# Analytical Model for Thermoregulation of the Human Body in Contact with a Phase Change Material (PCM) Cooling Vest

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Hotplate

^{−1}K

^{−1}, respectively.

#### 2.2. Sample Preparation

^{2}, respectively.

#### 2.3. Procedure

## 3. Heat Transfer Theory

#### 3.1. Basic Heat Equations

^{−1}K

^{−1}], k denotes the thermal conductivity in [Wm

^{−1}K

^{−1}] and Q

_{V}is a volumetric heat source term with units [W/m

^{3}]. Integrating over the layer thickness d, we obtain

_{1}corresponds to half the torso thickness.

^{rad}term in Figure 2 denotes the radiation heat flow, whereas q

_{i}is the heat flux due to conduction

_{i}[m

^{2}K/W] is the thermal resistance of layer i. If there are two adjacent insulation layers with resistances of, say, R

_{i,a}and R

_{i,b}then heat flux to both layers is still equal to q

_{i}and the effective thermal resistance of the combined layers can be expressed as the sum of the sub-layer contributions

#### 3.2. Human Cooling Model

^{rad}= h

_{r}(T

_{4}− T

_{a}) in which h

_{r}[Wm

^{−2}K

^{−1}] is the radiative heat transfer coefficient, T

_{4}is the temperature of the outer surface and T

_{a}is the mean radiant temperature. Similar to the inner insulation layer above, we can eliminate temperature T

_{4}and define an effective thermal resistance for the heat flux from layer 3 to temperature T

_{a}:

_{1}(t = 0) = T

_{1}

^{0}and T

_{3}(t = 0) = T

_{3}

^{0}as initial conditions:

_{0}= 0. Note that the source terms comprise all the heat generated or absorbed by the body:

_{1}. Details for determining the individual contributions can be found in thermophysiology literature [33].

#### Skin Temperature

^{skin}and which is part of the thermal resistance between the body core and PCM (inner insulation, layer 2). Since the heat fluxes through the skin and the rest of layer 2 are equal, we can calculate the temperature on the skin surface as

_{R}can be interpreted as the ratio of insulation resistance between the skin layer and the total of layer 2. For thin skin, f

_{R}is small and the skin temperature is close to the body core temperature, T

_{1}whereas in absence of other insulation the skin temperature matches that of the PCM, T

_{3}. In the following section, we will derive solutions for T

_{1}and T

_{3}in a few practically relevant cases.

#### 3.3. Solutions for T_{1} and T_{3}

#### 3.3.1. Solution 1: Human Body Cooling with Heat Production and Radiation; No PCM Layer

_{2}and R

_{4}, respectively, and radiation to temperature T

_{a}(see Figure 1). We thus have one transient thermal element with temperature T

_{1}and one insulation layer combination with effective resistance R

_{2}

^{eff}= R

_{2}+ R

_{4}+ 1/h

_{r}. Dividing by ρ

_{1}c

_{1}d

_{1}the differential equation then becomes

_{0}and Q’

_{1}are defined below Equation (10). The solution of Equation (9) with initial condition T

_{1}(t = 0) = T

_{1}

^{0}then becomes

_{1}can be regarded as the time constant which governs the thermal transients and depends on the values of the thermal insulations of layer 2 and 4 as well as on the radiation resistance 1/h

_{r}. Equation (10) can describe both the body heating due to internal heat sources and radiation from an elevated ambient temperature as well as the cooling of a body in a colder environment. In the steady state, radiation and heat generation are in equilibrium and T

_{1}approaches T

_{a}+ Q

_{1}/h

_{0}.

