Comparative study of droplet drying models for CFD modelling

chemical engineering research and design 8 6 ( 2 0 0 8 ) 1038–1048

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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd

Short communication

Comparative study of droplet drying models for CFD modelling Meng Wai Woo a,∗ , Wan Ramli Wan Daud a , Arun S. Mujumdar b , Meor Zainal Meor Talib a , Wu Zhong Hua b , Siti Masrinda Tasirin a a

Department of Chemical & Processing Engineering, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor DE, Malaysia b Mechanical Engineering Department, National University of Singapore, Kent Ridge Crescent, Singapore

a b s t r a c t CFD simulation is used to study wall deposition and agglomeration phenomena commonly encountered in industrial spray dryers. This paper initially provides a comparison of two drying kinetics models: Characteristic Drying Curve (CDC) and Reaction Engineering Approach (REA). Comparisons are made with experimental data with application to carbohydrate droplet drying obtained from past workers. These models were then extrapolated to actual drying conditions to assess their performance. The REA model predicts the progressive reduction in drying rate better than the CDC model for the carbohydrate droplets. A modified CDC model incorporating a convex falling rate produced better agreement than the conventional linear falling rate model. Further analysis showed that the REA model can be extended to simulate the particle surface moisture which may affect the agglomeration process. The proposed concept was compared with reported simulation results from a diffusion model which showed reasonable fit with data. © 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Spray drying; Single droplet drying; CFD sub-model

1.

Introduction

The key mass transport phenomenon in the spray drying process is the drying of individual atomized droplets in the chamber. Prediction of the drying process of individual droplets in CFD simulations of the spray dryer is important as it greatly affects three aspects of the modelling outcome. Firstly, it determines the final product moisture. In actual operation, this affects many aspects of the product quality, e.g. particle density, degradation and friability (Walton and Mumford, 1999). Therefore, efficient modelling of the drying phenomenon is essential in future modelling of product quality. Secondly, it affects the particle trajectory as the mass changes during drying. Lastly, the drying process predicts the condition of the drying particles during collision (Langrish and Kockel, 2001). The latter two aspects are important if CFD sim-

∗

ulation is to be used to study wall deposition or agglomeration in spray drying. Many single droplet drying models are available in the literatures. Neglecting empirical models (not the interest of this work), the droplet drying models can generally be divided into two groups: diffusion and lump parameter models. The first group takes into account and keeps track of the detail phenomena within the drying particle (e.g. moisture gradient, diffusion, internal bubble). Subsequently, the models translate these individual internal phenomena onto the internal moisture gradient which affects the overall drying rate (Sano and Keey, 1982; Adhikari et al., 2003). Although these models are ‘physically’ more realistic, solution of the associated partial differential equations in CFD might require high computational effort (Chen and Lin, 2005; Kieviet, 1997).

Corresponding author. E-mail address: [email protected] (M.W. Woo). Received 26 October 2007; Accepted 14 April 2008 0263-8762/$ – see front matter © 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cherd.2008.04.003

chemical engineering research and design 8 6 ( 2 0 0 8 ) 1038–1048

Nomenclature aW A Cp D Df EV h hm Hevap kf m Nu Pv Pr R Re Sc Sh t T X Subscript a cr d eq s sat wb ∞

equilibrium relative humidity droplet surface area (m2 ) specific heat capacity (J kg−1 K−1 ) droplet diameter (m) diameter of glass filament (m) apparent activation energy heat transfer coefficient (J m−2 s−1 K−1 ) mass transfer coefficient (m s−1 ) heat of evaporation (J k−1 ) glass thermal conductivity (W m−1 K−1 ) mass (kg) Nusselt number vapour concentration (kg m−3 ) Prandtl number Universal gas constant (J K−1 mol−1 ) Reynolds number Schimdt number Sherwood number time (s) temperature (K) moisture content (kg water/kg solid)

air critical droplet equilibrium solid saturated at surface wet bulb ambient

concept in which the lump model can be extended for this purpose.

2.

