**2. Theoretical background**

In solid drying processes, momentum, heat, and mass transport phenomena occur simultaneously. The study of these phenomena is important to define aspects of the mechanical and thermal design of a drying equipment. Particularly, in a fluidized bed dryer, the study of the fluid dynamics of the bed and the determination of minimum fluidization velocities and pressure drop in the bed have been the objective of many studies.

In some cases, the problem has been considered as a particle isolated from the rest of the particles; in others, the interaction with other particles is considered (a situation that always occurs in a fluidized bed). It is well known that in fluidized beds with irregular particles, turbulent flow occurs, in which case energy losses due to kinetic effects are more important than losses of viscous origin.

On the other hand, in the thermal design of a fluidized bed dryer, it is important to know the heat and mass transfer coefficients that occur at the particle-gas interface because with them the boundary conditions can be formulated in modeling problems of the drying process. Particularly, during the period of constant drying rate, in which the predominant drying mechanism is convective at the gas-particle interface.

#### **2.1 Transport of momentum in a fluidized bed**

In fluidized systems, it is necessary to perform analysis using constitutive equations, which are added to the momentum balance equations.

In the case of flows in porous matrices with low-velocity flows and low Reynolds numbers, the Darcy equation relates the resistive force between the fluid and the porous matrix with the superficial velocity of the fluid. On the other hand, for fluidized beds of large particles, a situation that occurs in a particulate biomass dryer, the resistant force depends on the viscous losses and the kinetic energy losses.

Therefore, in a fluidized bed, the equation that determines the pressure drops, developed by Ergun [6] establishes that:

$$\frac{\Delta p}{L} = \mathbf{150} \frac{\left(\mathbf{1} - \boldsymbol{\varepsilon}\right)^2}{\boldsymbol{\varepsilon}^3} \frac{\mu\_\mathbf{g} U}{D\_p} + \mathbf{1.75} \frac{\mathbf{1} - \boldsymbol{\varepsilon}}{\boldsymbol{\varepsilon}^3} \frac{GU}{D\_p} \tag{1}$$

In the case of fluidized beds with irregular particles (non-spherical), the Ergun equation must take into account the sphericity of the particle through the relationship 6/*ϕDp*, that represents the specific surface area of the particles *Sp* for arbitrary particles of similar size and expressed as the contact surface of particles per unit volume of solids.

Therefore, in the case of non-spherical particles, Ergun equation can be rewritten in terms of *Sp* as follows:

$$\frac{\Delta p}{L} = 150 \frac{\left(1 - \varepsilon\right)^2}{\varepsilon^3} \frac{\mu\_\text{g} U}{36} \text{S}\_p^2 + 1.75 \frac{1 - \varepsilon}{\varepsilon^3} \frac{GU}{6} \text{S}\_p \tag{2}$$

If the superficial velocity *U* is lower than the minimum fluidization velocity *Umf*, the bed remains in its rest condition and Eq. (2) allows determining the specific surface area *Sp* of the particles contained in the bed. For this, experimental data of the pressure drop as a function of superficial velocity must be available under the condition that *U* < *Umf*.

#### **2.2 Gas-particle convective heat and mass transfer in fluidized bed**

For the analysis of heat transfer between a spherical particle and a gas with relative velocity *U* (spherical liquid drop falling in a gas), in Ranz and Marshall [7] the following dimensionless equation is proposed:

$$Nu\_{\rm gp} = \frac{h\_{\rm gp} D\_p}{k\_{\rm g}} = 2.0 + 0.60 \left(\frac{\rho\_{\rm g} U D\_p}{\mu\_{\rm g}}\right)^{1/2} \left(\frac{C\_{\rm g} \mu\_{\rm g}}{k\_{\rm g}}\right)^{1/3} \tag{3}$$

In the analysis of heat transfer in fluidized beds, particularly at high fluidization velocities and Reynolds numbers, there are dimensionless correlations similar to Eq. (3). For example [8] proposes the following correlation for coarse particles (*Rep* > 100) and in a fixed bed condition:

$$Nu\_{\rm gp} = 2 + \mathbf{1.8} Pr\_{\rm g}^{1/3} Re\_{\rm p}^{1/2} \tag{4}$$

For low values of Reynolds number, this equation presents important deviations in relation to experimental results. However, in such cases, the experimental results correlate well with those predicted by the Kunii and Levenspiel equation, whose fundamental assumption is the existence of a plug flow regime:

$$Nu\_{\rm gp} = 0.03 Re\_p^{1.3} \tag{5}$$

for 0.1 < *Rep* < 100, under the assumption of a plug flow.

On the other hand, and in a similar way to what happens with the phenomenon of heat transfer, the analysis of the mass transfer in a forced convection regime between an isolated spherical particle and a gas in relative motion can be carried out using the experimental correlation of [9]:

$$Sh\_{\rm gp} = \frac{k\_{\rm gp} D\_p}{D\_v} = 2.0 + 0.60 \left(\frac{\rho\_{\rm g} U D\_p}{\mu\_{\rm g}}\right)^{1/2} \left(\frac{\mu\_{\rm g}}{\rho\_{\rm g} D\_v}\right)^{1/3} \tag{6}$$

for 0.6 < *Scg* < 2.7 and 2 < *Rep* < 800.

For fixed beds [7], also reported an expression for the calculation of the Sherwood number for large particles (*Rep* > 80) in liquid and gas systems. When applied to gaseous systems, the equation can be written as:

$$\text{Sh}\_{\text{gp}} = 2.0 + \text{1.8} \,\text{Re}\_{\text{p}}^{1/2} \text{Sc}\_{\text{g}}^{1/3} \tag{7}$$

In analogy with the phenomenon of heat transport, for flow regimes with low Reynolds numbers, Eq. (7) does not adequately predict the value of the Sherwood number and in these cases, it is recommended to use the following equations proposed by Richardson and Szekely [10].

*Experimental Investigation on Drying of Forest Biomass Particles in a Mechanically… DOI: http://dx.doi.org/10.5772/intechopen.113973*

$$\text{Sh}\_{\text{gp}} = 0.374 \,\text{Re}\_p^{1.08} \text{ for } 0.1 < \text{Re}\_p < 15 \tag{8}$$

$$\text{Sb}\_{\text{gp}} = 2.01 \,\text{Re}\_p^{0.5} \text{ for } 15 < \text{Re}\_p < 250 \tag{9}$$
