**2.3 Introducing the mathematical description of heat sources into the mathematical model**

After substituting *<sup>q</sup>*<sup>v</sup> in Eq. (1) by the expression for *<sup>q</sup>*v‐fw from Eq. (5) and of *<sup>∂</sup>ψ*ice‐fw *<sup>∂</sup><sup>τ</sup>* in this expression by *<sup>∂</sup>ψ*ice‐fw *<sup>∂</sup><sup>τ</sup>* � *<sup>∂</sup><sup>Т</sup> <sup>∂</sup>Т*, it is obtained that

*<sup>ρ</sup>*<sup>w</sup> *<sup>c</sup>*we‐fr‐bw � *<sup>K</sup><sup>ψ</sup>*�bw*L*cr‐ice

*<sup>ρ</sup>*<sup>w</sup> � *<sup>c</sup>*we‐fr‐bw<sup>∗</sup> �

�K�<sup>1</sup> .

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

*<sup>c</sup>*we‐fr‐bw<sup>∗</sup> <sup>¼</sup> *<sup>c</sup>*we‐fr‐bw � *<sup>K</sup>ψ*�bw*L*cr‐ice

freezing only of the bound water in it, J�kg�<sup>1</sup>

**the freezing of the bound water**

<sup>Ψ</sup>ice�bw <sup>¼</sup> *<sup>m</sup>*ice�bw

< 272.15 K, *m*nfw, according to the following equation:

*m*ice�bw þ *m*nfw

Eq. (13) and for *u*fsp from Eq. (23), it is obtained that

<sup>0</sup>*:*<sup>12</sup> <sup>þ</sup> *<sup>u</sup>*<sup>272</sup>*:*<sup>15</sup>

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

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

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

> *∂λ*wr *∂T*

wood, J�kg�<sup>1</sup>

depending on *T*:

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

Ψice�bw ¼ 1 �

*t* = 20°C.

**117**

*∂λ*wr *∂T*

*DOI: http://dx.doi.org/10.5772/intechopen.83772*

þ *λ*wp

*∂*2

Equation (19) can be represented, as follows:

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

þ *λ*wp

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

*∂*2

*<sup>∂</sup>ψ*ice‐bw *∂T*

*T r*ð Þ *; z; τ ∂z*<sup>2</sup> þ

�

*Modeling of the Energy for Bound Water Freezing in Logs Subjected to Refrigeration*

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

*∂λ*wp *∂T*

*∂*2

*T r*ð Þ *; z; τ ∂z*<sup>2</sup> þ

The effective specific heat capacity *<sup>c</sup>*we‐fr‐bw<sup>∗</sup> in Eq. (20) is equal to

heat capacity, which is formed by the release of the latent heat of the maximum possible amount of bound water during its crystallization in the

*T r*ð Þ *; z; τ ∂r*<sup>2</sup> þ

*<sup>∂</sup>ψ*ice‐bw

�K�<sup>1</sup>

where *c*we�fr�bw is the effective specific heat capacity of the wood during

**2.4 Mathematical description of the relative icing degree of logs caused from**

The relative icing degree of the logs, which is caused from the freezing of only the bound water in them, Ψice�bw, can be determined as a relationship of the mass of the ice formed by the bound water in 1 kg wood, *<sup>m</sup>*ice‐bw, to the sum of the mass of this ice and of the mass of the nonfrozen water in 1 kg wood at *u* ¼ *u*nfw and at *T*

> <sup>¼</sup> *<sup>u</sup>*fsp � *<sup>u</sup>*nfw *u*fsp � *u*nfw þ *u*nfw

Using experimental data of Stamm [30, 31], in [10] the following equation was

fsp is the standardized fiber saturation point at *T* = 293.15 K, i.e., at

fsp � <sup>0</sup>*:*<sup>12</sup> � exp 0½ � *:*0567ð Þ *<sup>T</sup>* � <sup>272</sup>*:*<sup>15</sup>

After substituting *u*nfw and *u*fsp in Eq. (22) by the expressions for *u*nfw from

suggested for the calculation of the fiber saturation point of the wood species,

*∂λ*wp *∂T*

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

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

*∂*2

*:*

1 *r :*

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

*T r*ð Þ *; z; τ ∂r*<sup>2</sup> þ

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

1 *r :*

þ

*:*

*<sup>∂</sup><sup>T</sup>* <sup>¼</sup> *<sup>c</sup>*we�fr�bw � *<sup>c</sup>*Lat�bwm*,* (21)

, and *c*Lat�bwm is the specific

<sup>¼</sup> <sup>1</sup> � *<sup>u</sup>*nfw *u*fsp

fsp � 0*:*001ð Þ *T* � 293*:*15 *,* (23)

fsp � <sup>0</sup>*:*001ð Þ *<sup>T</sup>* � <sup>293</sup>*:*<sup>15</sup> *:* (24)

