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The turbulence effect of the airflow on the calorific losses in foodstuff dryers
Renewable Energy 29 (2004) 661–674 www.elsevier.com/locate/renene
The turbulence effect of the airflow on the calorific losses in foodstuff dryers S. Youcef-Ali a,, J.Y. Desmons a, M. Daguenet b a
b
Laboratoire de Me´canique et d’Energe´tique, Universite´ de Valenciennes et du Hainaut-Cambre´sis, Le Mont Houy—59313 Valenciennes Cedex 9, France Laboratoire de Thermodynamique et Energe´tique Universite´ de Perpignan, 52, avenue de Villeneuve, 66860 Perpignan Cedex, France Received 18 June 2003; accepted 30 July 2003
Abstract The calorific losses through the sidewalls delay drying, particularly in the higher racks of the foodstuff dryers, which provide us information on total drying time. One notes a temperature gradient of the air during its circulation upwards, between the entry and the exit of the empty dryer. This gradient is more significant with the decrease of the air velocity. The heat transfer coefficient by forced convection between the air and the interior walls is often determined by using the average Nusselt number defined in flow of boundary layer on plane plate (laminar regime). In addition, the presence of the grills, on which slices of product are spread, causes turbulence which is not taken into account by this coefficient of Nusselt number. In this paper, we determine the heat transfer coefficient between the air and the sidewalls of a dryer experimentally. The latter contains grills on which thin layer slices of a dry material (wood) of the same geometry as the initial potato slices are spread. This leads us to determine an expression of the Nusselt number of the form Nu ¼ aReb Pr1=3 , without mass transfer. Thereafter, this correlation of the Nusselt number is introduced into the mathematical model, already suggested in our former work, which predicts the performances, in forced convection, of the foodstuff dryers. In the end, the influence of these losses is presented by means of a comparative study between various models, those that take into account and those that ignore these losses between two beds of slices. The results thus obtained are compared with the experiment in an application on potato drying. # 2003 Elsevier Ltd. All rights reserved. Keywords: Calorific losses; Drying; Forced convection; Heat transfer; Turbulence
Corresponding author. Tel.: +33-3-27-51-1960; fax: +33-3-27-51-1961. E-mail address: [email protected] (S. Youcef-Ali).
0960-1481/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2003.07.012
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Nomenclature Cpc Cpf Cpfs Cpv Cpprs eb ec el G Gs H hCp hrc hvfpi hvfpr hvv Hm Lv M mprs Rv Sfpr Sp t Ta T0 a Tc Tfs Tpi Tpr Va Vl vfs vv W
specific heat of water in food (J/kg K) specific heat of air (J/kg K) specific heat of dry air (J/kg K) specific heat of water vapour in the air (J/kg K) specific heat of dry solid (J/kg K) thickness of the wood (m) distance between two grills (m) thickness of the bed of slices at time t (m) air mass flow rate (kg/h) air mass flow rate per unit of dryer area (kg/s m2) total exchange coefficient through the walls of the dryer (W/m2 K) = kb/eb, conduction heat transfer coefficient in the wall of dryer (W/m2 K) radiation heat transfer coefficient between the wall and the sky (W/m2 K) convection heat transfer coefficient between the air and the wall (W/m2 K) convection heat transfer coefficient between the product and the air (W/m2 K) ¼ 5:67 þ 3:86vv , convection heat transfer coefficient caused by the wind (W/m2 K) global mass transfer coefficient (m/s) ¼ 4186ð597 0:56ðTpr þ 276ÞÞ, latent heat of vaporisation (J/kg) moisture content of product at time t (kg water/kg dry basis) dry solid mass in bed (kg) characteristic constant of water (J/(kg water K)) exchange surface between the product and the air in bed at time t (m2) exchange surface of the walls in the bed (m2) time (s) v ambient temperature ( C) v equivalent ambient air temperature ( C) v ¼ 0:0552ðTa þ 273Þ1:5 273, sky temperature ( C) v dry temperature of drying air ( C) v internal superficial temperature of the wall of dryer ( C) v surface temperature of the product ( C) volume of air in bed (m3) apparent volume of bed (m3) average air velocity between two beds (m/s) wind velocity (m/s) moisture content of the air (kg water/kg dry air)
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Dimensionless numbers Nu Nusselt number Pr Prandtl number Re Reynolds number Greek symbols e bed porosity at time t us relative humidity of air (%) k thermal conductivity of air (W/m K) kb thermal conductivity of wood (W/m K) l kinematic viscosity of air (kg/ms) qpr density of food (kg/m3) qprs ¼ qpr =ð1 þ MÞ, dry solid density (kg/m3) qf air density (kg/m3)
1. Introduction Generally, the theoretical models of the drying kinetics are experimentally validated in research laboratories. In industry, when one extrapolates numerical results, the effect of the calorific losses through the walls of the dryer becomes more noticeable. These losses occur along the sidewalls of the dryer, in other words, in the beds and between two beds of slices. On the one hand one notices, that the air velocity between two beds is lower than that in a bed, and on the other, the thickness of the bed of slices decreases gradually during the drying because of the shrinkage phenomenon. Then, it is possible to conclude that the calorific losses between the two beds of slices are not negligible compared to those evaluated in the beds. We thus take into account these calorific losses in the mathematical model that predict the performances in forced convection in foodstuff dryers [8]. Inside the rectangular channels and in forced convection, the correlations of the average Nusselt number of the exchange coefficient are generally given for small sections of channels, by taking its hydraulic diameter as characteristic length [2]. In habitat and in natural convection, the exchange coefficient by convection between the air and the interior walls is estimated at approximately 4.5 (W/m2 K) [5]. In our case and in forced convection, if we use the average Nusselt number Nu ¼ 0:66Re0:5 Pr1=3 , we would note that for a Reynolds number of 1400, the value of the exchange coefficient hvfpi is only 4.1 (W/m2 K). The Nusselt number is defined in Ref. [6], in a flow of boundary layer in plane plate (laminar regime), to calculate the convection heat transfer coefficient hvfpi between the air and the sidewalls of the dryer. We will thus use a Nusselt number, which will take into account the turbulence of the airflow due to the presence of the grills and the product spread on these grills in thin layer of slices. Indeed, the convective exchanges
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between the air and the sidewalls of the dryer increase, and the appropriate Nusselt number will keep the same form, Nu ¼ aReb Pr1=3 , where coefficients ‘‘a’’ and ‘‘b’’ will be determined. On the one hand, these coefficients are calculated in the presence of grills only, and on the other, in the presence of thin layer of dry wood slices similar to the product to be dried. The effect of the air turbulence has been studied in the case of the airflow around isolated bodies, in front of which are placed grills [3]. For these cylindrical forms of bodies, an increase in the coefficient of convective exchange going up to 50% has been noted. One finds similar effects for spheres. At the end of this work, we compare and discuss the results of the drying kinetics obtained by the various theoretical models: those that take into account the calorific losses between two beds and in the beds of slices (the Nusselt numbers used are those proposed in the presence of pierced grills and thin layers of dry wood slices) and those that do take into account these losses between two beds of slices through the sidewalls. We use potato as an example of product to dry.
