An improved model to analyze radiative heat transfer in flame-resistant fabrics exposed to low-level radiation

Abstract

An improved heat transfer model, based on the two-flux model, in a multilayer flame-resistant fabric system with an air gap was proposed. The developed model considered the thermal radiation by absorbing, transmitting, emitting and reflecting in porous fabrics. The predicted results of the new model were compared with the previous Beer’s law model and the experimental results, and were found to be in good agreement with the experimental ones. The aim of this study is to investigate the mechanism of radiant heat transfer in the multilayer fabric system and the effects of the optical properties of flame-resistant fabric on heat transfer in the fabric system. The numerical results demonstrated that the self-emission in multilayer fabric system increases not only the rate of thermal energy transferred to human skin during thermal exposure, but also the rate of thermal energy transmitting to the ambience during cooling. The fabric’s optical properties have a complex influence on the transmitted and stored energy in multilayer protective clothing. The finding obtained in this study can provide references for the improvement of the thermal protective performance of flame-resistant fabrics.

Keywords

radiative heat transfer flame-resistant fabric optical properties thermal protective performance two-flux model

Firefighters, when rescuing in fire situations, usually encounter multiple thermal hazards, such as flash fires, high-intensity thermal radiation and hot steam.^1–3 These environmental threats can cause first-degree to third-degree skin burns.⁴ Firefighters’ uniforms, which have higher thermal protective performance, contribute to reducing skin burn injuries.⁵ This is why firefighters are required to wear protective clothing that is characterized by flame retardance and thermal insulation. Flame-resistant fabrics are usually considered as a porous medium in the process of heat and mass transfer.⁶ The thermal radiation from the fire environment can be absorbed and scattered by the fabric system, and then transmitted to the skin surface.⁷ On the other hand, a fabric system with higher temperature can emit a lot of thermal radiation, for example, the thermal radiation emitted from the thermal liner can be absorbed by skin tissues.⁷ The optical properties of the fabric have an important influence on the thermal protective performance of firefighters’ protective clothing, especially in intense thermal environments where radiation is the key mode of heat transfer in a fabric system with air gaps.

In earlier studies, the majority of researchers focused only on conductive heat transfer in porous materials. The importance of radiant heat transfer was put forward by Vershoor and Greebler⁸ and Rowley et al.⁹ in the 1950s. Radiative heat transfer in a porous medium is quite complex, involving absorbing, emitting, scattering and transmitting.¹⁰ In previous radiative transfer models, researchers firstly used the simple conduction model to approximately simulate the radiant heat transfer.^11–14 The main reasons might be that these models were easily solved and understood. However, they were only suitable to the situation in which conductive heat transfer was treated as the main heat transfer process. In addition, they were not widely used because the thermal conductivity due to radiation was measured experimentally for different materials and different environmental conditions. Based on the above deficiencies, the two-flux model proposed by Schuster¹⁵ was used to simulate radiant heat transfer in fibrous materials that considered the absorptive and back-scattering property.¹⁶ At the end of the 20th century, Tong et al.¹⁷ proposed a spectral two-flux model in a porous medium accounting for thermal radiation with absorbing, emitting and scattering. Since then, the two-flux model has been widely used in modeling radiative heat transfer in fibrous materials by some researchers, such as Farnworth,¹⁸ Spinnler et al.,¹⁹ Bhattacharjee and Kothari²⁰ and Wan and Fan.²¹

In recent decades, heat and moisture transfer models in thermal protective clothing have witnessed rapid development, but most researchers have tended to use Beer’s law to model thermal radiation in flame-resistant fabrics. It is defined from Beer’s law that thermal radiation through the fabric system can be exponentially decayed due to the fabric’s absorption²² and calculated using the transmissivity of the fabric.⁷ Torvi⁷ experimentally calculated the extinction coefficient of NomexIIIA and Kevlar/PBI fabric popularly used in the protective clothing field. The study demonstrated that the radiative heat transfer in flame-resistant fabrics conformed to Beer’s law. On the basis of Beer’s law, Song et al.²³ employed a clothing numerical model to explain heat transfer in the “environment–clothing–human body”, and found that a fabric’s emissivity significantly affected heat transfer in air gaps between the fabric surface and skin. A cylindrical heat transfer model using similar methods was developed to evaluate the effect of body geometry on heat transmission.²⁴ In addition, some researchers used the radiative transfer equation (RTE) to simulate thermal radiation in a multilayer fabric system. For instance, Mell and Lawson²⁵ developed a heat transfer model in low-intensity thermal radiation by embedding the surface reflectivity of a multilayer fabric system. Fu et al.²⁶ used the simulation method to investigate the influence of moisture on radiant heat transfer under low-level radiation exposure. Moreover, the RTE in recent years was employed by Ghazy and Bergstrom^27–29 to study the effect of self-emission and absorption of air gaps on heat transfer, including the multiple air gaps model of the multilayer fabric system²⁸ and dynamical air gap models, owing to the body motion and thermal shrinkage of fabric.^27,29