Mathematical model

In this section, we would like to firstly review on the theoretical aspect of the two droplet drying models and the assumptions made by past authors along the way. Once these models are clarified, we would then proceed to make comparison with the limited experimental drying data available. It is noteworthy that both models do not take into account of physical traumas, such as inflation, rupture and blowholes, which might be experienced by a particle in harsh drying conditions. Such trauma will probably eject or cause sudden release of moisture from the particle. At the moment, there are no mathematical models which can sufficiently predict these complex particle morphologies at different operating conditions, but yet provide the simplicity to be implemented in CFD modelling. The drying models presented here assume that the mass transfer behave in a diffusion-limited fashion.

2.1.

Heat transfer model and coefficient determination

It is assumed that the temperature profile within the droplet is negligible. Although some authors chose to consider the internal temperature profiles (Farid, 2003), the models considered here was developed based on a lumped droplet temperature. Furthermore, it was already proven that the kinetic data which will be used for our application of the models later on follows this assumption (Adhikari et al., 2003, 2004). For each model, a common heat transfer model was adopted and takes the following form,

mCp On the other hand, lump parameter models consider overall droplet parameters without taking into account the diffusion gradient. Subsequently, these models require solving only algebraic equations, resulting in simpler computational effort which is desired for CFD modelling. Two such approaches utilizing the lump models recently in discussion are the Characteristic Drying Curve (CDC) and the Reaction Engineering Approach (REA) (Langrish and Kockel, 2001; Chen and Lin, 2005; Lin and Chen, 2005). This paper aims to make comparison between these two models with laboratory drying kinetic data to form a basis for our future CFD modelling work on carbohydrate droplets. On a separate note, as a trade off for simplicity, a drawback in the lump models is the inability to describe the particle surface moisture (hence the word ‘lump’). Determination of the particle surface moisture is important in the modelling of particle deposition or agglomeration. In past reports, workers investigating the general flow pattern and operation of the spray dryer often prefer the lump models or evaporating model due to its simplicity (Harvie et al., 2002; Kievet, 1997; Huang et al., 2003; L. Huang et al., 2004). However, in a modelling work on agglomeration in spray drying recently published, the comprehensive diffusion model was used as it was important to accurately predict the particle surface condition (Verdumen et al., 2004). As part of our larger project in modelling wall deposition, it will be useful if a lump model can be utilized but yet provides details on the particle surface moisture. Therefore, this paper also aims to present a novel

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dTd dX = hA(Ta − Td ) − Hevap mS dt dt + 0.5Df



hDf kf (Ta − Td )

(1)

The third part of the equation is to take into account of the heat transfer from the supporting glass filament in the experiments by Dr. B. Adhikari (Adhikari et al., 2003). The heat and mass transfer coefficient will then be calculated using the classical Ranz–Marshall correlations, Nu = 2 + 0.6Re1/2 Pr1/3

(2)

Sh = 2 + 0.6Re1/2 Sc1/3

(3)

2.2.

Characteristic Drying Curve

The CDC theory is based on the assumption of two distinct periods of drying, namely, the constant rate period which is then followed by the falling rate period (Langrish and Kockel, 2001). Using this model, the drying rate can be represented by the following equations, dX Ah (Ta − Twb ) =f dt mS Hevap f =

X − Xeq , Xcr − Xeq

f = 1,

X > Xcr

X ≤ Xcr

(4)

(5)

(6)

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(8)

where the EV is the apparent activation energy introduced, which is likely to be dependent on the ‘mean or ‘average’ moisture. In their past works, by using data sets of different drying temperatures, Chen and co-workers (Chen and Xie, 1997; Chen and Lin, 2005) suggests that the normalized relation of EV as a function of X is material specific for each product is independent of the drying condition. It is assumed that this can then be used as the fingerprint of the material to determine EV at different drying conditions and can be expressed in the

Case B

0.0001 3.5 4 95 1.5 0.5 0.00002 3.5 4 95 1.5 0.5 0.0023 2 1 95 1.0 Pure sucrose 0.0023 2.5 1 63 1.0 Pure sucrose 0.0023 2 1 95 1.0 Pure maltodextrin 0.0023 2.5 1 63 1.5 0.5

RT

Droplet diameter (m) %RH Air velocity (m s−1 ) Drying air temp (◦ C) Initial moisture (dry basis) Sucrose–maltodextrin ratio

