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A study for multi-layer skin burn injuries based on DPL bioheat model

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A Correction to this article was published on 11 September 2020

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Abstract

In this paper, multi-layer skin burn injuries are studied using the DPL bioheat model when skin surface is subjected to different non-Fourier boundary conditions. A skin made of three layers known as epidermis, dermis, and subcutaneous layer. These layers assumed to be homogeneous and each layer studied separately. The metabolic heat varies linearly with temperature. The diffusion and evaporation of water in the multi-layer of skin increases heat loss in the skin layer. To solve the BVP of hyperbolic PDE, the FELWG method has been used. The whole analysis presented in a non-dimensional form and the results are shown graphically. In a particular case, the result obtained is compared with the exact solution and is in good agreement. The effects of relaxation time, layer thickness, different temperature, and non-Fourier boundary condition are analyzed at the temperature of the tissue related to the burning of the skin, and the three layers are discussed in detail.

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Abbreviations

\(c_{\uprho}\) :

Specific heat/J kg\(^{-1}\,^{\circ }\)C\(^{-1}\)

k :

Thermal conductivity/W m\(^{-1}\,^{\circ }\)C\(^{-1}\)

t :

Time/s

T :

Temperature/\(^{\circ }\)C

\(D_{f}\) :

Coefficient of water diffusion in tissue/m\(^2~\text {s}^{-1}\)

\(M_{\mathrm{W}}\) :

The molar mass of water/18 g mol\(^{-1}\)

RH :

Relative humidity/%

P W :

Vapor pressure of water/Pa

r :

Space coordinate/m

ΔH vap :

Enthalpy of water vaporization/2408 J kg\(^{-1}\)

Δm :

Water vaporization rate from the skin surface/g m\(^{-2}~\text {s}^{-1}\)

\(\Delta {r}\) :

Body core distance from current tissue position/m

\(\delta c\) :

The average distance of the momentum boundary layer/m

\(\rho \) :

Density of skin/kg m\(^{-3}\)

\(R_{\mathrm{a}}\) :

Universal gas constant/8.314 J mol\(^{-1}\,^{\circ }\)C\(^{-1}\)

\(\tau _{\mathrm{q}}\) :

Phase lag of heat flux/s

\(\tau _{\mathrm{t}}\) :

Phase lag due to temperature gradient/s

\(T_{\mathrm{w}}\) :

Wall temperature at the boundary/\(^{\circ }\)C

H :

Coefficient of reference heat transfer/W m\(^{-2}\,^{\circ }\)C\(^{-1}\)

\(q_{\mathrm{w}}\) :

Reference heat flux/W m\(^{-2}\)

\(T_\mathrm{s}\) :

Ambient temperature/\(^{\circ }\)C

\(Q_\mathrm{v}\) :

Evaporation of water

\(Q_\mathrm{d}\) :

Diffusion of water

b :

Blood

c :

Core

a :

Air

f :

Diffusion of water

m :

Metabolic production

s :

Surface of skin

v :

Vaporization

W:

Water

\(F_{\mathrm{o}}\) :

Non-dimensional time

x :

Non-dimensional space coordinate

\(K_{\mathrm{i}}\) :

Kirchhoff number

\(B_{\mathrm{i}}\) :

Biot number

\(F_{\mathrm{ot}}\) :

Non-dimensional phase lag due to temperature gradient

\(F_{\mathrm{oq}}\) :

Non-dimensional phase lag of heat flux

\(P_{\mathrm{mo}}\) :

Non-dimensional coefficient of metabolic heat source

\(P_{\mathrm{f}}\) :

Non-dimensional coefficient of blood perfusion

\(\theta \) :

Non-dimensional tissue temperature

\(\theta _\mathrm{s}\) :

Non-dimensional fluid temperature

\(\theta _{\mathrm{w}}\) :

Non-dimensional wall temperature of the tissue

\(\theta _{\mathrm{b}}\) :

Non-dimensional blood temperature

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Acknowledgements

First author would like to thanks the CSIR, New Delhi, India, for the financial support under the JRF (09/013(0931)/2020-EMR-I) scheme and also to the Department of Mathematics (Institute of Science), Banaras Hindu University (BHU), Varanasi (U.P), India, for providing necessary facilities. We thank all anonymous reviewers for spending valuable time to give valuable comments so that our manuscript is improved in the present form.

