NATIONAL BUREAU OF STANDARDS REPORT

3186

A METHOD FOR DETERMINING

THE

DIFFUSION COEFFICIENT OF A POROUS MATERIAL

by

Samuel Mo Genensky

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Sinclair Weeks, Secretary NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director

THE NATIONAL BUREAU OF STANDARDS

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

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

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

cover of this report.

Electricity. Resistance and Reactance Measurements. Electrical Instruments. Magnetic

Measurements. Electrochemistry.

Optics and Metrology. Photometry and Colorimetry. Optical Instruments. Photographic

Technology. Length. Engineering Metrology.

Heat and Power. Temperature Measurements. Thermodynamics. Cryogenic Physics. Engines

and Lubrication. Engine Fuels. Cryogenic Engineering.

Atomic and Radiation Physics. Spectroscopy. Radiometry. Mass Spectrometry. Solid State Physics. Electron Physics. Atomic Physics. Neutron Measurements. Infrared Spectros- copy. Nuclear Physics. Radioactivity. X-Ray. Betatron. Nucleonic Instrumentation. Radiological Equipment. Atomic Energy Commission Radiation Instruments Branch.

Chemistry. Organic Coatings. Surface Chemistry. Organic Chemistry. Analytical Chemistry.

Inorganic Chemistry. Electrodeposition. Gas Chemistry. Physical Chemistry. Thermochemistry.

Spectrochemistry. Pure Substances.

Mechanics. Sound. Mechanical Instruments. Fluid Mechanics. Engineering Mechanics. Mass

and Scale. Capacity, Density, and Fluid Meters. Combustion Control.

Organic and Fibrous Materials. Rubber. Textiles. Paper. Leather. Testing and Specifications. Polymer Structure. Organic Plastics. Dental Research.

Aletallurgy. Thermal Metallurgy. Chemical Metallurgy. Mechanical Metallurgy. Corrosion.

Mineral Products. Porcelain and Pottery . Glass. Refractories. Enameled Metals. Concreting

Materials. Constitution and Microstructure.

Building Technology. Structural Engineering. Fire Protection. Heating and Air Conditioning. Floor, Roof, and Wall Coverings. Codes and Specifications.

Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering.

Electronics. Engineering Electronics. Electron Tubes. Electronic Computers. Electronic

Instrumentation. Process Technology.

Radio Propagation. Upper Atmosphere Research. Ionospheric Research. Regular Propagation

Services. Frequency Utilization Research. Tropospheric Propagation Research. High Frequency

Standards. Microwave Standards.

^Office of Basic Instrumentation ^Office of Weights and Measures.
NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1002 - 10-1029 March 30 , 195^ 3186

A METHOD FOR DETERMINING

THE

DIFFUSION COEFFICIENT OF A POROUS MATERIAL

by

Samuel Mo Genensky

Fire Protection Section

Building Technology Division

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

The publication, rep

unless permission is

25, D. C. Such pen

cally prepared If tt

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST)

on October 9, 2015.

n part, Is prohibited

indards, Washington

)ort has been speclflport for Its own use,
5

'

‘ 4
A METHOD FOR DETERMINING

THE

DIFFUSION COEFFICIENT OF A POROUS MATERIAL

Abstract

A mathematical analysis based upon Fick's law is developed

for a spherical body exposed to an atmosphere with a time

invariant moisture concentrationo An analytic solution is

obtained from which diffusion coefficients may be computed,,

1 . INTRODUCTION

The fire endurance of many structures is greatly dependent on

the amount of water adsorbed on the surface and within the

capillary pores of the material of which the structure is formed

This moisture content is a function of both the pore structure

of the material, its afinity for water and the moisture vapor

pressure present in the ambient atmosphere within which the

specimen is conditioned c Many studies have been made of the

equilibrium moisture content of woods and textile fibers, as well

as Portland cement pastes « However, little work appears to have

been done on concrete and as a result it seems worth while to

perform a few exploratory tests on representative concrete

specimens for the purpose of obtaining some indication of the

moisture content of concretes prepared in the same fashion and

dried to equilibrium under controlled ambient conditions =, In

planning these experiments it appeared that by proper selection

of specimen shape and method of test, it would be possible to

obtain moisture diffusion coefficients with very little additional effort o The present report outlines the mathematical

reasoning which appears to justify this phase of the proposed

worko

2o ASSUMPTIONS

It was proposed that the experiments would permit the

measurement or calculation of the following for the specimen

under examination:

Wq the weight of the specimen when completely dry

- 2 -

W]_ the equilibrium weight of the specimen exposed to an

atmosphere containing moisture at a constant concentration

Cqo

W2 (t) the weight of the specimen, a function of time t,

exposed to an atmosphere containing moisture at a constant concentration Cpo Wp = WpCtg), tQ being the

time the specimen is first exposed to this atmosphere

W2 = W2 (t) t^T the equilibrium weight of the specimen

exposed to the atmosphere containing moisture at a

' constant concentration Cpo T is the time at which

an increase in the specimen's weight can no longer be

detected by the experimental apparatus*

A mathematical model._based upon Fick's law which utilizes this

information ±0 obtaiir an estimate of the diffusion coefficient is

available for a spherical specimen of radius R, throughout which the

concentration of. the moisture depends upon the radial distance

from the center of the specimen and upon the time that the specimen

has been exposed to the atmosphere containing moisture at a

constant concentration Ci* This same analysis is equally applicable

to a hemispherical specimen whose flat surface is completely

insulated against moisture flow because here also the concentration

of .the moisture in the specimen again depends upon the radial

distance from the center of the specimen and upon the time* For

this model the solution is one-half that for the sphere*

No attempt is made to account for heats of adsorption or

desorption in the following analysis.

Mathematically the

1 3c _

2 0C

D 9t “ ^p2 r 0r

C = Cp

C = C

3

C 7

^ C50

3. ANALYSIS

problem becomes;

( 0 < r < R, t

(O^r^R, t

(r = R, t >

(r = O 5 t^

( 1 )

to ) ( 2 )

0

) (3)

0 ^ (^)

-3-

where r is the radial distance measured from the center of the

sphere o cmo

t is the time measured from the instant the specimen is

exposed to the atmosphere containing moisture at a constant

concentration C]_o seco

C=C(r,t) is the concentration of the moisture in the specimen and

is a function of both radial distant r and time t» gm/cm^

D is the diffusion constant for the specimeno cm2/sec»

C2= C2 is the concentration in gm/cm3 of the moisture in

the specimen when it is introduced into the atmosphere

containing moisture at a constant concentration Cp„

Wf—

q

EitrI

is the concentration in gm/cm^ of the moisture

the specimen when it reaches equilibrium in the

atmosphere containing moisture at a constant concentration

Cq.

No generality will be lost if to is taken equal to zero. Condition (4) guarantees that the cohcentration remains finite at

the center of the sphere in compliance with physical requirements.

„ W2-W0

3 1

Let V = V(r,t) = C(r,t) -C3 ( 5 ) Using (5) and recalling that

(1) - (4) become

to has been set equal to zero,

1 n ^ + 2di

D 9 t 3^ r (0-^r.s:R, t>0) (6)

V = C2 “ C3 (O^r^/R, t = 0)

1

1

( 7 )

V = 0 (r = R, t > 0 ) (8)

V 7

^ 007 (r = 0 , t;^ 0) ( 9 )

let V = V(r,t) = p(r) ®^(t) = ptr (10)

then II II and 3^2 -p-r' (11)

Substituting these expressions into (6) and multiplying the

resulting expression by yields:

1 « = ^

+ 2 ei

D ‘TT f r ^

(0<.r <R, t >0) ( 12 )

- 4-

Let equal where X is real (13)

Therefore = 0

‘ (IM-)

and thus where ^ is an arbitrary constant (15)

Now by (13)

^

+ 2

^ _ X2

p r p

(16)

and thus p" p P' p

= 0 (17)

Let S = rp (18)

Using (18), (17) becomes

S" + X^s = 0

, , , ^ , Cos Xr , ^ SinXr

and thus P- A p

+ B ^ where A and B are

arbitrary constants

(19)

(20)

Now since lim CosXr , _ a= ^

r->0 r —

for otherwise (9) would

contradicted.

be

V(r,t) = SlnXr „hereP=dB (21)

Now in view of (8) Sin^H = 0

and thus n=0, +1, +2, o = o

Therefore (21) may be written ~ flfot „2

V(r,t) = 2_ Pn®

n=-cx)

sdT r Sin R

r ^ tiEt ^2 jflrr

Z_ Qn©'’ r Qn"Pn”P-n

n-1

n = 1,2,

( 22 )

(23)

(24)

(25)

o o e

5-

Considering (7) and (25) and recalling Fourier series analysis

(C2 - C:,)r = 21 Qr ,Sinn1tr

n=l ^

(26)

and thus

2 r* R

£

hftr

R I Sin p dr (27)

Integrating (27)

^ _ 2(C2-C:^)R

Qn

Xzill

n

n+1

(28)

Thus

V(r,t) 2IC2-P3)R, ir

c?o

L

a=l

i^n+1 yr2pt 2 ^ ® r

2 ^ Sin-^/r (29)

Now the total moisture content of the sphere is found by integrating

V(r,t) over the volume of the sphere. Thus

-'o

R

4irl V(r,t)r2dr = f".^C(r,t) -> C3_/r2dr (30)

and

411^ C(r5t)r2dr = j1^r3c-

+8(C2-C3)R L

n=l

(-1)

n+1 r2 P R

n

„ . nttr , rSin^— dr

(31)

Integrating (31)

1

R

J C(r,t)r^dr

o = jTTR^C 3 + ^^ 2^03 )R^ ^ ^

e

It? Dt

r2

1V n=l (32)

- 6 -

and for a hemispherical specimen

2lKr''c(r,t)r2dr = flTR^C, + y 4> 3 J ^ Z- „2 r2

n=^l

^n2 ,

. " (33)

Now

rn

4irj C(r,t)r^dr is the weight of the moisture in the specimen

at time t (where t = 0 denotes the moment the specimen is placed

into the atmosphere containing moisture at a constant concentration

C2_)o Therefore

1''O

R

C(r,t)r2dr = W2(t) - W 0< t < (3V)

Recalling the definitions of C2 and

1+3iTR^C3 = W2

- t >T (35)

and

C2 = Wi - w

n

tt

(36)

Recalling that Z- (3*t)-(36), (33) becomes

1>2Dt„2 oo

W2(t) - Wo = W2 - w + Z(Wi-Wo)-(W2-Wol^ n2*

r2

-n

n=-l

(37)

and thus

W2( t) - ^

6

OG

]fJl - W2

^

ir^Dt ^2

e R^

(38)
4 y

nr^-

'7

• /;*" '

^''i'

' mA' ' ' *v,

> •'';-,-A ' >v*;
-7-

Let

o=»

vr- 1

n-1

irSot ^2

-rZ- ^

and (38) becomes

W2(t-) - W2

V^(0) - W2 recalling that = ¥2(0)0

( 39 )

(^0 )

Since W2(t), ¥2(0), ¥p, t and R are known, (^0) can thus be solved

for the diffusion coefficient Do Figure 1 which is a plot of

f

lt^Dt “tT^Dt

( ) against -rz;— has been included to ease the labor in the

r2

evaluation of D

1l"^Dt

For ^2

R2

,1T2Dt N

t

lr2pt

r2

% R^

<. 0,001

)

iy-2pt

r

2 and thus the simpler expression/|^2

approximation to

p -f^2r^^ -t (

* 2

' ) over that range

becomes a very good
>
FIG. I

RATIO

OF

UNACCOMPLISHED

ABSORPTION

TO

TOTAL

POSSIBLE

ABSORPTION

VS
I'i-ri I
THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

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

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

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

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

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

j>roblems; invention and development of devices to serve special needs of the Government; and the

development of standard practices, codes, and specifications. The work includes basic and applied

research, development, engineering, instrumentation, testing, evaluation, calibration services, and

various consultation and information services. A major portion of the Bureau’s work is performed

for other Government Agencies, particularly the Department of Defense and the Atomic Energy

Commission. The scope of activities is suggested by the listing of divisions and sections on the

inside of the front cover.

Reports and Publications

The results of the Bureau’s work take the form of either actual equipment and devices or published papers and reports. Reports are issued to the sponsoring agency of a particular project

or program. Published papers appear either in the Bureau’s own series of publications or in the

journals of professional and scientific societies. The Bureau itself publishes three monthly periodicals, available from the Government Printing Office: The Journal of Research, which presents

complete papers reporting technical investigations; the Technical News Bulletin, which presents

summary and preliminary reports on work in progress; and Basic Radio Propagation Predictions,

which provides data for determining the best frequencies to use for radio communications throughout

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

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

Publications.

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

the National Bureau of Standards ($1.25) and its Supplement ($0.75), available from the Superintendent of Documents, Government Printing Office. Inquiries regarding the Bureau’s reports and

publications should be addressed to the Office of Scientific Publications, National Bureau of

Standards, Washington 25, D. C.