Drying of latex films and coatings: Reconsidering the fundamental mechanisms

Progress in Organic Coatings 57 (2006) 236–250

Drying of latex films and coatings: Reconsidering the fundamental mechanisms Søren Kiil ∗ Department of Chemical Engineering, Technical University of Denmark, Building 229, DK-2800 Kgs. Lyngby, Denmark Received 26 June 2006; accepted 5 September 2006

Abstract The two existing theories describing drying of latex films or coatings are reconsidered. Subsequently, a novel mathematical drying model is presented, the simulations of which can match and explain experimental drying rate data of two previous investigations with latex films. In contrast to previous model studies, but in agreement with observations, simulations suggest that during the falling rate period of the drying process of a latex film, a porous skin of partly coalesced latex particles is indeed formed, which limits transport of water vapour from the receding air–liquid interphase to the surface of the film. The value of the effective diffusion coefficient of water vapour in the dry and partly coalesced layer (7 × 10−7 m2 /s at 19–24 ◦ C), the adjustable parameter of the model for the falling rate period, was found to be independent of initial wet film thickness (89–1322 ␮m), latex particle size (500–600 nm), initial polymer volume concentration (19–47 vol.%), and molecular weight of latex polymer (not quantified). Simulations also demonstrate that the transition from a constant to a falling drying rate in all cases takes place when the polymer volume concentration of the latex film is equal to that of hexagonal closest packed monodisperse spheres (74 vol.%). Consequently, the model has predictive properties and model inputs are only needed on the specific experimental (or field) conditions of interest. The effects on drying time of variations in relative humidity, wet film thickness, initial polymer volume concentration, and air flow velocity are simulated and analysed using the new model. © 2006 Elsevier B.V. All rights reserved. Keywords: Mathematical modelling; Paint; Water-based; VOC; Coalescence; Evaporation

1. Introduction A latex is a milky dispersion of nano-size polymer particles in water prepared using emulsion polymerization and stabilized using surfactants. The first latex was commercialized in 1946 [1] and today a large fraction of the latex polymers produced are utilised as binders in latex coatings [2]. Upon evaporation of water, at a temperature above the minimum film formation temperature, the latex particles, typically of size 50–500 nm, are first concentrated, then deform into rhombic dodecahedrons and fuse (coalesce), thereby forming a coherent and clear solid film or a colourful coating if pigmented. Increasing demands for organic solvent-free or at least solvent-reduced coatings in the industrial and domestic sector [3] have been the reason for much Abbreviations: HTMI, heat and mass transfer increase; PVC, polymer volume concentration (i.e. volume fraction of polymer in a latex film); WFT, wet film thickness ∗ Tel.: +45 45 25 28 27; fax: +45 45 88 22 58. E-mail address: [email protected]. 0300-9440/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.porgcoat.2006.09.003

research in latex coatings in the past decades (see for example, the extensive review by Steward et al. [2]). The shift to waterbased coatings for domestic use, besides the well-known wall paints, has partly been done in some countries (e.g. with exterior wood coatings and primers for indoor use), but many highperformance industrial organic solvent-based products (e.g. anticorrosive, antifouling and some automotive coatings) are more difficult to replace. The current penetration of water-borne coatings in the European industrial coating sector is around 20% [4]. A thorough understanding of the drying and film formation behaviour of latex films and coatings is useful in the design of novel or improved environmentally friendly coatings. Typical problems related to domestic use of latex coatings are [4,5]: too short “open time” (viscosity effect and defined as the period of time during which a painter can make corrections to a freshly applied wet paint film without leaving brush marks) and “wet edge time” (period of time during which no edge marks are produced when a freshly applied paint is lapped over a previously painted area). For industrial coatings, which are often applied in higher wet film thicknesses (perhaps 200–500 ␮m per layer) than

S. Kiil / Progress in Organic Coatings 57 (2006) 236–250

Nomenclature A C CP CR C* De,v Dv h hc hnc Hvap k kg,v kg,v l lo L M p p* Pr ReL Sc t T v W Ww xww X

film surface area exposed to air (m2 ) concentration (mol/m3 ) heat capacity (J/kg K) radian number for a black body (5.72 × 10−8 W/m2 K4 ) saturated water vapour concentration (mol/m3 ) effective diffusion coefficient of water vapour in partly coalesced dry layer (m2 /s) bulk phase diffusion coefficient of water vapour (m2 /s) heat transfer coefficient (convection or natural convection) (W/m2 K) average convective heat transfer coefficient (W/m2 K) heat transfer coefficient (natural convection) (W/m2 K) heat of vaporisation of water (J/kg) heat conductivity (W/m K) mass transfer coefficient (convection or natural convection) (m/s) average convective mass transfer coefficient (m/s) position in film (m) initial wet film thickness (m) horizontal length of film (m) molar weight (kg/mol) water vapour pressure (Pa) saturated water vapour pressure (Pa) Prandtl number for air flowing along drying film (ηCp /k) Reynold number for air flowing along drying film (ρvL/η) Schmidt number for air flowing along drying film (η/Dρ) time (s) temperature (K) air flow velocity (m/s) cumulative weight loss during drying (kg/m2 or g/cm2 ) total amount of water evaporated in the experiments of Eckersley and Rudin (kg) water mass fraction in latex film the fraction of the intial amount of water in the film that is still present in the wet film

Subscripts B bulk air F film (wet) i initial value L latex or latex particles o at the end of stage 1 p polymer (of which latex particles are made) s1 at the air/wet film interphase s1o at the air/wet film interphase at the end of stage 1 s2 at the air/dry film interphase

s2o sub v w 1 2

237

at the air/dry film interphase at the end of stage 1 substrate vapour liquid water stage 1 stage 2

Greek letters ε emissivity of wet latex film εp porosity of dry partly coalesced layer γo initial polymer volume concentration γs polymer volume concentration at the end of stage 1 (=0.74) η viscosity (kg/m s) ϕRH relative humidity ρ density (kg/m3 ) τ tortuosity factor

architectural coatings (30–100 ␮m per layer) to obtain optimal protection properties, the main disadvantages are expensive raw materials, slow drying and formation of weak films in humid conditions (such as in a dry-dock or other places near water), reduced flow and wetting properties compared to solvent-based systems, and potential water sensitivity during application and service life. In an attempt to avoid these problems, coalescing agents (i.e. various organic solvents) are usually added to latex films and coatings in limited amounts [5]. Despite the many years of investigation (more than 50 in 1999 according to [6]), there is no clear convergence with respect to the mechanisms underlying the drying of latex films and coatings. In this work it is the aim to analyse the previous mathematical models describing drying of latex films and coatings and to propose a novel model, which is in agreement with previous experimental data and observations. The phenomenon of coalescence will only be discussed where it is needed to explain drying observations. Thus, the mechanisms of coalescence and the potential subsequent cross-linking reactions and network formation, as discussed, for instance, in the thorough review by Taylor and Winnik [1], are not part of this work. However, as different definitions are used, it should be mentioned that the term “coalescence” in this work refers to the definition given by Steward et al. [2], which encompasses the entire process of compaction, deformation, cohesion, and polymer chain interdiffusion of the individual latex particles. To avoid further complications in the process of elucidating the drying mechanisms, there are no attempts in this work to model full latex coatings (i.e. pigmented latex dispersions with additives), but only latex films. 2. Previously proposed mechanisms and mathematical latex drying models According to reviews [2,5,7] and numerous other publications, basically two different theories have been proposed and