2. Theoretical analysis We use a mathematical model by adopting the method used beforehand [8] for dryers in forced convection (Fig. 1). In the present article, the calorific losses through the walls between two beds and in the beds of slices, in addition to the shrinkage phenomenon and the variation of the thermophysical properties of the product according to its moisture content and temperature are taken into account. The physical properties of the air, dependent on its temperature, are calculated according to the latter element.
Fig. 1. Layout of the experimental device.
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2.1. In the bed of slices For each bed of slices, the conservation equations of heat and mass are written as: Gs Va ðCpfs þ Cpv W Þ
dTfs ¼ hvfpr Sfpr ðTfs Tpr Þ hvfpi Sp ðTfs Tpi Þ dx
ð1 eÞqprs ðCpprs þ Cpc MÞ
dTpr Sfpr ¼ hvfpr ðTfs Tpr Þ dt Vl dM þ ð1 eÞqprs Lv dt
dW 1 dM ¼ ð1 eÞqprs dx Gs dt
ð1Þ
ð2Þ
ð3Þ
In Eq. (1), Tpi represents the average superficial temperature of the dryer internal wall. It is determined from pure thermal resistances (Fig. 2), according to the drying air temperature Tfs, equivalent ambient air temperature Ta0 and the thermal resistances. It is written as follows: Tpi ¼
hvfpi ðhcp þ hrc þ hvv ÞTfs þ hcp ðhrc þ hvv ÞTa0 hcp ðhrc þ hvv Þ þ hvfpi ðhcp þ hrc þ hvv Þ
ð4Þ
The tension of Thevenin 5 Ta0 is written: Ta Tc þ Ta hvv þ Tc hrc h h vv ¼ Ta0 ¼ rc 1 1 hvv þ hrc þ hrc hvv
Fig. 2. Electric diagram equivalent to the exchanges through the walls of the dryer.
ð5Þ
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The drying rate is given by the transfer model of Combes [1], according to the partial pressure [7]: mprs
dM Hm ðPsat ðTpr Þ Pv ðTfs ÞÞSfpr ¼ dt Rv ðTfs þ 273Þ
ð6Þ
with: Psat(Tpr) being the saturating water vapour pressure at the temperature Tpr of the surface of the product (in Pa) and Pv(Tfs) the partial pressure of the water vapour in the air (in Pa). 2.2. Between two beds of slices The calorific losses through the sidewalls of the drying chamber, between two beds of slices, seem to be more significant than those inside a bed, particularly when the later is thin. In this case, the distance between the two beds is larger than their thickness and the air velocity between the two beds is lower than that inside the beds. Knowing that the insulating walls of the dryer are of low thickness in wood, we can thus neglect the effects of thermal inertia. For a section of fluid thickness dx of cross-section As and perimeter Ps (Fig. 3), we can write the variation dQ during time dt of the heat quantity contained in this section in the form: dQ ¼ As dxdtqf Cpf dTfs ¼ Ps dxdt/ ¼ Ps dxdtHðTfs Ta0 Þ where Ta0 is the equivalent ambient air temperature and / being the outgoing flow of the section by the sidewalls. Finally, one will obtain: @Tfs mfs @Tfs Ps H þ ¼ ðT 0 Tfs Þ As qf Cpf a @t @x
Fig. 3. Exchanges between two beds of slices.
