# Drying and Heating Processes in Arbitrarily Shaped Clay Materials Using Lumped Phenomenological Modeling

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. The Physical Problem and Geometry

#### 2.2. Mathematical Modeling

- (a)
- The solid is homogeneous and with constant thermophysical properties;
- (b)
- Moisture content and temperature distributions inside the brick are uniform along the drying process;
- (c)
- Drying process occurs by heat and mass diffusion inside the solid and heat and mass convection and evaporation occur on its surface;
- (d)
- Dimensions of the brick are constant along the drying process.

#### 2.2.1. Mass Conservation Equation

_{1}and S

_{2}are the external and internal surface areas of the homogeneous solid (m²), respectively; $\dot{\mathrm{M}}$ is the generation of moisture (kg/kg/s); ${\mathrm{M}}_{\mathrm{e}}$ is the equilibrium moisture content on a dry basis (kg/kg) and t is the time (s).

#### 2.2.2. Thermal Energy Conservation Equation

#### 2.3. Applications to Drying of Industrial Ceramic Bricks

#### 2.3.1. Volume and Surface Area of the Brick

_{I}is the internal surface area (internal faces determined by the holes) and ${\mathrm{a}}_{\mathrm{h}}$ and ${\mathrm{a}}_{\mathrm{v}}$ are the height and width of a hole, respectively.

_{1}, a

_{2}, a

_{3}and a

_{4}, using the following equations:

#### 2.3.2. Thermophysical Properties of Air and Water

_{a}is the absolute humidity of the air, RH is the relative humidity of the air, ${\overline{\mathrm{MM}}}_{\mathrm{a}}=28.966$ kg/kmol is the molecular weight of the gas, ${\mathrm{R}}_{\mathrm{a}}$ is the universal constant of the gases, T

_{abs}is the absolute temperature in Kelvin and P is the pressure on Pascal.

#### 2.3.3. Estimation of Convective Heat and Mass Transfer Coefficients

#### Natural Convection

**(a) Vertical plate**

**(b) Horizontal plate (Top surface)**

**(c) Horizontal plate (bottom surface)**

**(d) Vertical Plate (Hole)**

**(e) Horizontal plate (Hole)**

_{o}is the absolute humidity of the air at the drying air temperature and x

_{bu}is the absolute humidity of the air at the wet bulb temperature of the drying air, given by the following:

_{wb}is the vapor pressure in the air at the wet bulb temperature of the air (which is equal to the saturation pressure of the water vapor at the wet bulb temperature) and ${\overline{\mathrm{MM}}}_{\mathrm{v}}$ is the molecular weight of the water vapor.

#### Forced Convection

**(a) Laminar flow (Re < 5 × 10**

^{5})**(b) Turbulent flow (5 × 10**

^{5}< Re < 1 × 10^{7})_{1}− 3a

_{3})/4] for the side plates.

#### Combined Natural and Forced Convection

#### 2.3.4. Cases Studied

#### Validation

- Initially, dimensions, mass, brick temperature, ambient temperature and relative humidity were measured.
- Following that, the bricks were placed inside the oven where drying was carried out. In this process, air temperature inside the oven was established at 100°C through the temperature controller.
- At previously established time intervals, the brick was withdrawn from the oven, making it possible to measure its temperature, mass and dimensions. In principle, measurements were made every 10 min for up to 30 min. Then, the measurements were made every 30 min, with the next measurements being made every 60 min until the mass was approximately constant.
- Soon after, the sample was subjected to drying for 24 h to obtain the equilibrium mass, and then for another 24 h, to obtain the dry mass.

_{1}, a

_{2}, a

_{3}and a

_{4}) were obtained.

_{1}, A

_{2}, k

_{1}and k

_{2}in Equation (65) was performed using the numerical method of Rosembrock and quasi-Newton using the Statistica

^{®}Software, with a convergence criterion of 0.001. Table 1 summarizes the values of these estimated parameters.