It is clear that the current radiative heat transfer models in flame-resistant fabrics involve the effects of absorbing, transmitting and surface reflecting, but the self-emission of flame-resistant fabrics has still not been investigated in heat transfer models. However, most flame-resistant fabrics widely used for firefighters’ protective clothing, such as NomexIIIA and Kevlar/PBI fabric,⁷ are characterized by high emissivity such that these fabrics can emit thermal energy under the high-temperature condition. Meanwhile, few studies have been conducted to evaluate the influence of the optical properties of flame-resistant fabrics on the thermal protective performance of protective clothing, regardless of the experimental study and numerical simulation.

Therefore, the aim of the paper was to improve the current radiant heat transfer model in the multilayer fabric system based on the two-flux model. The improved model considered the self-emission, absorption, transmission and reflection in order to more precisely simulate the radiative heat transfer in the fabric system. Combining with the Pennes bio-heat transfer model,³⁰ the impacts of optical properties of flame-resistant fabrics on the thermal protective performance and skin temperature were studied during and after the low-level thermal radiation.

Mathematical model

Typical firefighting protective clothing consists of an outer shell, a moisture barrier and a thermal liner. Figure 1 shows the firefighter protective clothing together with human skin and the air gap between the thermal barrier and the skin. Thermal energy is transferred by radiation from the heating source to the fabric, while a portion of this energy is transferred by both radiation and convection from the fabric to the ambient. In order to simplify the heat transfer in a multilayer fabric system, the related assumptions are as follows.

Considering the fabric’s thickness, which is much less than the width of the fabric, heat transfer is one-dimensional along the thickness of the fabric layers, ignoring the moisture transfer.^24,25

Conductive and radiative heat transfer is coupled to totally determine the temperature distribution in the multilayer fabric system. It is supposed that convective heat transfer only happens in the external surface of the outer shell.³¹

The incident radiation only penetrates through the outer layer of the fabric system, as almost 95% of the incident radiation is absorbed after a distance of three fiber diameters.⁷ Radiative heat transfer in the fabric system considers the effects of self-emission, absorption, transmission and reflection, without refraction and diffraction.

The thermal properties of the fabric are taken to be a function of temperature, but the thermal shrinkage and thermal reaction of the fabric layers are not considered because the temperature is too low to melt and degrade the flame-resistant fabrics under low-intensity thermal radiation.

Concerning the thin air gap (6.4 mm), the heat transfer within the air gap is considered as heat conduction and radiation, without convection heat transfer.³²

Figure 1.

Schematic diagram of heat transport in the “Heating source-Multilayer fabric-Air gap-Skin tissues” system.

Heat transfer model in the multilayer fabric system

According to the conservation of heat energy,³³ the heat transfer equations of fabric layers during radiant heat exposure and the cooling phase can be obtained, respectively, by

{(ρ C p) fab \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} (k fab (T) \frac{\partial T}{\partial x}) - \frac{\partial q rad - trans 1}{\partial x} - (\frac{\partial q_{rad}^{+}}{\partial x} - \frac{\partial q_{rad}^{-}}{\partial x}) 0 < t < t \exp (1 a) (ρ C p) fab \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} (k fab (T) \frac{\partial T}{\partial x}) - (\frac{\partial q_{rad}^{+}}{\partial x} - \frac{\partial q_{rad}^{-}}{\partial x}) t \exp < t (1 b)

where ρ_fab, (C_p) _fab and k_fab are the fabric layer density, specific heat and thermal conductivity, respectively, q_rad-trans₁ is the transmitting portion of incident radiant heat flux in the outer shell of the fabric system,

q_{rad}^{+}

and

q_{rad}^{-}

are the forward radiative heat flux and backward radiative heat flux in fabric system due to self-emission, respectively, and t_exp is the exposure duration. The fabric thermal conductivity (k_fab) is determined from the fiber-to-air fraction in the fabric³³ as follows

k fab (T) = V air % k air (T) + (1 - V air %) k fiber (T)

