ORNL-3494.txt

 

 

A

ORNL-3494 L.
UC-4 — Chemistry '\

TID-4500 (23rd ed.)

 

 

AN EXPERIMENTAL STUDY OF SORPTION OF
URANIUM HEXAFLUORIDE BY SODIUM

FLUORIDE PELLETS AND A MATHEMATICAL

ANALYSIS OF DIFFUSION WITH
SIMULTANEOQOUS REACTION

L. E. McNeese

 

  

OAK RIDGE NATIONAL LABORATORY

operated by

UNION CARBIDE CORPORATION

for the

U.5. ATOMIC ENERGY COMMISSION

 
 

——

 

e —— e

 

 

 

 

 

LEGAL NOTICE

This report was prepared os on cccount of Government sponsored work. Meither the United States,

ner the Commission, nor ony person acting on behalf of the Commission:

A. Mockes any warranty or ropresentation, expressed or implied, with respect to the sccwrocy,
completeness, or usefulness of the information contained in this report, or thot the use of
any informotion, opparatus, method, or process disclosed in this repert may not iniringe
privately owned rights; ar :

B. Assumes any lichilities with respect to the use of, or for damages resulting from the use of
any information, apparatus, method, or process disclosed in this report.

As used in the obove, "“‘person acting on behalf of the Commission"™ includes any employee or

controctor of the Commission, or employee of such centractor, to the extant that such employee

or contractor of the Commission, or employee of such controctor prepores, disseminotes, or
provides occess to, any information pursuont to his employment or contract with the Commissian,
or his employment with such contraetor.

 
y ¥

ORNL-349

Contract No. W-TLO5-eng-26

CHEMICAL TECHNOLOGY DIVISION

Unit Operations Section

AN EXPERIMENTAL STUDY OF SORPTION OF URANIUM HEXAFLUORIDE
BY SODIUM FLUORIDE PELLETS AND A MATHEMATICAL ANALYSIS

OF DIFFUSION WITH SIMULTANEOUS REACTION

L. E. McNeese

This report was prepared as a thesis and submitted to the Faculty of ,
the Graduate School of The University of Tennessee in partial fulfillment
of the degree of Master of Science in the Department of Chemical Engineering.

Date Issued

NOV 14 1363

OAK RIDGE NATIONAL LABORATORY
Oak Ridge, Tennessee
operated by
UNION CARBIDE CORPORATION
for the
U. S. Atomic Energy Commission
-t

ii

ACKNOWLEDGEMENT

The writer wishes to acknowledge the numerous helpful suggestions
of Dr. S. H. Jury of the Chemical Engineering Department of the
University of Tennessee and Dr. M. E. Whatley of the Chemical Technology
Division of the Oak Ridge National Laboratory.

Analyses of the sodium fluoride peilets and of the uranium .
hexafluoride-sodium fluoride complex were performed by members of the
Analytical Chemistry Division of Oak Ridge National Laboratory. The
porosimetry work on sodium fluoride pellets was done by P. G. Dake and
co-workers of the Special Analytical Service Group of the Oak Ridge
Gaseous Diffusion Plant. The measurements of the uranium concentration
profile for partially reacted pellets were done by H. W. Dunn of the
Analytical Chemistry Division of Oak Ridge National Laboratory. The
co-operation and care of these individuals is greatly appreciated.

The close attention to experimental detail and Fhe fine work of

J. Beams of the Chemical Technology Division of Oak Ridge National

Laboratory during the experimental phase of this study is appreciated.
This work was performed in conjunction with the Fluoride

Volatil?ty Process of the Chemical Technology Division, Oak Ridge

National Laboratory. The writer is grateful for the interest of

F. L. Culler, Jr., Division Director, M. E. Whatley, Section Chief, and

R. W. Horton, Group Leader, whose support made this work possible.
‘-

4

iid

ABSTRACT

The rembval of uranium hexafluofide frdm a gas stream containing
uranium hexafluoride and nitrogen by a single layer of sodium fluoride
pellets was investigated. Experimental data on the rate and extent of
sorption were obtained in the temperature range 29 to 100 degrees cen-
tigrade and uranium hexafluoride concentration range 0.57 to 10.9 mole
per cent uranium hexafluoride.