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studied over the years that attempt to explain latex film/coating drying and a potential coupling to film formation. The first is that initiated by Vanderhoff et al. [8], which consists of three consecutive stages. During the first, water evaporates from the latex at the same rate as for pure water or dilute surfactant solution. This stage ends when the particles come into irreversible contact with one another (closest packing of monodisperse spheres) with the remaining water filling the interstices. Further evaporation, during the intermediate or falling rate stage, causes the air-wet film interphase to contract between the packed spheres, thereby decreasing the air-wet film interfacial area and thus the rate of drying, until finally the particles coalesce. In the final stage, the small amount of water remaining in the film escapes by diffusion through capillary channels in the film or by diffusion through the polymer membrane itself. At the beginning of stage 3, the amount of water present in the film is very small and the air-wet film interfacial area has been substantially reduced. Therefore, the overall rate of drying is much smaller than in the first stage (and practically constant). In a later publication [9], Vanderhoff and co-workers set up a mathematical model of a drying latex film to explain experimental drying data (not the ones of [8], but new measurements at a temperature of about 25 ◦ C and an air velocity of about 1 m/s). The model includes convective mass transfer between the wet latex film and the surrounding air as well as detailed energy balances (incl. convection, conduction, and radiation) for the film, glass slide and platform on which the samples were exposed. It was assumed that the evaporative surface temperature was equal to that of the average film temperature. In general, the researchers find that model predictions are in good qualitative agreement with experimental data of drying fluxes. It is, however, somewhat difficult to analyse this as the transient experimental drying data and simulations are not done at exactly the same conditions and not plotted together in the same figures. Of the five experimental drying series provided, only two, contrary to expectations and the earlier results in [8], show a falling rate period and, for these two series, for some reason the gradual reduction of the drying rate begins as soon as the experiments are initiated. The researchers state that the experimental data presented in [9] are less accurately measured for the falling-rate period because the value of the “wetted area” (white parts) of the drying film (from which drying is assumed to take place only) is inaccurately estimated from photos taken by a camera placed above the drying latex film. It should be noted that the experimental data also shows a surprisingly rapid and large increase (50–100%) in the drying fluxes near the very end of the drying process, according to the authors due to an effect of starting length (i.e. the distance from the platform edge to a given position in the film) on the local convective mass transfer coefficient as the film dries from front to back in the direction of air flow. The researchers do, however, cite another researcher, who also observed such a phenomenon in similar experiments, for assuming that the increase is due to an underestimation of the film area assumed to be available for evaporation (“wetted area”) near the end of drying where the dried area becomes much larger than the remaining wetted area. Thus, non-uniform drying across the horizontal film surface is expected by the researchers to be important, but may

actually have led to an error being imposed on the experimental data. It is not clear from the reference, how the authors have included in the model the proposed decreasing air-wet film interfacial area to account for the gradual reduction in the evaporation rate observed. Also, not all parameter values (e.g. thermal conductivities, heat capacities, viscosities, heat of vaporization of water, and bulk diffusion coefficient of water vapor) used in the model are provided, preventing a careful examination of the simulations. The researchers find that, for their particular experiments, heat conduction from the substrate to the latex film contributes up to 50% of the total heat flow and that heat radiation from the surrounding walls to the evaporate surface may amount to 20–50% of that by convection. The second theory proposed is that of Croll [10], who suggests that drying of latex can be described using only two stages. At first, water evaporates from a boundary, which at some point retreats into the coating as the film dries, leaving a porous layer behind it. According to Croll, transport of water vapour through this outer porous layer is governed by percolation and not diffusion, meaning that the layer does not offer any mass transfer resistance unless the critical volume fraction of air in the layer falls below a certain value. This percolation limit is not reached before a sufficient amount of coalescence has taken place, according to Croll essentially after completion of drying. During stage 1, the reservoir of water in the wet layer in contact with the substrate supplies water to maintain a constant evaporation rate. As explained by Eckersley and Rudin [11], one must visualize a hypothetical boundary layer (termed the transition zone), with a water concentration gradient, to exist between the underlying wet film and the dry upper portion of the film. The evaporation rate begins to fall when the transition zone impinges on the substrate. In the second stage, termed the falling rate period, the thickness of the transition zone decreases until all the water has evaporated and the film is dry. Croll suggests that stage 2 in the theory is due to “a change taking place within the wet interior of the coating”, but there is no elaboration on this point. In a subsequent publication [12], Croll writes that in the falling rate period, the solids content of the coating influences the evaporation rate and it is suggested that the coating ingredients are concentrated at the evaporation front due to surface tension effects. In [12], a two-stage mathematical model of latex films and coatings is presented to quantify the drying mechanism proposed in [10]. Stage 1 of the model, in addition to a mass balance for water, includes an energy balance for the film with convection and conduction terms, whereas in stage 2 mass transfer of water is combined with heat effects through an adjustable parameter, ␣. In total, Croll’s model contains four unknown input parameters: the evaporation constant, α, the time at which the transition zone first encounters the substrate (i.e. the end of stage 1), Ts , the water concentration in the transition zone at time Ts , C(0), and the thickness of the transition zone at Ts , l(0). α is adjusted to a suitable value by matching model simulations to experimental data from the initial constant rate drying period (stage 1). Ts is estimated from the experimental drying data (weight loss versus time) by visual inspection. C(0) is calculated a priori as an arithmetic average of the water

S. Kiil / Progress in Organic Coatings 57 (2006) 236–250

content of the initial wet latex and the water content in the formulation when the solids contents are packed as well as they can be without latex coalescence. l(0) is evaluated using the experimentally measured increment of weight loss during the whole of the falling rate period (i.e. the parameter can be estimated once Ts is decided upon). Thus, to perform simulations of the falling rate period, Ts and l(0) need to be estimated for every new set of experimental conditions including different film thicknesses of the same latex. Croll also assumes that “the concentration that would be at equilibrium with the vapour pressure remote from the surface”, Crh , is zero and that the air at the top of the transition zone is unsaturated with water vapour. Croll [12] is able to simulate experimental data very well using the model described above. Contrary to the investigation in [9], Croll did not assume any non-uniform drying across the horizontal film surface. Croll [12] states that climatic conditions could be conceived that limit evaporation to such an extent that the latex particles coalesce before water escapes, but that there is no evidence of a very slow, diffusion controlled loss of water at any stage. Eckersley and Rudin [11], who investigated the applicability of Croll’s model conclude that the model is quite realistic, but that some experimental observations cannot be neglected. The researchers state that a coalesced layer does form at the surface during drying, but reason that it is apparently not sufficiently dense to influence the diffusion rate of water vapour from the wet part of the film to the bulk air. The above analysis of previous attempts to verify the mechanisms underlying latex film drying, in particular the falling rate stage, suggests that the true mechanisms have not been described. The model of Promojaney et al. [9] relies on the use of a decreasing interfacial area to account for the gradual reduction in drying rate observed experimentally, but the phenomenon is neither verified nor is it described how it was incorporated in the model. Furthermore, the experimental data for used for their model verification contradicts most other latex drying studies (e.g. [11,12]). Croll [12] is able to simulate experimental data very well on the basis of a hypothesis of the dry upper layer, formed in stage 2, not offering any mass transfer resistance to water vapour combined with an assumption of the existence of a so-called transition zone. However, apart from the hypothesis, the model also makes use of two adjustable parameters for the falling rate period that need to be estimated, for every new set of experimental conditions, from the experimental data

239

that the model tries to simulate, thereby making the mechanism verification questionable. Furthermore, as clearly pointed out by Eckersley and Rudin [11] a somewhat coalesced layer does form at the film surface during drying though these researchers support the hypothesis of this layer not offering any mass transfer resistance. 3. Novel latex film drying theory The rate of drying of latex films is assumed to be governed by heat and mass transfer and a two-stage drying mechanism is proposed. The first is a constant rate period, similar to that proposed by Vanderhoff et al. [8]. This first stage ends when the polymer volume concentration reaches that of hexagonal closest packing of spheres. Contrary to the previous theories, it is proposed that the so-called dry layer that forms in the second stage of the drying process, when latex particles are exposed to air, does limit the rate of water vapour mass transport. There will be effective diffusion in the dry layer and the (unknown) value of the effective diffusion coefficient of water vapour is determined by the degree of coalescence that has time to take place during stage 2. A third stage, stage 3, is not part of the drying and will not be treated here, but it should be mentioned that completion of coalescence and potential cross-linking reactions take place during stage 3. 4. Novel mathematical latex film drying model A model capable of describing the drying behaviour of latex films and based on the theory described above is presented here. The model takes into account convective heat and mass transfer, heat radiation exchange with surrounding surfaces, heat conduction from underlying substrates, water evaporation, and effective water vapour diffusion in the dry layer that forms during stage 2. The last part of the drying process (falling rate period) is schematically visualised in Fig. 1. The assumptions underlying the model development are: • No temperature gradient exists throughout the wet latex film (i.e. the film is isothermal at all times). • The partial pressure of water vapour at the air–liquid interphase is equal to the saturated vapour pressure at the prevailing temperature of the film.