(2)

where V_air% is the air content percentage contained in the fabric’s pores, which can be calculated using the density of fibers and air,³³ k_fiber and k_air are the fiber’s thermal conductivity and the thermal conductivity of the air, respectively, which are individually defined³³ below 700 K as follows

{k fiber (T) = 0.13 + 0.0018 (T - 300) (3 a) k air (T) = 0.026 + 0.000068 (T - 300) (3 b)

During the exposure and cooling phases, the thermal boundary conditions of differential equations (1a) and (1b) are written as, respectively, as³⁴

- k 1 \frac{\partial T}{\partial x} | x = 0 = - h conv, amb 1 (T | x = 0 - T amb) - σ ɛ 1 F shell - amb (1 - ɛ g) (T^{4} | x = 0 - T_{amb}^{4}) 0 < t < t \exp (4 a)

- k 1 \frac{\partial T}{\partial x} | x = 0 = - h conv, amb 2 (T | x = 0 - T amb) - σ ɛ 1 (T^{4} | x = 0 - T_{amb}^{4}) t > t \exp (4 b)

- k 3 \frac{\partial T}{\partial x} | x = L fab = q therm - skin - k air \frac{\partial T}{\partial x} | x = L fab 0 < t

(5)

where k₁ and k₃ are the outer shell and the thermal barrier thermal conductivity, respectively, h_conv,amb₁ and h_conv,amb₂ are the convective heat transfer coefficient between the outer shell and the ambient during the exposure and during the cool down, respectively, σ is the Stefan–Boltzmann constant, ɛ₁ and ɛ_g are the outer shell emissivity (0.9) and the emissivity of hot gases (0.02),⁷ respectively, T|_x_= 0 and T_amb are the surface temperature of the outer shell and the ambient temperature (300 K), respectively, F_shell-amb is the view factor between the outer shell and the ambient, L_fab is the multilayer fabric thickness and q_therm-skin is the radiation heat flux at the backside of the thermal liner that can be estimated by solving³⁵

q therm - skin = \frac{σ (T^{4} | x = L fab - T^{4} | x = L fab + L air)}{(\frac{1 - ɛ 3}{ɛ 3} + \frac{1}{F therm - skin}) + \frac{1 - ɛ skin}{ɛ skin}}

(6)

where ɛ₃ and ɛ _skin are the thermal liner emissivity (0.9)²⁸ and the skin surface emissivity (0.95),²³ respectively, L_air is the air gap thickness, and F_therm-skin is the view factor between the thermal liner and the skin surface, which can be obtained by reference to Su et al.³⁴

Heat transfer model in the air gap

The importance of the air gap between the fabric and the skin surface on the thermal protective performance of the clothing system has been proved by a large number of experimental^36,37 and numerical studies^27–29 under flames or radiant heat exposure. It is assumed that the air gap can absorb thermal radiation, but ignores the impacts of emitting and scattering of thermal radiation.³⁸ The conduction and radiation heat transfer within the air gap are coupled in the air gap. The governing equation is written as

(ρ c p) air \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} (k air (T) \frac{\partial T}{\partial x}) + \frac{\partial q rad - absorb}{\partial x}

(7)

where (c_p) _air is the air thermal conductivity and q_rad-absorb is the absorbed portion of the radiant heat flux from the thermal barrier to the sensor. The absorption of the incident thermal radiation (q_therm-skin) is given based on Beer’s law¹⁰

q rad - absorb = q therm - skin (1 - exp (- κ air x))

(8)

where κ_air is the absorption coefficient of the air gap (5 m⁻¹).²⁸

Skin bio-heat transfer model and burn prediction

A transient one-dimensional skin model is used to simulate heat transfer in the human skin, assuming heat conduction only within the skin and deeper layers. Different thermal properties for the epidermis, dermis and subcutaneous layers are considered in the skin model in order to account for the effect of blood flow in the dermis and subcutaneous layers on the heat transfer, which are described in ASTM F2731-11.³⁹ The energy equation for the human tissues is³⁹

