The objective of this paper is to report a study on the prediction of the steady-state thermal resistance of woven compression bandage (WCB) by using three different mathematical models. The experimental samples of WCB were 100% cotton, cotton–polyamide–polyurethane, and viscose–polyurethane. The bandage samples were evaluated at extensions ranging at 10–100%, with two- and three-layer bandaging techniques. Experimental thermal resistance was measured by thermal foot manikin (TFM) and ALAMBETA testing devices. The obtained results by TFM and ALAMBETA were validated and compared with the theoretical models (Maxwell–Eucken2, Schuhmeister, and Militky), and a reasonable correlation of approximately 78%, 92%, and 93% for ALAMBETA and 75%, 82%, and 83% for TFM, respectively, was observed.

#### Keywords

- Mathematical models for thermal resistance
- thermal foot manikin
- ALAMBETA
- woven bandage properties

Thermal manikins are devices by means of which it is possible to simulate heat exchange between humans and the environment [1, 2]. Human thermal comfort is defined as a condition of mind, which expresses satisfaction with the surrounding environment. High temperatures and humidity provide discomforting sensations and sometimes heat stress, which leads to reduce the body's ability to cool itself. Moreover, this discomfort reduces the productivity of workers and may lead to more serious health problems, especially for aged people [3].

Clothing comfort is an important factor in the stage at which people make their clothing selections [3]. The thermal resistance (_{ct}_{ct}

The measurement of clothing insulation with a thermal manikin is a dynamically balanced adjustment process, which initially depends on the central temperature inside the manikin. It means that continuous adjustment of heat flux makes the manikin skin temperature approach a constant temperature gradually under the heat diffusion. These adjustments limit the change in the manikin skin temperature of all parts to a range of ±0.5°C around the required temperature. The final state is that the manikin skin temperature is steady and very close to the constant temperature [9].

The main factors affecting testing of _{ct}_{ct}

Convection is a process in which heat is transferred by a moving fluid (liquid or gas). For example, air in contact with the body is heated by conduction, and then carried away from the body by convection [13].

A common method for removing water from textiles is convective drying. When the fluid starts at a constant temperature and the surface temperature is suddenly increased above that of the fluid, there will be convective heat transfer from the surface to the fluid as a result of the temperature difference (Δ

When people wear multilayer clothing ensembles in cold weather conditions or in hot environments, air spaces are present between the skin and the inner layer or between two adjacent layers. The _{ct}_{ct}_{ct}

The thermal resistance of fabrics can be calculated by means of experimental, analytical, and numerical methods [16, 17]. There are many models to be found within the textile engineering and heat transfer fields for the thermal resistance prediction. The preference of selection depends on the requisite precision and nature of the solution. Conductive heat transfer is the simplest way to illustrate mathematically and is often the key way of heat transfer [18].

Numerical solutions deal with materials of irregular shapes and properties, different types of heat transfer, and boundary conditions. Numerical methods also have the capability to achieve the maximum precision [19]. There are many commercially available softwares that allow users to solve their problems through numerical solutions. However, these methods are inherently more difficult and complicated, and in some conditions, simple methods prove to be more accurate at much less effort [20]. Thermal resistance is also predicted by using artificial neural networks and statistical models. Some researchers have predicted the thermal resistance of fabrics using mathematical approaches.

Schuhmeister suggested a relationship for the thermal conductivity prediction of fabrics by assuming one-third of fibers to be parallel and two-third in series with a homogeneous distribution in all directions [21]. Later, many researchers used Schuhmeister's model by assuming different ratios of series and parallel components [22, 23, 24]. Presently, Mansoor et al. [25, 26] have modified Schuhmeister and Militky models by combining the water and fiber filling coefficients for the prediction of thermal resistance of wet socks.

Das et al. [27] calculated the heat transfer through the fabric assemblies with the electric resistance and Fricke's law analogy by assuming them as cuboids packed with randomly oriented infinite fibers. Wie et al. [28] suggested a model for fabric thermal resistance prediction by assuming that heat passes through the fabric as a combination of fiber and air in series plus the air in parallel.

