**10. Mathematical model of cryotherapeutic device**

Given the variety of design solutions used in the manufacture of devices for WBC, the mathematical model should have the most generalized form. It is necessary to stop considering particular design features and focus on the fundamental issues. It becomes possible with a one-dimensional model of the WBC zone (**Figure 12**). The model considers the processes occurring in a volume unit of the WBC zone.

The surface of the patient's body 1 is cooled with a gaseous heat carrier 2 which fills the volume of the thermal fencing 3. The heat flux is removed from the patient's body surface *qHC*. The heat flux is supplied to the gas 2 from the surface of the thermal fencing 3 *qTI*. The movement of gas fluxes creates an additional source of the heat load *qGC*. To keep the gas temperature in the WBC zone at a constant level, it is necessary to ensure the removal of total heat *q*<sup>Σ</sup> to the cryostatting system. To form a one-dimensional model, it is necessary to relate all system indicators to the volume unit of the WBC zone. The specific heat transfer surface of the thermal insulation *f3* and the patient's body *f1* are determined taking into account the volume of the treatment cab *V3*:

$$f\_3 = F\_3 / V\_3; \ f\_1 = nF\_1 / V\_3. \tag{28}$$

**Figure 12.** *Heat fluxes in the WBC zone.*

*Technique and Technology of Whole-Body Cryotherapy (WBC) DOI: http://dx.doi.org/10.5772/intechopen.83680*

*Q*<sup>Σ</sup> ¼ *QHC* þ *QTI* þ *QGC:* (26)

*η<sup>H</sup>* ¼ *QHC=Q*Σ*:* (27)

*f* <sup>3</sup> ¼ *F*3*=V*3*; f* <sup>1</sup> ¼ *nF*1*=V*3*,* (28)

The energy efficiency of the installation design for WBC can be estimated by the

To estimate the expenditure of energy and select the optimal technology for WBC procedures, it is necessary to conduct a numerical experiment on a mathe-

Given the variety of design solutions used in the manufacture of devices for WBC, the mathematical model should have the most generalized form. It is necessary to stop considering particular design features and focus on the fundamental issues. It becomes possible with a one-dimensional model of the WBC zone (**Figure 12**). The model considers the processes occurring in a volume unit of the

The surface of the patient's body 1 is cooled with a gaseous heat carrier 2 which fills the volume of the thermal fencing 3. The heat flux is removed from the patient's body surface *qHC*. The heat flux is supplied to the gas 2 from the surface of the thermal fencing 3 *qTI*. The movement of gas fluxes creates an additional source of the heat load *qGC*. To keep the gas temperature in the WBC zone at a constant level, it is necessary to ensure the removal of total heat *q*<sup>Σ</sup> to the cryostatting system. To form a one-dimensional model, it is necessary to relate all system indicators to the volume unit of the WBC zone. The specific heat transfer surface of the thermal insulation *f3* and the patient's body *f1* are determined taking into account the volume

share of the useful load on the cryostatting system, the coefficient of thermal

efficiency:

*Low-temperature Technologies*

WBC zone.

**Figure 12.**

**150**

*Heat fluxes in the WBC zone.*

of the treatment cab *V3*:

matical model of a cryotherapeutic device.

**10. Mathematical model of cryotherapeutic device**

where *F*<sup>3</sup> means the total area of the internal surface of the cab's thermal fencing, *n* means the number of patients in the cab, and *F*<sup>1</sup> means the surface area of a patient's body.

The specific heat input from the patient's body and the thermal fencing is calculated considering the temperatures of their surfaces:

$$q\_{T1} = a\_{3-2} \, f\_3(T\_3 - T\_2); q\_{HC} = a\_{1-2} \, f\_1(T\_1 - T\_2), \tag{29}$$

where α3–<sup>2</sup> and α1–<sup>2</sup> means the heat transfer coefficients from the thermal fencing and the patient's body, respectively; *T*<sup>1</sup> and *T*<sup>3</sup> means the temperatures of surfaces of the body and the fencing; and *T*<sup>2</sup> means the temperature of the heat carrier gas.

Specific characteristics of devices designed for the implementation of GWBC and IWBC technologies have large differences. In multi-seat installations, the patient accommodation density is 0.4–0.7 person/m<sup>3</sup> , and the specific volume of free space *V0* is at least 98%. The specific surface area of the patient's body is 0.6–1.0 m2 /m<sup>3</sup> ; the thermal fencing area of the WBC zone is 2.4–3.0 m<sup>2</sup> /m<sup>3</sup> . Low compactness of accommodation of patients in multi-seat installations is necessary so that they can move from one low-temperature chamber to another.