*:* (22)

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

þ

(19)

(20)

$$\rho\_{\rm w} \cdot \varepsilon\_{\rm w\rm e\-fer\-fw} \frac{\partial T(r, z, \tau)}{\partial \tau} = \lambda\_{\rm w\rm r} \left[ \frac{\partial^2 T(r, z, \tau)}{\partial r^2} + \frac{1}{r} \cdot \frac{\partial T(r, z, \tau)}{\partial r} \right] + \frac{\partial \lambda\_{\rm w\rm r}}{\partial T} \left[ \frac{\partial T(r, z, \tau)}{\partial r} \right]^2 + \dots$$

$$\lambda\_{\rm wz} \frac{\partial^2 T(r, z, \tau)}{\partial z^2} + \frac{\partial \lambda\_{\rm wz}}{\partial T} \left[ \frac{\partial T(r, z, \tau)}{\partial z} \right]^2 + K\_{\rm w-fw} \rho\_{\rm w} L\_{\rm cr-ice} \frac{\partial \mu\_{\rm ice-fw}}{\partial \tau} \cdot \frac{\partial T(r, z, \tau)}{\partial T} \,. \tag{14}$$

For the freezing of the free water in the wood, Eq. (14) is transformed in the following form:

$$\begin{split} &\rho\_{\rm w} \left[ c\_{\rm we-fr-fw} - K\_{\rm y-fw} L\_{\rm cr-ice} \frac{\partial \nu\_{\rm ice-fw}}{\partial T} \right] \cdot \frac{\partial T(r, z, \tau)}{\partial \tau} = \lambda\_{\rm wr} \left[ \frac{\partial^2 T(r, z, \tau)}{\partial r^2} + \frac{1}{r} \cdot \frac{\partial T(r, z, \tau)}{\partial r} \right] + \\ &\frac{\partial \lambda\_{\rm wr}}{\partial T} \left[ \frac{\partial T(r, z, \tau)}{\partial r} \right]^2 + \lambda\_{\rm wz} \frac{\partial^2 T(r, z, \tau)}{\partial z^2} + \frac{\partial \lambda\_{\rm wz}}{\partial T} \left[ \frac{\partial T(r, z, \tau)}{\partial z} \right]^2. \end{split} \tag{15}$$

Equation (15) can be represented, as follows:

$$\begin{split} \rho\_{\text{w}} \cdot c\_{\text{we-fr}\text{-}\text{fw}^{\text{-}}} \cdot \frac{\partial T(r, \mathbf{z}, \tau)}{\partial \tau} &= \lambda\_{\text{w}\text{r}} \left[ \frac{\partial^{2} T(r, \mathbf{z}, \tau)}{\partial r^{2}} + \frac{1}{r} \cdot \frac{\partial T(r, \mathbf{z}, \tau)}{\partial r} \right] + \\ \frac{\partial \lambda\_{\text{w}\text{r}}}{\partial T} \left[ \frac{\partial T(r, \mathbf{z}, \tau)}{\partial r} \right]^{2} &+ \lambda\_{\text{wz}} \frac{\partial^{2} T(r, \mathbf{z}, \tau)}{\partial \mathbf{z}^{2}} + \frac{\partial \lambda\_{\text{w}\text{z}}}{\partial T} \left[ \frac{\partial T(r, \mathbf{z}, \tau)}{\partial \mathbf{z}} \right]^{2} . \end{split} \tag{16}$$

The effective specific heat capacity *<sup>c</sup>*we‐fr‐fw<sup>∗</sup> in Eq. (16) is equal to

$$c\_{\text{we-fr-fw}}, = c\_{\text{we-fr-fw}} - K\_{\text{y}-\text{fw}}L\_{\text{cr-ice}} \frac{\partial \nu\_{\text{ice-fw}}}{\partial T} = c\_{\text{we-fr-fw}} - c\_{\text{Lat-fw}} \tag{17}$$

where *c*we�fr�fw is the effective specific heat capacity of the wood during freezing only of the free water in it, J�kg�<sup>1</sup> �K�<sup>1</sup> , and *c*Lat�fw is the specific heat capacity, which is formed by the release of the latent heat of the free water during its crystallization in the wood, J�kg�<sup>1</sup> �K�<sup>1</sup> .