Fig. 1. Schematic illustration (cross-section view) of partly dried latex film during the falling rate period (stage 2). A porous skin, partly coalesced, is formed upon drying. The two moving boundaries, ls1 and ls2 , are indicated on the figure.

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• The liquid phase of the latex dispersion is assumed to be pure water (i.e. any effect of coalescent agents or surfactants on the drying behaviour is neglected). • The latex dries uniformly from the surface exposed (i.e. no lateral drying fronts are present). • Stage 1 ends when the polymer volume concentration of the film, ϕs , is equal to 0.74 (i.e. corresponding to that of closest packing of monodisperse spheres [2,13]). • Uniform distribution of latex particles in the wet part of the coating at all times (i.e. no particle gradient is present). Specifically for stage 2: • Some coalescence of the latex particles takes place prior to complete drying of the film. • The convective heat and mass transfer coefficients are constant throughout the entire drying process. • The dry layer does not contain any residual liquid water and the temperature in this layer is always isothermal and equal to the temperature of the underlying wet film. • The effective water vapour diffusion coefficient in the dry layer is constant (i.e. it is not a function of time, temperature (19–24 ◦ C), water vapour concentration, or degree of coalescence of the latex particles). • No water vapour (passing from the wet layer below) is adsorbed on or absorbed in the latex particles in the dry layer. • A pseudo-steady-state concentration profile of water vapour is established in the dry layer. • The latex films dry completely during stage 2. The validity of the above assumptions is addressed in a later section. 4.1. Stage 1 4.1.1. Moving air–liquid interphase A mass balance for water over the air–liquid interphase, stating that the rate of water vapour transport from the liquid surface to the bulk air must equal the rate of liquid water removal from the film, provides an equation for the rate of movement of this front dls1 kg,v Mw ∗ = (p (TF ) − pvB ) dt ρw RT v

(2)

and pvB =

h(TB − TF ) − Hvap ρw (dls1 /dt) dTF = dt

+ksub (dTsub /dl)|lo + εCR [TB4 − TF4 ] [ρw (lo (1 − γo ) − ls1 ) + lo γo ρp ]CPL

with TF (t = 0) = TB

(5)

Heat conduction from substrate to the latex film depends on substrate material(s) and thickness(es) used in a given experiment and may be calculated from transient heat balances over the substrates. However, substrate conditions are not known in any detail for the experimental set up of Croll [10,12] and Eckersley and Rudin [11], the data of whom will be used to verify the model, and therefore the heat conduction term is approximated by:    dTsub  ksub ksub (TB − TF ) = (6) dl lo lsub where lsub is the hypothetical distance from the latex filmsubstrate boundary to the point in the substrate where the temperature is equal to TB (assuming a linear temperature profile). The parameter (ksub /lsub ) is specific for an experimental setup and has to be estimated from measurements such as a steadystate temperature depression. Note, that Eq. (6) ensures that the heat conduction term is zero when TB = TF . The consequence of using the approximation in Eq. (6) is that the initial transient temperature development (temperature depression of a few degrees due to evaporation) is not described accurately. However, as shown later, this short initial temperature development to a steady-state temperature (wet bulb temperature) does not affect the overall drying behaviour and therefore will not influence the analysis of the underlying drying mechanisms. The film temperature increase during stage 2 is also not described very accurately, but also this temperature change is very small. The heat radiation term in Eq. (5) is derived based on an assumption of the area of the surroundings being much larger than that of the evaporating film, whereby only the emissivity of the wet film appears in the energy balance. 4.1.3. Weight loss and water mass fraction For the sake of comparison of simulations with experimental data, the following equation is needed for calculation of cumulative weight loss (kg/m2 or g/cm2 ) as a function of time W = ρw lo (1 − γo )(1 − X1 )

ϕRH p∗v (TB )

(4)

(1)

with ls1 (t = 0) = 0

from the surroundings to the film surface, and conduction from the underlying substrates to the latex film must equal the rate of evaporation and temperature change of the film

(3)

where

The mass transfer coefficient in Eq. (1), kg,v , may be for convective flow or for conditions of natural convection.

X1 =

4.1.2. Energy balance An energy balance for the film can be derived stating that heat transported by convection from the flowing air, radiation

(7)

(ls1o − ls1 ) + (lo − ls1o )(1 − γs ) lo (1 − γo )

(8)

(γs − γo )lo γs

(9)

and ls1o =

S. Kiil / Progress in Organic Coatings 57 (2006) 236–250

X1 is the fraction of the initial amount of water in the film that is still present in the film and ls1o is the position of the air–water interfacial area at the end of stage 1. The water mass fraction of the drying latex films can be calculated from: xww

ρw (1 − γo )X1 = ρw (1 − γo )X1 + ρp γo

(10)

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of spheres, γ s , now enters the equation: h(TB − TF ) − ΔHvap ρw (1 − γs )(dls1 /dt) +ksub (dTsub /dl)|lo + εCR [TB4 − TF4 ] [ρw (lo − ls1 )(1 − γs ) + lo γo ρp ]CPL

dTF = dt

(16)

The initial value of TF is given by the value of this variable at the end of stage 1, TF1 .

4.2. Stage 2 4.2.1. Moving air–liquid interphase In this case, a mass balance for water over the air–liquid interphase, stating that the rate of diffusion of water vapour through the dry layer must equal the rate of liquid water removal from the film, provides an equation for the rate of movement of this front:  De,v Mw dCv  dls1 (11) = dt ρw (1 − γs ) dl ls1 with the initial value of ls1 given by the value of this variable at the end of stage 1, ls1o . The effective diffusion coefficient of water vapour in the dry porous layer present in stage 2, De,v , accounts for the fact that not all of the area normal to the direction of the flux is available (i.e. void) for the molecules to diffuse, that the diffusion path is tortuous, and that the pores are of varying cross-sectional areas [14]. 4.2.2. Water vapour concentration profile A pseudo-steady-state mass balance for water vapour in the dry porous layer yields De,v

d 2 Cv =0 dl2

4.2.5. Weight loss and water mass fraction The following equation is needed for calculation of the cumulative weight loss (kg/m2 or g/cm2 ) as a function of time W = ρw lo (1 − γo )(1 − X2 )

(17)

where (lo − ls1 )(1 − γs ) lo (1 − γo )

X2 =

(18)

The water mass fraction of the drying latex films can be calculated from: xww =

ρw (1 − γo )X2 ρw (1 − γo )X2 + ρp γo

(19)

5. Solution technique The model is rendered dimensionless by introduction of dimensionless variables and solved using an integration routine such as that provided in [15]. When switching from the stage 1 to the stage 2 model, a very thin dry layer is assumed initially to start the calculations and avoid numerical problems.

