## Theory of microwave dehumidification

Van der Waals force (or energy sites) attract water molecules onto the surface of sold desiccant material during dehumidification (Fig. 1a). Adsorbed water can be removed by pressure swing 21,22 and thermal swing 9,23. It is the most energy-consuming process in dehumidification 24. On the other hand, attraction forces have electrostatic behavior, so oscillating dipole-structured water molecules with electromagnetic waves (microwaves) could detach from the surface faster and with less energy compared to the above-mentioned methods. Microwave-assisted desorption is an emerging method (Fig. 1b), where two desorption mechanisms are applied: the direct microwave effect on molecules (selective energy transport) and the thermal microwave effect18. It does not require heating of purged air stream to transport energy as in thermal swing; instead, the energy is transported directly to the water molecule 18. Microwaves are electromagnetic waves ranging from about 1 m to 0.001 m (with frequencies between 0.3 GHz and 300 GHz) 25, and like all electromagnetic waves, it obeys Maxwell's equation systems. The time-harmonic electromagnetic field can be represented by the following differential equation that is obtained from Maxwell equation systems by applying a frequency-domain approach:

$$\nabla \times \left(\nabla \times \overrightarrow{E}\right)-{k}_{0}^{2}\left({\epsilon }_{r,eff}^{ }\right)\overrightarrow{E}=0,$$

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where \(\nabla\) (nabla) is a vector differential operator, \(\overrightarrow{E}({E}_{x},{E}_{y},{E}_{z})\) is the vector field of an electric field, \({k}_{0}\) is wavenumber. \({\epsilon }_{r,eff}^{ }\) is effective complex permittivity, and it has real and complex components, as shown by the following equation:

$${\epsilon }_{r,eff}^{ }={\epsilon {\prime }}_{r,eff}^{ }+i{ \epsilon "}_{r,eff}^{ },$$

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where \({\epsilon }_{r,eff}^{"}\) is the real part of effective complex permittivity (dielectric constant), \({\epsilon }_{r,eff}^{"}\) is the imaginary part of effective complex permittivity (dielectric loss factor). In simulations, averaged microwave power consumed by the desiccant wheel is calculated according to the Poynting equation: \({P}_{mw}=V\pi f{\epsilon }_{0}^{ }{\epsilon }_{r,eff}^{"}{E}^{2}\), where *P**mw* is the microwave power, *V* is the desiccant wheel's volume, *f* is the microwave's frequency, \({\epsilon }_{0}^{ }\) is the free space permittivity. Another important parameter is the time-averaged vector field (\(\overrightarrow{S}\)), which showed the power flow and microwave direction. \(\overrightarrow{S}=\overrightarrow{E}\times \overrightarrow{{H}^{*}}\), where \(\overrightarrow{{H}^{*}}\) is the vector field of the magnetic field and complex conjugate. Silica gel was chosen among adsorbents and coated on a cellulose-based honeycomb structured wheel to achieve a high surface area per unit volume of the wheel. These materials (silica gel and cellulose) are almost transparent to microwave radiation; hence, microwave energy is solely focused on ejecting the water molecules from the pores of the adsorbent.

## Effective complex permittivity and penetration depth

The honeycomb-based adsorbent wheel permits airflow through its channeled voids. For accurate modeling, it is necessary to obtain the effective complex permittivity of the honeycomb wheel, which is a function of complex permittivity of air and desiccant materials (silica gel, binder, cellulose), that is 26:

$${\epsilon }_{r,eff}^{ }={f}_{op}{\epsilon }_{r,air}^{ }+{{f}_{cd} \epsilon }_{r,cd}^{ }={f}_{op}\left({{\epsilon }^{{\prime }}}_{r,air}^{ }\right)+{f}_{cd}({{\epsilon }^{{\prime }}}_{r,cd}^{ }+i{ \epsilon "}_{r,cd}^{ })$$

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where *f**op* is the volume fraction of air in the openings (honeycomb) and *f**cd* *=1-f**op* is the volume fraction of composite desiccant. The penetration depth of microwaves is also calculated with effective complex permittivity by the following formula 18:

$${D}_{p}=\frac{{\lambda }_{0}}{2\pi \sqrt{\left(2{\epsilon {\prime }}_{r,eff}^{ }\right)}}\frac{1}{\sqrt{\left(\sqrt{1+{\left(\frac{{\epsilon {\prime }{\prime }}_{r,eff}}{{\epsilon {\prime }}_{r},eff}\right)}^{2}}-1\right)}}$$

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## Multi-objective optimization of the microwave chamber