#### 3.3.2. Solution 2A: Human Body Cooling with PCM Layer, Heat Production and Radiation

_{1}and T

_{3}change with time. In the case of a human body in contact with a frozen phase change material that means that the PCM heats up and the human body slowly cools. In the next section, we discuss the case in which the PCM material is in its melting stage. We start with rewriting Equation (6) as

_{2}

^{eff}= R

_{2}and R

_{4}

^{eff}= R

_{4}+ 1/h

_{r}. The solution of the differential equations will have exponential terms of the form ${A}_{i}{e}^{-{\alpha}_{i}t}$ in which the coefficients A

_{i}and α

_{i}follow by substitution in Equation (11) in combination with the initial conditions T

_{1}(t = 0) = T

_{1}

^{0}and T

_{3}(t = 0) = T

_{3}

^{0}. The results consist of a term that describes the combined effect of conduction and radiation, T

^{c+r}, and a second term that deals with the combined source terms and radiation, T

^{Q+r}. For the overall temperature we can thus write:

_{a}. Remark that the above (and following) solutions all have the structure

_{1}and A

_{2}denote the magnitude of these terms. For the source terms in combination with radiation (Q + r) we obtain

_{3}/(h

_{1}+ h

_{3}) (see Supplementary Information). If the PCM layer is absent, β = 1.

#### 3.3.3. Solution 2B: Human Body Cooling with PCM during Melting Stage

_{3}[J/kg] denotes the latent heat of material 3. During melting the temperature in the material is constant and at its melting point T

_{3}

^{m}. Inserting this in Equations (6) and solving the equations then yields

_{1}is the time the PCM layer reaches its melting point. Note that this equation does not contain any radiation parameter which makes sense since the constant PCM temperature in fact blocks the transfer of outside radiation effects to layer 1. The net effect of radiation is that it speeds up the melting process, an effect which will be discussed in more detail for case 3B below. Equation (17) shows the effect of PCM melting on the body core temperature. Whether the body temperature increases due to the metabolic heat or decreases due to PCM cooling depends on the sign of the term between brackets. If Q’

_{1}/h

_{1}> T

_{1}(t

_{1}) − T

_{3}

^{m}then the body temperature increases, if it is smaller, it decreases.

_{1}− T

_{3}. Most relevant to estimate the cooling efficiency is its value during the PCM melting stage for which the PCM temperature is at its melting point. In addition, the core temperature will never deviate much from its initial value, so we may also replace T

_{1}with its initial value T

_{1}

^{0}to obtain as an estimate of the PCM cooling power during melting:

#### 3.3.4. Solution 3A: Hot Plate with PCM Transients and Radiation

_{1}is controlled at a fixed T

_{1}

^{hp}by applying a heat source of strength Q

_{1}= P

_{hp}. The hot plate experiment serves to determine experimentally how much heat is transferred from the PCM material through the insulation to the body in the ideal situation that the body temperature would remain constant. This allows, for example, to experimentally determine the insulation resistance R

_{2}

^{eff}and the time constant α

_{0}. Solving Equation (6) we obtain for the hotplate power, P

_{hp}and PCM temperature

_{3}gradually rises towards the hot plate temperature the generated cooling power, P

_{hp}, decreases during the experiment. The speed of this decrease is inversely proportional to h

_{3}+ h

_{4}and thus depends on the radiation parameter h

_{4}. In the setup of the experiment, the choice can be made to isolate the upper PCM surface. In that situation, h

_{4}= 0 and parameter h

_{3}can be extracted directly from the measurements. In addition, by putting the hotplate in an environment with ambient temperature T

_{a}= T

_{hp}, we can determine the sum of h

_{3}and h

_{4}and obtain the radiation parameter h

_{4}. It is also possible to do the hotplate experiment with a clothing layer in between the hot plate and PCM to measure the thermal resistance value of the clothing fabric. From the experimental data during the transient phases, the effect of the radiation and outer insulation is obtained whereas the thermal resistance of the inner layers follows from the cooling power at the melting plateau (see Equation (21) below).

