NATIONAL BUREAU OF STANDARDS REPORT

2720

SURFACE TEMPERATURES

WITH

THERMAL INDICATORS

by

S. M. Genensky

A. F. Robertson

Report to

Quartermaster Corps

Philadelphia Research & Development Laboratories

U, S. Army

U. S. DEPARTMENT OF COMMERCE

NATIONAL DUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Sinclair Weeks, Secretary

NATIONAL BUREAU OF STANDARDS

A, V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards is suggested in the following listing of

the divisions and sections engaged in technical work. In general, each section is engaged in specialized research, development, and engineering in the field indicated by its title. A brief description

of the activities, and of the resultant reports and publications, appears on the inside of the back

cover of this report.

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Department of Defense. • Office of Basic Instrumentation • Office of Weights and Measures.
NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1000-10-4803 August 3, 1953 2720

SURFACE TEMPERATURES

WITH

THERMAL INDICATORS

by

S. M. Genensky

and

A. F. Robertson

Fire Protection Section

Building Technology Division

To

Quartermaster Corps

Philadelphia Research & Development Laboratories

U. S. Army

Requisition No. 49-106-567-53

Project No. 7-12-01-001

The publication, reprii

unlesi permission Is ob 25. D. C. Such pertr

prepared IT that age

Approved for public release by the '* p"’’’'*’'*®'* Director of the National Institute of

Standards and Technology (NIST)

on October 9, 2015.

SURFACE TEMPERATURES

WITH

THERMAL INDICATORS

by

S. M. Genensky and A. F. Robertson

Abstract

At the request of the Quartermaster Corps,

analytical and numerical solutions have

been obtained for the surface temperature

rise of homogeneous, semi-infinite solids

and similar solids with one or two finite

lamina of other materials on the exposed

surface when heated by a time invariant

radiant energy source. By the use of these

two methods of calculation it has been possible to estimate the surface temperature

rise for skin and inert assemblies intended

to simulate skin when these are heated by

radiant energy which is totally absorbed at

the surface. It is shown that, as a result

of their thermal properties, the temperature

rise of the Quartermaster temperature indicators is very different from that to be

expected of skin under similar conditions.

1, INTRODUCTION

The Quartermaster Research and Development Laboratories

have developed thermal indicators and requested that an analysis be made of the extent to which these indicators with

various types of backing materials may be considered as simulating temperature changes on the surface of opaque skin when

exposed to thermal radiation.

2

This report presents some results of analytical and

numerical calculations which may be of assistance in evaluating the usefulness of these indicators. The calculations

have been made with the objective of estimating the maximum

surface temperature rise of homogeneous or laminated materials which have been exposed to a time invariant pulse of

radiant energy of one-half second duration. Calculations

have been performed with the use of thermal data furnished

by the Quartermaster Corps on various assemblies proposed

as representing:

a. Human skin

b. Homogeneous skin simulants

c. Quartermaster temperature

indicator assemblies

Most of the calculations have been made by means of an

analytical solution of the problem for the case of a finite

lamina in good thermal contact with a semi-infinite backing.

A few additional numerical solutions have been obtained by

means of SEAC, the National Bureau of Standards digital computer, for the more complex assembly of two finite lamina in

contact with a semi-infinite backing.

Because of limited funds available for the work, no

attempt has been made to obtain solutions for all the assemblies proposed. The solutions for the one, two, and three

body composite systems considered have been obtained for a

wider range of thicknesses and thermal properties than was

requested, in an effort to make the work more general.

2. THE PROBLEM AND WORK DONE

The work requested (1) was very broad. It included the

following "jobs":

1. Investigate the effect of fabric covering on the

temperature rise of skin or indicator and backing

exposed beneath the fabric.

2. Calculation of surface temperature rise of skin,

using the assemblies and thermal properties furnished.

(1) Philadelphia Quartermaster Research and Development Laboratories letter QMDP 4-23.112 RP , 30 March 1953? signed

by John M. Davies.

- 3 -

3. Calculation of the surface temperature rise of semiinfinite materials which have been proposed as skin

simulants.

Calculation of the surface temperature rise of a

variety of temperature indicator assemblies with

several different backing materials.

It was suggested that items 2 to ^ be calculated for an energy

pulse of one-half second duration. An investigation of item 1

was considered too complex in view of the funds available.

Work was, therefore, confined to items 2, 3 5 and 4 .

The thermal properties used in performing the calculations

are shown in Table 1. Most of these were obtained from the

Quartermaster Corps. (1)

Table 2 presents the eight different skin assemblies for

which calculations were requested (Job 2). The manner in which

the suggested assemblies were modified in solving for temperature rise by both analytical and SEAC calculations is indicated.

Table 3 presents the thermal properties of four finite

simple slab materials and one semi-infinite composite assembly

which have been proposed as simulating behavior of skin (Job 3 )«

For the finite specimens a number of thicknesses were proposed

for solution. However, for the half-second pulse used, all of

these can be considered as semi-infinite at the time when surface

temperatures are a maximum and thus one solution serves for all

thicknesses requested.

Table 4 presents the fifteen different indicator mounting

and assembly arrangements for which calculations were requested

(Job 4 ). As a result of discussions with Quartermaster officials,

it was decided to concentrate work on solution of indicators

having type C assembly. This table shows the manner in which

the six-layer system was simplified. The fact that the thermal

properties of the thin surface indicator layer were different

from those of the backing was neglected and a constant value of

k was assumed for the next four layers; the semi-infinite backing representing the sixth layer completes the system.
:--V > U > f

:\Ci rf.' /v * r, ;i .

• t

"i /1

,:,v

! / tffl

• i-!-. '^>?A/>•:

L.

I

“>

.

''V
- 4 - 3.

ANALYSIS AND RESULTS

As mentioned earlier, Loth an analytical solution of a

two material composite semi-infinite body, and a numerical

solution for a three material semi-infinite slab were obtained. The latter made use of SEAC. All solutions were

developed with the following assumptions:

1. The input energy pulse is zero at time zero and

equal to ^ between time zero and one-half second,

after which it is again zero.

2. The materials are opaque.

3 . The thermal properties of the bodies being heated

are constants.

4 . There are no losses from the heated body to the

surroundings and all of the incident energy is

absorbed.

5. At all solid interfaces both temperature and heat

flux are continuous functions of distance.

6. The bodies being heated are inert.

It was agreed that computations should be made for the

maximum temperature rise at the surface of the specimen.

This would occur at the end of the radiant exposure or onehalf second. It was hoped that an analytical solution might

be found or developed for the multi-layer specimens. Such a

solution was not found and appeared impractical to develop for

assemblies more complex than those of two materials.

The analytical solution for a finite surface layer on a

semi-infinite solid was developed as follows. Consider a semiinfinite solid having a surface layer of finite thickness B

in intimate contact with the remainder of the solid. The temperature within the solid is considered a function of depth x

and time t and may be written @(x,t). Initially the solid and

surrounding medium are assumed to be at temperature •©•(x,0) =

0 (-oo<x<oo). Further, the exposed face of the solid is subjected to a constant flux |) from time t>0 to t=T. the termination

of the experiment, and the temperature ^(x,t) as x tends toward

infinity is zero for all time t. The exposed solid has thermal

conductivity k]_ and thermal diffusivity ^ ^ and the backing or

- 5 -

underlying layer has thermal conductivity kp and thermal

diffusivityX 2* solid interface both the temperature and conductive flux are assumed continuous. Mathematically the problem may be expressed as follows:

39(x, t) e(x,t)

dx

(0<x<B) (0<t:^T)

-k-,59(0,t)

^ X

= f) (0« t;$T)

lim 9(B - x,t)

x-^o

= lim 0 (B + X, t

)

X ->o

( X > 0 , 0 $ t^ T )

lim X^o

. , d©(B+x,t) ^ lim ko-— 2 X-^ 0

^ dx (x>0, 0^ t<^T)

^©(x, t ) ^t

<y s(x,t)

^ 5x2

(B<x<oo) (0<t^T)

lim 0 (x, t

)

x->oo

-4* 0 for all t

^(x,0) = 0 ( - 00< X<00 )

and the solution of this problem, which of course assumes

linear heat flov/, may be expressed as follows:
, J -p- i

fr«-
- 6 -

0(x,t)

$

2B(n+l)-x

.

2

2B(n+l)-x . (2B{n+l)-x\ — erfcf1 2

r ^ ^ 2B(n+l)+Xf

+ 2 e L t a

2B(n+l)+x / 2B ( n+1 ) +x

J Spa '

>= J

P 2

” X

_ X

»r

(1)

for <0$x^B)

- 7 -

and

00

(j) 0 '5

"

^lir 2 E ®

V

(2n+l)B , x-B

”h V 1

2

^

—(2n+l)B — x-B _ , r— + 7=_erfc

s/^ *“ J^2

(2n+3)B x-B~1^

( 2n+l)B ^ x-B

\/^ . j 2fT ^ *# 2./T

“ “ (2n+3)B x-B

effc

1 2

(

(2n+3)B ^ x-B

j'El

\

2ft

B x-B

2

A

k

1

B ^ x-B

y^T 2 -

B x-B

erfc 1 2 I 2/r

for (E<x<oo)

8

where 6(x,t) is the temperature, a function of position x and

time t.

This equation served as the basis upon which all analyses involving hand and desk calculator computations were carried out.

Since this temperature rise is directly proportional to |), a more general solution is obtained by expressing the results of

calculations in terms of 9/(|) which for brevity we call "temperature rise". These computations were restricted to ascertaining

this surface temperature rise of various homogeneous and composite solids exposed to the time invariant energy pulse for onehalf second. It is to be emphasized that since the analytic

approach was confined to semi-infinite solids composed of at

most one finite layer in intimate contact with the rest of the

solid, the computations can at best be regarded as approximations

to the structures offered for analysis.

Further

s

= and \

= m/q

where

Q

and

k-| fjL o

For an energy pulse of one-half second, the maximum temperature rise occurs when x=0 and t=0.5» Substituting these values

in equation (1) gives;

- 9

-

e(o,o.5) _

2

(n+i)B \/ 2

00

^

J

erfo B(n+l)\f2 ^

Figures 1 to 7 and Table 5 present the results of the

calculations. An attempt has been made to furnish generalized

data by study of a wider range of variables than was requested

for the specific problems involved. All these figures present

the maximum surface temperature rise one-half second after exposure to a constant intensity radiant energy input pulse of

|) cal/sec cm^. In these figures the symbol^ is used to designate the thickness in inches of each of the individual layers.

Figures 1 through 4 and table 5 are for two material

composite bodies or for homogeneous materials and represent

the results of the analytical solution of the problem.

Figures 5 through 7 present results of the SEAC numerical

solution of the problem for three material composite bodies.

Data are presented for both surface and two subsurface interfaces as a function of time for the duration of the one-half

second exposure.

It should be remembered that most solutions presented

involve some simplifying assumptions and in all cases where the

indicator assembly has been studied the presence of the thin,

translucent indicator surface film has been completely neglected. This is justified because of its very small thickness and

the fact that it cannot be considered as opaque.

10

Table 5 gives surface temperature rise of various homogeneous solids. These calculations were based upon values

from Table 3*

Figure 4 is a plot of surface temperature rise of

various composite solids with the same surface layer and

various high conducting backings. The surface layer ^s

10x10 ^ inches thick and has a conductivity kp= 3xl0~^cal/cm

°C sec. The figure also indicates the surface temperature

rise of a homogeneous solid made of the surface material, and

contains a band which indicates the probable range in which

the surface temperature of human skin might fall.

4. DISCUSSION OF RESULTS

Figure 1 presents the results of the analytical solution

for simplified skin and indicator assemblies. It shows that

the indicator assemblies cannot be expected to demonstrate the

same thermal behavior as skin even when the conductivity of

the backing is varied over wide limits.

