Many researchers developed numerical models on heat and/or mass transfer through fabric/clothing toward firefighters [34-36,84,98,305]. In this context, four numerical heat and/or mass transfer models (Gibson model, Torvi model, Mell and Lawson model, and Song model) are very popular in the scientific community; these models are discussed in detail below.

Gibson model: The Gibson model was developed based on Whitaker’s [475] theory of heat and mass transfer through a porous medium. Gibson considered textile fabric as a hygroscopic multiphase porous media (a solid phase with a concentration of bound water, a free liquid water phase, and a gaseous phase of water vapors in air) to develop the heat and mass transfer model [475,476]. This model treats heat and mass transfer in three dimensions: conduction by all three phases, convection by only gas and liquid phases, and transformations among all three phases. In order to develop the model, Gibson applied continuity, linear momentum conservation, and energy conservation equations to fabric. Gibson’s developed model can be represented by Eq. (6.1). This equation can be rewritten as Eq. (6.2) by numbering each distinct phase, “1” for water, “2” for dry solid polymer, and “3” for inert air. In Eq. (6.2), (p)Cp = (p_{1})(cp)_{1} + (p_{2})(cp)_{2} + (p_{3})(cp)_{3}. Here, the bracketed () terms denote a volume averaged over all three phases [solid (ff) phase solid + liquid); liquid (в phase (liquid only); and gaseous (y) phase] or over the single phase as indicated by the subscript. The term e denotes the volume fraction of a single phase, v denotes air velocity, ST/St = change of temperature with respect to time, Cp/cp = specific heat,

K_{eff} is the effective thermal conductivity, Ah_{vap} is the heat of vaporization of the liquid phase, and Q_{l} is the heat of desorption from the solid phase. Here, m_{lv}, m_{sl}, and m_{sv }denote the mass flux desorbed from the solid to the liquid phase, desorbed from the solid to the gas phase, and evaporated from the liquid phase, respectively.

Gibson developed this model by considering various important parameters (eg, fabric phases, mass flux, heat of desorption) that affectheat and mass transfer through fabric thus affecting the thermal protective performance of fabric. This model was later extended by Gibson and Charmchi to model the heat transfer to a human body in contact with fabric [476]. Although these models have several good features, they are not properly valid in all situations. This is because Gibson [34] did not correlate the heat and mass transfer data predicted from his model with the data obtained from any laboratory-simulated experiment. Additionally, this model is inapplicable when the fabric is exposed to a high heat flux (approximately >10 kW/m^{2}). Hence, the Gibson model can be extended to treat high heat fluxes. In addition, the absorption of radiation and thermo-chemical processes into fabric during high heat flux can thoroughly be modeled. Moreover, the Gibson model can be applied/modified to understand unsteady capillary wicking of sweat-vapor through fabric structure; condensation and evaporation of sweat-vapor within fabric structure; the convective metabolic heat transfer through fabric; and impact of swelling and shrinkage of fibers and yarn (as they absorb/desorb liquid water or water vapor) on heat and mass transfer through fabric.

Torvi model: The Torvi model is directly based on a thermal protective performance experimental apparatus shown in Fig. 6.5 [36,350,477]. The test apparatus consists of a fabric heated from beneath by a Meker burner using premixed reactants (propane and air), a copper calorimeter test sensor mounted in an insulating block, and a completely enclosed air space with a thickness (S) of about 6 mm between the fabric and test sensor. The dimension of the apparatus (looking from above) is 150 mm x 150 mm; the fabric specimen is 100 mm x 100 mm, while the portion of the fabric directly exposed to the flame is 50 mm x 50 mm. This model accounts for heat transfer from the burner through a single-layered fabric to a copper sensor (firefighter’s body). In order to develop the model on heat transfer, Torvi accounted for convection and radiation in the air gap between the burner and the fabric; the conduction, absorbed radiation, and thermochemical reaction within the fabric; and the convection and radiation in the air gap

Fig. 6.5 Heat transfer mechanism through fabrics in the Torvi model.

