Modeling of the Two-Dimensional Thawing of Logs in an Air Environment

*Nencho Deliiski, Ladislav Dzurenda and Natalia Tumbarkova*

### **Abstract**

A two-dimensional mathematical model has been created, solved, and verified for the transient nonlinear heat conduction in logs during their thawing in an air environment. For the numerical solution of the model, an explicit form of the finite-difference method in the computing medium of Visual FORTRAN Professional has been used. The chapter presents solutions of the model and its validation towards own experimental studies. During the validation of the model, the inverse task of the heat transfer has been solved for the determination of the logs' heat transfer coefficients in radial and longitudinal directions. This task has been solved also in regard to the logs'surface temperature, which depends on the mentioned coefficients. The results from the experimental and simulative investigation of 2D nonstationary temperature distribution in the longitudinal section of poplar logs with a diameter of 0.24 m, length of 0.48 m, and an initial temperature of approximately –30°C during their many hours thawing in an air environment at room temperature are presented, visualized, and analyzed.

**Keywords:** heat conduction, modeling, logs, thawing, heat transfer coefficients, surface temperature

### **1. Introduction**

The duration and the energy consumption of the thermal treatment of frozen logs aimed at their thawing and plasticizing for the production of veneer in winter are very high [1–9]. For example, thawing and plasticizing of poplar and pine logs with an initial temperature of –10°C and moisture content of 0.6 kgkg<sup>1</sup> about 53 kWh<sup>m</sup><sup>3</sup> and 64 kWh<sup>m</sup><sup>3</sup> thermal energy, respectively, are needed [9].

In the specialized literature, there are few reports about the temperature fields subjected to thawing in agitated water or steam frozen logs [7–21], and there is very scarce information about research of the temperature distribution in frozen logs during their thawing in an air environment given by the authors only [22].

The computation of the temperature field in logs during their thawing in water or steam is carried out using mathematical models, which solve the so-called direct task of the heat transfer. This is the task when all variables in the model are known, and this allows computing the temperature field in the body [23, 24].

The computation of the temperature field in logs during their thawing in an air environment requires solving of the so-called inverse task of the heat transfer. This is the task when the model of the studied object and the experimentally obtained temperature field in it are known, but one or more variables in the model need to be determined during the solving and validation of the model [24].

• Along the longitudinal coordinate z on the logs' cylindrical surface during

In [8] solutions of Eq. (1) only at conductive boundary conditions for the case of autoclave steaming of logs aimed at their plasticizing in the production of veneer

An approach for solving Eq. (1) at much more complicated convective boundary

In **Figure 1** the three temperature ranges are presented, at which the process of the logs' thawing above the hygroscopic range is carried out, i.e., when *u* >*u*fsp. There thermophysical characteristics of the logs and of both the frozen and nonfrozen free and bound water in them during the separate temperature ranges are also shown. The information on these characteristics is very important for the

The mathematical descriptions of the thermal conductivities of nonfrozen wood, *<sup>λ</sup>*<sup>w</sup>‐nfr, and frozen wood, *<sup>λ</sup>*<sup>w</sup>‐fr, and also of the specific heat capacities of nonfrozen wood, *c*<sup>w</sup>�nfr, frozen wood, *c*<sup>w</sup>�fr, and frozen free and bound water in the wood, *c*fw and *c*bwm, have been suggested in [8, 9, 18] using the experimentally determined data in the dissertations by Kanter [25] and Chudinov [2] for their change as a function of *t* and *u*. According to the suggested in [8, 9, 18] approach, the wood thermal conductivity during thawing of logs with moisture content *u* above the hygroscopic range can be calculated with the help of the following equations for *λ*<sup>w</sup> *T*, *u*, *ρ*b, *u*fsp :

The coefficients γ and β in Eq. (5) are calculated using the next equations:

*Temperature ranges of the logs' thawing process at* u *>* u*fsp and thermophysical characteristics of the wood and*

*λ*<sup>w</sup> ¼ *λ*w0 � *γ* � ½ � 1 þ *β* � ð Þ *T* � 273*:*15 , (5)

*ρ*2

*v* ¼ 0*:*1284 � 0*:*013*u:* (7)

, (6)

fsp and 272*:*15K<*T* ≤ 423*:*15K

<sup>b</sup> <sup>þ</sup> <sup>1</sup>*:*<sup>015</sup> � <sup>10</sup>�<sup>3</sup>

γ ¼ 1*:*0, (8)

*ρ*b

**2.2 Mathematical description of the thermophysical characteristics of logs**

*<sup>λ</sup>*wrð Þ 0, *<sup>z</sup>*, *<sup>τ</sup>* ½ � *<sup>T</sup>*ð Þ� 0, *<sup>z</sup>*, *<sup>τ</sup> <sup>T</sup>*mð Þ*<sup>τ</sup>* (4)

*<sup>∂</sup><sup>z</sup>* ¼ � *<sup>α</sup>*wrð Þ 0, *<sup>z</sup>*, *<sup>τ</sup>*

*Modeling of the Two-Dimensional Thawing of Logs in an Air Environment*

conditions and verification of model (1) to (4) is considered below.

<sup>λ</sup>w0 <sup>¼</sup> *<sup>K</sup>*ad � *<sup>v</sup>* � <sup>0</sup>*:*<sup>165</sup> <sup>þ</sup> ð Þ� <sup>1</sup>*:*<sup>39</sup> <sup>þ</sup> <sup>3</sup>*:*8*<sup>u</sup>* <sup>3</sup>*:*<sup>3</sup> � <sup>10</sup>�<sup>7</sup>

• For nonfrozen wood at *u*> *u*<sup>272</sup>*:*<sup>15</sup>

*of the frozen and nonfrozen bound and free water in it.*

**Figure 1.**

**73**

thawing

*<sup>∂</sup>T*ð Þ 0, *<sup>z</sup>*, *<sup>τ</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.93177*

have been realized and graphically presented.

solving of the model given above.

The results from investigations of the temperature change subjected to thawing frozen logs only at conductive boundary conditions (i.e., at prescribed surface temperature) have been reported [2, 7–21].

The modeling and the multiparameter study of the thawing process of logs in air environment is of considerable scientific and practical interest. For example, as a result of such a study, it is possible to determine the real initial temperature of logs depending on their dimensions, wood species, moisture content, and the temperature of the air near the logs during their many days staying in an open warehouse before the thermal treatment in the production of veneer. The information about the real value of that immeasurable parameter can be used for scientifically based computing of the optimal, energy saving regimes for thermal treatment of each specific batch of logs.

This chapter presents the creation, numerical solving and validation of a two-dimensional nonlinear mathematical model of the transient heat conduction in frozen logs during their thawing at convective boundary conditions in an air environment. A validation of the models towards own experimentally determined 2D temperature distribution in poplar logs with a diameter of 0.24 m, length of 0.48 m, initial temperature of approximately –30°C, and moisture content above the hygroscopic range during their 70 h thawing at room temperature has been carried out.

During the validation of the model, the inverse task has been solved for the determination of the unknown logs' heat transfer coefficients in radial and longitudinal directions. This task has been solved also in regard to the logs'surface temperature, which depends on the mentioned coefficients.

### **2. Mechanism of 2D heat distribution in logs during thawing**

### **2.1 Mathematical model of the 2D temperature distribution in frozen logs during their thawing in an air environment**

In [8] the following common form of a model, which describes the 2D nonstationary temperature distribution subjected to thawing frozen logs in an air environment, has been suggested:

$$\begin{split} \rho\_{\text{we-1,2,3}} \cdot \rho\_{\text{w}} \frac{\partial T(r, \mathbf{z}, \tau)}{\partial \tau} = \lambda\_{\text{w}\tau} \left[ \frac{\partial^2 T(r, \mathbf{z}, \tau)}{\partial r^2} + \frac{1}{r} \frac{\partial T(r, \mathbf{z}, \tau)}{\partial r} \right] + \frac{\partial \lambda\_{\text{w}\tau}}{\partial T} \left[ \frac{\partial T(r, \mathbf{z}, \tau)}{\partial r} \right]^2 + \\ + \lambda\_{\text{wp}} \frac{\partial^2 T(r, \mathbf{z}, \tau)}{\partial \mathbf{z}^2} + \frac{\partial \lambda\_{\text{wp}}}{\partial T} \left[ \frac{\partial T(r, \mathbf{z}, \tau)}{\partial \mathbf{z}} \right]^2 \end{split} \tag{1}$$

with an initial condition

$$T(r, z, \mathbf{0}) = T\_{\text{w0-avg}} \tag{2}$$

and boundary conditions for convective heat transfer:

• Along the radial coordinate *r* on the logs' frontal surface during thawing

$$\frac{\partial T(r,\mathbf{0},\boldsymbol{\tau})}{\partial r} = -\frac{a\_{\rm wp}(r,\mathbf{0},\boldsymbol{\tau})}{\lambda\_{\rm wp}(r,\mathbf{0},\boldsymbol{\tau})}[T(r,\mathbf{0},\boldsymbol{\tau}) - T\_{\rm m}(\boldsymbol{\tau})] \tag{3}$$

*Modeling of the Two-Dimensional Thawing of Logs in an Air Environment DOI: http://dx.doi.org/10.5772/intechopen.93177*

• Along the longitudinal coordinate z on the logs' cylindrical surface during thawing

$$\frac{\partial T(\mathbf{0}, z, \tau)}{\partial z} = -\frac{a\_{\rm wr}(\mathbf{0}, z, \tau)}{\lambda\_{\rm wr}(\mathbf{0}, z, \tau)} [T(\mathbf{0}, z, \tau) - T\_{\rm m}(\tau)] \tag{4}$$

In [8] solutions of Eq. (1) only at conductive boundary conditions for the case of autoclave steaming of logs aimed at their plasticizing in the production of veneer have been realized and graphically presented.

An approach for solving Eq. (1) at much more complicated convective boundary conditions and verification of model (1) to (4) is considered below.

#### **2.2 Mathematical description of the thermophysical characteristics of logs**

In **Figure 1** the three temperature ranges are presented, at which the process of the logs' thawing above the hygroscopic range is carried out, i.e., when *u* >*u*fsp.

There thermophysical characteristics of the logs and of both the frozen and nonfrozen free and bound water in them during the separate temperature ranges are also shown. The information on these characteristics is very important for the solving of the model given above.

The mathematical descriptions of the thermal conductivities of nonfrozen wood, *<sup>λ</sup>*<sup>w</sup>‐nfr, and frozen wood, *<sup>λ</sup>*<sup>w</sup>‐fr, and also of the specific heat capacities of nonfrozen wood, *c*<sup>w</sup>�nfr, frozen wood, *c*<sup>w</sup>�fr, and frozen free and bound water in the wood, *c*fw and *c*bwm, have been suggested in [8, 9, 18] using the experimentally determined data in the dissertations by Kanter [25] and Chudinov [2] for their change as a function of *t* and *u*. According to the suggested in [8, 9, 18] approach, the wood thermal conductivity during thawing of logs with moisture content *u* above the hygroscopic range can be calculated with the help of the following equations for *λ*<sup>w</sup> *T*, *u*, *ρ*b, *u*fsp :

$$
\lambda\_{\rm w} = \lambda\_{\rm w0} \cdot \chi \cdot [\mathbf{1} + \boldsymbol{\beta} \cdot (T - 27 \mathbf{3.15})], \tag{5}
$$

$$\lambda\_{\rm w0} = K\_{\rm ad} \cdot \nu \cdot \left[ 0.165 + (1.39 + 3.8u) \cdot \left( 3.3 \cdot 10^{-7} \rho\_{\rm b}^2 + 1.015 \cdot 10^{-3} \rho\_{\rm b} \right) \right], \tag{6}$$

$$v = 0.1284 - 0.013u. \tag{7}$$

The coefficients γ and β in Eq. (5) are calculated using the next equations:

• For nonfrozen wood at *u*> *u*<sup>272</sup>*:*<sup>15</sup> fsp and 272*:*15K<*T* ≤ 423*:*15K

$$
\gamma = \mathbf{1}. \mathbf{0}, \tag{8}
$$

#### **Figure 1.**

*Temperature ranges of the logs' thawing process at* u *>* u*fsp and thermophysical characteristics of the wood and of the frozen and nonfrozen bound and free water in it.*

is the task when the model of the studied object and the experimentally obtained temperature field in it are known, but one or more variables in the model need to be

frozen logs only at conductive boundary conditions (i.e., at prescribed surface

This chapter presents the creation, numerical solving and validation of a two-dimensional nonlinear mathematical model of the transient heat conduction in frozen logs during their thawing at convective boundary conditions in an air environment. A validation of the models towards own experimentally determined 2D temperature distribution in poplar logs with a diameter of 0.24 m, length of 0.48 m, initial temperature of approximately –30°C, and moisture content above the hygroscopic range during their 70 h thawing at room temperature has been carried out. During the validation of the model, the inverse task has been solved for the determination of the unknown logs' heat transfer coefficients in radial and

longitudinal directions. This task has been solved also in regard to the logs'surface

temperature, which depends on the mentioned coefficients.