#### 3.3.5. Solution 3B: Hot Plate during PCM Melting with Radiation at the PCM Surface

_{3}is constant and the cooling power plateau is simply given by

_{m}is obtained by solving for the source term in the second part of Equation (6)

_{3}A t

_{m}= −E

_{3}

^{m}= −m

_{3}L

_{3}. The minus sign is to account for the fact that the PCM is not a heat source but a heat sink. Rearranging then gives for the duration of the PCM melting stage

## 4. Model Verification

#### 4.1. Comparison with Measured Cooling Power of PCM Ice Packs

^{−1}K

^{−1}for the TPU and silicone rubber, respectively, and for water a density of 1000 kg/m

^{3}, a solid and liquid heat capacity of 2100 and 4180 J kg

^{−1}K

^{−1}, respectively, and latent heat of melting of 334 J/g. To account for the air gap between the sample hotplate an initial resistance value of 3.8 10

^{−3}m

^{2}K W

^{−1}is assumed. A comparison with the predicted cooling power curves (dashed lines) shows that both the initial decay, the durations of the melting plateau and the subsequent exponential decay are reasonably well predicted. The root mean square error between measurements and predictions amounts to 4.7 W, which is about 10% of the absolute cooling power values. The values of the plateau values at the end of melting are plotted in Figure 4b together with their predictions. The agreement between measured and predicted cooling power at the melting plateau (Figure 4b) can be considered as good.

#### 4.2. Comparison with Data Reported in Literature

^{2}, [30]), followed by a 20-min recovery period (1 MET, 58 W/m

^{2}). Tests were performed using a cooling vest with and without Glauber salt-based PCM packs with an initial surface temperature of 15 °C. The mean radiant temperature and relative humidity were kept constant at 36 °C and 59%, respectively. For the modeling a radiation heat transfer coefficient of 14 W m

^{−2}K

^{−1}and an initial body temperature of 37 °C were used. The thermal properties used for the calculations can be found in Table 1.

^{2}KW

^{−1}. In Figure 5 we compare their predictions of the PCM temperature (dashed line with symbols) with our solutions for T

_{3}(full line). It is clear that both phase 1 (warming of the PCM to its melting temperature) and phase 2 (PCM melting) are accurately captured by our model whereas the last part (warming of the liquid PCM towards T

_{a}= 36 °C) is slightly over predicted.

_{1}− T

_{3}) and (T

_{a}− T

_{3}), respectively (see Equation (4)), the heat flow plateaus between 5 and 50 min reflect the PCM melting period. The peak in Wan’s body heat flow between 5 to 15 min is due to sweat condensation on the PCM inner surface which was taken into account in the numerical calculations but not in our model. Condensation thus does have an effect on the heat flow, but its contribution is limited. Comparing the heat flow calculations with those of Wan [23] shows a fairly good agreement over the full time scale.

_{1}

^{0}we can obtain a simple expression for the ratio r

_{HF}of heat lost to the environment with respect to that absorbed from the body and calculate the corresponding fractions (Equation (23)). As shown by the full lines in Figure 6b these results agree quite well with the simulation data of Kang [25]. The only difference appears in the case in which the extra insulation is absent (see also Figure 6a). It should be noted that by choosing a different heat transfer coefficient (25 instead of 14 Wm

^{−2}K

^{−1}) a much better fit would have been obtained.

## 5. Parametric Study

^{2}, initial body and mean radiant temperatures of 36 °C, a heat transfer coefficient of 14 W m

^{−2}K

^{−1}and a density of 500 kg/m

^{3}. The effective thermal resistance R

_{2}

^{eff}is the sum of the resistances of the skin, underwear and PCM pocket (see Table 1) and the extra air gap resistance of 0.038 m

^{2}KW

^{−1}mentioned in [23]. The parameters that we varied were the PCM thickness (obtained from m

_{3}= ρ

_{3}A

_{3}d

_{3}) and the PCM melting temperature, T

_{3}

^{m}.

## 6. Discussion, Limitations and Recommendations for Future Work

- The model considers one-dimensional planar heat transfer and thus neglects the effects of body curvature on the conduction and heat loss (or gain) to the environment. Since a curved body has more surface area per unit volume, this assumption underestimates the radiation contribution, and this effect gets larger for more curved body parts such as arms and legs. Since we are interested in the heat transfer near the PCM layer, the rule of thumb is that the model is valid if the PCM thickness is much less than that of (half of) the body, d
_{3}/d_{1}« 1. Note that in thermophysiology models the body is usually considered to consist of cylindrical elements [12,13,14]. This may be a good approximation for the arms and legs but could be less appropriate for the trunk area. - In order to simulate human thermoregulation in realistic conditions, heat loss by sweat evaporation, respiration and the effect of subcutaneous blood flow must be taken into account. As mentioned before, all these effects essentially act as an energy source or loss contributions in the body core layer and thus only change the numerical value of the Q
_{1}parameter in our model. The equations to model these energy loss terms are conveniently described in [33]. - In practice, only a part of the torso surface is covered with PCM packs. The effect of such a partial coverage on the core temperature can be estimated by a simple averaging process.

