Ind. Eng. Chem. Res. 2008, 47, 133-144
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Optimal Operating Conditions of Microwave-Convective Drying of a Porous Medium Patrick Salagnac,* Patrick Dutournie´ , and Patrick Glouannec Laboratoire d’Etudes Thermiques, Energe´ tiques et EnVironnement (LET2E), UniVersite´ de Bretagne-Sud, Centre de Recherche, B.P. 92116, 56321 Lorient Cedex, France
The objective of this study is to search for the optimal operating conditions needed to dry a porous medium via combined convective-microwave supplies. Design of experiments (DOE) methodology, based on numerical results, was used to obtain a behavioral reduced model that allows one to calculate the operating optimums quickly. Four specifications are studied: the drying time, the maximum of overpressure in the material, the energy balances of the process, and the material. The effects of all drying parameters (initial moisture, microwave power, air temperature, velocity, and humidity) are studied and presented in the form of response surfaces. From these results, a multiobjective optimization was performed and the Pareto optimal set of efficient solutions was evaluated. The methodology was validated by comparison with the experimental results. 1. Introduction Drying is an important step in the manufacturing of many products. However, it is very costly, in regard to both energy consumption and required manufacturing time. In this context, optimization of the thermal conditions for drying a porous medium is an important field of research and development in industry. Many studies that have been mainly focused on reducing capital and energy requirements and improving product quality can be found in the literature. For example, Dostie1 compared the drying performance of different processes using infrared, radio frequency, and convection heating. Chua et al.2 presented an overview of the intermittent drying of bioproducts and discussed the quality of dried bioproducts. Trelea et al.3 used a predictive optimal control algorithm for on-line control of a corn batch drying process, to minimize time and energy consumption. Hugget et al.4 presented a methodology to optimize a dryer globally, using neural networks and genetic algorithms. Other works can also be found on this topic in the scientific literature.5-9 Currently, there are many works in the literature that take product quality into account.9 This paper examines the microwave-convective drying of porous media. The microwave technologies are particularly advantageous for a drying operation. They offer several advantages, with regard to conventional drying, such as noncontact, volumetric, immediate, and significant energy input to the product. The high power density applied to the product can significantly decrease the drying time. However, microwaveconvective drying is an operation that can damage the material (due to important overpressures and thermal gradient in the sample). Many articles have presented work on combined microwave and convective drying. For example, Turner and Jolly10 used combined microwave and convection to dry a nonhydroscopic porous material (brick). McMinn et al.11 applied microwave-convective heating to dry pharmaceutical powders, and Sangra et al.12 presented a simulation of convectionmicrowave drying for a shrinking material. * To whom correspondence should be addressed. Tel.: (33) 02 97 87 45 16. Fax: (33) 02 97 87 45 00. E-mail address: patrick.salagnac@ univ-ubs.fr.
In our laboratory, we have been conducting studies on combined drying for many years. A drying pilot13 has been set up to conduct experiments that combine convection and radiant heating (microwave or infrared radiation). In parallel, a numerical program has been developed to simulate the thermal and hygroscopic behaviors of the material in our facility and was previously validated with experimental results.14 This work proposes a methodology based on experimental designs using simulated results to minimize energy resources and drying times (the scientific process is presented in Figure 1). It allows for the development of a behavioral model that estimates the respective influence of the parameters. From this model, it is possible to determine the optimal operating conditions easily. Actually, the design of experiments15 (DOE) methodology accelerates the learning of the inter-relationships of the process variables, determines what variables are critical to the process, and determines at what level these variables have a significant influence on the operation. In the first part of this paper, the experimental dryer and the numerical model are presented. Here, the number of operating parameters (air temperature, velocity, initial moisture, microwave energies, air humidity, etc.) is significant. Indeed, knowing their respective influences on the dryer performances gives important insight into the optimal conditions of work. The use of response surface methodology in optimization drying problems seems to be very useful when the optimization of a physical model is difficult and very time-consuming. The effects of all the parameters have been studied on the drying time, the maximum overpressure in the sample, the total energy used in the process, and the energy required to extract water from the product. A range of optimal operating conditions then is estimated, and some experimental tests have been performed on the experimental dryer. These tests are compared with DOE results and validate the used methodology. 2. Description of the Problem and Method 2.1. Experimental Section. Figures 2 and 3 show the experimental device. The experimental chamber is made of nonmagnetic stainless steel. The volume of the drying chamber is 38 dm3. A microwave applicator and an air-heating loop are coupled with this chamber (see Figure 2). The sample is placed
10.1021/ie070738q CCC: $40.75 © 2008 American Chemical Society Published on Web 11/29/2007
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Figure 1. Schematic of the global process.
