1.3.1 The relevant classical knowledge of physical chemistry
In the forthcoming sections, the ideas and the development behind the REA for modelling drying process will be briefly described. A short summary of the classic modelling approaches is given first, and the newer REA approach will be subsequently introduced.
To begin the discussion on the REA concepts, we need to outline physical chemistry principles of chemical reactions with a little originality on our part, in order to form the basis of the REA idea. The most prominent idea in reaction engineering is the expression of the chemical reaction rate. A chemical reaction rate of species A, involved in the reaction of two species (A and B) yields a product commonly expressed as:
−dcA
dt = kAcnAAcnBB, (1.3.1) where nAand nBare the orders of reactions associated with species A and B, respectively, kAis a rate constant, and CAand CBare the concentrations of species A and B, respectively.
This rate constant increases with temperature T, approximately increasing by two to
four times with a temperature increment of 10 K. The relationship between a reaction rate constant, k, and temperature, T, has been generically described using the famous Arrhenius equation (Fogler,1992):
−d ln kA
d T = EA
RT2, (1.3.2)
where EA is the activation energy of the reaction (J mol−1). This means that the value of ln kA change against temperature is proportional to the value of EA. The larger activation energy EAis, the more sensitive the reaction rate towards temperature change. EAcan be variable against temperature when multiple reactions are occurring simultaneously.
When the range of temperature is not large, EAmay be considered a constant. In this case, which is more commonly adopted in real life, the rate constant is expressed as:
kA= kAoe−EA/RT, (1.3.3)
where kAois a constant.
Molecular mechanisms of evaporation and condensation at free liquid surfaces under the vapour–liquid equilibrium are investigated with molecular dynamics computer sim-ulations for argon and methanol. Vapour molecules colliding with the surface are in the condition of almost complete capture for both fluids but, in the case of methanol, molec-ular exchanges strongly affect the evaporation–condensation rate (Matsumoto et al., 1995) (presumably water evaporation–condensation behaves similarly). Evaporation–
condensation is one of the fundamental processes in many fields of science and engi-neering. For decades, various experiments have been done to measure the absolute values of evaporation or condensation rate, but there still remains a lack of knowledge of the underlying molecular mechanisms. There is still a lot of room available to explore the fundamental aspects of evaporation–condensation. This phenomenon is further compli-cated by the presence of species other than water during the drying process. One could, however, understand intuitively that removal of water (in vapour) is an energetic process involving latent heat. One has to put in energy to ‘activate’ water molecules that could become free when the material is dried. When the water molecules are not associated with the solid or solute molecules and they stay in the bulk liquid domain, their inter-actions between one another would be as if there was no solid. Evaporating them into a gaseous form would be done mainly by providing energy sufficient to overcome the latent heat of water evaporation. Of course, another condition is that the vapour can be transported into the gas medium or into a vacuum. It is interesting to note that, since the condensation process is spontaneous and does not need to overcome a kind of activa-tion energy (an energy barrier), the activaactiva-tion energy for condensaactiva-tion is zero, and the activation energy of the evaporation process for pure water would be equal to the heat of reaction (the latent heat of water vaporisation); this is to say, the forward reaction activation energy is less than the reverse reaction activation energy, which is zero for condensation. This was the basic motive and idea in formulating the drying rate as a competition between evaporation and condensation (to be described later).
Introduction 17
1.3.2 General modelling approaches
Before REA there were three main general approaches in the literature to formulating drying models, summarised next (Chen and Xie,1997):
1. The concept of a characteristic drying rate curve (CDRC model), which recognises different drying stages; e.g. the constant rate period (which may also be called the
‘unhindered drying’ period, where the internal transport of the moisture does not affect the surface evaporation) and falling rate (‘hindered drying’) period(s).
2. The distributed-parameter models, mostly those based on volume averaging concepts, employing coupled heat and mass diffusion equations involving heat conductivities and mass diffusivities, etc.
3. The empirical models obtained entirely by simple or multivariable regression methods (often, a series of known time-dependence functions such as the Page model etc. have been used to simply correlate the weight loss over time).
There are many models in categories (1) and (3). Both (1) and (3) may be regarded as the ‘lumped drying models’ in that they do not need to solve for the spatially distributed moisture content and temperature. For (2), there have been a number of continuum-type mechanisms proposed and the corresponding mathematical models established.
These include effective liquid diffusion, capillary flow, evaporation–condensation, dual (temperature, water content gradient) and triple (temperature, water content and pressure gradient) driving-force mechanisms by Luikov (1986), another dual driving-force mech-anism by Philip and De Vries (1957) and De Vries (1958), and, finally, the dual-phase (liquid and vapour) transfer mechanism of Krischer as summarised by Fortes and Okos (1980).