 E  V

= exp −

Case A

Similarly, the same coupled energy balance Eq. (1) is used. The key component of this model is the fractionality relative to the interface saturation moisture content, , which should reach unity when the surface of the droplet is fully covered with liquid water. On the flip side, the fractionality should approach zero as the drying material reaches equilibrium moisture. Therefore, this fractionality is expected to be a function of moisture and temperature, which can be approximated by,

Case C

(7)

Table 1 – Cases considered in the simulations

dm hm A ( Pv,sat (Td ) − Pv,∞ ) = dt mS

0.0023 2.5 1 63 1.0 Pure maltodextrin

The main theory of the REA visualizes the drying process as an activation process in which an ‘energy’ barrier has to be overcome for moisture removal to occur (Chen and Xie, 1997). The drying rate can then be expressed in the following form,

0.0023 2 1 95 1.5 0.5

Case H Case G Case F Case E Case D

Reaction Engineering Approach

Parameters

2.3.

Case I

The constant rate period is identified as the period where the droplet moisture content is above the critical moisture. The critical moisture corresponds to the moisture content in which internal diffusion or formation of crust on the surface starts to hinder the process of moisture removal. In the constant rate period, the drying of the droplet is considered unhindered and is simply viewed as droplet evaporation. However, once the droplet reaches the critical moisture, the drying rate is considered to be progressively hindered by further moisture removal. Evident from Eqs. (4)–(6), it is assumed that there is a linear drop in drying rate proportional to the drop in moisture. From Eq. (5), we can also observe that for a fixed external condition, a lower Xcr will produce a longer period of unhindered drying. The CDC model is a lump model in which it simplifies the hindering effect of the internal mass transport on the drying rate with a linear relation relative to the material properties, Xeq and Xcr . While Xeq can be obtained from the isotherms of the material as a function of ambient air humidity and temperature, the Xcr is assumed to be constant independent of the drying condition. In reality, this might not be a fair assumption if the droplet experienced extreme contrasting initial drying rate (Fyhr and Kemp, 1998). It was observed for maltodextrin solution, that a higher initial moisture will result in a higher critical moisture and vice versa (Zbicinski and Li, 2006). However, these effects might be offset by the inherent characteristics of the CDC model in which lower Xcr will produce higher drying rate in the hindered region and vice versa (Fyhr and Kemp, 1998).

0.0023 2.5 1 63 1.5 Pure maltodextrin

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Table 2 – Parameters of the REA and CDC used Model

REA

Modified CDC

Material

Cases used for simultaneous fitting of the model parameters

Sucrose–maltodextrin

A and B

Sucrose

E and F C and D

Sucrose–maltodextrin

A and B

f

Sucrose

E and F

f

Maltodextrin

C and D

f =

c Ev = a e−b(X−X∞ ) Ev,∞

(9)

where a, b and c are parameters to be fitted. The ‘a’ parameter is approximately unity. These parameters can then be obtained by back-calculating the EV value at different moisture content from experimental data by combining and manipulating Eqs. (7) and (8),



(dm/dt)(1/hm A) + Pv,∞ Pv,sat

 (10)

From Eqs. (8) and (9), At high moisture content EV → 0, EV,∞

 →1

At low moisture content EV → 1, EV,∞

EV EV,∞ EV EV,∞ EV EV,∞

Maltodextrin

following form,

EV = −RT ln

Equation & Parameter Values

 →0

2.022

= e−0.892(X−X∞ )

2.816

= e−1.688(X−X∞ ) =

4.656 e−1.793(X−X∞ )

 X−Xeq 1.98 = Xcr −Xeq  X−Xeq 2.58 = Xcr −Xeq  X−Xeq 3.22 Xcr −Xeq

two cases of similar material type, but at different operating conditions. For the sucrose–maltodextrin mixture, parameters were determined by only simultaneously fitting Cases A and B. Similarly, parameters for pure maltodextrin were determined by only simultaneously fitting Cases C and D; for pure sucrose, only Cases E and F were used for the simultaneous fitting. Thus, the parameters determined as shown in Table 2 are derived specific to each material. The index for the modified CDC model, a modification of the linear model which will be introduced later-on in this paper, was also determined using this simultaneous fitting approach (Table 2). Similarly, the fitted index is expected to be material specific and only one index was used for each material irrespective of the operating condition. Simulations of these models were undertaken using Microsoft Excel spreadsheets. Simulation of the REA and CDC models adopted the finite difference method similar to those of Chen and Lin (2005). The aim of this exercise was to give an initial evaluation of the drying models in predicting the drying of the sucrose–maltodextrin droplets. This will form the basis for further evaluation in a changing ambient condition from a CFD simulation.

