Chemical Engineering and Processing 50 (2011) 687–693 Contents lists available at ScienceDirect Chemical Engineering and Processing: Process Intensification journal homepage: www.elsevier.com/locate/cep Flow pattern and heat transfer in agitated thin film dryer Sanjay B. Pawar a , A.S. Mujumdar b , B.N. Thorat a,∗ a b Department of Chemical Engineering, Institute of Chemical Technology (ICT), Nathalal Parekh Marg, Matunga (E), Mumbai 400019, India Department of Mechanical Engineering, National University of Singapore, Singapore 117576, Singapore a r t i c l e i n f o Article history: Received 24 July 2010 Received in revised form 6 April 2011 Accepted 9 April 2011 Available online 28 April 2011 Keywords: Scraped surface geometry SSHE Bow wave Conduction drying a b s t r a c t The design of agitated thin film dryer (ATFD) is difficult both mechanically and process engineering point of view. The present work describes the basic flow pattern in ATFD in terms of bow wave and its transformation along the dryer height. The effect of flow rate, jacket side heating medium temperature and speed of the rotor has been studied for a pilot scale ATFD. The effect of rotor speed was found less significant for water as a feed material than for sugar and ammonium sulfate solutions over the studied range of speed. The scraped side heat transfer coefficient was obtained using the penetration theory and its value was found in the range of 3000–7000 W/m2 ◦ C. © 2011 Elsevier B.V. All rights reserved. 1. Introduction Agitated thin film dryers (ATFDs) are widely used in chemical, pharmaceuticals and food industries to produce dry free flowing powder. The combination of short residence time, high turbulence and rapid surface renewal permits the agitated thin-film dryer to handle the heat-sensitive, viscous and fouling feed streams successfully. ATFD consists of two major assemblies: a heated body (stator) and a close clearance rotor as shown in Fig. 1. The process fluid enters in the unit tangentially above the heated zone and distributes evenly over the inner surface of wall by rotating action of the rotor. The rotor blades spread the feed over the entire heated wall and generate highly turbulent flow conditions in the thin layer of liquid. The feed progressively passes through the phases like slurry, paste, wet powder and finally powder of desired dryness as shown in Fig. 2. ATFD can be used as multi-utility equipment. ATFD is sometimes inadvertently referred to as ATFE, agitated thin film evaporator, when the desired product is a concentrated liquid. The major difference in the design of ATFD and ATFE lies in the respective hydrodynamic conditions which occur as a result of speed of the agitator, the clearance between the blade and inside body surface and so on. ATFD involves three phases which makes difficult to model the flow phenomena. ATFD consists of scraper/agitator with fixed or spring loaded blades. The design of ATFD is a strong function of process fluid and operating constraints of the process. Never- ∗ Corresponding author. Tel.: +91 22 24145616x2022. E-mail address: bn.thorat@ictmumbai.edu.in (B.N. Thorat). 0255-2701/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2011.04.005 theless, the greatest advantage is constant renewal of layer/film near the wall which helps in reducing the fouling and enhances the heat transfer coefficient. The heat transfer in such devices can be modeled using the penetration theory of heat transfer [1–3]. 2. Flow pattern in ATFD As discussed earlier, ATFD is a wiped surface assembly with the rotor having three or four rows of the blades. The shaft rotates with a given angular velocity and the blades lay out a thin film on the inner wall of the dryer. If the volumetric feed flow rate is relatively high enough, the film formed on the inner wall would be thicker than the clearance between the wiper blade and inner wall which results in a fillet/bow wave of liquid on blades front edge [4]. The flow can be either laminar or turbulent depending on the fluid properties and operating conditions. Several authors have studied the flow behavior of various feeds in the scraped surface geometry. Bott and Romero [5] observed a continuous fillet in front of the blade along the height of vertical scraped