(ρ c p) skin \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} (k skin \frac{\partial T}{\partial x})

(9)

where ρ_skin, (c_p) _skin and k_skin are the human tissue density, specific heat and thermal conductivity, respectively. The thermal boundary conditions can be given by³⁴

- k epi \frac{\partial T}{\partial x} | x = L fab + L air = q rad - trans 2 | x = L fab + L air - k air \frac{\partial T}{\partial x} | x = L fab + L air

(10)

T skin | x = L fab + L air + L skin = 306.65 K

(11)

where L_skin is the skin thickness and q_rad-trans₂ is the transmitted portion of the radiant heat flux in the air gap, shown in equation (12)³⁴

q rad - trans 2 = q therm - skin exp (- κ air x)

(12)

The temperature histories versus time in different skin layers are calculated by the above heat transfer model. Skin burn injury can take place when the basal layer or the dermal layer temperature reaches 44℃.^40,41 Henriques’ burn integral model⁴¹ is employed to predict times to receive skin burn, given by

Ω = \int_{0}^{t} P \exp (- \frac{Δ E}{RT}) d t

(13)

where Ω is a quantitative measure of burn damage at the basal layer or at any depth in the dermis, P is the frequency factor, ΔE is the activation energy, R is the universal gas constant, T is the absolute temperature at the basal layer or at any depth in the dermis and t is the total time for which T is above 317.15 K. These constants for calculation of Ω using equation (13) can be obtained in ASTM F2731-11.³⁹ When Ω reaches a value of 1 at the epidermis–dermis interface and the dermis–subcutaneous tissue interface, the corresponding times are treated as second- and third-degree burn times, respectively.

Radiant transfer model in the multilayer fabric system

In equation (1a), q_rad-trans₁ is the transmitting portion of incident radiant heat flux in the outer shell that is determined by subtracting radiative heat flux from the outer shell to the ambience from the radiative heat flux from the heating source to the outer shell. q_rad-trans₁ is written as follows according to Beer’s law¹⁰

q rad - trans 1 = q rad - inc exp (- κ 1 x)

(14)

where κ₁ is the absorption coefficient of the outer shell and q_rad-inc is the incident radiant heat flux at the surface of the outer shell, given by, respectively²⁵

κ 1 = In (\frac{1 - r 1}{τ 1}) / L 1

(15)

q rad - inc = \frac{F hs - shell σ ɛ hs (T_{hs}^{4} - T^{4} | x = 0) A hs}{A fab} (1 - r 1) - σ ɛ 1 F shell - amb (1 - ɛ g) (T^{4} | x = 0 - T_{amb}^{4})

(16)

where r₁ is the reflectivity of the outer shell (0.09), τ₁ is the outer shell transmissivity (0.1), A_hs and A_fab are the heating source area and the outer shell area, respectively, ɛ_hs is the heating source emissivity (0.98), T_hs is the heating source temperature (733.15 K)³⁹ and F_hs-shell is the view factor between the heating source and the outer shell, which can be obtained by reference to Su et al.³⁴

The multilayer fabric system is considered as a radiative participating medium due to fabric’s porosity. Although the fiber can absorb, emit and scatter thermal radiation, the extinction of fabric plays a main role in radiant heat transfer. The scattering portion of thermal radiation is usually ignored, as back-scattering and forward-scattering of radiation can balance out in terms of the structure of the fabric and one-dimensional heat transfer. By assuming thermodynamic equilibrium, we can use the two-flux model to study the effect of self-emission in the fabric system.^18,42,43 ${dq}_{rad}^{+}$ /dx and ${dq}_{rad}^{-} / dx$ can be determined

\frac{{dq}_{rad}^{+} (x)}{dx} = - κ 1 q_{rad}^{+} (x) + σ κ 1 T^{4} (x, t)

(17\rm a)

\frac{{dq}_{rad}^{-} (x)}{dx} = κ 1 q_{rad}^{-} (x) - σ κ 1 T^{4} (x, t)

(17\rm b)

Assuming the bounding surfaces of the fabric system are diffusely emitting and reflecting, the boundary conditions of equations (17a) and (17b) are

Model solution

The partial differential equations can be solved according to the finite difference method. Based on the Crank–Nicolson implicit scheme, a one-dimensional space coordinate and time coordinate are spanned.⁴⁴ These partial differential equations and the boundary conditions are discretized in order to obtain a non-linear tri-diagonal system. Due to nonlinearity that comes from the radiation boundary condition, the coupling conduction–radiation and the variation in the fabric thermo-physical and convective heat transfer coefficient with temperature, the Gauss–Seidel point-by-point iterative scheme is employed to solve these discrete equations. The calculated temperatures from the previous time step are used as initial values for the iteration loop of the next time step. The program is written in MATLAB version 8.3.