3.

Application of the mathematical models

3.2.

3.1.

Method and cases

The physical properties used for the application of the models are listed in Table 3.

In the works of Fyhr and Kemp (1998), theoretical comparison of the CDC and a diffusion model were made to a more comprehensive ‘rigorous’ model (as a reference), which took into account of the moisture transport mechanism, developed for porous and non-shrinking material. The CDC and diffusion model were then fitted to the drying curve generated with the ‘rigorous’ model. Keeping the fitted parameters constant, the models were then tested at different drying conditions. Adopting similar method, in this work, the CDC and REA were initially fitted to the experimental data available. Nine cases were considered as shown in Table 1. Experimental data for Cases A and B were provided by Dr. Benu Adhikari (University of Queensland). Cases C, D, E and F were used for comparison study on a wider range of materials. These four cases were extracted from published results (Adhikari et al., 2002). Cases G and H are extrapolated drying conditions used to gauge the application of the models to actual drying conditions. The last case, I, was used to verify a novel method in determining the surface moisture from the lump models. Experimental data from Cases A, B, C, D, E and F were used to determine the REA parameters. However, it is noteworthy that the parameters for the different materials used were determined separately. Parameters for each material were determined by simultaneously fitting data sets of only

Physical properties

4.

Results and discussion

4.1.

Comparison of the models to experimental data

Application of the CDC model requires the critical moisture to be assumed a priori. For small particles exhibiting short or minimal constant rate drying period, past work suggests utilizing the initial moisture as the critical moisture (Langrish and Kockel, 2001). The plot of the calculated drying rates from the available kinetic data, exhibiting non-existence of the initial constant rate period, supports this assumption (Fig. 1). Furthermore, it can be noted that there is an approximately linear falling rate which follows the basic assumption of the CDC model at least within the first 200 s of the experiment. The initial moisture content was used as the critical moisture content. On the other hand, application of the REA model requires assumption of the particle shrinkage characteristics. In this work, it was initially assumed that the effect of external conditions on the sucrose–maltodextrin particle shrinkage is similar to those of milk particles of linear correlation (Lin and Chen, 2004). Following this assumption, the following empirical equation was used to describe the particle shrinkage by

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Table 3 – Physical properties used for model fitting Properties a

Air thermal conductivity (W m−1 K−1 ) Water–airb diffusivity (m2 s−1 ) Air density (kg m−3 ) Viscositya of air, poise Densityc of droplet solution (kg m−3 )

Latent heat of vapourization (J kg−1 ) Specificd heat of sucrose–maltodextrin mixture (J kg−1 K−1 ) Specific heat of water (J kg−1 K−1 ) Universal gas constant (J K−1 mol−1 ) Droplet equilibrium moisture Glass thermal conductivity (W m−1 K−1 )

Values ◦

0.03135 (95 C) 0.02898 (63 ◦ C) 3.71E−05 (95 ◦ C) 3.16E−05 (63 ◦ C) 0.9504 (95 ◦ C) 1.0416 (63 ◦ C) 2.15E−5 (95 ◦ C) 2.01E−5 (63 ◦ C) 1170 (Cases A, B, G and H) 1218 (Cases C and D) 1225 (Cases E and F) 1161 (Case I) 1177 (Case I–sucrose) 2,261,452 3.98T + 1220.5

0.0087T2 − 0.3904T + 4181.6 8314 Refer to Adhikari et al. (2003) 1.09

Unless stated, the properties were obtained or the mathematical correlations regressed from data in Perry’s Chemical Engineer’s Handbook (Perry and Green, 1998). a b c

d

Fig. 2 – REA-predicted drying curves at different ‘b’ parameters for (a) Case A and (b) Case B.