surface heat exchanger for water as well as water–glycerol mixtures. Similarly, Abichandani et al. [6] have observed the fillet in front of the blade tip for horizontal geometry. Zeboudj et al. [7] have reported that the hydrodynamic conditions of the flow affects the film thickness and residence time of fluid elements in wiped film evaporator. McKelvey and Sharps [8] examined the velocity profile and flow structure of the bow wave in wiped blade evaporator and studied the dependence of blade clearance, film thickness and throughput. Komari et al. [9] studied the flow structure and mixing mechanism in the bow wave both theoretically and experimentally in a wiped film evaporator. The authors examined the effect of fluid 688 S.B. Pawar et al. / Chemical Engineering and Processing 50 (2011) 687–693 Bow wave (recirculating loop) for flow rate F1 Vapor Feed Shell Hot water out F1 < F2 Jacket Thin film formation on inner wall Blade Bow wave (recirculating loop) for flow rate F2 Hot water in Dry powder Fig. 1. Schematic representation of agitated thin film dryer. Fig. 3. Schematic representation of flow pattern in ATFD showing different fractions of bow wave at different flow rates. dimensions of bow wave depend on the feed flow rate and its physical properties. When the thin film flow is of concern in a scraped geometry, the flow can be distinguished in two sections: one is the bow wave which constitutes the major portion of feed and other is the thin film adhering to the inner wall (see Fig. 3). The concept of Reynolds number (Re) is well defined for a straight pipe to characterize the flow behavior of Newtonian fluid. Subsequently, it is also defined for other geometry configurations such as a stirred tank or an annulus based on the impeller tip speed and equivalent diameter concept. The rotational Reynolds number for an annular flow can be given by ReR = De (Rr ˝) (1) where De is the equivalent diameter which is given by De = 4 × rh = 4 × De = 4 × Fig. 2. Schematic representation of zone transformation in ATFD. (Ds2 − Dr2 )/4 = Ds − Dr Ds + Dr (2) (3) In the case of thin film flow, only the outer wall (stator) remains in contact with the fluid and the equivalent diameter can be written as De = 4 × viscosities up to 13 Pa s and found that from 70 to 90% of fluid flow in the device lies in the bow wave when the evaporator was equipped with vertically aligned blades. McKenna [10] presented a model for the design of a wiped film evaporator. The model considered the fluid transport and mass transfer aspects of devolatisation of polymer solution. The author observed a limiting rotational speed for mixing in WFE above which significant gain in the mass transfer was obtained at the expense of very large power consumption. From the above literature, it can be said that the flow pattern in ATFD is a combination of the rotational film flow induced by mechanical action of blades and an axial flow. The pitched blades move the fluid in the bow wave both tangentially and axially. The cross-sectional area of channel perimeter of channel in contact with fluid (Ds2 − Dr2 )/4 D2 − Dr2 = s Ds Ds (4) Thus, the rotational Reynolds number derived in the case of thin film would be higher as compared to that obtained for conventional annulus. 3. Heat transfer in ATFD 3.1. Penetration theory The penetration theory is the key to the analysis of heat transfer in a scraped surface geometry. Several authors have used the penetration theory in its original form or with some modification (use S.B. Pawar et al. / Chemical Engineering and Processing 50 (2011) 687–693 of correction factor) to model the scraped-side heat transfer coefficient [11–16]. The correlations based on the Nusselt number have been also developed considering the laminar and turbulent flow in such devices [17–19]. The heat transfer is controlled by conduction into a thin layer at the heat transfer surface and by the speed of mixing of this layer into the bulk of the fluid. The basic approach of heat transfer from hot wall which is scraped by a blade is provided by the penetration theory. The theory is based on conductive heat transfer. The mass flow rate and viscosity of the process medium have no influence on the heat transfer coefficient. The heat transfer process can be divided