**during their thawing in an air environment**

*∂*2

and boundary conditions for convective heat transfer:

*<sup>∂</sup><sup>r</sup>* ¼ � *<sup>α</sup>*wpð Þ *<sup>r</sup>*, 0, *<sup>τ</sup>*

*<sup>∂</sup>T r*ð Þ , 0, *<sup>τ</sup>*

þ*λ*wp *∂*2

environment, has been suggested:

*<sup>∂</sup>T r*ð Þ , *<sup>z</sup>*, *<sup>τ</sup>*

with an initial condition

*<sup>∂</sup><sup>τ</sup>* <sup>¼</sup> *<sup>λ</sup>*wr

*<sup>c</sup>*we‐1,2,3 � *<sup>ρ</sup>*<sup>w</sup>

**72**

**2. Mechanism of 2D heat distribution in logs during thawing**

*T r*ð Þ , *z*, *τ ∂r*<sup>2</sup> þ

*T r*ð Þ , *z*, *τ ∂z*<sup>2</sup> þ

**2.1 Mathematical model of the 2D temperature distribution in frozen logs**

In [8] the following common form of a model, which describes the 2D nonstationary temperature distribution subjected to thawing frozen logs in an air

> 1 *r :*

> > *∂λ*wp *∂T*

• Along the radial coordinate *r* on the logs' frontal surface during thawing

*<sup>∂</sup>T r*ð Þ , *<sup>z</sup>*, *<sup>τ</sup> ∂r*

þ *∂λ*wr *∂T*

*T r*ð Þ¼ , *z*, 0 *T*w0�avg (2)

*<sup>λ</sup>*wpð Þ *<sup>r</sup>*, 0, *<sup>τ</sup>* ½ � *T r*ð Þ� , 0, *<sup>τ</sup> <sup>T</sup>*mð Þ*<sup>τ</sup>* (3)

*<sup>∂</sup>T r*ð Þ , *<sup>z</sup>*, *<sup>τ</sup> ∂z*

*<sup>∂</sup>T r*ð Þ , *<sup>z</sup>*, *<sup>τ</sup> ∂r* <sup>2</sup>

<sup>2</sup> (1)

þ

The results from investigations of the temperature change subjected to thawing

The modeling and the multiparameter study of the thawing process of logs in air environment is of considerable scientific and practical interest. For example, as a result of such a study, it is possible to determine the real initial temperature of logs depending on their dimensions, wood species, moisture content, and the temperature of the air near the logs during their many days staying in an open warehouse before the thermal treatment in the production of veneer. The information about the real value of that immeasurable parameter can be used for scientifically based computing of the optimal, energy saving regimes for thermal treatment of each

determined during the solving and validation of the model [24].

temperature) have been reported [2, 7–21].

*Modeling and Simulation in Engineering - Selected Problems*

specific batch of logs.

$$
\beta = 3.65 \left( \frac{579}{\rho\_b} - 0.124 \right) \cdot 10^{-3} \,\text{.}\tag{9}
$$

The wood density, *ρ*w, which participates in Eq. (1), is determined above the

For solving of the 2D mathematical model given above, it is needed to have values for the heat transfer coefficients of the logs in radial and longitudinal direc-

As it was mentioned in the Introduction, the values of *α*wr and *α*wp can be computed by solving the inverse task of the heat transfer between the logs and

equations of the similarity theory, which are valid for the cases of heating of horizontally situated cylindrical bodies in conditions of free air convection [28]:

The calculation of *α*wr and *α*wp can be carried out with the help of the following

*<sup>α</sup>*wr <sup>¼</sup> Nur � *<sup>λ</sup>*<sup>a</sup>

*<sup>α</sup>*w<sup>р</sup> <sup>¼</sup> Nu<sup>р</sup> � *<sup>λ</sup>*<sup>a</sup>

� �<sup>0</sup>*:*<sup>25</sup> � Pra

Pra <sup>¼</sup> *<sup>w</sup>*að Þ *<sup>T</sup>*<sup>a</sup> *a*að Þ *T*<sup>a</sup>

Prs <sup>¼</sup> *<sup>w</sup>*að Þ *<sup>T</sup>*<sup>s</sup> *a*að Þ *T*<sup>s</sup>

In the accessible specialized sources, we did not find suitable mathematical descriptions of *λ*, *β*, *w*, and *а* of the air, depending on *T* and *φ*, which could be applied for the precise determination of *α*wr and *α*wp according to Eqs. (19)–(26). Our further study has shown that for solving the inverse task of the heat transfer between the logs and surrounding air, i.e., for the calculation of the heat transfer coefficients of the logs, which participate in the boundary conditions (3) and (4) of

For the usage of Eqs. (19)–(26), it is needed to have a mathematical description of the thermophysical characteristics of the air, λ, *β*, *w*, and *а*, depending on *T* and *φ*. The temperature of the air near the logs subjected to thawing during our experiments described below changes in the range from 243.15 to 303.15 K (i.е.,

С), and φ changes from 40–100% (see **Figures 2** and **3**). For the calculation of λa, *β*a, *w*a, and *а*a, the temperature of the air, *Т*a, must be used, but for the calculation of *w*<sup>s</sup> and *а*<sup>s</sup> in Eq. (26), the temperature of the surface

Prs

Prs

� �<sup>0</sup>*:*<sup>25</sup> " #, (21)

� �<sup>0</sup>*:*<sup>25</sup> " #, (22)

Nur <sup>¼</sup> *<sup>f</sup>* Grr ð Þ � Pra <sup>0</sup>*:*<sup>25</sup> � Pra

Nup ¼ *f* Grp � Pra

Grr <sup>¼</sup> *<sup>g</sup>* � *<sup>β</sup>*<sup>a</sup> � *<sup>L</sup>*<sup>3</sup> *w*<sup>2</sup> a

Grp <sup>¼</sup> *<sup>g</sup>* � *<sup>β</sup>*<sup>a</sup> � *<sup>D</sup>*<sup>3</sup> *w*<sup>2</sup> a

*ρ*<sup>w</sup> ¼ *ρ*<sup>b</sup> � ð Þ 1 þ *u :* (18)

*<sup>L</sup>* , (19)

*<sup>D</sup>* , (20)

� ½ � *T*mð Þ� *τ T*sð Þ*τ* , (23)

� ½ � *T*mð Þ� *τ T*sð Þ*τ* , (24)

, (25)

, (26)

hygroscopic range according to the following Equation [1–22, 27]:

*Modeling of the Two-Dimensional Thawing of Logs in an Air Environment*

*DOI: http://dx.doi.org/10.5772/intechopen.93177*

**2.3 Mathematical description of the heat transfer coefficients of logs**

tions, *α*wr and *α*wp, respectively, which participate in Eqs. (3) and (4).

surrounding air environment.

from –30°

**75**

С to 30°

of the logs,*T*s, has to be used.

the model, the following equations are suitable [29]:

• For frozen wood at *u* >*u*272*:*<sup>15</sup> fsp and 213*:*15K≤*T* ≤ 272*:*15K

$$\gamma = \mathbf{1} + \mathbf{0}.\mathbf{3}4 \mathbf{\color[rgb]{0,0,1}{1.0}} \mathbf{1}.\mathbf{15} (\mathbf{u} - \mathbf{u}\_{\text{fsp}}) \Big],\tag{10}$$

$$\beta = 0.002(u - u\_{\rm fsp}) - 0.0038 \left(\frac{579}{\rho\_{\rm b}} - 0.124\right). \tag{11}$$

The fiber saturation points of the wood species, *u*fsp and *u*272*:*<sup>15</sup> fsp , are calculated according to following Eqs. [9]:

$$u\_{\rm fsp} = u\_{\rm fsp}^{293.15} - 0.001(T - 293.15),\tag{12}$$

and consequently

$$
u\_{\rm fsp}^{272.15} = 
u\_{\rm fsp}^{293.15} + 0.021,\tag{13}$$

where *u*<sup>293</sup>*:*<sup>15</sup> fsp is the standardized fiber saturation point of the wood at *T* = 293.15 K (i.e., at *<sup>t</sup>* = 20°C), kg�kg�<sup>1</sup> , and *u*<sup>272</sup>*:*<sup>15</sup> fsp is the fiber saturation point at *T* = 272.15 K (i.e., at *<sup>t</sup>* <sup>=</sup> –1°C), kg�kg�<sup>1</sup> . At *t* = –1°C, the melting of the frozen bound water in the wood is fully completed, and the melting of the free water in the wood starts [22, 26].

The effective specific heat capacities of the logs during the pointed three ranges of the thawing process, *<sup>c</sup>*we‐1,2,3, which participate in Eq. (1), are equal to the following:

$$\text{First range}: \mathcal{c}\_{\text{we-1}} = \mathcal{c}\_{\text{w-fr}} + \mathcal{c}\_{\text{bwm}},\tag{14}$$

$$\text{Second range}: \mathcal{c}\_{\text{we-2}} = \mathcal{c}\_{\text{w-nfr}} + \mathcal{c}\_{\text{fw}},\tag{15}$$

$$\text{Third range}: c\_{\text{we-3}} = c\_{\text{w-nfr}}.\tag{16}$$

According to the suggested in [8, 9, 22] mathematical description, the effective specific heat capacities of the logs during their thawing can be calculated with the help of the following equations for *<sup>c</sup>*we‐1,2,3 *<sup>T</sup>*, *<sup>u</sup>*, *<sup>u</sup>*fsp � �:

$$c\_{\text{wc-1,2}} = \begin{cases} \left[1.06 + 0.04u + \frac{0.00075(T - 272.15)}{u\_{\text{fp}}^{2.15}}\right] \\ \cdot \frac{526 + 2.95T + 0.0022T^2 + 2261u + 1976u\_{\text{fp}}^{2715}}{1 + u} + \\ \cdot \frac{1 + u}{\text{u} + u} \\ \cdot \frac{\text{@u} \times 10^4 \left(u\_{\text{fp}}^{2715} - 0.12\right)}{\text{@u} \times u\_{\text{fp}}^{2715} \text{8213.15} \text{K} \le T \le 272.15 \text{K}} \quad (\text{1\* ''a n g e}) \\ \cdot \frac{2862u + 555}{1 + u} + \frac{5.42u + 2.95}{1 + u}T + \frac{0.0036}{1 + u}T^2 + \\ \cdot \frac{2862u + 555}{1 + u} + \frac{5.42u + 2.95}{1 + u} \\ \cdot \frac{u - u\_{\text{fp}}^{2715}}{1 + u} \\ \cdot \frac{2862u + 555}{1 + u} + \frac{5.42u + 2.95}{1 + u}T + \frac{0.0036}{1 + u}T^2 \\ \end{cases} \quad (17)$$
 
$$\begin{cases} \frac{2862u + 555}{1 + u} + \frac{5.42u + 2.95}{1 + u}T + \frac{0.0036}{1 + u}T^2 \\ \cdot \frac{2862u + 555}{1 + u} + \frac{5.42u + 2.95}{1 + u}T + \frac{0.0036}{1 + u}T^2 \\ \end{cases} \quad \text{(27)}$$

*Modeling of the Two-Dimensional Thawing of Logs in an Air Environment DOI: http://dx.doi.org/10.5772/intechopen.93177*

The wood density, *ρ*w, which participates in Eq. (1), is determined above the hygroscopic range according to the following Equation [1–22, 27]:

$$
\rho\_{\mathbf{w}} = \rho\_{\mathbf{b}} \cdot (\mathbf{1} + \mathbf{u}). \tag{18}
$$

### **2.3 Mathematical description of the heat transfer coefficients of logs**

For solving of the 2D mathematical model given above, it is needed to have values for the heat transfer coefficients of the logs in radial and longitudinal directions, *α*wr and *α*wp, respectively, which participate in Eqs. (3) and (4).

As it was mentioned in the Introduction, the values of *α*wr and *α*wp can be computed by solving the inverse task of the heat transfer between the logs and surrounding air environment.

