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4*lb

TRANSIENT HEAT AND MLOISTUJRE TRANSFER IX SKINTHRCIUGH THRM&LLY IRRADIATED CLOTH

Technical Report No. 8

f row

Fuels Research LaboratoryMa~ssachusetts Institute of Technology

Cambridge, Massachusetts

H. C. Hottel0G. C. wi~lliamsdG. H.'Miede1

C~rnrI 7ecember 26, 1961

Project Nos.* 7-12-01-002Cand 7.49-01-001

Con~tract No. ID 19-129-QI-LS92Quarterme ter ftesearcb and Engineering COOMnd

united .States Army

W NITPro Jeclt

Reproduction in whole or in part is A STIpermitted l'y the United S tates aewW

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TRANSIENT HEAT AND MOISTURE TRANSFE TO SKINTHROUGH THERMALLY InfADIATED CLOTH

Technical Report No. 8

from

Fuels Research LaboratoryMassachusetts Institute of Technology

Cambridge, MAssachusetts

H. C. HottelG. "C. V ill aa..

G. H. 'MiedelIV. T. Chen

V. P. Jensen

Dnceuber '2 1961

Project Moe. 7-12-01-002Cacm 7-99-01-001

Contract No. VA 19-129-QW-1592Qsarterwaster Research sad Bagineering Comand

United States Army

NIT ProjectDS 8556

Reproduction in %tole or in part ispermitted by the United States Government

r1

"TABLE OF CONTES

PAGE

7, INTRODUCTION 1

11. EXPMIA434L I

A. Skin simulajt 1B. Cloth Samples 2C. rolar Furnace 2D, Temperature Measurement 2

I11. FROCEDWUR 4

IV. EXPEIMENTAL RESULTS 4V. ThMEICrICAL - MY CLOIH OVE SKIN SIJULUTIII S' I 5

A. The Mathematical'Model 3B. General Solution 8

1. Application to other types of cloth including thosehaving direct transmittance 9

2. Application to relaxation period 103. Application to non-square save radiant source 10

C. Discussion of Results 11

1. Properties of the cloth and skin aimlsat i112. Comparison of the theoretical amalvsls with the

exp?,.aental data on the dry cloth - &ai gap -skin .mulant system 12

VI. THEORTICAL - MOIST CLOTH OVER SKIN SINUIANT SYSTM 14

'A. The Mathematical Model 14B. Method of Solution 17

1I Dimensionless formulation of the basic equations 172. Difference equations and solution algorithm 193, Remarks on the boundary conditiomn and variable

water vapor diffusivity 244. Summary of the computer prosram 255. Stability 286, Convergence 35T. Time-variant irrsdia-cion intensity 37. oun severity correlation 41

Co Discussion of Results 43

1. Numerical values used in the cowmitations 442. Constant irradiation - influece of transfer

cpefficients and diffusivity; comparlsonwith cxperim,,ts 45

3. Time-variant irradiation - influence of pulseshapes 48

4. Time-variant irradiation - comparison withexperimets 50

S. Remark on tte burn severity correlation 52

VII. buMMARY AND CONCLUSIONS

APPBN1IX A Nomenclature 54

APPENDIX B Method of Solution - Dry Cloth over SkinSimulant S)s ten 58

A. Opaque Cloth 58

1. Basic equations 582. Boundary conditions 593. Numerical solution - modified Schmidt technique 594. Stability and accuracy 62

B. Diathermanous Cloth 62

I' Basic ea,'-tions 622. Boundary condition at the front cloth surface 633. Boundary conditions in the air gap 644. Siaplified ganeral solution for diathermanous

cloth 64

APPEWTTIX C Solution Algorithm - Moist Cloth - SkinSimulant Sys ten 68

APPENDIX D Mass and Heat BOtance - Moist Cloth - Skin

Simulant System 73

APPENDIX B Computer Program Description 75

A. kunction vf Main- aid Subprograms 75

1. MAIN prozr-i% 752. SUDP 763. DOUBIN, DWB, T•B 764. PULSE 765. PARA 776. EG 777. CHAING 77

B. Machi.e Timc Consumption 77

APPSNDIX F Summary of Computer Solutions on Moist Cloth -Skin Siaulant System 92

APPENDIX G ft;&ilibrium Data on Moist Cloth 134

APPENDIX H Bibliography 137

J I, ; ~~. ... 1

/:

LIST OF FIGURES

FIGURE NO. PAGE

1.1 Dry-Cloth - Air Gap - Skin .imulant System 5

1.2 Illustration ro Finite-Difference Derivation 59

1.3 Illustration to Finite-Difference Derivation 60

1.4 Illustration to Finite-Difference Derivation 61

1.5 Skin Simulant No. 10 93

1.6 Skin Simulant No. li 93

1.7 Experimental Results on Dry Opaque (Deep Black, Black,Dark Gray) Cloth - Skin Simulant System 94

1.8 Experimental Resultq on Dry Medium Gray Cloth - SkinSimulant Sys tea 94

1.9 Experimental Results on Dry Light Gray Cloth - SkinSimulant System 95

1.10 Experimental Results on Dry White (With COC) ClothSkin Simulant System 95

1.11 Heat Transfer to Skin Simulant Through Moist OpaqueCloth 96

1.12 Heat Transfet to Skin Simalant Through MoistOihthqr-smos Cloth 96

1.1. Effect ot ;loth DiLthermancy on the Heat Transferto the Skin Simulant

1.14 Effect of Cloth Diathermancy on the Heat Transferto the Skin Simulant 97

1.15 Effect of Doubling the Thermal Conductance of Clothon the Heat Transfer to the Skin Simulant 96

1.16 Effect of Halving the Thickness of Cloth op theHeat Transfer to the Skin Simulant 98

1.17 iffect of U° and U.0 on the Heat Transfer to the SkinSimulant Thlough Dy Opaque Cloth 99

1.18-1.19 Tenmper-ture Response in the Skin Simulant Duringand After Irradiation

1.18 System: Dry Opaque Cloth in Contact with Skin Simulant 99

1.19 Systems Dry White Cloth in Contact with Skin Simulant 100

1,20 Comparison of Experimental Results on Dry Opaquerloth - Skin Simulant System with Theoretical Anulysis.Effect of U°, U and 1. 100

09i

FIGURE NO. PAGE

1.21 Comparison of Experimental Results on Dry M4edium GrayCloth - Skin Simulant System with Theoretical Analysis.Effects of U0 and UO 101

a ii-^2 Comparison of Experimental Results on Dry Light

Gray Cloth - Skin Simulant System with TheoreticalAnalysis. Effects of UO and U0 101

011.23 Comparison of Experimental Results on Dry White

(with CMC) Cloth - Skin Simulant System withTheoretical Analysis. -rfects of U0 and Ur 102

0 11.24 Comparison of Experimetatal Results on Dry Opaque

Cloth - Skin Simulant System with TheoreticalAnalysis. Effect of Residual Moisture in Cloth 102

1.25 Comparison of Experimental Results on Dry OpaqueCloth - Skin Simulant System with TheoreticalAnalysis. Effect of Thermal Conductance of Cloth. 103

1.26 Temperature in Clothed Skin Simulant Exposed to Non-Square Wave Radiation (Nuclear Explosion Pulse). 104

1.27 Comparison of Temperature Response in Clothed SkinSimulant Irradiated by Square-Wave Radiation andRadiation from a Nuclear Explosion. 104.

2.1 Slab Division for s - Cloth - Skin SimulantSys tem 20

2.2 Illustratior to Heat Boundary Ctndition at frontSurface of .15th, for Partially Dry Cloth 25

2.3 irradiation Pulse Shapeb 4;

2.4a-2.4b Convergence of Theoretical Solution. Influence ofM , AX 0 and ASO on Temperature Response of Skin;1mulant

2.4a Intensity I a 0.5 cal/cuasec 105

2.4b Intensity I1 a 1.0 cal/cmssec 106

2.4c Convergence of Theoretical Solution. Influence ofMA, U2 and AtSBon Temperature Response of Cloth.Intensity 10 W 0.5 cal/c.ssec 107

2.5 Tridiagonal Equation System for Solution of Equation

2.14 108

2.6 B6i/agonal Eqlation Syster for Solution of Equation22.14 109

2.7 Flow-Chart of Main Program. Moist Cloth - SkinSimulant System 110

2.8 Plow-Chart of Subprogram SUB. Moist Cloth -

Skin Simulant System 111

2.9 Make-up of Run Card Deck 112

F iGURE M). PAG..B2M10 Theoretical Solution fcc Moist Cloth - Skin Simulant

2:ysten (Opaque). Influence of Heat and Mass TransferCoe ff icien s. 113

2.11-3,212 Temperature Response of Moist Cloth - Skin SimulantSystem (Opaque). Comparison of Theoretical andBxperLwntal Results. Intensity Io a 0.83 cal/cmsec 113

2.13 Theoretical Solution for Moist Cloth - Skin SimulantSystem (Opaque). Influence of Fixed = VariableDif fusivity. 114

2.14 Theoretical Solution fcc Moist Cloth - Wk a SimulantSystem (Opaque). Influence of Transfer Coefficients 115

2.15-3,17 Temperavure Response of Moist Cloth - Skin SiaalantSystem (Opaque). Comparison of Theoretical andBxpet men ta L Results.

2.15 Intensity 1o0 0.5 cal/cmesec 11!

3.16 Intensity 1o0 1.26 cal/cassec 116

2.17 Intensity 10 a 3.96 cal/cm'sec 116

2.18 Affect of Intensity on the Temperature Response ofa Moist Cloth - Skin Simulant System. (TheoreticalSolution). Cloth a Opaque, 9.1% 1el. Hum'dity,Temperature a 30 0 C 117

2.19 Hffect nf Intensity on the Temperature Response ofa Moi. • 'loth - Skin Simulant System. (TheoreticalSolution). Cloth a White, W-49 9.1S Rel. Humidity,Temperature a 30C 1.17

2.20-3.21 Temperature Response of a Moist Opaque Cloth - SkinSimulant Systen. Influence of Irradiation PulseShape.

2.20 Pu4se Area a 0.545 cal/cmesec 118

2.21 Pulse Area a 1.09 cal/ca~sec 119

2,22-3X25 Temperature Response of a Moist Cloth - Skin SaulsatSystem (Opaque). Comparison of leoretical andExperimentak Results for Sqfare-ve Irradist ion.Mals Invensity Irota a 1.0 cal/-m'sec

2.22 Variation of Heat and Mass Transfer Coefficients atConstant Diffusivity D" 120

2.23 Variation of Heat and Mass Tratnfet Coefficients atConstant Diffusivity D" 120

2.24 Variation of Heat and Mass Transfir Coefficients atConstant Diffusivity W1' 121

2.25 Variable Diffusivity r 121

FIGURE NO. FAGE

2.26 -427 Temperatire Response of a Moist Cloth - Skin SimulvntSy3tem (Opaque). Comparison of Theoretical andExperimental Results for Square-Wave Irradiation.Max. Intensity, I c 1.4 ral/caosec

2.26 Constant Diffusivity D" 122

2.27 Variable Diffusivity D" 122

3.x-3.22 Complete Computer Solution for the Moist OpaqueCloth - Skin Simulant System

3.1 -3.7 10 a 0.83 dal/cm2 sec; Vaziation of Heat and MassTransfer Cnefficients 123 - 126

3,8.-3.16 Calculations with Irradiation Pulses of DifferentShapes and Amplitudes, Equal Area 126 - 130

3.17-3.22 Calculations with Square-Wave Irradiation at1oazx a 1.0 and 1.4 cal/cnesec and 4 sec. ExposureTime; 65% Rel. Humic.±ty. (Used for Compa:ison withExperiments). 131 - 133

LIST OP TABLE

PAGE

1.1 Optical Properties of Cloths (Integrated VaLies over Solar

Spec trum) 3

2.1 Thermqgwlear Pulse Intensity Distribution 40

B.L Cumput:r Program Listing - Moist Cloth - Skin SimulantSystem 79

G.1 Equilibrium Moisture Content of Cloth as a function ofTemperature and Relative Humidity 135

G.2 Gradients of Equilibrium Moisture Content of Cloth asFunctions of Tmperature and Partial Water Vapor Pressure 136

, S

PCREW(RD

This report was prepared by the 1luelz Research Laboratory of the

Massachusetts Institute of Technology under Contract No. DA-l9-29-C*4-

1592, Project Nos, 7-12-Q1-OOZC and 7-99-01-001, Quartermaster Research

and Eng~ineering Command, United Stlatei Army. The work was administered

under the direction of the Pioneering Division, Headquarters, Quartermaster

Research and.Engineering Command, United States Army.

This report covers work conducted fron December 1960- to October 1961.It incorporates a review of the underlying work donte under the associated

contract No. DA-i9-129-QM..454.

The authors gratefully acimawledge the Important contribution of

Professor P. L. Thibault Brian, Chemical Engineering Department,

Massachusetts Institute of Technology, to the theoretical analysis~and

also tne *timulating discussions with t4r. Cristobal. Bonifaz, research

assistant, on the mathematical side of the project. The computatinns "ere

carried, out on the IBM 709 digital computer of the MIT Computation Center,

Finally, the careful preparation of the text and graphs by Miss Sally Drew

is much appreciated.

AaSUACT

A study has been made of the action of cloth in protecting skin from

thermal injury resulting from exposure to high-intensity thermal radiation.

Methods have been developed to obtain temperature-time-depth data fur

a system simulating skin covered by a layer of dry or moist cloth. Bxperiso

mentally this has been accomplished by the use of a copper-air simlant

which accepts heat at the same rate as.human skin but develops a propor-

tionally stretched temperature profile.

The analysis of tim moist cloth-4kin simulsnt system involves some

simplifying assumptloe regarding the mode of moisture transport. It

Is applirable to low- and medium-intensity exposures constant or .variable

in time. The solution is not general but requires new computer calculations

for every *pecific case. The model yields reasonable estimates of the

skin taenperature response due to the unc'ertainty af a nmber of physical

parameters there is a margin of tolerance which could be narrowed by

further experiments combined with further calculations. 7he description

of the intermediate role of moisture flow as cootrihutbnK to tie heat

transport may still require refinseent; however, thmre is us yet no simplewmy to evaluate the bi r. of water vapor through the poree of a fabric.

The current method facilitates the inspection of the Influencce ues

temperature distribution In the skin of various system properties and the

surrounding conditions.

The skin enthalpy rise above a critical temperature level, a feasible

basis for correlating burn data, has been found to be very sensitive to

slight variations in the temperature response and deserves further testi'4.

Besides an account of recent research, this project incorporates a

review of the experimental work done earkier and of the mathematical

aodels with the methods for their solution.

L. IN7ROIXJICW-IO

Althcouh thermal injury to skin has been an important aedical problem for

many years, quantitative studies on the relation between the temperature history

and the production of cutaneous damage' did not begin until the forties by

Mantz and Henriques (5,,L5,.16, 17). The importance of knowing the tomperature

history of skin during its exposure to thermal energy was pointed out by

these authors. However, the problem of measuring the temperature rise in

skin remains unsolved. The difficulty lies in th" fact that the epidermis

of the skin whire the temperature history is essential to the analysis is only

80a thick, aid measurement of the temperature profile in such a thin laysr

is stillbeyond presently known techniques. In previous studies, (3) we

have developed skin sinulants made of copper and air with a stretch factor

(depth Wrinification) 28.4 times that of skin. Thus the temperature profile

in a skin-*yer 100p thick can be measured accurately by means of the simulant

over a distance of 2.84 sm.