In single-seat cryosaunas, the patient accommodation density reaches 2.0 persons/m<sup>3</sup> , the specific surface area of the patient's body is 3.2 m2 /m<sup>3</sup> , the thermal fencing area of the WBC zone is 6.4 m<sup>2</sup> /m<sup>3</sup> , and the specific free space volume is 84% [7]. High compactness of the patient accommodation is ensured by the fact that the patient does not move during the procedure; therefore, the cab size is comparable to the size of the patient's body.

The heat input with gas fluxes is determined by the intensity of convective mass transfer of warm gas to the volume of the WBC zone. The heat input by gas convection across the boundary of the WBC zone is determined from the expression:

$$\mathbf{g}\_{\rm GS} = \mathbf{c}\_{\rm p} \mathbf{g}\_{\rm G} (T\_{\rm g} - T\_{\rm 2}),\tag{30}$$

where *gG* means the specific transfer of the gas mass into the volume of the WBC zone, kg/(m3 sec); *cp* means the heat capacity of the gas, kJ/(kg K); and *Tg* means the temperature of the gas entering the WBC zone.

Large heat flows with gas fluxes are supplied into the WBC zone as patients enter and exit. For example, a multi-seat lock chamber and a cab of a single-seat cryosauna are filled with atmospheric air at the moment patients enter. 93 kJ/m<sup>3</sup> of heat enters the lock cab with atmospheric air. When the temperature recovers to the nominal level, the air density in the lock cab increases by 40%; this is accompanied by supplying additional air from the atmosphere, which contributes another 27 kJ/m<sup>3</sup> of heat. In one procedure, 120 kJ/m3 of heat transferred by gas convection enters the lock chamber.

The basis of the mathematical model of the WBC zone is a one-dimensional energy equation:

$$
\rho \frac{\partial h}{\partial \mathbf{r}} = \frac{\partial q\_{\mathbf{x}}}{\partial \mathbf{x}} + q\_{v^\*} \tag{31}
$$

where *qv* means the heat from internal sources:

$$q\_v = q\_\Sigma + q\_{HC} + q\_{T\bar{l}} + q\_{GC}.\tag{32}$$

#### *Low-temperature Technologies*

In the ideal case, *qv* = 0, since *q*<sup>Σ</sup> = �(*qHC* + *qTI* + *qGC*), i.e., the cooling system compensates for the heat input per volume unit of the WBC zone.

To account for material balance in the mathematical model of the WBC zone, the continuity equation is used:

$$
\frac{\partial \rho}{\partial \mathbf{r}} + \frac{\partial \mathbf{g}\_x}{\partial \mathbf{x}} = \mathbf{0}.\tag{33}
$$

Let us determine the specific values of the liquid nitrogen flowrate per

*LN* � � � � <sup>0</sup> *; Ga* <sup>¼</sup>

The results of the numerical experiment are summarized in **Table 3**. In the experiment on simulating the GWBC process, the time algorithm presented on the graph (**Figure 11**) was used; the nominal gas temperature in the main cab was 130 K. Energy indicators for the main and lock chambers were calculated. The energy efficiency of the technology was estimated by the total energy expenditures in the main and lock chambers. The results of the numerical experi-

The data in **Table 3** show that WBC procedures require the removal of large amounts of heat from the low-temperature zone. Specific heat input to the IWBC zone is *Q*<sup>Σ</sup> = 2012 kJ/m3. Given the short duration of the procedure (τmax = 180 sec), the mean heat load on the cooling system of the IWBC zone was 11.8 kW/m<sup>3</sup>

Considering that this heat load must be removed at a temperature level of 140 K, the estimated power of the cooling system of the low-temperature zone was

.

*Q*Σ, kJ/m<sup>3</sup> 2012 422 144 566 *Q*GC, kJ/m3 92 142 96 238 *Q*HC, kJ/m3 1427 246 33 279 *Q*TI, kJ/m3 493 33 14 47 *Q*Σ*=τmax*, kW/m3 11.8 2.76 1.18 3.94 η 0.71 0.58 0.23 0.49

<sup>5</sup> , kW/m3 136 11.04 4.72 15.76 *Q*5, kW/hour/m<sup>3</sup> 2.15 0.50 0.07 0.57 *Q*5*=τmax*, kW/m3 45.3 11.04 4.72 15.76 *GLN*, kg/m<sup>3</sup> 7.503 1.56 0.42 2.02 *G*LN*=τmax*, *g*/(sec�m3) 0.04 0.0029 0.009 0.0038

. Refrigerators of such power are rather expensive; therefore, it is economically feasible to use the nitrogen cooling system of the WBC zone. Specific

ð*<sup>τ</sup>*¼*τ*max *τ*¼0

**Cab Lock chamber Total**

*ga∂τ:* (39)

.