Most studies on _{ct}_{ct}_{ct}

The Maxwell–Eucken (ME) model (Eq. (2)) can be used to describe the effective thermal conductivity of a two-component material with simple physical structures. In Eq. (2), λ_{a}, λ_{polymer}, F_{a}, F_{polymer} are the thermal conductivities and volume fractions, respectively, and the subscripts represent the two components of the system. The effective thermal conductivity of the two-component material is λ_{fab} [34]. An emulsion is a dispersion of one liquid in another immiscible liquid. The phase that is present in the form of droplets is the dispersed phase and the phase in which the droplets are suspended is called the continuous phase. A number of effective thermal conductivity models require the naming of a continuous and a dispersed phase. In materials with exterior porosity, individual solid particles are surrounded by a gaseous matrix, and hence the gaseous component forms the continuous phase while the solid component forms the dispersed phase. For external porosity, and λ_{a} and λ_{polymer} are considered as the continuous and dispersed phases, respectively [35, 36].

_{polymer}_{polymer} are calculated based on Eqs (7) and (8).

Schuhmeister summarized the relationship between the thermal conductivity and structural parameters of a fabric using Eq. (3):
_{fab}_{polymer}_{a}_{polymer}_{a}

Militky summarized the relationship between the thermal conductivity and structural parameters of a fabric using empirical Eq. (6) and used the same steps for calculating _{s}_{p}

It is assumed that fabric density changes with wetting, which causes a change in the filling coefficient, porosity, and thermal conductivity of the fabrics. Based on these assumptions the following three equations were developed that will be used to find the fabric density, filling coefficient, and thermal conductivity for different moisture levels. The average thermal conductivity for different fibers (within socks) at different moisture levels are calculated based on Eq. (7) as follows:

_{fib}_{1} = First fiber filling coefficient

_{fib}_{2} = Second fiber filling coefficient

_{fib}_{3} = Third fiber filling coefficient

_{fib}_{1} = First fiber thermal conductivity

_{fib}_{2} = Second fiber thermal conductivity

_{fib}_{3} = Third fiber thermal conductivity

Filling coefficients for fiber and air are calculated as listed in Table 1, according to the following steps.

Filling coefficients calculation

_{fib} | |
---|---|

Content | % |

Weight | G |

Area | m^{2} |

Areal density | g/m |

Volumetric density | |

Filling coefficient |

Air filling coefficient (_{a}

The outputs of Eqs (7) and (8) are used as input for all the above models. Thermal conductivity of water and air is taken as 0.6 and 0.026 W/m/K while density of water is 1,000 Kg/m^{3}. The values of the different input parameters used in this study are listed in Table 2 [UllmannF.

Different fiber properties

^{3}) | ||
---|---|---|

Cotton | 1,540 | 0.5 |

Viscose | 1,530 | 0.5 |

Polyester | 1,360 | 0.4 |

Nylon 66 | 1,140 | 0.3 |

Polypropylene | 900 | 0.2 |

Wool | 1,310 | 0.5 |

Acrylic | 1,150 | 0.3 |

Experimental samples are 100% bleached cotton (BL-CO), cotton–polyamide–polyurethane (CO-PA-PU), and viscose–polyurethane (VI–PU) bandages. All _{ct}^{2}) and total thickness are performed on the extended state as illustrated in Table 3.

Specifications of woven bandage samples on thermal foot manikin

^{2}) | ||||
---|---|---|---|---|

Bleached cotton 100% | 100% cotton, two layers, 50% extension, 50% overlap | Two layers = 212.3 | One layer = 0.76 | |

Cotton 78%, polyamide 16%, polyurethane 6% | CO-PA-PU, two layers, 50% extension, 50% overlap | Two layers = 238.1 | One layer = 0.84 | |

Viscose 94%, polyurethane 6% | VI-PU, two layers, 50% extension, 50% overlap | Two layers = 220.7 | One layer = 0.79 | |

Bleached cotton 100% | 100% cotton, three layers, 50% extension, 66% overlap | Three layers = 317.5 | One layer = 0.76 | |