(12) 6. Estimation of model parameters

with boundary conditions Cv (ls1 ) = Cv ∗ (TF )

(13)

and

 dCv  = kg,v (Cv (ls2 ) − CvB ) De,v dl ls2

(14)

The mass transfer coefficient in Eq. (14), kg,v , may be for convective flow or conditions of natural convection. 4.2.3. Moving air-particle interphase The rate of movement of the air-particle interphase (i.e. the dry film surface) is given by: dls2 dls1 = (1 − γs ) dt dt

(15)

with the initial value of ls2 given by the value of this variable at the end of stage 1, ls2o = ls1o . The assumption underlying this equation is that the film is non-porous at the end of stage 2. 4.2.4. Energy balance The energy balance is very similar to that derived for stage 1 except that the polymer volume concentration of closest packing

The latex drying model requires a number of physical and chemical constants. These were taken from various literature sources and are available in Table 1. The average convective heat transfer coefficient over the length of the panel onto which the film is applied (to be used for simulation of the wind tunnel data of Croll [10,12]), hc , was calculated using the correlation of [16] for laminar flow along a flat plate (ReL < 3 × 105 and 0.6 < Pr) Nu =

hc L 1/2 = 0.664 ReL Pr1/3 kair

(20)

Table 1 Selected physical and chemical parameter values for the drying model at conditions used in Croll [10,12] and Eckersley and Rudin [11] Parameter (19–24 ◦ C)

Hvap CPL (latex, 19–50 vol.%) γs hc (22 ◦ C, 1.8 m/s, Croll conditions) kg,v (22 ◦ C, 1.8 m/s, Croll conditions) hnc (TB = 24 ◦ C, TF = 21.6 ◦ C, “still”, Eckersley and Rudin conditions) kg,v (TB = 24 ◦ C, TF = 21.6 ◦ C, “still”, Eckersley and Rudin conditions) ε (Distilled water, 19–24 ◦ C)

Parameter value 2480 kJ/kg 3201 J/(kg K) 0.74 9.3 W/(m2 K) 8.5 × 10−3 m/s 4.1 W/(m2 K) 3.0 × 10−3 m/s 0.98

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The average convective mass transfer coefficient over the length of the panel onto which the film is applied, kg,v (laminar flow) was calculated from a similar correlation (ReL <3 × 105 ) [16] Sh =

kg,v L 1/3 = 0.664 Re1/2 L Sc Dv

(21)

The value of the panel/film length (where air flows), L, to be used in the above correlations, was estimated from the information provided in Croll [12], that the total horizontal film area available for drying was 0.1 m2 . Assuming that L2 = 0.1 m2 gives a value of L of 0.316 m. Due to this uncertainty of the value of L, the same values for the convective heat and mass transfer coefficients were used in the relevant temperature interval of 19–22 ◦ C and the temperature used in the parameter estimation was that of the bulk air (and not the average of the bulk and film temperature, which is the usual practice). The heat transfer coefficient for natural convection (to be used for simulation of the data of Eckersley and Rudin [11]), hnc , was calculated using the correlation of [16] for air: hnc = b(TB − TF )m L3m−1

(22)

where b = 1.86 and m = 0.25. The value of L (10 cm) was taken as the diameter of a petri dish (in the reference is not mentioned the exact dimension, but the value agrees reasonably well with the exposed film areas that can be calculated from the data provided). No correlation was available for estimation of the value of kg,v under conditions of natural convection and it should therefore be determined by adjusting simulations to match experimental data (stage 1). Emissivity of water at 20–25 ◦ C was taken to be 0.98 [17]. The bulk phase diffusion coefficient of water vapour as a function of temperature can be estimated from [18] (valid in the interval 0–45 ◦ C with an error of less than 0.1 × 10−6 m2 /s): Dv = 21.2 × 10−6 (1 + 0.0071 T )

(23)

where T should be inserted in ◦ C. The saturated vapour pressure of water as a function of temperature can be calculated from the Antoine equation [19]: log(p∗v ) = A −

B T +C

(24)

where A = 8.10765, B = 1750.286, and C = 235. T must be inserted in ◦ C and p∗v is then in mmHg. Heat of vaporization of water was taken from [19]. Heat capacity of a latex film was taken from Croll [10] (47 vol.% polymer) and assumed to be independent of polymer volume concentration. The effective diffusion coefficient of water vapour in the dry porous layer can be expressed as [14]: εp De,v = Dv (25) τ where εp is the porosity of the dry layer and τ the tortuosity factor. The extent of coalescence that takes place during stage 2 is not known and consequently De,v cannot be calculated a priori (i.e. both εp and τ may vary as coalescence is occurring). To comply with this, De,v will be adjusted to match the experimental data (stage 2) and the value obtained will then be used to discuss to which extent coalescence has

taken place. Substrate heat conductivities are specific for the individual experimental setups and will therefore be discussed in Section 7. The remaining parameters of the model are readily available in standard compilations and will not be discussed any further. 7. Results and discussion To verify the latex drying model, the two independent experimental series obtained by Croll [10,12] and Eckersley and Rudin [11] were selected. Other, though less extensive, sets of experimental data are available (e.g. [8,20]). However, for the data to be useful for model verification, detailed information on wet film thicknesses, polymer volume concentration of the latex films, surface area exposed, and drying conditions applied are needed. This information is only available in Croll and Eckersley and Rudin. Furthermore, Croll has also provided measurements of temperature depressions during drying for a couple of latex films making these data sets almost complete and very useful. Difficulties in validating drying models are imposed by the fact that the rate of drying is very much dependent on the specific experimental conditions. Air temperature, wet film thickness, and relative humidity are well-defined, but the contribution to the drying rate of heat conduction through the substrate(s) and mass and heat convection from the surrounding air to the film surface are difficult (in many cases impossible) to calculate accurately a priori. Furthermore, it is only for relatively thick latex films (> about 200 ␮m wet film thickness) with polymer volume concentrations of more than about 40 vol.% that the duration of stage 2 (the falling rate period) is comparable to that of stage 1 and thereby allows a proper model verification of this part of the drying process. 7.1. Model verification using the data of Croll [10,12] Croll [10,12] has performed latex film drying experiments at conditions that may simulate outdoor drying (air speed velocity of 1.8 m/s and a temperature of 22 ◦ C). The same experimental weight loss data are presented in the two publications [10,12]; however, in Croll [12] measurements of temperature depressions during drying are also provided. Steady-state temperature depressions of 2.3–2.6 ◦ C for latex dry film thicknesses of 128 and 66 ␮m (latex solid volume is probably 47 vol.% though not stated explicitly) were found. Pure water showed a temperature depression of 2.6 ◦ C. In Fig. 2 comparisons of simulations and experimental data for four different wet film thicknesses and a polymer volume concentration of 47 vol.% are shown. A very good agreement can be seen. Two model parameters were adjusted (but kept the same for all film thicknesses), one for the constant rate drying period (ksub /lsub ) and one for the falling rate period (De,v ), all other parameters were estimated independently. The former was adjusted so as to obtain a steady-state temperature depression of 2.4 ◦ C and the latter to match simulations with the experimental data of the falling rate period in Fig. 2. It is not possible to calculate (ksub /lsub ) a priori as this would require information on the specific substrates below the films (type and thickness) used in the experiments and this

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The effective diffusion coefficient of a porous structure is given by [14]: De,v =

εp Dv τ

(26)

where εp is the porosity of the dry layer and τ the tortuosity factor. τ is defined as the ratio of the tortuosity and constriction factor [14]. The tortuosity is the actual distance a molecule travels between two points divided by the shortest distance between the two points, whereas the constriction factor for the pores accounts for any variation in the cross-sectional area normal to diffusion. The tortuosity factor is a function of the porosity and the pore size distribution. The extent of coalescence that takes place during stage 2 is unknown, and consequently De,v cannot be calculated a priori (i.e. both εp and τ may vary as coalescence is occurring). However, a crude estimation of εp is possible now that an estimated value for De,v is available. Using the simple relation between εp and τ, τ = 1/εp , suggested by Wakao and Smith [21], εp can be calculated from (assuming a constant diffusivity): εp De,v = ε2p = τ Dv

Fig. 2. Comparison of drying rate simulations (lines) with experimental data (symbols) of Croll [10,12]. PVC is the polymer volume concentration of the latex film initially and WFT is initial wet film thickness. Drying and film conditions are: TB = 22 ◦ C, ϕRH = 0.50, ρP = 1106 kg/m3 and an air velocity of 1.8 m/s. Adjustable parameters are: (ksub /lsub ) = 48 W/(m2 K) for stage 1 and De,v = 7 × 10−7 m2 /s for stage 2. Other parameters are provided in Table 1.