Huang (2005). International Critical Tables. Determined from our previous experiments using sucrose– maltodextrin and measurements on pure sucrose/maltodextrin (DE 9-12) solutions. Adhikari et al. (2003).

determining the ‘b’ parameter through trial and error, D X = b + (1 − b) Do Xo

(11)

A higher ‘b’ parameter, with a maximum value of unity, indicates smaller particle shrinkage to moisture loss ratio and vice versa. From Fig. 2a, there was not much difference in the predicted drying rate between using the b values of 0.7 and 0.9. However, from Fig. 2b, b value of 0.9 fits the experimental data better and this value is adopted for further analysis. The

Fig. 1 – Calculated drying rates from the droplet drying experimental data (Adhikari’s data).

Fig. 3 – Normalized activation energy and water content difference curve for sucrose–maltodextrin droplets. same shrinkage model and parameter were used for the CDC analysis (and also for the subsequent modified CDC). It is noteworthy that this value is relatively close to unity. The reason for this is because the sucrose–maltodextrin droplet actually forms folds and elongates as drying proceeds instead of just reducing in size as observed in the experiments (Adhikari et al., 2004). Past analysis on single droplet experiments suggests that any deviation in droplet sphericity will increase the mass or heat transfer area (Walton, 2004). Therefore, even if shrinkage occurs, the effective surface area might tend to be balanced-out as drying progresses. This consideration should be taken into account in future works when interpreting drying data of suspended single carbohydrate and sugar droplets. Serving as the ‘backbone’ for the REA model, the normalized activation energy and free moisture content curve of a sucrose–maltodextrin droplet was then determined by fitting the exponential model developed by past workers (Chen and Xie, 1997). The model exhibited good fit, R2 − 0.97, to the manipulated experimental data (Fig. 3). In fitting the exponen-

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Fig. 4 – Drying curves predicted by the two models tested (a) Case A and (b) Case B.

Fig. 5 – Fitting of modified CDC model for (a) Case A and (b) Case B.

tial model, normalized activation energy beyond value of 1 and any significant outliers due to the experimental fluctuations were removed. Fig. 4 shows the predicted drying curve by both models for Cases A and B. It is clear that the REA fits the experimental data very well. The only slight significant deviation is at the final stages of drying. This could be due to uneven shrinkage of the droplet towards the end of drying. On the other hand, the CDC grossly over predicted the drying rate. This over-prediction is due to the basic assumption of the CDC where the drying curve follows a prefix shape corresponding to a linear drop in drying rate. If we examine Fig. 1, the drying rates approximately resemble a linear drop only in the first 200 s. At later stages, changes in drying rate resemble more of a convex curve which slowly forms a plateau. This characteristic is likely to be due to the sudden formation of crust and explains why the CDC model grossly over-predicts the drying rate at the later stages. On the other hand, in view of the shape of the drying rate curve (Fig. 1), modifications to the CDC model as suggested by L. Huang et al. (2004) and L.X. Huang et al. (2004) might be more desirable. It was suggested that instead of a linear drop, the change in drying rate can be expressed in the form of a concave or convex curve which is a characteristic of the material. This can be incorporated by introducing an index for the calculated evaporation hindering factor found in Eq. (4) to give the rate of change of the total mass,

taneously (Table 2). This indicates that a convex falling rate might be more suitable for materials with such skin or crust formation tendency. In terms of the temperature profiles, there are slight deviations due to the mass changes (Fig. 6). The REA tends to predict a slightly higher initial temperature when compared to the CDC (and modified CDC). This is due to the lower initial drying rate as evident in Fig. 4a. On the other hand, the modified CDC over-predicts the middle stage of drying. However, the temperature profiles approach the experimental values towards the later stages of drying. When implemented in a CFD model,

f =

X − Xeq Xcr − Xeq

n (12)

where if n is less than unity, a convex falling rate will be produced. If n is more than unity, a concave falling rate will be observed. For the material under consideration, the latter seems more logical (Fig. 1). Fig. 5 shows the improvement by the modified CDC model which follows very closely to the experimental data. The index of 1.98 was determined by considering the best fit of both cases (Cases A and B) simul-

Fig. 6 – Temperature profiles predicted by the two models for (a) Case A and (b) Case B.

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Fig. 7 – REA-predicted drying curves at different ‘b’ parameters for (a) Case E and (b) Case F.