Figure 2. Schematic of the drying chamber.
in a parallelepiped-shaped Teflon crucible that is covered with an aluminum adhesive. The microwave radiations penetrate only through the upper side of the material, the air flux on which is funneled through a rectangular casing that is composed of a microwave-repellent material; the air temperature and velocity are controlled and recorded. Moreover, both the lateral and bottom faces of the crucible are insulated. This device allows one to approach adiabatic and impermeable conditions for surfaces that are not in contact with the drying air. The sample holder is placed on a tripod, itself resting on scales, which allows for constant monitoring of the product mass. The microwave generator operates at 2.45 GHz and has a nominal power of 1.2 kW. Electromagnetic radiation is spread through a guide that is equipped with radiating slits and situated on the backside of the chamber. This combination of chamber and guide results in a multimode applicator. This drying pilot allows one to monitor the external solicitations applied to the product and to follow the evolution of its temperature and mass. During the tests, the temperatures within the material are recorded by optical fibers. 2.2. Mathematical Model. A one-dimensional model was developed to simulate the hygrothermal behavior of a parallelepiped porous medium. The model was initially developed to study performances of combined drying (hot air, microwaves,
infrared). For this, a simple geometry and a homogeneous material were selected. Mass and heat transfers are assumed to be one-dimensional, and the material is assumed to be rigid and nonsaturated, with hygroscopic and capillary behavior. Figure 4 presents the boundary conditions applied to the product. The mathematical model for the heat and mass transfers is the generalization of the Whitaker approach,16 which was improved by Turner et al.,10 Moyne and Perre´,17 and Perre´ and Moyne.18 The model and the resolution are presented in the reported work of Salagnac et al.14 The used variables are temperature (T), mass moisture content (W), and the total pressure of the gaseous phase (Pg). The microwave input was calculated using the LambertBeer law.19 2.2.1. Governing Equations. The hygrothermal behavior of the material is described from mass and energy balances (dry air (denoted by “a”), vapor (denoted by “V”), and liquid (denoted by “l”)), in addition to the diffusion laws (Fick) and Darcy’s generalized equations, giving the mean filtration velocity fields of the liquid and gaseous phases. Capillary pressure (Pc) is calculated with the sorption isotherms (Kelvin’s relation). The effects of body forces and convection in the material are neglected. The liquid phase (free water and bound water) is supposed indissociable. Introducing a differential heat of sorption in the energy conservation equation takes into account the necessary energy input of the vaporization of bound water. Moreover, one assumes that the medium is homogeneous, the local thermodynamic equilibrium is achieved, the gaseous phase consists of a perfect gas, and there is no chemical reaction within the material. From the previous equations and assumptions, we obtain a set of three coupled and nonlinear partial differential equations (eqs 1, 2, and 3), which are functions of the three state variables (W, T, Pg). The moisture content balance is given as
Ind. Eng. Chem. Res., Vol. 47, No. 1, 2008 135
{[
∂T ∂W ∂ 1 W ∂W (DT + DTv ) + + (DW + l + Dv ) ∂t ∂x Fso l ∂x ∂x ∂Pg (DPl + DPv ) ∂x
]}
t t PMW,δx ) PMW,x - PMW,x+δ )
PtMW exp(-κx)[1 - exp(-κδx)] (11) ) 0 (1)
The equation for the total pressure of the gaseous phase (Pg) is given as
() ( ) ( ) (
∂Pg ∂T ∂W γ1 + γ2 + γ3 ) ∂t ∂t ∂t ∂Pg ∂T ∂W ∂ + DW + DPa DTa (2) a ∂x ∂x ∂x ∂x
)
The heat balance is expressed as
FCp
∂T ∂Τ ∂ -λ + K(∆Hv + ∆Hb) ) pMW + ∂t ∂x ∂x
(
)
(3)
where the phase change rate (K) is given by the relation
K)
(
)
∂Pg ∂T ∂W ∂ DTv + DW + DPv v ∂x ∂x ∂x ∂x
(4)
The phenomenological coefficients (D) are the diffusion coefficients of the liquid, vapor, and dry air, relative to temperature, mass moisture content, and pressure gradients. These different coefficients and the coefficients γ are described in the reported work of Salagnac et al.14 2.2.2. Boundary Conditions. The boundary conditions (depicted in Figure 4) are the same as those of the experimental survey. At the top (x ) 0), the pressure is considered to be equal to atmospheric pressure. The continuity of the heat flux and the mass flux are given by
-λ
∂T(0,t) h l(∆Hv + ∆Hb) ) + Fl V ∂x hc(T∞ - T(0,t)) + hr(Tw - T(0,t)) (5) h v + FlV h l) ) Fm -(FjgvV
(6)
Therefore, the power absorbed by the material is given as
PMW )
PMW,δx ∑ δx
(12)