into following two steps Hence, the heat transfer coefficient can be expressed as h(t) = Each blade scrapes a certain amount of fluid and accelerates it along the inner hot wall surface. At any given instant, the fluid pushed by the blade is partly in the form of film behind the blade and partly in the form of fillet in front of the blade. The penetration theory assumes the temperature equalization in bow wave/fillet after the film is scraped off. Hence it has been found that the penetration theory works well for low viscosity fluids. Goede et al. [19] modeled the heat transfer considering the regular turbulent model for pipe flow and the penetration theory for two different time periods. It is well known that the heat transfer depends on the flow regime. The values of heat transfer coefficient are usually low in the case of laminar flow because of low radial mixing and high in turbulent flow because of high radial mixing. Generally, the axial and rotational Reynolds numbers decide the flow regime in a scraped surface geometry and hence several authors have reported the heat transfer coefficient as a function of axial and rotational Reynolds number for laminar as well as transition regime [1]. Previously, the studies of heat transfer in scraped surface heat exchangers were based on the application of the penetration theory to calculate the overall heat transfer coefficient. It has been assumed that the scraper only removes the boundary layer, bulk mixing is ideal and the flow pattern is completely rotational. In order to describe the total heat transfer process with this relation, the time between two scraper passages should be smaller or at least equal to the time needed for full penetration of heat into the boundary layer/film. The temperature profile across the boundary layer can be given by Fourier equation as follows: ∂2 T ∂T =˛ 2 ∂t ∂z (5) with boundary conditions z = 0, T = Tw ; z → ∞ , T = TB and initial condition t = 0, T = TB for all z. Eq. (5) can be solved with given boundary conditions which results 2 T − Tw = √ TB − Tw √ z/(2 ˛t) 2 e−z dz (6) 0 The steady state conductive transport of heat can be described by Qw = −k dT dz (7) Combining Eq. (6) with Eq. (7), the heat flux can be expressed as dT Qw = −k = (TB − Tw ) dz atz=0 kCp t 0.5 kCp t (9) When the heat transfer mechanism between two scraper actions is described by the penetration theory, the time mean value of the heat transfer coefficient can be calculated from following equation h= 1 ts ts h(t) dt Combining Eqs. (9) and (10) results in kCp tsc h̄ = 2 (11) Expressing the time between two scraper actions in terms of number of blades (B) and the rotational frequency (n), tsc can be written as tsc = 1 nB (12) Combining Eq. (11) with Eq. (12), the scraped side heat transfer coefficient can be given by following equation hpen = 2 kCp nB (13) The complete derivation of Eq. (13) can also be found elsewhere [20,21]. The fluid passes through the ATFD in several forms such as liquid feed, paste, wet powder and dry powder at the dryer outlet. The scraped side heat transfer coefficient can be obtained by assuming the average properties throughout the dryer which are the function of moisture content. It can be assumed that the wet powder formed in the dryer lies in the form of thin layer adhering to the wall until the outlet of the dryer. The scraped side heat transfer coefficient decreases from top to bottom as the solid content tends to increase in the processing fluid along the height of the dryer. The ATFD is used to produce the powder from some specific type of feeds which generally have 10–30% solid content with viscosity in the range of 10–100 times that of water. Particularly ATFD is used for the drying of organic and inorganic salt solutions, pharmaceuticals and bulk drugs, dyes and pigments, etc. 3.2. Process-side heat transfer coefficient The heat transfer coefficient on the scraped side in ATFD can be estimated using the penetration theory. As the penetration theory does not depend on the viscosity, it can be used to calculate the heat transfer coefficient for slurry and paste type feed materials. The scraped side heat transfer coefficient depends on thermal conductivity, density and specific heat of the material. Now the question is how to model