AlllOl

I cations I

NAT'L INST. OF STAND & TECH R I.C.

AlllQS

NB5TR 80-1975

Analysis of Mathematical

Models of Integrating Monitoring

Devices

Robert J. Rubin

Polymer Science and Standards Division

Center for Materials Science

National Bureau of Standards

Washington, D.C. 20234

Final Report Covering the Period

February 1, 1976 through June 30, 1976

January 1980

U.S. DEPARTMENT OF COMMERCE

iTlONAL BUREAU OF STANDARDS

V y

.

national BUBIEAO or BTANSABD8

UalABT

NBSIR 80-1975

APR 6 1981

r)bt ucc. -

ANALYSIS OF MATHEMATICAL

MODELS OF INTEGRATING MONITORING

DEVICES

f)

'- 0

, U^h

no,

M$o

Robert J. Rubin

Polymer Science and Standards Division

Center for Materials Science

National Bureau of Standards

Washington, D.C. 20234

Final Report Covering the Period

February 1, 1976 through June 30, 1976

January 1980

U.S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Luther H. Hodges, Jr., Deputy Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technoiogy

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Analysis of Mathematical Models of Integrating Monitoring Devices

Robert J. Rubin

Polymer Science and Standards Division

National Bureau of Standards

Washington, D. C. 20234

Preface

The work reported here on analyzing the efficiency of models of personal

monitoring devices was supported by the Office of Air and Water Measurement

of NBS. The work was completed in 1976, but not reported in detail. Since

that time, interest in such personal monitoring devices has continued to

increase (see References 13-16). In addition, a group at NBS, which includes

Dr. Jimmie A. Hodgeson, is presently undertaking a study of such devices. Dr.

Hodgeson has encouraged the preparation and publication of this report.
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1 . INTRODUCTION

In this paper we analyze two mathematical models of a personal monitoring

device. The purpose of the device, which is to be worn by an individual, is to

indicate the level of exposure to various criteria air pollutants such as 0^, SO^j

NO

2

, CO during a time period which might range from 24 hours to a week or longer.

The goal of the analysis of the models is to determine how the quantity of

pollutant which is collected is related to the average pollutant concentration

and how material constants such as solubility and diffusion coefficients affect

the operation of the device.

The first model consists of a pollutant-permeable membrane exposed to the

atmosphere and backed by a reactive substrate which serves to trap any pollutant

that has diffused through the membrane. The quantity of pollutant collected is

related to the parameters of the model which in this case are: solubility

coefficient of the pollutant gas in the membrane, diffusion coefficient of the

pollutant gas in the membrane, membrane thickness, and the pollutant gas partial

pressure as a function of the time. Our analysis of this simple model (Model

I) has led us to consider a second model (Model II) in which the reactive substrate

is replaced by reactive sites which are distributed uniformly throughout the

membrane. Model 1 is treated in Section 2 and Model II is treated in Section 3.

The collecting efficiencies of these models are compared in Section 4 for "typical"

values of material parameters. We conclude there that the efficiency of Model

II is superior to that of Model I.

1
2. ANALYSIS OF BADGE MONITOR MODEL I

The first model of a "badge monitor which we consider consists of a permeable

membrane exposed to ambient air on one side and backed by a reactive substrate on

"Che other. When the monitor is exposed to air containing some time-varying amount

of a criteria air pollutant (e.g. 0^, SO^, NO^, CO), the pollutant can diffuse

through the membrane and subsequently react with or be trapped on the reactive

substrate. In Model I, we assme that the transport of the pollutant through the

membrane is adequately described by the one-dimensional diffusion equation:

C(pCyi^ __ ^

where C C 'tr') is the concentration of the dissolved pollutant in the membrane

at position iX. and time "i* expressed in the units, moles junit volume.

The one-dimensional configviration for Model I is shown in Fig. 1. The membrane

lies between the surfaces at Z^—

and , the latter surface being the

one exposed to the air and to a time varying partial pressure of pollutant, .

The "boundary conditions which the solution of Eq. (l) is assumed to satisfy at

the surfaces ~D(.~0 and ZC—Jl are:

(i) at , there must be equilibrium between the pollutant in the gas

phase and the pollutant dissolved in the membrane at

In Sq. (2) 3 is the solubility coefficient of the pollutant in the membrane

expressed in the same concentration units as ^ per unit pressure. Since ji

ve are interested in smaJ.1 partial pressures and low concentrations, we have

assumed the simplest fora for the solubility relation in Eq. (2).

2
( 3 )

(ii) at ^“0 , there is a similar condition. Since the substrate is

assumed to react or trap the gas leaving the membrane, the condition equivalent

to Eq. (2) is^

c(o,t)

In the initial condition, or state, of the membrane, the concentration of

dissolved gas is identically zero,

C(^,o) = 0 ( 4 )

The solution of Eq. (1) with the boundary conditions (2) and (3) for the initial

2 condition (4) is

(5)

The transport, or flux of dissolved gas from right to left across the plane sur£ace

at DC in Fig. 1 is given by the expression

( 6 )

where is the flux of gas molecules measured in moles per unit area per

unit time.

The total flux across the surface into the reactive substrate during the

time interval o<t£r is

<^(t) = J F{o,t)Jt

(7)

X

-0

J. 3>c

3
vhere ~iie tisie T corresponds to the total time of exposure to the pollutant. The

explicit relation between 4c r) and jplt) can be obtained by substituting

Zc. (5) in Eq. (T) and carrying out the indicated operations. The final result

is (see Appendix A for details)

( 8 )

It is also shown in Appendix A that in the expected mode of operation of the

monitor, the expression for the total amount of pollutant collected in the

substrate is given by the first term alone in Eq. (8)

or

vhere is the average partial pressure of the pollutant during the time T,

( 11')

4
The error 5 ^ ) which is made by neglecting the second term in Eq. (8)

<?=

X. (12)

is shown to be less than

il

T' 3p (13)

where is the maximum value of . By comparing the expression for

the error in Eq. (13) with the total amount of pollutant collected in the

substrate (lO) , it is seen that the error

^ is certainly negligible if

T» -^Vd when

^

characteristic time lyo

is the longest relaxation time associated with transient concentration

distributions in the membrane. In fact, the expression for the total flux in

Eq. (9) can be derived by assuming that at each instant there is a uniform

concentration gradient equal to 5

Thus the integrating or averaging characteristics of this model of a badge

monitor is extremely simple. The total amount of pollutant collected, according

to Eqs. (9) and (lO) is proportional to the time integral of the partial pressure

during the operating period T. In choosing a membrane material for the monitor,

the product DS , which is called the permeability, should be as large as possible

and the membrane thickness as small as possible. Typical values of these material

parameters will be discussed further in Section U.

3. ANALYSIS OF BADGE MONITOR MODEL II

In analyzing Model I in Section 2 , we found that the membrane should be as

thin as possible. That is, the sensitivity of the monitor becomes greater the

closer the reactive substrate is to the exposed surface. Clearly there is a practical

5
limit, with decreasing membrane thickness, at which the membrane becomes too

fragile. For this reason, we consider the modified badge monitor Model II where,

instead of a reactive substrate, reactive sites are distributed uniformly throughout the bulk of the membrane material. In this way, trapping sites are brought

as close as possible to the exposed surface of the membrane. The one-dimensional

configuration for Model II is shown in Fig. 2. The membrane again lies between

the surfaces at ^—O and the latter surface being the one exposed to the

air and a time varying partial pressure of pollutant, fit) . In our analysis we

assume, for simplicity, that the surface at is impermeable to gas diffusing

in the membrane. If both surfaces were exposed to the pollutant, i.e., if the

exposed surface area were doubled, then the amount of trapped pollutant would be

approximately doubled. In Model II, we assume that the transport of the pollutant

through the membrane is described by the modified diffusion equation:

(14)

and the companion equation

(15)

where is the concentration of the untrapped or unreacted gas at position

X. and time /I denotes the concentration of trapping sites at position

"X. and time

^ is the diffusion coefficient of the unreacted gas assumed

to be independent of the extent of the chemical reaction. It is evident from

the form of Eq. (15) that the chemical reaction involving the trapping sites is

assumed to be bimolecular and irreversible; and k is the rate constant of the

reaction. The boundary conditions ^which the concentration is assumed to

satisfy at the surfaces 'X'^Q and are:

(1) at i , there is equilibrium between the pollutant in the gas phase and

the unreacted pollutant dissolved in the membrane

6
clUyt) =

( 16 )

In Eq. (l6) it is assumed that the solubility constant of the gas in the membrane

is independent of the extent of the reaction between the gas and the reactive sites,

(ii) at CC^O ^ the condition of impermeability is equivalent to the

requirement that the flux of unreacted gas across the plane is zero, so

= o.

(IT)

As with Model I, the initial concentration of pollutant dissolved in the membrane

is assTimed to be zero.

cCx.O^^O ^

( 18 )

The initial concentration of reactive sites (which are immobile) is assumed to

be uniform^

nCx,oy — ^ d9)

Eqs. Cl^) and (15) are nonlinear; and although these equations have been

2 3 A 5 discussed in a number of different contexts, ’ ’ * a general solution for the boundary

and initial conditions (l6)-(l9) has not been obtained. However, a number of exact

solutions and approximate solutions have been obtained in various limiting cases.

We now consider some of these special cases.

7
(3.1) Limiting Case IIA

In the first case, we assume that the initial concentration of trapping sites,

, is so large compared to, , the concentration of dissolved pollutant

gas at the exposed surface, that ^ ^^^t^remains approximately unchanged during

the period of operation of the monitor. The diffusion equation (14) is effectively

linearized

( 20 ) X.

The foregoing reduction has been proposed by several investigators . The solution

of Eq. (20) with the boundary conditions (16) and (17) and the initial condition

(18) is^

c

( 21 )

Eq. (21) is an expression for the local concentration of dissolved pollutant which

has not reacted. In order to obtain the expression for the total amount of

pollutant which has reacted up to time T , first integrate both sides of Eq. (20)

with respect to X between the limits 0 and Jl to obtain

9?^ -Jk CCt) ( 22 )

8
where

P

^ Cit) c(X,-ir)(^x (23)

is the total amount of unreacted gas in the membrane at time per unit surface

area of the membrane. Then integrate both sides of Eq. (22) from O to Tand

rearrange terms to give the following expression for the total flux of pollutant

entering the membrane during the time interval 0-^"^^y”(per unit surface area

of the membrane)

r>T

V [ c/f = CLr)+^n. Cat)Jt.

(24)

Clearly Eq. (24) is a simple statement of conservation of molecules; and the

total amount of pollutant which has reacted up to time

~J' is the second term on

the right-hand side of Eq. (24),

T

f?(T) CLt)h (25)

Combining Eqs. (21), (23) and (25), one obtains (see Appendix B)

-f

K26)

9
As in the case of Model I, the first term on the right-hand side of Eq. (26) is

the dominant part of

( 27 ) vJiIf the relaxation time for diffusion is much smaller than the relaxation

time for reaction, ihrio) , then the expression for in Eq. (27) reduces to

(28)

a result independent of ^ , as should be expected. If we compare the amount of

pollutant collected in Model II, Eq. (27), with that collected in Model I, Eq. (9),

it is seen that the ratio

^( 0_ _ /

$[T) ^

// P ^ D

kn, t/p > /

(29)

can, in principle, exceed unity when f ly I . Whether the assumptions

on which Eq. (27) is based and whether either of the inequalities,

or > / , can be satisfied in practice is discussed further in Section 4.

(3.2) Limiting Case IIB

We next drop the assumption that the concentration of trapping sites is

unchanged and consider the solution of Eqs . (14) and (15) in the limit where the

rate of diffusion is very rapid compared to the rate of chemical reaction. This

limiting condition was introduced in obtaining Eq. (28) for ^Cr) . If the relaxation

time for diffusion is short compared to the relaxation time for chemical

reaction, and short compared to the time scale on which the partial pressure of

pollutant varies, then the concentration of dissolved pollutant is constant

10
throughout the membrane

C(’(,t) = S^l£) ^ V

Thus, in Eq. (15) there is no dependence of on X. and the solution of

Eq. (15) is

niT) = )0 „ J^Sj ( 30 )

The total amount of pollutant gas which has reacted with trapping sites in the

time interval O to "7” is

!?(T) = ^Ir?o-n(r)]

(31)

Clearly, the approximate expression for i?fr) in Eq. (28) can be obtained from

Eq. (31) if the exponent in (31) is small compared to unity.

(3.3) Limiting Case IIC

Now consider the opposite extreme to case (b) in which the trapping reaction

can be treated as instantaneous on the time scale of the diffusion process.

Since the reaction is treated as instantaneous, dissolved gas and trapping sites

cannot both be present in the same volume element of the membrane. Therefore,

there is a region next to the exposed surface in vdiich all trapping sites are

filled and in which additional pollutant gas is free to diffuse. This region

is separated from virgin membrane by a sharp plane boundary which moves at a

velocity determined by the rate at which dissolved pollutant gas diffuses to the

g moving boundary. Hermans has formulated the equations of motion of this moving

boundary problem and obtained a solution in the case where the partial pressure

of the gas is constant at the exposed surface. The diagrammatic representation

11
adopted by Hermans is shown in Fig. 3. A constant concentration of the diffusing

pollutant is maintained at ^ , c(o,±) = ^ t^o.

From this point, the concentration decreases and becomes zero at a point ^

The slope of the curve at this point is determined by the condition

(32)

The left-hand side of Eq. (32) specifies the momber of moles of dissolved gas

molecules diffusing to ^per unit area per unit time and the right-hand side is

the flux of molecules required per unit area per unit time to fill the trapping

sites as the boundary point ^moves to the right. The boundary condition (32)

can be transformed to

because

P (33)

-

1- 2^ Jt =0. ( 34 )

Consequently, Eqs . (14) and (15) with the appropriate boundary conditions and

initial equations have been replaced in the present limiting case by the simple

diffusion equation

PC

^

(35)

with the boundary conditions

C = O ,

( 37 )

12
and

D

' = n. 9t ( 38 )

and the initiaJ. condition

ci:?L^d) ~ o , o , (39)

%

Hermans shows that the solution to this problem is

-^n

J

where

' I' f

O

In Eq. (Uo) the quantity,

is a constant which is determined by the transcendental equation.

±e GU) = rr'"- nG .

ihl)

ih2)

ih3)

Thus the position coordinate, ^

, of the moving boundary increases proportional

to the square root of the time, according to Eq. (U2); and the constant

depends on the constant rr'^ SJb„ as shown in Fig. U. For small values

13
of 1^0 , Eq. (i+3) reduces to

(U1+)

or

(2 5yjo,Dt/K]„}’'t

Having determined the position of the moving boundary point T

as a function of the time ~t , we have also determined the total quantity of gas

trapped in the membrane at time "t under the constant partial pressure

exposure condition (36),

(i+5)

In the limit of small

R’(t) =

5>^Jr), , (^6) is simply

(i+6)

(hj)

The approximate expression Eq. (45) for the position of the interface was

obtained in the limit where is small compared to unity. It should

also be recalled that the exact solution from which (45) was obtained strictly

applies only if the partial pressure in the gas phase is a constant. Clearly,

this limitation of constant partial pressure of the pollutant is overly restrictive

for the application we have in mind. In order to assess the integrating characteristics of Model II when there is a time varying partial pressure, we have two

approaches available. First, we could undertake the numerical solution of

14
Eqs. (35), (37) and (39) for a "typical" partial pressure history ^(t) and

compare the position of the interface at time T with that obtained from Eqs.

(42) and (43) when the assumed constant partial pressure is the average.

T

f! - fJ fl-OJt.

Rather than develop this approach here^we adopt a second approach which involves

the approximate solution of Eqs. (35), (37) and (39). The remainder of this.

section is devoted to summarizing the physical nature of the approximation which

9 was apparently first used by Mott and Gurney in the case where the partial

pressure in the gas phase is constant.

Mott and Gurney assume that in Eq. (32) the concentration gradient is

linear over the whole range O , i.e..

^ (i ^

)

so that Eq. (32) can be replaced by

(48)

at ^ (49)

or

5 = (5 " (50)

-I a result obtained in Eq. (45) by assuming that is small compared to

unity. Booth^^ had made the point that although Eq. (48) is not exact, it is

approximately correct. Furthermore, he has shown that the error made in using

(48) is small provided that

r)o «

1

. (51)

15
This last condition is equivalent to the requirement that the concentration of

the diffusing gas is small compared to the concentration of trapping sites,

physically this requirement can be understood in terms of the following form

of Eq. (32)

n At . ( 52 )

In the time interval dt, the diffusive flux of gas molecules is — cJt

1 the boundary moves through a distance ci5 which is inversely proportional to H© .

Thus for larger relative values of , the boundary moves slower and there is

more time to establish the "steady-state" concentration distribution (48). We

propose that this argument can be carried one step further and that the same

approximation can be used when the partial pressure is time-dependent, namely

and

(53)

for 3 /|pttf)/r)^ 1

The expression for the position of the boundary which is obtained by integrating

Eq. (54) is

[

3 PS ^ /lOoJ''] (55)

Ultimately, the validity of the approximate expression for ^

in Eq. (55) can be

assessed by numerical Integration. For the present, we will use Eq. (55) to

16
determine the total amount of gas trapped in the membrane (per unit area) in

the time interval 0 to T

R (t) -

or

R(r') = [a.P5 >0oJ ( 56 )

According to Eq. (56), the collecting efficiency of badge monitory Model II is

optimized by choosing materials for which DS and n^ are as large as possible.

Typical values of these material parameters are given in Section 4.

17
4. CHARACTERISTICS OF MODELS I AND II

The performance characteristics of Model I are summarized in Eq. (9) and (10).

$(T) - Qi" \ f(t)k (9)

( 10)

p where is the number of moles of pollutant traped in the time interval

A

T seconds in the substrate backing of a square centimeter of membrane I centimeters

thick. The permeability, Q = DS , of the membrane is measured in the units: moles,

cm (thickness) per second per square centimeter of membrane area per atmosphere of

pressure difference across the membrane; and the time-dependent partial pressure,

^(t), of the pollutant gas is measured in atmospheres. There are two significant

aspects of the expression for ^(T) in Eq. (9). First, the integrating characteristics

of the model are ideally simple, i.e., the amount of pollutant collected is simply

proportional to the time integral of the partial pressure of the pollutant. This

time integral in turn is expressible as an average partial pressure , -jo, multiplied

by the time interval T^

T

The absolute amount of pollutant gas collected is controlled by the magnitude of the

coefficient Qi ^ in Eq. (9). In order to estimate the value of $(T) , we refer to

a recent compilation of permeabilities, , by Hwang, Choi, and Kammermeyer

H^K

In that reference, the permeability units of are: CnQ of gas at S.T.P.,

cm (thickness) per second per square centimeter of membrane area per cm of Hg

pressure difference across the membrane. In terms of expression in

11

Eq. (10) is

18
The expression for the number of moles collected in one day of exposure to an _8

average partial pressure of the pollutant of 5 x 10 atm (which is the maximum

allowable value of 50 ppb) is then

(ri\coC) 74

X (^Q^oo)[5x\o^')

;2.2_MOO

\SWnj

-S' / z. KIO ^ yYio)e^fcrT)^

A

Typical values of lie in the range

10~^° to 10~®

for gases such as N^, O^j CO^, If we assume this range of values to

be typical for the criteria pollutants^ then lies in the range

(57)

-IS-,-,

I.Skid i

It was noted in analysing Model I in Section 2 that the value of ^.(I)

inversely proportional to the thickness of the membrane. For that reason, we

considered a second model. Model II, in which the effective membrane thickness

was reduced to the greatest extent possible. In section 3, several approximate

expressions were obtained for R(T) , the number of moles of pollutant trapped per

square centimeter of membrane in the case of Model II. We confine our attention

to the result in Eq. (56)

,

or

(56)

(59)

where definitions of quantities in (56) and (59) are the same as in Eqs. (9) and

3

(10) and where X\ is the concentration of trapping sites in moles per cm . We now

o calculate for R(T) , an expression analogous to ^ in Eq. (57). In the case of

19
Model II, Eq. (59), the expression for the number of moles of gas collected in

_8

one day of exposure to an average partial pressure of pollutant of 5 x 10 atm is

R,

(r(\Ax)

For values of in the range

tlL/iX lo"^^ to lo”®

the range of values for is

5,4 xio n

‘4

to 54x10 mo)e^/c.rrK

In the most favorable circumstances, the largest value of which might be

achieved is of the order 10 ^ moles/cm? Thus the range of values for in

(60) is considerably more favorable than the range of values of | in Eq. (58)

It should be noted that in the version of Model II which lead to the formula for

R(T) in Eqs. (56) and (59) we succeeded in reducing the effective "thickness" of

the membrane. This "effective" thickness is given by Eq. (55) the formula for

the position ^ of the interface at time T

^ - [s Q

'f\o (55)

The range of values of the " thickness"

in Eq. (60) is

which corresponds to the range in ^(max)

—8 ~'/

5^X10

"

3r

-2

is as large as 10

to

“to c/y] if

20
APPENDIX A. DERIVATION OF EXPRESSION FOR $ (T) IN EQ. (8).

The total fliox in the time interval (O^-y") in moles per unit area

across the surface at is given in Eq. (7)

${T) = C) [

Jq 'z>c=c?

(Al)

where

^ n=(

- ^ f'^Vrur>/f

(A2)

^0

In order to avoid the formal divergence of series when (A2) is substituted in

(Al) , we proceed in the following manner. First, calculate

n-i Jo

by integrating by parts, and obtain

e

(/

T

5|pCA<lt^

0

"T _ py7V‘ Cr-t)

jti (A3)

The series which multiplies

^ , xM-b!

T

C? o

in Eq. (A3) has the value

(A4)

•t so the expression for ^ C reduces to

|pcr-t)e

"" %.(A5)

21
The total flux ^(T) can now be obtained by substituting Eq. (A5) in the expression

•T

qi = o

The final result is

§(t)= -l-PS^C-ir J^(T-t)e

^Dn\ t/jC'

dt.

(A6)

It is possible to obtain a crude but useful upper bound on the series in

Eq. (A6) by replacing each term by its absolute value and by replacing ^ (T-t)

by its maximum value The result is,

^T

VI)

(A7)

I 3p

Thus the contribution of the series in Eq. (A6) to the total flux collected

per square centimeter in the time interval T is at most a constant, independent

of the value of T.

22
APPENDIX B. DERIVATION OF EXPRESSION FOR R(T) IN EQ. (26).

The total amount of gas per square centimeter which has reacted with

trapping sites in the membrane in the time interval 0 to T is given in Eq. (25) ,

where C(t) is defined in Eq. (23)

C(t) =

j

cCx^t'jcJ:

and the expression for C given in Eq. (21)

cxt)= J

X ^^0 L J

with

(Bl)

(B2)

(B3)

(B4)

Substitute (B3) in (B2) and perform the integration over to obtain

<v, T \

"^

C(t) = 1

\ Jl ~x

v^=0 o «

(B5)

Next, substitute (B5) in the expression (Bl) for R(T) and integrate by parts to

obtain

R(T)=

I

^ \e

yn

II /#

23
(B7)

12 The first sum in Eq. (B6) is

—O vYl =o

(B8)

A crude upper bound can be obtained for the remaining sum in Eq. (B8) by using

a similar procedure to that used in deriving Eq. (A7) . Here too, the upper

bound is a constant, independent of T, so that for sufficiently large values

of T the second term can be neglected.

24
REFERENCES

1. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids , (Oxford

University Press, Oxford, 1973, 2nd edition) p. 103.

2. J. Crank, The Mathematics of Diffusion , (Oxford University Press, Oxford,

1975, 2nd edition).

3. S. G. Bankoff, Advances in Chem. Eng. , _5, Academic Press, New York, 1964.

4. J. C. Muehlbauer and J. E. Sunderland, Appl. Mechanics Reviews , 18 , 951 (1965).

5. L. I. Rubinshtein, The Stefan Problem , Trans, of Math. Monographs, Vol. 27,

American Mathematical Society.

6. See, e.g., P. V. Danckwerts, Trans. Faraday Society (London) 300 (1950).

7. Ref. 1, pp. 104, 134.

8. J. J. Hermans, J. Colloid Sci. 387 (1947).

9. N. F. Mott and R. W. Gurney, Electronic Processes in Ionic Crystals , (Oxford

University Press, Oxford, 1940) Chap. 8.

10. F. Booth, Trans. Faraday Society (London) 796 (1948).

11. S. T. Hwang, C. K. Choi, and K. Kammermeyer, Separation Sci. 9_, 461 (1974).

12. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products , (Academic Press, New York, 1965, 4th edition) p. 36.

13. L. H. Nelms, K. D. Reiszner, and P. W. West, "Personal Vinyl Chloride Monitoring

Device with Permeation Technique for Sampling", Analytical Chem. 994 (1977)

.

14. F. C. Tompkins and R. L. Goldsmith, "A New Personal Dosimeter for the Monitoring

of Industrial Pollutants", American Industrial Hygiene Assn. J. 371 (1977).

15. P. W. West and K. D. Reiszner, "Field Tests of a Permeation-Type Personal

Monitor for Vinyl Chloride", American Industrial Hygiene Assn. J. 3^, 645 (1978).

16. "Badges Monitor Worker Exposure to Gases", Chem. and Engineering News, July

9, 1979, p. 21.

List of Symbols

CLKit')

p

S

RV,t-)

t

= concentration of gas dissolved in membrane, moles cm

2 -1

= diffusion coefficient, cm sec

= partial pressure of gas at time t, atm.

-3

= solubility coefficient of gas in membrane, moles cm atm

-2 -1

= flux of diffusing gas, moles cm sec

= total flux of gas diffusing into substrate in the time

moles cm“2

.

= thickness of membrane, cm,

= concentration of trapping sites at position and

time t, moles cm”^.

= initial concentration of trapping sites, moles cm

-3

A

C[t)

rate constant of irreversible bi-molecular chemical

reaction between diffusing gas molecules and trapping

sites, moles“^ cm^ sec“^.

total amount of unreacted gas per unit area in membrane

at time t, moles cm^^

.

RU)- total amount of gas per unit area which has reacted

up to time T, moles cm“^.
I
List of Illustrations

Fig. 1 Diagram representing Model I.

Fig. 2 Diagram representing Model II.

Fig. 3 Diagram showing the regions of filled and empty trapping sites

on either side of the boundary at ^ as well as the concentration

profile of diffusing gas.

2 Values of z as a function of

transcendential equation (43)

.

Fig. 4 obtained from the
,^gji
0) 0 > CO

"is o CO

CO

0) 3

cc CO
I
X

IIo

Concentration

of

diffusing

gas

i'

FIGURE 4

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4. TITLE AMD SUBTITLE

Analysis of Mathematical Models of Integrating Monitoring

Devices

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Robert J. Rubin

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16. ABSTRACT (A 200-wonl or teas lactual aummary ol moat aighi/ieant information. If document inefudea a aignificant bibliography or

literature surwey, mention it heroa)

Two mathematical models of a personal monitoring device are analyzed. The

purpose of the device, which is to be worn by an individual, is to indicate tne

level of exposure to various criteria air pollutants such as O3 , SO2

, N02» CC

during the time per’od which might range from 24 hours to a week or longer. The

goal of the analysis of the models is to determine how the quantity of pollutant

which is collected ‘s related to the average pollutant concentration and how

material constants such as solubility and diffusion coefficients affect the

operation of the device.

17. KEY WORDS to fw#/r« antriea; e/phebetical order; cmpitatixe only the Ural letter ot the tiret key word unteee a proper name;

eeparmted by eemicolonw)

Diffusion; integrating device; permeation type; personal dosimeter; reaction;

time-dependent.

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