Fig. 9 – Comparison of the drying models for (a) Case E and (b) Case F.

such minimal differences are expected to be negligible due to the rapid drying. The three models were further tested on pure sucrose (Cases E and F) and maltodextrin (Cases C and D) droplets to cover a wider range of materials. Similarly, only considering uniform shrinkage, the shrinkage factor of 1 was found to give the best fit of both sucrose and maltodextrin (Figs. 7 and 8) although the effect of different shrinkage factors is very minimal. Even with this best fit, the REA under-predicted the latter stages of drying (Figs. 9 and 10). It is unclear on the reason

Fig. 10 – Comparison of the drying models for (a) Case C and (b) Case D.

Fig. 8 – REA-predicted drying curves at different ‘b’ parameters for (a) Case C and (b) Case D.

of this deviation. One possibility could be due to the relatively lower fitting of the sucrose and maltodextrin apparent energy curve which exhibits R2 of 0.82 and 0.90 respectively. Similar trend were observed in the apparent energy curve of each material where the curve is higher than the experimental data at the high moisture end (Fig. 3). This could have led to lower initial evaporation for the pure sucrose and maltodextrin; although the REA predicted very well for the sucrose–maltodextrin droplet. On the other hand, there might be certain morphologies such as particle shriveling not taken into account by the shrinkage model (Walton, 2004).

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Fig. 11 – Temperature profiles predicted by the two models for (a) Case E and (b) Case F.

The modified CDC fitted the experimental data slightly better than the REA for the four cases (Figs. 9 and 10). Index value for the modified CDC was determined as 2.58—sucrose and 3.22—maltodextrin by simultaneously fitting to experimental data of Cases E and F, and Cases C and D respectively. The higher value for maltodextrin is logical as it exhibits a higher tendency to form an outer skin. However, the index value for the sucrose–maltodextrin droplet which is expected to lie between these values is lower than that of the sucrose index. This is evidence that the formation of skin in carbohydrate droplets drying might be partly affected by the initial moisture of the droplet and should be further investigated. A more dilute solution might induce different arrangement of solid material at the surface. Arrangement of different solid components at the surface has been observed for milk particles (Kim et al., 2003). In all cases, the CDC model grossly over-predicted the drying rate. Due to this over-prediction, the CDC predicted relatively lower droplet temperature particularly at the beginning of the drying process (Figs. 11 and 12). In general, the temperature predictions followed the experimental data reasonably well except for Case E, where the models over-predicted the temperature profile. On the contrary, it is unclear why the CDC model which grossly over-predicted the drying rate, followed the experimental data better in this case (Fig. 12a). In general, it is clear that the REA follows the experimental data better when compared to the conventional linear drop CDC. On the other hand, the modified CDC provided better agreement with the experimental data, similar to those of the REA. This suggests that in addition to the linear falling rate proposed for small droplet which was deduced mainly from milk droplets (Langrish and Kockel, 2001), our analysis of the experimental data showed that carbohydrate droplets exhibit a convex falling rate. It is noteworthy that the critical moistures used in this work were maintained as the initial moisture. This was intended to implement a simplified version of the CDC proposed by Kockel

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Fig. 12 – Temperature profiles predicted by the two models for (a) Case C and (b) Case D.

and Langrish (2001) for further application in CFD. Therefore, the analysis and determination of the index is strictly specific to this simplified version of the model.

4.2.

Extrapolation to actual drying condition

Section 4.1 shows that the models predict relatively different drying curve profiles. It is unclear and will be interesting to gauge if the differences noticed in the comparison above will be significant under actual spray drying conditions. This work will extrapolate the models to such conditions. For this purpose, Cases G and H were considered (Table 1) as sucrose–maltodextrin droplet is of main interest in our work. Drying air of 95 ◦ C and 3.5% relative humidity was chosen. This air humidity chosen reflects on the local laboratory air condition (30 ◦ C, 80% RH) when heated to 95 ◦ C. Airflow velocity of 4 m s−1 was estimated from the pilot spray dryer available based on the geometry of the air inlet. The particle size of 20 ␮m in Case G is the mean particle size produced in our previous pilot scale experiments. However, droplet size in commercial operations might lie in the range of 10–300 ␮m. Therefore, Case H was introduced utilizing a droplet of 100 ␮m as an average value for comparison. In the simulation of these cases, the component of heat transfer from the glass filament was removed from the model. From Fig. 13, the models are capable of predicting similar and logical magnitude in drying time corresponding to the particle sizes. The linear drop CDC tends to predict a higher drying rate when compared to the REA and the modified CDC. This is particularly in the middle stages of drying. On the contrary, towards the end-stage, the three models approach each other. These imply that provided that there is sufficient residence time, the three models will tend to predict similar particle moisture content. This will translate to similar CFD moisture prediction for particles experiencing longer ‘free flowing’ residence time which avoided deposition. However, for particles