the physical properties with respect to solid content along the height of the dryer. The average values of thermal conductivity, density and specific heat are considered at saturation temperature for the given pressure (vacuum) condition. These properties can be written as kprocess = XL kLavg + (1 − XL )kSavg (14) process = XL Lavg + (1 − XL )Savg (15) Cp process = XL Cp Lavg + (1 − XL )Cp Savg (16) The heat transfer coefficient by the penetration theory is then calculated as follows, (8) (10) 0 (A) Only molecular conduction transfers the heat at the surface during the time between two scrapings. (B) At the end of scraping, the film at the surface gets perfectly mixed with the bulk flow. 689 hprocess-side = 2 kprocess process Cp nB (17) 690 S.B. Pawar et al. / Chemical Engineering and Processing 50 (2011) 687–693 Table 2 Dimensions of the agitated thin film dryer. Th TW Parameters Value Unit Diameter of shell Diameter of rotor Diameter of scraper Effective height Shell wall thickness Heat transfer area (HTA) Rotor speed range Gap clearance Number of blades 0.112 0.09 0.110 0.18 0.004 0.05 1–10 0.001 3 m m m m m m2 rps m – TB Jacket Side Process Side Z Wall Fig. 4. Schematic representation of conductive heat transfer across the wall. (corresponding to vacuum in the system) and averaged over the height of the dryer. For example, the density of ammonium sulfate is 1260 kg/m3 at 45 ◦ C and that of water is 990 kg/m3 . So, the density of 20% (by weight) ammonium sulfate solution would be (0.2 × 1260 + 0.8 × 990 = 1044) 1044 kg/m3 . As the water evaporates down the dryer height, its content in solution decreases relatively to ammonium sulfate. Further, for 20% water loss, the density would be (0.4 × 1260 + 0.6 × 990 = 1098) 1098 kg/m3 . As the whole process is operated at constant saturation temperature of 45 ◦ C, the dependence of physical properties over the temperature range is not considered here. 3.3. Overall heat transfer coefficient 4.2. Agitated thin film drying (ATFD) The thermal design of ATFD is similar to the conventional heat exchanger wherein the first step is to obtain the overall heat transfer coefficient (U) and then calculate the desired heat transfer area for a given capacity. In the present experimental setup the heat transfer area is fixed (0.05 m2 ) and it is required to calculate the overall heat transfer coefficient. The process side heat transfer coefficient is calculated using the penetration theory. The schematic representation of conductive heat transfer through the wall is shown in Fig. 4. The overall heat transfer coefficient in the ATFD can be represented by Eq. (18), The dimensions of the lab scale ATFD (Techno Force Solution Pvt. Ltd., Nasik, India) are as shown in Table 2. The experiments were performed to determine the evaporation rate in the lab scale ATFD. The experiments were performed according to 33 factorial designs with 27 numbers of runs by varying speed of the rotor, jacket-side heating temperature and flow rate of feed. The experimental conditions are reported in Table 3. The feed was supplied by reciprocating type feed pump (V.K. Pumps, Nasik, India) with a flow range of 0.5–3 kg/h. The whole system was operated under vacuum created by the rotary vane type vacuum pump (Model: vkc 6, Precise Vacuum System Pvt. Ltd., Nasik, India). The temperature of feed inlet, hot water inlet and outlet were determined by K-type thermocouple with 1 ◦ C accuracy. The temperature of vapor was determined using K-type thermocouple located in vapor line of ATFD. The vapor was condensed in the vertical condenser (Techno Force Solution Pvt. Ltd., Nasik, India) of 0.5 m2 heat transfer area. The chilled water of temperature 10 ◦ C was used to condensate the vapor to be then collected at the base of the condenser. The quantity of condensed water was measured and thus the evaporation rate was determined. The formation of bow wave in front of the blade was confirmed by measuring the hold-up using conventional start/stop method. Here, the hold-up is defined as the ratio of occupied volume of the solution to the annular volume available to flow. The hold-up for the studied flow rate was found to be greater than the actual volume between the gap of blades and the outer wall. 1 1 = U hprocess-side D so Dsi + 1 Dso ln(Dso /Dsi ) + hjacket-side 2kw (18) The metal wall resistance can be neglected because of the high thermal conductivity of wall material and small thickness. 