information is not available in detail (and as such not of a general interest). The effective diffusion coefficient of water vapour in the dry porous layer, De,v , was estimated to 7 × 10−7 m2 /s for all film thicknesses. It is interesting to compare this value with the bulk phase diffusion coefficient of water vapour in air (21 ◦ C), which is 2.44 × 10−5 m2 /s. Thus, the bulk phase diffusion coefficient is 35-times higher than the effective diffusion coefficient, suggesting that some coalescence has indeed taken place during the falling rate period (stage 2). This is in agreement with the environmental scanning microscopy observations by Eckersley and Rudin [11], which show that latexes do form a skin during drying, prior to the complete evaporation of water. Eckersley and Rudin suggest, similarly to Croll, that the skin probably remains sufficiently porous to allow unhindered water vapour transport, but they do also express their doubts in several places as to whether this is correct. The results of Fig. 2 indicate that the drying does become diffusion controlled in the falling rate period. The model of this work, contrary to that of Croll, does not rely on any visual inspection of the experimental data to determine the time for when stage 1 ends for different film thicknesses. Stage 1 simply ends when the polymer volume concentration is equal to γ s (which can be calculated a priori to 0.74 based on an assumption of hexagonal closest packing of monodisperse spheres), in agreement with [8].

(27)

giving a value of εp of 0.17. At the onset of stage 2, the value of εp is 0.26 (1 − γ s ) and because an average effective diffusion coefficient was used through stage 2 with εp = 0.17, it points to the fact that deformation of the particles at the end of stage 2 has reduced the pore volume by 35% (0.26 to 0.17) or more. Though simulations can predict drying in the falling rate period very well with a constant value for the effective diffusion coefficient, it is quite likely that detailed modelling of dynamic pore volume reduction will give even more precise predictions. However, from a practical point of view it is not worthwhile in terms of predicting drying. It may be of relevance in terms of film property development. The value of the effective diffusion coefficient (7 × 10−7 2 m /s) does not suggest that diffusion of water vapour could be through a polymer membrane (formed during coalescence) itself. In this case, diffusion coefficients of the order of 10−11 to 10−13 m2 /s are expected [22]. The fact that coalescence, to some extent, takes place during the falling rate period is also in agreement with the calculations of Visschers et al. [23]. Their analysis suggests that the capillary force associated with the receding air–liquid interphase can cause deformation of the particles as the interphase moves through the coating. It is worth noting about Fig. 2 (wet film thicknesses of 248 and 323 ␮m) that according to both simulations and experimental data, the falling rate period (stage 2) lasts about half the time of complete drying of the films. The falling rate effect is only minor for thin films (see for example, data for a wet film thickness of 89 ␮m in Fig. 2), but for the thick films it is quite important to include the diffusion limited stage (for a wet film thickness of 323 ␮m, the total drying time is prolonged by more than 20% due to the drying rate being reduced in stage 2). In other words, drying cannot be predicted based on evaporation of pure water alone under the conditions prevailing.

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Fig. 3. Simulated temperature depression, TB − TF , as a function of drying time for a latex film with initial wet film thickness of 323 ␮m (weight loss data for the coating is shown in Fig. 2). Drying conditions are those of Croll [10,12]: TB = 22 ◦ C, ϕRH = 0.50, γ o = 0.47, ρP = 1106 kg/m3 and an air velocity of 1.8 m/s. (ksub /lsub ) = 48 W/(m2 K) and De,v = 7 × 10−7 m2 /s. Other parameters are provided in Table 1.

In Fig. 3 the simulated temperature depression for the coating with an initial wet film thickness of 323 ␮m is shown. It can be seen that during stage 2 (from about 2000 s and onwards), the temperature of the wet part of the film begins to approach that of the bulk air because evaporation is slowed down due to water vapour mass transfer resistance in the dry layer. At the end of stage 2, the thickness of the wet film (for which the temperature depression is calculated) is negligible and the film temperature (now of completely dry film) equals that of the bulk air. Initially, the temperature depression rises from 0 to 2.4 ◦ C in about 80 s. The experimental data of Croll [12] shows a steady-state film temperature depression of this magnitude in 300–400 s (almost irrespective of film thickness) in good agreement with the measurements reported in [8], where a temperature depression of 7.4 ◦ C was established in about 400 s for a latex with a film thickness of 1000 ␮m. Though the exact conditions for Croll’s temperature depression experiments are not provided it seems clear that this initial transient part cannot be predicted well by the model. The reason for this, as discussed during the model development, is that a simplified term (as opposed to full energy balances for the substrates below the film) was used for the heat conduction from substrate to latex film (due to lack of knowledge of the substrate conditions). Fortunately, simulations and experimental data of stage 1 behaviour are not sensitive to this transient behaviour (most of the temperature change in this initial period happens during the first 100 s), which may be evidenced by looking at Fig. 2, where it can be seen that the rate of drying, for all practical purposes, is the same all through stage 1. For the same reasons as discussed above, the transient temperature development during stage 2 is also an approximation. However, it can be seen in Fig. 3 that the temperature rise during stage 2 is only 1.4 ◦ C (temperature depression decreases from 2.4 to 1.0 ◦ C) and a more elaborate calculation will only provide an insignificant improvement in the accuracy of the simulations. At steady-state conditions during stage 1, the fraction of heat transferred to the film by convection, conduction and radiation

Fig. 4. Comparison of drying rate simulations (lines) with experimental data (symbols) of Croll [10,12]. PVC is the polymer volume concentration of the latex film initially, WFT is initial wet film thickness, and HMTI stands for “heat and mass transfer increase”. Drying and film conditions are: TB = 22 ◦ C, ϕRH = 0.50, ρP = 1106 kg/m3 and an air velocity of 1.8 m/s. Adjustable parameters are: (ksub /lsub ) = 48 W/(m2 K) for stage 1 and De,v = 7 × 10−7 m2 /s for stage 2. Other parameters are provided in Table 1. Note that the 132 and the 263 ␮m simulations have been plotted with the same line appearance.

is 15, 76 and 9%, respectively. When performing simulations at other conditions than the ones considered here, typical temperature depressions must be measured, under the conditions of relevance, before the model can be used with a high accuracy. In Fig. 4 comparisons between simulations and experimental data of Croll [10,12] for an initial polymer volume concentration of only 19 vol.% are seen. The same values as for the simulations in Fig. 2 were used for the two adjustable parameters (ksub /lsub and De,v ). It can be seen that stage 2 hardly appears for these films because the dry porous layer will emerge late and be quite thin. Thus, these data are not suitable for validation of stage 2 simulations. Note, that stage 1 is not predicted accurately, the constant drying rate appears to be too low. According to Croll, the experimental conditions should be identical to the ones for the 47 vol.% films discussed above, but the only explanation that can be given is that the flow pattern must have changed around the films due to small variations in temperature or air flow velocity. Croll, in his model, has accounted for the difference by letting the evaporation rate constant, α, vary with film or coating type. It can be seen in Fig. 4 that increasing the convective heat and mass transfer coefficients in the present model by 33% leads to a good agreement with experimental data. This is a clear illustration of the caution one must take when analysing drying rate data.