Author information

Authors and Affiliations

  1. Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi, 221005, India

    Rajneesh Kumar Chaudhary & Jitendra Singh

  2. Department of Mathematical Sciences, IIT-BHU, Varanasi, 221005, India

    Kabindra Nath Rai

Authors
  1. Rajneesh Kumar Chaudhary
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  3. Jitendra Singh
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Correspondence to Jitendra Singh.

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In the original publication, few equations and figure captions were published incorrectly. The original version is corrected.

Appendix

Appendix

Second kind non-Fourier boundary condition

From Eq. (23), we have found the second kind non-Fourier boundary condition, i.e.,

$$\begin{aligned} \frac{\partial }{\partial {x}}\left[ \,\theta (x,F_{\mathrm{o}})+F_{\mathrm{ot}} \frac{\partial {\theta (x,F_{\mathrm{o}})}}{\partial {F_{\mathrm{o}}}}\right] \, =-K_{\mathrm{i}} \;\;\;\;at \;\;x=0. \end{aligned}$$
(63)

Taking the Laplace transform of Eq. (63), we obtained

$$\begin{aligned} \frac{\partial }{\partial {x}}\left[ \,\overset{\sim }{\theta }(x,s)+F_{\mathrm{ot}} \{s \overset{\sim }{\theta }(x,s)-\theta (x,0)\}\right] \, =-\frac{K_{\mathrm{i}}}{s} \;\;\;\;at \;\;x=0. \end{aligned}$$
(64)

Using initial condition \(\theta (x,0)=0\) in Eq. (64), we get

$$\begin{aligned}&\frac{\partial }{\partial {x}}\left[ \,\overset{\sim }{\theta }(x,s)+F_{\mathrm{ot}} s\overset{\sim }{\theta }(x,s)\right] \, =-\frac{K_{\mathrm{i}}}{s} \;\;\;\;at \;\;x=0, \end{aligned}$$
(65)
$$\begin{aligned}&(1+F_{\mathrm{ot}}s)\frac{\partial {\overset{\sim }{\theta }(x,s)}}{\partial {x}} \, =-\frac{K_{\mathrm{i}}}{s} \;\;\;\;at \;\;x=0, \end{aligned}$$
(66)
$$\begin{aligned}&\frac{\partial {\overset{\sim }{\theta }(x,s)}}{\partial {x}} \, =-\frac{K_{\mathrm{i}}}{s(1+F_{\mathrm{ot}}s)} \;\;\;\;at \;\;x=0. \end{aligned}$$
(67)

Inverse Laplace transform of Eq. (67) is

$$\begin{aligned} \frac{\partial {\theta (x,F_{\mathrm{o}})}}{\partial {x}} \, =-K_{\mathrm{i}} (1-e^{-\frac{F_{\mathrm{o}}}{F_{\mathrm{ot}}}}) \;\;\;\;at \;\;x=0. \end{aligned}$$
(68)

Third kind non-Fourier boundary condition

From Eq. (23), we have found third kind non-Fourier boundary condition, i.e.,

$$\begin{aligned}&\frac{\partial }{\partial {x}}\left[ \,\theta (x,F_{\mathrm{o}})+F_{\mathrm{ot}} \frac{\partial {\theta (x,F_{\mathrm{o}})}}{\partial {F_{\mathrm{o}}}}\right] \, -B_{\mathrm{i}} \theta (x,F_{\mathrm{o}})=-B_{\mathrm{i}} \theta _\mathrm{s} \;\;\;\;at \;\;x=0, \end{aligned}$$
(69)
$$\begin{aligned}&\frac{\partial }{\partial {x}}\left[ \,\theta (x,F_{\mathrm{o}})+F_{\mathrm{ot}} \frac{\partial {\theta (x,F_{\mathrm{o}})}}{\partial {F_{\mathrm{o}}}}\right] \, =-B_{\mathrm{i}} (\theta _\mathrm{s}-\theta (x,F_{\mathrm{o}})) \;\;\;\;at \;\;x=0. \end{aligned}$$
(70)

Taking the Laplace transform of Eq. (70), we obtained

$$\begin{aligned} \frac{\partial }{\partial {x}}[ \,\overset{\sim }{\theta }(x,s)+F_{\mathrm{ot}} \{s \overset{\sim }{\theta }(x,s)-\theta (x,0)\}] \, =-B_{\mathrm{i}} (\theta _\mathrm{s}-\theta (x,s)) \;\;\;\;at \;\;x=0. \end{aligned}$$
(71)