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By neglecting the temporal variation of the temperature in front of its space variation, permanent regime, the preceding equation becomes: @Tfs þ BTfs ¼ BTa0 @x
with
B¼
2ðLs þ ls ÞH Ls ls qf Cpf vfs
whose solution is of the form: Tfs ðx; tÞ ¼ AeBx þ Ta0 For x ¼ 0: A ¼ Tfs ð0; tÞ Ta0 The final equation is written as: Tfs ðx; tÞ ¼ ðTfs ð0; tÞ Ta0 ÞeBx þ Ta0
ð7Þ
when ec and el are the distance between two grills and the height of the bed slices, respectively (which decreases during drying because of the shrinkage phenomenon of the product), we can write for x ¼ ec el : Tfs ðec el ; tÞ ¼ ðTfs ð0; tÞ Ta0 ÞexpðBðec el ÞÞ þ Ta0
ð8Þ
Flow / having for expression: / ¼ HðTfs Ta0 Þ with
1 1 1 1 ¼ þ þ : H hvfpi hcp hrc þ hvv
3. Experimental device We have an indirect solar dryer, operating in forced convection, linked to a flat plate collector (Fig. 1); the system has neither storage nor air recycling. The initial quantity of product is 6.4 kg corresponding to 5 kg per unit of area of solar air flat plate collector. The product, which is cleaned and cut into slices of 3 mm thickness and whose average diameter is 4 cm, is distributed equally on the first three racks of the chamber. This drying unit is fed by air heated across a solar collector equipped with offset plate fins placed in the dynamic air vein under the absorber. The air is sucked up by a ventilator, which is fixed at the exit of the dryer in order to ensure an even distribution of air. During the first two hours of drying, the product is weighed every half an hour. After this, the weights are registered every hour, until they reach a state of equilibrium. At the same time, we record the dryand wet-bulb temperatures above and below the beds, the points of measurement being right below the latter. The drying chamber (wood thickness: eb ¼ 0:015 m) has an area of (0:875 0:50 m2) and a height of 0.90 m. Three experiments are performed without mass transfer: with a dryer completely empty and provided only with grills, a dryer containing a thin layer of dry wood slices above each pierced grill and a dryer containing a thin layer of dried potato slices above each grill.
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4. Results and discussion We noticed that the losses through the walls of the dryer have an influence on the drying kinetics. For that, we take into account the empty dryer (inside a v ventilated room of ambient temperature Ta ¼ 27:5 C) through which hot air, aspired by a ventilator, circulates upwards. The evolution of the air temperature during its passage through the dryer provided only with grills is represented in Fig. 4 according to the air mass flow rate. One notes the existence of a gradient air temperature, which is due to the calorific losses through the sidewalls of the dryer. This gradient is more significant with the decrease of the airflow rate. For each section perpendicular to fluid flow, we calculate the average air temperature from the values taken in several points of measurement. The equations of the smoothed curves of the temperature’s experimental points showed in Fig. 4 are used to calculate the convective exchange coefficient hvfpi. These equations, introduced into the mathematical model (Eq. (7)), enable us to determine coefficient B and then the coefficient of total exchange H. Thereafter, we determine the convective exchange coefficient hvfpi as well as the Nusselt number Nu according to the Reynolds number. Since the Prandtl number for the air varies very little ( 0.7), we only need to consider two ratios without dimension: the Nusselt number (hvfpiec/k) and the Reynolds number (vfsec/l), where the height of the rack ec is constant (15 cm). We propose the curve of Fig. 5 which gives the Nusselt number according to the Reynolds number. We then deduce the Nusselt number Nu ¼ 0:165Re0:9 Pr1=3 for an airflow along the dryer in the presence of grills. In Fig. 5, the results obtained for the first rack are represented, and we obtain exactly the same result for the other racks. This form of the Nusselt number is used in the mathematical model for the representation of the theoretical air temperature profiles in Fig. 4. In the algorithm
Fig. 4. Theoretical and experimental evolution of the air temperature in the dryer (provided only with grills) according to the air mass flow rate.
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Fig. 5. Evolution of the average Nusselt number according to the Reynolds number (dryer provided only with grills).
which allows the calculation of the theoretical values of these temperatures, we introduce the air temperature at the entry of the dryer Tfs obtained by the equation of the smoothed curve of the experimental points as input data: Tfs ¼ 0:0004G 2 0:1996G þ 56:065 ðR2 ¼ 0:9893Þ; v
where Tfs is in C and G in kg/h.The wind velocity then equals 1 m/s. We proceed in the same way to determine the Nusselt number by introducing thin layers of dry wood slices on the grills. We note that the curves keep the same pace (Fig. 6); on the contrary, the variation in temperature is much more significant than in the preceding case (dryer provided only with grills). Under the same conditions, the calorific losses through the sidewalls of the dryer are then more significant, because the presence of the slices causes more turbulence in the airflow.
Fig. 6. Theoretical and experimental evolution of the air temperature in the dryer (provided with grills and thin layers of dry wood slices) according to the air mass flow rate.
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Fig. 7. Evolution of the average Nusselt number according to the Reynolds number (dryer provided with grills and thin layers of dry wood slices).
Obviously, the values of the Nusselt number are must more significant. Represented in Fig. 7 according to the Reynold number, the Nusselt number is written as: Nu ¼ 0:65Re0:9 Pr1=3 . We note that coefficient ‘‘a’’ passes from 0.165 to 0.65, when one introduces thin layers of dry wood slices. The theoretical values of the temperatures of Fig. 6 are calculated by using the air temperature at the entry of the dryer Tfs obtained by the equation of the smoothed curve of the experimental points as input data: Tfs ¼ 0:00009G 2 0:0992G þ 57:324 ðR2 ¼ 0:9931Þ v
where Tfs is in C and G in kg/h. The use of thin layers of dried potato slices spread on the grills gives practically the same experimental curves of the air temperature profiles (Fig. 8) as in the case
Fig. 8. Theoretical and experimental evolution of the air temperature in the dryer (provided with grills and thin layers of dried potato slices) according to the air mass flow.