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7.2. Model verification using the data of Eckersley and Rudin [11] To further validate the model, another independent experimental series of drying data was selected. The data of Eckersley and Rudin [11] are useful because the initial wet film thickness, lo , of the films used can be calculated from the data provided, even though these values are not stated directly, using the following equation:  lo =

Ww A



1 final xwwi ρL

(28)

where Ww is the total amount of water evaporated in an experiment, A the film surface area exposed, and xwwi the initial water mass fraction of the film. The wet film thicknesses used by Eckersley and Rudin (954–1322 ␮m) are much higher, maybe too high for practical relevance, than the ones used by Croll (89–563 ␮m), but these thicknesses make them very suitable for testing the capabilities of the model. Unfortunately, temperature depression measurements are not available and so it has been assumed that the temperature depression of the constant rate period is identical to that measured by Croll (2.4 ◦ C) even though the films are thicker and the evaporation rates lower. A further complication is that the drying rate experiments of Eckersley and Rudin were done under approximated “still” conditions (in petri dishes placed in an inverted cardboard box), perhaps representative of indoor drying of latex coatings. The convective heat transfer coefficient under these conditions (natural convection) can be estimated a priori from a correlation, but not the mass transfer coefficient (at least no correlation was found in the literature). Thus, it was necessary to adjust the mass transfer coefficient so as to match simulations with the data of the constant drying rate period (stage 1). Drying conditions were the same for all experiments and therefore the same value for this parameter was used for all simulations. In Fig. 5, comparisons between simulations and experimental data for two polymers (L1 and L4) are shown. Note that the total drying times are now much longer (7–11 h) than the ones of Croll in Fig. 2 (less than 1 h). It is evident that a good match is obtained for all cases, though some deviation is seen for L1 for a film thickness of 960 ␮m. The latter is due to the uncertainty of the experimental data. Though experimental conditions are suppose to be exactly the same for the two samples of the same latex (11.629 and 10.275 g were used in the two cases) it can be seen that the experimental drying rates for stage 1 deviates. As the same value is used in the simulations for the convective mass transfer coefficient, a better agreement cannot be expected. Lowering of the value for the convective mass and heat transfer coefficients for L1 (960 ␮m) will lead to a perfect match with experimental data (both stages 1 and 2), but such a correction cannot be justified without more information on the uncertainties of the experimental setup. It is interesting that the value of the effective diffusion coefficient for water vapour in the dry layer, De,v , used in the simulations of Fig. 5, was the same in all cases and also the same value as used for the simulations of the experimental data of

Fig. 5. Comparison of drying rate simulations (lines) with experimental data (symbols) of Eckersley and Rudin [12] (conditions of natural convection). WFT is initial wet film thickness and L1 and L4 represent two different latex particles. Drying and film conditions are: TB = 24 ◦ C, ϕRH = 0.50, ρP = 1130 kg/m3 . Adjustable parameters are: (ksub /lsub ) = 15 W/(m2 K) and kg,v = 3.0 × 10−3 m/s for stage 1, and De,v = 7 × 10−7 m2 /s for stage 2. Other parameters are provided in Table 1.

Croll (7 × 10−7 m2 /s). The polymers used in the two studies are not identical. The mean particle size of the ones used by Croll is 500 nm and probably based on methyl methacrylate, butyl acrylate and acrylic acid (Rhoplex® AC-388). Those of Eckersley and Rudin are 580 (polymer L4) and 600 (polymer L1) nm, respectively, and based on methyl methacrylate, butyl acrylate and methacrylic acid. L1 was synthesized in the presence of chain transfer agent leading to a lower average molecular weight (not quantified) than that of L4 and should therefore, according to the researchers, coalesce during the drying period. The fact that the same value could be used for De,v in all simulations is important because it points to the fact that the diffusion resistance in the falling rate period is similar for drying of different latexes and may (though not a generally verified statement) be estimated a priori, keeping in mind that temperature effects have not been considered in this work. Had De,v been allowed to vary somewhat with latex type and film thickness, a truly perfect match of all data could be obtained, but from a mechanistic point of view, it seems more instructive to try and match all data using the same value for this parameter. In Fig. 6, simulations of water mass fractions of the latex films for L1 and L4 are compared with experimental data. It can be seen that simulations match experimental data quantitatively and

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Fig. 7. Comparison of water mass fraction simulations for different values of De,v (lines) with experimental data (symbols) of L4 from Eckersley and Rudin [11] (conditions of natural convection). The initial wet film thickness and PVC is 954 ␮m and 45 vol.%, respectively. Drying and film conditions are: TB = 24 ◦ C, ϕRH = 0.50, ρP = 1130 kg/m3 . Adjustable parameters for stage 1 are: (ksub /lsub ) = 15 W/(m2 K) and kg,v = 3.0 × 10−3 m/s. Other parameters are provided in Table 1.

Fig. 6. Comparison of water mass fraction simulations (lines) with experimental data (symbols) of Eckersley and Rudin [11] (conditions of natural convection). WFT is initial wet film thickness and L1 and L4 represent two different sizes of latex particles. Drying and film conditions are: TB = 24 ◦ C, ϕRH = 0.50, ρP = 1130 kg/m3 . Adjustable parameters are: (ksub /lsub ) = 15 W/(m2 K) and kg,v = 3.0 × 10−3 m/s for stage 1, and De,v = 7 × 10−7 m2 /s for stage 2. Other parameters are provided in Table 1.

that the characteristic sigmoidal shape, reported also in [8], is predicted, the latter being a direct consequence of the functional form of Eqs. (10) and (19) (the rate of change of X1 and X2 during drying are practically constant with dX1 /dt > dX2 /dt). Eckersley and Rudin [11] claim that their data shows that the statement of Vanderhoff et al. [8] (and also an assumption underlying the model of this work) that the polymer volume concentration at the start of stage 2, γ s , should always have a value of 0.74 (equivalent to a volume and mass fraction of water of 0.26 and 0.237, respectively), corresponding to closest packing of perfect spheres, is invalid. Thus, for Vanderhoff et al. [8] to be right, the polymer volume concentration should be independent of the polymer material and initial mass (or film thickness) and this was not found when analysing the data. The reason that this was not found is probably that the value of time and water mass fraction at the end of stage 1 was estimated by visual inspection of the experimental data. This may well be too inaccurate because the rate of drying is slowly reduced after the transition point from stage 1 to stage 2 and the slope of the water mass fraction versus time curve is fairly steep making the visual inspection difficult. As an example, Eckersley and Rudin estimated the time for completion of stage 1 for L1 and L4 for the two sets of data shown in Fig. 6 to 6.0 and 3.9 h, respectively,

and the corresponding water mass fractions to 0.212 and 0.282. Simulations by the model of this work, which match experimental data with a water mass fraction of 0.237 at the end of stage 1 in both cases, suggest the times to be 5.50 and 4.13 h. Thus, even though the differences in the values of time are not that large it is enough to reach the two different conclusions. In Fig. 7, the effect on simulations of variations in De,v can be seen for a selected case. Simulations are not very sensitive to the value of De,v . Increasing it by a factor of four from 3.5 × 10−7 to 14 × 10−7 m2 /s decreases the overall drying time from 8.49 to 6.59 h. This fairly low sensitivity towards De,v may explain why the experimental drying data of [10–12] can all be simulated with a constant value of De,v independent of initial wet film thickness, particle size, initial PVC, and molecular weight of the latex polymer: relatively large differences in the morphology of the dry layer (and thereby the diffusion rate of water vapour) are required to reveal themselves in the drying rate data. The importance of this is that simulations of drying behaviour for other types of latex systems and other conditions (disregarding temperature) than the ones considered in this work can probably all be performed with a reasonable accuracy with the value of De,v found in this work (7 × 10−7 m2 /s). Consequently, the model has true predictive properties and model inputs are only needed on the specific experimental (or field) conditions of interest. One may argue that the steady-state temperature depression for the Eckersley and Rudin data should be lower than the 2.4 ◦ C that Croll measured in his experiments because the drying rates are somewhat lower (assuming similar types of substrates). To investigate this, simulations, which ensured a temperature depression of less than 1 ◦ C and still matching the stage 1 evaporation rate data, were carried out for all four film thicknesses by varying the values of (ksub /lsub ) and kg,v , but keeping them the same in all simulations. In all cases, experimental data of the falling rate period was still matched with an accuracy comparable to that shown in Fig. 5 using a value of De,v of 7 × 10−7 m2 /s.