Using initial condition \(\theta (x,0)=0\) in Eq. (71), we get

$$\begin{aligned}&\frac{\partial }{\partial {x}}\left[ \,\overset{\sim }{\theta }(x,s)+F_{\mathrm{ot}} s\overset{\sim }{\theta }(x,s)\right] \, =\frac{-B_{\mathrm{i}} (\theta _\mathrm{s}-\theta (x,s))}{s} \;\;\;\;at \;\;x=0, \end{aligned}$$
(72)
$$\begin{aligned}&(1+F_{\mathrm{ot}}s)\frac{\partial {\overset{\sim }{\theta }(x,s)}}{\partial {x}} \, =\frac{-B_{\mathrm{i}} (\theta _\mathrm{s}-\theta (x,s))}{s} \;\;\;\;at \;\;x=0, \end{aligned}$$
(73)
$$\begin{aligned}&\frac{\partial {\overset{\sim }{\theta }(x,s)}}{\partial {x}} \, =\frac{-B_{\mathrm{i}} (\theta _\mathrm{s}-\theta (x,s))}{s(1+F_{\mathrm{ot}}s)} \;\;\;\;at \;\;x=0. \end{aligned}$$
(74)

Inverse Laplace transform of Eq. (74) is

$$\begin{aligned} \frac{\partial {\theta (x,F_{\mathrm{o}})}}{\partial {x}} \, =-B_{\mathrm{i}} (\theta _\mathrm{s}-\theta (x,F_{\mathrm{o}})) (1-e^{-\frac{F_{\mathrm{o}}}{F_{\mathrm{ot}}}}) \;\;\;\;at \;\;x=0. \end{aligned}$$
(75)

Non-Fourier symmetric condition

From Eq. (27), we have found non-Fourier symmetric condition, i.e.,

$$\begin{aligned} \frac{\partial }{\partial {x}}\left[ \,\theta (x,F_{\mathrm{o}})+F_{\mathrm{ot}} \frac{\partial {\theta (x,F_{\mathrm{o}})}}{\partial {F_{\mathrm{o}}}}\right] \,= 0 \;\;\;\;at \;\;x=1. \end{aligned}$$
(76)

Taking the Laplace transform of Eq. (76), we obtained

$$\begin{aligned} \frac{\partial }{\partial {x}}[ \,\overset{\sim }{\theta }(x,s)+F_{\mathrm{ot}} \{s \overset{\sim }{\theta }(x,s)-\theta (x,0)\}] \, =0 \;\;\;\;at \;\;x=1. \end{aligned}$$
(77)

Using initial condition \(\theta (x,0)=0\) in Eq. (77), we get

$$\begin{aligned}&\frac{\partial }{\partial {x}}[ \,\overset{\sim }{\theta }(x,s)+F_{\mathrm{ot}} s\overset{\sim }{\theta }(x,s)] \, =0 \;\;\;\;at \;\;x=1, \end{aligned}$$
(78)
$$\begin{aligned}&(1+F_{\mathrm{ot}}s)\frac{\partial {\overset{\sim }{\theta }(x,s)}}{\partial {x}} \, =0 \;\;\;\;at \;\;x=1,\end{aligned}$$
(79)
$$\begin{aligned}&\frac{\partial {\overset{\sim }{\theta }(x,s)}}{\partial {x}} \, =0 \;\;\;\;at \;\;x=1. \end{aligned}$$
(80)

Inverse Laplace transform of Eq. (80) is

$$\begin{aligned} \frac{\partial {\theta (x,F_{\mathrm{o}})}}{\partial {x}} \, =0 \;\;\;\;at \;\;x=1. \end{aligned}$$
(81)
Fig. 2
figure 2

Epidermis layer: Comparison of exact with approx solution

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Fig. 3
figure 3

Dermis layer:Comparison of exact with approx solution

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Fig. 4
figure 4

Subcutaneous layer: Comparison of exact with approx solution

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Fig. 5
figure 5

Epidermis layer: Effect of \(F_{\mathrm{oq}}=0.00696379\) and \(F_{\mathrm{ot}}=0\) on skin temperature with the first kind non-Fourier boundary condition