Whitaker (1977;1999) has proposed detailed transport equations to account for the macro- and micro-scale structures in biological materials. Three-phase (solid, liquid and vapour) conservations and their local volume-averaged behaviours are considered. The mechanisms for moisture transfer are largely the same as those proposed by Luikov (1975;1986) and Philip and De Vries (1957), except that the small-scale phenomena (local pores, pore channels, shells, voids, etc.) have been taken into account. This theory is based on a known (or pre-assumed) distribution of the macro-scale and micro-scale unit structures, which allow local volume averaging to be carried out. Pore-network models have become popular in recent years, as the concept of scale and multi-physics is expanding, e.g. coupling meso-scale problems and equipment scale problems (Perre,2011). When a local thermodynamic equilibrium is not attained, the time scales usually overlap. This is a real multi-scale configuration and challenging in terms of the great demand in computational power and mathematics. Several scales can be consid-ered simultaneously, ranging from simple exchanges between macroscopic phases to a comprehensive formulation, in which the time evolution of microscopic parameters and microscopic gradients are considered over a representative elementary volume accord-ing to a recent review by Perre (2011). These models are often mathematically highly involved.
For (3), the models usually have little physics explained and it is difficult to extract any fundamental information, though some of them have attempted to show some physical significance on somewhat weaker grounds. On the other hand, some accurate time functions under category (3), such as the Page model, can be used to fit the data points to generate accurate drying rate data sets for other purposes. Some models of a comprehensive nature will be described in detailed mathematical terms in later chapters when the authors compare the performances of REA in various practical cases against well-known models.
In modern times, expanding what we have already in category (2), the comprehensive modelling of drying in a spatially distributed manner (or, say, in a discrete manner) has frequently involved more pore-level information (pore-network models) and math-ematical techniques, which do not require volume-averaging procedures of some sort.
The respective pore networks can be used to systematically study the influence of the structure of a porous medium on drying kinetics. There are potentials of this discrete modelling for use as a virtual laboratory to improve our understanding of how struc-tures correlate with properties and how better products may be developed (Tsotas and Mujumdar,2007).
In general, several scales of problem are involved in drying, modified from those summarised by Tsotsas and Mujumdar2007:
The molecular scale (water molecules interact with each other and with the other species in the liquid or gas, and with the solid surfaces), the pore scale (the smallest entity for expressing transport phenomena within the drying particles or single bodies), the particle scale (single drying body can be identified; this larger scale can include rather ‘large particles’ such as wood boards stacked and dried in an industrial drying kiln), the particle-system scale (the equipment is designed and properly operated at this level; the interactions between the particles, the gas flow and the apparatus are considered at this scale), and finally the process-system scale (the drying system interacting with other engineering systems ensuring the proper operation of the entire production plant).
1.3.3 Outline of REA
Following the basic descriptions of physical chemistry in relation to the expression of chemical reaction rate as described earlier, the REA was proposed by the author (XDC) in 1996, and in the subsequent year, a couple of papers were published (Chen and Chen, 1997; Chen and Xie,1997; Chen and Pirini,2004). The idea was also partially inspired by two pieces of information: a paper published by Professor Brian Gray (1990) and a series of works on the long-established CDRC model (van Meel,1958; Keey,1978;
1992).
In 1990, a mathematical model for a wet-combustion system, the exothermically reactive (porous solid) system, which is also influenced by the presence of water, was published by Professor Gray, who was a Senior Professor of Mathematics at School of Chemistry in Macquarie University, Sydney, Australia. However, before that landmark, the role of water on the exothermic (solid) systems, such as spontaneous heating or spon-taneous combustion, had been proposed in several models from more of an engineering
Introduction 19
perspective (Chen,1991). However, the role of water as a direct participant in chemical reactions (in oxidation in particular) had not been considered quantitatively. In the paper by Gray, he added a term in the mass balance and energy balance, respectively, which accounts for the direct participation of water in chemical reaction (exothermic), in con-junction with the water effect through evaporation (liquid to vapour) and condensation (vapour to liquid).
In the same paper, Gray wrote the following equations to describe a wet-combustion system where the temperature of the (combustible) solid material is assumed to be uniform, as is the water content within the solid matrix:
d x
dτ = c(1− x) − exe−α/u− wxe−αw/u, (1.3.4) where x, u andτ are the dimensionless liquid water concentration, temperature of the (combustible) material, and time, respectively. Term x is defined as the current water content (g) divided by the initial water content (g) that is available in liquid form (in fact, it is the total water content in the system boundary), i.e. mw/mw,o. The model constants are represented byϕ in Equation (1.3.4); this system is also, in the theory of thermal ignition and combustion, known as the Semenov approach, signifying the uniformity of the variables throughout the material of concern (Bowes, 1984). This is a useful assumption, which paves the way for a large number of mathematical analyses that have considerable physical meaning relevant to practical conditions.