_{A}denotes the area fraction of the torso covered by PCM packs. The PCM temperature T

_{3}and calculated cooling times as derived here remain as they are.

_{3}terms in the equations above can be omitted. Next to that, simplifications can be made by linearizing the transient elements, as explained in the Suppl. Info. In addition, for cases where the thermal mass of layer 3 (the PCM layer) is much smaller than that of the body layer the β and δ parameters are close to unity and more simplifications are possible. The largest simplifications are obtained in cases where radiation is absent. In that case, there is only one exponential term needed. The solutions for the no-radiation case are given in the Supplemental Information.

## 7. Conclusions

- A set of closed-form equations is presented which do not need a numerical solver and directly relate cooling vest design parameters such as the PCM mass and melting temperature to its performance
- The model was able to reproduce cooling power data obtained from experiments with ice packs on a hot plate with constant surface temperature as well as PCM cooling data in various literature sources;
- The relation between the PCM performance and the PCM melting temperature and layer thickness is shown in a dedicated parametric study.

_{3}terms in the equations above. The hotplate solutions can also be used to describe the case of a thermal manikin with different clothing and PCM configurations in constant surface temperature mode. Moreover, of course, the solutions are not limited to a human body system and can also be used to estimate the effectiveness of cooling elements in contact with food or PCMs used in industrial processes or building environments. The equations are computationally simple and do not need a numerical solver, which allows them to be embedded in a spreadsheet or app as a simple means of estimating the layers of clothing needed to dress in a cold environment or the maximum work time in insulating garments while working in a hot environment.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A | Surface area of layer element (m^{2}) | ||

A_{1}, A_{2} | Parameters in solution for T (°C) | Greek symbols | |

C | Parameter in Equations (14) and (15) (s^{−1}) | α | Parameter in exponential function (s^{−1}) |

c_{1}, c_{3} | Specific heat (J kg^{−1} K^{−1}) | β | Parameter in Equations (13) and (15) (-) |

d | Layer thickness (m); d_{1} is half the torso thickness | γ | Parameter in Equations (13) and (15) (-) |

E | Energy (J) | δ | Parameter in Equations (13) and (15) (-) |

f | Fraction (-) | ρ | Density (kg m^{−3}) |

h_{i} | Parameter (s^{−1}) | Sub- and superscripts | |

h_{r} | radiative heat transfer coefficient (Wm^{−2}K^{−1}) | 0 | Solution at t = 0 |

H | Parameter (s^{−1}) | 1 | Body core layer |

k | Thermal conductivity (Wm^{−1}K^{−1}) | 2 | Insulation between body core and PCM layer |