4. Materials and methods 4.1. Materials Generally, the ATFD is used for the low viscosity solution and mostly the evaporating solvent is water, hence, water was preferred as a primary test material for the experimental runs. The experimental runs were carried out to determine the maximum/optimum evaporation capacity of the dryer for the system maintained at pressure of 133.34 Pa. After fixing the optimum flow rate and the jacket-side temperature, further experiments were carried out for sugar solution and ammonium sulfate solution (20%, w/w). The physical properties of all above mentioned solutions are given in Table 1. The properties of sugar solution and 20% (w/w) ammonium solution were derived at saturation temperature of 45 ◦ C 5. Results and discussion As previously described, the agitated thin film drying involves three phases (feed: liquid, product: solid, evaporated vapor-gas). Table 1 The physical properties of different feeds at saturation temperature (45 ◦ C). Properties Water Sugar solution (40%) 20% ammonium sulfate Density (kg/m3 ) Viscosity (kg/m s) Specific heat (J/kg ◦ C) Thermal conductivity (W/m ◦ C) 990 0.0006 4180 0.6 1060 0.00283 3800 0.51 1040 0.001 3500 0.35 S.B. Pawar et al. / Chemical Engineering and Processing 50 (2011) 687–693 691 Table 3 The operating conditions in ATFD for various systems. Feed Speed of rotor (rps) Water 40% sugar solution 20% aqueous ammonium sulfate 1–10 1–10 1–10 Rotational Reynolds number (ReR ) 22,544–184,020 4000–15,160 14,250–114,020 It is well known that the dealing with the three phase flow phenomena is a very complex problem. So preliminary, it was decided to go with the two phase flow phenomena using water as a feed instead of slurry or solution to determine the optimum parameters for the present dryer. Here, the processing unit can be termed as ATFE. Actually, there is no significant difference in ATFD and ATFE except for the speed of the rotor and the heat input. For the high viscosity solutions, ATFE can be operated at higher speed of the rotor (typically 25–30 rps) to overcome the laminar flow regime. For ATFD working with typically low viscosity solutions such a high speed of the rotor is not necessary, however, higher heat flux will be required. The experiments were carried out for the feed flow rate of 1–3 kg/h at different heating temperatures Twj (jacket side) and speed of the rotor. The hot water was used as a heating medium at the jacket side. The feed of 40 ◦ C was supplied by the feed pump at the top of the dryer over the feed ring. The vapor formed in the dryer was sucked by the vacuum pump through the vapor line. After several trials, it was found that the speed of the rotor did not make any significant increment in the evaporation rate. The evaporation rates are as shown in Fig. 5 for various operating parameters. It can be seen that the evaporation rate is below the 50% for the flow rate of 3 kg/h and Twj at 70 and 80 ◦ C as shown in Fig. 5A. The evaporation rate was found to be increased up to 70% for Twj at 90 ◦ C; whereas for the flow rate of 2 kg/h, the evaporation rate was found to be greater than 70% for Twj at 80 and 90 ◦ C as shown in Fig. 5B. The above results could be due to the incomplete mixing of thin film and bow wave in the case of relatively high flow rate and it is possible that there could be relatively high hold-up of the water in the form of bow wave in the evaporator at high flow rate. Further, for the flow rate of 1 kg/h, the evaporation rate was found to be greater than 90% at Twj of 80 and 90 ◦ C as shown in Fig. 5C. Based on this, it can be said that the Twj of 90 ◦ C is sufficient to evaporate the feed of 1 kg/h for given operating conditions of the system. After optimizing the system for water as a feed, further runs were carried out for sugar solution of viscosity 40◦ Brix (0.00325 Pa s). Slightly different results were obtained for this case as compared to the earlier system of water as a feed. The effect of speed of the rotor was found significant in this case and the evaporation rate was found to be increased from 40% for 1 rps to 56% for 5 rps