The sinulart has been used in evaluating the energy transport prncesses

in a cloth-air gap-skin sýtex by exposing the test assembly to high intensity

solar radiation. Rxle. mental temperature-time-depth data were thus obtained

on the simulant. This paper presents the results of these experiments aad

the uuthematical models proposed to describe the energy transport process

in such system. The presentation is divided into, two parts. The first part

deals with the dry cloth-skin system in which the Moat transfer by conduction

and radiation through the cloth layer is considered; the second part deals

with the moist cloth-skin system in which in addition to conduction and radiation,

diffusion of the moisture in the cloth layer also contributes importantly to

the heat transfer from the cloth to the skin.

II. =PFRi4EW~kL

A. SKIN SIMUIANT

The principal inuterest in the present experiments was the temperatt.e-

time-depth variation in the skin simulant. No attempt as made to measure the

temperatura in any part of the cloth. In previous studies (5, 12) it was

concluded that the complexity al the cloth structure made it impossible to

identify the cloth "Surfaces'* unambiguuusly. Consequently, little- signif icance

could be ascribe'1 to experimentally measured cloth temperatu~res.

A soldered fin-type skin simulant (skin similant #10) (3, 5) was used in

all "in-contact,* rt'ns. It consisted of sixteen fins, with twelve mutually

parallel and four perpendiculipr to the twelve to block the- exposed air space

and help to maintain more nearly iso~hezns in the Individual fins of the main

uimailant. The fins were copper sheets, each 3.175 cu. and 7.95 cm. x 0.0102 cm.

thick, bent into shallow channels as shown in Figure 1.5. The four guard fins

were bent to a 3-inch rudius arc in a specially aschined din. In oeder to

make simulants of this type 3yiunetrical, six channels f~ce one way ard six the

opposite way, i~e., all the flange toes faced'toward the center of the simulante

The fi ns were soldered to a top sheet of coyper lsving dimensions of 4.37 x

3.61 cm. x 0.0102 cm. thick, Five 0.0025 ca. disameter constantan wires were

soldered onto the central copper fin, forming the hot Junctions of the thermo-

couples. TIe five wires were led out, fran -the bottom and soldered to heavierconstants: leads 0#24 B. and S. gage). A common copper lead wire was soldered

to the copper sheet. The cold junction for the theymocouples consisted ofseveral massive cop;4r blocks, one for each cons tsetan lead and? another for the

copper lead wire. The Js f%. ions were located inside the constant temperature

box, where connections to the temperature recorder were made. A cylirz'!_z1simulant as shown in Figure 1.6 was also used in several experiments in which

the cloth sample was set 0.038 ca. from the top sur-face of the simulant.

B. CL07H SAMPLU

A series of six "standard" 9-cunce cotton sattee and 9-ounce Olive Green

uniform c~uoth supplied by the Quartermaster Research acd Development Command

at Natick, Karisachumetts, were used in the present investigation.

The optical properties of. the cloths (5, 6) have been determiuied bythe Naval Material Laboratory co~vering the wave-length range from 0.40 to

2.7 .xicrons. 7hese data have been Integrated over the solar asPeetrum; the

results are listed In Table 1.1.

Wooden embroidary riqs 5-incaes in diameter were wed to hold the cloth

samples. All cloth samples dere oven Jird at 1059C for 24 hours and then

2

cooled and r-onditioned in desiccators containing one of three items: (1)

indicating silica gel, (2) saturated sodium nitrate solution (65-66% relative

humidity at 15-300 C), and (3) saturated ammonium chloride solution (79.5%

relative humidity at 20-300C).

Table 1.1

OPTICAL PROPERTIES OF CLOT4S (INTBGRATED VALUE OVER SOLAR SPBCIRL'A)

Code No. Designated Color Reflectance Absomptance Transmittance*

QM-4 White with O.C 0.67 0.19 0.14

QM-5-65 Light gray with C1C 0.56 0.37 0.07

Q,.-6-40 Medium gray with C.C 0,50 0.47 0.03

QM-7-22 Dark gray with CMC 0.22 0.78 0

QM .8-12 Black with CNC 0.11 0.89 .0

QN-9-7 Deep black with CMC 0.09 0.91 0

•tN.3 Olive green 107 0.18 0.79 0.03'

C. SCLAR FtRLU.CB

The MIT Solat Fu-n-.e (5, 9,.10, 11) was used as the source of high

intensity radiation. It.tensity up to 6 cal/cma-sec. over a 5 cz a target

arep was obtainable with a clear sky.

D. T3M PERA%.RE MI-ASUREtr'NT

A four channel 150-1500 series Sanborn recorder equipped with "low-level"

preamplifiers was used to record the outputs of thermocoutples. The recorder

has a maximum sensitivity of IOCpv per cm (or approximately 2.50C for copper-

constantan thermocouples). The time constant is 0.03 sec.

Since the cloths were made from the same white 9 ox. cotton sateenstock, and it is unlikely that a simple dyeing process would changethe geometry of the fibers, it is reasonable to assume that thecloth construction provides complete geometrical blockage of theincident light, and that the transmittance depends on the degree ofopacity of the individual cloth fibers. These transmittance valuesshould therefore be regarded not as due to direct transmittancethrough holes but to diathermancy of the cloths.

3

III, PR.OCEURE

Cloth samples were prepared several days in advance and kept in their

respective desiccators until testing time. Then the sample was quickly

transferred from the desiccator to a constant humidity box attached on the

solar furnace. The constant humidity box was kept under the same condition

as the desiccator. Constant temperature was maintained in the box with

an air blower and a thermoregulatcr. After the skin simulant was placed

behind the sample, the box was closed and the air blower was allowed to

contitiue blowing for 10-15 minutes to insure constant temperature and

relative humidity in thea box. The air blower was stopped during the exposure

of the samplc to the addiation. "the sample was exposed for one to 30

seconds to the high intensity radiation ranging from 0.3 to 6.0 cal/cm2 -sec.

The length e; the exposure was controlled by a timing device which opened

and closed the diaphragm shutter. The intensity of radiation was regulated

by partially blocking off the primary mirrors of the solar furnace.

IV. EXPEUIMLTAL RESULTS

Experimental data ob ± med on six kinds of dry cloth are presented in

tie following figures in w.ich the normalized temperature rise is plott,.

against the exposure time in seconds on logarithmic papers with the three or

four skin depths as the parameters$ Figure 1.7 combines the result on three

opaque cloths (deco black, black, asid dark gray); Figure 1.8 shows the

result on the medium gray cloth; Figure I.€ shows the result on "e light

gray cloth; and Figure 1.10 shows the result on the white cloth.

The experiments on moist cloth over skin were design"e to study the

effect of the radiation intensity on the coupling effect of moisture transfer

or' heat transfer. Two types of cloths, one black and one white were used.

The effect of moisture transfer on heat transfer can be illustrated

by the following figures, in which the normalized temperature rise of the

first thermocouple in the skin simulant is plotted aFainst the exposure

time, with iitcnity as the paramoter. Figure 1.11 shows the resuLt on thi

opaque cloth and Pig. X12 shows the result on the white cloth. The complete

data may be found in reference MS).

4

V. THEORETICAL - lMlY CLOTH OVER SKIN SIMULANT SYST34

A. THE MATHEMATICAL MODEL

The general assumptions made in the aralysis are as follows. The

system under consit-ration is sketched below:

Air Lkin" Li Space Sinulant ----- "

Cloth

Figure 1.1

DRY CLOTH-AIR GAP-SKIN SDIUIANT SYT3I

1. Heat flow is unidirectional, occurring in the x-direction, from the

cloth through the air space into the skin simulant. The surface of the

cloth is irradiated by square wave radiation. It will be shown later that

the general solution can be applied to radiation of any pulse form.

2. The cloth kkyer "- considered to be a homogeneous solid, For cloths

which show no measurable transmittance (e.g., deep black, black, and dark

gray of the 9 oz. "standard" cotton sateen), radiant energy is absorbed at

the surface of the cloth. For diathermanous cloths, the radiant eaiergy

is assumed to penetrate into the interior of the cloth layer in accordance

with the Beer-Lawbert law:

Ix uI° exp (-yxc)

where Ix * intensity of radiation at a plane distant xc cm from the

irradiated surface.

I 0 intensity of radiation entering the cloth layer, oro

I 0 a(1 - r), cal/cma-sec.I a Inrident intensity of radia•ion.

-ly a extinction coefficient, cm , a characteristic constant of

the cloth mr the particular source of radiation in question.

If the measured "transmittance" of a diathermanous cloth of thickness

L cm is T, then exp(-yL) s T/(l-r), where r is the reflectance of the cloth.

The solution is applied specifically to the 9 ox. "standard" cotton

sateen; it could be equally well be applied to either diathermanous or

direct-transmitting cloths.

3. The physical properties of the cloth layer remin constant through-

out the exposure.

4. Any effects due to moisture and/or chemical reaction are negligible.

5. Heat losses from the front surface of the cloth layer by convection

ant re-radiation are characterized by a single overall heat transfer

coefficient Uo.

6. i!'At transfer from the cloth layer through the air gap to the simulant

is similarly characterized by an overall heat transfer coekficient U1 .

7. The skin simnulant is considered to be a homogeneous opaque solid

of infinite extent in its x-direction.

The validity of these assumptions must first be checked against

experimental data obtainee %, om an equivalent inanimte system. Further.-

more it should be born in mind that such an analysis will be of value ouuy

if the temperature history data can in combination with some postulated

mechanism of burning correlate the in vivo results.

The basic equations for the cloth layer are:

1. Opaque cloth

bT k as T

2. Diathermanous cloth

• , + c "pT IC (,.2)

The equation for the skin is

Cps -3x (1.3)3xS6

6

T1r surface bounda)'y equations are:

1. For opaque cloth

(a) Front surface

Sut ] ".4)

Ia Uo(Txu - T )kc ITI0 0 U(xc=O " surr) a k xc X.WO

(b) Back svrface and skin suri.aceAT•I Ui(T T . -k T

2. For diathermanous cloth

(a) Front surface

k T 1 . aUCT -T )(1.6)c xXCO surr

(b) Back surface and skin surface

"-k ;T + I exp(-YLc) M UU(T - T-)

+ I exp(-yL,) m as T (1.7)

These equations are solved by a finite-difference approximation method

which gives temperature as a function of the following five independent

variables; time, depth, Ui, U0 , and R, a property group. The solution is

expressed as a relation among the following dimentionless groups.kT

1. Tmperat-e group T c * CXoLC

2. Time group go~ • kc (&PC a

3. Depth group X°0 *c

U L(1.8)

4. Surface heat losses group U Co

"\ !7

U L0 0 c,

5. Air gap group Ui jC

4X0

6. T group -

Stated another way, the temperaturc-time-depth relation is a function

of the three other groups.

The details of the solution are given in Appendix 3.

B. GENIAL SOLUTION

The general solution consists of the temperature-time-depth relation as

a function of the following parameters:

1. U0 the dimensionless surface heat loss group0

2. U0 the di.mensionless air gap group

3. I' the physical property ratio group

For opaque cloth, only one set of solutions is needed; for diathermanous

cloth, a technique which uses the weighted mean of three sets of solutions,

each representing radialion absorption at a different plane in the system,

gives results within I to "/a of those obtained by a more rigorous method.

These three sets of solutions are as follow:

1. The case of incident radiant energy absorbed at the exposed cloth

surface,

2. The case of incident radiant energy absorbed at X0 a 0,75, andc

3. The case of incident radiant energy absorbed at the surface of the

skin sinulant.

By taking advantage of the additive nature of the solutions, the

temperature rise at any plane in any diathermanons-cloth problem can be

obtained as the sum of the temperature rises due to assoming that the

total energy absorption is lumped at three planes. The Beer-Lambert law

is used to calceulate absorption in the first hall of the cloth, the second

half, and the skin surface. These quantities are then trested as though

lumped at the front 3urface of the cloth where X O 0, at the plane X 0.75,c S

8

and at the skin surface, where X° 0.

1 1. Application to Other Types of Cloth Including those having Direct

Transmittance

As a corollary to the above additive principle, these three curves

can be -weighted in any fashion according to known distribution rule

of the incident radiation inside the cloth. For cloth having direct trans-

mittarce, an additional proportion of the Curve III solution is added. However,

it must be noted that the solution is based on an opaque skin simulant, hence

it is not applicable to diathermanous skin covered by a single open-weave

cloth layer.

For cloths of different thicknesses and thermal conductivities, the

general solutions are applicable. The only change is in che relationship

between t'.e actual temperature ( C) - time (sec) - depth(ca) and their

dimensionless groups. Thus doubling the thermal conductivity of the cloth

doubles the To, 8g0 , and X scales. Halving the thickness doubles the Toc a

and Zo scales, and quadruples the 0 0 ecale.

The general solutions also allow the prediction of the TO, 80 X°0 datas

at the skin surface, a d other depths where experimental data are not

available.

The following examples illustrate the effects of diathermancy, thickness,

thermal conductivity of the cloth, and the size of air gap and surface heat

losses on the To - 8° - X° relationship in the skin. The solutions arec s

shown in the following figures.

Figures 1.13, 1.14 show AT0 versus A6 at X: U 0, aid X0 0 0.05, 0.45

respectively. The solutions are plotted on Cartesian scale to demonstrate

the effect of diathermancy on the time lag between the initiation of the

exposure and the rise in temperature in the skin. These figures are self-

axplanatory.

Figure 1.15 illustrates the effect of doubling the value of k on thec

To - 0° - X0 relationship. Curves to the right are calculated fer whitec 3

(W4) cloth, which is highly diathermanous, the curves to the left for

opaq*n cloth. The different part kc plays in these two types of cloth is

interesting.

"9

Figure 1.16 illustrates the effect of doubling the thickness; for the

.3ame two types of cloth. It is seen that the variation in !thickness has

a much greater effect than thermal conductivity variation on the To - 0 -XC s

relationship.

Pigure 1.17 illustrates the relative importance of Uai o

TO - 00 - XO) it is seen that, as expected, U 0 is important at relativelyc a 1small 90 and U° becomes important at long e0 values.

C 0 c

2. App4ication to Relaxation Period.

Taking advantage again of the additive nature of the solution, theTP - 0o - XO relationship during the relaxation time, i.e., after the

c 5termination of the radiant source, can be obtained from the general solution

by adding at the terminution time to the general solution, a negative-

radiatAe.- )f the saw intensity.

Figures 1.15, 1.19 show the temperature-time curves at two depth

planes with radiation terminated at various dimensionless times, in the

case of dia thermanous white cloth and black opaque cloth, respectively.

These figures are self-explanatory, and it is interesting to note the time

when various depths of Jy skin reach their maximua temperatures.

3. Application to Non-Square Wave Radiant Source.

Even though the getieral solutions are based on a radiant source of

uniform intensity, solutions to radiant sources of any shape or duration can

he obtained frcom these general solutions. An example is shown in Figure 1.20.