between the fabric and sensor (Fig. 6.5). Based on Fig. 6.5, it is clear that radiation occurs in all three places. Therefore, Torvi represented the radiant heat flux (q_{rad1}) between (1) the burner and fabric, and (2) the fabric and firefighter’s body through Eq. (6.3) (where, a is the Stefan-Boltzmann constant; e_{g}, e_{f},and e_{b} are emissivities of the hot gases, the fabric, and the burner head, respectively; T_{g}, T_{f}, T_{a}, and T_{b} are the temperatures of the hot gases, the fabric’s outer side, the ambient air, and the burner head, respectively; F_{a} and F_{b} are view factors accounting for the geometry of the fabric with respect to the ambient air and the burner, respectively; and, A_{f} and A_{b} are the surface areas of the fabric and the burner head, respectively). Similarly, he represented the radiant heat flux (q_{rad2}) within the fabric by Eq. (6.4), where T_{s}, e_{s}, and A_{s} are the firefighter’s body temperature, emissivity, and surface area, respectively; and F_{s} accounts for the geometry of the fabric with respect to the firefighter’s body. By considering all these radiant heat fluxes (q_{rad}) along with conduction, thermo-chemical reaction, and absorption of incident radiation within the fabric, Torvi developed the temperature distribution equation in the fabric-air gap-sensor system. This temperature distribution equation can be represented by Eq. (6.5),where, C^{A }is a temperature-dependent “apparent” specific heat that incorporates latent heat associated with thermo-chemical reaction, T is the temperature, ST/St=change of temperature with respect to time t, k(T) is a temperature-dependent thermal conductivity, ST/S x=change of temperature with respect to fabric thickness x, and у is the extinction coefficient of the fabric.

By using this developed model, Torvi [350] conducted a parametric study. First, he selected Nomex-IIIA fabrics with a wide range of thickness, moisture regain, thermal conductivity, decomposition reaction temperature, and transmissivity. Then, he predicted the temperature rise of the selected fabrics using this developed model in an intensified flame exposure. He also selected a wide range of imposed flame temperatures, convective heat transfer coefficients, and microclimate sizes to predict the temperature rise of a typical Nomex-IIIA fabric. Subsequently, the same selected fabrics were tested in the laboratory using the apparatus shown in Fig. 6.5 to evaluate their temperature rise during flame exposures. It was found that the results predicted by Torvi’s model had a good correlation with the laboratory evaluated results. This demonstrates that Torvi’s model can be effectively used to predict the temperature rise of a fabric in a flame exposure without conducting any expensive, cumbersome laboratory experiments. This model can also be effectively used to understand the relationship between a fabric attribute and its temperature rise or thermal protective performance. Recently, this model has been improved by Sawcyn and Torvi [478] as well as Torvi and Threlfall [102]. By using these improved models, heat transfers in the air gap and after exposure to flame (during cooling) were predicted, and these predicted results were validated with the laboratory experimental results. Although Torvi’s model is widely applicable to understanding the impact of various parameters on heat transfer or thermal protective performance, it is clear that this model considers heat transfer in one dimension only (Fig. 6.5). It can further be extended to evaluate heat transfer in multiple dimensions. It may also be useful to model the convective heat transfer in a fabric. For example, it is notable that moisture within the fabric or in the air gaps plays an important role in heat transfer [167,302,349,408]. Therefore, heat conveyed by moisture within the fabric or in the air gaps also needs to be modeled. As Torvi developed the heat transfer model for only single-layered fabrics, this model can further be integrated by considering multilayered fabrics.

Mell and Lawson model: The Mell and Lawson model is an extension of the Torvi model. In this model, Mell and Lawson considered the heat transfer through multilayered fabrics consisting of a shell fabric (outer layer), a moisture barrier (middle layer), and a thermal liner (inner layer) [98]. As demonstrated in Torvi’s model, Mell and Lawson also observed the conduction and absorbed radiation inside exposed fabric in order to develop a model governing the conservation of energy. This model can be represented by Eq. (6.6), where p = fabric density, Cp = specific heat of fabric, STjSt = change of temperature with respect to time, Sq_{c}jSx = change of conductive heat flux through fabric with respect to fabric thickness, Sq_{R}jSx=change of radiative heat flux through fabric with respect to fabric thickness, and g = energy generated inside the tested fabrics. In this regard, Mell and Lawson acknowledged the incident heat fluxes on both sides of each layer of the fabric in order to account for interlayer heat flux due to the radiation reflected between fabric layers.