at Twj of 90 ◦ C. For the case of ammonium sulfate solution, the results of the thin film drying were similar to that obtained for the earlier system of water as a feed and hence they are not shown here. When applying the penetration theory, the process of heat transfer from wall to liquid layer can be considered as molecular conduction to the semi-infinite solid. Those who were suspicious of applicability of the penetration theory for highly viscous liquid, they have developed the Nusselt type correlations which are based on rotational Reynolds number and/or Prandtl number [1,22]. Latinen [21] and Trommelen et al. [12] have well described the penetration theory. The theory is independent of the velocity (mass flow rate) and the viscosity of the liquid. It has been reported that the penetration theory works well for low viscosity liquid such as water in turbulent regime (ReR > 6000). Actually the theory is based on the assumption of temperature equalization in bow wave immediately after scraping which might not happen in the case of high viscosity liquid. In contrast, the assumptions are quite realistic Axial Reynolds number (Rea ) 1.5–4.5 0.335–0.88 1.02–3.12 Prandtl number (Pr) 4.12 90.5 10 Fig. 5. % water evaporation at different wall temperatures: A: flow rate 3 kg/h, B: flow rate 2 kg/h and C: flow rate 1 kg/h. 692 S.B. Pawar et al. / Chemical Engineering and Processing 50 (2011) 687–693 Fig. 6. The comparison of heat transfer magnitude derived from the penetration theory for 20% ammonium sulfate, 40% sugar solution and pure water. in the case of wiped thin film flow rather than the full liquid flow such as in a scraped surface heat exchanger. With the hold-up of 15–20% in ATFD/ATFE, the complete mixing is possible even for the high viscosity liquid because of turbulence created by the rotor. Moreover, the penetration theory was found to be applicable to predict the heat transfer coefficients for the drying of sludge and paste type materials. The drying of sludge and paste type materials in the paddle dryer [23,24], Nara type paddle dryer [25] and LIST type knead dryer [26] were carried out under vacuum conditions. The authors have used the penetration theory to model the heat transfer from the wall to the processing material and they followed the methodology originally developed by Schlunder and Mollekopf [27] for free flowing particles. Hence, it can be said that the penetration theory can be used for sludge, paste or granular materials to predict the heat transfer coefficient under vacuum or in the presence of inert gas conditions [28,29]. In the case of ATFD, even if the feed is in the form of solution, rapid evaporation at the entrance results in deposition of solids along the inner wall which is immediately scraped off by the scraper and get mixed with the bow wave. Thus, the processing feed becomes slurry and passes subsequently into the form of paste and wet powder. From the above studies, it can be said that the penetration theory can be used to describe the behavior of slurry in a scraped geometry provided the properties of slurry are averaged. In the case of ATFD, wherein, the feed is in the form of solution and the product is powder (which comes from dissolved solid), it is necessary to consider the average properties of process fluid. These properties can be obtained by solving Eqs. (14)–(16) assuming the linear relationship of moisture content with respect to the height of the dryer. The heat transfer coefficients are determined by Eq. (17) for sugar solution and 20% ammonium sulfate. It is then compared with that obtained for pure water evaporation (Eq. (13)) which is shown in Fig. 6. It has been observed that the average heat transfer coefficient for 20% ammonium sulfate solution (80% water) shows lower Nusselt number than that obtained for pure water evaporation at the same operating conditions. This signifies that the change of phase (solution to wet powder) lowers the heat transfer along the height. This variation of the heat transfer coefficient along the height is the key point to determine the effective heat transfer area for the thermal design of ATFD. In the present work, it is assumed that the evaporation occurs on the interface of thin film only and any bubble formation is Fig. 7. The