The calculation of *α*wr and *α*wp can be carried out with the help of the following equations of the similarity theory, which are valid for the cases of heating of horizontally situated cylindrical bodies in conditions of free air convection [28]:

$$a\_{\rm wr} = \frac{\mathbf{Nu\_r} \cdot \boldsymbol{\lambda}\_{\rm a}}{L},\tag{19}$$

$$a\_{\rm wp} = \frac{\mathbf{N} \mathbf{u}\_{\rm p} \cdot \boldsymbol{\lambda}\_{\rm a}}{D},\tag{20}$$

$$\mathbf{Nu\_{r} = f \left[ \left( \mathbf{Gr\_{r} \cdot Pr\_{a} \right)}^{0.25} \cdot \left( \frac{\mathbf{Pr\_{a}}}{\mathbf{Pr\_{s}}} \right)^{0.25} \right],\tag{21}$$

$$\mathbf{Nu\_{p}} = f \left[ \left( \mathbf{Gr\_{p}} \cdot \mathbf{Pr\_{a}} \right)^{0.25} \cdot \left( \frac{\mathbf{Pr\_{a}}}{\mathbf{Pr\_{s}}} \right)^{0.25} \right],\tag{22}$$

$$\mathrm{Gr}\_{\mathfrak{r}} = \frac{\mathrm{g} \cdot \beta\_{\mathfrak{a}} \cdot L^{\mathfrak{z}}}{w\_{\mathfrak{a}}^2} \cdot [T\_{\mathfrak{m}}(\mathfrak{r}) - T\_{\mathfrak{s}}(\mathfrak{r})],\tag{23}$$

$$\text{Gr}\_{\text{p}} = \frac{\text{g} \cdot \beta\_{\text{a}} \cdot D^{\text{3}}}{w\_{\text{a}}^2} \cdot [T\_{\text{m}}(\tau) - T\_{\text{s}}(\tau)],\tag{24}$$

$$\text{Pr}\_{\mathfrak{a}} = \frac{w\_{\mathfrak{a}}(T\_{\mathfrak{a}})}{a\_{\mathfrak{a}}(T\_{\mathfrak{a}})},\tag{25}$$

$$\text{Pr}\_{\text{s}} = \frac{w\_{\text{a}}(T\_{\text{s}})}{a\_{\text{a}}(T\_{\text{s}})},\tag{26}$$

For the usage of Eqs. (19)–(26), it is needed to have a mathematical description of the thermophysical characteristics of the air, λ, *β*, *w*, and *а*, depending on *T* and *φ*. The temperature of the air near the logs subjected to thawing during our experiments described below changes in the range from 243.15 to 303.15 K (i.е., from –30° С to 30° С), and φ changes from 40–100% (see **Figures 2** and **3**).

For the calculation of λa, *β*a, *w*a, and *а*a, the temperature of the air, *Т*a, must be used, but for the calculation of *w*<sup>s</sup> and *а*<sup>s</sup> in Eq. (26), the temperature of the surface of the logs,*T*s, has to be used.

In the accessible specialized sources, we did not find suitable mathematical descriptions of *λ*, *β*, *w*, and *а* of the air, depending on *T* and *φ*, which could be applied for the precise determination of *α*wr and *α*wp according to Eqs. (19)–(26).

Our further study has shown that for solving the inverse task of the heat transfer between the logs and surrounding air, i.e., for the calculation of the heat transfer coefficients of the logs, which participate in the boundary conditions (3) and (4) of the model, the following equations are suitable [29]:

<sup>β</sup> <sup>¼</sup> <sup>3</sup>*:*<sup>65</sup> <sup>579</sup>

The fiber saturation points of the wood species, *u*fsp and *u*272*:*<sup>15</sup>

β ¼ 0*:*002 *u* � *u*fsp

*Modeling and Simulation in Engineering - Selected Problems*

*<sup>u</sup>*fsp <sup>¼</sup> *<sup>u</sup>*293*:*<sup>15</sup>

*u*<sup>272</sup>*:*<sup>15</sup>

, and *u*<sup>272</sup>*:*<sup>15</sup>

help of the following equations for *<sup>c</sup>*we‐1,2,3 *<sup>T</sup>*, *<sup>u</sup>*, *<sup>u</sup>*fsp

1*:*06 þ 0*:*04*u* þ

<sup>þ</sup>1*:*<sup>8938</sup> � 104 *<sup>u</sup>*272*:*<sup>15</sup>

<sup>þ</sup>3*:*<sup>34</sup> � <sup>10</sup><sup>5</sup> *<sup>u</sup>* � *<sup>u</sup>*272*:*<sup>15</sup>

@*u*> *u*272*:*<sup>15</sup>

2862*u* þ 555 <sup>1</sup> <sup>þ</sup> *<sup>u</sup>* <sup>þ</sup>

@*u*> *u*272*:*<sup>15</sup>

2862*u* þ 555 <sup>1</sup> <sup>þ</sup> *<sup>u</sup>* <sup>þ</sup>

@*u*> *u*272*:*<sup>15</sup>

�

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

8

fsp <sup>¼</sup> *<sup>u</sup>*<sup>293</sup>*:*<sup>15</sup>

is fully completed, and the melting of the free water in the wood starts [22, 26].

The effective specific heat capacities of the logs during the pointed three ranges of the thawing process, *<sup>c</sup>*we‐1,2,3, which participate in Eq. (1), are equal to the following:

According to the suggested in [8, 9, 22] mathematical description, the effective specific heat capacities of the logs during their thawing can be calculated with the

> 0*:*00075ð Þ *T* � 272*:*15 *u*272*:*<sup>15</sup> fsp

> > �

fsp &213*:*15K<sup>≤</sup> *<sup>T</sup>* <sup>≤</sup>272*:*15K 1st ð Þ range

fsp & 272*:*15K<*<sup>T</sup>* <sup>&</sup>lt; <sup>273</sup>*:*15K 2nd range � �

fsp & 273*:*15K≤*<sup>T</sup>* <sup>≤</sup>413*:*15K 3rd range � �

0*:*0036 <sup>1</sup> <sup>þ</sup> *<sup>u</sup> <sup>T</sup>*<sup>2</sup>

0*:*0036 <sup>1</sup> <sup>þ</sup> *<sup>u</sup> <sup>T</sup>*<sup>2</sup>

<sup>526</sup> <sup>þ</sup> <sup>2</sup>*:*95*<sup>T</sup>* <sup>þ</sup> <sup>0</sup>*:*0022*T*<sup>2</sup> <sup>þ</sup> <sup>2261</sup>*<sup>u</sup>* <sup>þ</sup> <sup>1976</sup>*u*272*:*<sup>15</sup>

1 þ u

fsp � 0*:*12 � �

5*:*42*u* þ 2*:*95 <sup>1</sup> <sup>þ</sup> *<sup>u</sup> <sup>T</sup>* <sup>þ</sup>

5*:*42*u* þ 2*:*95 <sup>1</sup> <sup>þ</sup> *<sup>u</sup> <sup>T</sup>* <sup>þ</sup>

fsp 1 þ *u*

" #

� �:

• For frozen wood at *u* >*u*272*:*<sup>15</sup>

according to following Eqs. [9]:

and consequently

(i.e., at *<sup>t</sup>* = 20°C), kg�kg�<sup>1</sup>

where *u*<sup>293</sup>*:*<sup>15</sup>

at *<sup>t</sup>* <sup>=</sup> –1°C), kg�kg�<sup>1</sup>

*<sup>c</sup>*we‐1,2,3 <sup>¼</sup>

**74**

ρb

γ ¼ 1 þ 0*:*34 1*:*15 *u* � *u*fsp

� � � <sup>0</sup>*:*<sup>0038</sup> <sup>579</sup>

� 0*:*124 � �

fsp and 213*:*15K≤*T* ≤ 272*:*15K

� <sup>10</sup>�<sup>3</sup>

ρb

fsp is the standardized fiber saturation point of the wood at *T* = 293.15 K

. At *t* = –1°C, the melting of the frozen bound water in the wood

First range : *<sup>c</sup>*wе‐<sup>1</sup> <sup>¼</sup> *<sup>c</sup>*<sup>w</sup>�fr <sup>þ</sup> *<sup>c</sup>*bwm, (14) Second range : *<sup>c</sup>*we‐<sup>2</sup> <sup>¼</sup> *<sup>c</sup>*<sup>w</sup>�nfr <sup>þ</sup> *<sup>c</sup>*fw, (15) Third range : *<sup>c</sup>*we‐<sup>3</sup> <sup>¼</sup> *<sup>c</sup>*<sup>w</sup>�nfr*:* (16)

�

fsp

exp 0½ � *:*0567ð Þ *T* � 272*:*15 1 þ *u*

þ

þ

9

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=

(17)

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

� � � � , (10)

� 0*:*124 � �

fsp � 0*:*001ð Þ *T* � 293*:*15 , (12)

fsp þ 0*:*021, (13)

fsp is the fiber saturation point at *T* = 272.15 K (i.e.,

*:* (9)

*:* (11)

fsp , are calculated

(RSME) between the calculated model and experimentally obtained results about

**3.1 Experimental research of the 2D temperature distribution in poplar logs**

For solving the inverse task of the heat transfer aimed at validation of the suggested above mathematical model, it is necessary to have experimentally obtained data about the 2D temperature distribution in logs during their thawing. That is why we realized such experiments using poplar (*Populus nigra* L.) logs with

In **Figure 4** the coordinates of four representative points of the logs, in which the 2D change in the temperature was measured and registered during the logs'

The automatic measurement and record of *t*m, *φ*m, and *t* in the representative points of the logs during the experiments was accomplished by Data Logger type

In **Figures 2** and **3**, the change in the temperature of the processing air medium,

*t*m, and in its humidity, *φ*m, and also in the temperature in four representative points of two poplar logs, named below as P1 and P2, respectively, during their

*Radial (left) and longitudinal (right) coordinates of four characteristic points for the measurement of the*

All curves of the experimentally obtained data on these figures are drawn using the licensed software HW4 of the Data Logger. The left coordinate axis on the figures is graduated at % of *φ*m, and the right one is

For the freezing of the logs before their thawing, a horizontal freezer was used with adjustable temperature range from –1 to –30°C. Sensors Pt100 with long metal casings were positioned in the drilled four holes of the logs. After 50 h separately freezing each logs, the freezer was switched off. Then its lid was opened,

and 70 h thawing of the log at room temperature was carried out.

HygroLog NT3 produced by the Swiss firm ROTRONIC AG.

the change of the temperature fields subjected to thawing logs.

*Modeling of the Two-Dimensional Thawing of Logs in an Air Environment*

**3. Experimental research of the thawing process of logs**

**during their thawing**

thawing, are shown.

graduated at °

**Figure 4.**

**77**

*D* = 0.24 m, *L* = 0.48 m, and *u* > *u*fsp [26].

*DOI: http://dx.doi.org/10.5772/intechopen.93177*

separate 70 h thawing is presented.

C of *t*.

*temperature in logs subjected to thawing.*

#### **Figure 2.**

*Experimentally determined change in tm, φm, and t in four points of the studied poplar log P1 during its 70 h thawing.*

#### **Figure 3.**

*Experimentally determined change in tm, φm, and t in four points of the studied poplar log P2 during its 70 h thawing.*

• In the radial direction on the cylindrical surface of the logs

$$a\_{\rm wr} = \mathbf{1.123} [T(\mathbf{0}, z, \tau) - T\_{\rm m}(\tau)]^\ast \tag{27}$$

• In the longitudinal direction on the frontal surface of the logs

$$a\_{\rm wp} = 2.56[T(r, \mathbf{0}, \tau) - T\_{\rm m}(\tau)]^\times \tag{28}$$

where *x* is an exponent, whose values are determined during the solving and validation of the model through the minimization of the root-square-mean error (RSME) between the calculated model and experimentally obtained results about the change of the temperature fields subjected to thawing logs.

### **3. Experimental research of the thawing process of logs**

### **3.1 Experimental research of the 2D temperature distribution in poplar logs during their thawing**

For solving the inverse task of the heat transfer aimed at validation of the suggested above mathematical model, it is necessary to have experimentally obtained data about the 2D temperature distribution in logs during their thawing. That is why we realized such experiments using poplar (*Populus nigra* L.) logs with *D* = 0.24 m, *L* = 0.48 m, and *u* > *u*fsp [26].

In **Figure 4** the coordinates of four representative points of the logs, in which the 2D change in the temperature was measured and registered during the logs' thawing, are shown.

For the freezing of the logs before their thawing, a horizontal freezer was used with adjustable temperature range from –1 to –30°C. Sensors Pt100 with long metal casings were positioned in the drilled four holes of the logs. After 50 h separately freezing each logs, the freezer was switched off. Then its lid was opened, and 70 h thawing of the log at room temperature was carried out.

The automatic measurement and record of *t*m, *φ*m, and *t* in the representative points of the logs during the experiments was accomplished by Data Logger type HygroLog NT3 produced by the Swiss firm ROTRONIC AG.

In **Figures 2** and **3**, the change in the temperature of the processing air medium, *t*m, and in its humidity, *φ*m, and also in the temperature in four representative points of two poplar logs, named below as P1 and P2, respectively, during their separate 70 h thawing is presented.

All curves of the experimentally obtained data on these figures are drawn using the licensed software HW4 of the Data Logger. The left coordinate axis on the figures is graduated at % of *φ*m, and the right one is graduated at ° C of *t*.