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Fig. 14 – Prediction of the surface moisture by the proposed model for maltodextrin droplets.

Fig. 13 – Extrapolation of the drying models to (a) Case G and (b) Case H. deposited during its trajectory or escaped the dryer having insufficient drying time, the predicted moisture might be very different. Consider an example in comparing the CDC and REA model. If a 20 ␮m particle impacts the wall at 0.1 s, the difference in the predicted moisture is as large as 0.17 kg water/kg solid. This might affect prediction of the sticky condition of the particles if the average moisture is used, as stickiness is moisture and temperature sensitive. On the other hand, due to the rapid moisture evaporation in the chamber, droplets tend to form a relatively drier outer layer in the initial stages of drying regardless of the internal moisture. This implies that if the particle surface moisture is used for computation of surface stickiness, the error in predicting stickiness will be minimized when compared to using the average moisture. Usage of the outer layer moisture will also prove to be more accurate in modelling particle stickiness as this phenomenon is governed by the surface condition (Adhikari et al., 2003, 2004). However, one drawback of current lump drying models is the inability to determine the particle surface moisture; hence the name ‘lump’. On the contrary, to the best of the author’s knowledge, only solution of the partial differential diffusion equations and complex analysis were able to give this description. In Section 4.3, we will attempt to fill this gap by introducing a new calculation method which is an extension of the REA model.

4.3. A new method to compute particle surface moisture As mentioned in the previous section, this new computation method is extended from the REA model. If we return to Eq. (7), is considered as the fraction of the surface vapour saturation. Following this theory, this fraction decreases as drying progresses due to the increase in evaporation activation energy. The increase in activation energy is due to the reduction in water content at the surface due to limitation by the internal diffusion.

It is also logical to view that the reduction in the fractionality is directly due to the changes in surface moisture and not on the average moisture. However, the fractionality was correlated as a function of average moisture as measurements of the surface moisture is, at the moment, purely impossible. Therefore, the fractionality can actually be viewed as a function of both the average and surface moisture. As a close approximation, the fraction can then be expressed as the ratio of the surface moisture to the average moisture taking the following form, = aW ×

Xsurf Xaverage

(13)

The equilibrium relative humidity term was introduced as hygroscopic materials like carbohydrates tend to exhibit a lower vapour pressure from saturation at corresponding surface moisture content and temperature. Taking the average moisture at any instance and expressing the relative humidity in terms of the surface moisture content, we can then determine the surface moisture content as the fractionality is pre-calculated from the original drying model. If the relative humidity is a complex function of surface moisture, for example, Guggenheim–Anderson–de Boer (GAB) equation, the complete equation can be solved by any numerical method. In our case, we utilized the Microsoft Excel Solver. In order to validate this theory, comparisons were made with past data. Although drying kinetics for different droplets are abundant in the literature, data on surface moisture is limited as it is currently not experimentally feasible. Comparisons can only be made with simulation results of more comprehensive diffusion models. Therefore, surface moisture content of pure maltodextrin droplets reported by Adhikari et al. (2003) was taken as a basis. The drying condition for this surface moisture simulation is as shown in Case I. The REA fingerprint previously obtained for Cases C and D were used. Even with the under-prediction of the REA model (Fig. 10), we proceeded to simulate the surface moisture. Fig. 14 shows that the predicted surface moisture follows the same trend as those predicted by the diffusion model; a sudden drop in moisture followed by a plateau as a solid outer layer is formed. The difference between the simulation by the diffusion model and our model is only significant in the initial region of drying. Similar simulations were undertaken for pure sucrose droplets. Cases E and F were considered to construct the fingerprint while the surface moisture simulation case resembles those of Case I. The comparison of the simulation results is shown in Fig. 15. Even for pure sucrose droplets, similar trend was produced and the deviation from the diffusion model is

