#### 2.1. Heat Transport Modeling

The heat conducted in a solid may be described by a linear relationship between the temperature gradient and the heat flux, i.e., Fourier’s law:

where

$k$ (W/m·K) is the thermal conductivity of the solid, i.e., in the present study the porridge and the skin. The general heat balance for bioheat transfer in the skin may be expressed by Pennes bioheat equation:

where

$\mathsf{\rho}$ (kg/m

^{3}) is the skin density,

$C$ (J/kg·K) is the skin specific heat,

$t$ (s) is the time,

${W}_{b}$ (m

^{3}/m

^{3}·s) is the blood perfusion rate,

${\mathsf{\rho}}_{b}$ (kg/m

^{3}) is the blood density,

${C}_{b}$ (J/kg·K) is the blood specific heat,

${T}_{b}$ (K) is the supplied blood temperature,

${Q}_{met}$ (W/m

^{3}) is the metabolic heat production, and

${Q}_{ext}$ (W/m

^{3}) is heat supplied from an external heat source. In the present work,

${Q}_{ext}$ represents the heat supplied to the skin from the hot porridge or after porridge removal, the heat loss from the skin surface to the ambient air and to tempered water.

For short duration heat exposures, Ng and Chua [

21] concluded that blood perfusion does not influence the extent of burns significantly. Lipkin et al. [

22] found that about 20 s is needed for the skin to increase the blood flow. This was also recently concluded by Fu et al. [

9]. In the present work, up to 60 s heat exposure is analyzed. Blood perfusion and metabolic heat production therefore needs to be included in the modeling. The blood perfusion and metabolic heat production models presented by Rai and Rai [

23] were therefore included in the bioheat equation, i.e., Equation (2).

Porridge is experienced to wet human skin completely. The external heat source, i.e., the hot porridge, was therefore assumed to be in perfect contact with the skin surface. Since the diameter of the arm/thigh is much larger than the heat penetration depth, the system may be considered as a one-dimensional flat surface. This allows for studying heat flow in one dimension, i.e., only dependent on the depth (x-dimension). Since a moderate temperature increase is studied in the present work, it is assumed that the thermal skin properties are temperature independent.

The heat transfer model is shown in

Figure 1a for the scald heating, in

Figure 1b for removed porridge, and in

Figure 1c for the possible final skin surface cooling by tempered water.

In the present work, it is assumed that the application of porridge to the skin surface happens instantaneously at $t$ = 0, and that the porridge temperature at that moment is uniform. At $t$ > 0, heat is conducted from the hot porridge into the skin, while some heat is also lost convectively from the hot porridge to the ambient air. This case is then compared to cases where the hot porridge after some time is suddenly removed, exposing the skin surface to ambient air. For some cases, the skin is also finally exposed to tempered water for post scald cooling.

To correctly determine the external heat flux to the skin surface, the heat equation must also be solved for the hot porridge layer in contact with the skin surface. For simplicity, it is conservatively assumed that the porridge wets the skin completely, i.e., there is no heat transfer barrier like air bubbles between the porridge and the skin. The representative values for thermal conductivity, density, and specific heat of rice and milk products given by [

20] was used for modeling porridge heat transport in the present work, i.e.:

The outer porridge surface at temperature

${T}_{\mathrm{p},\mathrm{s}}$ (°C) will be convectively cooled by the ambient air at temperature

${T}_{\mathrm{air}}$ (°C). This heat loss may be expressed by

where

${h}_{\mathrm{air}}$ (W/m

^{2}·K) is the porridge surface to air convective heat transfer coefficient. The cooling of the skin when exposed to ambient air after porridge removal was calculated by Equation (3), where the porridge surface temperature (

${T}_{\mathrm{p},\mathrm{s}}$) was substituted with the skin surface temperature,

${T}_{\mathrm{s},\mathrm{s}}$ (°C). For the cases where the skin was cooled by water at temperature

${T}_{\mathrm{w}}$ (°C), and convective heat transfer coefficient

${h}_{\mathrm{w}}$ (W/m

^{2}·K), the heat loss was similarly given by

In the present work, ${h}_{\mathrm{air}}$ and ${h}_{\mathrm{w}}$ were estimated to be 10 W/m^{2}·K and 600 W/m^{2}·K, respectively. The cooling water temperature was set to 15 °C.