The pulse shape and, duration is that of a 1.46 Megatoa nuclear explosion

(2, 3, M). The procedure is illustrated by the sketch at the lower right

corner of Figure 1.26. Radiation of uniform intensity is added to the

solution in a step-wise fashion until it reaches the maximm intensity, then

negative radiation is added to the solution in the sume way, (shown as

double cross-hatched area) and net radiation received by the system is shown

as the cross-hatched area. The dimensionless temperature rise group

expressed in terms of the maximum intensity, I.., is plotted against

dimensionless time, 400 for various depths in skin as shown in Figure 1.26.C.

In Figure 1.27, the temperature history at a dimensionless skin depth

of 0.05 for the nuclear explosion is compared with that of square wave

10

radiation of the sa,,e energy, delivered in 4.5 and I dimensionless time units.

It is to be noted that the theoretical solution is solved by dividing

the cloth into 6 increments and the modulus, AX02/A49 , is chosen as 2;C C

thus each time increment represents 0.0139 dimensionless tite units wh:ich

in turn is equal to about 0.034 sec. for the 9 oz cotton sateen. If the non-

toniform pulse is to be delivered in a very short time, say 0.1 sec, the present

general solution would only give a very crude approximation of the pulse shape

(i.e., in three time increments). But the situation can be easily improved

by increasing the modulus or by decreasing the size of each increment.

C. DISCUSSION OF REULMS

I. Properties of the Cloth and Skin Simulant

For the optical properties of the cotton sateen cloth tike values listed

in Table 1.1 have been used in the evaluation of exPeriments as well as in

the calculations.

Literature values for the specific heat of cotton fabrics vary between

0.32 (7) and 0.35 cal/gm-q, (2). A value of 0.34 cal/gm0 C was used in the

calculations.

In determining the heat conductivity of cloth by the conventi-nn' s+tcadv-

state method, it is difficult to assign a proper effective thickness to the

cloth in a situation involving both radiation and conductive heat transfer.

A similar problem is posed Lt. the measurement of density. These difficulties

are avoided by combining the thickness term with both the density and thermal

conductivity. For the former, a value of (pL)c a 0.0273 * 2% gm/cma was

measured by a method described in (4). The group (k/L)c bas to be estimated

and improved by a trial-and-error method also explained in (4), which uses an

adjustment of theoretical to experimental temperature responses. This method

should give a better approximation of kc for our purposes than the steady-

state method. However, it presupposes correct values of the heat transfer

coefficients U0 and UP which are not known precisely. Therefore the influence

of a possible error in k will have to be tested (see V, C, 2).

The value of (k/L) for the 9 OZ standard cotton sateen was determined

to be around 3.76 x 10 cal/cm sec°C. If L is taken as 0.043 cm (measuredc

11

with a micro-t:ter), then the above value corresponds to k - 1.62 x 10-4

cal/cm sec C, which is close to the value kc a 1.5 x 10 cal/cm sec C indicated

by other literature.

For the skin simulant (no. 10) used in the experiments the values-3 0 -4k 3 1.035 x 10 cal/cm sec C and (kCp)s is 8.65 x 10 cal*/cmnsec°Ca were

measured. These differ slightly from the properties of average white skin-3

agreed upon at an AFSWP conference (1), which are ks a 1.065 x 10- C4l/ca sec C

az (]kta) 8.5 x 10-4 cala/cm'sec°0Ca.

2. Comparison of the Theoretical Analysis with the Experimental Data

on the Dry Cloth - Air Gap - Skin Simulant System

Using the physical properties mentioned above, the experimental data

(figures 1.7 - 1.10) are transformed into their dimensionless form and replotted

in Figures 1.73 - 1.23, ior ccmparison with two sets of theoretical results.

The dotted lines are based on U0 • 0.1 and U? - 2.0 (corresponding to a0 1 -4

surface heat transfer coefficient U° = 376 x 10 cal/cm~sec°C and an air-4 0

gap of width 0.008 ca with an air conductivity of 0.6 x 10 cal/cm see C).

The solid lines represent Uo a 0.2 ard U0 a 5.0.

The theoretical solution can be expected .o be influenced by the heat

transfer coefficients, the ti.-.mal conductivity of cloth, the transmittance

and residual moisture of cloth.

Figures 1.20 - 1.23 show that the effect of the temperature rise in the

skin simulant of changing the assumed valles of U 0 and U0 decreases with0 i

rizirn diAthermancy, as can be expected fron the fact that the contribution

of conduction to the heat transfer through the cloth dWAnishes.

For the diathermanous cloth, the agreement with the range o& the two

sets of calculations is quite good. The values for the opaque cloth, however,

are better approximated by the solid curves than the dotted ones at short

exposure times. The apparent necessity of assigning higher values to the

heat transfer coefficients than those representing a stagnant air layer and

natural convection at the front surface (case of U° a 0.1 and Uoa 2.0)

(McAdams, 14), may be explained by the Lact that under rapid transient heating

tte transid-t heat transfer coefficients can be considerably higher than the

values o'tained under steady-state conditions (8, 13, 18). in additions

12

movement of air in the pores of the cloth under a high temperature gradient

may also contribute to the effective heat transfer coefficient.

Further possible influences are shown by Figures 1.20 - 1.24. Assumption

of a small transmittance of the opaque cloth (although not revealed by

spectrometer measurements) say raise the temperature response of the skin

simulant considerably at low exposure times (Figure 1.20). A similar effect

can be caused by a low residual moisture content of the "dry" cloth, whose

desorption, migration and subsequent condensation on the surface of the

relatively cool skin simulant will contribute to the energy transport.

Figure 1.24 illustrates the variation due to 1.8% residual moisture content,

calculated according to the moist cloth theory.

Last, the effect of a variation in the thermal conductivity of the cloth

has been studied. In Figure 1.25 (k/L) is raised by 30% over the valuec

previously used. Here the dimensionless heat transfer cz'efficients are left

constant (at 0.1 and 2.0), which implies that their dimensional equivalents

U° and UP. too, are increased by 30% over their original values for naturalconvection and a stagnant air gap. This latter fact alone accounts for arise in temperature. Thus if there is a contribution on (kL)c to the

temperature rise at al. it must be rather small.

To reveal the influence of (k/L)c further, some experiments have been

carried out with a cylindrical skin simulant placed at a distance of 0.038 ca

behind the cloth. These tests clearly indicate that the heat transfer through

the stagnant air sp.ce in our sys tem is much higher than can be predicted byan increased thermal conductivity in conjut.'tion with the stagnant-air value

o0 Ui.

Summarizing, it may be asid that while slight effects may be due to

transmittance and residual moisture in the dry cloth as well as its thermal

conductivity, they appear unlikely. The major influence on the theoretical

temperature response of the system is most probably effected by the choice

of proper heat transfer coefficients. It seems necessary to base the

theoretical solution on values of U and Ui higher than those related to

steady-state natural convection and a stagnant air space. A rigorous

confirmation of this would require further experimental investigation.

13

VI. THEORETICAL - MOIST CLO'Th OVER SKIN STMULANT SYSTEM

A. THE MATIII•4ATICAL MODEL

Based on the assumption that a linear relationship exists between the

physical properties of the cloth fibers and that of water, an attempt was

made to correlate the moist cloth data with the theoretical solution for

the dry cloth-skin system using the calculated "pseudo" physical properties.

The calculated theoretical tmperature history was for lower than the experimental

data. This suggested that the Moisture in the cloth vaporized when exposed to

high intensity radiatiou with part of the water vapor penetrating through the

air gap and condensing on the cold skin simulant surface, thus resulting in

a high temperature rise. A successful mathematical model must, therefoie

include the effect of moisture transport on the heat transfer process. *ur

proposed model assumes the following:

1. Mass transfer of moisture is caused by the molecular diffusion rd

bulk flow of vapor through the gas voids surrounding the cloth fibers, out

not by liquid diffusion on the surface of or through pores in the clot., fibers.

2. The local moisture content (called "regaine in textile technol ,gy)

of the cloth is at all tiien in equilibrium with the partial pressure .1

water vapor at the temperatL.e of the local gas id the voids, i.e., th.':,-..

of equilibration is much faster than the vapor diffusion process.

3. Sensible heat transfer in the vapor phase is negligible compared

with the heat conduction through the cloth and with the heat effects of

adsorption and desorption of water on the cloth fiber.

4. The volume of air flowing through the cloth is negligible compared

with that of the water vapor due to desorption and adsorption of waRsr to

and from the cloth fiber, ie., air can be considered as a stagnnt layer

through which water vapor diffuses.

5. Other assumptions are the same as those for the dry cloth problem. An

energy balance around a differential element of the cloth layer givest

k-T (I-f ) a am (2.1)

14

or

6T amr(- "2. a)

where Pc a Po(I - f) * bulk density of cloth, and the continuity equation for

water can be written as:

gD •Y1l= PC a ~

rx[ (I -?i1 3 'Ti] NJ 3M W (2.2)

Since the bulk flow due to a total pressure gradient is neglected in

this model, the total pressure is assumed to be constant at I atm., and the

mole fraction of water vapor, yI can be replaced by the fractional partial

pressure p/W. Aquation (2.2) can be written ass

a PC3M (2.2a)

where

(I - -1y IT

p m ass density of the gas phase, pgm/cm

A third relationship can be outained from assumption (2), thus

M ( 0 (T,p)

and

(i*Ip -I (r ~ (2..3)

Substituting (2.3) into (2.1a) ana (2.2a),e liminating bp/b@ from the twoequations and making the assumption that I T • we obtain,

3? k a*T? Ffm' s (2.4

is

andX am

c> J 0 " [- T J a2T (2.5)

Equations (2.4) and (2.5) represent the two equations describing the heat

and .'=s5 transfer in the cloth layer.

For the skin simulant, we have,

aT k aSTG"C . •' (2.4a)

where the subscript s denotes skin simulant.

The bounds-,! conditions ares

(1) At the front surface of the cloth:

(a) L10 (p =0 -Psurr afD~" ac x1 u (2.6)(.,'(b) 10 - U (T ~ ~Ts _cT " (3.7)

(2) In the air gap bet .en the cloth and the skin s3aulant, neglecting

the heat capacity of the air:

(a) Material balance:

K i(Pxc.Lc pX6. 0) - 4-' D alealI (2.5)

(b) Energy balances

- T- K (p -p

cj xz.L i XCULg x~sO

U ( . - ) -(T(p p )

a .. (2.,9)

16

• okI xs7

B. METHOD OF SOLUTION

1. Dimersionless Formulation of the Basic Equations

Unlike the case of the dry cloth-skin system, the moist-clkth problem

does not lend itself to a generalized solution in dimensionless form. Not

only is its solution non-linear with respect to irradiation intensity, but

a general formulation would also require analytical expression of the

hygroscopic equilibrium behavior of the cloth, so far only available in

tabulated form. To aescribe this equilibrium, as will an the water vapor

pressure curve, a number of dimensionless parameters would have to be

introduced btsides those needed for the heat and mass transfer coefficients,

diffusivity, and the surrounding conditions.

The solution method developed is therefore not general, but is specific

with resaect to water vapor as the vaporizable component, the initialmoisture content, the initial temperature aid the irradiation intensity;

i.e., the funtions relating vapor pressure and moisture content are dealt

with in absolute terms. Nevertheless, the calculations are carried out indimensionless form, which not only provides an organizational aid but also

alsows for examinat~on of the degree of nonlinearity as exhibited by the

results.

The transformation groups for time, space, temperature, pressure and

the transfer and diffusion, coefficients are listed below:

To a dimansionless temperature group.oc

X 0 --. dimensionless distance $roup, measured from thec front surface of the cloth.

0 x kxs. . A . E dimensionless skin-depth group.

Lc k

0a k cO dimensionless time group.

UoL~U0 .• -.s surface convective and radiant heat losses group.0 kc

17

/

UO cU0 k convective heat transfer through air gap group.cT C property-ratio group.

0 pXpO a - ZA dimensionless partial water vapor pressure group.

PO ,(P is a convenient base pressure value, takentoe bOO,. Psutr the equilibrita partial pressureof water vapor at time sero.)

MoPo10 0 0 " surface heat losses group due to moisture removal.

0

M1i~o10 - heat transfer group due to moisture transfer through

0 th sair gap,

-- 02 heat transfer group due to moisture diffusion.LCII

Using the above transformations, the differential equations (2.4, 2.4a, 2.5)

become:

(a) For the cloth layer:

C z

ago axos aC C

utlre

D.fD°

LWDLC O£

(r~ F)T 'PADOLcCc I° I ,MIT")

C 0 lFp T Cc (•)T J

(b) For the skin:

neo AxoVS

The boundary conditions (2.6, 2.. •.9) are transforet• into:

(a) At the outer stwface of the cloth:

Mass flux: P = O (f(D°Ao)) " o x= (2.6)'10.0 0 x0 Ixu

C cj C

Energy flux: 1 -I U- To T (2.7)'0" O a X o .0

(W) In the air gaps

Mass fIux: p0 PO .Xfo(DO,•) .* ] (2.8)t.0- 1 0.0o azj'x0e

Energy f lu - - o

3-0cl -0 I -1 P 0 1 4-0

U O U TP )Tr(Po K )

" ' - X°=O .1 X.0-0

zJ 80. 0

2. Difference Auations and Solution Algorithm

(a) Finite difference notation

In order to solve the system (2.490 2,4&'9 2.5), it is rewritten I.n the

form of finite-difference equations. The variables involved now are lumpedquantities, their values being assigned to discrete points in space and time,

separated by intervals AX° (cloth), ALs (skin) and A 00 (Figure 2.1). The

subscripts J and n denote, respectively, a point in space and tim. In

particular, the subscript n denotes "present time": n-1 refers to "past time",

/

arn n*1 to "future time".

1 ± T TkII I I I1 I I701 T2 T0 ý6 TS010

P1 P2 3 P4 5 P6 7 PS I ISI I I i i I I II I I I I ,l i I Ii

I I I I I g I I ,I I j I i I I a

•z I I, .-.. •

12 , I I i aiX1 1 1 2 5 0

Figure 2.1

SLAB DITIlION FOR MOIST CLTH-SKIN SI•UANT SYSTBi

The parti•al derivatives, e r. of t.le temperature diet, ibution, in finitedifference notation take the following fort:

T. T 0 - 2T 0 + TobaTPI J-1*, a J+jIn

xo o 0 (AO)a80 • g AST°

abbr.: ...4a. n T (2.10)(A)om xxJ, n

c

_oj0 T0 n o -T 0 . ,

be 0 x 0 y 0 Ago ATJ uc n br ~ m (2.110 0 ~ ab A 0 OrTj~f

Thus, the governing difference equations are (with the abbreviations intro-

duced above):

20

(1) For the cloth layer:

- . - *B (2.4+)"

Al' - - (2.5)"ago (•3)a1, (4,•t`)s

C0 C

(2) For the skins

and the boundary conditions become:

(-.) itAhe front surface of the cloths

0A' a1Mass flux: P 0 -(f(D°/ic 0 )) Apo (2,6)"

Heat flux: I -U 0 To m 7'0 (2.7)"0 xcSO - 0 (x0mo

(2) In the air IPP.