#### **Figure 4.**

• In the radial direction on the cylindrical surface of the logs

**Figure 2.**

*thawing.*

**Figure 3.**

*thawing.*

**76**

• In the longitudinal direction on the frontal surface of the logs

where *x* is an exponent, whose values are determined during the solving and validation of the model through the minimization of the root-square-mean error

*Experimentally determined change in tm, φm, and t in four points of the studied poplar log P2 during its 70 h*

*Experimentally determined change in tm, φm, and t in four points of the studied poplar log P1 during its 70 h*

*Modeling and Simulation in Engineering - Selected Problems*

*<sup>α</sup>*wr <sup>¼</sup> <sup>1</sup>*:*123½ � *<sup>T</sup>*ð Þ� 0, *<sup>z</sup>*, *<sup>τ</sup> <sup>T</sup>*mð Þ*<sup>τ</sup> <sup>x</sup>* (27)

*<sup>α</sup>*wp <sup>¼</sup> <sup>2</sup>*:*56½ � *T r*ð Þ� , 0, *<sup>τ</sup> <sup>T</sup>*mð Þ*<sup>τ</sup> <sup>x</sup>* (28)

*Radial (left) and longitudinal (right) coordinates of four characteristic points for the measurement of the temperature in logs subjected to thawing.*

### **3.2 Mathematical description of the air medium temperature during logs' thawing**

The change shown in **Figures 2** and **3** air medium temperature *T*<sup>m</sup> during the logs' thawing with correlation 0.98 and root-square-mean error, *σ* < 1.5°C, has been approximated with the help of the software package Table Curve 2D by the following equation:

$$T\_{\mathbf{m}} = \frac{a + c \cdot \boldsymbol{\tau}^{0.5}}{\mathbf{1} + b \cdot \boldsymbol{\tau}^{0.5}},\tag{29}$$

These figures show the positioning of the knots of the calculation mesh and four representative points, in which the nonstationary 2D distribution of the temperature in the longitudinal section subjected to thawing log has been calculated. The mesh has been built on ¼ of the longitudinal section of the log due to the fact that this ¼ is mirrored symmetrical towards the remaining ¾ of the same section. Taking into consideration Eqs. (5) and (6), it can be written that

*Calculation mesh and representative points T1,T2,T3, and T4 on ¼ of the longitudinal section subjected to*

*Modeling of the Two-Dimensional Thawing of Logs in an Air Environment*

*DOI: http://dx.doi.org/10.5772/intechopen.93177*

*<sup>K</sup>*wp*=*wr <sup>¼</sup> *<sup>K</sup>*wp

suitable for programming in FORTRAN, has the following form [7, 30]:

*<sup>∂</sup>T r*ð Þ , *<sup>z</sup>*, *<sup>τ</sup> <sup>∂</sup><sup>τ</sup>* <sup>≈</sup>*<sup>c</sup>*

The discrete finite-difference analogue of the left-hand part of Eq. (1), which is

*n*

Taking into account Eqs. (30), (31), and (32), the discrete analogue of the right-

*<sup>∂</sup>T r*ð Þ , *<sup>z</sup>*, *<sup>τ</sup> ∂r* <sup>2</sup>

Δ*r*

*<sup>i</sup>*,*k*�<sup>1</sup> � <sup>2</sup>*T<sup>n</sup>*

*K*wr

we�1,2,3 � *<sup>ρ</sup>*<sup>w</sup>

þ

*<sup>i</sup>*,*<sup>k</sup>* <sup>þ</sup> *<sup>T</sup><sup>n</sup> i*,*k*þ1 <sup>Δ</sup>*z*<sup>2</sup> <sup>þ</sup> *<sup>β</sup>* � *<sup>T</sup><sup>n</sup>*

*<sup>i</sup>*,*<sup>k</sup>* � <sup>273</sup>*:*<sup>15</sup> �

<sup>þ</sup> *<sup>β</sup>* � *<sup>T</sup><sup>n</sup>*

*<sup>i</sup>*�1,*<sup>k</sup>* � *<sup>T</sup><sup>n</sup> i*,*k*

Δ*r*<sup>2</sup> <sup>2</sup>

<sup>¼</sup> *<sup>λ</sup>*w0r � *<sup>γ</sup>* � <sup>1</sup> <sup>þ</sup> *<sup>β</sup>* � *<sup>T</sup><sup>n</sup>*

<sup>λ</sup>w0r <sup>¼</sup> *<sup>K</sup>*wr � *<sup>v</sup>* � <sup>0</sup>*:*<sup>165</sup> <sup>þ</sup> ð Þ� <sup>1</sup>*:*<sup>39</sup> <sup>þ</sup> <sup>3</sup>*:*8*<sup>u</sup>* <sup>3</sup>*:*<sup>3</sup> � <sup>10</sup>�<sup>7</sup>

<sup>λ</sup>w0p <sup>¼</sup> *<sup>K</sup>*wp � *<sup>v</sup>* � <sup>0</sup>*:*<sup>165</sup> <sup>þ</sup> ð Þ� <sup>1</sup>*:*<sup>39</sup> <sup>þ</sup> <sup>3</sup>*:*8*<sup>u</sup>* <sup>3</sup>*:*<sup>3</sup> � <sup>10</sup>�<sup>7</sup>

*c*we�1,2,3 � *ρ*<sup>w</sup>

hand part of Eq. (1) has the following form:

*∂λ*wp *∂T*

*<sup>i</sup>*,*<sup>k</sup>* � <sup>273</sup>*:*<sup>15</sup> �

*<sup>∂</sup>T r*ð Þ , *<sup>z</sup>*, *<sup>τ</sup> ∂r*

þ *∂λ*wr *∂T*

*<sup>∂</sup>T r*ð Þ , *<sup>z</sup>*, *<sup>τ</sup> ∂z* <sup>2</sup>

> 1 ð Þ� *i* � 1 Δ*r*

� *Tn <sup>i</sup>*�1,*<sup>k</sup>* � *<sup>T</sup><sup>n</sup> i*,*k*

*Tn*

1 *r :*

*<sup>i</sup>*,*<sup>k</sup>* <sup>þ</sup> *<sup>T</sup><sup>n</sup> i*þ1,*k* Δ*r*<sup>2</sup> þ

*λ*wr *∂*2

**Figure 6.**

*thawing log.*

þ*λ*wp *∂*2

� *Tn*

**79**

*T r*ð Þ , *z*, *τ ∂r*<sup>2</sup> þ

> *T r*ð Þ , *z*, *τ ∂z*<sup>2</sup> þ

<sup>þ</sup> *<sup>λ</sup>*w0p � *<sup>γ</sup>* � <sup>1</sup> <sup>þ</sup> *<sup>β</sup>* � *<sup>T</sup><sup>n</sup>*

*<sup>i</sup>*�1,*<sup>k</sup>* � <sup>2</sup>*T<sup>n</sup>*

and using the coefficient

*λ*<sup>w</sup> ¼ *λ*w0r,p � *γ* � ½ � 1 þ *β* � ð Þ *T* � 273*:*15 (30)

(31)

(32)

*ρ*2

*ρ*2

*T<sup>n</sup>*þ<sup>1</sup> *<sup>i</sup>*,*<sup>k</sup>* � *<sup>T</sup><sup>n</sup> i*,*k*

<sup>b</sup> <sup>þ</sup> <sup>1</sup>*:*<sup>015</sup> � <sup>10</sup>�<sup>3</sup>

<sup>b</sup> <sup>þ</sup> <sup>1</sup>*:*<sup>015</sup> � <sup>10</sup>�<sup>3</sup>

*ρ*b

*ρ*b

<sup>Δ</sup>*<sup>τ</sup> :* (34)

þ

*:*

(35)

*<sup>i</sup>*,*k*�<sup>1</sup> � *<sup>T</sup><sup>n</sup> i*,*k*

Δ*z*<sup>2</sup> <sup>2</sup> (33)

whose coefficients are equal to:


and *τ* is the sum of the time of logs' freezing, equal to 50 h = 180,000 s, and the current time of the subsequent thawing of the logs, s.

Eq. (29) was used for solving Eqs. (3) and (4) of the model.

### **4. Numerical solution of the mathematical model of the logs' thawing process**

The mathematical descriptions of the thermophysical characteristics of the logs and also of *T*<sup>m</sup> considered above were introduced in the mathematical model (1) to (4). An explicit form of the finite-difference method was used for solving of the model without any simplifications [7, 8].

#### **4.1 Presentation of the model in a form suitable for programming**

### *4.1.1 Presentation of Eq. (1) of the model*

The presentation of Eq. (1) of the model suitable for programming discrete analogue has been carried out using the given in **Figures 5** and **6** coordinate system.

**Figure 5.**

*Positioning of the knots of 2D calculation mesh on ¼ of longitudinal section of a log subjected to thawing (left) and calculation mesh for solving of the model (right).*

*Modeling of the Two-Dimensional Thawing of Logs in an Air Environment DOI: http://dx.doi.org/10.5772/intechopen.93177*

#### **Figure 6.**

**3.2 Mathematical description of the air medium temperature during logs'**

The change shown in **Figures 2** and **3** air medium temperature *T*<sup>m</sup> during the logs' thawing with correlation 0.98 and root-square-mean error, *σ* < 1.5°C, has been approximated with the help of the software package Table Curve 2D by the

*<sup>Т</sup>*<sup>m</sup> <sup>¼</sup> *<sup>a</sup>* <sup>þ</sup> *<sup>c</sup>* � *<sup>τ</sup>*0*:*<sup>5</sup>

and *τ* is the sum of the time of logs' freezing, equal to 50 h = 180,000 s, and the

**4. Numerical solution of the mathematical model of the logs' thawing**

and also of *T*<sup>m</sup> considered above were introduced in the mathematical model (1) to (4). An explicit form of the finite-difference method was used for solving of

The presentation of Eq. (1) of the model suitable for programming discrete analogue has been carried out using the given in **Figures 5** and **6** coordinate system.

*Positioning of the knots of 2D calculation mesh on ¼ of longitudinal section of a log subjected to thawing (left)*

**4.1 Presentation of the model in a form suitable for programming**

The mathematical descriptions of the thermophysical characteristics of the logs

• For log P1: *а* = 293.3637194, *b* = �0.00236425, *c* = �0.69281743.

• For log P2: *a* = 299.2738855, *b* = �0.00245303, *c* = �0.73047119.

Eq. (29) was used for solving Eqs. (3) and (4) of the model.

current time of the subsequent thawing of the logs, s.

*Modeling and Simulation in Engineering - Selected Problems*

the model without any simplifications [7, 8].

*4.1.1 Presentation of Eq. (1) of the model*

*and calculation mesh for solving of the model (right).*

<sup>1</sup> <sup>þ</sup> *<sup>b</sup>* � *<sup>τ</sup>*0*:*<sup>5</sup> , (29)

**thawing**

following equation:

**process**

**Figure 5.**

**78**

whose coefficients are equal to:

*Calculation mesh and representative points T1,T2,T3, and T4 on ¼ of the longitudinal section subjected to thawing log.*

These figures show the positioning of the knots of the calculation mesh and four representative points, in which the nonstationary 2D distribution of the temperature in the longitudinal section subjected to thawing log has been calculated. The mesh has been built on ¼ of the longitudinal section of the log due to the fact that this ¼ is mirrored symmetrical towards the remaining ¾ of the same section.