The results of this study indicate that the rate controlling
mechanisms are: transfer of uranium hexafluoride across a stagnant gas
film surrounding the pellet, diffusion of gaseous uranium hexafluoride
in the pores of the pellet, and diffusion of uranium hexafluoride
through a layer of uranium hexafluoride-sodium fluoride complex covering
the unreacted sodium fluoride in the interior of the pellet. The
crystalline density of the complex, UF6-2NaF, was determined to be
4,13 grams per cubic centimeter which indicates that incomplete reaction
of the sodium fluoride will occur for pellets in which the initial
volume void fraction is less than 0.807.

A useful model was devised to represent the sorption of uranium
hexafluoride by a single pellet of sodium fluoride, and differential
equations based on the model were written. A general method of solution
of the partial differential equation describing simultaneous diffusion
and irreversible reaction with variable diffusivity and reaction rate
constant was derived for systems in which a steady-state type of
solution is valid. The experimental data were correlated on the basis

of the model with a root-mean-square error of 9.5 per cent for all
'

iv
points. The resulting computer code and associated data may be used

for design of sorber systems such as fixed and moving beds.
&

v

TABLE OF CONTENTS

CHAPTER \ PAGE
I. INTRODUCTION. « ¢ ¢ ¢ & o o 4 o o o o o o o o o o s o s o & 1
IT. REVIEW OF THE LITERATURE. . . ¢ « « o « + v ¢ & « + e o o 3
Uranium Hexafluoride-Sodium Fluoride System . . . . . . . 3

Uptake of Gases by Solids « « ¢« + ¢« v ¢ v 4 ¢ 4 ¢ o =« » « 6
Diffusion of Gases Through Porous Solids. . . . . . . . . 10

? Simultaneous Diffusion and Irreversible Reaction. . . . . 12
Mass Transfer to the External Surface of the Pellet . . . 15

IIT. MATHEMATICAL MODEL«. + + &« ¢ &« & & « o o o o o o o « + + » + 19

IV. METHOD OF SOLUTION OF SIMULTANEOUS DIFFUSION AND REACTION

EQUATION + ¢ ¢ = o « « & o « o o o o o o o o o o o o « « o 32

V. MATERIALS AND EQUIPMENT + «. « « o s ¢ ¢ o s s s o o o« « » o« ho

MAteridals « « o v o« v s o o o o o o o o o s s e o o o+ . ko

EQUIPmMENnt « « v o o 4 o ¢ 4 4 4 o 4 4 v s e e e e e e .. ke
VI. EXPERTMENTAL PROCEDURE. « « « + « v « v o v v v v v v v .. L5
‘ Differentiai-Bed Studies. + + + &+ 4 4 4 4 4 s s s . . . . b5
: Determination of Density of Complex . . . . . . . . . . . LT

VII. EXPERIMENTAL RESULTS. . « + « &« ¢ o ¢« o « 4 o + & +» « « « . ko
Differential-Bed Studies. . . « +« ¢« « v & & « + o + « « . U9
Examination of Partially Reacted Pellets. . . . . . . . . 56
Density of Complex. . . . . . . . « . « . « . . . . . . . 59

VIII. ANALYSIS AND DISCUSSION OF RESULTS. + « « v + o « o « o « . 65
Diffefential-Bed Data + « + + o o 4 e 4 v s e 4 s e 4 e . B5