Mass flux p 0 - P 0o -(f(D /L)) . (2.8)-0. 3k.0 • x 0.01 20)

or, if one assumes condensation to take place to the plane 1/2 AZý above the

skin surface and accordingly reduces the diffusional resistance across the

air gap (l/X 0) by that through hialt a skin ircrements

F0 •PO ( (f(DK (2.Sa)"

Where

' .1 2

PO, . P0 in the 6-alice model of Figure 2.1.X80 X: /2 9L

0 0

21

12

Ax00

Hea flp 7P- ((P9)

8 x x.o (2.9)

(b) Backward difference method

In solving difference equations the structure of the computing algorithm

as well as the stability arnd accuracy of the solution depend on the choice

of subscripts assigned to each of the deriwatives. For example, in the case

of the J~eat conduction equation,

taking the second derivative in space at present time results in the "explicit"

for mulationr

Ju~ 'j~n J+l.n * Jn+1 i.E

of To at time n Is known.

If instead one chooses to take the second derivative at future time,

one arrives at the so-called "backward difference" eqmtions

To? .TO 00 Tjullm+l JIn+l +Tj+l.n~l lpJ.n~l TJ.n

which at each point (J,n~l) presents the three unknowia Tjel.ln., T" n÷2,

T and is also called "implicit". 'TJ•l,n.J.

Besides these methods, there are others involving both present and

future time values in the evaluation of the next distribution (18). The

formulation mentioned first is explicit in that the single unknown quantity

22

at each point (J,n) is readily obtainable. However, this method is known to

possess a restriction on the solut.on stability. The second, backward

difference formulation with three apknowns at (J,n) requires setting up a

system of as many simultaneous equations as there are space points. Since

the equations deal with subsequent spa'e points, the unknowns appear in

advancing positions and thus form a "tridiagonal set of equations which can

be solved by recursion after application of the boundary conditions. This

procedure entails more computational work than the explicit method, but in the

case of the heat conduction equation it is known to produce an unc.-ditionally

stable splution. A backward difference method was chosen, ton, to solve the

problem presented by equations (2.4", 2,4a, 2.5"). An investigation ofthe stability of the particular algorithm used has been carried out aid is

described in chapter VI, B, 5.

Whenever one of the equations is used to calculate implicitly the distri-

bution of either var.bI*.Atluture tb* (subscript n~l), the distributionof the other variable is its last known distribution. Furthermore, the

two equations are used alterrstingly in a pair of time steps, such that in thefirst step one first selves foe T, then for P, and in the second step, first

for P, then for T.

a. Solve (2.4)" and (2.4a for for all Jg and P,0 knownJnnl kno Jwb. Solve (2.5)" for PO for all J; To anj P, known

Jn2J.

d. Solve (2.4)" an] 2.4a)" for To foe all J; 1 Tno and Pn0 known

(2.12)

then replace n by n.2.

Details of the computing algorithm are described in Appendix C.

The .jain difference in solving for the tmperature or pressure distri.-

bution is that the coefficients U awd D in equation (2.5)', are dependent

on temperature and pressure. their present tmae values are determined at

each time step preceding the solution of the equation set. This is done bymeans of a double interpolation (with entricA T and p) of tabulated values

23

f or (OM/WT)p and (Bp/aT)M, which are based on equilibrium data fco moist cloth

obtained by Urquhart and Williams (22).

3. Remarks on the Boundary Conditions and Variable Water Vapor Diffusivity

(a) Moisture condensation of the skin surface

The boundary condition (2.8)" relates the mass transfer at the innerocloth surface to the vapor diffusion through the air gap. In it, ? s0 is

the water vapor pressure at the skin surface, or, more precisely, the saturated

vapor pressure at the temperature of the skin surface, since the skin is

taken to have no absorptivity for moisture. In calculating the pressure

distribution, the above relation is used first, implying the assumption

that there is condensed Moisture on the skin. If this assumption is incorrect,

(as revealed by a test of the pressure gradient at the air gap and a record

of the moistnr- condensed), the pressure calculation is repeated once, with

a simpler boundary condition stating that the moisture flux through the inner

cloth surface is rerot

P, a P8 (in the 6-slice model of Figure 2.1)

A distinction has to be made in this context between the phases b and c of the

alternating scheme (2.12), a. n.iugh both phases -apply the same basic eqi÷'-.-z,

In phase b, the skin surface temperature at "future time" T-OO0 Itis0, n~liS known and thus also P o, .. In phase c, when n+2 means future time,

this is not true. The "plesent time" value T;s.Oa.n1l is therefore chosen as

a trial value. If the ensuing calculation of teperatures in phase d does

not confirm it closely enough, an improved value for the skin surface temperature

is chosen and phases c and d are repeated until convergemce is achieved.

(b) Treatment of dry cloth slices

At sufficiently high irradiation intensity, the evapocation of water

and the moisture flow to the cooler interior of the system will eventually

cause parts of the cloth to become dry, starting from the outer surface. A

slice of the cloth is called dry when either its temperature rises above

some fixed level (i.e., 180°C) or the relative humidity in it drnps below

a minimum (0%). The dry slices are now excluded from the pressure calculation

based on the moistuts equilibrium relationship. If, however, the equation

24

. , , . . . ' ,

for the temperature calculation (2.4)" is to remain unaltered, some meaningful,

if very low, pressure'values must ae assigned to the "dry" slices. The storage

capacity of these slices for moisture can be considered negligible as long as

they stay "dry". Therefore at these points the pressure values should not

contribute to the temperature response as exp~ressed by equations (2.4)"; the

second derivative of pressure may be co-sidered to be zero, which turns these

equations into pure heat conduction equations. In other words, the pressure

profile is to .have a linear section, which in equivalent to stating that

the values in question are only determined by the diffusion from the known

vapor pressure in the first moist slice to the outside surface,

If m is the number of dry cloth slices, then P1 and P+ 2 0 re related by

a modification of the boundary condition (2.6) which takes into account the

lengthe.e.1 path of diffusion to the boundary:

No dry slices:

2fD 0 K 0(26)"P*l 2 2fD .0 c 2.)0

0 c

m slices dry:

2f D o to -[ Ax 0Pol P - 0 0pO =~ Zf. -1 € (2.6)"

a 2fD + (2m + 1x 0 AzX

0

(See Pigure 2.2 below)

Accordingly, the pressures in the "dry" part of the cloth are found by

d interpolation.

24a

I.,

dry dry

, I I° . !

IlD c . Ic

C CO

Figure 2.2

(c) The diffuslvit7" Of water vapor theough air is given n.• a funtlton of

temper~t tur eST 3/2

The effective coefficient .ef .•..ffusion lD' can be calculated f~rom 9 :l4ii•

an average temperatwre (e.g., 5O°C)a

17" vav....(a.l3)(1 - iT)""

where £ i~s the tortuosit.• k'actct.

The prOgram allows fct the use of either a fixed value of D" or a

variable one, to 1w determined from the local vapor pressure and temperature

at each cloth slice and at each ti~e. step.

4. Sumasry of the Computer Program

(a) Input

The program requires four groups of input data. The first type of data

describes the moisture-temperature-water vapor pressure relationship for the

25

I't

moist cloth. The functions

a)* f(T,p)

and

( f(Tp)

arm tables with entries T and p. Their values are also listed in Table G.2.

They are based on original data obtained by Urquhart and Williams, (22), which

are compiled in (5).

"The second group of data contains constants describing the properties of

.,e physical system:

U0

i ufD° "

o omax

(L/k)A

(-/k)c

Tcrit(kQp)€

The third group comprises constants referring to the properties of the

mathematical modtl :

A tolerance for the rial value of the skin surfact temperature rise

MA -(AX)

4X;

4x0

The number of double time steps

26

An indicator for constant or variable water vapor diffusivity

The number of single time steps between printouts

The number of cloth slices

A time mark for optional change of M

An alternate value for hA

Last, a number of parameters define "experimental" conditions, namely the

surrounding temperature and relative humidity, the irradiation intensity, its

duration and pulse shape:

Psurr

Ts-rr

An indicator for the type of irradiation pulse shape

I

The pulse width

The trapezoidal pulse shape parameter

(b) Output

2hw results of computations are presented ass

The dimension)et. temperature distribution. in and at the cloth andskin simulant

The dimensionless pressure distribution in and at the cloth

as functions of dimensionless time.

As a by-product, the values of the variable coefficients in equation (2.5)'

are given as well as measures for the moisture loss due to evaporation at

the cloth surface and condensation on the skin simulant.

A record of the number of cloth slices having dried out is kept, and

optionally the value of ihe maximum ctitic-l enthalpy assumed by the skin

is indicated.

(c) Program description

A common experience with any complete description of a s~seable computer

program is that it tends to become as involved as the program itself appears

at first glance. It is hoped, however, that the computing process is explained

sufficiently by the description of the algorithm which forms the basic part

27

, , '"" I " II

of the program (Appendix C) along with diagrams showing the flow of control

through its mainportions, a commentary on the purpose of the subroutines

used and the program listings, all of which are included in Appendix E.

The program in form of symbolic and binary cards may be obtained upon request

from the files of the Fuels Research Laborptory, Chemical Engineering Department,

Massachusetts Institute of Technology, Cambridge 39, Massachusetts.

S. SlIbility

A main distinction between the formulation of a differential equation

in analytical or finite-difference terms arises from toe fact that the latter

formulation may be satisfied not only by one true solution but also by a

number of so-called extraneous solutions contributing to the total calculated

function. If these are not to disturb the solution of the problem, the

condition t.w be imposed on. them is that their amplitudes decay with time,

i.e., that values of an extraneous solution u at a space point i at successive

time steps n and n*l have an amplitude ratio or growth factor • • 1.

S Iultn

This is the condition of stability of the solution process.

Whether a finite-differen4e method has the property of (unconditional

or conditional) stability is determined by the structure of the particular

computing algorithm chosen (see VI, B, 2), in conjunction with the coefficients

of the differential equations. An investigation of the stability problem

associated with the alternating explicit-implicit scheme used for the moist

cloth-skin simulant system follow below *. It results in a simple criterion

for stability which is dependent only on the four coefficients of the equation

sys tem.

In the derivatioths below, a nomenclature exclusive to this chapter ix

used$ T, P and the coefficients X, L. •, I only retain their meaning. The

superscripts denoting "dimensionless' are dropped, aid for the derivatives,

the abbrevia tioia (2.10, 2.11) of chapter VI, 3, 2 are used.

SThis study was done with the kind assistance of Professor P. L. Thibault

Brian, Department of Chemical Engineetring, M.I.T.

2S

Temporarily, the system is assumed to be linear, i.e., to posses$ constantcoefficients T, g, E, CD. The eqmtions in question are

Ta A'Txx * 3( (3.1)

Va - 6T *x 5P x(3.2)

Assuming T and P to conasist of separable functions.

T (• () f (3.3)

P(X9,Z)f s (s) (3.4)

the system becomes:

Xf • (r"/r) * I (s"/r) a f' (3.5)

Ef • (ru/s) * *g (s-/s) a g' (3.6)

If the variables are truly separated, then the functions

(r"/r), . ',s),W("'r), an (s"Is)

must be iulependent of x.

Setting (rt/r) a - i*•a ). we find

(x) a(x) (3.7)

we now solve foa f(9) and '(9),

Af(o) *4() () (3.8)

*f(9) &Ag (9) s 5(O) (3.9)

Prom (3.10)

f'(e) " ) (6)_

( U - (3,10)

(g)

29

Subatiuting into (3.9) we find:

Therefore,

ith ((C) - Clel1 0 + C * 2 9

with

Analogously, the solution f•,e,( is.

€() ceiA1 I ce'20

3 4@The total solutij then iLe,

ST " 9 (40erl + OPA2 )(cce mk: * c6 ain mkx) (3.13)

" (c 3 Olf * C 4 eno)(c cOS *kx * c &an uz) (3,13)

T and P are wave functions in distance, Their "growth" with tive depends onthe four coefficients as follows

if iv > T and P decoy with timeiLf B• - rc T and P see independent ofr tint

t iD < T and P increase with tineSince only the first case Is pbYsically Ieaningful, a finite-differenceprocedure to solve for T and P moat also provide solutions that decay with

time,

We deflito, o?

TJon T (xaJ*4x, #".AO)

J'n a P(x*eJeAx, $".,a)

30

In the finite-difference notation, we introduce as usual,

r" a r J.1 - 2rj * rj1

It follows, for the functions introduced in (3.5, 3.6)

(r"/r) ie(J-l )Ax .2 e JAx + *e(J+l)Ax

*i0JAx

(r"/r) eiuz - 2 + eihx u -4sin*((•i,2) (3.14)

and likewise,

(r"/r), (re/8), We/0), (50/4) a-4sina(w•/2)

Introducing the abbreviations,

c o - 4i.sae( /)

41-A . sing(uAX/Z)y 4(Ax) *p - -4i A. . si 1n(s•x/2)

(AX)PI- 4- sin*(uAx)

a - - 4D------ . sin(w•)

(As )o

we rewite. the equation system in the sequence to te used for its solution

(see V1. 3, 2):

" an~l " i n .1 anl"f 31a

" "- 2+l ' Snnl "- m (3,16b)

"i 0+*1 " 6gn*2 a $"*a " -n*l (3,16c)

" "it n,2- P"v*23 0 lfn2 -" 3nl (3,16d)

This represents a double time step scheme in the second halt of which the

sequence of operations is inverted (tihu "alternating"). We are Interested

31

in determining its behavior over one entire double time step, i.e., in finding

a relation between fn+2 and fn and likewise between gn+2 and gn*

From (3.16a) is extracted,

f * fn - Oan (3.17)

Substituting this into (3.16b) yields,

Gn+l (1 ; + ÷I +) " n?()"n (3.+))

Substituting these two expressions into (3.16c) results in

- y(2 + *)fn - (6 * l)*(s + )gn+2

y(2 8) 1 + a3 gn 0 (3,19)

and into (3.1l6) in

(I # 0*f2 - fa (I ÷ l ))*n+2 + g ga 0 (3.20)

Having elimenated all functirno values at time n.l, we now actually consider

the double time step as one pe.,od, rewirting the last equations.

-Y(2 + S)f - ( * 1)(6 * 1)g*+l

+ P [ (2 + 6) * I + ,g 0 (3.19s)

(l. )f f -f , (l*S)0g *g 5 0 (3.20a)

Next a relation is to be found foc the time behavior of one of the

solutions, i.o., '0)) expressed as a growth factor

Thus the functions f are to be eliminated first; this is done most easily

by substituting f from (3.19s) into (3.20s), hereafter also fn 1 from the

32

same equation, having increased the index by 1. The result is,

(1 + M)"(1 -* - I Ivo)8 + Oy( 2 + 6)(2 * 6) (+ U* 8)) +"U" " "c". (3.21)

The stability condition . the function g is

It is not necessary to go through the same derivation to find a corresponding

relation for the growth factors associated with the function f. Consider the

double time step to go from n*l to n+3, comprising the solution of equations

(3.16c,d), i.e., n+l to n+2, followed by that of iquations (3.16a,b), i0e.,

n+2 t- n+3.