_{1}, B

_{2}, k

_{3}and B

_{3}was performed using the Quasi-Newton numerical method using the Statistica

^{®}Software, with a convergence criterion of 0.0001. Table 2 summarizes the values of these estimated parameters.

#### Studied Cases

## 3. Results and Discussion

#### 3.1. Validation

_{M}= 0.00509741 (kg/kg)

^{2}and ${\overline{\mathrm{S}}}_{\mathrm{M}}^{2}=$ 9.10252 × 10

^{−5}(kg/kg)

^{2}were obtained. For this physical situation, the following convective mass transfer coefficients were obtained hm

_{1}= 6.46958 × 10

^{−7}m/s and hm

_{2}= 6.13473 × 10

^{−7}m/s.

_{θ}= 0.346251426 (°C/°C)

^{2}and ${\overline{\mathrm{S}}}_{\mathsf{\theta}}^{2}=$ 0.006183061 (°C/°C)

^{2}were obtained. For this physical situation, the following convective heat transfer coefficients were obtained: hc

_{1}= 6.89281 W/m

^{2}K and hc

_{2}= 6.53606 W/m

^{2}K. These low values of the convective heat transfer coefficient are typical of a physical situation of natural convection. In fact, for this case, a ratio Gr/Re

^{2}= 10705.4 was obtained, which is much greater than 1.0, justifying that the effect of natural convection is much higher than forced convection.

#### 3.2. Analysis of the Drying Process

#### 3.2.1. Effect of Relative Humidity of Drying Air

#### 3.2.2. Effect of Drying Air Speed

_{1}, hm

_{2}, hc

_{1}and hc

_{2}are small, typical of a natural mass and a thermal convection problem. This can be confirmed by the high values of the relationship between the Grashof and Reynolds numbers ($\frac{\mathrm{Gr}}{{\mathrm{Re}}^{2}}\gg 1$), in such a way that the effects of forced convection are negligible. However, for a speed between 1.0 and 8.0 m/s, the process occurs by combined natural and forced convection, confirmed by the values of the relationship between the Grashof and Reynolds numbers that are in the range of $0.23\le \frac{\mathrm{Gr}}{{\mathrm{Re}}^{2}}\le 14.89$.

_{1}) and heat (hc

_{1}) are much higher than the values of these parameters in the holes, where natural convection predominates.

#### 3.3. Limitations of the Proposed Model

- (a)
- In physical situations involving the effect of moisture removal on solid heating (simultaneous heat and mass transport);
- (b)
- In physical situations involving heterogeneous solids;
- (c)
- It does not allow for the determination of temperature gradients and the moisture content inside the material; therefore, analysis of internal hydric and thermal stresses during the drying process is not possible.

## 4. Conclusions

- (a)
- The phenomenological mathematical modelling based on a lumped analysis to predict the mass and heat transfers in hollow ceramic bricks, was adequate, with small deviations from the experimental data on the average moisture content and temperature of the product over the process;
- (b)
- The lower the relative humidity is and the higher the speed of the drying air is, the faster the brick loses moisture and raises its temperature, which can cause defects in the post-drying product, reducing its quality for the firing step;
- (c)
- The heat and mass transfer coefficients on the external surface and on the brick hole are different, being higher on the external surface, especially under a forced convection condition;
- (d)
- The convective mass transfer coefficient on the external surface of the brick varied from 1.89 × 10
^{−7}m/s in the drying condition at 100 °C, 70% and 0.1 m/s (natural convection) up to 20.90 × 10^{−7}m/s at 100 °C, 70% and 8.0 m/s (combined and natural convection). The convective mass transfer coefficient in the brick hole varied from 1.83 × 10^{−7}m/s in the drying condition at 100 °C, 20% and 0.1 m/s (natural convection) up to 2.30 × 10^{−7}m/s at 100 °C, 70% and 8.0 m/s (natural convection). - (e)
- The convective heat transfer coefficient on the external surface of the brick varied from 4.13 W/m
^{2}K in the drying condition 100 °C, 70% and 0.1 m/s (natural convection) to 45.55 W/m^{2}K at 100 °C, 70% and 8.0 m/s (combined and natural convection). The convective heat transfer coefficient in the brick hole varied from 3.98 W/m^{2}K in the drying condition 100 °C, 70% and 0.1 m/s (natural convection) to 5.02 W/m^{2}K at 100 °C, 70% and 8.0 m/s (natural convection). - (f)
- Based on the maximum final moisture content of 3 to 4% (dry basis) after drying, commonly used in the industry, the temperature of the brick, the position of the brick with a hole perpendicular to the air flow, the drying time in these moisture content conditions and the experimental results (in terms of brick quality during drying) presented by the author of [8], T = 100 °C, RH = 50% and v = 0.1 m/s are proposed as the optimal drying conditions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Latin Letters | ||