Application of Data to Sorber Design. . « . « . « + « « . T8
CHAPTER

Vi

Discussion of EXror . ¢ ¢ ¢ ¢ ¢ o o o o o o o o s o s

IX. CONCLUSIONS AND RECOMMENDATIONS . -« « « « « & & « o« « o &

Conclusions .« + « ¢ ¢ ¢ ¢ 4 ¢ v & 4 o s o ¢ & o o o o o
Recommendations . « « &+ « ¢« ¢ ¢ ¢ ¢ o & o « o o o o+
LIST OF REFERENCES. . . « ¢ & ¢ ¢« ¢« & o & o s o o o o o &
APPENDICES
A. Temperature of Pellet During Sorption . . . . . . .
B. Viscosity of Uranium Hexafluoride-Nitrogen Mixtures
C. Bulk Diffusivity of Uranium Hexafluoride. . . . . .
D. Mean Free Path of Uranium Hexafluoride in Mixtures
of Uranium Hexafluoride and Nitrogen. . . . . . .
E. Properties of the Sodium Fluoride Pellets . . .
F. Confiergence Characteristics of Numerical Method . .
G. Heat Transfer-Characteristics of Differential-Bed
and Gas Preheater . . « « « « ¢ ¢ v ¢ ¢« s o o o
H. Computer Code . . + . + « « - v v v v v v v 0 o b
I. Original Data . . . . . . e c e e s e e e

LIST OF SYMBOLS « + ¢ ¢ ¢ ¢ « o o &+ 4 o s o o o & s o s

- PAGE

80
81
81
82
8k

91
95

101
104
108

11k
116
121

122

.

v
n)

o

=of

TABLE

IT.

III.

IvV.

VI.

VII.

VIII.

IX.

Experimental Results
with 2.62 Mole Per
Experimental Results
with 2.35 Mole Per
Experimental Results
with 0.57 Mole Per
Experimental Results
with 2.45 Mole Per
Experimental Results
with 8.51 Mole Per

Experimental Results

vii

LIST OF TABLES

from

Cent

from

Cent

from

Cent

from

Cent

from

Cent

from

Differential-Bed Runs
Uranium Hexafluoride.
Differential-Bed Runs
Uranium Hexafluoride.
Differential-Bed Runs
Uranium Hexafluoride.
Differential-Bed Runs
Uranium Hexafluoride.
Differential-Bed Runs
Uranium Hexafluoride.

Differential-Bed Runs

PAGE
at 29°C
c e e e .. 50
at 50°C

at 100°C
e e e 4« . 53
at 100°C
c o« s« e 54
Showing

Variation of Effective Capacity for Uranium Hexafluoride

with Temperature.

&« e s« .« 55

Weight Gain and Exposure Data for Preparation of Uranium

Hexafluoride-Sodium Fluoride Complex at 100°C . . . . . . 62

Uranium Content of Samples of the Complex Before and After

Exposure to Toluene . .

e e e oos Bh

Viscosity of Uranium Hexafluoride-Nitrogen Mixtures in the

Temperature ‘Range 29 to 100°C and Uranium Hexafluoride

Concentration Range 0.57 to 8.5 Mole Per Cent at One

Atmosphere. . . .

- X
TABLE

x.

XI.

. XII.

viii

Diffusivity of Uranium Hexafluoride in Mixtures of Uranium

Hexafluoride and Nitrogen at Atmospheric Pressure in the

Temperature Range 29 to 100°C .. « . . . . .

PAGE

. 100

Mean Free Path of Uranium Hexafluoride in Uranium Hexafluoride-

Nitrogen Mixtures in the Temperature Range 29 to 100°C

and Composition Range 0.5 to 8.5 Mole Per Cent Uranium

Hexafluoride at Atmospheric Pressure.

Composition of Sodium Fluoride Pellets Before and After

Fluorination at 400°C for One Hour.

. 103

. 105
&

FIGURE

a)

10.

ix

LIST OF FIGURES

PAGE
Spherical Shell Used in Derivation of Diffusion Equation
for Variable Reaction Rate, Diffuéivity, and Volume
Void Fraction. . . . ; =
Crystalline and Broken Layers»of Complex vaering
Unreacted Sodium FIUOTide. « « o o o « o v o o o 0 v o 28

Typical One-eighth-inch Right Circular Cylindrical Sodium
Fluoride Pellets « « v ¢ & ¢ ¢ o v o o o o o« o o & + « « k41
Flow Diagram for Equipment Used in the Study of Sorption
of Uranium Hexafluoride by Sodium Fluoride . . . . . . . L3
Sorption Vessel Used in Differential-Bed Studies. . . . . . L6
Axially-Sectioned Sodium Fluoride Pellets Containing