*rLN* þ *cLN Tg* � *T*″

ment on simulating the GWBC process are summarized in **Table 3**.

**Indicators IWBC GWBC**

/m<sup>3</sup> 3.2 0.62 0.62

/m<sup>3</sup> 6.4 2.4 2.4 *V*0, % 0.84 0.97 0.97

<sup>Σ</sup> , kW/m3 33 7.43 3.97

The expenditure of electrical energy and liquid nitrogen flowrate for cooling

*gLN* <sup>¼</sup> *<sup>q</sup>*<sup>Σ</sup>

*Technique and Technology of Whole-Body Cryotherapy (WBC)*

*DOI: http://dx.doi.org/10.5772/intechopen.83680*

procedure:

45.3 kW/m<sup>3</sup>

*f*1, m<sup>2</sup>

*f*2, m<sup>2</sup>

*qmax*

*Nmax*

**Table 3.**

**153**

Features of WBC zone

Heat input to the WBC zone

*Energy features of devices for IWBC and GWBC.*

liquid nitrogen flowrates are 7.503 kg/m<sup>3</sup>

So, the transfer of heat by the thermal conductivity of gas is small; expression (30) is simplified and can be transformed by replacing the derivatives with differential approximants:

$$
\rho \frac{\Delta h}{\Delta \tau} = q\_v; h' = h + \frac{q\_v \Delta \tau}{\rho}. \tag{34}
$$

The numerical solution of the continuity (Eq. (33)) allows to take into account the input of gas mass to compensate for the change in density:

$$\mathbf{g}'\_{\mathbf{x}} = \mathbf{g}\_{\mathbf{x}} + \frac{(\rho' - \rho)\Delta\mathbf{x}}{\Delta\mathbf{r}}.\tag{35}$$

Eqs. (29), (34) and (35) allow to analyze the processes occurring in the WBC zone during the implementation of individual or group technology. To perform a computational experiment, it is necessary to adopt an algorithm for changing the temperature of the cooling gas for IWBC and GWBC.

Formulation of a temperature algorithm for the IWBC process is relatively simple. let us take the time of filling the zone with a cryogenic gas (**Figure 11**) τ<sup>I</sup> = 20 sec, τII = 150 sec, and τIII = 10 sec and the gas temperature in the isothermal phase II *T*II = 140 K. The specific heat transfer surfaces of heat sources are *f*<sup>1</sup> = 3.2 m<sup>2</sup> /m<sup>3</sup> , and *f*<sup>3</sup> = 6.4 m<sup>2</sup> /m<sup>3</sup> ; the specific free space volume is 84% [7]. When simulating the IWBC process, heat fluxes from various sources (28) and (29) are calculated, and the integral heat input is determined:

$$\mathbf{Q\_{HC}} = \mathbf{f\_2} \int\_{\mathbf{r}=0}^{\mathbf{r}=\mathbf{r\_{max}}} q\_{HC} \partial \mathbf{r}; \mathbf{Q\_{TI}} = \mathbf{f\_3} \int\_{\mathbf{r}=0}^{\mathbf{r}=\mathbf{r\_{max}}} q\_{TI} \partial \mathbf{r}; \mathbf{Q\_{GC}} = \int\_{\mathbf{r}=0}^{\mathbf{r}=\mathbf{r\_{max}}} q\_{GC} \partial \mathbf{r}.\tag{36}$$

By Eqs. (26) and (27), the total heat load on the cooling system and the coefficient of thermal efficiency are calculated. It is assumed that the cooling system covers all types of heat load, so the specific power of the refrigerator electric drive can be determined by the heat load and the value of the coefficient of performance at the current temperature level:

$$N\_{\mathfrak{F}} = q\_{\mathfrak{L}} / e\_{\mathfrak{F}} ; \varepsilon\_{\mathfrak{F}} = f\left(T\_{\mathfrak{g}}\right), \tag{37}$$

where ε<sup>5</sup> means the coefficient of performance and the ratio of the heat removed to expenditure of energy in the refrigerator at the temperature level of 140 K, ε<sup>5</sup> = 0.25 W/W.