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of the dry wood slices. If the theoretical profiles are represented by taking into account the air temperature at the entry of the dryer Tfs ¼ 56:077e0:0017G (Tfs in v C and G in kg/h) and for a Nusselt number Nu ¼ 0:65Re0:9 Pr1=3 , one can see that the curves are close to the experimental results. Thus, the effect of the slices of dry wood and dried potato on the turbulence of the airflow is almost the same. The majority of the authors do not take into account the calorific losses through the walls of the dryers. In order to underline their influence, we represent in Fig. 9a–c the theoretical and experimental drying kinetics of potato (of initial total mass 6.4 kg) on the first three grills of the drying chamber and cut into 3 mm thick v slices. The conditions of the drying air are: G ¼ 122 kg/h, Tfs ¼ 50 C and us ¼ 20:6%. In the first figure (Fig. 9a), the model does not take into account the losses through the sidewalls of the dryer between the beds of slices but only those in the beds by using the Nusselt number Nu ¼ 0:66Re0:5 Pr1=3 for the calculation of the convective exchange coefficient hvfpl [8]. On the contrary, the mathematical model takes into account those in the beds and between the beds of slices by using the Nusselt number Nu ¼ 0:165Re0:9 Pr1=3 (Fig. 9b) and Nu ¼ 0:65Re0:9 Pr1=3 (Fig. 9c), allowing the calculation of the convective exchange coefficient hvfpi. We note a conspicuous improvement of the theoretical curves which move close to the experimental curves in particular for the two last racks, and when one uses the Nusselt number which is calculated when the air is disturbed by the presence of pierced grills and dry wood slices (Fig. 9c). Table 1 groups together the experimental and theoretical results of drying time of the product in the first three racks. It compares the theoretical results with the experimental values: – 1st model does not take into account the losses through the walls of the dryer between the two beds of slices, – 2nd model takes into account the losses through the dryer walls between and in the beds of slices with a Nusselt number Nu ¼ 0:165Re0:9 Pr1=3 , – 3rd model takes into account the losses through the walls of the dryer between and in the beds of slices with a Nusselt number Nu ¼ 0:65Re0:9 Pr1=3 . The drying time, for each rack, represents the moment reached by hygroscopic equilibrium between the air and the product, that is to say a value of 0.05 (kg water/ kg db) of the equilibrium product’s moisture content calculated according to the v conditions of the drying air (Tfs ¼ 50 C and us ¼ 20:6%), in the case of potato [4]. For the three racks, the average relative variation concerning the drying time, which is calculated from the experimental and theoretical results of the first model, varies between 3.85% and 11.45%, whereas for the second model, the values are lower (between 2.60% and 4.17%) and much lower for the third model (between 0.52% and 4.29%). Concerning the product of the last rack in particular and when we improve the model, on the one hand by taking into account the calorific losses along the walls and on the other, by improving the Nusselt number which determines the convective exchange coefficient between the air and the walls of the
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Fig. 9. Comparison between the theoretical and experimental drying kinetics of potato in the first three v racks of the dryer (G ¼ 122 kg/h, Tfs ¼ 50 C and us ¼ 20:6%). (a) The model does not take into account the calorific losses between two beds, with Nu ¼ 0:66Re0:5 Pr1=3 . (b) The model takes into account the calorific losses between and in the beds of slices, with Nu ¼ 0:165Re0:9 Pr1=3 . (c) The model takes into account the calorific losses between and in the beds of slices, with Nu ¼ 0:65Re0:9 Pr1=3 .
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Table 1 v Comparison between the experimental and the theoretical drying times (G ¼ 122 kg/h, Tfs ¼ 50 C and us ¼ 20:6%) Rack
1 2 3
Time in min
Theoretical time in min
Experimental
1st model
Dt/t (%)
2nd model
Dt/t (%) 3rd model
Dt/t (%)
720 780 960
650 750 850
9.72 3.85 11.46
690 805 935
4.17 3.11 2.60
4.17 4.29 0.52
690 815 955
dryer, one can notice that the theoretical drying time is quasi-close to that calculated in the experiment.
5. Conclusion We have proposed a form of the Nusselt number for the calculation of the average coefficient of forced convective exchange between the air and the interior walls of a dryer and by taking into account the existence of the pierced grills as well as a thin layer of dry wood slices on these grills in the absence of mass transfer. These grills cause the disturbance of the airflow and the creation of turbulence, which increases the convective exchanges by increasing the calorific losses outward. A good estimate of these losses by using the most appropriate Nusselt number improves the theoretical model of drying kinetics. Indeed, a comparative study between two theoretical models—those that take into account the losses through the walls of the dryer between two beds of slices and those that do not—underline the improvement of theoretical drying times. By comparing the latter with the experimental results, one notes, in particular for the last rack, that the relative variation decreases from 11.46% to 0.52%, which is not negligible. We have given an experimental approach of the Nusselt number Nu ¼ 0:165Re0:9 Pr1=3 when the airflow is disturbed by the presence of pierced grills. This Nusselt number form can be generalized for air in forced turbulent flow inside the channels of parallelepipedic form, with established profiles of temperature and velocity and for a Reynolds number (17 < Re < 1500). Another form of the Nusselt number Nu ¼ 0:65Re0:9 Pr1=3 can be used in the theoretical models, which determines the drying kinetics of the vertical dryers with pierced grills in the same conditions.
References [1] Daguenet M. Les Se´choirs Solaires: the´orie et pratique. UNESCO; 1985. [2] Lallemand A. Convection force´e interne. Engineer Tech 1991;A1541:5–11. [3] McAdams WH. Transmission de chaleur, 2. Pairs: Dunod; 1961.
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[4] McLaughlin CP, Magee TRA. The Determination of Sorption Isotherm and the Isosteric Heats of Sorption for Potatoes. J Food Eng 1998;35(3):267–80. [5] Recknagel, Spenger, Ho¨nmann. Manuel pratique du ge´nie climatique, 2. Pyc-Edition; 1986. [6] Sacadura JF. Initiation aux transferts thermiques, Cast INSA of Lyon. Technique and Documentation: Paris. 1980. [7] Sembiring M. Contribution a` l’e´tude des performances technico-e´conomiques d’un se´choir pour fruits et le´gumes a` chauffages partiellement solaire. PhD Thesis. University of Poitiers. France. 1990. [8] Youcef-Ali S, Moummi N, Desmons JY, Abene A, Messaoudi H, Le Ray M. Numerical and experimental study of dryer in forced convection. Int J Energy Res 2001;25(6):537–53.