Uranium Hexafluoride Sorbed at 50°C. « . « + « « « + « .+ 57

Axially-Sectioned Sodium Fluoride Pellets Containing

‘ Uranium Hexafluoride Sorbed at 100°C . . . . . . . . . . 58
Typical Solid-Phase Uranium Hexafluoriae Concentration
Profile for Pellets Reacted at 100°C . . . . . . . . . . 60
Comparison of Experimental and Model-Predicted Data Showing
| Variation of Pellet Loading with Time at 100°C and 0.57
.Mble Per Cent Uranium Hexafluoride . . . . . . . . . . . 70O
Comparison of Experimental and Model-Predicted Data Showing
>Variation of Pellet Loading With Tifie at 100°C and 2.45

Mole Per Cent Uranium Hexafluoride . « . ¢« . . . + . « « T1
FIGURE PAGE
11. Comparison of Experimental and Model-Predicted Data Showing

Variation of Pellet Loading wifh Tifie at 100°C and 8.51

Mole Per Cent Uranium Hexafluoride. . . . . . . . . . . . 72
12. Comparison of Experimental and Model-Predicted Data Showing

Variation of Pellet Loading with Time at 50°C and 2.35

Mole Per Cent Uranium Hexafluoride . . . . . . . . . . . 73
13. Comparison of Experimental and Model-Predicted Data Showing

Variation of Pellet Loading with Time at 29°C and 2.62

Mole Per Cent Uranium Hexafluoride. + + « « ¢ ¢« « « « « « Th
1k. Comparison of Experimentally Determined and Model-Predicted

Results on Variation of Effective Pellet Capacity with

Temperature o« + + « o o o s o o o « s o s 2 o« o« o o « « « 5
15. Calculated Values of Effective Pellet Capacity for Pellets

having an Initial Void Fraction of 0.45 Showing Effect

of Pellet Surface Area. . . . « . « . . . + ¢« + . o« . TT
16. Calculated Values of Effective Pellgt Capacity for Pellets

having an Initial Void Fraction of 0.45 and a Surface

Area of 0.86 Square Meter per Gram. . + « « o« & RN
17. Time Variation of Temperature Difference Between Pellet

Surface and Gas Stream. .« « « « + + =« + 4+ ¢« s « o + « « . Oh
18. Porogram of Sodium Fluoride Pellets Treated with Fluorine

for One Hour at 4O0°C +« ¢ ¢« + & o ¢ &+ « « o o« o« « « « « 2107

19. Variation of Convergence Ratio with Number of Shells and

Dimensionless Parameter © « « « « « « « + « + + + o« » .« 113
1

CHAPTER I
INTRODUCTION

The study reported here concerns the determination of the rate
controlling mechanisms for the removal of uranium hexafluoride from
flowing streams of uranium hexafluoride in nitrogen by cylindrical
pellets (one-eighth-inch right circular) of sodium fluoride.

Uranium hexafluoride reacts reversibly with sodium fluoride to
form a solid complex. The reversible character of the reaction makes
it attractive as a means for separating uranium hexafluoride from other
gases and/or as an alternative to low-temperature cold trapping for
collecting uranium hexafluoride.

An important application of the uranium hexafluoride-sodium
fluoride system is in the Oak Ridge National Laboratory's Fluoride
Volatility Process for recovery of uranium from irradiated nuclear
fuels.lLE During the final step of the process, a molten fluoride salt
containing dissolved uranium tetrafluoride and fluorides of fission
products and corrosion products is contacted with gaseéus elemental
fluorine at 500 to 600°C (degrees centigrade). The gas stream leaving
the fluorinator consists of a mixture of uranium hexaflfioride, unreacted
fluorine; fission product fluorides, and corrosion product fluorides.
Differences in the decomposition pressures of the fluorides that form
sodium fluoride complexes are exploited during alternate sorption and
desorption of the uranium hexafluoride in fixed beds of sodium fluoride

to produce a uranium hexafluoride product of high purity.
2

Information on the effects of various system parameters on the
rate of.sorption is needed so that sodium fluoride sorbers may be
designed for a wide range of operating conditions..