For the instantaneous values of the calculated power of the system refrigerator, the specific expenditure of energy for cooling the IWBC zone per procedure is calculated:

$$Q\_5 = \int\_{\tau=0}^{\tau=\tau \text{max}} N\_5 \partial \tau \tag{38}$$

*Technique and Technology of Whole-Body Cryotherapy (WBC) DOI: http://dx.doi.org/10.5772/intechopen.83680*

In the ideal case, *qv* = 0, since *q*<sup>Σ</sup> = �(*qHC* + *qTI* + *qGC*), i.e., the cooling system

To account for material balance in the mathematical model of the WBC zone,

So, the transfer of heat by the thermal conductivity of gas is small; expression (30) is simplified and can be transformed by replacing the derivatives with differ-

*Δτ* <sup>¼</sup> *qv; h*<sup>0</sup> <sup>¼</sup> *<sup>h</sup>* <sup>þ</sup> *qvΔτ*

The numerical solution of the continuity (Eq. (33)) allows to take into account

*<sup>x</sup>* <sup>¼</sup> *gx* <sup>þ</sup> *<sup>ρ</sup>*ð Þ <sup>0</sup> � *<sup>ρ</sup> <sup>Δ</sup><sup>x</sup>*

Eqs. (29), (34) and (35) allow to analyze the processes occurring in the WBC zone during the implementation of individual or group technology. To perform a computational experiment, it is necessary to adopt an algorithm for changing the

Formulation of a temperature algorithm for the IWBC process is relatively simple. let us take the time of filling the zone with a cryogenic gas (**Figure 11**) τ<sup>I</sup> = 20 sec, τII = 150 sec, and τIII = 10 sec and the gas temperature in the isothermal

simulating the IWBC process, heat fluxes from various sources (28) and (29) are

*<sup>τ</sup>*<sup>¼</sup>ð*<sup>τ</sup>*max

*qTI∂τ; QGC* <sup>¼</sup>

*τ*¼0

*N*<sup>5</sup> ¼ *q*Σ*=ε*5*; ε*<sup>5</sup> ¼ *f Tg*

to expenditure of energy in the refrigerator at the temperature level of 140 K,

the specific expenditure of energy for cooling the IWBC zone per procedure is

*Q*<sup>5</sup> ¼

where ε<sup>5</sup> means the coefficient of performance and the ratio of the heat removed

For the instantaneous values of the calculated power of the system refrigerator,

ð*<sup>τ</sup>*¼*τ*max τ¼0

By Eqs. (26) and (27), the total heat load on the cooling system and the coefficient of thermal efficiency are calculated. It is assumed that the cooling system covers all types of heat load, so the specific power of the refrigerator electric drive can be determined by the heat load and the value of the coefficient of performance

phase II *T*II = 140 K. The specific heat transfer surfaces of heat sources are

/m<sup>3</sup>

*qHC∂τ; QTI* <sup>¼</sup> *<sup>f</sup>* <sup>3</sup>

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> <sup>0</sup>*:* (33)

*<sup>ρ</sup> :* (34)

*Δτ :* (35)

; the specific free space volume is 84% [7]. When

*<sup>τ</sup>*<sup>¼</sup>ð*<sup>τ</sup>*max

*qGC∂τ:* (36)

*τ*¼0

� �*,* (37)

*N*5*∂τ* (38)

compensates for the heat input per volume unit of the WBC zone.

*ρ Δh*

the input of gas mass to compensate for the change in density:

*g*0

temperature of the cooling gas for IWBC and GWBC.

calculated, and the integral heat input is determined:

, and *f*<sup>3</sup> = 6.4 m<sup>2</sup>

*<sup>τ</sup>*<sup>¼</sup>ð*<sup>τ</sup>*max

*τ*¼0

at the current temperature level:

*∂ρ ∂τ* þ *∂gx*

the continuity equation is used:

*Low-temperature Technologies*

ential approximants:

*f*<sup>1</sup> = 3.2 m<sup>2</sup>

/m<sup>3</sup>

*QHC* ¼ *f* <sup>2</sup>

ε<sup>5</sup> = 0.25 W/W.

calculated:

**152**

Let us determine the specific values of the liquid nitrogen flowrate per procedure:

$$\mathbf{g}\_{LN} = \frac{q\_{\Sigma}}{\left[r\_{LN} + c\_{LN}\left(T\_{\mathbf{g}} - T\_{LN}^{'}\right)\right]'};\\\mathbf{G}\_{a} = \int\_{\tau=0}^{\tau=\tau\_{\text{max}}} \mathbf{g}\_{a} \,\partial\tau. \tag{39}$$

The results of the numerical experiment are summarized in **Table 3**. In the experiment on simulating the GWBC process, the time algorithm presented on the graph (**Figure 11**) was used; the nominal gas temperature in the main cab was 130 K. Energy indicators for the main and lock chambers were calculated.