Experimental details

Materials

All the selected fabrics are popularly used in the firefighter protective clothing field, including the outer shell, moisture barrier and thermal liner. The basic specifications of the testing samples are listed in Table 1. The fabric’s density was calculated by the fabric’s mass, which was tested according to standard ASTM D3776. In accordance with the procedures in standard ASTM D1777-96, the thickness of test specimens were measured under a pressure of 1 kPa. Differential scanning calorimeter (NETZSCH DSC 204 F1) tests were employed to measure thermal conductivity and specific heat of the fabrics, but the thermal conductivity of the outer shell was obtained from Torvi.⁷ Single-layer fabrics can be assembled into a multilayer fabric system.

Table 1.

Properties of the fabrics used in the numerical simulation

Layer	Component	Density (kg/m³)	Thickness (mm)	Specific heat at 300 K (J/(kg K))	Thermal conductivity at 300 K (W/(m K))
Outer shell	100% Nomex	342	0.6	1570	0.047
Moisture barrier	80% Nomex/ 20% Kevlar (PTFE)	122	0.9	1160	0.034
Thermal liner	100% M-aramid	123	2.2	1350	0.035

PTFE: polytetrafluoroethylene.

Testing apparatus and procedures

The stored thermal energy test (SET) apparatus is widely used to investigate the thermal protection of firefighter turnout composites against low-level radiation heat. The test apparatus employed in this study is MTN-P292-08 (Measurement Technology Northwest, Seattle WA, USA), in accordance with the ASTM F2731-11 test standard.³⁹ The schematic of the test apparatus is shown in Figure 2. The testing specimen is exposed to a nominal radiant heat flux of 8.5 kW/m² produced by a black ceramic thermal flux source. The heat flux rise versus time at the back of the specimen is measured by the sensor assembly, which consists of a water-cooled Schmidt-Boelter thermopile-type sensor and a sensor housing. The sensor assembly can be in contact with the specimen or with a spacer introducing a 6.4 mm air gap between the fabric and the sensor assembly.

Figure 2.

Schematic of the test apparatus with an air spacer.³⁹

The testing specimen stays in the non-exposure position for insulating from the radiant heat source before testing. The combined sensor assembly and specimen holder can be moved between the non-exposure position and the heating source by controlling the electronically triggered transfer tray. As soon as the transfer tray moves over the heating source, the data acquisition system begins collecting data. At the end of 300 s of exposure, the transfer tray is moved away from the heating source in order to simulate the cooling process of the testing specimen. The data collection sensor continues to record data for 200 s. According to the above experimental procedures, three specimens per fabric system are measured under the thermal radiation of 8.5 kW/m² for 300 s and the cooling phase for 200 s, and the results are given as mean values.

Results and discussion

Model validation

In order to verify the developed model in the multilayer clothing system without an air gap and with an air gap, the predicted times to second-degree burn and third-degree burn are compared with the experimental results, as shown in Table 2. The reliability analysis method is employed to explain the experimental precision. The intra-class correlation coefficient is 0.996 and the coefficient of variation is from 1.23% to 4.07%, indicating that the repeatability and the precision of the test is acceptable. Comparing to the experimental results shown in Table 2, the predicted deviations for the multilayer clothing system without an air gap are, respectively, 19.24% and 18.07%. Similarly, the predicted deviations of times to second- and third-degree burn in the multilayer clothing system with an air gap are 9.95% and 10.43%, respectively. The reason why the burn times for inserting an air gap model are closer to the experimental value might be that the developed model ignores the convective heat transfer in the air gap of 6.4 mm. Although the convection heat transfer can be ignored in the air gap model when the air gap thickness ranges from 0 to 6.4 mm,³² the importance of natural convective heat transfer in the air gap obviously increases when the air gap thickness is close to 6.4 mm.

Table 2.

Experimental and predicted results of times to second-degree burn and third-degree burn.