After substituting *<sup>q</sup>*<sup>v</sup> in Eq. (1) by the expression for *<sup>q</sup>*<sup>v</sup>‐bw from Eq. (6) and of *<sup>∂</sup>ψ*ice‐bw *<sup>∂</sup><sup>τ</sup>* in this expression by *<sup>∂</sup>ψ*ice‐bw *<sup>∂</sup><sup>τ</sup>* � *<sup>∂</sup><sup>Т</sup> <sup>∂</sup>Т*, it is obtained that

*<sup>ρ</sup>*w*c*we‐fr‐bw *<sup>∂</sup>T r*ð Þ *; <sup>z</sup>; <sup>τ</sup> <sup>∂</sup><sup>τ</sup>* <sup>¼</sup> *<sup>λ</sup>*wr *∂*2 *T r*ð Þ *; z; τ ∂r*<sup>2</sup> þ 1 *r : <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> ∂r* <sup>2</sup> þ *λ*wp *∂*2 *T r*ð Þ *; z; τ ∂z*<sup>2</sup> þ *∂λ*wp *∂T <sup>∂</sup>T r*ð Þ *; <sup>z</sup>; <sup>τ</sup> ∂z* <sup>2</sup> <sup>þ</sup> *<sup>K</sup>*<sup>ψ</sup>�bw*ρ*w*L*cr‐ice *<sup>∂</sup>ψ*ice‐bw *∂τ* � *T r*ð Þ *; z; τ <sup>∂</sup><sup>T</sup> :* (18)

For the case of freezing the bound water in the wood, Eq. (18) is transformed in the following form:

*Modeling of the Energy for Bound Water Freezing in Logs Subjected to Refrigeration DOI: http://dx.doi.org/10.5772/intechopen.83772*

$$\rho\_{\rm w} \left[ \varepsilon\_{\rm we-fr-bw} - K\_{\rm w-bw} L\_{\rm cru-ice} \frac{\partial \nu\_{\rm ice-bw}}{\partial T} \right] \cdot \frac{\partial T(r, z, \tau)}{\partial \tau} = \lambda\_{\rm wr} \left[ \frac{\partial^2 T(r, z, \tau)}{\partial r^2} + \frac{1}{r} \cdot \frac{\partial T(r, z, \tau)}{\partial r} \right] + \varepsilon\_{\rm wr} \left[ \frac{\partial T(r, z, \tau)}{\partial z} \right] + \varepsilon\_{\rm wr} \left[ \frac{\partial T(r, z, \tau)}{\partial z} \right] \tag{19}$$
 
$$\frac{\partial \lambda\_{\rm wr}}{\partial T} \left[ \frac{\partial T(r, z, \tau)}{\partial r} \right]^2 + \lambda\_{\rm wp} \frac{\partial^2 T(r, z, \tau)}{\partial z^2} + \frac{\partial \lambda\_{\rm wp}}{\partial T} \left[ \frac{\partial T(r, z, \tau)}{\partial z} \right]^2. \tag{10}$$

Equation (19) can be represented, as follows:

**2.3 Introducing the mathematical description of heat sources into the**

*T r*ð Þ *; z; τ ∂r*<sup>2</sup> þ

*<sup>∂</sup><sup>τ</sup>* � *<sup>∂</sup><sup>Т</sup>*

*∂*2

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

> *<sup>∂</sup>ψ*ice‐fw *∂T*

*T r*ð Þ *; z; τ ∂z*<sup>2</sup> þ

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

*∂*2

�

After substituting *<sup>q</sup>*<sup>v</sup> in Eq. (1) by the expression for *<sup>q</sup>*v‐fw from Eq. (5) and of

For the freezing of the free water in the wood, Eq. (14) is transformed in the

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

*∂λ*w*<sup>z</sup> ∂T*

*∂*2

*T r*ð Þ *; z; τ ∂z*<sup>2</sup> þ

The effective specific heat capacity *<sup>c</sup>*we‐fr‐fw<sup>∗</sup> in Eq. (16) is equal to

*<sup>∂</sup>Т*, it is obtained that

1 *r :*

<sup>þ</sup> *<sup>K</sup>*<sup>ψ</sup>�fw*ρ*w*L*cr‐ice

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

*T r*ð Þ *; z; τ ∂r*<sup>2</sup> þ

*<sup>∂</sup>ψ*ice‐fw

where *c*we�fr�fw is the effective specific heat capacity of the wood during freez-

After substituting *<sup>q</sup>*<sup>v</sup> in Eq. (1) by the expression for *<sup>q</sup>*<sup>v</sup>‐bw from Eq. (6) and of

*<sup>∂</sup>Т*, it is obtained that

1 *r :*

<sup>þ</sup> *<sup>K</sup>*<sup>ψ</sup>�bw*ρ*w*L*cr‐ice

For the case of freezing the bound water in the wood, Eq. (18) is transformed in

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

�K�<sup>1</sup>

which is formed by the release of the latent heat of the free water during its

*T r*ð Þ *; z; τ ∂r*<sup>2</sup> þ

�K�<sup>1</sup> .