Figure 2 curve B shows the surprisingly constant temperature rise of the surface of a composite body when the conductivity of the 3x10~3 in. thick surface layer is varied over wide

limits. > The kp c of the backing is maintained constant at

17»5x10” cal^/cm °C^sec. Data presented in table 5 indicate

that the probable temperature rise of skin §. is likely to be

or 2

^ between the values of 20 and 35 ^ These temperature

cal

limits correspond to kpc products of about 5t and l6 cal^/cm^°C^

sec for homogeneous materials.

The importance of the k^c product may be appreciated when

it is considered that:

= 2 BT (txT)

k \J 'W ^

is the solution of the analytic problem presented earlier when

X = 0 , kj_ = k

2

, and ^

p

“

2

'
,T .

m

',11 I
11

This may be written as:

e(Q,t)

2

When t = 0.5 sec, we have

9(0,0. 5)

An examination of figure 3 leads to the conclusion that

a composite semi-infinite solid with surface layer twenty or

more thousandths of an inch thick might well be considered a

thermally homogeneous semi-infinite solid having the conductivity and diffusivity of the surface material. This statement

holds with reasonable accuracy for the range of conductivities

and diffusivities found in table 1 listed under indicator

materials. It might be argued that figure 3 was based upon a

composite semi-infinite solid with k^_> k2 and thus the conclusion drawn here is not necessarily applicable to the semiinfinite solids of figure 1 where, for the most part, k^< k^.

However, observe that the kpc of the surface layer of composite

solids used in constructing figure 1 are less than those used

in the construction of figure 3 and thus the equation derived

in the previous paragraph indicates that, if the semi-infinite

solid were composed solely of the outer material, the surface

temperature of the solids of figure 1 would be higher than

those of figure 3* This means that heat finds more resistance

in the surface layers of solids of figure 1 than those of figure

3 . Consequently, it is not unreasonable to expect that a twentythousandths inch layer of the surface materials in figure 1 may

be considered semi-infinite for the purpose of calculating surface

temperature rise.

The data presented in figure h were obtained in an attempt

to "drag down" the indicator surface temperature rise by the use

of highly conductive backings. It is evident that this objective

was not obtained because of the rather thick indicator assembly

and its low kp c product. It therefore appeared desirable to

provide an indicator backing having different thermal properties.

12

The suggested change does not appear too feasible, as the

absorbent properties of the paper appear essential for proper

operation of the indicator.

Figures 5> 6, and 7 show temperature rise for three material composite bodies as obtained on SEAC by numerical methods.

This method of solution provides time-temperature rise data for

the surface and two material interfaces. The solution for skin,

figure 5) indicates that only the first two layers are involved

in the initial transient heating process and therefore the surface temperature rise at 0.5 sec is comparable with data of figure

1, curve c, for a conductivity of 9xl0~^cal/cm° Csec for the backing. It will be observed that the SEAC solution provides a surface temperature rise about 20^ greater than was found by analytical methods.

No method is readily available for checking the accuracy

of the other two SEAC solutions presented in figures 6 and 7»

It is likely that the error will be considerably less than the

figure mentioned above because a finer grid was used in these

computations. It does, though, seem hardly likely that the

almost perfect check observed for surface temperature rise for

plastic in figure 7? with the similar data for polyethylene in

figure 1, represents the true condition. It is likely than an

error of or more percent is present in both figures 6 and 7»

5. CONCLUSIONS

The following conclusions appear to be justified from the

work which has been accomplished:

1. An analytic solution has been obtained for the surface

temperature rise of a two material composite semiinfinite body during heating by a time invariant radiant

energy pulse.

2. A code has been devised to permit use of SEAC for

solution of surface temperature rise when two finite

lamina of different materials are placed on the

surface of a semi-infinite solid.

3» The two methods of solution have been used to calculate

the surface temperature rise of a wide variety of

homogeneous, two layer and three layer bodies after

- 13 -

one-half second exposure to a time invariant radiant

energy which is totally absorbed at the exposed surface.

4. It is shown that the Quartermaster temperature indicator assemblies have thermal properties which prevent their thermal behavior from simulating that of

skin when subjected to the same thermal radiation

exposure.

Thermal

Properties

of

Skin

and

Indicator o0

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OJ1—

cdoo . • • • •

lo

CM loO

CJ 0 0-lr\0 lo sOMD

H OCMMD PO-4- 0 O 1—1 1—1 rH O O 1—1 O 1—It—

H H \ o o o o o o o o o o o o \ CM o o o o o o o o o o o o ao • • • • 1 o a O CM CJNPO C\I (—1 I—

0-4-

a o \DsO POO PO PO-4- MD CM lo CM CM O • • • • • « e 0 0 • • • I—

( 1 cd 0 o oa0 0 0 0

Cm c 0 olo 0 o loCO-4- H CM CM lOlOPOOO CM PO « o_i O O O H O O O O O O O O a O O O O O O O O O O O O a CJ O O O O o o o o o o O O o \ • • • • 0 •

f-i 1—

cdo o oo

tuO I—

Ojo

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N.

0-0 l-O0-lr\ H loj- ^OU^

POa 0 POlo CM 1—1 t—

o CO 00 OCO I 1 00 CO ONJ- O CJN lovo o • 9 9 • cd • 9 tiD •H 1 1 1—1 p0 0

-p o S4 cd cd 0 0 S p (—

I 1 1—1 •H O pq 0 0

cd

a P -p > S4 4:J •H p 0 0 o cd (4 •H -P •H C\j 0 •H 1—

-p o o 0 P 0 Oh

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<4 0 o a o cd •H -p 0 0 0

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a,

Table 2.

Some of the Suggested Calculations of Skin Temperatures

and Manner in which Assemblies were Changed

for Actual Calculations

Item Number Proposed As Calculated

Thickness k Thickness k

1

in.

3x10"^

80

80

160

cgs

5x10”^

9

4

11

in,

3x10“^

00

cgs

5x10"^

9

Ka) Seac

Solution

3x10"^

80

80

160

5x10“^

9

4

11

3x10 “3

80

00

5x10"^

9

4

2 3x10-3

80

80

00

5x10-4

9

4

11

3x10~3

00

5x10“4

9

3 3xl0~^

80

80

00

5x10"^

25

4

11

3x10"3

00

5xio""*-

25

4 3x10"3

80

80

00

5x10"^

25

10

11

3x10"3

00

5x10"^

25

5 3x10“^

80

80

00

-4

5x10 ^ 25

10

25

3x10"^

00

-4

5x10 ^ 25

6 3x10"^

80

80

oo

10x10"^

25

10

25

3x10'3

00

10x10"^

25

7 3x10-3

40

80

00

5x10"^

11

40x10"3

oo 9x10"^4

8 3x10-3

8o

5x10-^ 80x10-3

00

9x10-^

4
• •' #>

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Reference Table 2

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* 25x10 "^ 0.7 19.1

** 8 0.7 33.7

Polyethylene 8 0.5 39.5

Wood 3 0.2 103.0

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THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March

3, 1901, as amended by Congress in Public Law 619, 1950. These include the development and

maintenance of the national standards of measurement and the provision of means and methods

for making measurements consistent with these standards; the determination of physical constants and properties of materials; the development of methods and instruments for testing materials,

devices, and structures; advisory services to Government Agencies on scientific and technical

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the world. There are also five series of nonperiodical publications: The Applied Mathematics

Series, Circulars, Handbooks, Building Materials and Structures Reports, and Miscellaneous

Publications.

Information on the Bureau’s publications can be found in NBS Circular 460, Publications of

the National Bureau of Standards ($1.00). Information on calibration services and fees can be

found in NBS Circular 483, Testing by the National Bureau of Standards (25 cents). Both are

available from the Government Printing Office. Inquiries regarding the Bureau’s reports and

publications should be addressed to the Office of Scientific Publications, National Bureau of Standards, Washington 25, D. C.
VNBS