Like Torvi [36], Mell and Lawson [98] also selected a set of fabrics to predict change of temperature with respect to time by using the model shown in Eq. (6.6). At the same time, they also tested the selected fabrics in the laboratory to evaluate their temperature changes with respect to time. It was identified that the results predicted from the model were close to the experimentally evaluated results. Hence, the model developed can be used to accurately predict the change of temperature with respect to time for fabrics or the thermal protective performance of fabrics. This model can be further extended to treat heat transfer in multiple dimensions. The convective heat transfer in fabric as well as heat conveyed by moisture within the fabric or in air gaps can also be extensively studied in the future. Recently, Lawson, Mell, and Prasad [479] improved their originally developed model shown in Eq. (6.6) by considering wet fabrics and other thermal properties of fabrics such as thermal conductivity and thermo-optical attributes (transmittance, reflectance, and absorptance). These improved models were also validated. These improvements have significantly expanded the capabilities of the original model and provided a more robust tool for economically predicting thermal protective clothing performance.

Song model: Song’s model regarding heat transfer through clothing (made from single-layer fabric) is mainly based on the PyroMan flash fire manikin system (Fig. 6.6) [35,369]. This manikin model assumed that convective heat only affects the surface of the fabric, whereas, radiative heat could penetrate through the fabric [36]. Based on these assumptions, the energy balance model is represented in Eq. (6.7), where p_{fab} is density of the fabric; T = temperature of the fabric; Cp_{fab} is the specific heat of the fabric; t=exposure time; ST/St = change of fabric temperature with respect to time; x=fabric thickness; k_{fab} is the thermal conductivity of the fabric; and ST/Sx = change of fabric temperature with respect to fabric thickness. The у (extinction coefficient of the fabric) and q_{rad} (incident radiant heat flux) in Eq. (6.7) can be further represented by Eqs. (6.8), (6.9), respectively (where, т = transmissivity; L_{fab} = fabric thickness; о = Stefan-Boltzmann constant; e_{g} = emissivity of the hot gases; T_{g} = temperature of the hot gases; T_{fab} = temperature of the outside surface of the fabric; e_{fab} = emissivity of the fabric; T_{amb} = temperature of the ambient air; and F_{fab-amb} = view factor accounting for the geometry of the fabric relative to the ambient air). In the boundary condition (x=0 and t > 0 for the outside surface of the fabric), the thermal conductivity of the fabric (k_{fab}) will be as per Eq. (6.10), where, q_{rad} = incident radiant heat flux and q_{conv} = convective heat flux between the fabric and hot gases. Here, q_{conv} can be represented by Eq. (6.11), where, h = convective heat transfer coefficient and the subscript fl, g, and fab refer to the burner flame, the hot gases from the burner, and outside surface of the fabric, respectively. Furthermore, the thermal conductivity of the fabric at the inside (x=L_{fab} and

t> 0) can be represented according to Eq. (6.12). In Eq. (6.12), q_{air},_{rad} is the energy transfer by radiation across the air gap between the fabric and firefighter’s body (Fig. 6.5). The q_{air},_{rad} can be represented by Eq. (6.13) (where, a = Stefan-Boltzmann constant; T_{fab} = temperature of inside surface of the fabric; T_{skin} = temperature of the human skin; A_{skin} = surface area of the human skin; A_{fab} = surface area of the fabric; e_{fab} = fabric emissivity; F_{fab-skin} = the view factor accounting for the geometry of the fabric relative to the human skin; e_{skin} = skin emissivity). Additionally, q_{air},_{cond}/_{conv} is the thermal energy transfer by conduction/convection from the fabric to the human skin across the microclimate air gap (Fig. 6.6). The q_{air},_{cond}/_{conv} can be represented by Eq. (6.14). In Eq. (6.14), h_{air gap} is the heat transfer coefficient of air due to conduction and natural convection in the air gap; the h_{air gap} is shown in Eq. (6.15) (where, Nu is the Nusselt number; k_{air}(T) is the thermal conductivity of the air; and L_{airgap} is the thickness of the air gap).

Fig. 6.6 Heat transfer mechanism through fabrics of firefighters’ protective clothing in Song’s model.