*<sup>∂</sup><sup>τ</sup>* � *<sup>∂</sup><sup>Т</sup>*

*∂*2

*<sup>∂</sup>T r*ð Þ *; <sup>z</sup>; <sup>τ</sup> ∂z* <sup>2</sup> *∂λ*w*<sup>z</sup> ∂T*

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

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

þ *∂λ*wr *∂T*

*<sup>∂</sup>ψ*ice‐fw *∂τ* �

*∂*2

*:*

1 *r :*

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

*T r*ð Þ *; z; τ ∂r*<sup>2</sup> þ

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

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

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

> 1 *r :*

þ

*:*

*<sup>∂</sup><sup>T</sup>* <sup>¼</sup> *<sup>c</sup>*we�fr�fw � *<sup>c</sup>*Lat�fw*,* (17)

, and *c*Lat�fw is the specific heat capacity,

þ *∂λ*wr *∂T*

*<sup>∂</sup>ψ*ice‐bw *∂τ* �

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

*T r*ð Þ *; z; τ <sup>∂</sup><sup>T</sup> :* þ

(18)

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

þ

(14)

þ

(15)

(16)

**mathematical model**

*Low-temperature Technologies*

*<sup>∂</sup><sup>τ</sup>* in this expression by *<sup>∂</sup>ψ*ice‐fw

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

þ *λ*wz

*∂*2

Equation (15) can be represented, as follows:

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

þ *λ*wz

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

*∂λ*w*<sup>z</sup> ∂T*

*<sup>∂</sup>ψ*ice‐fw

*<sup>ρ</sup>*<sup>w</sup> � *<sup>c</sup>*we‐fr‐fw

following form:

*T r*ð Þ *; z; τ ∂z*<sup>2</sup> þ

*<sup>ρ</sup>*<sup>w</sup> *<sup>c</sup>*we‐fr‐fw � *<sup>K</sup>ψ*�fw*L*cr‐ice

*<sup>ρ</sup>*<sup>w</sup> � *<sup>c</sup>*we‐fr‐fw<sup>∗</sup> �

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

ing only of the free water in it, J�kg�<sup>1</sup>

crystallization in the wood, J�kg�<sup>1</sup>

*<sup>∂</sup><sup>τ</sup>* in this expression by *<sup>∂</sup>ψ*ice‐bw

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

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

*∂λ*wp *∂T*

*<sup>c</sup>*we‐fr‐fw<sup>∗</sup> <sup>¼</sup> *<sup>c</sup>*we‐fr‐fw � *<sup>K</sup>ψ*�fw*L*cr‐ice

*∂λ*wr *∂T*

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

*λ*wz *∂*2

*∂λ*wr *∂T*

*<sup>∂</sup>ψ*ice‐bw

*<sup>ρ</sup>*w*c*we‐fr‐bw

the following form:

*T r*ð Þ *; z; τ ∂z*<sup>2</sup> þ

*λ*wp *∂*2

**116**

$$\begin{split} \rho\_{\text{w}} \cdot c\_{\text{we-fr-bw}^{\*}} \cdot \frac{\partial T(r, z, \tau)}{\partial \tau} &= \lambda\_{\text{wr}} \left[ \frac{\partial^{2} T(r, z, \tau)}{\partial r^{2}} + \frac{1}{r} \cdot \frac{\partial T(r, z, \tau)}{\partial r} \right] + \\ \frac{\partial \lambda\_{\text{wr}}}{\partial T} \left[ \frac{\partial T(r, z, \tau)}{\partial r} \right]^{2} &+ \lambda\_{\text{wp}} \frac{\partial^{2} T(r, z, \tau)}{\partial z^{2}} + \frac{\partial \lambda\_{\text{wp}}}{\partial T} \left[ \frac{\partial T(r, z, \tau)}{\partial z} \right]^{2} . \end{split} \tag{20}$$

The effective specific heat capacity *<sup>c</sup>*we‐fr‐bw<sup>∗</sup> in Eq. (20) is equal to

$$\mathcal{L}\_{\text{we-fr-bw}^\*} = \mathcal{c}\_{\text{we-fr-bw}} - K\_{\text{y}-\text{bw}} L\_{\text{cr-dec}} \frac{\partial \mathcal{y}\_{\text{ice-bw}}}{\partial T} = \mathcal{c}\_{\text{we-fr-bw}} - \mathcal{c}\_{\text{Lat-bw}\nu} \tag{21}$$

where *c*we�fr�bw is the effective specific heat capacity of the wood during freezing only of the bound water in it, J�kg�<sup>1</sup> �K�<sup>1</sup> , and *c*Lat�bwm is the specific heat capacity, which is formed by the release of the latent heat of the maximum possible amount of bound water during its crystallization in the wood, J�kg�<sup>1</sup> �K�<sup>1</sup> .