Spacing	Time to second-degree burn (s) (SD)		Time to third-degree burn (s) (SD)
Spacing	Contact	6.4 mm	Contact	6.4 mm
Experimental results, t₁	103.4 (3.87)	163.8 (2.02)	159.3 (6.48)	287.5 (4.09)
Predicted results, t₂	83.5	146.8	130.5	264.2
Relative error, \|t₂− t₁\|/t₁ (%)	19.24	9.95	18.07	10.43

The improved model and the previous model³⁴ are both applied to calculate temperatures of the epidermis–dermis interface for the multilayer fabric system without and with an air gap exposed to a radiative heat flux of 8.5 kW/m² for 300 s and a cooling period for 200 s. Figure 3 shows the experimental results and the numerical results from the new and previous models. It can be seen that the temperature of the epidermis–dermis interface experiences an increasing tendency during the exposure, and continues to go up for several seconds during the cooling due to the discharging effect of the stored energy in the multilayer fabric system. After the end of the discharging process, the temperature begins to sharply decline, as the ambient temperature is 300 K during the cooling. Therefore, it can be concluded that these numerical results from the improved model and the previous model have a similar trend to the experimental results. However, the temperature rise predicted by the two models during heat exposure is marginally higher compared with that of the experimental measurement. This is because, as reported by Sawcyn and Torvi,⁴⁵ the center temperature of the fabric system is higher than the edge temperature of the fabric system when the fabric system is exposed to the heating source. Thus, the thermal energy from the central portion of the fabric system can be transferred to the surrounding portion due to the temperature difference. However, the developed models ignore the multi-dimensional heat transfer in the multilayer fabric system. Moreover, the tendency of deviation between numerical simulation and experimental measurement are in good agreement with the numerical results of Ghazy and Bergstrom⁴⁶ where the single-layer fabric was exposed to flash fire.

Figure 3.

Comparison of predicted and experimental results of temperature histories on the epidermis–dermis interface for a multilayer system without an air gap (a) and with an air gap (b).³⁴

In addition, the temperature rise predicted by the new model is relatively higher than that of the previous model during the exposure, while the difference between the two models gradually reduces after the exposure. The improved model considers the effect of self-emission in the multilayer fabric system on heat transfer. It is believed that the thermal radiation traveling through a participating medium in the direction of x gains energy by emission.¹⁰ Therefore, the self-emission in the multilayer fabric system not only contributes to the heat transferring to human skin during the exposure, but also increases the rate of thermal energy transmitting to the ambience during the cooling. Meanwhile, the influence of the fabric’s self-emission has an increasing tendency with the increment of the temperature in the fabric system. Therefore, the aim of this paper is to study the mechanism of radiative heat transfer and the effects of the optical properties of fabrics by employing the improved radiant heat transfer model.

Radiant heat transfer in the multilayer fabric system

In order to investigate radiant heat transfer in the multilayer clothing system, predictions from the established model were performed for the duration of 300 s radiant heat exposure and 200 s cooling, with the introduction of an air gap. Figure 4 shows the variation in the radiative heat flux of the fabric system over time and location. It indicates that the radiative heat flux in different locations of the outer shell declines with the increment of exposure time as the temperature of the outer shell gradually increases during the exposure. The radiative heat flux through the surface of outer shell has the largest descending rate. This is because the outer shell with higher temperature can release thermal energy to the low-temperature environment. In addition, the radiative heat flux remains stable after the exposure of around 100 s. The main reason for this is that the fabric system can stop storing thermal energy owing to the limited storing capacity.³⁴ It is also found from Figure 4 that the radiative heat flux shows a descending trend from the outer shell surface (0 m) to the backside of the outer shell (0.0006 m) due to the absorption of thermal radiation in the fabric system. The radiant heat flux at the backside of the outer shell during the exposure is greater than zero, indicating that thermal radiation can penetrate through the outer shell. Even though there is no the incident radiative heat energy entering into the fabric system after the exposure, the fabric system still has radiative heat transfer owing to the effect of self-emission in the fabric system. Therefore, it can be concluded that radiative heat transfer, considering the self-emission, absorption, transmission and reflection, can significantly affect the heat transfer in the multilayer fabric system during the exposure and the cooling.

Figure 4.

Spatial distribution of the radiative heat flux in the outer shell versus time.