Solving Equation (2) numerically makes it possible to successively alter the boundary conditions for heat transport to the skin, i.e. by (i) conduction from the hot porridge in the first period,

${t}_{\mathrm{p}}$ (s), (ii) air cooling for a period

${t}_{\mathrm{a}}$ (s) after removing the porridge, and finally (iii) cooling the skin by tempered water at

${t}_{\mathrm{w}}$ (s). The skin temperature was initially, i.e., at

$t=0$, set to 37 °C [

11] for all depths 0 ≤

$x$ ≤

$\Delta $, where

$\Delta $ (m) represents the modeling domain size. The domain included the skin layers shown in

Table 1. For the period of contact with a hot porridge layer of thickness

$L$ (m), the boundary condition to the air surface at

$x=-L$ is given by:

At time

${t}_{\mathrm{p}}$, the hot porridge is instantaneously removed, and the skin surface at

$x=0$ is then exposed to ambient air, with the following boundary condition:

For selected cases, at time

${t}_{\mathrm{w}}$, the final cooling of the skin surface by tempered water is started, with the following boundary condition:

The inner skin surface boundary condition, i.e., at

$x=\Delta $, is for simplicity given as the contact with an adiabatic surface, i.e.,

The domain size (depth of the skin) must be sufficiently large to limit any influence of the finite dimensions. A depth ∆ >

$2\sqrt{at}$ [

24] would normally be sufficient to ignore the influence of the finite domain size, where

$t$ (s) is the time, and

$a$ (m

^{2}/s) is the thermal diffusivity given by

In the present work, the muscle layer was included in the calculation domain, even though this was not needed to comply to the ∆ > $2\sqrt{at}$ criterion.

The literature values for skin layer thicknesses and thermal properties vary. A comprehensive summary of skin properties relating to scalds was presented by Johnson et al. [

11]. Their reported values vary somewhat from those reported by Millington and Wilkinson [

25]. A typical value for the skin epidermis layer in these references is 60 μm. Since the epidermis thickness varies with the location even on one extremity, and children are known to have thinner skin, 40 μm and 50 μm was also studied in the present work.

For numerical stability during the modeling, the Fourier number must satisfy

where

$\Delta t$ (s) is the numerical integration step length and

$\Delta x$ (m) is the numerical layer thickness. A C++ computer program was used to solve Equation (2) for the boundary conditions involved in a series of cases explained in detail in the results section. A layer thickness

$\Delta x$ = 10 μm and a numerical step length

$\Delta t$ = 2 × 10

^{−4} s complied with Equation (9), also for the high thermal diffusivity muscle layer.

#### 2.2. Skin Damage Modeling

Burns and scalds were originally classified as first-degree, second-degree, and third-degree burns, according to burn severity. This classification system is still popular among laypeople. Medical personnel do, however, classify burns based on the injury depth. This includes four categories [

19]: (1) Superficial (S) burns confined to the epidermal layer and characterized by slight edema and fast healing; (2) Superficial partial-thickness (SP) burns that extend into the outer part of the dermal layer and result in moderate edema but little or no scarring (typically less than 1 mm burn depth); (3) Deep partial-thickness (DP) burns that extend well into the dermal layer and are slow to heal (typically 1 mm or greater burn depth) and result in hypertrophic scarring; and (4) Full-thickness (FT) burns that extend through the entire dermis requiring skin grafting (burn depths typically greater than 2 mm). Category S burns correspond to the commonly used term first-degree burns. Category SP and DP burns correspond to second-degree burns, and category FT burns correspond to third-degree burns.

Cell injury develops due to excessive skin temperatures and resulting protein breakdown. Collagen is one of the main proteins involved. The cell injury can be calculated to evaluate the burn severity at certain depths, and is expressed by the damage index

$\mathsf{\Omega}$ [

26]:

where

${C}_{0}$ and

${C}_{\tau}$ represent the number of undamaged cells prior to and after the heat exposure, respectively. A damage index of 0.1 corresponds to 90% of the cells still being undamaged. A damage index of 1.0 indicates that only 36% of the cells are still undamaged. The rate of the developing skin injury can be calculated by an activation energy-based model, originally developed by Henriques [

27]:

where

$P$ (1/s) is the pre-exponential frequency factor,

$\Delta E$ (J/mol) is the activation energy,

$R$ (8.314 J/mol·K) is the molar gas constant, and

$T$ (K) is the absolute temperature. Literature data for these damage parameters vary considerably [

11]. In the present work, the original data presented by Henriques and Moritz [

6,

27], i.e.,

$P$ = 3.1 × 10

^{98} 1/s and

$\Delta E$ = 6.28 × 10

^{8} J/mol, was used for modeling the skin injury.

The total cell damage is obtained by integrating Equation (11) over the time interval for which the basal layer temperature is above the injury threshold temperature, i.e., 43.0 °C:

Since the damage index is calculated as an integral, it is often referred to as the damage integral. A damage integral

$\mathsf{\Omega}$ = 0.53 at the basal layer was reported by Ye and He [

28] as the limit for

superficial burns and

$\mathsf{\Omega}$ = 1.0 as the limit for

superficial partial-thickness burns. The numerical integration of Equation (12) was in the present work done in parallel to the temperature modeling, i.e., in the previously mentioned C++ program.