The turbulence effect of the airflow on the calorific losses in foodstuff dryers S. Youcef-Ali a,, J.Y. Desmons a, M. Daguenet b a
b
Laboratoire de Me´canique et d’Energe´tique, Universite´ de Valenciennes et du Hainaut-Cambre´sis, Le Mont Houy—59313 Valenciennes Cedex 9, France Laboratoire de Thermodynamique et Energe´tique Universite´ de Perpignan, 52, avenue de Villeneuve, 66860 Perpignan Cedex, France Received 18 June 2003; accepted 30 July 2003
Abstract The calorific losses through the sidewalls delay drying, particularly in the higher racks of the foodstuff dryers, which provide us information on total drying time. One notes a temperature gradient of the air during its circulation upwards, between the entry and the exit of the empty dryer. This gradient is more significant with the decrease of the air velocity. The heat transfer coefficient by forced convection between the air and the interior walls is often determined by using the average Nusselt number defined in flow of boundary layer on plane plate (laminar regime). In addition, the presence of the grills, on which slices of product are spread, causes turbulence which is not taken into account by this coefficient of Nusselt number. In this paper, we determine the heat transfer coefficient between the air and the sidewalls of a dryer experimentally. The latter contains grills on which thin layer slices of a dry material (wood) of the same geometry as the initial potato slices are spread. This leads us to determine an expression of the Nusselt number of the form Nu ¼ aReb Pr1=3 , without mass transfer. Thereafter, this correlation of the Nusselt number is introduced into the mathematical model, already suggested in our former work, which predicts the performances, in forced convection, of the foodstuff dryers. In the end, the influence of these losses is presented by means of a comparative study between various models, those that take into account and those that ignore these losses between two beds of slices. The results thus obtained are compared with the experiment in an application on potato drying. # 2003 Elsevier Ltd. All rights reserved. Keywords: Calorific losses; Drying; Forced convection; Heat transfer; Turbulence
Corresponding author. Tel.: +33-3-27-51-1960; fax: +33-3-27-51-1961. E-mail address: [email protected] (S. Youcef-Ali).
0960-1481/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2003.07.012
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Nomenclature Cpc Cpf Cpfs Cpv Cpprs eb ec el G Gs H hCp hrc hvfpi hvfpr hvv Hm Lv M mprs Rv Sfpr Sp t Ta T0 a Tc Tfs Tpi Tpr Va Vl vfs vv W
specific heat of water in food (J/kg K) specific heat of air (J/kg K) specific heat of dry air (J/kg K) specific heat of water vapour in the air (J/kg K) specific heat of dry solid (J/kg K) thickness of the wood (m) distance between two grills (m) thickness of the bed of slices at time t (m) air mass flow rate (kg/h) air mass flow rate per unit of dryer area (kg/s m2) total exchange coefficient through the walls of the dryer (W/m2 K) = kb/eb, conduction heat transfer coefficient in the wall of dryer (W/m2 K) radiation heat transfer coefficient between the wall and the sky (W/m2 K) convection heat transfer coefficient between the air and the wall (W/m2 K) convection heat transfer coefficient between the product and the air (W/m2 K) ¼ 5:67 þ 3:86vv , convection heat transfer coefficient caused by the wind (W/m2 K) global mass transfer coefficient (m/s) ¼ 4186ð597 0:56ðTpr þ 276ÞÞ, latent heat of vaporisation (J/kg) moisture content of product at time t (kg water/kg dry basis) dry solid mass in bed (kg) characteristic constant of water (J/(kg water K)) exchange surface between the product and the air in bed at time t (m2) exchange surface of the walls in the bed (m2) time (s) v ambient temperature ( C) v equivalent ambient air temperature ( C) v ¼ 0:0552ðTa þ 273Þ1:5 273, sky temperature ( C) v dry temperature of drying air ( C) v internal superficial temperature of the wall of dryer ( C) v surface temperature of the product ( C) volume of air in bed (m3) apparent volume of bed (m3) average air velocity between two beds (m/s) wind velocity (m/s) moisture content of the air (kg water/kg dry air)
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Dimensionless numbers Nu Nusselt number Pr Prandtl number Re Reynolds number Greek symbols e bed porosity at time t us relative humidity of air (%) k thermal conductivity of air (W/m K) kb thermal conductivity of wood (W/m K) l kinematic viscosity of air (kg/ms) qpr density of food (kg/m3) qprs ¼ qpr =ð1 þ MÞ, dry solid density (kg/m3) qf air density (kg/m3)
1. Introduction Generally, the theoretical models of the drying kinetics are experimentally validated in research laboratories. In industry, when one extrapolates numerical results, the effect of the calorific losses through the walls of the dryer becomes more noticeable. These losses occur along the sidewalls of the dryer, in other words, in the beds and between two beds of slices. On the one hand one notices, that the air velocity between two beds is lower than that in a bed, and on the other, the thickness of the bed of slices decreases gradually during the drying because of the shrinkage phenomenon. Then, it is possible to conclude that the calorific losses between the two beds of slices are not negligible compared to those evaluated in the beds. We thus take into account these calorific losses in the mathematical model that predict the performances in forced convection in foodstuff dryers [8]. Inside the rectangular channels and in forced convection, the correlations of the average Nusselt number of the exchange coefficient are generally given for small sections of channels, by taking its hydraulic diameter as characteristic length [2]. In habitat and in natural convection, the exchange coefficient by convection between the air and the interior walls is estimated at approximately 4.5 (W/m2 K) [5]. In our case and in forced convection, if we use the average Nusselt number Nu ¼ 0:66Re0:5 Pr1=3 , we would note that for a Reynolds number of 1400, the value of the exchange coefficient hvfpi is only 4.1 (W/m2 K). The Nusselt number is defined in Ref. [6], in a flow of boundary layer in plane plate (laminar regime), to calculate the convection heat transfer coefficient hvfpi between the air and the sidewalls of the dryer. We will thus use a Nusselt number, which will take into account the turbulence of the airflow due to the presence of the grills and the product spread on these grills in thin layer of slices. Indeed, the convective exchanges
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between the air and the sidewalls of the dryer increase, and the appropriate Nusselt number will keep the same form, Nu ¼ aReb Pr1=3 , where coefficients ‘‘a’’ and ‘‘b’’ will be determined. On the one hand, these coefficients are calculated in the presence of grills only, and on the other, in the presence of thin layer of dry wood slices similar to the product to be dried. The effect of the air turbulence has been studied in the case of the airflow around isolated bodies, in front of which are placed grills [3]. For these cylindrical forms of bodies, an increase in the coefficient of convective exchange going up to 50% has been noted. One finds similar effects for spheres. At the end of this work, we compare and discuss the results of the drying kinetics obtained by the various theoretical models: those that take into account the calorific losses between two beds and in the beds of slices (the Nusselt numbers used are those proposed in the presence of pierced grills and thin layers of dry wood slices) and those that do take into account these losses between two beds of slices through the sidewalls. We use potato as an example of product to dry.