#### **2.4 Mathematical description of the relative icing degree of logs caused from the freezing of the bound water**

The relative icing degree of the logs, which is caused from the freezing of only the bound water in them, Ψice�bw, can be determined as a relationship of the mass of the ice formed by the bound water in 1 kg wood, *<sup>m</sup>*ice‐bw, to the sum of the mass of this ice and of the mass of the nonfrozen water in 1 kg wood at *u* ¼ *u*nfw and at *T* < 272.15 K, *m*nfw, according to the following equation:

$$\Psi\_{\rm ice-bw} = \frac{m\_{\rm ice-bw}}{m\_{\rm ice-bw} + m\_{\rm nfw}} = \frac{u\_{\rm fsp} - u\_{\rm nfw}}{u\_{\rm fsp} - u\_{\rm nfw} + u\_{\rm nfw}} = 1 - \frac{u\_{\rm nfw}}{u\_{\rm fsp}}.\tag{22}$$

Using experimental data of Stamm [30, 31], in [10] the following equation was suggested for the calculation of the fiber saturation point of the wood species, depending on *T*:

$$
u\_{\rm fsp} = \boldsymbol{u}\_{\rm fsp}^{293.15} - 0.001(T - 293.15),\tag{23}$$

where *u*<sup>293</sup>*:*<sup>15</sup> fsp is the standardized fiber saturation point at *T* = 293.15 K, i.e., at *t* = 20°C.

After substituting *u*nfw and *u*fsp in Eq. (22) by the expressions for *u*nfw from Eq. (13) and for *u*fsp from Eq. (23), it is obtained that

$$\Psi\_{\rm ice-bw} = 1 - \frac{0.12 + \left(u\_{\rm fsp}^{272.15} - 0.12\right) \cdot \exp\left[0.0567(T - 272.15)\right]}{u\_{\rm fsp}^{293.15} - 0.001(T - 293.15)}.\tag{24}$$

and

$$\frac{\partial \Psi\_{\rm{kcr-bw}}}{\partial T} = -\frac{\left\{0.0567 \left(u\_{\rm{fsp}}^{22.15} - 0.12\right) \cdot \exp\left[0.0567(T - 272.15)\right]\right\} \cdot \left[u\_{\rm{fsp}}^{293.19} - 0.001(T - 293.15)\right]}{\left[u\_{\rm{fsp}}^{293.15} - 0.001(T - 293.15)\right]^2}$$

$$+\frac{0.001 \cdot \left\{0.12 + \left(u\_{\rm{fsp}}^{22.15} - 0.12\right) \cdot \exp\left[0.0567(T - 272.15)\right]\right\}}{\left[u\_{\rm{fsp}}^{293.15} - 0.001(T - 293.15)\right]^2}.\tag{25}$$

Mathematical descriptions of the specific heat capacities *c*w�nfr, *c*w�fr, *c*fw, and *c*bwm and also of the thermal conductivity of nonfrozen and frozen wood, *λ*w, have been suggested by the first co-author earlier [9, 10] using the experimentally determined data in the dissertations by Kanter [32] and Chudinov [2] for their

These relations are used in both the European [5–10] and the American specialized literature [11–16] when calculating various processes of the wood

*Modeling of the Energy for Bound Water Freezing in Logs Subjected to Refrigeration*

Using Eqs. (26)–(28), Eqs. (16) and (20) can be united in the following equation, with the help of which, together with the initial condition (2) and the boundary conditions (3), and (4) the 2D change of the temperature in subjected

*∂*2

For the convective boundary conditions (3) and (4) of the logs' freezing process,

*T r*ð Þ *; z; τ ∂r*<sup>2</sup> þ

> *∂λw<sup>р</sup> ∂T*

1 *r :*

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

*<sup>α</sup>*wr‐fr <sup>¼</sup> <sup>2</sup>*:*56½ � *<sup>T</sup>*ð Þ� <sup>0</sup>*; <sup>z</sup>; <sup>τ</sup> <sup>T</sup>*<sup>m</sup>�frð Þ*<sup>τ</sup> <sup>E</sup>*fr*,* (30)

*<sup>α</sup>*wp‐fr <sup>¼</sup> <sup>1</sup>*:*123½ � *T r*ð Þ� *;* <sup>0</sup>*; <sup>τ</sup> <sup>T</sup>*<sup>m</sup>�frð Þ*<sup>τ</sup> <sup>E</sup>*fr *,* (31)

*<sup>Q</sup>*fr‐bw‐total <sup>¼</sup> *<sup>Q</sup>*fr‐bw � *<sup>Q</sup>*Lat‐bw*,* (32)