L | Latent heat of PCM (J/kg) | 3 | PCM layer |

m_{i} | Mass of layer i (kg) | 4 | Outside insulation |

P | Cooling power (W) | a | ambient |

q | Heat flux (W/m^{2}) | eff | effective |

Q | Heat production per unit area (W/m^{2}) | HF | Heat flow |

Q’ | Q/(ρcd) (K/s) | hp | hotplate |

Q_{V} | Volumetric heat production (W/m^{3}) | i | Layer number |

R | Thermal resistance (m^{2}K W^{−1}) | l | liquid |

t | Time (s) | m | melt |

T | Temperature (°C) | R | Resistance contribution |

x | Thickness coordinate (m) | rad | Radiation contribution |

s | solid | ||

Abbreviations | skin | Skin Layer | |

PCM | Phase change material |

## References

- Mokhtari Yazdi, M.; Sheikhzadeh, M. Personal cooling garments: A review. J. Text. Inst.
**2014**, 105, 1231–1250. [Google Scholar] [CrossRef] - Smolander, J.; Kuklane, K.; Gavhed, D.; Nilsson, H.; Holmer, I. Effectiveness of a light-weight ice-vest for body cooling while wearing fire fighter’s protective clothing in the heat. Int. J. Occup. Saf. Ergon. JOSE
**2004**, 10, 111–117. [Google Scholar] [PubMed] - Teunissen, L.P.J.; Wang, L.-C.; Chou, S.-N.; Huang, C.-H.; Jou, G.-T.; Daanen, H.A.M. Evaluation of two cooling systems under a firefighter coverall. Appl. Ergon.
**2014**, 45, 1433–1438. [Google Scholar] [CrossRef] [PubMed] - Butts, C.L.; Smith, C.R.; Ganio, M.S.; McDermott, B.P. Physiological and perceptual effects of a cooling garment during simulated industrial work in the heat. Appl. Ergon.
**2017**, 59, 442–448. [Google Scholar] - House, J.R.; Lunt, H.C.; Taylor, R.; Milligan, G.; Lyons, J.A.; House, C.M. The impact of a phase-change cooling vest on heat strain and the effect of different cooling pack melting temperatures. Eur. J. Appl. Physiol.
**2013**, 113, 1223–1231. [Google Scholar] - Song, W.; Wang, F. The hybrid personal cooling system (PCS) could effectively reduce the heat strain while exercising in a hot and moderate humid environment. Ergonomics
**2016**, 59, 1009–1018. [Google Scholar] - Ouahrani, D.; Itani, M.; Ghaddar, N.; Ghali, K.; Khater, B. Experimental study on using PCMs of different melting temperatures in one cooling vest to reduce its weight and improve comfort. Energy Build.
**2017**, 155, 533–545. [Google Scholar] [CrossRef] - Li, W.; Liang, Y.; Liu, C.; Ji, Y.; Cheng, L. Study of ultra-light modular phase change cooling clothing based on dynamic human thermal comfort modeling. Build. Environ.
**2022**, 222, 109390. [Google Scholar] [CrossRef] - ASTM F2371-10; Standard Test Method for Measuring the Heat Removal of Personal Cooling System Using a Sweating Heated Manikin. ASTM International: West Conshohocken, PA, USA, 2010.
- Ciuha, U.; Valenčič, T.; Mekjavic, I.B. Cooling efficiency of vests with different cooling concepts over 8-hour trials. Ergonomics
**2021**, 64, 625–639. [Google Scholar] [CrossRef] - Gao, C.; Kuklane, K.; Holmér, I. Cooling vests with phase change material packs: The effects of temperature gradient, mass and covering area. Ergonomics
**2010**, 53, 716–723. [Google Scholar] [CrossRef] - Gao, C.; Kuklane, K.; Holmér, I. Cooling vests with phase change materials: The effects of melting temperature on heat strain alleviation in an extremely hot environment. Eur. J. Appl. Physiol.
**2011**, 111, 1207–1216. [Google Scholar] [CrossRef] [PubMed] - Stolwijk, J.A.J.; Hardy, J.D. Temperature regulation in man—A theoretical study. Pflüger’s Arch. Für Die Gesamte Physiol. Des Menschen Und Der Tiere
**1966**, 291, 129–162. [Google Scholar] [CrossRef] - Fiala, D.; Lomas, K.J.; Stohrer, M. A computer model of human thermoregulation for a wide range of environmental conditions: The passive system. J. Appl. Physiol.
**1999**, 87, 1957–1972. [Google Scholar] [CrossRef] [PubMed] - Tanabe, S.-I.; Kobayashi, K.; Nakano, J.; Ozeki, Y.; Konishi, M. Evaluation of thermal comfort using combined multi-node thermoregulation (65MN) and radiation models and computational fluid dynamics (CFD). Energy Build.