In Equation (1.3.4), the first term on the right-hand side represents the condensation process, denoted by subscript c, and the second term evaporation, denoted by e. The last term on the right-hand side represents the consumption of water due to the water-induced or -involved chemical reaction (exothermic), the wet oxidation, denoted by w. When we remove this wet-oxidation term, an inert system, we have:
d x
dτ = c(1− x) − exe−α/u. (1.3.5) Equation (1.3.5) represents the water exchange (condensation less evaporation) between the moist material and the environment/surrounding. (1− x) signifies a conservation of the ‘total water content in the domain of interest’. The evaporation term is considered to be first-order as far as water ‘reactant’ is concerned. The most important description of the evaporation term is used in the Arrhenius dependence function (a well-known function in physical chemistry), i.e. e−αu. Theα in the context of Gray’s analysis denotes the dimensionless latent heat of water vaporisation. This may not have been completely new, even at that time, but certainly was a great approach to describe the physical picture, as we tried to understand moisture movement in and out of a porous solid matrix. Gray (1990) and Gray and Wake (1990) took full advantage of this simple and effective formulation to fruitfully explore the wealth of the behaviours in the wet-combustion system. In any case, a steady state can be attained for the system described by Equation (1.3.5) as:
c(1− x) = exe−α/u. (1.3.6)
T x
1
0
Figure 1.9 Schematic illustration of the effect of temperature on final liquid water content (qualitatively derived from Equation1.3.6).
Rearrange to give:
1− x x = e
c
e−α/u, (1.3.7)
which essentially suggests that as the material gets hotter, the liquid water content in the system reduces more. The activation energy of the evaporation ‘reaction’ was assumed to be the latent heat of water vaporisation. This has good intuitive ground-ing (Chen,1998). As the dimensionless temperature u gets infinitely large, x becomes zero (see Figure 1.9). This system, however, does not seem to capture the physics behind when water content in an environment becomes zero; even when tempera-ture is moderate, the liquid water content inside the material can also become zero.
In other words, this system might have neglected another dimension of the drying system.
On the practical side, a drying kinetics approach, the CDRC model, had been employed in order to design dryers and optimise drying operations for improving energy efficiency. In his developing understanding of drying operations and the fun-damentals of moisture transfer, van Meel (1958) postulated that, when working with convective batch dryers, a single characteristic drying curve could be deduced for a moist material. This model reflects the nature of the drying rate curve(s) shown in Figure 1.10(a). It is empirical but has been successful in correlating the drying kinet-ics of small particles. The model assumes that, for any given sample water content, a unique relative drying rate exists. This rate is relative to the initial unhindered dry-ing rate and is independent of the external drydry-ing conditions (refer toFigure 1.10(b)).
These conditions include the temperature, humidity and pressure of the drying gas.
The model also implies that a region exists where, for a period of time, the rate remains unhindered.
The physics behind what is shown inFigure 1.10(a) has been described scientifically by Perre, Remond and Turner (2007). For non-deformable materials (negligible shrinkage) such as building materials and natural mineral products (including fragmented rocks), the relationship between porosity and moisture content is obvious. As drying proceeds, the moisture content is simply replaced by drying gas. For highly deformable products such as food, the moisture removal is related to both volume reduction and porosity. It is necessary to know whether the loss of moisture turns into volume change or into an increase in porosity (Perre and May, 2001). Considering these factors one can interpret
Introduction 21
Critical point (1,1)
Unhindered region Hindered
region
Equilibrium point
0 φ
ξ
Cooling down
Warming up
Critical average water content Equilibrium
water content Falling
drying flux period
Drying flux (kg.m--2.g--1 )
Constant drying flux period
Average content (X) Xc
X∞ 0.0
(a)
(b)
Figure 1.10 (a) Drying flux versus average water content X ; (b) the CDRC (characteristic drying rate curve).[Reprinted from Chemical Engineering Science, 9, D.A. van Meel, Adiabatic convection batch drying with recirculation of air, 36–44, Copyright (2012), reprinted with
permission from Elsevier.]
the results on constant drying rate, which should be the constant drying flux, properly.
The mass exchange surface area is an important parameter involved here.
Here, the relative drying rate is defined as:
ξ = N Nc
, (1.3.8)
whereξ is the relative drying rate, N is the instantaneous drying rate (best defined to be the drying flux, kg m−2s−1) and Ncis the drying rate at the critical condition (i.e. when the drying regime is in transition between the unhindered rate and falling rate periods;
at the critical water content, Xc).