Cotton 78%, polyamide 16%, polyurethane 6% | CO-PA-PU, three layers, 50% extension, 66% overlap | 356.3 | One layer = 0.84 | |

Viscose 94%, polyurethane 6% | VI-PU, three layers, 50% extension, 66% overlap | 329.4 | One layer = 0.79 |

Thermal manikins are designed to simulate the human body's heat exchange and its interaction with the surrounding environment. There are two types of testing on thermal manikin, practically called “nude and clothed Manikin” [2]. Experimental samples consist of three types of WCBs, which as porous materials should provide thermal comfort to enable air permeability, heat transfer, and liquid perspiration out of the human body. Mercerized cotton socks were used to cover TFM as underwear to measure _{ct}_{0} for all measured samples to ensure more stabilization and steady conditions before measuring _{ct}_{ct}

However, the same tests were confirmed on ALAMBETA at the same extension and number of layers using special tensioning frame, as shown in Figure 3, but using the free air convection system [41].

_{ct}on TFM and ALAMBETA respectively

The stabilization process of the TFM could be achieved after 20 min – at least – then the final testing step takes place [10]. Finally, _{ct}_{ct}_{0} as a reference value, as illustrated in Figure 4, and it can be calculated using Eq. (9) as follows:
_{ct}^{2}.°C/W), or (m^{2}.°C.W^{−1})_{s}_{a}^{2}), and _{ct}_{0} is the clothed TFM dry resistance (m^{2}/°C/W).

For comparison, all bandage types were wrapped on TFM and ALAMBETA at the same extension level ranging from 10% to 100% using both two and three layers of bandages [10]. Figure 5 illustrates that _{ct}

The same analysis is proved for the three layers of bandages as demonstrated in Figures 6 and 8. Moreover, the effect of higher applied tension by three-layer bandaging appeared as the third main factor that decreases the volume of the air layers between the adjacent bandage layers.

_{ct}results on TFM and ALAMBETA with three theoretical models

The experimental results of _{ct}_{ct}_{max}_{dyn}_{steady}

The transient heat flow is shown in Eq. (10), whereas the steady-state heat flow is shown in Eq. (11) as follows:
^{0.5}/m^{2}/K^{1}), Δ _{1} − _{2} is the temperature difference between the two convection surfaces, and _{ct}_{ct}

The experimental evaluation of _{ct}_{ct}

The obtained results of _{ct}_{ct}_{0} and the corresponding values of _{ct}_{ct}

#### Different fiber properties

^{3}) | ||
---|---|---|

Cotton | 1,540 | 0.5 |

Viscose | 1,530 | 0.5 |

Polyester | 1,360 | 0.4 |

Nylon 66 | 1,140 | 0.3 |

Polypropylene | 900 | 0.2 |

Wool | 1,310 | 0.5 |

Acrylic | 1,150 | 0.3 |

#### Filling coefficients calculation

_{fib} | |
---|---|

Content | % |

Weight | G |

Area | m^{2} |

Areal density | g/m |

Volumetric density | |

Filling coefficient |

#### Specifications of woven bandage samples on thermal foot manikin

^{2}) | ||||
---|---|---|---|---|

Bleached cotton 100% | 100% cotton, two layers, 50% extension, 50% overlap | Two layers = 212.3 | One layer = 0.76 | |

Cotton 78%, polyamide 16%, polyurethane 6% | CO-PA-PU, two layers, 50% extension, 50% overlap | Two layers = 238.1 | One layer = 0.84 | |

Viscose 94%, polyurethane 6% | VI-PU, two layers, 50% extension, 50% overlap | Two layers = 220.7 | One layer = 0.79 | |

Bleached cotton 100% | 100% cotton, three layers, 50% extension, 66% overlap | Three layers = 317.5 | One layer = 0.76 | |

Cotton 78%, polyamide 16%, polyurethane 6% | CO-PA-PU, three layers, 50% extension, 66% overlap | 356.3 | One layer = 0.84 | |

Viscose 94%, polyurethane 6% | VI-PU, three layers, 50% extension, 66% overlap | 329.4 | One layer = 0.79 |

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