Effect of optical properties of flame-resistant fabric

The radiative heat transfer in fibrous material plays an important role in thermal energy distribution in the fabric system,^7,18 especially for radiant heat exposure. Thus, this section of the paper investigates the impacts of optical properties (such as absorptivity, reflectivity, emissivity and transmittivity) of the outer shell on skin temperature and heat transfer based on the developed model, aiming to improve the protective performance of flame-resistant fabrics in radiant heat exposure.

Effects of the fabric’s transmissivity and absorptivity

The absorptivity is equated to the emissivity in each layer of fabric according to Kirchhoff's Law. The sum of the absorptivity, transmissivity and reflectivity is 1. It is assumed that the fabric’s reflectivity remains at a constant value (0.09). Figure 5 illustrates the prediction of temperature histories on the epidermis–dermis interface for a multilayer fabric system with an air gap under various transmissivities and absorptivities. For the selected transmissivity range of 0.01–0.31, the model predicts an overall increase in the temperature of the epidermis–dermis interface as the transmissivity of the fabric increases. At the beginning of the exposure, the temperature differences among different transmissivities are insignificant, as the fabric system can store the thermal energy and reduce the heat transfer rate. The differences gradually increase with the increment of the exposure duration. However, the differences after the exposure begin to decrease because the larger transmissivities also increase the radiative heat transfer between the outer shell and the ambient due to the self-emission of the outer shell.

Figure 5.

Temperature histories of the epidermis–dermis interface under different transmissivities.

The effect of different transmissivities on the incident heat flux is shown in Figure 6. Obvious differences can be found for various transmissivities during radiant heat exposure. The incident heat flux through the outer shell increases with the increment of the fabric’s transmissivity. Due to the constant of fabric reflectivity, the fabric’s absorptivity or emissivity decreases when the transmissivity of fabric increases. The fabric system with the lesser emissivity can reduce the thermal energy emitting the ambience. In contrast, the larger transmissivity can contribute to the thermal energy transferring into the fabric system, thus accelerating skin burn injuries. During the cooling phase, the differences between incident heat flux are insignificant because the incident heat flux is equal to the thermal exchange between the outer shell and the ambient, without the heating source. Under these circumstances, the fabric’s transmissivity has a dual character in the heat transfer. For one thing, the larger transmissivity can increase the radiative heat transfer rate between the fabric system and the ambient. For another, it can reduce the emission of thermal energy in the fabric system.

Figure 6.

Influence of the fabric’s transmissivity on the incident heat flux in the fabric system.

Effects of the fabric’s reflectivity and absorptivity

In terms of the constant transmissivity (0.01), the effects of different reflectivities and absorptivities on the heat transfer in the multilayer fabric system are studied using the developed model. The fabric system is exposed to an 8.5 kW/m² radiant heat exposure for 300 s and a cooling for 200 s. Figure 7 shows the variation in the temperature of the epidermis–dermis interface versus time under various reflectivities. It is obvious that the fabric’s reflectivity has an important influence on the skin temperature. The temperature difference among different reflectivities gradually increases, while the exposure time increases. The maximum difference of skin temperature is when the exposure time is around 305 s. After the end of radiative heat exposure, there is a quicker descending rate in the amount of temperature for less reflectivity. Combining with Figure 5, we can find that the increment of transmissivity can increase the rate of heat transfer in the fabric system, while the larger reflectivity decreases the heat transfer rate.

Figure 7.

Temperature histories of the epidermis–dermis interface under different reflectivities.

Figure 8 describes the influence of the fabric’s reflectivity on the incident heat flux in the fabric system. The fabric’s reflectivity can obviously affect the incident heat flux before the steady state, including the exposure and cooling phase. The numerical results have a contrary trend compared with the results of Figure 6. With regard to the same extent of variation for the transmissivity and the reflectivity, the effect of the fabric’s reflectivity on the heat transfer is more important than that of the fabric’s transmissivity. This is because the incident heat flux can be reduced by reflecting more thermal energy, but having a constant heat transfer rate when the fabric system achieves a steady state (see Figure 8). In contrast, the transmissivity barely changes the incident heat flux in the outer shell before a steady state, while it can increase the rate of thermal energy transmitting to the fabric system (see Figure 6). It can be concluded that the incident thermal energy mainly depends on the fabric’s reflectivity, while the heat transfer rate in the fabric system is determined by the transmissivity of the fabric.