In this study, a determination was made of the rate-controlling
steps in the sorption of uranium hexafluoride from a stream of uranium
hexafluoride and nitrogen by sodium fluoride in the form of one-eighth-
inch right circular cylindrical pellets at atmospheric pressure. The
temperature range covered was 29 to 100°C; the uranium hexafluoride

concentration range was 0.57 to 10.9 mole per cent.

 
5

CHAPTER II

REVIEW OF THE LITERATURE

The removal of uranium hexafluoride from a flowing stream of

uranium hexafluoride in nitrogen by sodium fluoride pellets is believed

to involve some or all of the following processes.

w)

1.

- 5.

Transfer of uranium hexafluoride from the gas stream to the
external surface of the pellet.

Transfer of uranium hexafluoride from the external surface of
the pellet to the interior of the pellet by diffusion of
gaseous uranium hexafluoride in the pores of the pellet.
Adsorption of uranium hexafluoride on the internal surface

of the pellet.

Diffusion of adsorbed uranium hexafluoride or gaseous uranium
hexafluoride from the internal surface of the pellet through
a layer of uranium hexafluoride-sodium fluoride complex to
underlying sodium fluoride.

Reaction of ufanium hexafluoride with sodium fluoride.

A discussion of the pertinent information from the literature is given

below.

Uranium Hexafluoride-Sodium Fluoride System

The reaction of uranium hexafluoride with sodium fluoride was

noted first by Ruff and Heinzelman in 1911.

>0 Grosse in 19&126 reported

that hydrogen fluoride was necessary for the reaction, which resulted
4
in the formation of a ternary complex. Subsequent study of the complex
formed in the temperature range 30 to 100°C Ey Martin et al. in 195157
showed the composition UF6-5NaF and indicated decomposition of theA
complex at 450°C yielding uranium hexafluoride and fluorine. It was
also concluded that the presence of hydrogen fluoride is not necessary
for obtaining a reaction between uranium hexafluoride and sodium
fluoride.

Cathers et al. in 195712 studied the formation and decomposition
reactions of the complex and concluded that the reaction involved an
equilibrium between gaseous uranium hexafluoride, solid sodium fluoride,
and the solid complex which was given the formula UF6-3NaF although it
was noted that some preparations had a composition nearer to 2UF6'5NaF.
The decomposition pressure of the complex was measured in the tempera-

ture range 80 to 360°C and conformed to the equation

log10 p = 10.88 -~ 5.09 x 103/T ) (2)
where
p = decomposition pressure in millimeters mercury,
T = temperature in degrees Kelvin.

Use of the Clausius-Clapeyron relation with the decomposition pressure
relation yielded the heat of sorption for the complex: -23.2 kilocal-
ories per mole. A study was made of the rate of decomposition by two
alternative reactions reported by Martin et al., the first of which was
found to yield fluorine and a white complex in which uranium had the

valence of plus five, and a subsequent reaction which was found to yield

7
>
a green complex in which uranium had the valence of plus four, the
result of the liberation of additional fluorine. Based on the rate
constant for the first decomposition reaction, these reactions can be
neglected in the present study in which the temperature range is below
100°C.
Worthington 'in 195763 studied the rate of reaction between pure
gaseous uranium hexafluoride (at a pressure of fifty-six millimeters of
mercury ) and finely divided sodium fluoride in the temperature range
80 to 150°C. No estimates of particle size or surface area were given.
At a given temperature, the reaction followed the logarithmic rate law.
The final composition contained slightly fiore uranium than would corre-
spond to the fofmula UF6-3NaF. The average rate increased with tempera-
ture up to 130°C, after which a marked decrease was observed. In view
of the e#perimental method used (admission of uranium hexafluoride to
an evacuated chamber contéining a thin layer of sodium fluoride), the
temperature control of the sample was undoubtedly poor in the early
stages of sorption; it is believed that this is the origin of the
decrease in average reaction rate above 130°C.