2. Theoretical analysis We use a mathematical model by adopting the method used beforehand [8] for dryers in forced convection (Fig. 1). In the present article, the calorific losses through the walls between two beds and in the beds of slices, in addition to the shrinkage phenomenon and the variation of the thermophysical properties of the product according to its moisture content and temperature are taken into account. The physical properties of the air, dependent on its temperature, are calculated according to the latter element.
Fig. 1. Layout of the experimental device.
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2.1. In the bed of slices For each bed of slices, the conservation equations of heat and mass are written as: Gs Va ðCpfs þ Cpv W Þ
dTfs ¼ hvfpr Sfpr ðTfs Tpr Þ hvfpi Sp ðTfs Tpi Þ dx
ð1 eÞqprs ðCpprs þ Cpc MÞ
dTpr Sfpr ¼ hvfpr ðTfs Tpr Þ dt Vl dM þ ð1 eÞqprs Lv dt
dW 1 dM ¼ ð1 eÞqprs dx Gs dt
ð1Þ
ð2Þ
ð3Þ
In Eq. (1), Tpi represents the average superficial temperature of the dryer internal wall. It is determined from pure thermal resistances (Fig. 2), according to the drying air temperature Tfs, equivalent ambient air temperature Ta0 and the thermal resistances. It is written as follows: Tpi ¼
hvfpi ðhcp þ hrc þ hvv ÞTfs þ hcp ðhrc þ hvv ÞTa0 hcp ðhrc þ hvv Þ þ hvfpi ðhcp þ hrc þ hvv Þ
ð4Þ
The tension of Thevenin 5 Ta0 is written: Ta Tc þ Ta hvv þ Tc hrc h h vv ¼ Ta0 ¼ rc 1 1 hvv þ hrc þ hrc hvv
Fig. 2. Electric diagram equivalent to the exchanges through the walls of the dryer.
ð5Þ
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The drying rate is given by the transfer model of Combes [1], according to the partial pressure [7]: mprs
dM Hm ðPsat ðTpr Þ Pv ðTfs ÞÞSfpr ¼ dt Rv ðTfs þ 273Þ
ð6Þ
with: Psat(Tpr) being the saturating water vapour pressure at the temperature Tpr of the surface of the product (in Pa) and Pv(Tfs) the partial pressure of the water vapour in the air (in Pa). 2.2. Between two beds of slices The calorific losses through the sidewalls of the drying chamber, between two beds of slices, seem to be more significant than those inside a bed, particularly when the later is thin. In this case, the distance between the two beds is larger than their thickness and the air velocity between the two beds is lower than that inside the beds. Knowing that the insulating walls of the dryer are of low thickness in wood, we can thus neglect the effects of thermal inertia. For a section of fluid thickness dx of cross-section As and perimeter Ps (Fig. 3), we can write the variation dQ during time dt of the heat quantity contained in this section in the form: dQ ¼ As dxdtqf Cpf dTfs ¼ Ps dxdt/ ¼ Ps dxdtHðTfs Ta0 Þ where Ta0 is the equivalent ambient air temperature and / being the outgoing flow of the section by the sidewalls. Finally, one will obtain: @Tfs mfs @Tfs Ps H þ ¼ ðT 0 Tfs Þ As qf Cpf a @t @x
Fig. 3. Exchanges between two beds of slices.