Figure 8.

Influence of the fabric’s reflectivity on the incident heat flux in the fabric system.

In addition, the transmissivity has an insignificant effect on the rate of storing thermal energy, as shown in Figure 9. In contrast, the fabric’s reflectivity can change the storing thermal rate (see Figure 10), thus affecting the incident heat flux during transient heat transfer. However, after the end of the storing energy process, the incident heat flux reaches a constant value. The optical properties of the fabric system do not change the capacity of storing thermal energy, as the capacity mainly depends on the structure and type of fabric system, the exposure duration, the air gap size and the moisture content within fabrics.^47,48 So, the curves of different reflectivities have an intersection in Figure 10. The differences among different reflectivities at the beginning of the exposure are dependant on the incident heat flux. The trend that the rate of storing energy increases with the decrease of reflectivity is changed after the intersective time because the fabric system only stores the same amount of thermal energy. Although the optical properties of fabric do not affect the capacity of storing thermal energy with the fabric system, the fabric’s reflectivity can change the rate of storing energy during the exposure and the rate of discharging energy during cooling.

Figure 9.

Influence of the fabric's transmissivity on the storing energy rate in the fabric system.

Figure 10.

Influence of the fabric reflectivity on the storing energy rate in the fabric system.

Conclusion

An improved heat transfer model in a multilayer flame-resistant fabric system with an air gap was developed by using the classical two-flux model. The improved model considered the thermal radiation with absorbing, transmitting, emitting and reflecting in flame-resistant fabric. A human skin burn model was incorporated into the heat transfer model in the fabric system in order to provide a convenient way of predicting the skin burn injuries and assessing the thermal protective performance of firefighters’ clothing. The predicted results of the new model were compared with the previous Beer’s law model and the experimental results, and found to be in good agreement with the experimental ones.

The aim of this study was to investigate the mechanism of radiant heat transfer in the multilayer fabric system and the effects of the optical properties of flame-resistant fabric on the heat transfer. The conclusion can be made that the self-emission in the multilayer fabric system increases not only the rate of thermal energy transferring to the human skin during exposure, but also the rate of thermal energy transmitting to the ambience during cooling. The fabric’s optical properties have a complex influence on the transmitted and stored energy in multilayer protective clothing. For one thing, the larger transmissivity can increase the radiative heat transfer from the fabric system to the ambience, but reduce the emission of thermal energy in the fabric system. Also, it barely changes the incident heat flux before a steady state. For another thing, the incident heat flux can be reduced for the larger reflectivity, but has a constant heat transfer rate when the fabric system reaches a stable state. With regard to the same extent variation for the transmissivity and the reflectivity, the role of the reflectivity is more important than that of the transmissivity, as the incident thermal energy mainly depends on the fabric’s reflectivity, while the heat transfer rate in the fabric system is determined by the transmissivity of the fabric. In addition, although the optical properties of fabric do not affect the capacity of storing thermal energy within the fabric system, the fabric’s reflectivity can change the rate of storing energy during exposure and the rate of discharging energy during cooling. The developed model can be used to efficiently design the optical properties of flame-resistant fabrics to improve the thermal protective performance of protective clothing.

While a dry heat transfer and one-dimensional model was used in this research, future work is also planned to develop the heat and moisture transfer model in fire environments. The fabric system can be wetted by hot steam or hot water from a hose spray and dew or rains.⁶ Also, human skin can sweat a lot under high-temperature and high-intensity operations.⁴⁹ Therefore, the development of a heat and moisture transfer model helps to simulate the actual fire-ground, which can then be used to investigate a wider range of design variables of protective clothing and exposure conditions.

The air gap size in this model was restricted from 0 to 6.4 mm, while the average air gap width in protective clothing is approximately 25 mm in real wearing situations.⁴⁸ The convective heat transfer plays an important role in heat exchange between the fabric system and the skin surface as the air gap size increases. Therefore, further developments should be conducted to model the convective heat transfer in the air gap, coupled with thermal radiation and conduction.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Nature Science Foundation (grant number 51576038), the Fundamental Research Funds for the Central Universities (grant number 16D110713), Donghua University PhD Thesis Innovation Funding (grant number 16D310701) and the Open Funding Project of the National Key Laboratory of Human Factors Engineering (grant number SYFD150051812K).

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