Optimization was conducted, minimizing the reflected microwave power and unheated areas to increase heating performance within the honeycomb wheel. Three multi-objective optimizations with the weighted sum of objectives were carried out as three shapes were considered, namely, (i) the first is a rectangular block chamber with a pyramidal horn-shaped side, (ii) the second is a rectangular block, and (iii) a cylindrical chamber. The control variables were denoted by a,b,c,d and displayed in Fig. 6a. Global optimum value was obtained with random initial control values within the constraining range. Moreover, the optimization was constrained by the chamber's geometry, with wheel dimensions kept constant at a radius of 0.224 m and height of 0.4 m (Fig. 6b). For mathematical modeling, the following assumptions were used: 1) The complex permittivity and the effective complex permittivity of honeycomb material are homogeneous and isotropic; 2) The perforated metal sheet was assumed to have the same reflective characteristics as the non-perforated one due to the much smaller perforation hole diameter (4 mm) than the wavelength of microwaves (124 mm). For the design of the waveguide and chamber, Eq. (1) was solved to obtain the electric field (V/m) subjected to the boundary conditions. At the entrance of the waveguide (from the magnetron), the electric field of the x-direction is designed according to Eq. (7) whilst the corresponding values in the y and z directions are zero 27:

\({E}_{x}=\sqrt{{P}_{in}/{P}_{mode}} \text{*}sin\left(\right(w-y)/w\text{*}pi)/w,\) \({E}_{y}=0\), \({E}_{z}=0.\) (7)

Equation (7) would satisfy the microwave irradiation classified as the TE10 mode under the industrial standard waveguide (WR340) at a frequency (*f*) of 2.45 GHz. The assumption of the perfect electrical conductor was applied for all walls and perforated sheets, where tangential components of the electric field were set equal to zero:

$$\overrightarrow{n}\times \overrightarrow{E}=0$$

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.

The computation region consists of two domains (Fig. 6b) because air (gray) and the desiccant wheel (red) have different effective complex permittivity. The Nelder-Mead algorithm was used for optimization calculations. Nelder–Mead algorithm is a nonlinear optimization method that uses the simplex concept. At each iteration, a new vertex is defined by the four operations known as reflection, expansion, contraction, and shrinkage. The value of the objective function at n + 1 vertex of a simplex is calculated as it is moved toward the minimum point28. Two objective functions were defined such as reflected power ratio and low electric field ratio:

\({f}_{p}={w}_{c}*{P}_{refl}/{P}_{in}\) and \({f}_{low}={\int }_{Vhmpz}^{ }{E}_{low}dV/{V}_{hmpz}\) (9)

where the weighting coefficient (\({w}_{c}\)) was equal to 5 as usually 20% of microwave power was reflected back.

$${E}_{low}=\left\{\begin{array}{c}1, if\left({E}_{norm}<{E}_{threshold}\right);\\ 0, \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e},\end{array}\right.$$

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where \({E}_{threshold}\) was equal to 3000 V/m, which was discovered from the authors' waveguide experiment. A low electric field ratio is needed to maintain uniform heating by microwaves. Control variables and their bounds for all cases are presented below:

Case-1: 0.5[m] > h > 0.005[m]; Case-2: 0.2[m] > a > 0, 0.25[m] > b > 0; Case-3: 0.2[m] > c > 0.

COMSOL Multiphysics computational platform was used to perform optimization. The system of equations was solved with FGMRES Iterative Solver, which uses the restarted flexible generalized minimum residual method. Mesh built of (minimum) 2251507 tetrahedral domain elements and 73048 triangular boundary elements.

## Experimental apparatus

Desiccant (silica gel) captures moisture from the air (Fig. 1a). Then, the moisture in the desiccant is desorbed by microwaves (Fig. 1b). The key feature of microwaves is that they can oscillate water molecules and desorb from the adsorbent's surface (silica gel). The lab-scale pilot microwave dehumidification system is illustrated in Fig. 7a, and its schematic diagram is illustrated in Fig. 7b. A microwave generator (Fricke und Mallah, Germany) was used to generate the microwaves. Frame and equipment were grounded with protective grounding to prevent users from high voltage electrical hazards. Two modes were considered: the mode without heat recovering and the mode with heat recovering from outlet air. Temperatures and differential pressure readings were logged continuously by software Labview and Agilent 34970A for both modes. The desiccant wheel rotating motor speed was set to the desired value, running only during desorption. Figure 7b demonstrates a setup diagram. The study performed the following procedure without heat recovery: Air damper-1 and air damper-3 were opened, and air damper-2 and air damper-4 were closed, letting the air bypass the heat-recovering device. Then, the honeycomb-structured desiccant wheel was saturated with moisture at constant relative humidity and temperature at a regular airflow rate until the inlet and outlet temperatures were the same: the same temperature and humidity indicated equilibrium conditions. Consequently, microwaves were switched on for the preset time and preset power from the control panel; the desorption process finishes when the outlet humidity ratio becomes lower than the inlet humidity ratio. Mode with heat recovery is similar to mode without heat recovery; When the inlet and outlet temperatures became the same, air damper-1 and air damper-3 were closed, and air damper-2 and air damper-4 were opened to recover heat from outlet air.