The Reaction Engineering Approach (REA) for Modeling Droplet Drying Behavior
To introduce the Reaction Engineering Approach (REA) concept in mathematical terms, based on the conventional transport phenomena theory, we write the vapor flux at the boundary (solid-gas) as follows:
where pvs is the vapor concentration at the interface of solid and gas (kg • m~3) and RHS is the relative humidity of the air (or gas in general) at the solid-gas boundary, which may be defined as:
During the drying process, this surface relative humidity RHS is reducing but is an unknowm quantity.
Equations (2.2.1) and (2.2.2) thus show a significant contrast between them. Equation (2.2.2) defines the instantaneous drying flux as a “fraction” of the maximum possible; Equation (2.2.1) follows the widely accepted mass transfer expression, w'ith the first term in the bracket on the right-hand side of the equation being the product of RHS and saturated vapor concentration at the interface. The difficulty of using Equation (2.2.2) is how one can express RHS as a function of some known quantities.
The concept of the REA has incorporated the more conventional approach to expressing mass transfer at the boundary, i.e., Equation (2.2.1). It is an application of the chemical reaction engineering principle to establish a workable function for RHS. In this approach, evaporation is modeled as zero order kinetics with activation energy, while condensation is described as the first order wetting reaction with respect to drying air vapor concentration without activation energy (Chen and Chen, 1997; Chen and Xie, 1997). The REA approach employs the Arrhenius equation in the evaporation term, which has an “origin” in the paper by Gray (1990), but is very different overall from what was proposed by Gray (Chen, 2008). The REA approach offers the advantage of being expressed in terms of simple, ordinary differential equations w'ith respect to time. This negates the complications arising from use of partial differential equations (Chen, 2008). The REA does need accurate experimental data to determine model parameters, and an accurate equilibrium isotherm and surface area measurement. It can be showm later that the REA accommodates a natural transition because of the smooth activation energy as a function of moisture content (Chen, 2008). When activation energy for evaporation becomes higher than the latent heat of water evaporation, free water should have already been removed (Chen, 2008).
As a lumped approach (REA was initially proposed as a lumped parameter model), the drying rate (flux multiplied by surface area) of materials can be expressed as:
where ms is the dried mass of thin layer material (kg), X is the moisture content on dry basis (kg • kg"1) and X is the mean water content on dry basis (kg • kg"1), t is time (s), pvs is the vapor concentration at the interface (kg • m~3), pvb is the vapor concentration in the drying medium (kg • m"3), /?,„ is the mass transfer coefficient (m • s'1) and A is the surface area of the material (m2). The mass transfer coefficient (hj is determined based on established Sherwood number correlations or established experimentally for the specific drying conditions involved (Incroper and Dewitt, 1990). Equation (2.2.3) is basically correct for all cases of water vapor transfer from a porous solid. In other words, there is no assumption of uniform water content in this approach, even though it was started with the mean water content. The surface vapor concentration (pvs) can be correlated in terms of saturated vapor concentration of water (p, sa,) by the following equation (Chen and Chen, 1997; Chen and Xie, 1997):
where Д£j, represents the additional difficulty of removing moisture from materials on top of the free water effect. It was thought that it would be excellent (and indeed lucky) to be able to relate ДEv to the average water content of the material. In other words, this Дis ideally to be moisture content (X) dependent. T is the temperature of the material being dried (K). For a small temperature range, say from 0°C to just over 100°C, pvsa, (kg • nr3) can be estimated using the following equation (Chen, 1998):
where Kv was found to be 1.61943 x 10s (kg • s_1) and Ev was found to be 38.99 kJ • mol"1. Ev is similar to the latent heat of water vaporization illustrating the physics involved (Chen, 1998). This is in line with the idea that evaporation is an activation process while condensation is not. The activation energy of the pure water evaporation reaction is equivalent to the value of the latent heat of water evaporation, as suggested earlier, based on classic physical chemistry.
For a wider temperature range, one can use the following, which correlates to the entire range of the data (0°C to about 200°C) summarized by Keey (1992):
where Tis temperature (K) based on the given data (Putranto etai, 2010) (Figure 2.1.8).
FIGURE 2.1.8 Saturated water vapor concentration in air under 1 atm (Equation 1.21).
The mass balance is then expressed as:
For small objects such as particles or thin layer materials, the material temperature T is approximately the same as the surface temperature Ts. Basically, this happens when the Chen-Biot number is sufficiently small (Chen and Peng, 2005; Chen, 2007; Chen and Mujumdar, 2008). In this case, uniform temperature can be assumed throughout the material being dried such that one only needs to couple Equation (2.2.7) with a lumped energy balance to “govern” the drying process.
The activation energy (AE,) is determined experimentally by rearranging Equation (2.2.7) as follows:
where dX/dt is determined from experimental data on weight loss. It has been found, based on practical experiences of using the REA, that drying experiments to generate the REA parameters need to be conducted where the air (or gas) humidity is very low in order to cover the widest range of A£j, versus X. The dependence of activation energy on moisture content can be normalized as:
where £ is a function of moisture content difference and AE„oo is at its maximum when the moisture concentration of the sample approaches relative humidity and temperature of the drying air:
Xx is the equilibrium moisture content corresponding to RHX and Tx, which can be related to one another by the equilibrium isotherm (Keey, 1992). It is worth noting again that, so far, the experiments to achieve the relevant equation (Equation 2.2.9) have been conducted under very dry air conditions so the final water content attained is usually fairly low.
For the same material, the same initial water content and the same initial sample size (sometimes different sizes do not matter), the relationship (Equation (2.2.9)) may be viewed as unique, as many experimental results obtained under the above conditions but different drying conditions for the same material produced more or less the same trend quantitatively (Chen, 2008). This aspect will be shown later in various applications described in the forthcoming chapters.
The REA parameters for the drying of a material can be obtained from one good drying experiment and can then be applied to other different drying conditions (different drying air temperatures and air velocities) if the normalized activation energy would collapse to the same profile in these cases. However, the REA parameters should be generated from the material with same initial moisture content, since the activation energy has been found to be dependent on initial moisture content as well (Chen, 2008).
In many scenarios tested involving different materials, Equation (2.2.9) holds - a very pleasant outcome indeed. Of course, other forms of the equation are possible, and one should be worried if there was a temperature dependence function, or a material structure parameter became involved.
When the temperature of the moist material being dried does not vary much within itself, a uniform temperature may be considered (more quantitative assessment of this assumption can be found in a later part of this book that describes the modified Biot number and modified Lewis number). This leads to:
where f represents the mean temperature of the material (K). For droplets and particles, this is satisfied basically.
This has allowed us to present the REA energy balance in a “lumped capacitance” (Incropera and Dewitt, 1990) for the material being dried:
where the mass of the material being dried is expressed as: