A1110D TTMbbS NBS

publications

NBSIR 81-2320

A Heat Transfer Analysis of

Scald Injury

Center for Consumer Product Technology

National Engineering Laboratory

U.S. Department of Commerce

National Bureau of Standards

Washington, DC 20234

July 1981

Final Report

BATIOWAX BURBAU OB RTAKDAJUJe

LIBRART

AUG 1 8 1981

NBSIR 81-2320

06/60

A HEAT TRANSFER ANALYSIS OF

SCALD INJURY

/?

Robert L. Palla, Jr.

Center for Consumer Product Technology

National Engineering Laboratory

U S. Department of Commerce

National Bureau of Standards

Washington, DC 20234

July 1981

Final Report

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

ii

ABSTRACT

Numerical solutions for skin tissue temperature

during scald injury events are obtained and utilized in

conjunction with a thermal injury criterion, to predict

critical exposure levels for various heated fluids. A

one-dimensional tissue model of the type used by Love is

employed to determine the initial tissue temperature

distribution. The bio-heat equation for tissue heat

transfer is then solved via an implicit finite difference

technique, subject to convective heating and cooling at

the surface.

The sensitivity of the critical exposure level to

variations in tissue properties and convective heating

coefficients is investigated. Thermal injury thresholds

are presented for various fluids along with bounds to

reflect uncertainty in assumed tissue properties. The

results obtained are in good agreement with existing

experimental scald injury data.

iii

PREFACE

This report is a product of a short term research effort sponsored

by the National Bureau of Standards, Center for Consumer Product Technology

(CCPT) . The objective of this research is a more thorough understanding of

thermal injury from heated fluids — a prerequisite for the development of

improved product standards.

The work reported herein, dealing with the heat transfer aspects of

thermal injury, was performed in the Product Performance Engineering Division

of CCPT. Companion work in the area of thermal injury criteria was conducted

concurrently in the Product Safety Division of CCPT, and provided the basis

for quantifying thermal injury in the present study.
iv

TABLE OF CONTENTS

Page

Abstract ii

Preface iii

List of Tables v

List of Figures v

Nomenclature vi

1. Introduction 1

2. Thermal Injury Model 2

2.1 Tissue Heat Transfer 2 2.2 Initial Tissue Temperature Profile -5

2.3 Thermal Injury Criterion 8

3. Parameter Values 9

3.1 Tissue Properties 9

3.2 Heat Transfer Coefficients . 11

4. Numerical Simulations 12

4.1 Sensitivity Analysis for Tissue Properties . . 14

4.2 Effect of Heat Transfer Coefficients ..... 16

5. Conclusions 17

6. References 19

Tables 24

Figures 26

Appendix A - Derivation of the Bio-heat Equation .... 32

Appendix B - Finite Difference Formulation for Tissue

Heat Transfer During Scalding 41

Appendix C - Development of the Thermal Injury Criterion 49
V

LIST OF TABLES

Table 1. Typical Properties for Components of Skin

Table 2. Heat Transfer Coefficients for Various Fluids

LIST OF FIGURES

Figure 1. Injury Threshold for Water Determined

Experimentally by Moritz

Figure 2. Infinite Slab Model of Skin Tissue and the

Initial Tissue Temperature Distribution

Figure 3. Tissue Temperature Profiles for Two Fluid Exposures

Figure 4. Injury Threshold Predicted by Subject Model for

Base Case Values

Figure 5. Effect of Thermal Conductivity on the Injury

Threshold

Figure 6. Effect of Blood Perfusion Rate on the Injury

Threshold

Figure 7. Effect of Initial Temperature Profile on the

Injury Threshold

Figure 8. Effect of Initial Air Temperature on the

Injury Threshold

Figure 9. Threshold Range for Extreme Combinations of

Thermal Properties

Figure 10. Initial Temperature Profile for Combinations

of Properties

Figure 11. Injury Threshold for Exposure to Various Fluids
vi

A

A

B

c P

C

C

AC

D

DA

DB

E

AF

h

H

j

k

K

Z

L

m

n

n

E

P

Pr

NOMENCLATURE

coefficient matrix

2

area, m

dimensionless parameter, defined by Equation 15

specific heat, J/kg K

thermal capacity, J/K

3 concentration of substance in blood, kg/m

3 concentration of substance in volume element, kg/m

body part diameter, m

thickness of tissue in Region A, m

thickness of tissue in Region B, m

activation energy, cal/mol

3 rate of blood flow into/out of volume element, m /sec

2 convective heat transfer coefficient, W/m K

parameter h/k, m

^

flux density of blood, m /m sec

thermal conductivity of tissue, W/mK

reaction rate constant, dimensionless

blood flow vector

distance from the tissue surface to the region at

which T. = T , m

t a*

dimensionless parameter, defined by Equation 12

total number of molecules in system

number of molecules having an energy greater than E

number of steric events required for a reaction to

occur, 3.1 x 1098

fluid Prandtl number, dimensionless
vii

q

Q

AQ

qo R

R

t

T

t

1

t

2

At

V

AV

W

x,y,z

X

AX,AY,AZ

Y

a

0

v

5

p

3 metabolic heat generation rate, W/m

quantity of substance in an organ, kg

quantity of substance in differential element, kg

dimensionless parameter, defined by Equation 14

gas constant, 2 cal/K mol

thermal resistance, K/W

time, sec

temperature, Q C

time during heating at which T = 44,0 **0, sec

time during cooling at which ^ 44.0 °C, sec

solution time step, sec

free stream velocity of fluid, m/sec

3 volume of element, m

3 perfusion rate, kg/m sec

parameter / w^c^^/k , m^

space coordinates in cartesian system

dimensionless parameter, defined by Equation 13

dimensions of volume element, m

distance from core towards tissue surface, m

thermal diffusivity, m / sec

temperature difference °C

kinematic viscosity of fluid, m /sec

dimensionless parameter, defined by Equation 11

3 density, kg/m
viii

4

>

X

Q

Subscripts

a arterial blood

(a) pertaining to Type (a) blood vessels

A node region nearest to tissue surface

AL last node in Region A

b blood

(b) pertaining to Type (b) blood vessels

b+ surface capillaries with flow in positive x direction

b- surface capillaries with flow in negative x direction

B node region furthest from tissue surface

B1 first node in Region B

BL last node in Region B

f fluid

volumetric blood perfusion rate, sec

parameter ^/P^Cp^» m^/sec

damage integral, 1.0 for partial thickness burn

n nth node

p due to perfusion

t tissue at a depth of 0.01 cm

v venous blood

x+ positive x direction

x- negative x direction

00 evaluated at free stream conditions
Superscripts

t present time

t+1 future time

1 . INTRODUCTION

The most common cause of thermal injury in both adults and

children is scalding — tissue damage resulting from exposure to

heated fluid. Studies of the epidemiology and treatment of scald and

other burn injuries are numerous in the literature; however, only a

few studies attempt to define critical exposure levels, that is, the

temperature of the scalding fluid and the duration of exposure just

sufficient to cause thermal injury. Most notable are the studies of

Moritz et al. [1] and Diack et al. [2] dealing with the establishment

of a thermal injury "threshold" which defines critical exposure levels

for water (see figure 1), and similar studies by Stoll [3-5],

Weaver et al. [6], and Takata [7], dealing with injury from radiative

heating. These quantitative studies of thermal injury are based on

small samples of laboratory animals or human volunteers subjected to

specific cases of thermal insult and therefore do not reflect the wide

spectrum of potentially hazardous fluid exposures, including variations

in fluid types and flow velocities. When used in conjunction with

nervous system response times, such information can be used to establish

"safe" temperature levels for fluids in the home and in industry.

The present paper explores the heat transfer aspects of the scald

phenomenon as a means of extending existing experimental studies to a

wide variety of fluid types and scald conditions and to exposure times

shorter than those reported in the literature. The basic differential

equation utilized in this study is the bio-heat equation first applied

by Pennes [8] in his studies of tissue temperatures. The equation is
2

solved subject to convective boundary conditions via a finite difference

technique, and used to determine skin tissue time-temperature histories

resulting from exposure to a variety of hypothetical scald conditions.