Taking into consideration Eqs. (5) and (6), it can be written that

$$
\lambda\_{\rm w} = \lambda\_{\rm w0r,p} \cdot \chi \cdot [1 + \beta \cdot (T - 27 \mathbf{3.15})] \tag{30}
$$

$$\lambda\_{\rm w0r} = K\_{\rm wr} \cdot v \cdot \left[ 0.165 + (1.39 + 3.8u) \cdot \left( 3.3 \cdot 10^{-7} \rho\_{\rm b}^2 + 1.015 \cdot 10^{-3} \rho\_{\rm b} \right) \right] \tag{31}$$

$$\lambda\_{\rm w0p} = K\_{\rm wp} \cdot v \cdot \left[ 0.165 + (1.39 + 3.8u) \cdot \left( 3.3 \cdot 10^{-7} \rho\_{\rm b}^2 + 1.015 \cdot 10^{-3} \rho\_{\rm b} \right) \right] \tag{32}$$

and using the coefficient

$$K\_{\rm wp/wr} = \frac{K\_{\rm wp}}{K\_{\rm wr}} \tag{33}$$

The discrete finite-difference analogue of the left-hand part of Eq. (1), which is suitable for programming in FORTRAN, has the following form [7, 30]:

$$
\sigma\_{\text{we}-1,2,3} \cdot \rho\_{\text{w}} \frac{\partial T(r,z,\tau)}{\partial \tau} \approx c\_{\text{we}-1,2,3}^{n} \cdot \rho\_{\text{w}} \frac{T\_{i,k}^{n+1} - T\_{i,k}^{n}}{\Delta \tau}. \tag{34}
$$

Taking into account Eqs. (30), (31), and (32), the discrete analogue of the righthand part of Eq. (1) has the following form:

$$\begin{split} \lambda\_{\text{wr}} & \left[ \frac{\partial^2 T(r, z, \tau)}{\partial r^2} + \frac{1}{r} \cdot \frac{\partial T(r, z, \tau)}{\partial r} \right] + \frac{\partial \lambda\_{\text{wr}}}{\partial T} \left[ \frac{\partial T(r, z, \tau)}{\partial r} \right]^2 + \\ & + \lambda\_{\text{wr}} \frac{\partial^2 T(r, z, \tau)}{\partial z^2} + \frac{\partial \lambda\_{\text{wr}}}{\partial T} \left[ \frac{\partial T(r, z, \tau)}{\partial z} \right]^2 = \left\{ \lambda\_{\text{w0}} \cdot \mathbf{y} \cdot \left[ 1 + \boldsymbol{\beta} \cdot \left( T^u\_{i,k} - 273.15 \right) \right] \right\}. \\ \cdot \left[ \frac{T^u\_{i-1,k} - 2T^u\_{i,k} + T^u\_{i+1,k}}{\Delta r^2} + \frac{1}{(i-1) \cdot \Delta r} \cdot \frac{T^u\_{i-1,k} - T^u\_{i,k}}{\Delta r} \right] + \boldsymbol{\beta} \cdot \left( \frac{T^u\_{i-1,k} - T^u\_{i,k}}{\Delta r^2} \right)^2 + \\ & + \left\{ \lambda\_{\text{w0}} \cdot \mathbf{y} \cdot \left[ 1 + \boldsymbol{\beta} \cdot \left( T^u\_{i,k} - 273.15 \right) \right] \right\} \cdot \frac{T^u\_{i,k-1} - 2T^u\_{i,k} + T^u\_{i,k+1}}{\Delta \sigma^2} + \boldsymbol{\beta} \cdot \left( \frac{T^u\_{i,k-1} - T^u\_{i,k}}{\Delta \sigma^2} \right)^2. \end{split} \tag{35}$$

After alignment of Eq. (34) with Eq. (35) and taking into account Eq. (33), at Δ*z* = Δ*r*, it is obtained that Eq. (1) is transformed into the following system of algebraic equations:

where according to Eqs. (28) and (29)

*DOI: http://dx.doi.org/10.5772/intechopen.93177*

*longitudinal coordinate*

The variable *Gn*

and *Т<sup>n</sup>*

**81**

where

*αn*

*Modeling of the Two-Dimensional Thawing of Logs in an Air Environment*

*Тn*

the following final form, suitable for programming in FORTRAN:

*Т<sup>n</sup>*þ<sup>1</sup> 1,*<sup>k</sup>* <sup>¼</sup> *<sup>T</sup><sup>n</sup>*

1,*<sup>k</sup>* in Eq. (42) is equal to

*αn*

in the calculation environment of Visual FORTRAN Professional.

<sup>m</sup> is calculated according to Eq. (41).

1,*<sup>k</sup>* <sup>¼</sup> <sup>Δ</sup>*<sup>r</sup>* � *<sup>α</sup><sup>n</sup>*

*<sup>λ</sup>*w0r � *<sup>γ</sup>* � <sup>1</sup> <sup>þ</sup> *<sup>β</sup>* � *<sup>T</sup><sup>n</sup>*

wr <sup>¼</sup> <sup>1</sup>*:*<sup>123</sup> *<sup>T</sup><sup>n</sup>*

*Gn*

where according to Eq. (27)

**4.2 Input data for solving of the model**

presented in **Figures 2** and **3**, respectively.

logs during the experiments were as follows:

they were during the experimental research.

*K*wr = 1.48, and *K*wp = 2.88 [7, 9] have been used.

• For log P1: *<sup>t</sup>*w0-avg <sup>=</sup> �29.7°C, *<sup>ρ</sup>*<sup>b</sup> = 359 kg�kg�<sup>1</sup>

• For log P2: *<sup>t</sup>*w0-avg <sup>=</sup> �28.0°C, *<sup>ρ</sup>*<sup>b</sup> = 364 kg�kg�<sup>1</sup>

wp <sup>¼</sup> <sup>2</sup>*:*<sup>56</sup> *<sup>T</sup><sup>n</sup>*

*4.1.4. Presentation of the equation of boundary condition in the model along the*

<sup>m</sup> <sup>¼</sup> *<sup>a</sup>* <sup>þ</sup> *<sup>c</sup>* � ð Þ *<sup>n</sup>* � <sup>Δ</sup>*<sup>τ</sup>* <sup>0</sup>*:*<sup>5</sup>

Analogously, the boundary condition (4) of the logs' thawing process obtains

2,*<sup>k</sup>* <sup>þ</sup> *<sup>G</sup><sup>n</sup>*

<sup>1</sup> <sup>þ</sup> *<sup>G</sup><sup>n</sup>* 1,*k*

1,*<sup>k</sup>* � *<sup>T</sup><sup>n</sup>*þ<sup>1</sup> <sup>m</sup>

wr

1,*<sup>k</sup>* � *<sup>T</sup><sup>n</sup>* m

The numerical solving and verification of the model (1) to (4) has been realized

The initial temperature, *t*w0-avg; basic density, *ρ*b; and moisture content, *u*, of the

As it was mentioned above, the duration of the freezing and duration of the

The model was solved with step Δ*r* = Δ*z* = 0.006 m along the coordinates *r* and *z*, with step Δτ = 6 s [8, 23], and with the same initial and boundary conditions, as

subsequent thawing of the logs were equal to 50 and 70 h, respectively.

During the solving of the model, mathematical descriptions of the

thermophysical characteristics of poplar sapwood with *u*<sup>293</sup>*:*<sup>15</sup>

Using own software package in that environment, computations were carried out for the determination of the 2D nonstationary change of *t* in the representative points of the logs P1 and P2, whose experimentally registered temperature fields are

*<sup>i</sup>*,1 � *<sup>T</sup><sup>n</sup>* m *<sup>х</sup>*

, (40)

*:* (42)

<sup>1</sup> <sup>þ</sup> *<sup>b</sup>* � ð Þ *<sup>n</sup>* � <sup>Δ</sup>*<sup>τ</sup>* <sup>0</sup>*:*<sup>5</sup> *:* (41)

1,*<sup>k</sup>* � <sup>273</sup>*:*<sup>15</sup> , (43)

*<sup>х</sup>* (44)

, and *<sup>u</sup>* = 1.44 kg�kg�<sup>1</sup>

, and *<sup>u</sup>* = 1.78 kg�kg�<sup>1</sup>

fsp <sup>¼</sup> <sup>0</sup>*:*35 kg�kg�<sup>1</sup>

.

.

,

$$T\_{i,k}^{n+1} = T\_{i,k}^{n} + \frac{\lambda\_{\text{w0r}} \cdot \boldsymbol{\gamma} \cdot \Delta \boldsymbol{\tau}}{c\_{\text{w0r},2,3}^{n} \cdot \rho\_{\text{w}} \cdot \Delta \boldsymbol{\tau}^{2}}.$$

$$\left\{ \begin{aligned} &\left[\boldsymbol{\mathbbm{1}} + \boldsymbol{\beta} \cdot \left(T\_{i,k}^{n} - 273.15\right)\right] \cdot \begin{bmatrix} T\_{i-1,k}^{n} + T\_{i+1,k}^{n} + K\_{\text{wp}/\text{wr}} \left(T\_{i,k-1}^{n} + T\_{i,k+1}^{n}\right) - \\ - \left(\boldsymbol{2} + 2K\_{\text{wp}/\text{wr}}\right) T\_{i,k}^{n} + \frac{1}{i-1} \left(T\_{i-1,k}^{n} - T\_{i,k}^{n}\right) \end{bmatrix} + \\ &+ \boldsymbol{\beta} \cdot \left[\left(T\_{i-1,k}^{n} - T\_{i,k}^{n}\right)^{2} + K\_{\text{wp}/\text{wr}} \left(T\_{i,k-1}^{n} - T\_{i,k}^{n}\right)^{2}\right] \end{aligned} \right\}. \tag{36}$$

The term <sup>1</sup> *<sup>r</sup>* in the right-hand part of Eq. (1) is represented as <sup>1</sup> ð Þ� *<sup>i</sup>*�<sup>1</sup> <sup>Δ</sup>*<sup>r</sup>* in Eq. (35). According to the requirements of FORTRAN [7, 30], the knots of the calculation mesh, which are situated on the log's surfaces, are denoted by numbers *i* = 1 and *k* = 1 along the coordinate axes *r* and *z*, respectively.

The temperature in these surface knots is calculated with the help of Eqs. (38) and (42) given below. Since Eq. (36) calculates the temperature in the knots, which are located inside the logs, i.e., in the knots with *i* ≥ 2 and *k* ≥ 2, the denominator of the term <sup>1</sup> *<sup>i</sup>*�<sup>1</sup> in this equation is always greater than zero.

It can be noted that the effective specific heat capacities of the log during the pointed above three ranges of its thawing process (see **Figure 1**), *<sup>c</sup>*we‐1, *<sup>c</sup>*we‐2, and *<sup>c</sup>*we‐3, which are unitedly represented as *<sup>c</sup>*we‐1,2,3 in Eq. (36), are computed according to Eq. (17) separately for each knot of the calculation mesh.

#### *4.1.2 Presentation of the equation of the initial condition in the model*

The initial condition (2) of the model of logs' thawing process obtains the following discrete finite-difference form:

$$T\_{i,k}^0 = T\_{\text{w0-avg}} \tag{37}$$

where *T*w0-avg is the experimentally determined average mass temperature of the log at the beginning of the thawing process, K.

### *4.1.3 Presentation of the equation of boundary condition in the model along the radial coordinate*

The boundary condition (3) of the logs' thawing process obtains the following final form, suitable for programming in FORTRAN:

$$T\_{i,1}^{n+1} = \frac{T\_{i,2}^n + G\_{i,1}^n \cdot T\_{\text{m}}^{n+1}}{1 + G\_{i,1}^n} \,. \tag{38}$$

The variable *G<sup>n</sup> <sup>i</sup>*,1 in Eq. (38) is equal to

$$G\_{i,1}^{n} = \frac{\Delta r \cdot a\_{\rm wp}^{n}}{\lambda\_{\rm w0p} \cdot \chi \cdot \left[1 + \beta \cdot \left(T\_{i,1}^{n} - 273.15\right)\right]},\tag{39}$$

*Modeling of the Two-Dimensional Thawing of Logs in an Air Environment DOI: http://dx.doi.org/10.5772/intechopen.93177*

where according to Eqs. (28) and (29)

$$a\_{\rm wp}^n = 2.56 \left[ T\_{i,1}^n - T\_{\rm m}^n \right]^x,\tag{40}$$

where

After alignment of Eq. (34) with Eq. (35) and taking into account Eq. (33), at Δ*z* = Δ*r*, it is obtained that Eq. (1) is transformed into the following system of

*<sup>i</sup>*þ1,*<sup>k</sup>* <sup>þ</sup> *<sup>K</sup>*wp*=*wr *<sup>T</sup><sup>n</sup>*

*i*,*k* þ

1 *i* � 1

*<sup>i</sup>*,*k*�<sup>1</sup> <sup>þ</sup> *<sup>T</sup><sup>n</sup>*

*<sup>i</sup>*,*<sup>k</sup>* <sup>¼</sup> *<sup>T</sup>*w0‐avg (37)

*<sup>i</sup>*,1 � <sup>273</sup>*:*<sup>15</sup> � � � � , (39)

*:* (38)

*Tn*

*i*,*k*þ1 � �� 3 7 5þ 9 >>>>>=

(36)

>>>>>; *:*

ð Þ� *<sup>i</sup>*�<sup>1</sup> <sup>Δ</sup>*<sup>r</sup>* in Eq. (35).

*<sup>i</sup>*�1,*<sup>k</sup>* � *<sup>T</sup><sup>n</sup> i*,*k*

� �

algebraic equations:

*i*,*k* þ

<sup>1</sup> <sup>þ</sup> *<sup>β</sup>* � *<sup>T</sup><sup>n</sup>*

<sup>þ</sup>*<sup>β</sup>* � *<sup>T</sup><sup>n</sup>*

The term <sup>1</sup>

the term <sup>1</sup>

*coordinate*

The variable *G<sup>n</sup>*

**80**

*cn*

*<sup>i</sup>*,*<sup>k</sup>* � <sup>273</sup>*:*<sup>15</sup> � � � � *:*

*<sup>i</sup>*�1,*<sup>k</sup>* � *<sup>T</sup><sup>n</sup> i*,*k* � �<sup>2</sup> <sup>þ</sup> *<sup>K</sup>*wp*=*wr *<sup>T</sup><sup>n</sup>*

*λ*w0r � *γ* � Δ*τ*

we‐1,2,3 � *<sup>ρ</sup>*<sup>w</sup> � <sup>Δ</sup>*r*<sup>2</sup> *:*

*Modeling and Simulation in Engineering - Selected Problems*

*Tn*

2 6 4

� �<sup>2</sup> h i

*k* = 1 along the coordinate axes *r* and *z*, respectively.

*<sup>i</sup>*�<sup>1</sup> in this equation is always greater than zero.

*4.1.2 Presentation of the equation of the initial condition in the model*

to Eq. (17) separately for each knot of the calculation mesh.

following discrete finite-difference form:

log at the beginning of the thawing process, K.

final form, suitable for programming in FORTRAN:

*Gn*

*Т<sup>n</sup>*þ<sup>1</sup> *<sup>i</sup>*,1 <sup>¼</sup> *<sup>T</sup><sup>n</sup>*

*<sup>i</sup>*,1 <sup>¼</sup> <sup>Δ</sup>*<sup>r</sup>* � *<sup>α</sup><sup>n</sup>*

*<sup>λ</sup>*w0p � *<sup>γ</sup>* � <sup>1</sup> <sup>þ</sup> *<sup>β</sup>* � *<sup>T</sup><sup>n</sup>*

*<sup>i</sup>*,1 in Eq. (38) is equal to

*<sup>i</sup>*�1,*<sup>k</sup>* <sup>þ</sup> *<sup>T</sup><sup>n</sup>*

� 2 þ 2*K*wp*=*wr � �*T<sup>n</sup>*

> *<sup>i</sup>*,*k*�<sup>1</sup> � *<sup>T</sup><sup>n</sup> i*,*k*

*<sup>r</sup>* in the right-hand part of Eq. (1) is represented as <sup>1</sup>

According to the requirements of FORTRAN [7, 30], the knots of the calculation mesh, which are situated on the log's surfaces, are denoted by numbers *i* = 1 and

The temperature in these surface knots is calculated with the help of Eqs. (38) and (42) given below. Since Eq. (36) calculates the temperature in the knots, which are located inside the logs, i.e., in the knots with *i* ≥ 2 and *k* ≥ 2, the denominator of

It can be noted that the effective specific heat capacities of the log during the pointed above three ranges of its thawing process (see **Figure 1**), *<sup>c</sup>*we‐1, *<sup>c</sup>*we‐2, and *<sup>c</sup>*we‐3, which are unitedly represented as *<sup>c</sup>*we‐1,2,3 in Eq. (36), are computed according

The initial condition (2) of the model of logs' thawing process obtains the

where *T*w0-avg is the experimentally determined average mass temperature of the

*4.1.3 Presentation of the equation of boundary condition in the model along the radial*

The boundary condition (3) of the logs' thawing process obtains the following

*<sup>i</sup>*,2 <sup>þ</sup> *<sup>G</sup><sup>n</sup>*

<sup>1</sup> <sup>þ</sup> *<sup>G</sup><sup>n</sup> i*,1

wp

*<sup>i</sup>*,1 � *<sup>T</sup><sup>n</sup>*þ<sup>1</sup> <sup>m</sup>

*Т*0

*Tn*þ<sup>1</sup> *<sup>i</sup>*,*<sup>k</sup>* <sup>¼</sup> *<sup>T</sup><sup>n</sup>*

8 >>>>><

*:*

>>>>>:

$$T\_{\rm m}^{n} = \frac{a + c \cdot (n \cdot \Delta \pi)^{0.5}}{1 + b \cdot (n \cdot \Delta \pi)^{0.5}}.\tag{41}$$

### *4.1.4. Presentation of the equation of boundary condition in the model along the longitudinal coordinate*

Analogously, the boundary condition (4) of the logs' thawing process obtains the following final form, suitable for programming in FORTRAN:

$$T\_{1,k}^{n+1} = \frac{T\_{2,k}^n + G\_{1,k}^n \cdot T\_{\text{m}}^{n+1}}{1 + G\_{1,k}^n} \,. \tag{42}$$

The variable *Gn* 1,*<sup>k</sup>* in Eq. (42) is equal to

$$G\_{1,k}^{n} = \frac{\Delta r \cdot \alpha\_{\rm wr}^{n}}{\lambda\_{\rm w0r} \cdot \chi \cdot \left[1 + \beta \cdot \left(T\_{1,k}^{n} - 273.15\right)\right]},\tag{43}$$

where according to Eq. (27)

$$a\_{\rm wr}^{\rm n} = \mathbf{1.123} \left[ T\_{1,k}^{\rm n} - T\_{\rm m}^{\rm n} \right]^{\rm x} \tag{44}$$

and *Т<sup>n</sup>* <sup>m</sup> is calculated according to Eq. (41).

### **4.2 Input data for solving of the model**

The numerical solving and verification of the model (1) to (4) has been realized in the calculation environment of Visual FORTRAN Professional.

Using own software package in that environment, computations were carried out for the determination of the 2D nonstationary change of *t* in the representative points of the logs P1 and P2, whose experimentally registered temperature fields are presented in **Figures 2** and **3**, respectively.

The initial temperature, *t*w0-avg; basic density, *ρ*b; and moisture content, *u*, of the logs during the experiments were as follows:


As it was mentioned above, the duration of the freezing and duration of the subsequent thawing of the logs were equal to 50 and 70 h, respectively.

The model was solved with step Δ*r* = Δ*z* = 0.006 m along the coordinates *r* and *z*, with step Δτ = 6 s [8, 23], and with the same initial and boundary conditions, as they were during the experimental research.

During the solving of the model, mathematical descriptions of the thermophysical characteristics of poplar sapwood with *u*<sup>293</sup>*:*<sup>15</sup> fsp <sup>¼</sup> <sup>0</sup>*:*35 kg�kg�<sup>1</sup> , *K*wr = 1.48, and *K*wp = 2.88 [7, 9] have been used.

### **4.3 Inverse determination of the heat transfer coefficients during solving of the model**

The model (1) to (4) was solved with various values of the exponent *x* in Eqs. (27) and (28). The computed by the model change of *t* in the four representative points of the logs with each of the tested values of the exponent *x* during the thawing was compared mathematically with the corresponding one experimentally registered change of *t* in these points with an interval of 15 min.

The aim of this comparison was to determine the values of *x*, which ensure the best compliance between the computed and experimentally registered temperature fields in subjected to thawing logs.

As a criterion of the best compliance, the minimum average value of RSME, *σ*avg, was used, which is equal to

$$\sigma\_{\text{avg}} = \sqrt{\frac{\sum\_{n=1}^{N} \sum\_{p=1}^{P} \left(t\_{p,n}^{\text{comp}} - t\_{p,n}^{\text{exp}}\right)^2}{P \cdot (N-1)}},\tag{45}$$

exponent *x* in Eqs. (27) and (28), which were obtained during the solving of the

**Figures 7** and **8** present the calculated change in *α*wr and *α*wp during the studied

**Figures 9** and **10** present the calculated change in *t*<sup>m</sup> and also in the logs'surface

It can be seen that with the decrease of the difference between *t*<sup>m</sup> and *t*<sup>s</sup> during

�K�<sup>1</sup> for P1 and from 1.9 to 1.2 W�m�<sup>2</sup>

�K�<sup>1</sup> for P1 and from 4.5 to 2.8 W�m�<sup>2</sup>

�K�<sup>1</sup> for P2.

�K�<sup>1</sup> for P2.

the logs' thawing, the heat transfer coefficients on **Figures 7** and **8** gradually

inverse task, *x* = 0.22 for log P1 and *x* = 0.20 for log P2.

temperature *t*<sup>s</sup> and *t* of 4 representative points of the studied logs.

*Modeling of the Two-Dimensional Thawing of Logs in an Air Environment*

thawing process of the logs P1 and P2, respectively.

*Calculated change in αwr and αwp of the log P2 during its 70 h thawing.*

*Experimentally determined and calculated change in tm, ts, and t in four points of the log P1 during its 70 h*

decrease, as follows:

**Figure 8.**

**Figure 9.**

*thawing.* **83**

• At *<sup>α</sup>*wr: from 2.3 to 1.0 W�m�<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.93177*

• At *<sup>α</sup>*wp: from 5.1 to 2.2 W�m�<sup>2</sup>

where *t* comp *<sup>p</sup>*,*<sup>n</sup>* and *t* exp *<sup>p</sup>*,*<sup>n</sup>* are the computed and experimentally registered temperatures in the representative points; *p* is the number of the representative points of the logs, *р* = 1, 2, 3, 4, i.e., *Р* = 4 was inputted into Eq. (45); *n* is the number of the moments of the thawing process, (*n* = 1, 2, 3, … , *N* = τthaw)/(150Δτ) = 252,000 s/900 s = 280, because of the circumstance that the comparison of the computed values of *t* with experimentally registered values in the same points was made with an interval of 15 min = 900 s = 150Δτ.

For the calculation of *σ*avg, a software program in the calculation environment of MS Excel was prepared. At τthaw = 70 h = 252,000 s, RSME has been calculated with the help of the program simultaneously for a total of *N*�*P* = 1120 temperature–time points during the thawing of each log.

It was determined that the minimum values of RSME overall for the studied four representative points are equal to *σ*avg = 1.37°C for log P1 and to *σ*avg = 1.34°C for log P2. These minimum values of *σ*avg correspond to the following values of the

**Figure 7.** *Calculated change in αwr and αwp of the log P1 during its 70 h thawing.*

*Modeling of the Two-Dimensional Thawing of Logs in an Air Environment DOI: http://dx.doi.org/10.5772/intechopen.93177*

exponent *x* in Eqs. (27) and (28), which were obtained during the solving of the inverse task, *x* = 0.22 for log P1 and *x* = 0.20 for log P2.

**Figures 7** and **8** present the calculated change in *α*wr and *α*wp during the studied thawing process of the logs P1 and P2, respectively.

**Figures 9** and **10** present the calculated change in *t*<sup>m</sup> and also in the logs'surface temperature *t*<sup>s</sup> and *t* of 4 representative points of the studied logs.

It can be seen that with the decrease of the difference between *t*<sup>m</sup> and *t*<sup>s</sup> during the logs' thawing, the heat transfer coefficients on **Figures 7** and **8** gradually decrease, as follows:


**Figure 8.** *Calculated change in αwr and αwp of the log P2 during its 70 h thawing.*

**Figure 9.**

*Experimentally determined and calculated change in tm, ts, and t in four points of the log P1 during its 70 h thawing.*

**4.3 Inverse determination of the heat transfer coefficients during solving**

The model (1) to (4) was solved with various values of the exponent *x* in Eqs. (27) and (28). The computed by the model change of *t* in the four representative points of the logs with each of the tested values of the exponent *x* during the thawing was compared mathematically with the corresponding one experimentally

The aim of this comparison was to determine the values of *x*, which ensure the best compliance between the computed and experimentally registered temperature

As a criterion of the best compliance, the minimum average value of RSME, *σ*avg,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*P* � ð Þ *N* � 1

tures in the representative points; *p* is the number of the representative points of the logs, *р* = 1, 2, 3, 4, i.e., *Р* = 4 was inputted into Eq. (45); *n* is the number of the moments of the thawing process, (*n* = 1, 2, 3, … , *N* = τthaw)/(150Δτ) = 252,000 s/900 s = 280, because of the circumstance that the comparison of the computed values of *t* with experimentally registered values in the same points was made with

For the calculation of *σ*avg, a software program in the calculation environment of MS Excel was prepared. At τthaw = 70 h = 252,000 s, RSME has been calculated with the help of the program simultaneously for a total of *N*�*P* = 1120 temperature–time

It was determined that the minimum values of RSME overall for the studied four representative points are equal to *σ*avg = 1.37°C for log P1 and to *σ*avg = 1.34°C for log

P2. These minimum values of *σ*avg correspond to the following values of the

exp *p*,*n* � �<sup>2</sup>

*<sup>p</sup>*,*<sup>n</sup>* are the computed and experimentally registered tempera-

vuut , (45)

registered change of *t* in these points with an interval of 15 min.

P*<sup>N</sup> n*¼1 P*<sup>P</sup> <sup>p</sup>*¼<sup>1</sup> *<sup>t</sup>* comp *<sup>p</sup>*,*<sup>n</sup>* � *t*

*σ*avg ¼

*Modeling and Simulation in Engineering - Selected Problems*

*Calculated change in αwr and αwp of the log P1 during its 70 h thawing.*

exp

an interval of 15 min = 900 s = 150Δτ.

points during the thawing of each log.

