*1.1. Theoretical Background*

#### 1.1.1. The Freeze-Drying Processes

Similar to other dehydration processes, FD includes simultaneous heat and mass transfer. A simple but conceptual explanation of the phenomenological processes taking place during a typical like-berry-fruit FD operation is described in Figure 1.

**Figure 1.** Like-berry-fruit freeze-drying system and its basic phenomenological description.

A product, initially completely frozen, is placed into a freeze dryer chamber. Then, the chamber pressure is lowered under the water triple point, and heat is supplied by mean of a heated shelf and/or the surrounding air temperature (primary drying or sublimation). As heat flows through the dried layer through heat transfer, sublimation latent heat is delivered, and a sublimation interface recedes leaving, resulting in a thicker porous layer of dried material, which acts as a resistance to heat transfer, as well as water vapor flow, toward the product surface through mass transfer, and finally, to the condenser trap, where water is separated before gases are expelled by the vacuum pump. Both heat and the mass transfer rate are the essential causes that make the FD process slow down. In FD, if heat supplied to the sublimation interface is equal to the latent heat associated with the sublimation rate of ice (m × ΔHS = q), the saturated pressure of sublimation (Pi) and its corresponding temperature (Ti) become stable, and the sublimation proceeds normally. However, if the heat supplied to the sublimation interface is not enough, the sublimation rate will drop; conversely, if the total resistance to vapor diffusion is too large (RT), then Pi and Ti may rise. As a result, frozen water will melt and collapse, and/or bust may take place [20,21]. During berry-fruit FD processing, in addition to the dry-layer resistance (Rdl), the skin operates as an important barrier (Rsk) that further extends the

processing times. Additionally, the low skin permeability can generate explosion or bust problems, as vapor cannot escape at the needed rate so as not to provoke a pressure lift just under the skin [5].

#### 1.1.2. Mass Transfer Resistance

The estimation of RT through experimental and theoretical approaches would explain and describe the phenomenological process associated with the berry busting process. The interaction e ffects between the blueberry fruit characteristics, such as mass transfer area (A) the product mass transfer resistance (RT), and the freeze-drying processing conditions, have been studied by using the semiempirical MTM procedure, which was originally developed to assess the temperature of the sublimation interface within the product and the dried-layer mass transfer resistance. The method has a large body of literature information [22–27]. The principle of MTM is based on the flow of water vapor from the product chamber to the condenser being momentarily interrupted during the primary drying time. The MTM is based on a rapid increase in the chamber pressure by closing the freeze-dryer butterfly valve (Figure 2). During this perturbation process, the chamber pressure will rapidly increase due to the continued sublimation of ice. Since the composition of the vapor phase in the chamber is nearly all water vapor, sublimation will stop when the chamber pressure reaches the vapor pressure of ice at the sublimation interface (diving force ΔP = 0); consequently, the pressure rise will cease. The dynamics of this pressure rise process can be theoretically described by equation (1), which is derived from the phenomenological heat and mass balance over a system, defined as the void volume of the product chamber [28]:

$$\mathbf{P(t)} = \mathbf{P}\_{\mathbf{i}} - (\mathbf{P}\_{\mathbf{i}} - \mathbf{P}\_{0}) \mathbf{Exp}\left(\frac{-\text{NART}\_{\mathbf{s}}}{\mathbf{M}\_{\mathbf{H}\oplus\mathbf{0}}\mathbf{VR}\_{\mathbf{T}}}\mathbf{t}\right) + \ 0.0465\mathbf{P}\_{\mathbf{i}}\mathbf{AT} \left[1 - 0.811\mathbf{Exp}\left(\frac{-0.114}{\mathbf{L}}\mathbf{t}\right)\right] + \mathbf{\lambda}\mathbf{t} \tag{1}$$

where P(t) is the system pressure (void volume of the freeze-drying chamber), Pi is the interphase pressure of the product, P0 is the initial pressure of the system, N is the number of fruit units, MH2O is the water molecular weight; V is the void volume of the freeze drier system, R is the universal gas constant, TS is the shelf temperature, A is the mass transfer area per fruit unit, RT is the total area normalized product resistance to mass transfer, ΔT is the temperature gradient across the frozen layer—normally fixed at 1 K, L' is the geometric product characteristic—ice thickness, and X is the linear term in equation (1) associated with external air infiltration. Tang et al. [28] generated simulations of pressure rise curves to analyze the relative contribution to the chamber pressure rise (P(t)), of the many terms in Equation (1), concluding that a resistance-dominated period, which takes place in the first 5 to 10 s of the pressure rise, is mainly expressed by the first exponential term, so the others terms are nearly neglected as part of the summation.

#### **2. Materials and Methods**