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Fig. 6
figure 6

Dermis layer: Effect of \(F_{\mathrm{oq}}=0.0140766\) and \(F_{\mathrm{ot}}=0\) on skin temperature with the first kind non-Fourier boundary condition

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Fig. 7
figure 7

Subcutaneous layer: Effect of \(F_{\mathrm{oq}}=0.00791667\) and \(F_{\mathrm{ot}}=0\) on skin temperature with the first kind non-Fourier boundary condition

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Fig. 8
figure 8

Epidermis layer: Effect of \(F_{\mathrm{oq}}=F_{\mathrm{ot}}=0.00696379\) on skin temperature with the first kind non-Fourier boundary condition

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Fig. 9
figure 9

Dermis layer: Effect of \(F_{\mathrm{oq}}=F_{\mathrm{ot}}=0.0140766\) on skin temperature with the first kind non-Fourier boundary condition

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Fig. 10
figure 10

Subcutaneous layer: Effect of \(F_{\mathrm{oq}}=F_{\mathrm{ot}}=0.00791667\) on skin temperature with the first kind non-Fourier boundary condition

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Fig. 11
figure 11

Epidermis layer: Effect of lagging on skin temperature with the first kind non-Fourier boundary condition at \(F_{\mathrm{o}}=0.5\)

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Fig. 12
figure 12

Dermis layer: Effect of lagging on skin temperature with the first kind non-Fourier boundary condition at \(F_{\mathrm{o}}=0.5\)

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Fig. 13
figure 13

Subcutaneous layer: Effect of lagging on skin temperature with the first kind non-Fourier boundary condition at \(F_{\mathrm{o}}=0.5\)

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Fig. 14
figure 14

Epidermis layer: Lagging behavior on skin during different temperature with the first kind non-Fourier boundary condition at \(F_{\mathrm{o}}=0.5\)

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Fig. 15
figure 15

Dermis layer: Lagging behavior on skin during different temperature with the first kind non-Fourier boundary condition at \(F_{\mathrm{o}}=0.5\)

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Fig. 16
figure 16

Subcutaneous layer: Lagging behavior on skin during different temperature with the first kind non-Fourier boundary condition at \(F_{\mathrm{o}}=0.5\)

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Fig. 17
figure 17

Epidermis layer: The effect on skin thickness between temperature and time with the first kind non-Fourier boundary condition

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Fig. 18
figure 18

Dermis layer: The effect on skin thickness between temperature and time with the first kind non-Fourier boundary condition

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Fig. 19
figure 19

Subcutaneous layer: The effect on skin thickness between temperature and time with the first kind non-Fourier boundary condition

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Fig. 20
figure 20

Epidermis layer: Effect of lagging on skin during hot and cold temperature

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Fig. 21
figure 21

Dermis layer: Effect of lagging on skin during hot and cold temperature

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Fig. 22
figure 22

Subcutaneous layer: Effect of lagging on skin during hot and cold temperature

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Fig. 23
figure 23

Epidermis layer: Ki effect on skin

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Fig. 24
figure 24

Dermis layer: Ki effect on skin

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Fig. 25
figure 25

Subcutaneous layer: Ki effect on skin

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Fig. 26
figure 26

Epidermis layer: Bi effect on skin

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Fig. 27
figure 27

Dermis layer: Bi effect on skin

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Fig. 28
figure 28

Subcutaneous layer: Bi effect on skin

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Fig. 29
figure 29

Epidermis layer: Effect of temperature on skin during non-Fourier generalized boundary condition

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Fig. 30
figure 30

Dermis layer: Effect of temperature on skin during non-Fourier generalized boundary condition

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Fig. 31
figure 31

Subcutaneous layer: Effect of temperature on skin during non-Fourier generalized boundary condition

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Fig. 32
figure 32

Epidermis layer: Effect of lagging on skin temperature with the first kind non-Fourier boundary condition

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Fig. 33
figure 33

Dermis layer: Effect of lagging on skin temperature with the first kind non-Fourier boundary condition

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Fig. 34
figure 34

Subcutaneous layer: Effect of lagging on skin temperature with the first kind non-Fourier boundary condition

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Chaudhary, R.K., Rai, K.N. & Singh, J. A study for multi-layer skin burn injuries based on DPL bioheat model. J Therm Anal Calorim 146, 1171–1189 (2021). https://doi.org/10.1007/s10973-020-09967-3

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