The rate of reaction between pure uranium hexafluoride at ninety
millimeters mercury pressure and sodium fluoride in the form of powder,
crushed pellets, and pellets was investigated by Massoth et al. in
195859 in the temperature range 24 to 68°C. The reaction with powder
having a surface area of 0.33 square meters per gram followed the para-

bolic law after a loading of 0.6 grams of uranium hexafluoride per gram

of sodium fluoride had been established. Insufficient data were
6
available to establish the rate law at lower loadings.

The reaction with crushed pellets followed the logarithmic law
until the loading reached 1.9 grams of uranium hexafluoride per gram of
sodium fluoride, after which reaction in accord with the parabolic law
was observed. It was concluded that the thickness of the film of com-
plex at the onset of the parabolic law was the same in both cases, based
on measurement of the particle size of the materials. An increase in
reaction rate was observed as the sorption temperature was raised for
both powdered sodium fluoride and crushed pellets. The data on the
sorption rate with whole pellets (one-eighth-inch right circular
cylinders) scattered badly and few conclusions can be drawn. A rapid
initial reaction was observed, after which sorption stopped at a loading
of about one gram of uranium hexafluoride per gram of sodium fluoride.
An inverse effect of temperature on the maximum loading was noted.

53

Studies in progress by Katz”” with sodium fluoride powder having
a surface area of 7.0 square meters per gram show that the composition
of the complex is UF6°2NaF. It is believed that the lower extent of
reaction noted in previous studies with low-surface-area powders was
due to the buildup of a thick film of complex on the outside of the
individual particles. This film was five to ten times as‘thick as the

film on the higher surface area material and caused a low rate of

sorption.

Uptake of Gases by Solids
The uptake of gases by solids may be divided into two types:

that of adsorption, where the gas is retained on the surface of the
T
solid, and that of sorption wherein the interior of a nonporous solid
is penetrated. Study of adsorpFion, both theoretically and experimen-
tally, has been widespread, while only rudimentary data are available on
the somewhat more complicated process of sorption. -

“Adsorption is further divided into physical (van der Waals)
adsorption and activated adsorption (chemisorption). In physical
adsorption the adsorbed gas is held at the solid surface by relatively
weak forces comparable to van der Waals forces in a gas. This type of
adsorption is similar to the condensation of a pure vapor in that in
both processes the rate is almost instantaneous. A second similarity
is that the quantity of heat released on adsorption is approximately
the latent heat of vaporization (five to ten Rilocalories per mole) of
the adsorbing material. Also, physical adsorption is observed only at
temperatures near or below the boiling point of the adsorbing material,
whereas chemisorption commonly occurs at temperatures far above the
boiling point of the material being adsorbed.

Chemisorption more closely resembles chemical reaction -than con-
densation. The bonding forces between the gas and the solid are
normally stronger than those in physical adsorption; this is reflected
in the heat of adsorption, which is usually greater than ten kilocal-
ories per mole but less than the heat of reaction for typical chemical
reactions, which is about one hundred kilocalories per mole. Chemi-
sorption normally proceeds at a rate which is lower than that of
physical adsorption, and in most cases, an activation energy is observed

as in most chemical reactions.
8

Adsorption normally results in the deposition of a monolayer or,
at most, a few molecular layers of the adsorbing gas whereas sorption
often results in complete reaction of the original solid. As pointed
out by Cabrera gg_gl:,B if the lattice constants of the reacting.solid
and the solid product differ by more than about ten per cent; cracking
and degradation of the product film may occur. M.cBainlLl discusses
numerous examples where an initially érystalline material is reduced to
a powder during sorption of a gas or liquid. Katz33 observed that the
sorption of approximately three moles of hydrogen fluoride per mole of
_sodium fluoride results in complete disintegration of the pellets with
a tenfold increase in surface area. It is believed that sorption of
hydrogen fluoride results in a decrease in the size of the initially
crystalline particles originally in the pellet.

‘Most of the data on the rate of sorption of gases by finely
divided solids can be represented by one of three common relations: the
linear law, the parabolié law, and the logarithmic law. The linear law,
which predicts a constant rate of reaction for slab geometry, .is
observed in cases where the sorption rate is controlled by the rate of
reaction betwéen the reacting solid and gas. The parabolic law predicts
that the rate of sorption is inversely proportional to the thickness of
reaction product through which the gas must diffuse. The parabolic rate
is thus independent of reaction rate and is the rate of diffusion of
reactant.