The time-temperature histories are then evaluated via a thermal injury

criterion to provide a quantitative measure of predicted scald severity.

Through this approach, critical exposure levels for different fluids

and flows are established.

The sensitivity of the solution (thermal injury threshold) to

different assumptions is also considered. These include: estimated

values for the tissue properties and convective heat transfer

coefficients, and the effect of the initial temperature profile. The

final results are new curves defining critical exposure for fluids other

than water, and bounds on the injury threshold curve for water to reflect

uncertainty in the assumed properties of tissue.

2. THERMAL INJURY MODEL

2.1 Tissue Heat Transfer

Most of the existing theoretical analyses of heat transfer in

living tissue have been based on the so-called bio-heat transfer equation,

first applied by Pennes [8] and since used by Wissler [9-11], Chato

and Shitzer [12-16], Love et al. [17], Rao et al. [18], and others.

The equation is:*

- !£ = V

2

T - (T-T ) + £

. ( a 3t k a k

Symbols are defined in the Nomenclature Section.
3

This equation, derived in Appendix A, was obtained by establishing

heat balance on a differential strip element and by assuming that blood

enters the element at some fixed arterial temperature, T^, and is

brought to the temperature of the tissue, T, in the capillaries.

The idealization of the thermal convective transport due to blood flow

has been critically examined by several investigators [19-21]. For

this analysis, the simple model is assumed to be adequate for indicating

the relative importance of blood flow and other physiological phenomena.

A scald event is rather short-lived — typically less than

5 seconds. During this time, the temperature gradient is steep near

the tissue surface but diminishes rapidly with increasing tissue depth.

As a result, the thickness of tissue affected is quite small with respect

to the linear dimension associated with the exposed tissue surface area.

A one-dimensional analysis is thus justified. Also, in view of the

wide range in reported values for thermal properties of the skin and

the relatively small differences from layer to layer, the skin can be

considered a semi-infinite, single layer material. Using the arterial

temperature as a reference point, and defining W = /~w~c~^ / k, the

one-dimensional equation may be written:

Ili = £* . „2 e + 1

a 3t . 2 k

dx

( 2 )

Thermal injury, which occurs with increasing severity as tissue

temperature is raised beyond 44 °C, is the result of damage inflicted

during each of two periods — the heating period, when the skin is
4

exposed to flowing fluid, and the cooling period immediately following

removal of the body part from the fluid flow. Hence, in order to

account for all tissue damage, Equation 2 must be solved in two parts

to obtain the complete time-temperature history.

The analytical solution for the entire scald event is quite

involved. Consider, for example, the solution for the simplest

case — an initially uniform tissue temperature and sudden convective

heating at the surface by a constant temperature fluid. The solution

to Equation 2 without the heat generation term is:

= y—r— [ (H-W) e Wx

erfc (

— W /at )+(H+W) eWx

erfc (

00 2(H -W ) 2 /at 2 /at

- 2H eHx+(H _w2)at

erf c (—-— + H /at)] (3)

2 /at

where H = h/k . Note: Equation 3 is valid only for the heating portion

of the scald event. An analytical solution describing the entire

event — rapid heating with a high convective coefficient, h, and gradual

cooling with a low h —- is considerably more difficult to obtain, and this

is even before considering an initial temperature distribution in the

tissue.

To avoid the analytical complexity dictated by the scald problem,

a numerical implicit finite-difference solution of Equation 2 was

utilized. This solution, described in detail in Appendix B, requires
5

as inputs the number and spacing of temperature nodes, tissue thermal

properties and fluid exposure times. Also required are convective

coefficients and fluid temperatures for the initial (pre-scald),

heating, and cooling periods of the scald event. Based on this

information, the initial temperature as well as the complete timetemperature profile at each of the specified nodes is obtained. The

result gives excellent agreement with Equation 3 for the degenerate

case.

2.2 Initial Tissue Temperature Profile

The temperature of the skin tissue immediately prior to exposure

to heated fluid has a pronounced effect on the outcome of the

exposure — the warmer the tissue initially, the greater the temperatures

experienced throughout the heating and cooling periods, and the greater

the threat of scalding. That the initial temperature is indeed important

can be illustrated by considering some typical temperature levels. The

core or deep body temperature is approximately 37 °C, while the surface

temperature under indoor ambient conditions is about 32 °C. Hence a

5 °C differential exists from core to surface. Now, the tissue

temperature at which thermal injury begins to occur is 44 °C, or 7 degrees

above core temperature — the same order of magnitude as the temperature

differential in the tissue.

In the simplest case, the tissue temperature profile can be assumed

flat, and assigned a value somewhere between typical surface and core

or arterial temperatures — 32 °C to 37 °C, respectively, A more
6

realistic approach is to consider the tissue as a slab having a fixed

arterial temperature at some depth below the skin surface, and convective

heat loss at the skin surface. Such an approach has been utilized

successfully by Love et al. [17] in medical thermography studies and

will be reviewed here.

For the boundary conditions, the origin of the system is chosen to

be at a depth L below the surface of the skin such that the local

' temperature is equal to the arterial temperature (see figure 2)

Also, the slope of the temperature at this point is equal to zero.

At the surface of the skin, heat conducted to the surface is assumed

equal to heat lost to ambient air via convection. Thus the boundary

conditions are given as

T = T at Y = 0, (4) a

* ~ =0 at Y = 0, and (5)

§

’ - £

< T-TJ at Y = L . (6)

Here, Y is measured from core towards surface as shown in figure 2.

Rewriting the governing equation and boundary conditions in terms of

dimensionless parameters results in a less cumbersome form:

d

2 —Z 2 2 — - m £ = m q (7)

dx2 °

E = 0 at X = 0 ( 8 )
7

o dX at X = 0 (9)

d£

dX

= B [1-5(1)] at X = 1 ( 10)

where the dimensionless variables, 5, X, m, qQ

, and B, are defined as

follows

T -T/T -T

a a oo (11)

X = Y/L (12)

L / Vpb / k (13)

(14)

i = hL/k . (15)

The solution to Equation 7 may be obtained by applying boundary conditions

(8) and (10):

5(X) = qQ

cosh (mX) + {[B-Bq

Q

cosh (m) + qQ

B - qQ

m sinh (m) ]

/

[m cosh (m) + B sinh (m)]} sinh (mX) - q

o (16)
8

The dimension L can be eliminated through application of boundary

condition (9) to Equation 16. The result is

i

q = q ^ sinh (m) + q cosh (m) - 1 . o o d o

(17)

By assigning values to the parameters q and ^ = / w, c ' k / h, the value

o a b pb

of m is determined by iterative solution of Equation 17. Core depth L can

then be obtained from Equation 13. Substituting Equation 17 into

Equation 16 yields the following simplification

£(X) = q [cosh (mX)-l] . (18)

o

This expression defines the initial tissue temperature profile. Its

dependence on all tissue parameters including w^ and h should be noted.

2.3 Thermal Injury Criterion

Once the time-temperature history of skin tissue is obtained from

the solution of Equation 2, the extent of thermal injury can be quantified

by applying an appropriate injury criterion to the history. In this

way, the severity of the corresponding fluid exposure is assessed.

Several thermal injury criteria have been reported in the

literature [6,7,22-25], All of these have the same form mathematically,

that is, injury is proportional to an integral of an Arrhenius rate

expression; however, significant differences exist in reported values for
9

the empirical rate constants P and E. The criterion used in the present

study is that originally proposed by Henriques [22] and recently

modified by Wu [24]:

ft = P • / exp [-E/RT

t

] dt . t. (19)

Here, T represents the time-varying tissue temperature at a depth of

0.01 cm, and t^ and t

2

denote the times at which the temperature rises

above and falls below 44 °C (the temperature at which injury begins to

occur)

The rationale for selecting this criterion over other criteria

reported in the literature is provided in Appendix C, along with a

development of the basic criterion form.

3 . PARAMETER VALUES

The values assigned to the tissue and convective parameters will

affect the initial tissue temperature profile as well as the thermal

response of the skin during the actual scald event. Thus it is important

that these values be carefully chosen.

3.1 Tissue Properties

Tissue properties were selected based on a review of the literature.

Typical values and ranges of values, obtained from Refs. [3,14,17,18,26-32],

are presented in Table 1. For the more easily quantified tissue properties,
10

such as density and specific heat, little disagreement exists between

reported values. However, determination of other properties, including

tissue thermal conductivity and blood perfusion, is the subject of much

ongoing research and cited values differ widely.

Values for the base case were assumed to be those most frequently

cited in the literature. For tissue conductivity and specific heat,

_3 values of 0.42 W/mK and 4.2 x 10 J/kg K were used respectively.

3 Blood perfusion rate and specific heat were taken to be 0.4 kg/m sec

_3 and 3.9 x 10 J/kg K, while a metabolic heat generation rate of

3 420 W/m was assumed. Because of uncertainty in the values of the

latter three properties, their influence on the thermal injury threshold

was investigated through a sensitivity analysis.

Tissue thermal conductivity has a range of at least two to one,

as seen in Table 1. When the process of sweating comes into play, the

effective thermal conductivity has a range of about four to one [31].

To determine the effect of this wide variation on the injury threshold,

the base case value was increased and decreased by a factor of two.

New injury thresholds corresponding to each extreme were then obtained

and compared to the threshold established using the base case value.

As shown in Table 1, reported values for blood perfusion rate, w^,

vary even more substantially — from 0.1 to 7.0 kg/m sec in the resting

state. Variation is due to differences in body region and measurement

techniques. In addition, though, vaso-dilat ion and constriction also

affect the blood perfusion rate. Local heating can bring about vessel

vaso-dilat ion in which the blood flow through the skin can increase
11

to as much as 100 times the minimum rate [31]. Similarly, large

variations in metabolic heat generation rate, q, are possible.

However, since oxygen supplied via blood perfusion is necessary to

support the metabolic reaction, a relationship between blood perfusion

and metabolic heat generation can be assumed. Following Rao et al. [18]

the ratio l/w

^

C

p^ was taken to be constant. The base case values for

q, w^, and c ^ then yield a ratio value of 0.27 — close to the

suggested value of 0.3. Keeping this ratio constant fixes the value

of q for each w^ and thus eliminates the need to consider separately

the sensitivity of the solution to variations in w^ and q. Instead,

the effect of these parameters can be determined simultaneously.

Calculations were performed for one-tenth and ten-times the base case

3 3 values of w^ and q, that is, w^ = 0.04 kg/m sec, q = 42 W/m ; and

3 3

w^ = 4.0 kg/m sec, q = 4200 W/m .

3.2 Heat Transfer Coefficients

The convective heat transfer coefficient for the pre-scald,

steady-state condition was based on the empirical relationship developed

by Rao et al . [18]

,

h = [6.74 + 12.1 /V] W/m2

K

where V is the air velocity in m/sec. For natural circulation of room

2 air, an h value of 13 W/m K was obtained.

( 20)
12

Values for h during heating were obtained from empirical

relationships reported in the heat transfer literature for a cylinder

in crossflow. The convective coefficient was taken to be that at the

stagnation point since this represents maximum heat flux. For

stagnation point heat transfer, Squire [33] gives:

h = 1.14 k/D (VD/v) 0,5

(Pr)°*

4

. (21)

Values of h for fluids which typically cause thermal injury

(namely, water, oil, and air) are given in Table 2. They range from

2 2 12,500 W/m K for high velocity water to 13 W/m K for low velocity air.

The greater scald potential of water is obvious due to its much higher

heat transfer coefficient h. The h value for the forced convection

fluid heating in the experiments of Moritz [1] was assumed, in this study,

to be 12,500 W/m K; this value was used as the base case. Values for

flowing oil and hot air were also used to show the effect of different

fluids on the injury threshold.

For the cooldown portion of the scald event, a value of h

representative of cooling by low velocity air and evaporation was

2 selected. This value was 21 W/m K.

4. NUMERICAL SIMULATIONS

Using base case parameter values and the initial tissue

temperature distribution defined by Equation 18, tissue time-temperature

histories at a tissue depth of 0.01 cm are shown in figure 3. These are

I

II
13

for a one second exposure to 70 °C water and a five second exposure

to 60.2 °C water. It can be noted that the cooling period accounts

for the majority of the time spent above the 44 °C damage temperature.

However, since the tissue temperatures experienced during cooling are

significantly lower than during heating, damage inflicted during the

cooling period only amounts to about 20 percent of the total. More

details on the breakdown of tissue damage are provided in Appendix C.

Application of the thermal injury criterion. Equation 19, to a

time-temperature history yields a quantitative measure of scald

severity, ft. Defining ft = 1 at the onset of irreversible thermal

injury, the locus of critical exposure times and corresponding critical

fluid temperatures can be determined iteratively. This locus, termed

the "injury threshold", is a function of the particular exposure

conditions (h values) and assumed tissue properties.

The thermal injury threshold generated using base case tissue

properties and convective coefficients is shown in figure 4. Excellent

agreement between the predicted threshold and limited experimental

results of Moritz and Henriques can be noted. The new threshold

extends to exposure times as short as 0.1 second and water temperatures

above 100 °C. It indicates that exposure to boiling water might be

tolerated if the duration of exposure is less than 0.2 seconds. This

prediction takes into account the damage inflicted during the cooling

period following 0.2 seconds.
14

4.1 Sensitivity Analysis for Tissue Properties

The uncertainty in the predicted injury threshold of figure 4 due

to errors in assumed tissue properties can be estimated by considering

the previously noted ranges in properties. Increasing the tissue

thermal conductivity by a factor of two results in a shift of the

threshold towards more conservative exposures (for the same exposure

time lower fluid temperatures are permissible) . Lower fluid temperatures

are permissible because with increased tissue conductivity, heat penetrates

the tissue much more readily. Reductions in injurious fluid temperature

of 10 °C and 5 °C are predicted for 0.1 second and 1.0 second exposures

respectively as shown in figure 5. Conversely, reducing the tissue

conductivity by a factor of two results in an injury threshold shift

of the same magnitude but in the opposite direction.

A similar effect on the injury threshold is observed when considering

the variation of blood perfusion and metabolic heat generation rates

(both are varied simultaneously such that the ratio q/w^c^k remains

constant). Increasing the rates by a factor of ten results in a shift

towards more conservative exposures while decreasing the rates by a

factor of ten results in an opposite effect. The range in the

threshold is shown in figure 6. The shift in the threshold depicted in

figure 6 is counter to what one would expect intuitively if the blood

perfusion was only thought of as a tissue cooling mechanism. The more

significant effect of the blood flow and heat generation, however, is

in determining the initial tissue temperature profile. With high blood

perfusion and heat generation rates, higher initial tissue temperatures
15

will be realized and thus more conservative exposures should be predicted,

as they are in figure 6.

The significance of the initial tissue temperature is illustrated

in figure 7 in which the thermal injury threshold is shown for different

initial temperatures. For an initially flat temperature profile at

32 °C, the threshold coincides with that for the base case. This result

is expected because the initial tissue surface temperature predicted by

Equation 18 for base case values is 32 °C, and because the tissue depth

on which the criterion (Equation 19) is based is only 0.01 cm below the

surface. The effect of an initially flat profile at 37 °C is much

different, the threshold for this case being shifted towards more

conservative exposures as expected. This shift is similar to the

one depicted in figure 6 for high blood perfusion and heat generation

rates.

The influence of the initial ambient air temperature on the injury

threshold is shown in figure 8. The threshold is shifted upward and

downward for lower and higher ambient air temperatures respectively.

For both figues 7 and 8, shifts in the injury threshold are due purely

to changes in the initial tissue temperature profile, since the tissue

properties and heating and cooling convective conditions are unaltered.

The maximum possible deviation in the injury threshold would occur

with simultaneous, additive-type errors in each tissue property.

Although such an occurrence is unlikely, the cumulative effect can alter

the threshold dramatically. The effect of a worst case combination of

parameters is shown in figure 9. Here the actual threshold is bound by
16

an upper and a lower limit. The upper limit represents a combination

of low tissue thermal conductivity, low blood perfusion and metabolic

heat generation rates and low initial air temperature. Conversely, the

lower limit represents a combination of high values for these parameters.

Maximum errors of as much as 40 °C for exposures of 0.1 second duration

are possible, diminishing to less than 5 °C for exposures lasting

5 seconds or longer. The contribution of the initial tissue temperature

profile to this wide range is apparent from consideration of figure 10.

Comparison of tissue surface temperatures in figure 10 reveals a

difference in initial tissue temperatures of 15 °C between the upper

limit and the lower limit cases. Despite the wide bound on the injury

threshold shown in figure 9, for typical applications a much narrower

band is expected.

4.2 Effect of Heat Transfer Coefficients

An important aspect of the injury model is that the effect of

different fluid types and flow conditions can be evaluated. The

injury threshold for exposures to flowing water, oil, and air are

shown in figure 11 using base case tissue properties and convective

heating coefficients of 13000, 2500, and 46 W/m K respectively.

For all three fluids, base case values for the initial and cooling

2 period convective coefficients were used — 13 and 21 W/m K respectively.

As shown in figure 11, an order of magnitude difference exists between
17

the injurious fluid temperatures for air and water. The difference

between oil and water is not as substantial, but is still about a

factor of two for 0.1 second exposure. For exposure times greater

than five seconds, the difference in injurious fluid temperatures is

minimal, injury here being controlled by tissue conduction rather than

convection at the surface.

5. CONCLUSIONS

Analyses of tissue heat transfer during scald events, when applied

in conjunction with an appropriate thermal injury criterion, provide

a means of extending the results of existing experimental studies, and

establishing critical exposure levels for different fluids.

One of the most important considerations in the heat transfer

analysis is the determination of the initial tissue temperature, which

in turn is dependent on tissue properties and pre-scald convective

conditions. For short exposure times, the initial temperature has a

major influence on tissue temperature excursions, and therefore strongly

affects the thermal injury threshold. With increasing exposure times,

however, the effect of the tissue temperature transient on thermal injury

diminishes, resulting in a reduction in sensitivity to variations in

tissue properties and initial conditions. For fluid exposure times greater

than 5 seconds, properties and initial conditions appear to have little

or no influence on the injury threshold.
18

Aside from fluid temperature, the convective heat transfer coefficient

for the scalding fluid is undoubtedly the most crucial factor in assessing

scald threat. Critical fluid temperatures can easily differ by a factor

of two when one considers the wide range of fluid properties and possible

flow conditions. The current approach for establishing critical exposure

levels is directly applicable to any fluid and flow of concern.
19

6.

REFERENCES

1. Moritz, A. R. , Henriques, F, C., Jr., Studies of Thermal

Injury: II, The Relative Importance of Time and Surface

Temperature in the Causation of Cutaneous Bums, Am. J. Path.

23:695, 1947.

2. Diack, A. W. , Schulty, R. D., et al., Technique for Quantifying

Low Temperature Bums, JSR, Vol. 4, No. 6, June 1964.

3. Stoll, A. M. , Greene, L. C., Relationship Between Pain and Tissue

Damage Due to Thermal Radiation, J. Appl. Physiol., Vol. 14, No. 3,

May 1959.

4. Stoll, A. M. , Chianta, M. A., Bum Production and Prevention

in Convective and Radiant Heat Transfer, Aerospace Medicine,

Vol. 39, No. 10, October 1968.

5. Stoll, A. M., Chianta, M. A., Method and Rating System for

Evaluation of Thermal Protection, Aerospace Medicine, Vol. 40,

No. 11, November 1969.

6. Weaver, J. A., Stoll, A. M., NADC Memo Report 6708, 1967,

7. Takata, A., Development of Criterion for Skin Bums, Aerospace

Medicine, Vol. 45, No. 6, June 1974.

8. Pennes, H. H, , Analysis of Tissue and Arterial Blood Temperatures in

the Resting Human Forearm, J. Appl. Physiol., Vol. 1, No. 2, August 1948.
20

9.

Wissler, E. H., A Mathematical Model of the Human Thermal System,

Bulletin of Mathematical Biophysics, Vol. 26, 1964.

10. Wissler, E.H., An Analysis of Factors Affecting Temperature Levels

in the Nude Human, In: Temp era ture-Its Measurement apd Control in

Science and Industry, J. D. Hardy, Ed., Vol. 3, Pt. 3, 1963.

11. Wissler, E. H., Steady-State Temperature Distribution in Man,

J. Appl. Physiol., Vol. 16, No. 4, 1961.

12. Shitzer, A., Chato, J. C., Analytical Solutions to the Problem of

Transient Heat Transfer in Living Tissue, Trans. ASME, Journal of

Biomechanical Engineering, Vol. 100, November 1978,

13. Chato, J. C., Shitzer, A., Analytical Prediction of the Heat

Transfer from a Blood Vessel Near the Skin Surface Cooled by a

Symmetrical Strip, Trans. ASME, Journal of Engineering for Industry,

February 1975.

14. Chato, J. C., and Shitzer, A., On the Dimensionless Parameters

Associated with Heat Transport within Living Tissue, Aerospace

Medicine, Vol. 41, No. 4, April 1970.

15. Shitzer, A., Chato, J. C. , Analytical Modeling of the Thermal Behavior

of Living Human Tissue, paper Cu 3.9 presented at the 4th International

Heat Transfer Conference, Paris-Versailles , 1970.

16. Chato, J. C., A Method for the Measurement of the Thermal Properties

of Biological Materials, Thermal Problems in Biotechnology, ASME

Symposium Series, 1968.
21

17. Rao, P. P., Francis, J. E. et al.. Comparison of Analyses and

Experiment in Transient Thermographic Studies, AIAA Paper

No. 75-712 presented at the AIAA 10th Thermophysics Conference,

Denver, Colorado, May 27-29, 1975.

18. Love, T. J., Francis, J. E. et al., A Comparison of Predicted Skin

Temperatures with Thermographic Measurements, presented at the

Fifth Symposium on Temperature, Washington, D.C., June 21-24, 1971.

19. Wulff, W. , The Energy Conservation Equation for Living Tissue, IEEE

Trans., Vol. BME-21, No. 6, November 1974.

20. Klinger, H. G., The Description of Heat Transfer in Biological

Tissue, Proceedings Conference on Thermal Characteristics of Tumors:

Applications in Detection and Treatment, New York Academy of Sciences,

March 14-16, 1979.

21. Chen, M. M., Holmes, K. R. , Microvascular Contributions in Tissue

Heat Transfer, Proceedings Conference on Thermal Characteristics of

Tumors: Applications in Detection and Treatment, New York Academy

of Sciences, March 14-16, 1979.

22. Henriques, F. C., Jr., Studies of Thermal Injury: V, The Predictability

and the Significance of Thermally Induced Rate Processes Leading to

Irreversible Injury, Arch. Path., 43:489, 1947.

23. Fugitt, C. E., A Rate Process of Thermal Injury, Armed Forces

Special Weapons Project, AFSWP-606, December 6, 1955.
22

24. Wu, Y. C., A Modified Criterion for Predicting Thermal Injury,

Report in Process, National Bureau of Standards, Washington, D.C.

25. Moussa, N. A., McGrath, J. J. et al.. Kinetics of Thermal Injury

in Cells, Trans. ASME, Journal of Biomechanical Engineering,

Vol. 99, August 1977.

26. Lawrence, J. C., Bull, J. P., Thermal Conditions Which Cause Skin

Burns, Engineering in Medicine, J. Mech. E. , 1976.

27. Rushmer, R. F., Buettner, J. K. et al.. The Skin, Science, Vol. 154,

No. 3747, October 21, 1966.

28. Jain, R. K. , Effect of Inhomogenetics and Finite Boundaries on

Temperature Distributions in a Perfused Medium, with Application to

Tumors, Trans. ASME, Journal of Biomechanical Engineering, Vol. 100,

November 1978.

29. Wu, Y. C., Control of Thermal Impact for Thermal Safety AIAA Journal,

Vol. 15, No. 5, May 1977.

30. Reay, D. A., Thiele, F. A. J., Heat Pipe Theory Applied to a

Biological System: Quantification of the Role of the "Resting" Eccrine

Sweat Gland in Thermoregulation, J. Theor. Biol., Vol. 64, 1977.

31. Richardson, P, D., Whitelaw, J. H., Transient Heat Transfer in Human

Skin, Journal of the Franklin Institute, Vol. 286, No. 3, September 1968,

32. Stoll, A. M., Heat Transfer in Biotechnology, in Advances in Heat

Transfer, Vol. 4, 1967.

33. Squire, H. B., Modern Developments in Fluid Dynamics 3rd Edition,

Vol. 2, Oxford: Clarendon Press, 1950.
23

34. Weinbaum, S., Jiji, L. M. , A Two Phase Theory for the Influence of

Circulation on the Heat Transfer in Surface Tissue, ASME Paper

No. 79/HT-72, presented at the Winter Annual Meeting, New York, NY,

December 2-7, 1979.

35. Perl. W., Heat and Matter Distribution in Body Tissues and the

Determination of Tissue Blood Flow by Local Clearance Methods,

J. Theor. Biol., Vol. 2, 1962.

36. Wu, Y. C., Rate of Cell Damage in Thermal Injury, NBS-CCPT Report

to the Consumer Product Safety Commission, Washington, D.C.
TABLE 1

24

<u

4-1

ffl OS

O 4-i r-» • »—

•H

O rH

to > pH

u o e •H 1 o >-• to X r co • • 4) 3 i rH o J= 4J-. <> rH H >4-1% <U

•pi

B to

Q • <c

•pt -X

01 CNJ >> iw u • o 1 *H o

Cde > 'w' CO

to 1 rH 1 4-1 4J HU • 1 CO co CM I c <u a 1 VO • • 1 <u 3 • o O co H c o1 a o CO S o co o • u o uo

>4-1

to

01 4-» •pt Cd CNI • Os

4-1 a) Cl • • u X 1 CO

01

O PO v>

a

a pH | CM o 1 vO • CM

<4-1

X 1 CO • PH <r U |t»2 1 I « i fH 0) t-j 00 CO o CO co

to a. |-X • I 8 U CO co CO

•H vO

a. 1o H rH c

0) o

•at".

X

PS I O CM • • I pH pH II

u~i o O pH o o o

U-l o iii iii

CO pH c o i H to o H e 1 S B s •pt rH 0)

a 3 3 c CO u CO O o o

4J a> a) *H 01 XI

X 3 pO •pt to c s •a o O >4-4 cd 4J 4J u u •H E JS 4J o O 14 u to to 4J O Oh <D 3 CO rH pH 0) a) 0) t-l w u W O W co M P- S X 0)

CM

ClI \0

CM» a) CO 3

t~t CO

CO

10

H

0)U

uc (1) 0) cH

a) CJ <H *4

a> o

ctf o- 6O o u <H

<H

CO c a)

<D 3

iH

cd cd H > I I I I MU'-'

Calculated

from

other

reported

thermal

properties

Most

representative

approximate

value
25

TABLE 2

Heat Transfer Coefficients for Various Fluids

Fluid

Fluid

Velocity

m/sec

Convective

Coef f icient

w/m^K

Water 0.3 4000

3.0 12500

Oil 0.3 800

3.0 2500

0.3 13

3.0 40

Air
INJURIOUS

FLUID

TEMPERATURE

°C

65.0

60.0

65.0

70.0

76.0

26

FIGURE 2. INFINITE SLAB MODEL OF SKIN TISSUE AND THE INITIAL TISSUE TEMPERATURE

DISTRIBUTION
INJURIOUS

FLUID

TEMPERATURE

C

TISSUE

TEMPERATURE

AT

0.01

CM

60.0

80.0

100.0

120.0

140.0

160.0

JO.C

35.0

40.0

45.0

50.0

55.0

60.0

65.0

27

9

S

LEGENO = 1 SECOND AT 70.0 C 0-5 SECONDS AT 60.2 C

31 nr—

33.6 H.0 TIME (SECONDS)

FIGURE 3. TISSUE TEMPERATURE PROFILES FOR TWO FLUID EXPOSURES

15.0
(INJURIOUS

FLUID

TEMPERATURE

°C

INJURIOUS

FLUID

TEMPERATURE

C

40.0

60.0

60.0

100.0

120.0

140.0

160.0

40.0

60.6

60.0

100.0

00.0

HO.O

28

FIGURE 6 . EFFECT OF BLOOD PERFUSION RATE ON THE INJURY THRESHOLD
29

FIGURE 7. EFFECT OF INITIAL TEMPERATURE PROFILE ON THE INJURY THRESHOLD

O
INITIAL

TISSUE

TEMPERATURE

°C

^

INJURIOUS

FLUlO

TEMPERATURE

C

20.0

25-0

30.0

35.0

40.0

q

40.0

60.0

60.0

100 0

120.0

140.0

30

3 i 1 -

LEGEND

0 LOWER LIMIT BASE CASE

6 LPPER LIMIT x MORITZ TEST OATA a LMCERTAINTr 1

%

1

t

- b

1 a

i

\

-

10

'1 icf itf FLUID EXPOSURE TIME (SECONDS)

9. THPESHOLD RANGE "OR EXTREME COMBINATIONS OF THERMAL PROPERTIES

LEGEND 0= LOWER LIMIT a- BASE CASE A= UPPER UMIT

\

\ •\

\ \

/

/

//

/

/

/ /

0.0 6.0 10.0 15.0 20.0 25.0 TISSUE DEPTH FROM SURFACE (CM)

FIGURE 10. INITIAL TEMPERATURE PROFILE FOR COMBINATIONS OF PROPERTIES
31

.FLUID EXPOSURE TIME (SECONDS)

FIGURE 11. INJURY THRESHOLD FOR EXPOSURE TO VARIOUS FLUIDS
32

Appendix A

Derivation of the Bio-heat Equation

The vascular architecture in living tissue is complex, with dramatic

variations in the number, size, spacing, and flow velocity associated

with blood vessels throughout the tissue. This complexity poses a

fundamental difficulty in the detailed modeling of the thermal interaction

between arteries, veins, and perfused tissue.

Although some detailed tissue heat transfer analyses have been

conducted [34], additional measurements are still needed to verify the

more complex thermal models. For the purpose of this study, a "crude"

approximation to the tissue heat transfer problem was therefore utilized.

The approximation, to be derived here, combines a local description of

- the tissue temperature gradients with a global description of the thermal

convective effects of the moving blood, which treats the blood phase as

a volumetric isotropic heat source.

The temperature field in a continuous, homogeneous medium with

constant thermal conductivity can be described by the well-known Fourier

heat conduction equation

pc = k V

2T + q . (Al)

p dt
33

If this equation is to apply to a material such as skin tissue,

the thermal properties would become "effective" properties, incorporating

all the physiological phenomena peculiar to tissue, including blood

perfusion and metabolic heat generation. Rather than treat tissue heat

transfer using this "lumped" approach, however. Equation A1 can be

modified to separately account for these phenomena. To do this,

additional terms must be introduced. These terms are developed through

consideration of a small volume of tissue, AV, perfused by blood, and

application of Fick's Principle.

The vascular network in a tissue is idealized into a system of

circulatory loops, as shown in figure Al. Each loop consists of three

functional components denoted arteriole, capillary, and venule (A, C, and V

in figure Al respectively). The capillary is that portion of the loop in

which all exchange (of a substance or heat) between the flowing blood and

the surrounding tissue is assumed to take place. The arteriole is the

portion of the loop in which blood flows toward the capillary and,

conversely, the venule is the portion in which blood flows away from the

capillary. It is assumed that no exchange between blood and tissue takes

place in these two portions. Although strictly speaking, heat transfer

in the arteriole and venule are not negligible, the diameter of a capillary

is so minute relative to its length (5 x 10 ^ cm versus 0.1 cm) that as

a heat exchanger its effectiveness is near unity; it is therefore credited

for all heat transfer.

The substance of interest, heat in our case, can enter and leave the

volume element by perfusion in either of two ways: (a) entry through
34

arterioles terminating within the volume and exit through venules

starting from within the volume element and (b) entry or exit through

capillaries intersecting the surface of the volume element. Vessels

of Type (a) are those in which the capillary is enclosed within the

element, such as I, II, and IV of figure Al. Type (b) vessels comprise

the remainder (III, V, VI, and VII) and are referred to as surface

capillaries. The heat transfer contributions of both vessel types

are different and must be dealt with separately.

Type (a) Vessels

The contribution of Type (a) vessels can be derived via Fick’s

Principle of Perfusion for an Organ. For the element AV, Fick's

equation may be written:

9 AC . dt J p(a)

<p Ca a

<J> C v v

(A2)

where AC is the concentration of substance in the volume element in

3 -1

kg/m ,

cf> is the blood perfusion rate in sec , and C is the concentration

3

of substance in the blood, kg/m • The subscripts a and v denote the arterial

and venous conditions respectively. For the substance of interest, heat,

concentration is equivalent to thermal energy per unit volume, or the

product of density, specific heat, and temperature. Hence, for heat flow.

Equation A2 becomes

pc -I7

-) = p c , (cj> T - <f> T ) p dt f v b pb a a v v

P (a)

(A3)
35

In the absence of blood accumulation or depletion in AV, <(> a = <J> v

= <J>.

Also, based on the existing idea that exchange in a capillary occurs

predominantly in its initial portion, the venous temperature can be

assumed equal to the local tissue temperature, that is, Tv

= T.

3 Incorporating these approximations and letting = p^, kg/m sec,

Equation A4 can finally be written

pc

9T\

8t ; p(a)

= w, c , b pb

(T - T) a

(A4)

Fick's perfusion equation comprises only contributions from

Type (a) vessel loops in which the capillaries are contained entirely

within the volume element. Type (b) vessel loops have capillaries which

intersect the surface of the element. The contribution of these vessels

to the rate of change in thermal energy of the element is in addition

to that given by Equation A4.

Type (b) Vessels

To derive the effect of surface capillaries, consider the flow of

blood in the capillaries intersecting the right hand AY AZ face of AV

(figure Al) . Letting j x+

denote the flux density of blood in the

positive x direction (path III) and j^_ denote that in the negative x

direction (path VI) , the net rate of inflow of substance across the

face is

(jJ C. - j . C,,) AY AZ x- b- J xrf b+

(A5)
36

where C^_ and C^

+

denote the average concentrations of substance in

the respective vessels. In thermal energy notation, concentration

is again equivalent to the product of density, specific heat, and

temperature. Equation A5 then takes the form

Since the heat exchange in a capillary occurs predominantly in its

initial portion, it is in near-equilibrium with its surroundings along

most of its length. T^_ and T^

+

can, therefore, be assumed equal to the

local tissue temperature T. Letting j = j -j » the net rate of heat

X Xt x inflow through the right AY AZ face is

The difference in flux between the left and right hand AY AZ faces may

be expressed as the product of the partial of Equation A7 with respect

to x and the length of the element in the x direction, AX. The result

is

(jJ p, c , T, - j , Pu e , Tu_i) AY AZ . x— b pb b- J x+ K b pb b+ (A6)

-p, c . (j T) AY AZ . b pb J x(A7)

3 ( j T)

-p, c , — AX AY AZ . b pb dx

(A8)

Similar terms can be derived for flows through the AX AZ and AX AY

faces (y and z flow directions respectively). The sum of these terms

represents the rate of heat gain of the element AV due to all surface
37

capillaries. This contribution may be written

9t 9(3 T) 3(jT) 3(j T)

pC

p Jt* , . “ ~Pb

C

pb ( 3^

+

3y

+

9z ^ P (b)

(A9)

= "

Pb

C

pb V(jT) ‘

Collecting terms on the left hand sides of Equations Al, A4, and A9

yields the complete partial differential equation for tissue heat transfer:

3T 7 pc

p “ kV T - pb

cpb V(JT) + w

b

cpb ( VT) + < • (A10)

That the effect of the surface capillary term is small can be shown by

considering the one-dimensional steady state case with zero metabolic

heat generation. Letting 9 = T-T

a and X

= k/p^ Cp^> Equation A10 becomes:

d 9 . d9 n x 72" s

- 4,9 = 0 • dx

(All)

The solution to Equation All subject to the boundary conditions

9 (x = 0) = 9

9 (x = 00 ) = real
38

is

-r

+ 4x * - j x

9 = 0 e

s

2X

(A12)

The maximum possible j exists when all the capillaries are uniformly

aligned and the blood flows in the x direction, as in vessels IV, V, and VI

of figure Al. In this case, the density vector j is

j x

Z(p (A13)

where Z is a vector of the magnitude of the average capillary length

and in the direction of the capillary net blood flow [35].

The relative insignificance of the surface capillary effect can

be seen by introducing this expression into Equation A12 and comparing

the magnitudes of the two terms under the square root sign. The ratio of

the first term to the second is

Z

2

<t> 4X

(A14)

Assuming approximate values for these parameters [35]:

Z = 0.001 m,

(f> = 0.01 sec \ and

—7 2

X - 1.2 x 10 m /sec
39

2 gives £ <f>/4x = 0.02. It thus appears that the surface capillary effect

is insignificant. This same conclusion can be arrived at by assuming

that the capillaries are sufficiently random in arrangement and direction

that they exhibit no preferred direction.

Neglecting the contribution of surface capillaries. Equation A10

reduces to its final form:

where a is the thermal diffusivity of tissue. This is the so-called

bio-heat equation first applied by Pennes [8] to determine the temperature

distribution in the human forearm.

St 9 pcpot

^ = kV T + w,bpba c , (T -T) + qn (A15)

or alternatively:

(A16)
40

FIGURE AT. VOLUME ELEMENT OF TISSUE INTERSECTED BY VARIOUS

PATHS OF CAPILLARY BLOOD FLOW [35]
41

Appendix B

Finite Difference Formulation for Tissue

Heat Transfer During Scalding

The scald problem requires that the t ime-temperature history of

the skin tissue at a depth of approximately 0.01 cm be known over the

complete scald event — heating and cooling — so that the severity

of scalding can be assessed via a thermal injury criterion. Because

the tissue is subject to an initial temperature distribution as well as

to different sets of convective boundary conditions for the heating

and cooling periods, a finite difference solution is the most expedient

means of obtaining the required temperature profile.

The finite difference formulation used in this study is an implicit

type. An energy balance for each selected node in the tissue is written.

The resulting set of equations for the "future" temperature of each node

is then assembled in matrix form and solved simultaneously via matrix

techniques.

Governing Equation

The governing partial differential equation for tissue heat transfer

is the bio-heat equation, described in detail in Appendix A. This equation

is the Fourier heat conduction equation with an additional term,

-w^c^^(T-T

a

) , to account for heat transport by perfusing blood. Here w^

and c ^ refer to the perfusion rate and specific heat of blood, while

T and T^ are the local tissue temperature and fixed arterial temperature

respectively.
42

For this analysis, one-dimensional heat flow is assumed. Also,

all thermal properties throughout the tissue, including blood flow

parameters, are considered constant.