The values of the reflection coefficient (R) and attenuation coefficient (κ) are dependent on the dielectric properties (dielectric permittivity and the relative loss factor).19 2.3. Operating Parameters. The operating conditions of the study are presented in Figure 3. The air is introduced at a temperature Text with a relative humidity Hr. This fresh air is mixed with the air being recycled and heated at the desired temperature T∞, which is then introduced in the drying chamber with velocity V and air humidity r. The material has an initial dry basis moisture fraction W0 and an initial temperature that is fixed at 30 °C. It is dried by convection and receives energy up to the core of the material, which is induced by microwaves. A part of the outflow air is recycled. The humidity of the air is almost constant at the inlet of the drying chamber. 2.4. Material Characteristics. The material for this study is an autoclaved cellular concrete. The samples are 10 cm × 10 cm sections with a thickness of 2.5 cm. Cellular concrete was chosen because of its industrial production, which guarantees reproducible physical characteristics. The hygroscopic material porosity is 0.8. The sorption isotherms are presented in the reported work of Bellini.21 The specific heat at constant pressure, thermal conductivity, and density of the dry product are 840 J kg-1 K-1, 0.15 W m-1 K-1 and 450 kg/m3, respectively. The dielectric permittivity (′r) and the relative loss factor (′′r) are closely related to the moisture content:22
′r ) 0.927 + 21.61W - 10.605W2 + 14.981W3 ′′r ) 0.124 + 0.802W
(13a) (13b)
3. Design of Experiments (DOE) Methodology
with
( )(
F m ) km
PatmMv Patm - Pv∞ ln RTf Patm - Pv(0,t)
)
(7)
The mass coefficient (km) is determined using the Colburn analogy.20 The bottom side of the sample (x ) e) is adiabatic and impermeable:
h g ) 0 and FlV hl ) 0 FjggV
(8)
∂T h l(∆Hv + ∆Hb) ) 0 + Fl V ∂x
(9)
-λ
2.2.3. Source Term. To calculate the microwave power transmitted to the material according to the incident microwave power, the Lambert-Beer law19 is used; this is given by
PtMW ) (1 - R)PiMW
(10)
Thus, the absorbed power on a thickness δx of material is given by the following equation:
The major advantage of DOE is to plan and conduct experiments to extract the maximum amount of information from the collected data in the smallest number of experimental runs. The basic idea is to change all relevant factors simultaneously over a set of planned experiments and then connect and interpret the results using mathematical models. In our case, the high nonlinearity of the process requires the development of a quadratic model. To estimate this model, a classical full factorial design is not sufficiently effective. Therefore, we have chosen to use a central composite design,15 which consists of the following parts: (i) a full factorial design; (ii) experiments at the center of the design (all variables xi ) 0); and (iii) experiments for xi ) (R, with R g 1. The central composite design is used with the following parameters xi: W0, T∞, r, V, and PtMW. Table 1 shows the values used in the calculations and the standardized levels of the variables. We have selected four response factors to estimate the dryer performances: (1) The drying time (td). This is the time required to attain Wmin (Wmin ) 0.03).
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Figure 3. Schematic of the experimental device; the energy and mass balances of the dryer are indicated.
Figure 4. Schematic depiction of the boundary conditions and spatial discretization. Table 1. Independent Variables: Experimental Design Levels Standardized Levels variables, xi
-R
-1
0
+1
+R
initial moisture, W0 (kg/kg) T∞ (°C) relative humidity, r (× 104 kg water/kg dry air) air velocity, V (m/s)
0.05 20 0
0.2 30 17
0.5 55 52.5
0.8 80 88
0.95 90 105
0.25
1
2.5
4
4.75
(2) The specific moisture extraction rate (SMER). This is the energy required to extract a unit of water from the product (expressed as the ratio between the evaporated mass of water and the energy injected in the material).
(W0 - Wmin)mso SMER ) QCV + QMW
(14)
(
)
QHE QtMW 1 + (W0 - Wmin)mso ηHE ηMW
(17)
QHE ) (1 - τ)m˘ Cp(T∞ - Text)td + τQCV
(18)
and QtMW, which is the microwave energy transmitted to the material, is given as
td
∫ hC S(T∞ - TS)dt
(15)
0
where QMW is given by
QMW )
Q hp )
where QHE is given as
with
QCV )
and PMW is the microwave power absorbed in the material. (3) The maximum of overpressure in the material. This factor, denoted by ∆P, is the pressure difference between the maximum pressure obtained within material and the atmospheric pressure (the latter of which is 101 325 Pa). (4) The normalized energy consumption. Denoted by Q h p, this factor is the total energy used in the process per kilogram of water evaporated, as defined by Figure 3:
∫0
td
PMW dt
(16)
QtMW )
∫0t
d
PtMW dt
(19)
The efficiency of the air heater (ηHE ) 0.3) takes into account the wall heat losses and the energy fan cost. The effective efficiency19 of the microwave generator and multimode chamber
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Figure 5. Drying time versus microwave power for (a) different temperatures (at an air velocity of V ) 2.5 m/s) and (b) different velocities (at T∞ ) 55 °C), with an initial moisture of W0 ) 0.5 kg/kg and a relative humidity of r ) 5.25 × 10-3 kg/kg.
is 0.3. The recycling rate (τ ) 0.6) has been chosen to have an almost-constant air humidity in the process (a humidity increase of