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Simulations are much more sensitive to variations in kg,v than in (ksub /lsub ). 7.3. Parameter study The model has now been verified and can be used to study the influence of important process parameters on drying time. The relevant parameters are: relative humidity, initial wet film thickness and polymer volume concentration, air flow velocity, and bulk air and substrate temperature. The latter cannot be varied as the temperature dependency of the mechanical properties of the latex polymers used by Croll [10,12] and Eckersley and Rudin [11] are unavailable making simulations unreliable because the mechanical properties may well affect the degree of coalescence taking place during stage 2. More experiments are needed to include this parameter. The influence of the other four parameters has been investigated. The influence of relative humidity (ϕRH ) on the total drying time can be seen in Fig. 8. For values of ϕRH below 50–60%, the drying time is not seriously affected (doubles from 28 to 58 min when ϕRH increases from 0 to 50%). Increasing ϕRH above 60%, however, prolongs the drying time substantially (increases by a factor of four from 1.2 to 4.6 h when ϕRH is increased from 60 to 90%). Going higher, from 90 to 99%, increases the drying time by a factor of 10 from 5 h to 48 h. These results are in good agreement with practical recommendations of applying latex coatings below a relative humidity of 80%. Apart from the long drying time (with associated potential dirt pick up) at high relative humidities, coalescence agents, present in the coating, may evaporate before the latex particles have come close enough too fuse leading to weak films [5]. The temperature depression during stage 1 is 0.05 and 4.86 ◦ C for a relative humidity of 99 and 0%, respectively. The effect of film thickness on total drying time is shown in Fig. 9 (PVC kept constant). For film thicknesses between 1 and

about 100 ␮m, drying time doubles when the film thickness is doubled. For higher film thicknesses, drying time increases by more than a factor of 2 when the film thickness doubles. The reason for this is that the time of the falling rate period makes up a larger fraction of the total drying time the higher the film thickness (also evidenced by the data plotted in Fig. 2). For a film thickness of 10 and 1000 ␮m the fraction is 30 and 58%, respectively. This also means that the temperature depression is reduced from 2.4 to 0.5 ◦ C (in 2.3 h) during the falling rate period of the 1000 ␮m film, whereas the reduction is only from 2.4 to 2.2 ◦ C (in 28 min) for the 10 ␮m film. In Fig. 10, the influence of the initial polymer volume concentration (γ o ) on the total drying time is provided. There is really not a strong effect of this parameter, drying time decreases from

Fig. 8. Simulated total drying time as a function of relative humidity (ϕRH ) for a latex film with initial wet film thickness of 323 ␮m (weight loss data for the coating is shown in Fig. 2). For a relative humidity of 99%, the total drying time is about 48 h. Drying conditions are: TB = 22 ◦ C, γ o = 0.47, ρP = 1106 kg/m3 and an air velocity of 1.8 m/s. (ksub /lsub ) = 48 W/(m2 K) and De,v = 7 × 10−7 m2 /s. Other parameters are provided in Table 1.

Fig. 10. Simulated total drying time as a function of initial polymer volume concentration (γ o ) for a latex film with initial wet film thickness of 323 ␮m. Drying conditions are those of Croll [10,12]: TB = 22 ◦ C, ϕRH = 0.50, ρP = 1106 kg/m3 and an air velocity of 1.8 m/s. (ksub /lsub ) = 48 W/(m2 K) and De,v = 7 × 10−7 m2 /s. Other parameters are provided in Table 1.

Fig. 9. Simulated total drying time as a function of wet film thickness (lo ) for a latex film. Drying conditions are those of Croll [10,12]: TB = 22 ◦ C, γ o = 0.47, ϕRH = 0.50, ρP = 1106 kg/m3 and an air velocity of 1.8 m/s. (ksub /lsub ) = 48 W/(m2 K) and De,v = 7 × 10−7 m2 /s. Other parameters are provided in Table 1.

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Fig. 11. Simulated total drying time as a function of air flow velocity (v) for a latex film with initial wet film thickness of 323 ␮m. Drying conditions are those of Croll [10,12]: TB = 22 ◦ C, ϕRH = 0.50, ρP = 1106 kg/m3 . (ksub /lsub ) = 48 W/(m2 K) and De,v = 7 × 10−7 m2 /s. Other parameters are provided in Table 1.

1.35 to 0.93 h when γ o is increased from 10 to 70 vol.%. At high values of γ o , there is less water present than at low values (initial wet film thickness is kept constant at 323 ␮m). However, the duration of stage 2 (falling rate period) makes up a larger fraction of the total drying time for large values of γ o than for low values of γ o (where stage 2 hardly appears) leading to only a small difference in drying time. Stage 2 lasts about 1 and 90% of the total drying time for γ o equal to 10 and 70, respectively. Note, that the final dry film thickness will be very different for the two cases (32 and 226 ␮m, respectively). Finally, in Fig. 11, the effect on total drying time of air flow velocity is shown. It can be seen that decreasing the air flow velocity below about 2 m/s substantially increases the drying time. At air flow velocities above 2 m/s, and in particular above 4 m/s, total drying time becomes almost independent of this parameter. One of the reasons for this is that the drying process, in the falling rate period, which last about half the total drying time, essentially becomes controlled by effective diffusion of water vapour in the dry layer and this subprocess is not influenced by air flow velocity. Only the drying rate of stage 1 is influenced. Furthermore, a higher air flow velocity, though increasing the heat and mass transfer coefficients, also leads to a larger temperature depression (0.8 and 4 ◦ C for an air flow velocity of 0.1 and 15 m/s, respectively), which reduces the evaporation rate. 8. Sensitivity analysis A sensitivity analysis of the model at a bulk temperature of 22 ◦ C, a relative humidity of 50%, and an initial wet film thickness of 323 ␮m (base case), with respect to all the physical and chemical parameters included in the model development was performed. The two adjustable parameters of the model (ksub /lsub ) for stage 1 and De,v for stage 2, are not included here as their values were discussed in a previous section. The following parameters were identified to influence model simulations

(total drying time) when varied ±20% around their estimated values (maximum relative deviations from the base case are provided in parenthesis): gas phase mass transfer coefficient (11%), saturated vapour pressure of bulk air (26%), heat of vaporization of water (4%), and polymer volume concentration at the end of stage 1 (17%). Of these parameters, only the gas phase mass transfer coefficient is not known with a high accuracy. However, even with an error of up to 11% in the latter value, model predictions are still in good quantitative agreement with experimental data. Thus, the important phenomena to quantify are external mass transfer, effective diffusion of water vapour in the partly coalesced layer, and heat conduction from the substrate to the film/coating. External heat transfer and radiation need not be included in the model with a high accuracy (for the conditions considered here) because simulations are practically insensitive to the presence of these terms in the equations. 9. Validation of model assumptions The latex drying model is based on a number of simplifying assumptions that have not all been verified in the above discussions. Here, the most important ones are readdressed. It was assumed that the wet film remains isothermal at all times. This was verified by a worst case calculation, assuming that all heat removed due to evaporation has to be conducted through the film. Using a maximum evaporation rate from the data of Croll, a wet film thickness of 323 ␮m and values of heat conductivities for water and rubber (taken as being representative of latex polymer) from the literature, a maximum temperature difference of 0.21 K across the film was found indicating that energy is rapidly conducted in the wet film. Such a small temperature difference does not justify including intrafilm temperature effects. Lateral drying fronts were neglected. The potential presence of these has been described by [6] and Croll [12] actually mentions that his films dried faster on the upstream edge and at the sides (reveals itself by the formation of clear areas on the milky film surface). The effect is thought to have a very small influence only on the overall drying behaviour (especially in the experiments of Croll because his film surface areas were 0.1 m2 each, which should be enough to make edge effects negligible). Eckersley and Rudin [11] also mention that the effect was neglected in their treatment of raw data. The polymer volume concentration of the film at the end of stage 1 was assumed to be that of closest packing of monodisperse spheres (i.e. 0.74 as shown in [24]). This is clearly the upper limit of the value as it represents the ideal case of perfect packing of all particles and uniform particle size [8]. As the real values for the systems considered in this work are not known, it seems like a reasonable assumption to use the value for the ideal case. Slightly more accurate values for the effective diffusion coefficient of water vapour in the dry layer may perhaps be obtained by including the effect (i.e. the potential error of not including the effect is lumped into the value of the effective diffusion coefficient). The assumption of pseudo-steady-state concentration profile of water vapour in the dry layer was verified by the fact that