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By neglecting the temporal variation of the temperature in front of its space variation, permanent regime, the preceding equation becomes: @Tfs þ BTfs ¼ BTa0 @x
with
B¼
2ðLs þ ls ÞH Ls ls qf Cpf vfs
whose solution is of the form: Tfs ðx; tÞ ¼ AeBx þ Ta0 For x ¼ 0: A ¼ Tfs ð0; tÞ Ta0 The final equation is written as: Tfs ðx; tÞ ¼ ðTfs ð0; tÞ Ta0 ÞeBx þ Ta0
ð7Þ
when ec and el are the distance between two grills and the height of the bed slices, respectively (which decreases during drying because of the shrinkage phenomenon of the product), we can write for x ¼ ec el : Tfs ðec el ; tÞ ¼ ðTfs ð0; tÞ Ta0 ÞexpðBðec el ÞÞ þ Ta0
ð8Þ
Flow / having for expression: / ¼ HðTfs Ta0 Þ with
1 1 1 1 ¼ þ þ : H hvfpi hcp hrc þ hvv
3. Experimental device We have an indirect solar dryer, operating in forced convection, linked to a flat plate collector (Fig. 1); the system has neither storage nor air recycling. The initial quantity of product is 6.4 kg corresponding to 5 kg per unit of area of solar air flat plate collector. The product, which is cleaned and cut into slices of 3 mm thickness and whose average diameter is 4 cm, is distributed equally on the first three racks of the chamber. This drying unit is fed by air heated across a solar collector equipped with offset plate fins placed in the dynamic air vein under the absorber. The air is sucked up by a ventilator, which is fixed at the exit of the dryer in order to ensure an even distribution of air. During the first two hours of drying, the product is weighed every half an hour. After this, the weights are registered every hour, until they reach a state of equilibrium. At the same time, we record the dryand wet-bulb temperatures above and below the beds, the points of measurement being right below the latter. The drying chamber (wood thickness: eb ¼ 0:015 m) has an area of (0:875 0:50 m2) and a height of 0.90 m. Three experiments are performed without mass transfer: with a dryer completely empty and provided only with grills, a dryer containing a thin layer of dry wood slices above each pierced grill and a dryer containing a thin layer of dried potato slices above each grill.
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4. Results and discussion We noticed that the losses through the walls of the dryer have an influence on the drying kinetics. For that, we take into account the empty dryer (inside a v ventilated room of ambient temperature Ta ¼ 27:5 C) through which hot air, aspired by a ventilator, circulates upwards. The evolution of the air temperature during its passage through the dryer provided only with grills is represented in Fig. 4 according to the air mass flow rate. One notes the existence of a gradient air temperature, which is due to the calorific losses through the sidewalls of the dryer. This gradient is more significant with the decrease of the airflow rate. For each section perpendicular to fluid flow, we calculate the average air temperature from the values taken in several points of measurement. The equations of the smoothed curves of the temperature’s experimental points showed in Fig. 4 are used to calculate the convective exchange coefficient hvfpi. These equations, introduced into the mathematical model (Eq. (7)), enable us to determine coefficient B and then the coefficient of total exchange H. Thereafter, we determine the convective exchange coefficient hvfpi as well as the Nusselt number Nu according to the Reynolds number. Since the Prandtl number for the air varies very little ( 0.7), we only need to consider two ratios without dimension: the Nusselt number (hvfpiec/k) and the Reynolds number (vfsec/l), where the height of the rack ec is constant (15 cm). We propose the curve of Fig. 5 which gives the Nusselt number according to the Reynolds number. We then deduce the Nusselt number Nu ¼ 0:165Re0:9 Pr1=3 for an airflow along the dryer in the presence of grills. In Fig. 5, the results obtained for the first rack are represented, and we obtain exactly the same result for the other racks. This form of the Nusselt number is used in the mathematical model for the representation of the theoretical air temperature profiles in Fig. 4. In the algorithm
Fig. 4. Theoretical and experimental evolution of the air temperature in the dryer (provided only with grills) according to the air mass flow rate.
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Fig. 5. Evolution of the average Nusselt number according to the Reynolds number (dryer provided only with grills).
which allows the calculation of the theoretical values of these temperatures, we introduce the air temperature at the entry of the dryer Tfs obtained by the equation of the smoothed curve of the experimental points as input data: Tfs ¼ 0:0004G 2 0:1996G þ 56:065 ðR2 ¼ 0:9893Þ; v
where Tfs is in C and G in kg/h.The wind velocity then equals 1 m/s. We proceed in the same way to determine the Nusselt number by introducing thin layers of dry wood slices on the grills. We note that the curves keep the same pace (Fig. 6); on the contrary, the variation in temperature is much more significant than in the preceding case (dryer provided only with grills). Under the same conditions, the calorific losses through the sidewalls of the dryer are then more significant, because the presence of the slices causes more turbulence in the airflow.
Fig. 6. Theoretical and experimental evolution of the air temperature in the dryer (provided with grills and thin layers of dry wood slices) according to the air mass flow rate.
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Fig. 7. Evolution of the average Nusselt number according to the Reynolds number (dryer provided with grills and thin layers of dry wood slices).
Obviously, the values of the Nusselt number are must more significant. Represented in Fig. 7 according to the Reynold number, the Nusselt number is written as: Nu ¼ 0:65Re0:9 Pr1=3 . We note that coefficient ‘‘a’’ passes from 0.165 to 0.65, when one introduces thin layers of dry wood slices. The theoretical values of the temperatures of Fig. 6 are calculated by using the air temperature at the entry of the dryer Tfs obtained by the equation of the smoothed curve of the experimental points as input data: Tfs ¼ 0:00009G 2 0:0992G þ 57:324 ðR2 ¼ 0:9931Þ v
where Tfs is in C and G in kg/h. The use of thin layers of dried potato slices spread on the grills gives practically the same experimental curves of the air temperature profiles (Fig. 8) as in the case
Fig. 8. Theoretical and experimental evolution of the air temperature in the dryer (provided with grills and thin layers of dried potato slices) according to the air mass flow.
