the elemental principle of the methods of formation of GBs is to deposit droplets of aqueous suspension containing EPNs over a layer of a mixture of material (carrier and adjuvants) and then the penetration of the liquid carrier happened [5, 9]. After that, the materials are mixed and compacted in different ways (i.e. agitation, eccentric rotational motion and compaction by rolling, among others). These are reasons to optimize the capillary penetration of aqueous suspension into the porous layer; the initial moisture content and moisture evaporation are critical factors in the design of granules for storage and transport of EPNs for biological applications [3, 50]. Following this approach, in [56] was developed a method to quantitatively describe the evaporation effect on radial capillary penetration of liquids in thin porous layers.

#### 3.4.7. Natural convection in cavities

The case study of Prakash et al. [32] is moisture migration in a rectangular cavity (2 m in height and 1 m in diameter) with half the cavity filled with silica gel and their paper presents a general method of solving heat and moisture transfer by analysing a two layer system with a fluid overlying a hygroscopic porous medium where turbulence in the fluid layer affect the natural convection flow. Also, these approaches are of interest for applications of formulation of EPNs if the phenomenon is scaled adequately to small containers in which GBs are deposited for storage, transportation and commercialization, because they permit air in the container to contact the exposed surfaces of the packages for oxygen exchange with EPNs [58, 59]. The model is based in equations for fluid flow and heat transfer and equations for moisture migration in porous media. The model equations were discretized using the control volume formulation and solved using the SIMPLE algorithm. The model is capable of simulating flow only when turbulence in the porous medium can be considered to be negligible. However, this would not be the case for porous media of high permeability. In order to overcome this limitation, the authors suggest that a turbulence model for the porous medium needs to be incorporated into the existing model.