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the time constant for diffusion was much smaller than the time constant for the rate of change of the water vapour concentration. The rate of movement of the air/dry layer interphase during stage 2 is based on an assumption of the dry layer being nonporous at the end of drying. This is not correct because it was shown in Section 7 that a partly coalesced, and thereby porous, layer is formed. However, the effect of this assumption on simulations is very small because the interphase does not move much during stage 2: for an initial wet film thickness of 323 ␮m at the conditions of Croll [12] it moves 55 ␮m. The true number is lower, probably in the range of 25 ␮m, but this difference is not enough to justify an attempt to use a more accurate calculation. The thickness of the dry layer increases from 0 to 151 ␮m during stage 2 (WFT of 323 ␮m). It was assumed that the convective heat and mass transfer coefficient (air to wet film) is constant throughout the entire drying period. This may not be entirely correct as the dry layer formed in stage 2 will be somewhat separating the flowing air from the wet film surface. This potential dynamic effect has not been verified. Heat radiation was included using an approximated equation. However, the effect of this term is very small and it can actually be neglected. During stage 2, it was assumed that no residual liquid water is present in the dry layer. Some authors (e.g. [23]) have argued that liquid bridges may be formed between the particles contributing to the gradual coalescence, but in terms of drying of the entire film, the amount present as liquid bridges is negligible (see for example, the experimental data of Fig. 2). The presence of coalescent agents or surfactants in the films used by Croll and Eckersley and Rudin, and their effect on drying was neglected due to lack of detailed information. This assumption was not confirmed. 10. Practical use of the drying model Drying time is very important when designing novel or improved industrial latex coatings. To use the novel mathematical model for practical drying cases, the following procedure is recommended: If a high accuracy is needed, measure the average steady-state temperature depression of the wet film/coating using a thermocouple junction under the external conditions of relevance (bulk temperature, relative humidity, type and temperature of substrate, air flow velocity and perhaps film thickness) and use this value to “calibrate” the energy balance (estimation of the substrate heat conduction term, ksub /lsub ) so that the correct steady-state temperature depression is predicted (heat and mass transfer coefficients may be estimated based on the correlations provided in this article or in [16]). If a lower accuracy can be tolerated then it may be sufficient to use a temperature depression of 2–3 ◦ C as found by Croll, though larger temperature depressions can occur (setting ksub equal to zero, corresponding to no heat conduction from the substrate to the film, for an intial wet film thickness of 323 ␮m and otherwise the conditions of Croll leads to a temperature depression of 5.64 ◦ C). To avoid having to determine (ksub /lsub ), it is necessary to include detailed energy balances, with estimations of material input parameters, for the

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substrates below the film or coating. This has to be done for every new set of relevant substrate conditions and it is probably too elaborate for most practical situations. The falling rate period needs no inputs (unless bulk air temperature varies greatly from 19 to 24 ◦ C) because the value of the effective diffusion coefficient of water vapour in the partly coalesced layer apparently has a universal value of about 7 × 10−7 m2 /s (at temperatures of 19–24 ◦ C) as shown in an earlier section. Note, that the effects on drying of the presence of pigments, additives, and coalescent agents in the films/coatings are not included in the model and may influence simulations (in Croll [10,12] experimental drying data of a commercial latex coating is available). Also, simulations can probably not be performed with accurate results at substantially different bulk air/substrate temperatures before temperature effects on the coalescence behavior of a given latex has been mapped by experiments. The fact that water vapour diffusion (and thereby drying time) can be severely limited in the falling rate period, especially for coatings applied in a thick layer, suggests that drying and perhaps coalescence properties may be manipulated with novel formulations aimed at influencing this part of the drying. 11. Conclusions A mathematical model for a drying latex film was developed. Experimental drying data and observations can be explained and quantitatively estimated by the model. Contrary to previous interpretations, simulations suggest that a partly coalesced layer, which limits water vapour transport in the dry layer during drying, is indeed formed. The same value for the effective diffusion coefficient of water vapour in the dry layer could be used for all the simulations of this work in order to match experimental data, thereby eliminating the need for an adjustable parameter for the falling rate period. It remains to be investigated if the value is truly universal (i.e. approximately valid for all types of drying latex films) and the effects of the presence of pigments, extenders, and additives in the film must be quantified and included in the model before it can be used for full commercial latex coatings. The temperature dependency of the effective diffusion coefficient also needs to be quantified. It was demonstrated that the transition from stage 1 (constant rate period) to stage 2 (falling rate period) takes place when the polymer volume concentration of the film is equal to that of closest packing of perfect spheres as originally suggested by Vanderhoff et al. [8]. Elucidation of the drying mechanisms and mathematical modelling (also exemplified in [25] for another type of coating) may support the development of efficient water-based coatings for industrial purposes, thereby reducing the need for organic solvents. References [1] [2] [3] [4]

J.W. Taylor, M.A. Winnik, JCT Res. 1 (3) (2004) 163. P.A. Steward, J. Hearn, M.C. Wilkinson, Adv. Col. Int. Sci. 86 (2000) 195. EU-Directive 2004/42/EC, 2004. S. Gupta, Asia Pacific Coat. J. (April issue) (2005).

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[5] A. Overbeek, F. B¨uckmann, E. Martin, P. Steenwinkel, T. Annable, Prog. Org. Coat. 48 (2003) 125. [6] A.F. Routh, W.B. Russel, AIChE J. 44 (9) (1998) 2088. [7] Z.W. Wicks, F.N. Jones, S.P. Pappas, Organic Coatings—Science and Technology, second ed., Wiley, 1999. [8] J.W. Vanderhoff, E.B. Bradford, W.K. Carrington, J. Polym. Sci.: Symp. No. 41 (1973) 155. [9] N. Pramojaney, G.W. Poehlein, J.W. Vanderhoff, Drying 80 (2) (1980) 93. [10] S.G. Croll, J. Coat. Tech. 58 (734) (1986) 41. [11] S.T. Eckersley, A. Rudin, Prog. Org. Coat. 23 (1994) 387. [12] S.G. Croll, J. Coat. Tech. 59 (751) (1987) 81. [13] F. Dobler, T. Pith, M. Lambla, Y. Holl, J. Coll. Interf. Sci. 152 (1) (1992) 1. [14] H.S. Fogler, Elements of Chemical Reaction Engineering, Prentice-Hall, Englewood Cliffs, New Jersey, 1999. [15] J. Villadsen, M.L. Michelsen, Solution of Differential Equation Models by Polynomial Approximation, Prentice-Hall, Inc., New Jersey, 1978.

[16] Perry’s Chemical Engineers Handbook, seventh ed., McGraw-Hill, 1997. [17] Institute for Computational Earth System Science: www.icess.ucsb.edu. Accessed June 2006. [18] Holsoft’s Physics Resources Pages: http://physics.holsoft.nl/physics/ ocmain.htm. Accessed June 2006. Their data is based on Gates, D.M. Biophysical Ecology, Springer Verlag, New York, 1980, 611 pp. [19] J.M. Smith, H.C. Van Ness, Introduction to Chemical Engineering Thermodynamics, fourth ed., 1987. [20] D.P. Sheetz, J. Appl. Polym. Sci. 9 (1965) 3759. [21] N. Wakao, J.M. Smith, Chem. Eng. Sci. 17 (1962) 825. [22] Diffusion Polymers.Com (independent agency): www.diffusion-polymers. com. Accessed June 2006. [23] M.V. Visschers, J. Laven, R. van der Linde, Prog. Org. Coat. 31 (1997) 311. [24] S.T. Eckersley, A. Rudin, J. Coat. Tech. 62 (780) (1990) 89. [25] S. Kiil, C.E. Weinell, M.S. Pedersen, K. Dam-Johansen, Ind. Eng. Chem. Res. 40 (2001) 3906.