a_{1}, a_{2}, a_{3}, a_{4} | Dimensions of the brick holes | (m) |

${\mathrm{A}}_{\mathrm{C}}$ | Total surface area | (m^{2}) |

${A}_{L}$ | Lateral area | (m^{2}) |

A_{I} | Internal surface area | (m^{2}) |

${\mathrm{a}}_{\mathrm{h}}$ | Height of a hole | (m) |

${\mathrm{a}}_{\mathrm{v}}$ | Width of a hole | (m) |

${\mathrm{c}}_{\mathrm{p}}$ | Specific heat of the brick | (J/kgK) |

${\mathrm{c}}_{\mathrm{v}}$ | specific heat of the vapor phase | (J/kgK) |

${\mathrm{c}}_{{\mathrm{p}}_{\mathrm{a}}}$ | Specific heat of the air | (J/kgK) |

${\mathrm{c}}_{\mathrm{w}}$ | specific heat of water in the liquid phase | (J/kgK) |

${\mathrm{D}}_{\mathrm{va}}$ | diffusion coefficient of water vapor in the air | (m^{2}/s) |

ERMQ_{M} | Quadratic deviations for moisture content | (kg^{2}/kg^{2}) |

ERMQ_{θ} | Quadratic deviations for temperature | (°C^{2}/°C^{2}) |

Gr | Grashof number | (---) |

g | Gravitational acceleration | (m/s^{2}) |

${\mathrm{h}}_{\mathrm{fg}}$ | Latent heat of water vaporization | (J/kg) |

${\mathrm{h}}_{\mathrm{c}1}$ | External convective heat transfer coefficients | (W/m^{2}K) |

${\mathrm{h}}_{\mathrm{c}2}$ | Internal convective heat transfer coefficients | (W/m^{2}K) |

${\mathrm{h}}_{\mathrm{m}1}$ | External convective mass transfer coefficients | (m/s) |

${\mathrm{h}}_{\mathrm{m}2}$ | Internal convective mass transfer coefficients | (m/s) |

$\mathrm{Le}$ | Lewis number | (---) |

Lc | Characteristic length | (m) |

${\overline{\mathrm{MM}}}_{\mathrm{v}}$ | molecular weight of the water vapor | (kg/kmol) |

$\dot{\mathrm{M}}$ | Generation of moisture | (kg/kg/s) |

${\mathrm{M}}_{\mathrm{e}}$ | Equilibrium moisture content | (kg/kg, d.b) |

${\mathrm{M}}_{0}$ | Initial moisture content | (kg/kg, d.b) |

$\overline{\mathrm{M}}$ | Average moisture content | (kg/kg, d.b) |

${\overline{\mathrm{MM}}}_{\mathrm{a}}$ | Molecular weight of the gas | (kg/kmol) |

n | Number of experimental points | (---) |

$\hat{\mathrm{n}}$ | Number of fitted parameters | (---) |

Nu | Nusselt number | (---) |

Pv_{wb} | Vapor pressure in the air at the wet bulb temperature | (Pa) |

Pr | Prandtl number | (---) |

Pvs | Saturation vapor pressure | (Pa) |

Patm | Atmospheric pressure | (Pa) |

P | pressure | (Pa) |

$\dot{\mathrm{q}\text{}}$ | Heat generation per unit volume | (W/m³) |

${\mathrm{R}}_{\mathrm{a}}$ | Universal constant of the gases | (J/molK) |

Ra | Rayleigh number | (---) |

Re | Reynolds number | (---) |

Rx | Width | (m] |

Ry | Height | (m] |

Rz | Length | (m] |

RH | Relative humidity | (%] |

S_{1} | Internal surface area | (m²] |

S_{2} | Enternal surface area | (m²] |

${\overline{\mathrm{S}}}_{\mathrm{M}}^{2}$ | Variance for moisture content | (kg^{2}/kg^{2}) |