**of the model**

fields in subjected to thawing logs.

was used, which is equal to

comp *<sup>p</sup>*,*<sup>n</sup>* and *t*

where *t*

**Figure 7.**

**82**

directions. This problem has been solved also in regard to the logs'surface temper-

• *<sup>σ</sup>*avg = 1.37°C for log P1 with *<sup>ρ</sup>*<sup>b</sup> = 359 kg�m�<sup>3</sup> and *<sup>u</sup>* = 1.44 kg�kg�<sup>1</sup>

• *<sup>σ</sup>*avg = 1.34°C for log P2 with *<sup>ρ</sup>*<sup>b</sup> = 364 kg�m�<sup>3</sup> and *<sup>u</sup>* = 1.78 kg�kg�<sup>1</sup>

coefficients subjected to thawing logs decrease gradually, as follows:

The following minimum values of the average RSME total for the temperature change in four representative points in each of the studied logs have been obtained:

During the solving of the inverse task, it was determined that the heat transfer

Good adequacy and precision of the model towards the results from extensive own experimental studies allow for the carrying out of various calculations with it, which are connected to the nonstationary temperature distribution in logs during their thawing in an air environment. For example, as a result of such calculations, it is possible to determine the real initial temperature of logs depending on their dimensions, wood species, moisture content, and the temperature of the air near the logs during their many days staying in an open warehouse before the thermal

The information about the real value of that immeasurable parameter is needed for scientifically based computing of the optimal, energy saving regimes for thermal

The model of the logs' thawing process can be applied also in the software for controllers used for advanced model predictive automatic control [20, 21, 32] of this treatment. The approach for solving of the inverse task of the heat transfer in this chapter could be further applied in the development and solving of analogous models, for example, for the calculation of the temperature fields during freezing or thawing processes of different wooden and other capillary porous materials.

This document was supported by the APVV Grant Agency as part of the project, APVV-17-0456, as a result of work of authors and the considerable assistance of the

�2

�s �1

�K�<sup>1</sup>

�K�<sup>1</sup> for P1 and from 1.9 to 1.2 W�m�<sup>2</sup>

�K�<sup>1</sup> for P1 and from 4.5 to 2.8 W�m�<sup>2</sup>

.

.

�K�<sup>1</sup> for P2.

�K�<sup>1</sup> for P2.

ature, which depends on the mentioned coefficients.

*DOI: http://dx.doi.org/10.5772/intechopen.93177*

*Modeling of the Two-Dimensional Thawing of Logs in an Air Environment*

• At *<sup>α</sup>*wr: from 2.3 to 1.0 W�m�<sup>2</sup>

• At *<sup>α</sup>*wp: from 5.1 to 2.2 W�m�<sup>2</sup>

treatment in the production of veneer.

treatment of each specific batch of logs.

*a* temperature conductivity, m<sup>2</sup>

*<sup>c</sup>* specific heat capacity, J�kg�<sup>1</sup>

*g* acceleration of gravity, *g* = 9.81 m�s

Gr Grashoff's number of similarity

Nu Nusselt's number of similarity Pr Prandtl's number of similarity

**Acknowledgements**

APVV agency.

**Nomenclature**

*D* diameter, m

*L* length, m

**85**

*R* radius: *R* = *D*/2, m

#### **Figure 10.**

*Experimentally determined and calculated change in tm, ts, and t in four points of the log P2 during its 70 h thawing.*

Using the obtained change in the heat transfer coefficients, the change in the logs'surface temperature during the thawing, *t*s, has been calculated by the model (refer to **Figures 9** and **10**).

The comparison to each other of the analogical curves in **Figures 2** and **9**, and also in **Figures 3** and **10**, shows good conformity between the calculated and experimentally determined changes in the very complicated temperature fields of the studied logs during their thawing.

During our extensive simulations with the model (1) to (4), we established good qualitative and quantitative compliance between computed and experimentally determined temperature fields of logs from numerous wood species with different moisture content above the hygroscopic range [31].

The overall RSME for the studied four representative points in the logs does not exceed 5% of the temperature ranges between the minimal and maximal temperatures of each log during its thawing.

### **5. Conclusions**

This chapter describes the creation, solving, and validation of a 2D nonlinear mathematical model for the transient heat conduction subjected to thawing frozen logs in an air environment.

The mechanism of the heat distribution in logs during their thawing has been described by a 2D equation of heat conduction at convective boundary conditions. For the numerical solving of the model with the help of explicit form of the finitedifference method, a software package has been prepared in the calculation medium of Visual FORTRAN Professional developed by Microsoft.

A validation of the model towards our own experimentally determined 2D temperature distribution in poplar logs with a diameter of 0.24 m, length of 0.48 m, and initial temperature about –30°C during their 70 h separate thawing at room temperature has been carried out.

During the validation of the model, the inverse problem has been solved for the determination of the logs' heat transfer coefficients in radial and longitudinal

directions. This problem has been solved also in regard to the logs'surface temperature, which depends on the mentioned coefficients.

The following minimum values of the average RSME total for the temperature change in four representative points in each of the studied logs have been obtained:


During the solving of the inverse task, it was determined that the heat transfer coefficients subjected to thawing logs decrease gradually, as follows:


Good adequacy and precision of the model towards the results from extensive own experimental studies allow for the carrying out of various calculations with it, which are connected to the nonstationary temperature distribution in logs during their thawing in an air environment. For example, as a result of such calculations, it is possible to determine the real initial temperature of logs depending on their dimensions, wood species, moisture content, and the temperature of the air near the logs during their many days staying in an open warehouse before the thermal treatment in the production of veneer.

The information about the real value of that immeasurable parameter is needed for scientifically based computing of the optimal, energy saving regimes for thermal treatment of each specific batch of logs.

The model of the logs' thawing process can be applied also in the software for controllers used for advanced model predictive automatic control [20, 21, 32] of this treatment. The approach for solving of the inverse task of the heat transfer in this chapter could be further applied in the development and solving of analogous models, for example, for the calculation of the temperature fields during freezing or thawing processes of different wooden and other capillary porous materials.

### **Acknowledgements**

Using the obtained change in the heat transfer coefficients, the change in the logs'surface temperature during the thawing, *t*s, has been calculated by the model

*Experimentally determined and calculated change in tm, ts, and t in four points of the log P2 during its 70 h*

The comparison to each other of the analogical curves in **Figures 2** and **9**, and

During our extensive simulations with the model (1) to (4), we established good qualitative and quantitative compliance between computed and experimentally determined temperature fields of logs from numerous wood species with different

The overall RSME for the studied four representative points in the logs does not exceed 5% of the temperature ranges between the minimal and maximal tempera-

This chapter describes the creation, solving, and validation of a 2D nonlinear mathematical model for the transient heat conduction subjected to thawing frozen

The mechanism of the heat distribution in logs during their thawing has been described by a 2D equation of heat conduction at convective boundary conditions. For the numerical solving of the model with the help of explicit form of the finitedifference method, a software package has been prepared in the calculation

A validation of the model towards our own experimentally determined 2D temperature distribution in poplar logs with a diameter of 0.24 m, length of 0.48 m, and initial temperature about –30°C during their 70 h separate thawing at room

determination of the logs' heat transfer coefficients in radial and longitudinal

During the validation of the model, the inverse problem has been solved for the

medium of Visual FORTRAN Professional developed by Microsoft.

also in **Figures 3** and **10**, shows good conformity between the calculated and experimentally determined changes in the very complicated temperature fields of

(refer to **Figures 9** and **10**).

**Figure 10.**

*thawing.*

the studied logs during their thawing.

tures of each log during its thawing.

**5. Conclusions**

**84**

logs in an air environment.

temperature has been carried out.

moisture content above the hygroscopic range [31].

*Modeling and Simulation in Engineering - Selected Problems*

This document was supported by the APVV Grant Agency as part of the project, APVV-17-0456, as a result of work of authors and the considerable assistance of the APVV agency.

### **Nomenclature**


**Superscripts**

**Author details**

Nencho Deliiski<sup>1</sup>

**87**

1 University of Forestry, Sofia, Bulgaria

2 Technical University in Zvolen, Slovakia

provided the original work is properly cited.

\*Address all correspondence to: deliiski@netbg.com

272.15 at 272.15 K, i.e., at –1°C 293.15 at 293.15 K, i.e., at 20°C

*DOI: http://dx.doi.org/10.5772/intechopen.93177*

*n* current number of the step Δτ along the time coordinate during solving

\*, Ladislav Dzurenda<sup>2</sup> and Natalia Tumbarkova<sup>1</sup>

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

of the model: *n* = 1, 2, 3, … , *N* = τthaw/Δτ

*Modeling of the Two-Dimensional Thawing of Logs in an Air Environment*


### **Subscripts**


*Modeling of the Two-Dimensional Thawing of Logs in an Air Environment DOI: http://dx.doi.org/10.5772/intechopen.93177*

### **Superscripts**

*r* radial coordinate: *0* ≤ *r* ≤ *R*, m

medium, W�m�<sup>2</sup>

C *<sup>u</sup>* moisture content, kg�kg�<sup>1</sup> = %/100 *w* kinematic viscosity coefficient, m<sup>2</sup>

*Modeling and Simulation in Engineering - Selected Problems*

*z* longitudinal coordinate: 0 ≤ *z* ≤ *L/*2, m

*σ* root-square-mean error (RSME), °

�K�<sup>1</sup> *β* coefficient of the volume expansion of the air, K�<sup>1</sup> *<sup>λ</sup>* thermal conductivity (for wood or air), W�m�<sup>1</sup>

�s �1

C

�K�<sup>1</sup>

*α* heat transfer coefficients between log's surfaces and the surrounding air

avg average (for mass temperature of logs or for root-square-mean error) b basic (for wood density, based on dry mass divided to green volume)

*i* current number of the knot of the calculation mesh in the direction along

*k* current number of the knot of the calculation mesh in longitudinal direc-

m medium (for temperature of the air environment near the logs during

C

C

1,2,3 1st, 2nd, 3rd (for temperature ranges of the logs' thawing process)

Δ*r* step along the coordinates *r* and *z* for solving of the model, m Δ*τ* step along the time coordinate for solving of the model, s

bwm maximum possible amount of the bound water in the wood

the log's radius: *i* = 1, 2, 3, … , 21 = (*R*/Δ*r* + 1)

their thawing process) p parallel to the wood fibers

w-fr wood with frozen water in it w-nfr wood with fully liquid water in it w0p parallel to the wood fibers at °

w0r radial direction of wood at °

& and simultaneously with this

we wood effective (for specific heat capacity)

tion of the logs: *k* = 1, 2, 3, … , 41 = (*L*/2/Δ*r* + 1)

*T* temperature, K *t* temperature, <sup>o</sup>

*x* exponent, �

*<sup>ρ</sup>* density, kg�m�<sup>3</sup>

*φ* relative humidity, %

*τ* time, s

**Subscripts**

a air

bw bound water

comp computed exp. experimental fr freezing fre end of freezing fsp fiber saturation point

fw free water

r radial direction

0 initial or at 0°C

@ at

**86**

s surface thaw thawing w wood


## **Author details**

Nencho Deliiski<sup>1</sup> \*, Ladislav Dzurenda<sup>2</sup> and Natalia Tumbarkova<sup>1</sup>


\*Address all correspondence to: deliiski@netbg.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

[1] Vorreiter L. Holztechnologisches Handbuch. Vien: Fromm; 1949. p. 2080

[2] Chudinov BS. Theoretical research of thermo physical properties and thermal treatment of wood [thesis for DSc.]. Krasnojarsk, USSR: SibLTI; 1966 (in Russian)

[3] Kollmann FF, Côté WA Jr. Principles of Wood Science and Technology. I. Solid Wood. Berlin, Heidelberg, New York: Springer-Verlag; 1984. p. 592

[4] Shubin GS. Drying and Thermal Treatment of Wood. Moscow: Lesnaya Promyshlennost, USSR; 1990. p. e337 (in Russian)

[5] Požgaj A, Chovanec D, Kurjatko S, Babiak M. Structure and Properties of Wood. 2nd ed. Bratislava: Priroda a.s; 1997. p. 485 (in Slovak)

[6] Trebula P, Klement I. Drying and Hydro-Thermal Treatment of Wood. Slovakia: Technical University in Zvolen; 2002. p. 449 (in Slovak)

[7] Deliiski N, Dzurenda L. Modelling of the Thermal Processes in the Technologies for Wood Thermal Treatment. Slovakia: Technical University in Zvolen; 2010. p. 224 (in Russian)

[8] Deliiski N. Transient heat conduction in capillary porous bodies. In: Ahsan A, editor. Convection and Conduction Heat Transfer. Rijeka, Croatia: InTech Publishing House; 2011. pp. 149-176. DOI: 10.5772/21424

[9] Deliiski N. Modelling of the Energy Needed for Heating of Capillary Porous Bodies in Frozen and Non-frozen States. Saarbrücken, Germany: Lambert Academic Publishing, Scholars' Press; 2013. p. 116. Available from: http:// www.scholars-press.com//system/cove rgenerator/build/1060

[10] Steinhagen HP. Computerized finite-difference method to calculate transient heat conduction with thawing. Wood and Fiber Science. 1986;**18**(3): 460-467

defrosting of the wood. Wood Research.