<sup>3</sup>*:*<sup>6</sup> � <sup>10</sup><sup>6</sup> � *<sup>Т</sup>*<sup>w</sup>�avg � *<sup>T</sup>*w0 *:* (33)

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

*:*

þ

(29)

, and *<sup>Q</sup>*Lat‐bw

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

þ *λw<sup>р</sup>*

• at cylindrical surface of the horizontally situated logs

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

*T r*ð Þ *; z; τ ∂z*<sup>2</sup> þ

the following relations for the heat transfer coefficients are most suitable [33]:

where *E*fr is determined during the validation of the nonlinear mathematical model of the freezing process through minimization of the root square mean error (*RSME*) between the computed by the model and experimentally obtained transient

**2.6 An approach for modeling of the energy needed for freezing the bound**

The total energy consumption, which is needed for freezing the bound water in 1 m<sup>3</sup> of logs, *<sup>Q</sup>*fr‐bw‐total, can be calculated according to the following equation:

where *<sup>Q</sup>*fr‐bw is the energy needed for the phase transition of the bound water

It is known that the energy consumption for heating 1 m<sup>3</sup> of wood materials, *Q*w, with an initial mass temperature *T*w0 to a given average mass temperature *T*<sup>w</sup>�avg is

is the latent thermal energy of the bound water in 1 m<sup>3</sup> of logs released in them

from liquid to solid state in 1 m<sup>3</sup> of logs subjected to freezing, kWh�m�<sup>3</sup>

.

*<sup>Q</sup>*<sup>w</sup> <sup>¼</sup> *<sup>c</sup>*<sup>w</sup> � *<sup>ρ</sup>*<sup>w</sup>

*∂*2

change as a function of *t* and *u*.

*DOI: http://dx.doi.org/10.5772/intechopen.83772*

to freezing logs can be calculated:

*<sup>ρ</sup>*<sup>w</sup> � *<sup>c</sup>*we‐fr‐gen1*,* <sup>2</sup>*,*<sup>3</sup>

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

• at frontal surface of the logs

during its crystallization, kWh�m�<sup>3</sup>

determined using the following equation [2, 5, 10]:

temperature fields in logs subjected to freezing.

thermal treatment.

*∂λ*wr *∂T*

**water in logs**

**119**

The sign "minus" in the right parts of Eq. (25) reflects mathematically the circumstances that the icing degree Ψice�bw decreases with an increasing temperature *Т*.

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

In **Figure 1**, the temperature ranges are presented, at which the refrigeration process of logs is carried out when *u* > *u*fsp. The thermophysical characteristics of the logs and of both the frozen free and bound water in them have also been shown. The information on these characteristics is very important for solving the mathematical model given above.

During the first range from *T*w0 to *T*fr�fw, only a cooling of the logs with full liquid water in them occurs. During the second range from *T*fr�fw to *T*fr�bwm, a further cooling of the logs occurs until reaching the state needed for starting the crystallization of the free water. During this range also, the phase transition of this water into ice is carried out. The second range is absent when the wood moisture content is less than *u*fsp.

During the third range from *T*fr�bwm to *T*<sup>w</sup>�fre�avg, a further cooling of the logs is carried out until reaching the state needed for starting the crystallization of the bound water. During this range also, the phase transition of the bound water into ice is gradually performed.

The generalized effective specific heat capacities of the logs, *<sup>c</sup>*we‐fr�gen, during the above pointed three ranges of the freezing process above the hygroscopic range, are equal to the following:

$$\mathbf{I.Range}: \mathcal{c}\_{\text{we-fr}-\text{gen1}} = \mathcal{c}\_{\text{w-nfr}} \tag{26}$$

$$\text{II.Range}: c\_{\text{we-fr-gen2}} = c\_{\text{w-nfr}} + c\_{\text{fw}} - c\_{\text{Lat}-\text{fw}} \tag{27}$$

$$\text{III.Range}: \mathcal{c}\_{\text{we-fr-gen3}} = \mathcal{c}\_{\text{w-fr}} + \mathcal{c}\_{\text{bwm}} - \mathcal{c}\_{\text{Lat-bwm}}.\tag{28}$$


#### **Figure 1.**

*Temperature ranges of the logs' refrigeration process above the hygroscopic range and thermophysical characteristics of the wood and the frozen water in it.*

*Modeling of the Energy for Bound Water Freezing in Logs Subjected to Refrigeration DOI: http://dx.doi.org/10.5772/intechopen.83772*

Mathematical descriptions of the specific heat capacities *c*w�nfr, *c*w�fr, *c*fw, and *c*bwm and also of the thermal conductivity of nonfrozen and frozen wood, *λ*w, have been suggested by the first co-author earlier [9, 10] using the experimentally determined data in the dissertations by Kanter [32] and Chudinov [2] for their change as a function of *t* and *u*.