${\overline{\mathrm{S}}}_{\mathsf{\theta}}^{2}$ | Variance for temperature | (°C^{2}/°C^{2}) |

t | Time | (s or min) |

T_{abs} | Absolute temperature | (K) |

${\mathrm{T}}_{\mathrm{f}}$ | Film temperature | (K or °C) |

Ts | Plate temperature | (K or °C) |

T∞ | Fluid temperature | (K or °C) |

${\mathrm{V}}_{\mathrm{T}}$ | Volume of the solid brick (with the holes) | (m³0 |

${\mathrm{V}}_{\mathrm{F}}$ | Volume of the holes | (m³) |

V | Volume | (m³) |

x_{a} | Absolute humidity of the air | (kg/kg) |

x_{o} | Absolute humidity of the air at the drying air temperature | (kg/kg) |

x_{bu} | Absolute humidity of the air at the wet bulb temperature | (kg/kg) |

Greek Letters | ||

$\mathsf{\alpha}$ | Thermal diffusivity of the air | (m^{2}/s) |

$\mathsf{\beta}$ | Thermal expansion coefficient | (K^{−1}) |

$\mathsf{\nu}$ | Kinematic viscosity | (m^{2}/s) |

${\mathsf{\rho}}_{\mathrm{s}}$ | Specific density of the dry solid | (kg/m^{3}) |

${\mathsf{\rho}}_{\mathrm{a}}$ | Air density | (kg/m^{3}) |

${\mathsf{\rho}}_{\mathrm{u}}$ | Specific density of the wet solid | (kg/m^{3}) |

$\overline{\mathsf{\theta}}$ | Instantaneous temperature | (K or °C) |

${\overline{\mathsf{\theta}}}_{\infty}$ | Temperature of the external medium | (K or °C) |

${\overline{\mathsf{\theta}}}_{0}$ | Initial temperature of the solid | (K or °C) |

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**Figure 4.**Brick model with the dimensions that characterize the brick and holes, and the location of the brick temperature measurement.

**Figure 5.**Comparison between the predicted and experimental average moisture content during the drying process of industrial hollow ceramic bricks.

**Figure 6.**Comparison between predicted and experimental surface temperature (vertex) during the drying process of industrial hollow ceramic bricks.

**Figure 7.**Average moisture content of industrial ceramic brick as a function of time for different relative humidity of the drying air.

**Figure 8.**Temperature at the vertex of the industrial ceramic brick as a function of time for different relative humidity of the drying air.

**Figure 9.**Average moisture content of industrial ceramic brick as a function of time for different drying air speeds.

**Figure 10.**Temperature at the vertex of the industrial ceramic brick as a function of time for different drying air speeds.

**Table 1.**Parameters of Equation (65) obtained after fitting to experimental data of average moisture content [8].

T (°C) | Parameters | R | Proportion of Variance | |||
---|---|---|---|---|---|---|

A_{1} (-) | k_{1} (min^{−1}) | A_{2} (-) | k_{2} (min^{−1}) | |||

100 | 4.875507 | −0.008383 | −3.827964 | −0.007881 | 0.998297496 | 0.996597890 |

**Table 2.**Equation (66) parameters obtained after adjustment to experimental moisture content data [15].

T (°C) | Parameters | R | Proportion of Variance | |||
---|---|---|---|---|---|---|