*DOI: http://dx.doi.org/10.5772/intechopen.93177*

*Modeling of the Two-Dimensional Thawing of Logs in an Air Environment*

[27] Hrčka R. Model in Free Water in Wood. Wood Research. 2017;**62**(6):

[28] Marin E, Calderon A, Delgado-Vasallo O. Similarity theory and

[29] Telegin AS, Shvidkiy BS, Yaroshenko UG. Heat- and Mass Transfer. Moscow: Akademkniga; 2002.

[30] Dorn WS, McCracken DD.

Numerical Methods with FORTRAN IV: Case Studies. New York: John Willej &

[31] Tumbarkova N. Modelling of the freezing and thawing processes of logs and their energy consumption [PhD Thesis]. Bulgaria: University of Forestry in Sofia; 2019. p. 198 (in Bulgarian)

p. 456 (in Russian)

Sons Inc.; 1972. p. 451

[32] Hadjiski M, Deliiski N, Grancharova A. Spatiotemporal parameter estimation of thermal treatment process via initial condition reconstruction using neural networks. In: Hadjiski M, Atanasov KT, editors. Intuitionistic Fuzziness and Other Intelligent Theories and their Applications. Cham, Switzerland: Springer International Publishing AG; 2019. pp. 51-80. DOI: 10.1007/978-3-

319-78931-6

dimensionless numbers in heat transfer. European Journal of Physics. 2009;**30**: 439. DOI: 10.1088/0143-0807/30/3/

831-837

001/meta

2013;**58**(4):637-650

[19] Deliiski N, Dzurenda L, Tumbarkova N, Angelski D. Computation of the temperature conductivity of frozen wood during its defrosting. Drvna Industrija. 2015;

**66**(2):87-96. DOI: 10.5552/

10.1016/j.ifacol.2015.12.056

processing. Cybernetics and

[22] Deliiski N, Tumbarkova N.

intechopen.84706

9781483159430

Russian)

2016-0013

**89**

[20] Hadjiski M, Deliiski N. Cost oriented suboptimal control of the thermal treatment of wood materials. IFAC-Papers. 2015;**48**(24):54-59. DOI:

[21] Hadjiski M, Deliiski N. Advanced control of the wood thermal treatment

Information Technologies, Bulgarian Academy of Sciences. 2016;**16**(2): 179-197. DOI: 10.1515/cait-2016-0029

Numerical solution to two-dimensional freezing and subsequent defrosting of logs. In: Iranzo A, editor. Heat and Mass Transfer—Advances in Science and Technology Applications. London: IntechOpen; 2015. p. 20. DOI: 10.5772/

[23] Whitaker S. Fundamental Principles of Heat Transfer. Oxford OX3 OBW: Pergamon Press; 1977. p. 574. eBook

[24] Kozdoba LA. Methods for Solving of

Non-linear Heat Transfer Tasks. Moscow: Nauka, USSR; 1975. p. 228 (in

[25] Kanter KR. Investigation of the Thermal Properties of Wood [Thesis]. Moscow: MLTI, USSR; 1955 (in Russian)

[26] Deliiski N, Tumbarkova N. A methodology for experimental research of the freezing process of logs. Acta Silvatica et Lignaria Hungarica. 2016; **12**(2):145-156. DOI: 10.1515/aslh-

drind.2015.1351

[11] Steinhagen HP. Heat transfer computation for a long, frozen log heated in agitated water or steam – A practical recipe. Holz als Roh- und Werkstoff. 1991;**49**(7–8):287-290. DOI: 10.1007/BF02663790

[12] Steinhagen HP, Lee HW, Loehnertz SP. LOGHEAT: A computer program of determining log heating times for frozen and non-frozen logs. Forest Products Journal. 1987;**37**(11/12): 60-64

[13] Steinhagen HP, Lee HW. Enthalpy method to compute radial heating and thawing of logs. Wood and Fiber Science. 1988;**20**(4):415-421

[14] Khattabi A, Steinhagen HP. Numerical solution to two-dimensional heating of logs. Holz als Roh- und Werkstoff. 1992;**50**(7–8):308-312. DOI: 10.1007/BF02615359

[15] Khattabi A, Steinhagen HP. Analysis of transient non-linear heat conduction in wood using finite-difference solutions. Holz als Roh- und Werkstoff. 1993;**51**(4):272-278. DOI: 10.1007/ BF02629373

[16] Khattabi A, Steinhagen HP. Update of "numerical solution to twodimensional heating of logs". Holz als Roh- und Werkstoff. 1995;**53**(1):93-94. DOI: 10.1007/BF02716399

[17] Deliiski N. Modelling and automatic control of heat energy consumption required for thermal treatment of logs. Drvna Industrija. 2004;**55**(4):181-199

[18] Deliiski N. Computation of the wood thermal conductivity during

*Modeling of the Two-Dimensional Thawing of Logs in an Air Environment DOI: http://dx.doi.org/10.5772/intechopen.93177*

defrosting of the wood. Wood Research. 2013;**58**(4):637-650

**References**

Russian)

(in Russian)

Russian)

DOI: 10.5772/21424

rgenerator/build/1060

**88**

1997. p. 485 (in Slovak)

[1] Vorreiter L. Holztechnologisches Handbuch. Vien: Fromm; 1949. p. 2080

[2] Chudinov BS. Theoretical research of thermo physical properties and thermal treatment of wood [thesis for DSc.]. Krasnojarsk, USSR: SibLTI; 1966 (in

*Modeling and Simulation in Engineering - Selected Problems*

[10] Steinhagen HP. Computerized finite-difference method to calculate transient heat conduction with thawing. Wood and Fiber Science. 1986;**18**(3):

[11] Steinhagen HP. Heat transfer computation for a long, frozen log heated in agitated water or steam – A practical recipe. Holz als Roh- und Werkstoff. 1991;**49**(7–8):287-290. DOI:

10.1007/BF02663790

[12] Steinhagen HP, Lee HW,

Loehnertz SP. LOGHEAT: A computer program of determining log heating times for frozen and non-frozen logs. Forest Products Journal. 1987;**37**(11/12):

[13] Steinhagen HP, Lee HW. Enthalpy method to compute radial heating and thawing of logs. Wood and Fiber Science. 1988;**20**(4):415-421

Numerical solution to two-dimensional heating of logs. Holz als Roh- und Werkstoff. 1992;**50**(7–8):308-312. DOI:

[15] Khattabi A, Steinhagen HP. Analysis of transient non-linear heat conduction

solutions. Holz als Roh- und Werkstoff. 1993;**51**(4):272-278. DOI: 10.1007/

[16] Khattabi A, Steinhagen HP. Update

[17] Deliiski N. Modelling and automatic control of heat energy consumption required for thermal treatment of logs. Drvna Industrija. 2004;**55**(4):181-199

[18] Deliiski N. Computation of the wood thermal conductivity during

[14] Khattabi A, Steinhagen HP.

in wood using finite-difference

of "numerical solution to twodimensional heating of logs". Holz als Roh- und Werkstoff. 1995;**53**(1):93-94.

DOI: 10.1007/BF02716399

10.1007/BF02615359

BF02629373

460-467

60-64

[3] Kollmann FF, Côté WA Jr. Principles of Wood Science and Technology. I. Solid Wood. Berlin, Heidelberg, New York: Springer-Verlag; 1984. p. 592

[4] Shubin GS. Drying and Thermal Treatment of Wood. Moscow: Lesnaya Promyshlennost, USSR; 1990. p. e337

[5] Požgaj A, Chovanec D, Kurjatko S, Babiak M. Structure and Properties of Wood. 2nd ed. Bratislava: Priroda a.s;

[6] Trebula P, Klement I. Drying and Hydro-Thermal Treatment of Wood. Slovakia: Technical University in Zvolen; 2002. p. 449 (in Slovak)

[7] Deliiski N, Dzurenda L. Modelling of

[8] Deliiski N. Transient heat conduction in capillary porous bodies. In: Ahsan A, editor. Convection and Conduction Heat Transfer. Rijeka, Croatia: InTech Publishing House; 2011. pp. 149-176.

[9] Deliiski N. Modelling of the Energy Needed for Heating of Capillary Porous Bodies in Frozen and Non-frozen States.

Saarbrücken, Germany: Lambert Academic Publishing, Scholars' Press; 2013. p. 116. Available from: http:// www.scholars-press.com//system/cove

the Thermal Processes in the Technologies for Wood Thermal Treatment. Slovakia: Technical University in Zvolen; 2010. p. 224 (in [19] Deliiski N, Dzurenda L, Tumbarkova N, Angelski D. Computation of the temperature conductivity of frozen wood during its defrosting. Drvna Industrija. 2015; **66**(2):87-96. DOI: 10.5552/ drind.2015.1351

[20] Hadjiski M, Deliiski N. Cost oriented suboptimal control of the thermal treatment of wood materials. IFAC-Papers. 2015;**48**(24):54-59. DOI: 10.1016/j.ifacol.2015.12.056

[21] Hadjiski M, Deliiski N. Advanced control of the wood thermal treatment processing. Cybernetics and Information Technologies, Bulgarian Academy of Sciences. 2016;**16**(2): 179-197. DOI: 10.1515/cait-2016-0029

[22] Deliiski N, Tumbarkova N. Numerical solution to two-dimensional freezing and subsequent defrosting of logs. In: Iranzo A, editor. Heat and Mass Transfer—Advances in Science and Technology Applications. London: IntechOpen; 2015. p. 20. DOI: 10.5772/ intechopen.84706

[23] Whitaker S. Fundamental Principles of Heat Transfer. Oxford OX3 OBW: Pergamon Press; 1977. p. 574. eBook 9781483159430

[24] Kozdoba LA. Methods for Solving of Non-linear Heat Transfer Tasks. Moscow: Nauka, USSR; 1975. p. 228 (in Russian)

[25] Kanter KR. Investigation of the Thermal Properties of Wood [Thesis]. Moscow: MLTI, USSR; 1955 (in Russian)

[26] Deliiski N, Tumbarkova N. A methodology for experimental research of the freezing process of logs. Acta Silvatica et Lignaria Hungarica. 2016; **12**(2):145-156. DOI: 10.1515/aslh-2016-0013

[27] Hrčka R. Model in Free Water in Wood. Wood Research. 2017;**62**(6): 831-837

[28] Marin E, Calderon A, Delgado-Vasallo O. Similarity theory and dimensionless numbers in heat transfer. European Journal of Physics. 2009;**30**: 439. DOI: 10.1088/0143-0807/30/3/ 001/meta

[29] Telegin AS, Shvidkiy BS, Yaroshenko UG. Heat- and Mass Transfer. Moscow: Akademkniga; 2002. p. 456 (in Russian)

[30] Dorn WS, McCracken DD. Numerical Methods with FORTRAN IV: Case Studies. New York: John Willej & Sons Inc.; 1972. p. 451

[31] Tumbarkova N. Modelling of the freezing and thawing processes of logs and their energy consumption [PhD Thesis]. Bulgaria: University of Forestry in Sofia; 2019. p. 198 (in Bulgarian)

[32] Hadjiski M, Deliiski N, Grancharova A. Spatiotemporal parameter estimation of thermal treatment process via initial condition reconstruction using neural networks. In: Hadjiski M, Atanasov KT, editors. Intuitionistic Fuzziness and Other Intelligent Theories and their Applications. Cham, Switzerland: Springer International Publishing AG; 2019. pp. 51-80. DOI: 10.1007/978-3- 319-78931-6

**91**

**Chapter 5**

Method

**Abstract**

pump designs.

**1. Introduction**

equations, control volume method

*Cemil Koyunoğlu*

Innovations in Heat Pump Design

Dynamics with Control Volume

Mathematical modeling of the heat pump as a result of continuity, momentum,

and energy equations is obtained. To solve these equations numerically, the problem is divided by a finite number of control volumes. Then the differential equations in these control volumes integrated and converted into algebraic equations. The importance of computational fluid dynamics in Industry 4.0 applications is to make current applications more efficient in heat pump applications. In this study, the book section is composed of the application of computational fluid dynamics by the control volume method using Ansys fluent program, which will benefit readers from industry 4.0 perspective, especially in energy efficiency issues according to the volume method of controlling correct heat

**Keywords:** heat pump design, industry 4.0, computational fluid dynamics, energy

The ever increasing demand for energy and the depletion of energy resources accelerate the search for new energy resources. The effects of global warming based on excessive fossil fuel consumption and problems in meeting the energy demand enabled the energy to become the main agenda item of the world as a current problem. In developed countries where energy is the main agenda item, frequent sessions are held on the balance of energy supply and demand. In particular, the search for cheap electricity, which the increasing population and industry needed, has increased the requirement for renewable energy resources. Hydraulic, solar, wind, and geothermal energy are important renewable energy sources because of

At the same time, geothermal energy, which is used in the technology of soil air source heat exchanger and which is mentioned in renewable energy sources, is extensively used in topics such as cooling as well as heating homes, electricity

their importance, easy-to-find, and cheap production [1–3].

generation, tourism, and heating greenhouses [4].

Using Computational Fluid

### **Chapter 5**