These relations are used in both the European [5–10] and the American specialized literature [11–16] when calculating various processes of the wood thermal treatment.

Using Eqs. (26)–(28), Eqs. (16) and (20) can be united in the following equation, with the help of which, together with the initial condition (2) and the boundary conditions (3), and (4) the 2D change of the temperature in subjected to freezing logs can be calculated:

$$\begin{split} \rho\_{\text{w}} \cdot \varepsilon\_{\text{we-fr-gen1},2,3} \frac{\partial T(r,z,\tau)}{\partial \tau} &= \lambda\_{\text{wr}} \left[ \frac{\partial^2 T(r,z,\tau)}{\partial r^2} + \frac{1}{r} \cdot \frac{\partial T(r,z,\tau)}{\partial r} \right] + \\ \frac{\partial \lambda\_{\text{wr}}}{\partial T} \left[ \frac{\partial T(r,z,\tau)}{\partial r} \right]^2 &+ \lambda\_{\text{wp}} \frac{\partial^2 T(r,z,\tau)}{\partial z^2} + \frac{\partial \lambda\_{\text{wp}}}{\partial T} \left[ \frac{\partial T(r,z,\tau)}{\partial z} \right]^2 . \end{split} \tag{29}$$

For the convective boundary conditions (3) and (4) of the logs' freezing process, the following relations for the heat transfer coefficients are most suitable [33]:

• at cylindrical surface of the horizontally situated logs

$$a\_{\rm wr\cdot fr} = 2.56[T(\mathbf{0}, z, \tau) - T\_{\rm m-fr}(\tau)]^{E\_{\rm fr}},\tag{30}$$

• at frontal surface of the logs

$$a\_{\rm wp-fr} = \mathbf{1.123} [T(r, \mathbf{0}, \tau) - T\_{\rm m-fr}(\tau)]^{E\_{\rm fr}},\tag{31}$$

where *E*fr is determined during the validation of the nonlinear mathematical model of the freezing process through minimization of the root square mean error (*RSME*) between the computed by the model and experimentally obtained transient temperature fields in logs subjected to freezing.

#### **2.6 An approach for modeling of the energy needed for freezing the bound water in logs**

The total energy consumption, which is needed for freezing the bound water in 1 m<sup>3</sup> of logs, *<sup>Q</sup>*fr‐bw‐total, can be calculated according to the following equation:

$$Q\_{\text{fr-bw-total}} = Q\_{\text{fr-bw}} - Q\_{\text{Lat-bw}} \tag{32}$$

where *<sup>Q</sup>*fr‐bw is the energy needed for the phase transition of the bound water from liquid to solid state in 1 m<sup>3</sup> of logs subjected to freezing, kWh�m�<sup>3</sup> , and *<sup>Q</sup>*Lat‐bw is the latent thermal energy of the bound water in 1 m<sup>3</sup> of logs released in them during its crystallization, kWh�m�<sup>3</sup> .

It is known that the energy consumption for heating 1 m<sup>3</sup> of wood materials, *Q*w, with an initial mass temperature *T*w0 to a given average mass temperature *T*<sup>w</sup>�avg is determined using the following equation [2, 5, 10]:

$$Q\_{\rm w} = \frac{c\_{\rm w} \cdot \rho\_{\rm w}}{3.6 \cdot 10^{6}} \cdot \left(T\_{\rm w-avg} - T\_{\rm w0}\right). \tag{33}$$

and

*<sup>∂</sup>*Ψice�bw *∂T* ¼ �

þ

*Low-temperature Technologies*

temperature *Т*.

0*:*0567 *u*272*:*<sup>15</sup>

<sup>0</sup>*:*<sup>001</sup> � <sup>0</sup>*:*<sup>12</sup> <sup>þ</sup> *<sup>u</sup>*272*:*<sup>15</sup>

mathematical model given above.

content is less than *u*fsp.

ice is gradually performed.

are equal to the following:

*characteristics of the wood and the frozen water in it.*

**Figure 1.**

**118**

fsp � 0*:*12 � �

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

n o

fsp � 0*:*12 � �

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

fsp � 0*:*001ð Þ *T* � 293*:*15

n o

The sign "minus" in the right parts of Eq. (25) reflects mathematically the circumstances that the icing degree Ψice�bw decreases with an increasing

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

The information on these characteristics is very important for solving the

In **Figure 1**, the temperature ranges are presented, at which the refrigeration process of logs is carried out when *u* > *u*fsp. The thermophysical characteristics of the logs and of both the frozen free and bound water in them have also been shown.