prediction of heat transfer magnitude for 20% ammonium sulfate along the height of the dryer (H/H0 ) by the penetration theory (flow rate 1 kg/h and Twj = 90 ◦ C). Table 4 The overall heat transfer coefficients for different types of materials (data adapted from Perry’s Chemical Engineers’ Handbook). Process side fluid Jacket side fluid Agitation U (W/m2 ◦ C) Solution Slurry Paste Lumpy mass Powder (5% moisture) Steam Steam Steam Steam Steam Double scrapers Double scrapers Double scrapers Double scrapers Double scrapers 990–1190 910–990 710–850 425–545 230–290 suppressed by scraper. The feed behaves as a paste type material throughout the dryer. The heat transfer coefficient along the height of the dryer for 20% ammonium sulfate is shown in Fig. 7. The heat transfer coefficient was found in the range of 5500–300 W/m2 ◦ C along the height from top to bottom. The heat transfer coefficient is likely to change with respect to the percentage of dissolved solids in the feed. The properties and fraction of the dissolved solids in the feed solution can greatly affect the heat transfer coefficient along the height and hence its average value. The overall heat transfer coefficients (U) for various types of feed are shown in Table 4. The values reported in Table 4 are taken from Perry’s Chemical Engineers’ Handbook [30] to provide the idea of the range of U for various types of process fluids. 6. Conclusions The only way to identify the flow regime in ATFD is the bow wave in front of the blade which occurs because of the low axial Reynolds number. The bow wave forms in front of the blade irrespective of the viscosity of fluid and flow regime. Thus it is difficult to express the flow regimes in the agitated thin film dryer. Hence, the rotational Reynolds number has been defined to characterize the thin film flow in ATFE/ATFD. The assumptions of the penetration theory are quite realistic in the case of wiped thin film flow rather than the full liquid flow in a scraped geometry. The theory can be used to model the conductive heating from the wall to the wet particles in a scraping geometry provided the wet particles form thin film over the scraping wall. The dependence of the penetration theory over the speed of the rotor (n0.5 ) signifies that the heat transfer coefficient increases with an increase in the speed of the rotor. However, in actual practice after some optimum speed of the rotor, no significant increment in the heat transfer coefficient was found for the studied feed materials. S.B. Pawar et al. / Chemical Engineering and Processing 50 (2011) 687–693 Appendix A. Nomenclature B Cp Ds Dso Dsi De Dr F1 , F2 h h H H0 (H/H0 ) k kw L n Q rh Rr rps t tsc T TB Th Tw Twj XL U V z Z number of blades specific heat capacity of the feed (J/kg K) diameter of shell (m) outer diameter of shell (m) inner diameter of shell (m) equivalent diameter (m) diameter of the rotor (m) mass flow rates (kg/s) heat transfer coefficient (W/m2 ◦ C) time mean value of the heat transfer coefficient (W/m2 ◦ C) varying height of the dryer (m) constant height of the dryer (m) dimensionless height of the dryer (m) thermal conductivity of the feed (W/m ◦ C) wall thermal conductivity (W/m ◦ C) effective height of the dryer (m) speed of the rotor (rev/s) heat required (W) hydraulic radius (m) radius of the rotor (m) revolutions per second time (s) time between two scraper actions (s) temperature (◦ C) bulk temperature (◦ C) average temperature of jacket side (◦ C) wall temperature (◦ C) temperature of hot water used in jacket side (◦ C) mass fraction of solvent (w/w) overall heat transfer coefficient (W/m2 ◦ C) axial velocity (m/s) thickness of boundary layer (m) co-ordinate axis (as shown in Fig. 4) Dimensionless numbers Pr Prandtl number (Cp /k) Reynolds number (Dsi V/) Rea ReR rotational Reynolds number (De (Rr )process /) Nu Nusselt number ((hprocess-side Dsi )/kprocess ) Greek letters ˛ thermal diffusivity = (k/Cp ) (m2 /s) density of the feed (kg/m3 ) viscosity of the feed solution (kg/m s) ˝ angular velocity (rad/s) Subscripts avg average e equivalent jacket jacket side h hot water side liquid L process process side pen penetration r rotor solid S s sc w 693 stator scraper wall References [1] H. 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