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Fig. 15 – Prediction of the surface moisture by the proposed model for sucrose droplets. only in the first 200 s of drying. In Fig. 15, the predicted average moisture was included and it is clear that the duration in which the deviation occurs is only in the initial 20% of the total drying time. Therefore, as the three sets of data discussed in Figs. 14 and 15 are all purely simulation results, it is unclear which model will more accurately represent the actual surface condition. Current experimental methods are not feasible to measure the surface moisture content of evaporating droplets. However, the deviation between the two models is only in the initial stages of drying and both models exhibited similar trend in the particle surface moisture. When the moisture removal is rapid in the actual drying condition, it is expected that the deviation will cause non-significant effect. More work should be undertaken to validate this method or provide further ‘refinement’ to the method if necessary. On top of that, development in the diffusion model should be undertaken hand-in-hand as currently, this would prove to be the only validation method.

4.4.

Overall performance of models tested

The REA seems to follow the curvature of the experimental data better than the linear drop CDC for sucrose–maltodextrin droplets. One advantage of the REA model is the minimal assumption on the shape of the drying curve. However, a drawback is that relatively wide range of experimental data is required to effectively implement the REA. It is uncertain on the reason behind the under-prediction for the sucrose and maltodextrin droplets. On the other hand, the modified CDC model by Huang follows even closer to the experimental data than those of the REA. However, this model requires an additional index constant to be arbitrarily fitted. Selection of the initial moisture as the critical moisture is specific to the simplified CDC model considered for CFD application (Langrish and Kockel, 2001). Apart from that, our analysis also showed that the REA model can be extended to compute the particle surface moisture. The proposed concept will be very powerful as it allows the lump model to compute the surface moisture profile which is conventionally only feasible by solving complex PDE in a diffusion model. This will provide an alternative in CFD modelling as the lump model, which is normally preferred due to its low computational requirement (Chen and Lin, 2005), can then capture the capability of the complex diffusion model. Contrary to the controlled drying condition in this study, the actual condition in the drying chamber will be very hostile and rapidly changing. It will be interesting to compare the three models in a CFD environment to gauge their per-

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formance in actual drying conditions. This will be undertaken in our future work on sucrose–maltodextrin droplets. It is noteworthy that these comparisons were derived mainly from the cases considered in this study. The elongation and formation of folds on the suspended droplets might have affected the accuracy of the models. Furthermore, the cases considered were all undertaken from the same experimental rig which makes the data vulnerable to any possible systematic errors. Therefore, results obtained here and the proposed model should be applied with caution. Apart from that, the models presented assume and comparisons made hitherto to a diffusion-limited shrinking morphology. At intense drying rates found in a spray dryer, a particle might experience other morphologies such as inflation or formation of an internal bubble. Further comparison work, if possible, should be undertaken at such particle morphologies from single droplet experimental data. On a separate note, for the carbohydrate droplets studied here, the diffusion model by Adhikari et al. (2003) still follows the presented experimental data better over the lump models considered here. However, the focus of this work is strictly on the comparison of lump models for CFD modelling purpose and hence, their simulation results were not included. Interested readers on the diffusion model can refer to the publications cited.

5.

Conclusion

Performance of the REA, CDC and modified CDC model were evaluated for the drying of carbohydrates. The REA and modified CDC follows the experimental data better when compared to the conventional linear drop CDC suggested for spray drying. Further analysis showed that the REA has the potential to be extended to compute the particle surface moisture. This concept will be useful for CFD modelling as the lump model which requires less computational resources can now be utilized to produce simulations which is conventionally only feasible by more comprehensive diffusion models. Comparison should be further undertaken in a CFD environment or different particle morphologies to gauge the effectiveness of these models and how the results reported here will translate in an actual drying chamber.

Acknowledgements This work was funded by the Ministry of Science, Technology and Innovation of Malaysia under a Science Fund grant: 03-0102-SF0046. The authors would like to thank Dr. Benu Adhikari of University of Queensland for providing the drying kinetic data for Cases A and B and his useful discussions on utilizing these data.

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