"- a -fngl "On f f

" n Yfnl - 8n~l a gn*l - In origiual double stepa - 6*

"- YfnUl " a8n+2 * gn2 n Suez n/(2.12)

" 6n.2 " •.2 • *3 " nol new double step

" n t'hn3 " OgnU2 n fn+3 " fn*2 n+1 "n+3

" Yfn+3 - Og n+? a g n÷3 -"n+3

If for the moment we replace

Tby PP byTtby P'*by 40*by Y''r byA'

f by g*g by f*

then the system nel to n*3 is formally identical with the original one. There-

fore we know fram the foregoing discussion the quantities a*, b', c* that will

33

7r

be associateo with the equation

g" ,*." .• c" -Og (3.2k)

go

£

or because of the r eplacement definition

f fe!** re÷ eb*2 c' 0 (3.22a)

where a*, b*, c* are expressed in terms of a* ... 8*" Resubstituting S* G S

etc., we obtain the -erms in (3.200) in terms of the original quantities 1 .0# 0,

only to find that they are identical with those in the growth factor relation

(3.19) far g. "his could be expected from the fact that the equation system

and the alglrithn far its solution are symmetric with respect to T and P

(or f and g).

It remains to be seen under what conditions the stability criterion,

identical for f and g, is satisfied, i.e.,

_ 5 1 where at * bt c • 0 from equation (3.21)

This can be transformed as i1,.ows:

Ii *-(b/2a) . t < 12

resulting in

-c•a * b

Using these terms as expressed in (3.21) we have,

- 1:S (1 *Co)sCl * G)- (-*( O)X

-Oy(2 * S)(2 * 6) a ( *)

From this fol•lo:

34

-1 ; (1 * 26 * ,)(l + 28 60) - (I + 26 G*) - +y(• * ,)(2 0 8)

- (1 * 26 * 8 '

or

O < (2M * 2 )(32 * 6 S ) G Py(2 6 6)(2 + 8)

which reduces to

Thus the condition of stability is satisfied whenever • / KS . 1.

So far, we have assumed the coefficients X, 9, U, and D to be constant.

Since in fact d and I are evaluated anew at each time step by table inter-

polation, and N can be either variable or held constant, one would have to

ascertain -.hat, their values always fulfill the stability condition. A

special subroutine was therefore written into an intermediate version of

the computer program to test the criterion, which indeed was never violated

during computations.

6. Conversence

The question of co vwrgenct of th* finite difference solution method

bears'upon two aspects of che problem:

(a) Ultimate uniqueness of the finite difference solution

(b) Bmpnse of machine time versus accuracy

One is interested in determining the convergence of the finite-difference

solution, wnem the difference net is refined, to the "true solution" that

satisfies the finite difef~ence system. (The true response of the physical

system, as obtained experimentally, is valuable foe comparison but not

decisive for the consideration of convergence, since the mathematical

abstraction implies simplifying assumptions). Unfortunately, no analytical

solution of the system in question is available by which the accuracy of

the calculated one could be determined. All one may ask for it ultimate

convergence of the solution to a unique result, when finer steps in time and

3S

space are taken. In the absence of general methods for r3tablishing such

behavior for coupled equations with coefficierts depending on the response

functions, a somewhat experimental approach was used to investigate it.

If one expresses the terms occurring in the finite difference equations

in Taylor series forms, then the total truncation error for the two functions

To and Po comprises terms containing AS0 , (4€)* and (AX)*. It may be

expected that the truncation error diqitiishes as these factors are reduced,

as it does in simplec, linear systems so long as the coudition for stability

is obeyed. This latter requirement is in fact fulfilled by our system.

In order to observe convergence rather than prove it rigorously, calcul- L

lations have been carried out to test the influence of different values of

the intervals t Alto and A60. First, the resulting response curves are

expected to appear in the same sequence as the zspective values of A1°C ps

and/or A0°. Second, the' rate of change of these curves Is expected to

become smaller as successively smaller finite differences are used, i.e.,

they should converge against a final function. Figures 2.7a,b illustrate

the results of these calculations, i.e., the temperature response of the

skin obtained at constant irradiation intensities of I s 0.5 and

1.0 cal/cmssec and dimensionless step widths,

1 =/2, 1/6 and 1/18,C

* 1/10, 1/30, 1/90 (resp.)

A u 0.0031 ... 0.25

For comparison, the temperature responses of the cloth are also depicted

for the case of Ieas * 0.5 cal/cmasec, (Figure 2.7c).

Inspection of the calculated results shows that the curves for the skin

temperature lie in the expected order with respect to each other. (Note

that the sequence my be ascending or descending a" the truncation error

my change its siga with time).

A similar behavior was observed for the cloth temperatures, at least

over a time range until so a 2.0 (Figure 2.7c).

Furthermore, it appears that successive refinement of £1l: a and

36

I

A0O results in a progressively smaller variation of the curves. For Ioa.

0.5 cal/cm2 sec the case of AX0 U 1/1 was not computed. At I a 1.0 cal/camsecc

however, the ref inement of interv'als in fact leads to convergence in that

the response obtained with Ae a 1/6 and AGO * 0.00556 very precisely repro-c

duces that obtained with AZ0 a 1/18 and AGO • 0.0031, within the low time

range until &0 a 1.0.

The agreement found from these trial calculations certainly has to be

taken cum grano salis. In particular, the comparison with the case of finest

"step widths ha not been carried out over a very large range of time, first

since this alone would have required excessive amoints of machine time, and

second, because one can hardly expect a perfect reproduction over such a

range from a system with slight built-in discontinuities, such as the effect

of cloth slices drying out. (The approximative treatment of those cloth

slices, which might introduce slight deviations, mast make itself felt more

strongly in the 18-slice model than with 2 or 6 slices, simply since the

condition applies to a larger number of such slices as time goes on).

For practival purposes one has to mtae a comparison between the accuracy

of the computed solution and the e xpen&e of machine time required to obtain

it&, The amount of comp•t. time used per time step strongly depends on the

fineness of the slices into which the continuum Is conceptually divided,

and on the length of time intervals, since the former affects the size of

the equation system to be solved, while the latter is imediately related

to the progression on the real time scale. The attempt at finding experimentally

the degree to which a somewhat "unique " or final solution is gained by

refining the spsae-time mosh leads one to believe that the 2-slice model

does not provife sufficient accuracy, producing deviations of up to 101

from the 6-slice case. On the other hand, not much accuracy Is gained by

preferring 18-alJcea over 6. As far as computer time Is concerned, this is

a fortunate balance. The actual amounts of computer time required per second

of real time are given in Appendix B.

7. Time-variant Irradiation Intensity

In the earlier studies of a mathematical model to describe the moist

cloth-skin system the intensity of irradiation was assumed to be constant.

37

This represents in fact the case of the experiments carried out at MIT; here

an exposure shutter was kept open for a certain time interval and then

closed rapidly, thus forming a square wave of irradiation.

The situation is different in the case of a thermonuclear explosion,

where the intensity of irradiation is time-variant, exhibiting a very

pronounced peak value and a lon docaying section. (An analytical expression

for this intensity distribution was given by Batter,.(23).

Thus the temperature-pressure-tiae response of the system will be quite

different; in particular it may be expected that an irradiation of constant

intensity much smaller than the peak i"tensity of a thermonuclear pulse

will produce a given temperature rise in the skin. At constant irradiation,

temperature rises monotonously and the moisture migrates to the cooler

interior of the system, partly condensing on the skin surface, and although

the moisture on the skin may later vaporize, there is no reversal of heat

flow during irradiation. With an irradiation pulse of finite duration and

energy, on the other hand, the temperature rise will eventually be followed

by a decay, first in the cloth, later also on the skin surface and in the

interior of the skin, Furthermore, it can be expected that there will be

a reversal of the -heat flow %nd with it will be associated a reversal of

the moisture flow.

In order to show these differences between the responses to constant

and time-variant irradiation as well as to allow&a coaiparison of theoretical

results with those obtained from experiments that actually involved a square

wave, the mathematical method has been set up to accept a variable irradiation

intensity as a boundary co.•dition.

This facility allow* foc four cases of intensity distribationst

Step function

Square wave

Trapezoidal wave

Thermonuclear explosion pulse

It is introduced as followas

It.e irradiation intensity affect* only the heat boundary conditionat the

38

outer cloth surface

Io Uo(Tc=0 T OT-Ko ]jj (2.7)

or in dimensionless finitc-difference terms:

I - u° . To &1o (2.7)"0 pu XF0 hX

Note that elsewhere in the formulation of the dimensionless system I is

used also, but merely as a reduction factor for dimensionless notation. If

one st..ll retains some characteristic value of intensity, such as the maximum

of a pulse distribution,. as a eference value, one is free to introduce

into the boundary condition a term expressing the relative irradiation intensity

Io tO / l ,011 " -U° ' T O =O 0 . T o- ~ " 0; - 0T

This value is computed at each time step, preceding the solution .f-r A.,e

temperature distribution.

(a) Description of irradiation pulses

The maximu irradiation intensity is prescribed in absolute terms

(cal/cmasec). Since the prograu used a dimencionless time scale- throughout,

all characteristic extensions of pulses in tine are prescribed in dimensionless

terms, (Figure 2.3).

For the square wve, one quantity r suffices. The time scale of thechahtrapesoidal pulse requires two quantities: S defining the end of its first

ramp, and 9 the end of the pulse.char

39

I0 0lot I I

oax maxOR"x n Iomax -

a0 6 0 a 0 90 O 0 90Bchar er char char

Figure 2.3

IRRADIATION PULSE SHAPES

7he thermonuclear explosion pulse requires one characteristic value 6hlocating the peak of the distribution, and a normalized shape function (20)

indicated in Table 2.1 (see fo:. example, Figure 2.20).

Table 2.1

TIRNOM•NLMEAR PULSE INTENSITY DIS2RIBUTION

Reg"om Relative intensity Energy delivered at time 800/9char Io o)/I (cal/calsec)

0 - 0.3 o.S6S(B~O° ° ) *o~ •O *° 0.25•char charon0 - 0.3 650 = - g° 08

O/char ) char tsS0 1.65 1.,-72 0. 34 7 1 [o0.52•4 o•* .(-aar. COX char char

0

"charar

so 00

1.4- 1.4 10.4342( j -. 1'". .00 r2.696 & char -,J5char LOS) I.. char

*Cý - I

h at

40

For comparison, the program verifies the areas of pulses, i.e., the radiation

energies delivered.

(b) Remark on the squiare wave

In computing the r esonse to a square wave input, a slight difficulty L5

sometimes encountered at the end of the pulse. The sudden. termination of

irradiation leaves in the above mentioned boundary condition only the tern

for the heat losses. Since a surface tempersture val~t is taken to govern

the outward heat flow during an entire time step, the size of time steps may

have to be reduced at this point so as not to simulate too large a heat

Loss during the beginning of the relaxation period. At a later time, when

the sutface temperature has dropped, the original intcrval size can be

resumed. The occurrence of this disturbing effect depends on whether a

sufficiently high surface temperature has been reached at the end of the

pulse. A convenient way of changing the time.step size by altering the

modulus (Ah 0)8/60° Is provided in the program.

Me effect described.does not occur in the cases of a trapesoidal or

thermonuclear pulse shape, whore the process is sufficiently smooth.

8. Burn Severitz Correlation

Different ways of rt.ting the temperature response of the irraaSated

cloth-skin system, to the severity of a resulting burn have been delt with

in former studies, (3, 5). Among these, a correlation based on the concept

of a critical base temperature and a constant skin eathalpy increase appeared

most promising. It states that a burn of given severity iseffected by a

certain minimum value of that part of the skin euthalpy which lies above

a critical temperature level, (see sketch below).

T (0C)

Tcr it

X ( s41(c)

41

This criterion has been tested earlier in particular by comparing University

of Rochester data on mild 2# burns with the results of skin simulant

experiments. Here the mximua of the critical enthalpy (i.e., the shaded

area in ths above plot) was taken to occur at the time at which the measured

temperaturp closeet to the skin simulant surface reached a maximum, (effected

by a square wave of Irradiation). Temperature distributions were plotted,

extrapolated towards the skin simulant surface and. their area above Tcrit

measured. A constant thermal energy increase of 0.89 - 0.93 cal/cms abovea critical temperature of 53 0 C was found to agree with most of the University

of Rochester data.

The same procedure can be applied to temperature distributions obtained

from theoretical calculations. It is more practical, however, to include thisevaluation in the computing program, as the entire temperature history isavailable to it and in particular because this provides a more precise way

of finding that distribution which yields the maximum value in time of the

critical entha t py:

X •a - rit >? T d X f (- ) ( 4 .1 )

(MOen a pulse of irradiati, a applied, the maximum value need not be

associated with taw maximum skin surface temperature, as the temperatures

in the interior of the skin simulant may still continue to rise while theirradiation intensity and the skin simulant surface temperature begin to

decrease).

The computer program described In this report is set up to calculate

by means of a Simpson-rule integration the maximum critical enthalpy attainedat any time during the process. The quantities (Ico) and T are to he

a critprescribed as input parameters.

The temperature distribution in the skin depends strongly on the

assumptions made for the heat and mass transfer coefficients, which can only

be defined within a certain tolerance. Since the burn correlation ietsgral

represents only the top part of the area under a temperatuze proiile, it

NIold be kept in mipi that its value is int tuenced even note by the ctioice

42

of these coefficients. Nevertheless, with this restrictiont it appears

useful to obtain in this direct way the information regarding the burn

correlation that the theoretical solution is able to supply. Moreover,

the facility can be used to find the maximum total heat content of the skin

simulant if the critical temperature 3s given the value of the surrounding

temperature.

C. DISCUSSION OF RESULTS

hrliet studies (S) have shown that the theoretical solutions obtained

for. a system involving diathermanous cloth tend to agree better with

experimental observations than those found fao an opaque cloth layer. This

is true for the case of dry cloth as well as for that of moist cloth, and

may be explained by the direct admittance of part of the heat received,

into the i-terior of the system. The pcrpose of extensive computer calculations

devoted to the opaque cloth situation was fourfold:

First, the comergence of the maserical solution with refinement of

the difference-net wa tested. The results of these calculations r e discussed

in Chapter VI, 3, 6.

Next, a number f comnputations was designed to show the influence of

the transfer coefficient-. Io, Us, Ko, Ki and the water vapor diffubLvay D"

on the theoretical response as well as* its agreemnt with experimental

results for constant irradiation levels I - 0.$ - 3.9 cal/coasec, 30C

and 80 relative humidity.

A further a&Im ms to imestigate the type of response obtained under

time-variant irradiation o0 various pulse shapes ad equal total energy,

at maximm intensity levels I a 0.3 - 1.0 cal/cnsset, 30°C and M

relative humidity, using one set of transfer coafficiemta.

A last group of calculations was carried met io order to test the

agreement between a member of experimental results from squate-wave, short

exposure irradiatinn and the theoretical sooltioesobtained rtth *dpntical

irradiation, temperature and relative humidity conditions, (ro0s• a 1.0,

1.4 and 2.6 cal/ca.sec; 19gC; 63 relative humidity). The transfer coefficients

43

and the diffusivity were varied in these tests. Purthermore, so-e theoretical

results for the diathermanous cloth case in comparison wit; experimental

curves are presented.

1. Nunerical Values used in the Computations

The values ot the parameters used are summarized below. A complete

reference, in particular on the estimation of the transfer coefficients,

is given in (5).

(a) Properties of the cloths

PO a 1.57 gm/cm for cellulose

PCLc a 0.027O gm/cm5 for * os. standard cotton sateen

Lc a 0.043 ca (measured)f • 1,- ( c /P• - 0.394

C a 0.34 cal/gmCkc w 1.62 # 10-4 cl/cm sec C, based on (L/kz) a 266 cm see C/cat1Ir a 0.105 pig/ca sec ms ag at an average temperature of 50°C and

with a tortmselty fartur of X * 1w2.4 (see VI, D, 3)

(b) Properties of the skin simulant: see V' C. 1.

Trans fer Coefficientss

The following estimates ere used as a basis-Surface heat transfer coefficient:

A dimensionless value 11 a 0.1 ws used, ccrrespondirg to0

U w 18.6 * 10" cal/cmssec°C

Air gap beat transfer coefficients

U: 5.0, corresponding to Ua I 3.76 103 cal/cmasec°C

Surface moisture ,ransfer coefficients

K* a 0.23agm/cosec - Mg

Air Sap moisture transfer coefficient trffective)s

x a 14.4pgm/cmec m HS

The variations applied to the transfer coefficients are discussed in the

following sections.

Heat of desorptiont

X a 6.0 cal/jm emO

44

In all of the computations, the 6-alice model was used with a modulus MA 5and a skin/cloth property ratio group * S.

2. Constant Irradiation - Influence of Transfer Coefficients and

Diffuaivityl Comparison with 9xperiments

The estimates of the parameters Uo0 i, X. o, K and D" have been

established in (5). Considerable tolerance has to be allowed for these

values, particularly regarding the effective water vapor diffusivity V'

and the mass transfer coefficients. D" is subject to an uncertainty due to

the tortuosity factor which may tange between I and 0.2 or less. It also

enters into the derivation of the mass transfer coefficient for the air

gap. The theoretical results vary rather strongly with changes in the

choice of all these parameters, easpecially at long exposure times.

In tart, once it has been shown that the experimental results can be

reproduced closely by a calculation, the comparison between theoretical

and experimental responses may be looked at as a method to find values

of the coefficients rather than as a check on the validity of the analytic

model.

The following discussion concerns calculations involving a surrounding

teraperture of 30OC and i ; lative humidity of M0g.

Fimure 2.10 shon the dimensionless skin simluant temperature response

at simulated depths X° a 0.05 and 0.15 as a function of dimensionless time.

At an irradiation intensity 1n a 0.83 5 s/caes, the transfer coefficients

are varied through the ranges

Us a 0.1...0.02S or U u (3.76 .#o694)010 4 csl/clse

U0 S.0.,.0 0 0 ot UL * (1,6 *.*#3?.2)*104 cal/camsee

0 - 0.2 ... 0.05 Vs/csmm HS

X- 14.4 ... 3.6 ugu/ctm HU

The effective diffusion coefficient DV is held at a constant value of

DO a 0.105 p"A/cn sec -m Hg (except wht*r indicated to be variable), It

Is seen that te, I aM U have the strongest influence, while the surface

45

heat trrnsfer coefficient only affects the solution at longer exposure tines.

Figures 2.11 and 2.12, based on Figure 2.10, present a comparison withthe experimental response, determined also at 1° a 0.83 cal/ctasec. Two

cases having different transfer coefficients are given. The experimental

curves describe the temperature at simulsted depths X* a 0.028, 0.038 aid

0.122 ca (thermocouple locations).

The theoretical curves referring to these sae depth values have been

obtained from cross-plots of the calculated data. It appears that a not too

drastic alteration from the original estimates for the transfer coefficients

(by factors 1/2 for Ui and 1/4 for U0 , K , and X I) leads to quite good

agreement with the measured temperatures.

Figure 2.13,also at 1 0 0.83 cal/cm, shows the variation introduced

when variab1,t diffusivity values DP', based on equation (2.13), are used in

the calculations instead of a fixed average value. This does not seen to

improve the agreement in the above case; in other examples, however, which

are discussed later (VIC, 4) the refinement has indeed led to a better

approximation, kick means a different combination of transier coefficients

may be needed here for closer agreement.

In Figure 2.14 the inflIte=ce of an increased value of U and decreased

ones for Us, 1o and KI is shma again for an irradiation intensity 1° -

0.5 cal/coesac.

Figures 2.15, 2.16, and 2.17 present a comparison with the experimental

response at intensities 0.5, 1.26, and 3.96 cal/camsec. The irregular

appearance of the experimental curves at 1 a 1.36 stens from a partly

scorched ;loth layer. The cases of 1o a 0.5 and 1.26 show rather good

agreement, with the theoretical curves being slightly too steep at tonger

times. At I1 a 3,%# the position of the calculated curves is high from the

beginning. It appears that in all of these cases the application of a

variable diffusivity (according to equation 3.13), with transfer ciwfficicnts

in the range indicated on the figures, would have yielded a considerably

better reproduction of the experimental response.

A comparison between the temperature responses of the moist and dry,

opaque cloth systems is drawn in Figures 1.11 (experimental) and 3.18

46

(theoretical). The two figures are not precisely analogous in that they

refer to slightly different skin simulant depths (xs a 0.028 cm and to

XA 0.05 0 xs - 0.014 cm). Yet they show that the theoretical response for

the larger intensities lacks the prnnounced change in slope occurring in

the experimental moist cloth curves prior to their intersection with the

curve for the dry cloth. In the logarithmic plots, too, this discrepancy

is displayed hy the slopes of tho theoretical curves which are too steep at

longer times.

The role of moisture in the system contributes greatly to the heat

transport towards the skin simulant, but only initially. Eventually, the

pressure gradients in the cloth reverse their direction (Which can be read

from the numerical results), leaving conduction as the only mode of heat

transmission into the interior of the system. An inspection of the above

figures sugg-sts that this, effect is not simulated strongly enough by the

theory. The reason may be,*ght in the neglection of bulk flow of water

vapor, in our model, but it is also possible that refinement is needed for

the treatment o( completely dry parts of the cloth which so far involves the

assumption of a partially linear pressure profile.

Deviations between theory and experiments as discussed above are less

striking in the case of a I. thermanous (moist or dry) cloth. for whirN

an analogous comparison based on earlier studies (5) is given in Figures

1.12 and 2.19.

So far, the theoretical results have been judged by the response at

small skin depths relatively close to each other (0.028 and 0.03$ cm). At

larger depths (0.122 cm) a deviation is common to most cases, which cannot

be neglected. It may be accounted for by the fa-t that the value used

exclusively for the skin/rloth property ratio ItIi(bCp) /CkCp)c c 5.0)

is the one based on the dry cloth system and need not hold exactly true for

the moist cloth situation; this would distort the sisulated skin depth

scale. As yet the effect has not been tested.

The water vapor diffueiy ity D" has not been given alternate (constant)

values In the calculations, since it affects the results mainly in the shost-

time range, while the major deviations of the calculated responses still

occur at longer times.

47

The preceding discussions concern only the predominant parameters among

those not known precisely. Their influences at least can be established

by the use of the computinj program. The effects of further variations,

e.g., of the tortuosity factor occurring both in D' and XL, of the cloth

heat conductivity k and 1, as well as the surrounding temperature and

relative humidity can likewise be determined, requiring a large but not

excessive amount of computer time. In order to define a narrower range

for the first group of parameters (U0, Uit ro, rX, D") however, not only

a considptable number of calculatlons would be needed but also a larger

set of experiments repeated under each condition.

3. Time-Variant Irradiation - Influence of Pulse Shamp

A set of calculations has been carried out in order to observe

qualitatively th' response variation with irradiation pulses of different

shapes. In order to provide soca common basis for comparison, the areas of

the p41ses (i.e., the total irradiation energies) are kept the same within

each of two groups. The first group is based on the total energy delivered

by a thermonuclear pulse distribution with peak intensity I O 0.5 cal/cuasec

and peak-t.ime ch a 0.5 until 6 0 5..0 this energy equals 0.545 cal/cm'sec

(its dimension stems from the -a- of a dimensionless time scale but absolute

intensity). The second group useýs a thermonuclear pulse with peak intensity

I • 1.0 cal/cmasec as a basis, with an area of 1.09 cal/cmssec until

e 5.0.

Square and trapezoidal " ,s at different irradiation intensity levels

&4J the thernonuclear pu~se are used s input form. The latter, fat conven-

ience, shall be referred to as the "true pulse. The other conditions assumed

and comon to all of these calculations ares surrounding temperature 30°C.reoative humidity 80%, and transfer coefficiants and constant water vapor

diffusivity as stated in VI, C, 1.

The results of the first group of runs are showm in Pigure 2.20# The

four cases presented are the true pulse with peak at 0.5 cal/cmasac. a

square wave distribution with intensity level I o 0.5 cal/cmsec, a

square and trapesoidal wave both with I a 0.3 cal/cmsew; these distri-

butions are indr;.jted in the upper part of Figure 2.20. The loet part

48

of the graph shows the respective temperature responses of the skin simulant

at simulated depths Xu 0 0.0167 and 0.15. In order to permit a comparison,s

the temperatures are given in reduced, but not dimensionless form, i.e., as

value of To -. I oX (- AT. (kjL) ).

It is not surprising that the square wave with I a 0.5 cal/cm2 sec

yields a higher temperature response than the true pulse of same intensity

level. However, the latter response shows values still considerably lower

thar those produced by the square and trapezoidal waves at 60% of its peak

intensity, (0.3 cal/cm sec) near the skin simulant surface as well as in its

interior.

Figure 2.21 shows the results of thesecond group of calculations, comparing

the true pulse function of saxioun level 1.0 cal/cResec with trapezoidal

waves at 1I3 a 1.0 and 0.6 cal/cm2sec, a square wave with I 0 0.6 and

0.4 cal/cmasec, and the case of constant irradiation intensity 1.0 cal/cdasec.

The temperature response To Io is given far one depth value X° a 0.0167.

As in the first group, the square and trapesoidal waves not only at 0.6 Dut

even at 0.4 cal/cnssec lead to much higher skin iimulant temperatures than

the true pulse. This can be attributed to the fact that the latter irradiation

distribution yields a faster rise of the cloth surface temperattire and of the

water vapor pressure neat dj e cloth surface than square ant 1rapesoii. iaves

at lower intensity levels, thus increasing the beat and moisture losses at

the system surface. Furthermore, the reveral of the pressure gradient

in the cloth takes place at an earlier time with the true pulse so that during

the section of decaying irradiation intensity the conductive heat transfer

to the ehin sinulant is counteracted by the moisture transport directed

outward.

In none of the cases described do the temperature gradients in cloth and

skin reverse their direction, so that within the calculated time range the

decay of temperatures in the skin sioulant is only due to the mass transfer

from its surface to the cloth. Naturally, the temperatures in its interior

keep rising for some time while the surface l smparature has begun to drop.

These conditions can be read clearly from the numericaL restits; they

may only be stated here in passing since a complete representation of the

49

results would require too much space. Reference is made though to the logarithmic

plots of the skin simula~it temperature respenses of all the cases, given0for simulated depths up to X 0.95 in Pigures 3.8 through 3.16, Appendix P.

4. Time-Variant Irradiation - Comparison with Experiments

It is clear from pr',vious discussions that a rather wide range of

experiments is required to serve as a yardstick, if the capacity of the

mathematical anslysis to model outr physical system is to be examined. The

number of physical coefficients which are subject to an estimation only

underlines this necessity, In the effort to come to a closer agreement

between theoretical predictions and experimental evidence, the judgement of

how well the model reproduces the situation and the *improvement" of those

parameters are intertwined, thus making the judgment somewhat relative.

The examples for a theoretical approximation of the measured response

described before have left some uncertainty regarding which of the five

most important parameters are to be altered and to what d4gree. They applied

to a constant irradiation input, i.e., to cases in which the entire precess

develop* in one direction. A ne rigid test for the model may be formed

by a pulse input, where the direction oi events ultimately is reversed,

even to the extent that part of the cloth layer will dry out first and later

attain moisture again.

Some calculations serving such a test have been executed, with conditions

matching those of experiments. All of then were done with a surrounding

tempet-ature of 190C, and 653 relative humidity.

The first group of these computations applied to a square wave irradiation

of intensity level 1.0 cal/cmesec and 4 seconds exposure time (1O a 1.62),char

In the calculations presented by Figures 2.22 through 2.24, constant values

DR w 0.105 pgagcm sec m g HS and U0 a 0.1 (U • 3.76 • 10-4 Cal/calmec°C) are0 0

used, while the other tra-wfer coefficients are varied in the ranges

-osU1 - 2.5 - 10.0 or Ul - (9.3 - 37.2) - 10"3 cal/cm sec C

K a 0.05 - 0.2 agm/cm'sec - HV

a 1 3.6 - 14.4 "ag/cmssec M Hg

(The combinations are indLcted on the plots).

so

.- /

Neither the positions nor the shapes of the curves referring to 3 simulated

skin depths appear quite satisfactory. Additional trial run. in the aboveran$e of parameter values (not presented here) havy not yielded any better

agreement. It is to be noted that the cloth is recorded by the program

to become dry throulhout after 00 a 1.6 (4w4 seconds, L.e., at the end of

the pulse). In the cases quoted first, half of the cloth is dry already

at Go" 0.95. Obviously the test is hampered by tha fact that 65% relative

humidity may be low enough to produce a situation where the moist-cloth

theory finally applies -only to a very small section of the cloth, namely

Just the last xli'e out of six slices, while the dry part is manipulated

according to the procedure described in V, 8, 3.

It is suspected that the water vapor diffusivity must not be excluded

from our parameter trial procedure. In fact, one calculation carried out

with vsriaole diffusivity (according to equation 2.13) and the "orilinal"

values for the transfer coefficients (VM, C, 1) yields much better agreement

between theory and experiment* if not beyond the maximum of the temperature

functions, (Figure 2.25). It seems that further improvement can be achieved

hy not too severt variations of the heat u& as** transfer coefficients from

their original estimated values, while using variable diffusivity.

Considering ihe requ.-.ed expense of machine time howevo&, the effort

had to be limited to a few trials.

The situation looks more favorable in a second pair of calculations

involving a square wave of intensity 0os a 1.4 cal/cussee and two seconds

exposure time,

In the case recorded in Figure 2.26, a constant diffusivity DO.

0.10 pgm/cm s&c ma 1% is used, while Figure 2.27 depicts the response

when D" is variable. For Uo,U 9, 10 ,h the original estimtes are used. Again

it appears that the assumption of a fixed average diffusivity should not be

made. As before, the cloth dries out rapidly while irradiated, until only

two out of six slices are left moist (contrasted with one in the previous

expeple, which may explain the discrepancy in agreement with the experiment

there). In the subsequent relaxation period, the cloth becomes moist through-

out, nourished fro, the condensed moisture on the skin simulant. surfaee.

S1

Thus considering the fact that the computation has to treat the system

as a composite dry-moist system during almost the entire period, the results

mentioned last could be considered remarkable. While the reversal of moisture

flow can be detected readily from the numerical results, the temperature

graoient in the cloth and skin simulant - except in the outermost half-

slice of the cloth - does not change its direction, so that the only outward

heat flow in the system interior during the relaxation period is constituted

by the action of the moisture transport. This obbervati3n holds true for

all pulse responses computed so far.

5. Remark on the Burn Severity Correlation

In the calculations discussed above, the integral (4.1) designed tn

serve as a basis for the correlation of burn severity has been evaluated.

As expected, its maximum occurs at some time aftcr the surface temperature

of the skin similant is highest. The results also show that the value of

the integral varies more than do the temperatures obtained when transfer

coefficients etc. are changed, For example, while the skin simulant temper-

atures at and near the surface vary by shout 9% between the curves of figures

2.22 and 2.24, the integral hianges its value by about 247.. This indicates

that a fairly high reliability of the calculated temperature profiles is

required to define the critic,. skin enthalpy clorely enough, if it ta o,

be applied to a correlation with experimental burn data.

VII. SUW94ARY AND CONCLUS1ION

The temperature response of a system composed of a skin simulant

covered by dry or moist cloth and exposed to radiative heating has been

studied experimentally and theoretically.

The mathematical analysis adequately describes the behavior of the

dry. cloth-skin simulant system. The response to constant or time-variant

irradiation can be constructed on the basis of gen~eral solutions both for

opaque and diathermanous cloth.

The theoretical model for the moist cloth-skin simulant system is

based on molecular diffusion ut water vapor as the only mode of moisture

transfer, this& neglecting bulk flow of water vapor -and other simplifying

52

assumptions. It is designed for an opaque cloth layer but could be altered

to apply to diathermanous cloth as well by a method analogous to that

developed for the dry cloth case. The theoretical formulation is not

general but requires new computer calculations for any set of 3pecific

conditions. Either constant or time-variant irradiation intensities are

acceptable as boundAry conditions; the latter may be a square wave,

trapezoidal wave or a thermonuclear explosion pulse distribution. It also

takes into account the complete drying of part of the cloth layer which can

always be expected in reality.

This model yields solutions which agree fairly well with experimental

results both for constant and time-varying irradiation. Uncertainties are

inherent mainly in the values of heat and mass transfer coefficients and the

water vapor diffusivity. Both further experiments and calculations extending

over a ra.?-a of these parameters could serve to determine better estimates

for their values and thus imprnve the capability of the computing system

for predicting the system behavior more accurately,

With the above reso;vations the variation in the temperature responses

to irradiation of different distribution forms can be estaolished clearly.

The model can serve as a tool for inspection of the influences caused not

only by the transfer co* "'i ients, but by ot.>er conditions s'uch as .a..

surrounding temperature, relative humidity and properties of the cloth.

Th7 numerical results permit the observation of phenome-a which cannot be

detected experimentally, such as the direction and rate of moisture flow

and the process of drying out of the cloth.

Ilfe solution includes a measure for the "critical enthalpy" of the cloth,

which can be related to the severity of burns. This criterion is most

sensitive to slight varltiors of the theoretical temperature response and

deserves further testing.

The study of the influences on the heat transfer by chmical reactions

in the Irradiated cloth, by the structure of the fabric and moisture-retaining

additives still are open paths fcr further research into the protection

f in burns0

33

APPENDIX A

NOMENCIATURE

S Coefficient in partial differential tquation

I Coefficient in partial differential equation

C Mess heat cappcity, cal/pom°C

S Coefficient in partial differential equation

D Diffusion coefficient 5 lop n mol/cm sec

D' Average diffasiohicoefficient 2 D/(l-y av) gn aol/cm sec

D" Corrected Mass diffusion coefficient through cloth - gD'/i/W

pgm/cm sec mm Hg

0 Diffusivity, caa/sec

5 Coefficient in partial differential equation

f Volume fraction of voids in cloth, dimensionless

g Tortuosity factor for the diffusion through cloth

I Intensity of incident ra " tion, cal/cmnsec

10 Intensity of unreflected radiation, cal/cassec; constant, unless specified

as time-variant

I 00M Characteristic intensity level of unreflected radiation, cal/cmssec;

(used whenever I specified as time-variant)0

Ix Intensity of radiation at depth x, cal/cmesec

k Thermal conductivity, cal/ca sec0 C

K Mass (moisture) transfer coefficients, Pgm/cPasec°C

L Thickness, ca

M .ioisture content of cloth, wt. H@0/100 wt. dry cloth

R Molecular weight

MA Modulus

54

p Partial pressure, -m Hg

P Base pressure used in connection with dimensionless transformation,0

"mmHg

P0 Dimensionless partial pressure

R Universal gas constant, casum HS/g mol°X

T Property ratio- 6x / Ao !

t Time .sec

T Temperature, 0C except when specified as OK

T Base temperature, °C

u Function

U Overull heat transfer coefficient, cal/cassec°C

w Right hand sides of tridiagonal equation systen for finite difference

solution

x Distance or depth, cm

Z° Dimeosionless depth

y Mole fraction, dime.-ts'nless

Greek:

SThermal diffusivity of cloth

j Leftt hand side. coefficients of tridiagonal equation system for finite

difference solution

y Extinction coefficient, ca"I

A Finite increment

qr light hand sides of tridiagonal equation systes for finite difference

solution

* Exposure or irradiation time, sec

X Heat of desorption, cal/agn (nuaerically negative)

W Total pressure, -i Jg

5,

I4 Left hand side coefficients of bidiagc''e, equation system fcc finite

difference solution

P Mass density, pgjc/w.f

P0 Mass density of cellulose or solid, b-m/cma

p Molal density, gX aol/cmI

X Summation

1 Transmittance

Function

Subscript-,;

c Cloth

char Characteristic abscissa -n'iriadiation pulse

cond Moisture condensation

crit Critical value (tinperaturt)

er Characteristic abscissa In trapezoidal irradiation pulse

evap Moisture evaporation

i Air gap between cloth and skin simulant except when used to denote i

components

J Depth planes

n Time increments

o Cloth surfa, - unless spe, tied otherwise

SSkin SiAular;

surr Surroundinrs

A, a, ... Depth planes

5@

first subscript : spac.

second subscript tim

Superscripts:

0 Dimensionless groups

117

APPENDIX B

METHOD OF SOLUTION - IMY CLOTH OVER SKIN SIMULANT SYSTIN

A. OPAQUE CLOTH

With the dimensional transformations (1.8) introduced in the main section

a;..: with the finite-difference notations:

&PTo -2T* *Toa

Am°-nn i° J+.n&Xoa 0 (aK 0)s (Ax')s

C C j.1 C C

To T6• ro me .nj l " J.n

the problem of the npaque-cloth-okin system (equations 1.1- 1.3) can be

stated in dimensionless ter" a followas

I. Basic Bq'mtions:

(a) For the cloth layers

AT° 43TO

as* (p.1)oC C

(b) For the skin simulants

- -- (1.3)'4,0 (4XO)S

udiere A60 a Ag o ASe if the modulus (&XO)*/It0 Ia taken to be the sam InC 0both the cloth aid skin simulant, which implies that the ratio of apace

intervals becomest

5.

0AX c tC~O

A~~zR=I (1.9)

x )c

2. Boundary Conditions:

(a) At the outer surface of ihe ctoths

(l/Ub) (- TO ., S(1.4)'

CX° 0 (I/U0)

(b) At the inner surface of the cloth sad the skin surfaces

ATOTOO TOOAI: 0.1 (lA W ATj

3. Numerical Solution -Modified Schmidt Technique

Cloth ad skin simulant are divided into slices of width AX* aOn 0C

respectively, The reference planes foe temperature are located at the center

of the slices and also halt a slice beyond the surfaces of the cloth and

skin simmlant (see Figure 1.2). This convention is known to produce a better

appcoximation to the true %--ution than taking the surfaces as reference

planes (14, 19, 20). A scheme having two cloth slices is used below for

demonstration.

I*-- a

c!

a a

"I I I

0 0 0.$ 1.0 o 0 011 0, -

CaVM SKIN aIM LANT

59

a II

The calculation of a temperature distribution starts with the temperature

value at plane A, for which the boundary condition (1.4)' can be written &at

*(lU~)- SA A? + TO (a initially zero) (.)TAnl (/Uo). "z/2 c B

0 C

where (To 4 a and To 0ibis relation is illustrated by

Figure 1.3.

Next, the temperature values inside the cloth are computed according to

equation (M)' , which may be written as:

(O)a

CT0 -~ T, fP r0 C -T To) 0 ýBCn Tn B" A'n 0 i",n+l TD.n)C'o

0 4

This relation car be simpified considerably if the modulus (AX )/Ao 0

is taken to bto 2, as proposed by Schmidt:

*A.na C~n3,m*l 2

or generally,Tj n I . T o', nz Tol.

where J may denote any slice b. t ii the cloth. In particular, this fcarmULation

allow for a graphical interpretation of the process, (Figure 1.3).

A a C D B FGHIg j ! ag

l: J i ll ,I I i I,.o! I '

vo

* LI

1: . 1 .o0 LC X oFigure A .3

60

At the planes above the inner cloth surface and the skin surfsme (W 1nd

B for the 2-slice model) the tempera ures at time n*l are obtained1 by

means of the boundary condition (1.5)1, which ii• reordered form yields:

Cn Fn) 2 +

D~n.1

and similarly (1.5)"

(TO -TO ) A-* C. a P.m s8 T-01,n.10 F,M

Io

Ui 2

These reletiong can also be verified graphically if the planes X; I andC

X0 a 0 are .ep.,rated by a distance lAJ, (Fe. z.4)1 t:, temperaturedistribution throughout the system can then be dranm contimnuusly.

C D 11

I I

I i 'I J I

C C I I

C I

1o .0 X; 0

FJigure 1.4

61

The conduction of heat into the interior of the skin is governed by

equation (1.3)'. Using the modulus (AX0 )*/AO0 . 2 for the skin simulant,

the same simple formulation is obtained as for the cloth equation:

T 0 'J-l.n JTln (13)"J'n+l 2

It is to be noted here that the method outlined above is a modification of

the one originally proposed by Schmidt (19, 20). In applying the boundary

condition (1.4), Schmidt's method properly states that at time o, T o 0X noTo 0 at the outer clcth surface. However, in the next step, thesu rr

Calculation of 0T°B' all the space between the planes A and B is implicitly

assumed to have the heat capacity of cloth. In the modified technique

described, this error is partly compensated for by the assumption (in

boundary condition 1.4") of T0 ! To • 0 at time 0 and the neglection5,0 surr

of heat capac.ty in the half-slice ftom the surface to plane B, which

leads to a closer approxim t ion to the true solution.

4. Stability and Accuracy

It is known for this particular type of computing algorithm that the

solution is both stable And convergent with tefinement of the intervals

if the modulus is larger then or equal to 2, (18).

Comparitive solutions us. -, the 2- and 6-slice models and a modulus

N a 2 (in order to retain the simplicity of the computing scheme) have

shown that the accuracy is improved only in the low-time range (P I:S I,

corresponding approximately to 0° < 2.5 Arc of real time) by the finer mesh.

Therefore the calculations are best started using the 6-alice model and

then continued with the 2-slice model.

Fuihier details of the method are described in (4).

8. DIATHUMAPOJS CWTH

1. Basic Equations

The basic equations governing the case of diathermanous cloth can

be transformed into the following dimensionless and finite-difference fnrmi

62

(a) For the cloth layer:

AT0 A'T0 A (e' c) (1.2) 'U o

(b) For the skin s3nulant:

(identical with the opaque cloth case)

AT0 A- - - (1 .3)'as

C

again with A$ a s A.: - ant , 0 M.um the only new term0

sA(e YL,.cX thintroduced is a heat sourceat each cloth increment.

in ui.ing the finite-difference model, it is usu&ed that the amount

of radiant ,-ergy absorbed in the center of each slice of the cloth is

distributed according to the Beer-Lanbert law. Inspection of this distribution

shows this to be justified for clothof high isiathermncy, where the radiant

energy decreases nearly linearly across the cloth. With a cloth of low

diathermancy, however, tei decay of intbmsity is much stronger bear the

exposed surface of the cloth. Therefore a better assumption is that the

fraction of incident radiat'on absorbed by the first cloth slice is shmerwed

at the front surface (aislowous to the opaque cloth case).

The boundary condition at the front surface thus appears different

according to either assumption.

2. Boundary Condition at the Front Cloth Surfae

(a) High diathersmancy

X:-O € 7-.

00(b) LOW diathermncy

IA' .YCI~) (70 To1 O r o S a " n; ( l . 6 b ) '

UýU 0C x 00

63

Also for low diathermancy, the radiation term disappears from the equation

for th-, firs. increment:

A~o a~o(1.2a)'

A90 -0 ( 0X20

3. Boundary Conditions in the Air Gap

(a) Between the back surface of the cloth and the air gaps

T0 T0 TOO -a"- "U ig" " (1. 7.)'

(b) Between the air gap and the skin simulant:

1•° I*YC) C. 0. T00 )

.. TJ ' - ~uO z~l(1.7b)t

.ý Jr-ON i

The evaluation of this pr~blem follows the same path as that of the opaqueA(@TLCX:)o

case, except that the i.sdependent variable has to be evaluated

for each cloth increment.

As a consequence, the solu Lon will be specific with respect to tpe

value yL *

4. Simplified General Solution for Diathermanous Cloth

Since the system is linear, however, it is in fact possible to acrive

at a generalized solution eliminating the need to calculate anew f(c

different diathormancies.

7he response to any one intensity can be considered to be the weighted

sum of a numbes of solutions, in each of which a unit of radiant energy is

assuen* Oo be abeorbed in one particular cloth increment only. The weighting

factors to be applied then are the fractions of the incident radiation that

are associated with each increment by the given distribution, i.e., A(e'YL )

Thus as many basic solutions are required as there are planes in

64

which absorption is assumed, (which needs not coincide with the number of

conceptual slices in the cloth). In particular, a method was proposed in

which absorption is taken to occur at three locations only, resulting in

three basic curves or solutions:

Curve I represents the case of alsorption of a unit of radiant energy

at the exposed surface of the cloth, which is equivalent to the opaque

cloth solution.

Curve II represents the case of unit absorption at X° 0 0.75, and

Curve III represents that of absorption at the surface of the skin

simulant. fat this method, the equations in dinite difference form

beccmes

Curve 1: identical with opaque cloth case.

Curve Y1:

1. Far the c~otk layers

0(a) Any increment other than that assc•ciated with X C 0.75

h .° (1.209A9e (Ax O)v

(b) 77w clct.i increment %tose centerline ia at x, a 0,75

"To I (1,2b) fo (AX O)s AXc°

c c

2. Fee the skin simulants

- 0 (1.3)T'Wo (AX:la

3, Dwudary coaditions at the front surface of the cloth#

SAT° • (•.0 l~strr)(l.6a)'

'S

"~~~~~~~~~A 1 0

4, Boundary conditions in the air gaps

0T U o (Too. --To

Ao Ut . x0.O) (1.7b)'

0 0

Curve Ill:

1. For all increments of the cloth layer:

•T°=. 'r°(1 .2)'A• 4STO

e0* (42x0)3C

2. For the skin sifulant:

3. Boundary condition at the front surface of the cloth:

17 U.0(Too -To (I6)0 aXC*O surr

4. Boundary conditior.% zi the air gaps

(a) Between the back surface of the cloth and the air gaps

l X u I o 1 Two - T:o ) (l.7a)'AX~ c a-i T- xsftO

(b) Between the air Sap ard the surface of the skin simulant:

o ...a I (1.7b)l

The total response resulting from any particular diAthermancy can be

obtained from contributtions of the three curves by applying to then a

weighting factor according to the fractions of radiant energy itworbed at

the three discrete locations:

6'

w

or< X0.0. 5T0 L j [.0YU0c I T I *o 1.0 " T1

C c

00oX 1.10

[eYLcXc rX 0' L (1.9)

c

A solution in which radiant energy is assumed to be absorbed in each

cloth slice (according to equation 2') can be expected to be more accrat¢.

than the scheme described above. Calculations of such a solution for the

6.-slice model, however, have shotm that the gain in accuracy is in fact small

and would not Justify the increase in computing Wfort over that required

by the general three-curve solution method.