The energy efficiency of the technology was estimated by the total energy expenditures in the main and lock chambers. The results of the numerical experiment on simulating the GWBC process are summarized in **Table 3**.

The data in **Table 3** show that WBC procedures require the removal of large amounts of heat from the low-temperature zone. Specific heat input to the IWBC zone is *Q*<sup>Σ</sup> = 2012 kJ/m3. Given the short duration of the procedure (τmax = 180 sec), the mean heat load on the cooling system of the IWBC zone was 11.8 kW/m<sup>3</sup> . Considering that this heat load must be removed at a temperature level of 140 K, the estimated power of the cooling system of the low-temperature zone was 45.3 kW/m<sup>3</sup> . Refrigerators of such power are rather expensive; therefore, it is economically feasible to use the nitrogen cooling system of the WBC zone. Specific liquid nitrogen flowrates are 7.503 kg/m<sup>3</sup> .


#### **Table 3.**

*Energy features of devices for IWBC and GWBC.*

#### *Low-temperature Technologies*

The energy indicators of the GWBC zone are much lower (**Table 3**). The specific heat input is *Q*<sup>Σ</sup> = 566 kJ/m3 , the mean heat load on the cooling system is 15.6 kW/ m3 , and the power of the cooling system is determined by the sum of the inflows of heat into the main cab and lock chamber.

**11. Conclusion**

less than 2.4 kg/m<sup>2</sup>

**Other declarations**

**Author details**

**155**

Andrew Zaitsev and Ruslan Polyakov

provided the original work is properly cited.

.

Russian Federation (Project 11.4942.2017/6.7).

The performed analysis of the healthcare and energy efficiency of the two options for the implementation of the WBC technology allows us to reasonably give preference to individual procedures that not only combine high healthcare efficiency with relatively low expenditure of energy but also to a greater extent correspond to the traditional principle of individuality of therapeutic techniques.

*Technique and Technology of Whole-Body Cryotherapy (WBC)*

*DOI: http://dx.doi.org/10.5772/intechopen.83680*

The effectiveness of WBC technology depends on the choice of the duration of

the case of using nitrogen cooling system, the cryoagent consumption should be not

The research was supported by the Ministry of Education and Science of the

Alexander Baranov\*, Oleg Pakhomov, Alexander Fedorov, Vladimir Ivanov,

Saint Petersburg National Research University of Information Technologies,

© 2019 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,

Mechanics and Optics, Saint Petersburg, Russian Federation

\*Address all correspondence to: abaranov@corp.ifmo.ru

, and in

contact with cryogenic gas. The minimum duration of WBC procedure at the optimum gas temperature (130°С) is 120 s. Meanwhile, one should remove 440 kJ/m<sup>2</sup> with an average intensity of at least 2.4 kW/m<sup>2</sup> and spend not less than 1.7 kg/m2 of liquid nitrogen on heat removal. The electric drive of the cooling system of WBC zone should have an average power of at least 9.3 kW/m<sup>2</sup>

Due to the low compactness of the accommodation of patients in the treatment area, the GWBC thermal efficiency coefficient was 0.49. Under conditions of a single-seat installation, the thermal efficiency coefficient was 0.71, which indicates a more rational expenditure of energy. This is clearly illustrated by the histogram of the structure of the heat load on the cooling system of the IWBC and GWBC zones (**Figure 13**).

In single-seat installations, the heat storage capacity of the thermal fencing makes a significant contribution to the heat load, due to which the share of heat removed from thermal insulation reaches 24%. At the beginning of each procedure, a single-seat cab is filled with atmospheric air, which heats the inner surface of the thermal insulation. When implementing the GWBC technology, the heat load from the insulation is insignificant means of 9%, but the convective heat supply is 24%. The negative impact of convective heat transfer is determined by a large share of the free space in the low-temperature zone.

The data in **Table 3** do not allow giving an unambiguous preference for a particular technology. This is due to the fact that all indicators are related to the volume unit of the WBC zone, while the technological task of the process is to cool the surface of the patient's body shell. If we calculate the specific heat load values and the expenditure of energy for cooling a unit of the shell surface (**Table 4**), the advantages of the IWBC technology become indisputable. According to all energy indicators, the IWBC technology is 1.5 times more efficient than the GWBC process.

**Figure 13.**

*The structure of the heat load on the cooling system zones IWBC and GWBC.*


#### **Table 4.**

*Energy features of devices for IWBC and GWBC.*
