Defining a procedure for predicting the duration of the approximately isothermal segments within the proposed drying regime as a function of the drying air parameters

One of the main disadvantages of the recently reported method, for setting up the drying regime based on the theory of moisture migration during drying, lies in a fact that it is based on a large number of isothermal experiments. In addition each isothermal experiment requires the use of different drying air parameters. The main goal of this paper was to find a way how to reduce the number of isothermal experiments without affecting the quality of the previously proposed calculation method. The first task was to define the lower and upper inputs as well as the output of the “black box” which will be used in the Box-Wilkinson’s orthogonal multi-factorial experimental design. Three inputs (drying air temperature, humidity and velocity) were used within the experimental design. The output parameter of the model represents the time interval between any two chosen characteristic points presented on the Deff – t. The second task was to calculate the output parameter for each planed experiments. The final output of the model is the equation which can predict the time interval between any two chosen characteristic points as a function of the drying air parameters. This equation is valid for any value of the drying air parameters which are within the defined area designated with lower and upper limiting values.


Introduction
Even though drying of porous material has been investigated for decades, it is still a current and actual topic of researchers in many scientific areas, e.g. chemical engineering, civil engineering and soil science. Physics and engineering have provided basics principles, with which this aspect of science can be additionally examined and discussed. A comprehensive understanding of the way in which water is transported from within the porous medium up to its surface, during drying can lead to many technical innovation and energy savings. In order to properly solve heat and mass transfer problems both the transport in air and in the porous material has to be modeled. This can be achieved at different complexity levels in both media. Calculation techniques (models) which are commonly used can be classified into four major groups: diffusion [1][2][3], receding front [4,5], macroscopic continuum models for coupled multiphase heat and mass transport in porous materials [6,7] and pore network models [8]. The conjugate modeling degree in each drying model can be determined by the way in which the heat and mass transport in air are accounted for in the calculation procedure.
The procedure for setting up the non isothermal drying regime, that is consistent with the theory of moisture migration during drying, was recently reported [9]. In order to properly apply this procedure it is necessary to firstly determine the change of effective moisture diffusivity vs. moisture content or drying time (Deff -MR or Deff -t curve) for each isothermal experiment, since these plots represents a good indicator for evaluation and presentation of the overall mass transport property of moisture during isothermal drying. In other words all possible mechanisms of moisture transport and their transition from one to another during isothermal drying, within a clay roofing tile, are visible on previously mentioned plots. Detailed information regarding the procedure for identification and quantification of moisture transport and their transition during drying can be found in reference [10]. Optimal drying regime is consisted of five isothermal segments. Durations of previously mentioned drying segments were detected from the relevant Deff -MR curves.
One of the main disadvantages of the reported method, for setting up the drying regime based on the theory of moisture migration during drying, lies in a fact that it is based on a large number of isothermal experiments. In addition each isothermal experiment requires the use of different drying air parameters. The main objective of this study was to find a way how to reduce the number of isothermal experiments without affecting the quality of the previously proposed calculation method. In order to complete this task and to find a mathematical equation, which can predict the time interval between any two chosen characteristic points (duration of each characteristic drying segments) as a function of the drying air parameters, within the defined area designated with lower and upper limiting values of input parameters, the Box-Wilkinson's orthogonal multi-factorial experimental design was used.

Materials and Methods
The raw material, used in this study, was obtained from the roofing tile producer "Potisije Kanjiža".
Its detail characterization was reported in the study [11]. The raw material was first dried at 60 0 C and then crashed down in a laboratory perforated rolls mill. After that simultaneously it was moisturized and milled in a laboratory differential mill, first with a gap of 3 mm and then of 1 mm. Laboratory roofing tile samples 120 × 50 × 14 mm were formed, from the previously prepared clay, in a laboratory extruder "Hendle" type 4, under a vacuum of 0.8 bar. Drying experiments were performed on previously formed roofing tile samples in laboratory recirculation dryer. The mass of the samples and their linear shrinkage were continually monitored and recorded during drying. The accuracies of these measurements were 0.01 g and 0.2 mm. Drying air parameters were regulated inside the dryer with accuracies of ±0.2 °C, ±0.2 % and ±0.1 % for temperature, humidity and velocity, respectively.
The response function in the Box-Wilkinson's orthogonal multi-factorial experimental design is presented in a form of the equation (1).
This equation corresponds to a surface in a multidimensional space, called the response surface. The space in which the response surface exists is called the factorial space. In the general case, when k factors are covered, equation (1) describes the response surface in k +1 measurement space. This function is usually defined as a polynomial expression (2).
The methodology, valid for isothermal experiments, presented in the [10] was used to calculate the functional dependence of the effective diffusivity with moisture content (Deff -MR), to divide obtained curves in segments and to identify all possible mechanisms of moisture transport. Obtained data were analyzed and used to predict the response y (duration of the proposed non isothermal drying segments).
Parameters x1, x2..., xi represents the independent variables or factors (drying air velocity, temperature and humidity). The parameters b0, bi, bij represent the regression coefficients. When the regression coefficients are determined and the dependence defined by equation (2) is established the resulting equation is called a mathematical model. The adequacy of experiment reproducibility is checked using the Kohren criteria [12], while the adequacy of the model is checked using the Fisher Experimental conditions presented at table 1 were used in the present study. Each experiment was repeated 2 times. Drying air parameters which were maintained in each proposed non-isothermal drying regimes are presented in table 2.

Results and discussion
Drying segments along with mechanisms that can take place in them according to the reference [10] are summarized at table 3. All possible mechanisms of moisture transport and their transition from one to another during the constant and the falling drying period up to the "lower critical" point F, for isothermal experiments, were identified on the corresponding Deff -MR curves and are summarized in table 4. The procedure for setting up the non isothermal drying regime, that is consistent with the theory of moisture migration during drying (see table 2) was based on the principle of controlling the mass transport during the drying process and has demanded to divide the drying process into 5 segments. In each of these segments approximately isothermal drying conditions were maintained.
The main functions of the first drying segment was to restrain the moisture transport (evaporation), through the boundary layer between material surface and the bulk air, and to heat the ceramic body to the temperature of the drying air. During the second drying segment external (surface evaporation) and internal transport (of liquid water from the ceramic body up to the surface) should be increased and simultaneously harmonized in such way that the drying surface remains fully covered by a water film. The main function of the third segment is to provide the conditions that will lead to the fact that partially wet surfaces provide a constant rate of drying. Within the fourth drying segment, the liquid transport originating from the pores which are near or just below the "dry" patches on the surface, and are still in the funicular state, has to be simultaneously harmonized with the liquid flow originating from the surface "wet" patches.
Duration of the first drying segment was equal to the time interval detected in the corresponding isothermal experiment, from the beginning up to the characteristic points C. Duration of the second drying segment was equal to the time interval detected in the corresponding isothermal experiment, between the characteristic points C and D. Duration of the third drying segment was equal to the time interval detected in the corresponding isothermal experiment, between the characteristic points D and E. Duration of the fourth drying segment was equal to the time interval detected in the corresponding isothermal experiment, between the characteristic points E and F. Duration of the fifth segment was limited to 90 minutes. Table 3. Possible drying mechanisms according to reference [10].  (3) and is presented as an example. The data necessary for model valuation are presented in table 6. It can be seen that the evaluated models coefficients are very accurate and precise.