B_{1} (°C) | B_{2} (°C/Log_{10}(min)) | B_{3} (min) | k_{3} (-) | |||

100 | −2.86969 | 15.41788 | 118.38213 | 2.234665 | 0.984632771 | 0.969501694 |

**Table 3.**Experimental parameters of air and dimensions of hollow ceramic bricks used in the experiments [8].

Air | Brick | ||||||||
---|---|---|---|---|---|---|---|---|---|

T (°C) | RH (%) | v (m/s) | R_{x}(mm) | R_{y}(mm) | R_{z}(mm) | a_{1}(mm) | a_{2}(mm) | a_{3}(mm) | a_{4}(mm) |

100 | 1.8 | 0.10 | 92.8 | 198.0 | 202.0 | 11.7 | 9.41 | 8.74 | 8.0 |

**Table 4.**Experimental parameters of air and brick for drying test [8].

Air | Brick | t (h) | ||||
---|---|---|---|---|---|---|

T (°C) | M_{o} (d.b.) | M_{f} (d.b.) | M_{e} (d.b.) | θ_{0} (°C) | θ_{f} (°C) | |

100 | 0.16903 | 0.00038 | 0.00038 | 26.1 | 93.2 | 12.3 |

Brick | ||||||
---|---|---|---|---|---|---|

k (W/mK) | ρ_{u}(kg/m ^{3}) | ρ_{s}(kg/m ^{3}) | c_{p}(J/kgK) | S_{1}(mm ^{2}) | S_{2}(mm ^{2}) | V (mm ^{3}) |

0.833 | 1754.88 | 1889.95 | 545.00 | 134,651.775 | 226,514.720 | 1,734,026.095 |

Case | Air | ||||
---|---|---|---|---|---|

${\mathbf{T}}_{\mathbf{\infty}}(\xb0{C})$ | RH (%) | v (m/s) | T_{wb} (°C) | x_{0} (kg/kg) | |

1 | 100 | 20 | 0.1 | 62.46 | 0.15550 |

2 | 100 | 30 | 0.1 | 70.48 | 0.26660 |

3 | 100 | 40 | 0.1 | 76.76 | 0.41470 |

4 | 100 | 50 | 0.1 | 81.98 | 0.62210 |

5 | 100 | 60 | 0.1 | 86.45 | 0.93330 |

6 | 100 | 70 | 0.1 | 90.38 | 1.45200 |

7 | 100 | 70 | 0.5 | 90.38 | 1.45200 |

8 | 100 | 70 | 1.0 | 90.38 | 1.45200 |

9 | 100 | 70 | 3.0 | 90.38 | 1.45200 |

10 | 100 | 70 | 5.0 | 90.38 | 1.45200 |

11 | 100 | 70 | 8.0 | 90.38 | 1.45200 |

**Table 7.**Heat transfer coefficients and convective mass for different relative humidity of the drying air.

Drying Air Condition | Mass Transfer Coefficient | Heat Transfer Coefficient | Gr/Re^{2}(-) | ||||
---|---|---|---|---|---|---|---|

T (°C) | RH (%) | v (m/s) | hm_{1}(m/s) | hm_{2}(m/s) | hc_{1}(W/m ^{2}K) | hc_{2}(W/m ^{2}K) | |

100 | 20 | 0.1 | 4.10 × 10^{−7} | 3.92 × 10^{−7} | 5.91 | 5.66 | 6037.75 |

100 | 30 | 0.1 | 3.46 × 10^{−7} | 3.32 × 10^{−7} | 5.53 | 5.31 | 4694.73 |

100 | 40 | 0.1 | 2.94 × 10^{−7} | 2.83 × 10^{−7} | 5.19 | 4.99 | 3663.88 |

100 | 50 | 0.1 | 2.59 × 10^{−7} | 2.50 × 10^{−7} | 4.85 | 4.67 | 2820.57 |

100 | 60 | 0.1 | 2.22 × 10^{−7} | 2.14 × 10^{−7} | 4.50 | 4.34 | 2107.97 |

100 | 70 | 0.1 | 1.89 × 10^{−7} | 1.83 × 10^{−7} | 4.13 | 3.98 | 1488.59 |

Drying Air Condition | Mass Transfer Coefficient | Heat Transfer Coefficient | Gr/Re^{2}(-) | ||||
---|---|---|---|---|---|---|---|