67

. , , i i I I I II-

APPENDIX C

SOLUTION ALG ITIM - MOIST CLOTH - SKIN SINULANT SYST94

In the following description the cloth is assumed to be conceptually

divided into six slices of width AX0 and the skin is assumed to have anc

infinite number of slices AX° (figure 2.1).5

The T°- and PO values associated with these are taken at their aid-

points. Points .-8 and 9 are located above the cloth and skin surfaces

at distances AX 0 / and 0AX/ , respectively.C a

With the derivatives written out and the unknowns separated to the

left hand sides, the equations governing the four phases of the alternating

solution scheme are:

-CT 2 -. MA)0 -+XTA"Tc - (Jn+ J+l.n+lj-l,n.l "M *A )T,- A"T° "

!(P 2P 0 -N T,0 (2.14)A. ~n J~n *j+l,n MA j'n

1Po . M) Aon~l Pojln.

C (74 2T Z 0 + T0 )-.0 (2.15)J-nn+l J,n+l J+J.l,n~l

15P0 -(~ ) 0 + 6P0J.4,n*2 -(' MA pJ,n+2 j~l,n.2'(B -o -T +2• T,,:,(2,n16)

J-l-n+n Jn÷l- J*•.n A(

J '-l,n+2 + 2 MA)T Jn+2 J.1,n.2

J.. 1n+2 - in*2 J+l,n2) MA TJDn~l

ism--

Both skin and cloth are treated in the same calculating system, such that

equations (2.15) and (2.16) are only evaluated for points J in the cloth.

Whenever in equations (2.14) and 2.17) J denotes a point in the skin,

the pressure term is dropped and the space interval AX a replaced by AX 0

As an example, the first phase of the alternating solution scheme

shall be described. Equation 2.14 is applied to the entire set of points

in the cloth, which is verified identically by lines 2 ... 6 of the system

Figure 2.5. At the cloth and skin surfaces, the boundary conditions are

to be taken into account. Thus as J 0 1, the unknown T1 na l is temporarily

substituted by means of Tn and the boundary condition 2.7"12,u~l

1 -U 0 (T0 no AT0

equival6nt to

0 U -2o* T-0 (2.7a)"2.U*Ax Uh1.3 2,n*l

0 c 0oC

which modifies both the left and right hand sides of the first equation

(line 0 - line I in Figure 2.5).

The temperatures T*, ... To near the surfaces confining the7,n~l lo~n~l

air gap are related through boundary condition (2.9)", which can be - ritten

as, 0 0(T..[... ..... [ (2.9T)"

T,nO l * •x° • rR.

U 0 2

*&As well AS

-T T )AX0 Us Ax *a [P? IRI* 7n~l 1O.n~l a + J- -- (2. 9b)"

O~n~lI AX a 0 0,n~l

Ii a

After reordering, these relations match the general schom applied in the

69

System (Figure 2.5) and fOr, lines 7 ind a.Last, equation (2.4a)" :Is applied t, all of the points in the skin,which is considered semi-jifin'te. At the start, i.e., at time zero,the equations are written do-*, faz a Sll, arbitrary number of pointsin the skin, since initially T!,,-, 4 0- In this Way, a tridiagonal systemof equations isi formed. The expiessions for all of the cceCt(cientsappearing in the middle diagonal and for the right hand sides are listed

in Figure 2.5.

Next, the system is soived by successive elimination. Dividingthe first equation by a factor I s I and adding to the second equation,one obtains s

To."•2 "l/))•13,n÷1 4 ,o~l

92

• -((W/ 2) w2

DiviJing this by • (2 - 1/VI) and adding to the third yields,

-(3 /V2 TO a TO-"•3 4,n'l S,n~lm3

"" ((w2 /' 2 ) 3 ) " 3

etc.This elimination scheme, carried throughout the system, reaults in abidiaaonal set of equations (Figure 2.6), the coefficients and right handsides of which follow the relationst

S"i. '.• -.2 ... 6, 9

ai" wi 1 /V , i a 2 . @ 9 ...Equations (2.9a)" and (2.9b)* are brought to match toic bidiagonal pattern

70

by means of a linear combination of the bth, 7th, and 8th lines, resulting

in

(,6Q - l)(l - S) + (l - Q)6 •6 ( - Q)

V V 6TS - 1I8 6 V6 7'(S "'1)

w6 (Q - S) _ 6 [- S)P - (1 - Q)SRJ

7 6 S(l - Q)

0 o0

.*SR r17+ _po W/,6 6 (5 - 9)

where(2AqJ•) * Ax0 o Axeq-5 C(2/U*) +AX0 4xO(2/U ) * Ax AX0

s. uo (Ax* + Ax),*S •

xoo

Ax0 AX0

QH -oK(27J•) + X°AXc "

Sit- (4x:/2 + I/U*)rK

The bidiagonsl sys ten is solved by backward substitution, yielding the

future time temperature values Starting with the last equations

in tim interior of the skin, we get

00 * ,M÷in.l •-.,,n.l WMA w N-I

w /VJ'n.l J.l,n.l " J-1J-1

71

until j = 2. To results from the boundary condition (2.7)". At thepoint 14.2 in the skin simulant, the temperature is zero. After each time

s tep, the number of points N and thus the size of the equation sys ten is

increased by one, accounting for the propagation of the tepexature

profile into the skip simulant.

The three remaining phases Of tU alternating scheme (2.12), namely

those based on equations (2.15 - 2.17), in principle follow the same path

as outlined above. In treating the implicit evaluation of future time

pressures, the elimination of Pn0 is done by means of the boundary

condition (2.8)". Die boundary condition (2.9)" for the pressure in the

air gap enters into the 7th equation of the tridiagonal system, which

has a constant size corresponding to the number of cloth slices.

72

APPENDIX D)

MASS AND HEAT BALANCE - MOIST CL07H - SKIN SIMULANT SYST13M

A means of testing the internal consistency of the computation is

provided by the indication of the (cumulative) amounts of moisture lost

due to avaporation at the front surface of the cloth and condensation at

the skin 31mulant surface.

A Mass balance can be set Up as follows:

evaporation condensationl :loth, firal aMcloth, initial

From equation (2.6)" follows~

The sum of the su~fACe prOLaure Values, taken ove all a past time steps,

is recorded as

"SURF LCISS' 2 P.~

with 69 69 - (L8 pC/k)c .n:M a (AX 0)*/AO0 9 the surface loss of Av.i~ure

is

Mevap 0PK oO) (L~'pCA)0 . (SURF LOSS) (uigaIHg/cm$)

c (53,*)

The moisture flux through the Inner cloth surface Is given by:

moisture flux I P 1' p e P.J o.2)(air gap) 0 1

(takiuig the 6-alice model as an example)

The program evaluates

73

Thus the total amount of moisture condensed is given by

Mcond a A 0 (o..,AJ ( (CciD) ( aO/c,,) (5.2.)cond 0 1(AX0c)'

The remaining moisture content of the cloth is constructed as

I

4cloth, final a PcLci= X c(TiP (pgnHvO/cx2) (3.3)

(i counting the number f -cloth slices)

A heat balance may also be established; however, the integrals

required (over the surface temperature gradients and over the enthalpy

fluxes due to moisture evaporation) are not evaluated by the program and

would have to be calculated from the printed results.

74

APPENDIX E

COMPUTER PROGRAM DBSCRIPTION

A. FUNCTION OF MAIN- AND SUBPROGRAM

The MAIN-program and subroutines are written in FORTRAN - and FAP-

language and can be run at any IBM 709 - installation (16K or 321) under

FMS control. The explicit subroutines aret

SUBP

DOUBIN

.DTAD, TAB

PULSE

TARA

C194NGE

F2R4

All other subroutines are FMS 709 library programs.

1. 14AIN Program

The flow of control is shown in Figure 2.7.

Any number of data set, can be nandled in one run.

First, the equilibrium data for cloth are tead in, to be used for

all data sets.

Next, a set of data is read. The calculation starts with the deter-

minatirm of the relative irradiation intensity followed by phase (a) of

the alterrtting scheme (2.12), whaich establishes the distribution of

temperatures. Based on the new temperature values, the variable coefficients

C and D of equation (2.5)' are found by interpolation of the tabulated

data.

Hereafter, phase (b) is solved for the pressure profile, assuming

that the boundary condition (2.6)' is valid, which implies condensation

of moisture on the skin surface. A test for this condition follows. Lf

it shows no condensation, the pressure calculation is repeated assuming

75

this subroutine determines the relative irradiation intensity Io(Go)/Iomaxand records it.

5. PARA

The input parameters pertaining to irradiation intensity and pulse

shape are interpreted and printed in dimensionless and absolute terms.

6. VZ

This subroutine computes, at the end of a run, the total irradiation

energy received in the cases of a step function or a thermonuclear pulse,

(equations of Table 2.1).

7. CHUGE

With CALL CHANGE(l), the purpose of this subprOgr&a is threefolds

1. To record the number of dry cloth slices.

2. Optional: to determine the maximum of the heat content of the skin

above a given critical temperature level, attained at any time (equation 4.1).

3. Optional: to vary the size of time steps at prescribed dimensionless

times by means of altering the modulus X.. Any number of such changes is

possible. The use of this procedure lies in the facility to reduce the

time step size near the enm c a square wave of irradiation, and to increase

it at a later time again. - Any rew value for MA must be a multiple of the

original one. The functions 2 and 3 are only activated by special data

cards.

3. ."OIINE ThLB CONSUWP7ON

The amount of machine time used per time step depends mainly on the

number of cloth slices, since the determination of pressure profiles

requires most of the work. The approximate expense of computer time is

given below:

No. of Cloth Slices Machine Time / Double Time Step (Sec,)

2 2

6 2- 26

16 4 -

77

Using the 6-slice model with a modulus 14 A 5.0, a doubLe time step is2A9° x 0.0111, which corresponds to 2e9 w 0.0272 sec (with 0,40 w

(LWX/k) a 2.45). The ratio of the micnine time scale to the real time

scale is,

machine time . ?n 05real tine

in other wrds, a second of real time requires 1.2 - 1.6 minutes of computer

time,

74%

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APPENDIX F

SUMMARY OF COMPUT] SOLUTIONS (iN MOIST CLOTH - SKIN SIMULANT SYSM4The theoretical results for the temperature response of the moistopaque cloth - skin s3mulanZ system are summarized in the foylowingdiagrams. The listings of the num-icagl -.olutions and the master diagramsare in the files of the Fuels Research Laboratory, Department of Chaeical

Engineering, MIT.

For reference see List of Figures.

92

4' �.

ii q B

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Ui'-p. -- z.U I

aI a

- �

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- z

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24-

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0.7

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0.

0 1.0 2.0 W 4A) &0 6.0 7.0 60 to 10.0 110 120 123

Oimm..eImaU T.um, 0Fit USl EFFECT OF CLOTH OIATHERUANCY ON THE HEAT TRANSFER TO THE SKIM SIULANT

s I If0.6-Q 20 &.0

0..0.7-

.r .42

Dwssiles SwM Depth

0o.4

)(1-0.45

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OimoensAssuo 7,4Ir,eFlO 1.14 EFFICT OF CLOT" DIAT'IEftUAW.Y ON THE HEAT TRANSFER TO THE $91N SIMAULANT

97

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1. 002 002

Its

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fie:.:11 IFPIECT Of 4LISTE H,0 SO CLOT11 ift *H ; CATTRASFE TOTht W1111SIMULANT

JUT W

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FIG JISCMPARSONOF EPERIENTL RESLT$ON DRY OP4Ec

'10

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*bass for'4.

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cal. prsw

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ag -- I ALi~D(press*")Oak". 0, Gsposmpefa s

a " & J ~ s o s t s " o 1 1 0

cus v-moit, lsss" asil

Rum",W

FIG 2.0 FLOW-CHART OF SUPROUTINE SUBPMOIST CLOTH-SKIN SIMULANT SYSTEM

= 6" "use"pat, Colo. NMI00110110 d-ble it N.I.-

9 lattem .0 t.3 14

u4c. ""I -cwj*fw cloth L.11.0 -- Mit.4for skis

xx temp. ciaocaUslate skis ff.&.WLAIcoefficia-le ewe caff I call. ImiCWI-(W else 4 1 R.N.S.

we cle"t 04-Ay".2.96-a&woaiootlm focurelve sotmý

10.2 skirocalc ' it"Sed total 'P.Ces go

t Andic. samose)r g4dwc* 14 00.6161-3.16

Cale. act2.10J.- sdial "at

nista.tom4f*A twist W"26

(de 3

Project clock P*4,.til LOUjkL_ is W of to a I-P.ties was

boom" er cow. modest" Joel volimC1.411 A" 4IAd CRIT ? We mort.

gGic- GUUL' tomp, rise sidespt4we lost live step

rysom"tioe me. flow An ver cf 14call. DATL Ledle. 30 .1

AXS case. rodsce grid

0".4"tj.u-&.a Adams PAMp"40", is So"vo.$"44ýt to bevedery

Sis msac& - a asest ISOgetursive lots. - #sen 70 ad PIP06"tion (cc ".11A nT, log am stop

moor o".61sta. 95-4

is P83"A ties for to ="Miss@to .11 Is (ord"wo) (aso

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AD I clnI= IUU

* MS 06oftite 8 m card)

FIG 2.9 MAKE-UP OF RUN CARD DECK

112

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FIG 220 TEMPRATURE ot5poeis" Of A MOIST oftow Q.O~jffSAM9 SIUIAAT SYSTEM IWLU6PFCC Of IR~IWADMTIO PICuRAUf P~gf AftA 0.546 cgI/cm* so

go -- 4-

-*o I \

.4 ...-.I -/ .. .................... \ .--, ....... ..., -.... ...................- ,* I

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Om*moss Tim 0,

F I ,2.2I TEM 1 "ATIJIMfE rI4 I ~ 9 OF A MS I T OS U CLOTY"-

S SIMIJALAT SYSTEMi. INFLUENC OF IRRAIATION PULSESK*M; PVLS9 ARA 1.0 tol /c.lefe

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APIENn!X G

MUILI1RIUM DAM% ON MOIST CLODI

This dat4 is used in the numerical solution of transient heat ar4

moisture transfer through cloth.

134

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APPENr A H

BIBLIOGRAPHY

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137

12. Ho leong, E., and Williams, M. E., "Surface Tenperatures in a Two-Layer Air-Spaced Slab System Irradiated from One Side", Technical:,eport No. 4, DIC Project No. 6797, Fuels Research Laboratory,MIT, Cambridge, Massachusetts, 1956.

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