Boundary conditions for the problem are: convection at the tissue

surface, and a constant arterial temperature at some finite depth below

the surface. The initial condition is the tissue temperature distribution

defined by Equation 18 in the text.

Node Configuration

Thermal injury criterion requires that temperature at a depth of

0.01 cm be known, but we also want the temperature at the deepest node

to remain constant during the heating and cooling event. Division of the

tissue into two regions facilitates this. In the first region. Region A

of figure Bl, nodes are closely spaced such that one node may lie at a

depth of 0.01 cm. In the second region, Region B, a sufficiently large

depth, DB, may be selected such that the temperature of the last node, BL,

remains constant throughout the scald event. The node spacing in

Regions A and B, AX^ and AX^ respectively, may be chosen to minimize the

number of nodes required.

Node Equations

An energy balance must be written for each node in the tissue. For

the nth node, the general form of the energy balance or node equation is:
43

t+1 t+1 t+1 t+1

t+1

(Tn

- T ) C = At [ n n

T _ - T n-1 n

R i n-1 ,n

t+1

T - T

n+1 n R ,

' n,n+l

(Bl)

-w

b

cpb

AV

n

(Tn

* V + t

AV

where the thermal capacitance G = pc^AAX. Thermal resistance, R, for

convection between the surface node and ambient air is given by

R = 1/hA (B2)

while for conduction between the surface node and all interior nodes it

is given by

R = AX/kA . (B3)

The temperature at most tissue nodes may be expressed using the same

form of Equation Bl; however, different equations are needed for the

surface node and the two nodes at the Region A-Region B interface. In

all, a total of five different forms of Equation Bl are necessary. Each

form applies to similar types of nodes; these are:

1) surface node

2) all interior nodes in Region A

3) last node in Region A

4) first node in Region B

5) all interior nodes in Region B .
44

The temperature of the last node in Region B (Node BL) is a specified

constant temperature, namely, the arterial temperature.

The particular equations for the future or updated temperature

for each of the node types are as follows:

Surface Node (Node 1)

t+1 v. AX. t+1 0

..

r , . 2At . k , A., , _ r 2aAt,

T

1 ^ +

pc AX

+

AX

+ W

b

C

b 2 + T2 ^ 2

1 p A A AX

a

T + — (w c, T + q) + T

pc AX t oo pc b b a 1

P A p

(B4)

All interior Nodes in Region A (Nodes A)

t+1 A. t+1

aAt aAt 1+T

A

[1 + 2 ffr +

pc

w

b

C

b^^‘

AX, ax

a

P

t+1

+ T T- A+l L aAt , At

rl =

AX,

(w, c,T + q) + T. . pc b b a A

(B5)

i
45

Last Node in Region A (Node AL)

t+1

. _a4t_.

t+1 „ ttAt 2aAt

2

] AL 2 AX

a

(AX

a+

AX

b

) pc

A A

(B6)

+ T [- — ] = — (w,c, T + q) + T at . Bl AX. (AX.+AX^) pc b b a AL

A A B p

First Node in Region B (Node Bl)

[- aV + T^V + w - A°-Al ?7V v +

al 1 ax

b

(ax

a

+ax

b

)

j

' X B1 ’ ax

b

(ax

a

+ax

b

) AX 2 pc

B

(B7)

t+1 a . + T r_ = -Ai.

B1+1

ax

2 pc

p D

KVa + 1) + TB1

All Interior Nodes in Region B (Nodes B)

t+1

t

b-i l- aAt

t+1

] + T [1 + 2

AX,

» . w,c. At

aAt , b b - 2 B ax

b

2 -P

t+1

+ T [

B+l 1 aAt , At

AV

1 ft KVa + + T

i

(B8)
46

For the last node in Region B (Node BL) select depth sufficiently

deep that = T

&

. Rearranging and replacing parameter groups with

constants. Equations B4 and B8 may be expressed:

t+1 t+1 t T 1 + Cl T 0 = C2 T, + C3 T + C4 (B9)

1 2 1 oo

,

t+1 t+1 t+1 t C5 T

a_ x + T

a + C5 T

a+1 = C6 T

a

+ C7 (BIO)

t+1 t+1 t+1 t c8tal-i

+ tal +c9tbi =cio Tal + cii (BID

t+1 t+1 t+1 t

C12 Tal + T

B1 + C13 Tb1+1 = C14 Tb1 + C15 (B12)

t+1 t+1 t+1 t C16 T+ T + C16 T = C17 + C18 . (B13)

d—Jl Jd dt! Jd

Equations B9 through B13, along with the requirement that T = T , may

15 Li cl now be assembled in matrix form

AX = b (B14)

where A is a tri-diagonal matrix of C coefficients, X is the vector of

updated temperatures (time t+1) , and b is the vector of known temperature

(time t) and constants. The assembled matrices are shown in figure B2.

The X vector can be solved by inverting the A matrix once and multiplying

by b each time step or by use of other matrix techniques.
i! n j

l

SURFACE

REGION A

(DA CTD, )

REGION B

(DB cm )

axa

I AL ! B1 • •

AXa axb

2 2

B

axb

BL

CORE

FIGURE Bl. TISSUE NODE CONFIGURATION
FIGURE

B2.

ASSEMBLED

MATRICES

FOR

IMPLICIT

FINITE

DIFFERENCE

SOLUTION

48

IT 1 <_> +

, LO 00 00 00 r— p— r— p—

ro ix rx rx O o o o o

<_> C_> l_> o

+ + + + + + + + +

-M LO +-> LO -M IX +-> 00 4-> CTt

4-> 4-> (XI +-> ro 4-> 1— 1— i— 1— l—

1— 1— 1— i— o IX IX

C\J LO LO LO 1

1 1 1 r— ro o

1 O O O o (_> (_> o o i—

i

II

1 r—

f— r— P— P— f— C” r— r— + + 4“ 4- + + 4* + 4- + o +-> 5= 4-> CM +-> ro +J xT -M LO +-> LO +J IX +-> C0 4-> CTl M r—

i

1— 1

1

h- 1— 1— 1— 1— 1— i—

1

• o LO

P—

C_> o LO o r—o

o LO f— LO r— r— o o

O ro r— LO o

r— r— o C_J O CTl r— LO o O r—

c_> O LO r— CM o

C_> r~o

o LO CO o

o o

O LO LO o

o O

LO O

O <_>

LO o

! O

1
49

Appendix C

Development of the Thermal Injury Criterion

The living cells in the skin are composed primarily of proteins

These proteins possess disulfide linkages, salt bridges, hydrogen

bonds, or combinations of these to form the structural network [36].

All of these chemical bonds are subject to breakage via exposure to

thermal energy in excess of the bond strength. When breakage occurs,

the protein is inactivated or destroyed. It is recognized that cellular

death follows the thermal inactivation of a critical fraction of the

protein molecules in the cells of the skin. The point at which tissue

damage is observed on a macroscopic scale is not distinct, however, because

skin proteins vary in their sensitivity to heat. For example, collagen

is not appreciably degraded at temperatures below 71 °C, whereas certain

enzyme systems are sensitive to much lower temperatures [26]. The

situation is further complicated by the fact that skin cells adjacent to

the obviously heat coagulated tissue observed in many burns may not

become completely necrotic until several days after injury [26],

Thermal Injury Model

It has been postulated by Henriques [22] and others that the

reaction leading to the death of tissue cells conforms to that of most

chemical and physical rate processes. The kinetics of a rate process

depends on the total energy content of the constituents involved. If

this energy content is less than a certain critical value, known as the
50

I activation energy, the process cannot take place. If, on the other hand,

the energy content is greater than this critical value, the process

takes place and proceeds at a rate proportional to the fraction of

the constituents which possess an energy at least equal to the

activation energy. This fraction is given by a form of the

Maxwell-Bolt zman energy distribution law:

. : >

t n—E = exp [- E/RT] . (Cl)

The rate at which the reaction proceeds can be described as:

dn

dt

Kn , (C2)

In addition to the requirement that energy content be greater than

activation energy, steric or spatial requirements for the molecular

reaction must also be met before the reaction proceeds. Letting P equal

the number of the necessary steric events occurring in one second , and

the factor exp [- E/RT] the probability that the particular molecules

affected by the steric events also possess the required reaction energy,

we may write the reaction rate as the product, or:

= - Kn = P exp [- E/RT] . (C3)
51

The number of molecules which will react in a given time is obtained

by integrating Equation C3 with the result:

t 2 n = / P exp [- E/RT] dt . (C4)

4

Knowing the time-temperature history of skin tissue resulting from a

scald event , the number of damaged cells and hence the severity of

thermal injury could be assessed via Equation C4. But a detailed

knowledge of each of the constituents involved in the reaction, including

activation energies, rate constants, etc., would first be necessary in

order to evaluate P and E.