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of the dry wood slices. If the theoretical profiles are represented by taking into account the air temperature at the entry of the dryer Tfs ¼ 56:077e0:0017G (Tfs in v C and G in kg/h) and for a Nusselt number Nu ¼ 0:65Re0:9 Pr1=3 , one can see that the curves are close to the experimental results. Thus, the effect of the slices of dry wood and dried potato on the turbulence of the airflow is almost the same. The majority of the authors do not take into account the calorific losses through the walls of the dryers. In order to underline their influence, we represent in Fig. 9a–c the theoretical and experimental drying kinetics of potato (of initial total mass 6.4 kg) on the first three grills of the drying chamber and cut into 3 mm thick v slices. The conditions of the drying air are: G ¼ 122 kg/h, Tfs ¼ 50 C and us ¼ 20:6%. In the first figure (Fig. 9a), the model does not take into account the losses through the sidewalls of the dryer between the beds of slices but only those in the beds by using the Nusselt number Nu ¼ 0:66Re0:5 Pr1=3 for the calculation of the convective exchange coefficient hvfpl [8]. On the contrary, the mathematical model takes into account those in the beds and between the beds of slices by using the Nusselt number Nu ¼ 0:165Re0:9 Pr1=3 (Fig. 9b) and Nu ¼ 0:65Re0:9 Pr1=3 (Fig. 9c), allowing the calculation of the convective exchange coefficient hvfpi. We note a conspicuous improvement of the theoretical curves which move close to the experimental curves in particular for the two last racks, and when one uses the Nusselt number which is calculated when the air is disturbed by the presence of pierced grills and dry wood slices (Fig. 9c). Table 1 groups together the experimental and theoretical results of drying time of the product in the first three racks. It compares the theoretical results with the experimental values: – 1st model does not take into account the losses through the walls of the dryer between the two beds of slices, – 2nd model takes into account the losses through the dryer walls between and in the beds of slices with a Nusselt number Nu ¼ 0:165Re0:9 Pr1=3 , – 3rd model takes into account the losses through the walls of the dryer between and in the beds of slices with a Nusselt number Nu ¼ 0:65Re0:9 Pr1=3 . The drying time, for each rack, represents the moment reached by hygroscopic equilibrium between the air and the product, that is to say a value of 0.05 (kg water/ kg db) of the equilibrium product’s moisture content calculated according to the v conditions of the drying air (Tfs ¼ 50 C and us ¼ 20:6%), in the case of potato [4]. For the three racks, the average relative variation concerning the drying time, which is calculated from the experimental and theoretical results of the first model, varies between 3.85% and 11.45%, whereas for the second model, the values are lower (between 2.60% and 4.17%) and much lower for the third model (between 0.52% and 4.29%). Concerning the product of the last rack in particular and when we improve the model, on the one hand by taking into account the calorific losses along the walls and on the other, by improving the Nusselt number which determines the convective exchange coefficient between the air and the walls of the
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Fig. 9. Comparison between the theoretical and experimental drying kinetics of potato in the first three v racks of the dryer (G ¼ 122 kg/h, Tfs ¼ 50 C and us ¼ 20:6%). (a) The model does not take into account the calorific losses between two beds, with Nu ¼ 0:66Re0:5 Pr1=3 . (b) The model takes into account the calorific losses between and in the beds of slices, with Nu ¼ 0:165Re0:9 Pr1=3 . (c) The model takes into account the calorific losses between and in the beds of slices, with Nu ¼ 0:65Re0:9 Pr1=3 .
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Table 1 v Comparison between the experimental and the theoretical drying times (G ¼ 122 kg/h, Tfs ¼ 50 C and us ¼ 20:6%) Rack
1 2 3
Time in min
Theoretical time in min
Experimental
1st model
Dt/t (%)
2nd model
Dt/t (%) 3rd model
Dt/t (%)
720 780 960
650 750 850
9.72 3.85 11.46
690 805 935
4.17 3.11 2.60
4.17 4.29 0.52
690 815 955
dryer, one can notice that the theoretical drying time is quasi-close to that calculated in the experiment.
5. Conclusion We have proposed a form of the Nusselt number for the calculation of the average coefficient of forced convective exchange between the air and the interior walls of a dryer and by taking into account the existence of the pierced grills as well as a thin layer of dry wood slices on these grills in the absence of mass transfer. These grills cause the disturbance of the airflow and the creation of turbulence, which increases the convective exchanges by increasing the calorific losses outward. A good estimate of these losses by using the most appropriate Nusselt number improves the theoretical model of drying kinetics. Indeed, a comparative study between two theoretical models—those that take into account the losses through the walls of the dryer between two beds of slices and those that do not—underline the improvement of theoretical drying times. By comparing the latter with the experimental results, one notes, in particular for the last rack, that the relative variation decreases from 11.46% to 0.52%, which is not negligible. We have given an experimental approach of the Nusselt number Nu ¼ 0:165Re0:9 Pr1=3 when the airflow is disturbed by the presence of pierced grills. This Nusselt number form can be generalized for air in forced turbulent flow inside the channels of parallelepipedic form, with established profiles of temperature and velocity and for a Reynolds number (17 < Re < 1500). Another form of the Nusselt number Nu ¼ 0:65Re0:9 Pr1=3 can be used in the theoretical models, which determines the drying kinetics of the vertical dryers with pierced grills in the same conditions.
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[4] McLaughlin CP, Magee TRA. The Determination of Sorption Isotherm and the Isosteric Heats of Sorption for Potatoes. J Food Eng 1998;35(3):267–80. [5] Recknagel, Spenger, Ho¨nmann. Manuel pratique du ge´nie climatique, 2. Pyc-Edition; 1986. [6] Sacadura JF. Initiation aux transferts thermiques, Cast INSA of Lyon. Technique and Documentation: Paris. 1980. [7] Sembiring M. Contribution a` l’e´tude des performances technico-e´conomiques d’un se´choir pour fruits et le´gumes a` chauffages partiellement solaire. PhD Thesis. University of Poitiers. France. 1990. [8] Youcef-Ali S, Moummi N, Desmons JY, Abene A, Messaoudi H, Le Ray M. Numerical and experimental study of dryer in forced convection. Int J Energy Res 2001;25(6):537–53.