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# Numerical models

Woodcock [492,493] numerically analyzed the metabolic heat and sweat-vapor transfer through regular clothing into the ambient environment. His research identified that metabolic heat and sweat-vapor can easily transfer from human bodies to clothing. He assumed that the metabolic heat and sweat-vapor transfer to the ambient environment is equivalent to heat (Hd) and evaporative-heat (He) transfer, respectively. He further assumed that the heat (Hd) and evaporative-heat (He) transfer processes are mutually independent. Hence, total metabolic heat transfer (Ht) from the clothed human body to the ambient environment is a combination of Hd and He (Eq. 6.22). Here, the Hd can transfer by convection or radiation based on Newton’s Law of Cooling (Eq. 6.23), or Stefan-Boltzmann Law (Eq. 6.24), respectively. Additionally, the Hd can transfer by conduction, especially when a clothed human body comes in contact with a relatively cold external surface (Eq. 6.25); furthermore, the He can be calculated by Eq. (6.26).

where Hconv = metabolic heat transfer by convection; h = convective metabolic heat transfer coefficient of the ambient air; A = surface area of the clothed human body; Tc = temperature of the clothed human body; and Ts = temperature of the ambient environment.

where Hrad = metabolic heat transfer by radiation; e = thermal emissivity of the clothed human body; а = Stefan-Boltzmann constant; A = surface area of the clothed human body; Tc = temperature of the clothed human body; and Ts = temperature of the ambient environment.

where Hcond = metabolic heat transfer by conduction; k=thermal conductivity of contacted external surface; A = surface area of the clothed human body; dT = temperature difference between the clothed human body and contacted external surface; and dl = medium length through which thermal energy passes.

where He = metabolic heat transfer by evaporation; X = heat of vaporization; m = permeance coefficient of the skin; ADu = DuBois area; Pc = partial vapor pressure at the clothed human body; and Ps = partial vapor pressure at the ambient environment.

Since Woodcock’s [492,493] research, many researchers have numerically modeled the metabolic heat and sweat-vapor transfer through fabric or clothing [306-311,432,490]. For example, Ogniewicz and Tien [309] developed a numerical convective and diffusive metabolic heat transfer model along with phase change due to condensation and evaporation; Farnworth [307] developed a time-dependent dynamic numerical model that included metabolic heat transfer by conduction and radiation as well as sweat-vapor transfer by diffusion; Li and Holcombe [308] further introduced a dynamic metabolic heat and sweat-vapor transfer numerical model of clothing in interaction with the two-node human thermoregulation models of Gagge, Stolwijk, and Nishi [118]; Fan and Cheng [311] developed a detailed dynamic numerical model on metabolic heat and sweat-vapor transfer, and they also compared the model-predicted results with experimental results to validate the model; Wu and Fan [432] numerically developed a metabolic heat and sweat-vapor transfer model of multilayered clothing; and Voelkar et al. [310] developed a numerical model on the metabolic heat balance of clothing by considering various parameters (Eq. 6.27).

where Cci = heat capacity of the clothing; cl/dt = change in the heat capacity of clothing with respect to time; Qc,skin-cl and Qe,skin_cl = metabolic heat transfer from human body to clothing through convection and evaporation, respectively; Qc cl-env, Qe cl-env, and Qr cl-env = metabolic heat transfer from clothing to the ambient environment by convection, evaporation, and radiation, respectively; and Qs cl-env = possible heat gain due to solar radiation.

Although the above models can numerically analyze the metabolic heat and sweat- vapor transfer behavior, these models are only applicable to regular clothing. In this context, it is notable that the configuration of regular clothing and protective clothing is quite different. Protective clothing comprises different fabric materials (eg, moisture barrier, thermal liner) and structures (eg, layered structure) than regular clothing. Hence, the above models can only be partially applicable to protective clothing. This reflects that the modeling on metabolic heat and sweat-vapor transfer through protective clothing needs more careful attention. With this in mind, Gonzalez et al. [494] attempted to develop a numerical model on metabolic heat and sweat-vapor transfer through chemical protective clothing. Similarly, Holmer [454] constructed a model that considered sweat-vapor transfer from the human body via chemical protective clothing. However, these models were developed considering the natural ambient environment. As firefighters’ working environments are comprised of different thermal exposures with varying intensities, these situations may severely affect the metabolic heat and sweat-vapor transfer behavior of their clothing. To date, no research has been conducted to develop the numerical metabolic heat and sweat-vapor transfer models through firefighters’ protective clothing by considering their working thermal environments. Less research attention is given because it is very difficult to accurately simulate firefighters’ working environments in the laboratory to develop a model on metabolic heat and sweat-vapor transfer [495].

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