Furthermore, Song developed another set of models by using single-layered fabric- based clothing with added cotton underwear for realistic purposes. In this model, an extra cotton fabric layer (underwear) was added in the clothing with an assumption that no air gap exists between a firefighter’s body and cotton underwear. It was also assumed that the thermal properties of an underwear cotton fabric remain constant during a very short time exposure. The heat transfer through underwear fabric can be represented by Eq. (6.16), where, p_{underwear} = density of underwear fabric; Cp_{underwear} = specific heat of underwear fabric; and k_{underwear} = thermal conductivity of underwear fabric. In the boundary condition (x=L_{fab}+L_{gap} and t > 0 for the outside surface of the underwear cotton fabric), the thermal conductivity of the underwear fabric can be represented by Eq. (6.17). In Eq. (6.17), q_{airrad} is the thermal energy transfer by radiation from protective clothing (fabric) to the underwear fabric across the microclimate air gap. The q_{air},_{rad} can be represented by Eq. (6.18), where, a = Stefan-Boltzmann constant; T_{fab} = temperature of the inside surface of the protective fabric; T_{underwear} = temperature of the inside surface of the underwear fabric; A_{underwear} = surface area of the underwear fabric; A_{fab} = surface area of the protective fabric; e_{fab} = emissivity of the protective fabric; F_{fab-underwear} = the view factor accounting for the geometry of the protective fabric with respect to the underwear fabric; and ?_{underwear} = emissivity of the underwear fabric. Additionally, q_{a}i_{r},_{cond}/_{conv }is the thermal energy transfer by conduction/convection from protective clothing (fabric) to the underwear fabric across the microclimate air gap. The q_{air},_{cond}/_{conv} can be represented by Eq. (6.19). In Eq. (6.19), h_{air gap} is the heat transfer coefficient of air due to conduction and natural convection in the air gap; the h_{airgap} is shown in Eq. (6.20) [where, Nu is the Nusselt number; k_{air}(T) is the thermal conductivity of the air; and L_{airgap} is the thickness of the air gap]. Moreover, at the inside surface of the fabric (x=L_{fab}+L_{gap} +L_{underwear} and t > 0), the thermal conductivity of the fabric is shown in Eq. (6.21). As it is assumed that the underwear fabric is in direct contact with human skin without any air gap, the conductive heat transfer occurs only at the interface of the fabric and skin.

By using the above-developed models, Song quantified the heat transfer through fabrics. These heat transfer values were used to evaluate the time required to generate burn injuries. For validation of the models, Song’s study also compared the results predicted from the model with experimentally evaluated results, and he found a positive association between predicted and experimented results. Hence, these models are valid and accurate to use for burn injury prediction. Furthermore, he also conducted the parametric study mentioned in Torvi’s model. Here, he used clothing made by Kevlar/PBI and Nomex fabric with different thermo-physical aspects (thickness, thermal conductivity, volumetric heat capacity, emissivity, and transmissivity) to evaluate their burn prediction (%) under a 4 s flame exposure with a heat flux of 2 cal/cm^{2} in the laboratory. Additionally, clothing with different sizes, designs, and microclimate sizes were also used to evaluate their thermal protective performances in terms of second- and third-degree burn percentages. At the same time, he also predicted the burn percentage of the selected fabrics from the models developed. He acknowledged a strong correlation among the results obtained from the laboratory experiments and the model predictions. Thus, he stated that the models can be used effectively to understand the relationship between second-degree burns and fabric/garment attributes. Although Song’s model extensively studied heat transfer through clothing, this model only evaluated the heat transfer through single-layer fabric-based clothing. The heat conveyed by moisture within the fabric or in the air gaps was not considered while developing the model. To overcome these shortcomings, Chitrphiromsri and Kuznetsov [38], and Song, Chitrphiromsri, and Ding [305] analyzed heat and moisture transport through multilayered protective fabrics under flash-fire conditions. In their study, the impact of heat and moisture transport on the performance of firefighters’ protective clothing was analyzed. Here, a partial differential equation (energy balance equation) was developed on temperature distribution in a multilayered fabric by combining the Gibson’s and Torvi’s models; in order to develop the differential equation, thermal properties of all fabric phases (solid, liquid, gaseous) were tested based on the relation given by Gibson. Additionally, the radiative heat transfer through fabric was tested based on the model given by Torvi. This temperature distribution equation was also validated and successfully applied by Song, Chitrphiromsri, and Ding [305]. However, Song, Chitrphiromsri, and Ding’s [305] models do not consider the convective heat transfer in the gas phase due to a pressure difference that can arise either due to body or air movement. They also assumed that no free water exists on the surface of the human skin or inside the fabric layer. This assumption may not always be true. In future, these shortcomings can be taken into account in order to develop a better model for heat transfer through clothing. It can also be further extended to treat heat transfer in multiple dimensions.