**2002**, 34, 637–646. [Google Scholar] [CrossRef] - Potter, A.W.; Blanchard, L.A.; Friedl, K.E.; Cadarette, B.S.; Hoyt, R.W. Mathematical prediction of core body temperature from environment, activity, and clothing: The heat strain decision aid (HSDA). J. Therm. Biol.
**2017**, 64, 78–85. [Google Scholar] [CrossRef] - Mokhtari Yazdi, M.; Sheikhzadeh, M.; Borhani, S. Modeling the heat transfer in a PCM cooling vest. J. Text. Inst.
**2015**, 106, 1003–1012. [Google Scholar] [CrossRef] - Mokhtari Yazdi, M.; Sheikhzadeh, M.; Dabirzadeh, A.; Chavoshi, E. Modeling the efficiency and heat gain of a phase change material cooling vest: The effect of ambient temperature and outer isolation. J. Ind. Text.
**2016**, 46, 436–454. [Google Scholar] [CrossRef] - Yazdi Motahareh, M.; Sheikhzadeh, M.; Chavoshi Seyed, E. Modeling the performance of a PCM cooling vest considering its side effects. Int. J. Cloth. Sci. Technol.
**2015**, 27, 573–586. [Google Scholar] [CrossRef] - Hamdan, H.; Ghaddar, N.; Ouahrani, D.; Ghali, K.; Itani, M. PCM cooling vest for improving thermal comfort in hot environment. Int. J. Therm. Sci.
**2016**, 102, 154–167. [Google Scholar] [CrossRef] - Itani, M.; Ghaddar, N.; Ghali, K.; Ouahrani, D.; Chakroun, W. Cooling vest with optimized PCM arrangement targeting torso sensitive areas that trigger comfort when cooled for improving human comfort in hot conditions. Energy Build.
**2017**, 139, 417–425. [Google Scholar] [CrossRef] - Itani, M.; Ouahrani, D.; Ghaddar, N.; Ghali, K.; Chakroun, W. The effect of PCM placement on torso cooling vest for an active human in hot environment. Build. Environ.
**2016**, 107, 29–42. [Google Scholar] [CrossRef] - Wan, X.F.; Wang, F.M.; Udayraj. Numerical analysis of cooling effect of hybrid cooling clothing incorporated with phase change material (PCM) packs and air ventilation fans. Int. J. Heat. Mass Tran.
**2018**, 126, 636–648. [Google Scholar] [CrossRef] - Xu, P.; Kang, Z.; Wang, F.; Udayraj. A Numerical Analysis of the Cooling Performance of a Hybrid Personal Cooling System (HPCS): Effects of Ambient Temperature and Relative Humidity. Int. J. Environ. Res. Public Health
**2020**, 17, 4995. [Google Scholar] [CrossRef] - Kang, Z.; Udayraj; Wan, X.; Wang, F. A new hybrid personal cooling system (HPCS) incorporating insulation pads for thermal comfort management: Experimental validation and parametric study. Build. Environ.
**2018**, 145, 276–289. [Google Scholar] [CrossRef] - Wan, X.; Fan, J. A new method for measuring the thermal regulatory properties of phase change material (PCM) fabrics. Meas. Sci. Technol.
**2009**, 20, 025110. [Google Scholar] [CrossRef] - Ying, B. Assessing the performance of textiles incorporating phase change materials. Polym. Test.
**2004**, 23, 541. [Google Scholar] - Ghali, K.; Ghaddar, N.; Harathani, J.; Jones, B. Experimental and Numerical Investigation of the Effect of Phase Change Materials on Clothing During Periodic Ventilation. Text. Res. J.
**2004**, 74, 205–214. [Google Scholar] [CrossRef] - Elson, J.; Eckels, S. An objective method for screening and selecting personal cooling systems based on cooling properties. Appl. Ergon.
**2015**, 48, 33–41. [Google Scholar] [CrossRef] - Al-Mujahid, A.; Zedan, M.F. Transient heat-conduction response of a composite plane wall. Wärme-Und Stoffübertragung
**1991**, 26, 33–39. [Google Scholar] [CrossRef] - Hu, Y.; Huang, D.; Qi, Z.; He, S.; Yang, H.; Zhang, H. Modeling thermal insulation of firefighting protective clothing embedded with phase change material. Heat Mass Transfer.
**2013**, 49, 567–573. [Google Scholar] [CrossRef] - Carslaw, H.S.; Jaeger, J.C. Conduction of Heat in Solids; Oxford University Press: Oxford, UK, 1947. [Google Scholar]
- ASHRAE. Thermal Comfort Conditions; ASHRAE Standard: Peachtree Corners, GA, USA, 2005. [Google Scholar]
- Grabowski, A.; Farley, C.T.; Kram, R. Independent metabolic costs of supporting body weight and accelerating body mass during walking. J. Appl. Physiol.
**2005**, 98, 579–583. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**(