During the first range from *T*w0 to *T*fr�fw, only a cooling of the logs with full liquid water in them occurs. During the second range from *T*fr�fw to *T*fr�bwm, a further cooling of the logs occurs until reaching the state needed for starting the crystallization of the free water. During this range also, the phase transition of this water into ice is carried out. The second range is absent when the wood moisture

During the third range from *T*fr�bwm to *T*<sup>w</sup>�fre�avg, a further cooling of the logs is carried out until reaching the state needed for starting the crystallization of the bound water. During this range also, the phase transition of the bound water into

The generalized effective specific heat capacities of the logs, *<sup>c</sup>*we‐fr�gen, during the above pointed three ranges of the freezing process above the hygroscopic range,

*Temperature ranges of the logs' refrigeration process above the hygroscopic range and thermophysical*

<sup>I</sup>*:*Range : *<sup>c</sup>*we‐fr�gen1 <sup>¼</sup> *<sup>c</sup>*<sup>w</sup>�nfr*,* (26)

II*:*Range : *<sup>c</sup>*we‐fr‐gen2 <sup>¼</sup> *<sup>c</sup>*<sup>w</sup>�nfr <sup>þ</sup> *<sup>c</sup>*fw � *<sup>c</sup>*Lat�fw*,* (27) III*:*Range : *cw<sup>е</sup>*‐fr‐gen3 <sup>¼</sup> *<sup>c</sup>*<sup>w</sup>�fr <sup>þ</sup> *<sup>c</sup>*bwm � *<sup>c</sup>*Lat�bwm*:* (28)

� exp 0½ � *:*0567ð Þ *T* � 272*:*15

h i<sup>2</sup> *:*

fsp � 0*:*001ð Þ *T* � 293*:*15 h i<sup>2</sup>

� exp 0½ � *:*0567ð Þ *T* � 272*:*15

� *<sup>u</sup>*293*:*<sup>15</sup>

fsp � 0*:*001ð Þ *T* � 293*:*15 h i

(25)

The multiplier 3.6�10<sup>6</sup> in the denominator of Eq. (37) ensures that the values of *<sup>Q</sup>*<sup>w</sup> are obtained in kWh�m�<sup>3</sup> , instead of in J�m�<sup>3</sup> .

Based on Eq. (33), it is possible to calculate the energy *<sup>Q</sup>*fr‐bw according to the equation

$$Q\_{\rm fr\cdot bw} = \frac{c\_{\rm bw\cdot avg} \cdot \rho\_{\rm w}}{3.6 \cdot 10^6} \cdot (272.15 - T\_{\rm w\cdot fr\cdot avg}),\tag{34}$$

where according to [8, 9]

$$\begin{aligned} c\_{\text{bw-avg}} &= 1.8938 \cdot 10^4 \left( u\_{\text{fp}}^{272.15} - 0.12 \right) \cdot \frac{\exp \left[ 0.0567 \left( T\_{\text{w-fr-avg}} - 272.15 \right) \right]}{1 + u} \\ &\quad \text{@} \, T\_{\text{w-fr-avg}} \le T \le 272.15 \text{K}, \end{aligned} \tag{35}$$
 
$$\begin{aligned} T\_{\text{w-fr-avg}} &= \frac{1}{\mathcal{S}\_{\text{w}}} \left[ \iint\_{\mathcal{S}\_{\text{w}}} T(r, z, \tau) \, \text{dS}\_{\text{w}} \, \copysuit T\_{\text{w-fr-avg}} \, \text{\textdegree} \, T(r, z, \tau) \le 272.15 \text{K}, \end{aligned} \tag{36}$$

and the area of ¼ of the longitudinal section of the log subjected to freezing, *S*w, is equal to

$$\mathcal{S}\_{\mathbf{w}} = \frac{D \cdot L}{4}.\tag{37}$$

**Figure 2.**

**Figure 3.**

**121**

*temperature in logs subjected to refrigeration.*

*DOI: http://dx.doi.org/10.5772/intechopen.83772*

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

*Modeling of the Energy for Bound Water Freezing in Logs Subjected to Refrigeration*

*Experimentally determined change in tm, ϕm, and t in four points of the studied logs P1 (above) and S1 (below)*

*with D = 0.24 m and L = 0.48 m during their 30 h refrigeration.*

Based on Eq. (33), it is possible to calculate also the energy *<sup>Q</sup>*Lat‐bw according to the equation

$$Q\_{\text{Lat-bw}} = \frac{c\_{\text{Lat-bw}} \cdot \rho\_{\text{w}}}{\text{3.6} \cdot 10^6} \cdot (272.15 - T\_{\text{w-fr-avg}}),\tag{38}$$

where the specific heat capacity, which is formed by the release of the latent heat of the bound water, *c*Lat�bw, is calculated according to Eq. (21) and the density *ρ*<sup>w</sup> is calculated according to Eq. (9).