T (°C) | RH (%) | v (m/s) | hm_{1}(m/s) | hm_{2}(m/s) | hc_{1} (W/m^{2}K) | hc_{2}(W/m ^{2}K) | |

100 | 70 | 0.1 | 1.89 × 10^{−7} | 1.83 × 10^{−7} | 4.13 | 3.98 | 1488.59 |

100 | 70 | 0.5 | 1.89 × 10^{−7} | 1.83 × 10^{−7} | 4.13 | 3.98 | 59.54 |

100 | 70 | 1.0 | 7.44 × 10^{−7} | 2.30 × 10^{−7} | 16.22 | 5.02 | 14.89 |

100 | 70 | 3.0 | 12.81 × 10^{−7} | 2.30 × 10^{−7} | 27.93 | 5.02 | 1.65 |

100 | 70 | 5.0 | 16.53 × 10^{−7} | 2.30 × 10^{−7} | 36.02 | 5.02 | 0.59 |

100 | 70 | 8.0 | 20.90 × 10^{−7} | 2.30 × 10^{−7} | 45.55 | 5.02 | 0.23 |

Case | Air | Brick | ||||
---|---|---|---|---|---|---|

${\mathbf{T}}_{\mathit{\infty}}(\xb0\mathbf{C})$ | RH (%) | v (m/s) | M (kg/kg) | θ (°C) | t (min) | |

1 | 100 | 20 | 0.1 | 0.03239 | 89.65 | 333.33 |

2 | 100 | 30 | 0.1 | 0.03385 | 90.34 | 383.33 |

3 | 100 | 40 | 0.1 | 0.03398 | 91.31 | 450.00 |

4 | 100 | 50 | 0.1 | 0.03502 | 91.60 | 500.00 |

5 | 100 | 60 | 0.1 | 0.03502 | 92.30 | 583.33 |

6 | 100 | 70 | 0.1 | 0.03644 | 92.58 | 666.67 |

7 | 100 | 70 | 0.5 | 0.03644 | 92.58 | 666.67 |

8 | 100 | 70 | 1.0 | 0.03825 | 92.21 | 283.33 |

9 | 100 | 70 | 3.0 | 0.03167 | 93.56 | 216.67 |

10 | 100 | 70 | 5.0 | 0.03782 | 92.30 | 158.33 |

11 | 100 | 70 | 8.0 | 0.03024 | 93.86 | 150.00 |

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## Share and Cite

**MDPI and ACS Style**

Lima, E.S.; Delgado, J.M.P.Q.; Guimarães, A.S.; Lima, W.M.P.B.; Santos, I.B.; Gomes, J.P.; Santos, R.S.; Vilela, A.F.; Viana, A.D.; Almeida, G.S.;
et al. Drying and Heating Processes in Arbitrarily Shaped Clay Materials Using Lumped Phenomenological Modeling. *Energies* **2021**, *14*, 4294.
https://doi.org/10.3390/en14144294

**AMA Style**

Lima ES, Delgado JMPQ, Guimarães AS, Lima WMPB, Santos IB, Gomes JP, Santos RS, Vilela AF, Viana AD, Almeida GS,
et al. Drying and Heating Processes in Arbitrarily Shaped Clay Materials Using Lumped Phenomenological Modeling. *Energies*. 2021; 14(14):4294.
https://doi.org/10.3390/en14144294

**Chicago/Turabian Style**

Lima, Elisiane S., João M. P. Q. Delgado, Ana S. Guimarães, Wanderson M. P. B. Lima, Ivonete B. Santos, Josivanda P. Gomes, Rosilda S. Santos, Anderson F. Vilela, Arianne D. Viana, Genival S. Almeida,
and et al. 2021. "Drying and Heating Processes in Arbitrarily Shaped Clay Materials Using Lumped Phenomenological Modeling" *Energies* 14, no. 14: 4294.
https://doi.org/10.3390/en14144294