In view of the difficulty in specifying the number of cells or

even the molecular species involved in the injury process, a different

approach is generally taken in applying the rate process concept of

Equation C4. The approach is as follows.

If the reaction leading to thermal injury does indeed conform

to that of most chemical and physical rate processes , the rate at which

damage is inflicted should be given by an expression of the same form

as Equation C3. Replacing n in Equation C3 by the parameter ft, an

arbitrary measure of "damage," the damage rate is

dft dt

= P exp [- E/RT

t

]

. (C5)
52

Integration of Equation C5 gives the total "damage" inflicted over an

interval

t

2 n - /

P exp [- E/RT ] dt . (C6)

h

Defining = 1 at the level of inflicted damage just sufficient to destroy

cells in the basal layer of the epidermis (partial thickness burn), P

and E are then evaluated numerically from experimental observations.

Determination of Empirical Constants

Several researchers have investigated the relationship between the

time-temperature history of skin tissue and resulting thermal injury.

For convective heat transfer, the original and most widely noted work

is that of Moritz et al. [1], This work involved the study of the

relationship between times of exposure to flowing hot water at various

temperatures and the occurrence of thermal injury. Their experimental

results are presented graphically in figure 1 of the text. Each point

along this line represents a different set of scald conditions; however,

the result of exposure to these conditions is the same, that is, destruction

of cells in the basal layer of the skin or £2 = 1. It should be noted

that this curve is valid only for fluid flows similar to that used by Moritz.

In determining the empirical constants P and E, Henriques [22] assumed

tissue temperature, T , time-invariant and integrated Equation C6 from

zero to the end of the heating period (neglecting the cooldown portion of

the scald event) . Through these assumptions and the selection of a constant

value of 1.0 for the scald threshold. Equation C6 becomes:
53

£n t = E/R • 1/T - £n P . (C7)

A plot of Equation C7 for the test data of Moritz showed a linear

98 relationship below T = 55 °C from which the constants P = 3.1 x 10

and E = 150000 were determined. The result is a good fit of theory

and experimental data for times greater than five minutes, but

increasing error for short times typical of scalding (less than five

seconds)

Evaluation of Henriques' criterions over the cooling as well as the

heating portion of the scald event does not offer much better agreement.

Selecting five different exposure conditions from the original data set of

Moritz, table Cl, the corresponding time-temperature history of skin

tissue can be generated using the solution technique described in the text.

Evaluating Henriques' criterion over these profiles results in the

9, values given in table C2. These values indicate that Henriques' injury

criterion even when evaluated over the entire heating and cooling profile

is still lacking for short time exposures. Table C2 values also show that a

considerable amount of thermal injury inflicted during a scald event occurs

in the cooldown period — 25% for one second exposures.

Fugitt [23] attempted to resolve the discrepancy between theory and

data for short times through an extension of Henriques' criterion. He

proposed a two-component thermal degradation process and fitted the original

test data with a second line for T greater than 55 °C. This yielded values

45

for P and E of 5 x 10 and 70,800 respectively for T

fc greater than 55 °C.

Unfortunately, this analysis also neglects the cooling portion and does not

offer much improvement over the original criterion.
54

Research similar to that or Moritz and Henriques was also conducted

for thermal radiation injury [3-7]. Criteria of the same form as

Henriques' was used to describe this type of thermal insult. Weaver et al. [6]

used the damage rates:

2.185 x 10

124

exp [-93534. 9/T

fc ] for 44<T

t <50 °C

and

1.823 x 10

51

exp [-39109 . 8/T ] for T > 50 °C .

Takata [7] found that the damage rates which best fit his test data were:

4.322 x 1064 exP [-50,000/Tt

l for 44 < T

fc < 50 °C

and

9.389 x 10

104

exp [-80,000/T

t

] for T

fc ^50 °C .

Again, as shown in table C2, agreement between criteria and experiment is

not good.

Wu [24] most recently utilized an averaging technique to obtain a

tissue temperature-dependent expression for activation energy E. The

result

E = 150000 - 122(T

t - 53) for T

t > 53.0 °C
55

gives good agreement between the injury criterion and the original test

data (see table C2) . This modification was adopted for use in the

present study. The complete thermal injury criterion is then:

t

2

ft = P • / exp [-E/RT ] dt (C8)

t l

where

ft = the damage integral, 1.0 for partial thickness burn,

P = a constant, 3.1 x 10^ sec \

E = activation energy, cal/mol,

= 150,000 for T 53.0 °C,

= 150,000 - 122 (T - 53.0) for T > 53.0 °C,

R = gas constant, 2 cal/K mol,

T^ = tissue temperature at a depth of 0.008 cm, K,

t^ = time at which T

fc = 44.0 °C during heating, sec, and

t 2 = time at which T

fc = 44.0 °C during cooling, sec.

Note that Henriques used a tissue depth, x, of 0.008 cm (the thickness of

2 the epidermis) and thermal diffusivity, a, of 0.00071 cm /sec, then

worked with the quantity x/2 /oT = 0.15. Here we have let x = 0.01 cm

2 and a = 0.001 cm /sec. The corresponding quantity x/2 /oT is only slightly

different from Henriques '
56

TABLE Cl

Exposure

Time

(sec)

Fluid

Temperature

(°C)

Corrected Fluid

Temperature* ,

f

(°C)

1 70 70.0

2 65 65.0

5 60 60.2

10 58 57.5

15 56 56.3

Based on the non-linear best fit:

-0.3928

T

f = 49.08 + 20.91 x t
57

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NBS-1 14A REV. 2 -8C ' U.s. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR 81-2320

2. Performing Organ. Report No. 3. Publication Date

July 1981

4. TITLE AND SUBTITLE

A Heat Transfer Analysis of Scald Injury

5. AUTHOR(S)

Robert L. Palla, Jr.

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8 . Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP

10. SUPPLEMENTARY NOTES

j Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bl iography or literature survey, mention it here)

Numerical solutions for skin tissue temperature during scald injury events are obtained and utilized in conjunction with a thermal injury criterion, to predict

critical exposure levels for various heated fluids. A one-dimensional tissue

model of the type used by Love is employed to determine the initial tissue

temperature distribution. The bio-heat equation for tissue heat transfer is

then solved via an implicit finite difference technique, subject to convective

heating and cooling at the surface.

The sensitivity of the critical exposure level to variations in tissue properties

and convective heating coefficients is investigated. Thermal injury thresholds

are presented for various fluids along with bounds to reflect uncertainty in

assumed tissue properties. The results obtained are in good agreement with existing

experimental scald injury data.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Bioheat equation; burn; criterion; critical exposure; fluid; heat transfer; injury

threshold; scald; thermal injury

13. AVAILABILITY

| X| Unlimited

| | For Official Distribution, Do Not Release to NTIS

f | Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

[__) Order From National Technical Information Service (NTIS), Springfield, ’'A. 22161

14. NO. OF

PRINTED PAGES

65

15. Price

$ 8.00

U S COMM- D C 6043-P80