The characteristic moisture content (dimensionless water content) is defined by the following equation:
= X− X∞
Xc− X∞, (1.3.9)
where X is the average water content (on dry basis) at any time t, Xis the equilibrium water content (on dry basis) and Xc is the critical water content. The drying rate is normalised to pass through the critical point and the equilibrium point, denoted by points (1,1) and (0,0), respectively inFigure 1.10(b).
According to Keey (1992), the characteristic curve method is attractive since it leads to a simple lumped-parameter expression for the drying rate, in the following form:
N = ξ Nc. (1.3.10)
This expression has been used extensively as the basis for understanding the behaviour of industrial drying plants. Because of the simplicity of the parameters used, this has been employed in some industrial applications.
Basically, the general form of the CDRC (Keey,1992) is expressed as follows:
ξ = f X
, if X ≤ Xc; ξ = 1, if X > Xc. (1.3.11) There are a number of successful applications of this approach, especially for materials of small dimensions such as particles or thin layers (Keey,1992). Keey (1992) has further specified that a unique characteristic curve can be established at Kirpichev numbers less than 2 or, in effect, when the material is thinly spread and permeability to moisture is high (i.e. the material has a large moisture (vapour) diffusivity).
The Kirpichev number is given as:
Ki = Ncδ
ρsXoDv,eff, (1.3.12)
whereδ is the thickness of the sample (m), ρsis density of the dry solid (kg m−3), Xo
is the initial water content (on dry basis) (kg kg−1) and Dv,eff is the effective vapour diffusivity (m2s−1). According to Keey (1992), in many cases the drying curve can be fitted using a simple algebraic equation over a limited moisture content range of interest by:
ξ =
X− X∞ Xc− X∞
, (1.3.13)
where j is a parameter dependent on the relative difficulty of removing moisture from a material. For example, j was found to be about 0.5 for cellulosic fibres (Langrish,2008).
Equation (1.3.10) is a nice, simple and user-friendly expression, if accurate enough, for the material of concern. The author was first exposed to the idea of the CDRC during his Ph.D. study at Canterbury University in Christchurch, New Zealand. The exposure of the author to CDRC was significant, due to the physical presence of the well-known advocate of the CDRC approach (and indeed one of the greatest in drying research and development), Professor Roger Keey at Canterbury University, at the time. During the
Introduction 23
same period of time, a visiting scholar from China, Professor Yuan Wu, was working on the application of a variation of the CDRC approach in through-drying of wool in Keey’s laboratory (which was later published in Chemical Engineering Science, Wu and Keey,1995). However, being young at the time, the author did not fully appreciate the usefulness of the CDRC approach, which only became apparent later in the author’s research career in drying.
According to the understanding gained so far, there are aspects of CDRC that need to be noted:
(1) The critical water content(s) have to be determined experimentally, which are known to be dependent upon drying conditions (temperature, humidity and velocity).
(2) The quality of results of data reduction is not good, as in some cases the data points can be quite scattered (for instance, with some large relative errors from the mean in the falling rate period(s)).
(3) In addition, the mass flux (Nc) calculations are sometimes based on the wet-bulb temperatures (Twb) for the gas phase.
Aspect (3) is illustrated as follows:
Nc= hm
ρv,sat(Ts)− ρv,∞
, (1.3.14)
where the material surface temperature Ts(K) may not be exactly the same as the wet-bulb temperature Twb(K) and hmis the mass transfer coefficient (m s−). Here,ρv,sat(Ts) is the saturated water vapour concentration at the surface temperature Ts(kg m−3) and ρv,∞is the water vapour concentration in the environment or in the drying gas medium (kg m−3).
A CDRC model is usually obtained from laboratory experiments under constant external conditions, with moist materials of similar form and size to that of inter-est in the real industrial dryer situation. As mentioned earlier, one also needs to note that the CDRC model must have accurate measurement, choice or prediction of the constant rate and the maximum rate, Nc. An accurate estimation of the critical water content as a function of the external drying conditions must be made, as well as the ability (or, rather, lack of it) to reduce effective data for the relative drying rates.
A CDRC model is usually obtained from laboratory experiments under constant external conditions, with moist materials of similar form and size to that of inter-est in the real industrial dryer situation. As mentioned earlier, one also needs to note that the CDRC model must have accurate measurement, choice or prediction of the constant rate and the maximum rate, Nc. An accurate estimation of the critical water content as a function of the external drying conditions must be made, as well as the ability (or, rather, lack of it) to reduce effective data for the relative drying rates.