**a**) Hotplate design consisting of top aluminum plate, heating elements and control electronics; (

**b**) Hotplate with silicone rubber insulation layer, ice pack and styrene cover.

**Figure 3.**Temperature profiles for human body with PCM. The thick black line indicates the initial temperature profile.

**Figure 4.**Measured cooling powers of ice packs on hotplate (

**a**) Comparison of experiments (full lines) and predictions (dash-dotted lines); (

**b**) Cooling power at melting plateau versus insulation layer thickness. Symbols: measured values, dashed line: predictions.

**Figure 5.**Predicted temperature profiles compared with literature data: (

**a**) Measured body core temperature T

_{1}(dashed lines, Song [6]) and analytical results (full lines) for the case without PCM cooling (black lines) and with PCM (red lines); (

**b**) PCM temperature T

_{3}as predicted by Equations (12)–(15) (full line) compared with numerical data from Wan [23] (dashed line with triangles).

**Figure 6.**(

**a**) Heat absorbed by the PCM from the body (green lines) and environment (red lines); Full lines: analytical model, dashed lines: numerical [23]; (

**b**) Effect of extra insulation of PCM outer surface on body cooling efficiency. Full lines: analytical solution Equation (23), dashed lines [25].

**Figure 7.**Results of parametric study: (

**a**) Cooling power plateau values, Equation (18); (

**b**) Cooling time, Equation (22). The numbers in the legend indicate the PCM thickness in mm.

**Table 1.**Thermal properties used as input for human PCM cooling model. Data from [23] unless indicated otherwise. Thermal resistances are calculated as the ratio of thickness and thermal conductivity.

Property | Layer 1 | Layer 2 | Layer 3 | Layer 4 | |||
---|---|---|---|---|---|---|---|

Body Core [14] | Skin [14] | Underwear | PCM Pocket | Air Gap | PCM, Na_{2}SO_{4} | Outer Layer | |

Thermal conductivity [Wm^{−1}K^{−1}] | 0.42 | 0.47 | 0.0614 | 0.0317 | 0.025 | 0.6 | 0.0592 |

Specific heat [J/kg] | 3800 | - | 1340 | 1340 | - | 3600 | 1210 |

Density [kg/m^{3}] | 1085 | - | 317 | 392 | - | 500 | 607 |

Thickness [mm] | 123 | 2.0 | 0.356 | 0.16 | 5 | 19.5 | 0.338 |

Melting temperature [°C] | - | - | - | - | - | 21 | - |

Melting enthalpy [kJ/kg] | - | - | - | - | - | 144 | - |

Thermal resistance [m^{2}KW^{−1}] | - | 0.0048 | 0.0058 | 0.0050 | 0.20 | - | 0.0057 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jansen, K.M.B.; Teunissen, L.
Analytical Model for Thermoregulation of the Human Body in Contact with a Phase Change Material (PCM) Cooling Vest. *Thermo* **2022**, *2*, 232-249.
https://doi.org/10.3390/thermo2030017

**AMA Style**

Jansen KMB, Teunissen L.
Analytical Model for Thermoregulation of the Human Body in Contact with a Phase Change Material (PCM) Cooling Vest. *Thermo*. 2022; 2(3):232-249.
https://doi.org/10.3390/thermo2030017

**Chicago/Turabian Style**

Jansen, Kaspar M. B., and Lennart Teunissen.
2022. "Analytical Model for Thermoregulation of the Human Body in Contact with a Phase Change Material (PCM) Cooling Vest" *Thermo* 2, no. 3: 232-249.
https://doi.org/10.3390/thermo2030017