The logarithmic law, known also as the Elovich equation, is less

well understood. Since the sorption of uranium hexafluoride by sodium
9

fluoride was observed to follow this law, it will be considered in
greater detail. Numerous attempts have been made to provide a theoret-
ical basis for the law, which states that the sorption rate decreases
exponentially with the quantity sorbed. Evans in 19#322 derived the
logarithmic rate law in consideration of the rate of oxidation of zinc
where cracking of the oxide layer was believed to occur. This mechanism
is believed to be applicable to the sorption of uranium hexafluoride on
sodium fluoride and will be discussed in Chapter III. Taylor in 195255
derived the logarithmic law for chemisorption leading to a monolayer.
He assumed that a certain number of active sites are produced at the
onset of sorption, that the sorption rate is proportional to the number
of active sites, and that the number of active sites decays bimolecu-

5T

larly. Trapnell in 1955 showed that a logarithmic law would be
followed if the activation energy increased linearly with the degree

of surface coverage for a monolayer buildup if the coverage is not near
completion. The uranium hexafluoride loading due to a monolayer of
uranium hexafluoride on sodium fluoride of the type used in this study
is 0.004 gram of uranium hexafluoride per gram of sodium fluoride, which
is approximately one per cent of the loading which is observed. For
this reason, these derivations of the logarithmic law are not considered
pertinent to the present discussion. Freund in 195"{2LF examined data

for the sorption of hydrogen on various oxides and found good agreement
between values of constants in the logarithmic law determined experimen-

tally and values calculated from a relation obtained by Sutherland et al.

in 195552 for the case in which the rate of sorption was controlled by
10
55

Knudsen flow in a porous material. Landsberg in 1955 reviewed the
literature on the logarithmic rate law for chemisorption and the oxida-
tion of mgtals.and gave a derivation of the law. The assumption was
made that the rate of adsorption was proportional to the surface density
of adsorption sites which were initially present or were generated
during sorption by such processes as the diffusion of adsorbed material
away from the ;urface. Although the basic idea on which the derivation

is based is certainly plausible, its use is limited by its indefinite

nature.

Diffusion of Gases Through Porous Solids

Gases may be transported through porous solids by bulk diffusion,
Knudsen diffusion, or a combination of the two. Bulk diffusion occurs
in a pore when the mean free path of the diffusing gas molecules is
small compared to the radius of the pore, so that ifi most collisions
the gas molecule collides with another gas molecule. When the mean free
path of the diffusing gas molecule is the same as or larger than the
radius of a pore, the gas molecule will collide more often with the
pore wall than with other gas molecules, and Knudsen diffusion will
occur. As discussed in Chapter III, only bulk diffusion is believed
important in this study.

The effective diffusivity of a gas being transported within a
porous solid by a diffusive mechanism is less thah the normal diffu-
sivity of the gas. Numerous efforts have been made to relate the ratio

of effective diffusivity to normal diffusivity and characteristics of
11
the porous solid. Maxwell in 1873ho considered the solid to be composed

of uniform spheres and obtained the relation:

 

D
BE'= 3 ?ee ’ (2)
where
De = effective diffusivity,
D = normal diffusivity, .
€ = volume void fraction.

Bruggeman in 19356 extended the range of validity of Maxwell's expres-
sion to higher values of ¢ by the use of a continuum model which yielded

the result:

UlmU

.52 - - (3)

Buckingham in 190&,T on the basis of data on the rate of diffu-
sion of oxygen and carbon dioxide through soil, suggested the relation

D :
S (1)
Masamune in 1962,38 working with large pore silver cat&lyéts‘in.which
only bulk diffusion was present; found that the Buckingham relation
represented the data better than the Maxwell or the Bruggeman relations.

58

Wakao in 1962 derived an expression for the effective diffusivity in

a solid containing both macro- and micropores; it reduces to Equation
() for the case of bulk diffusion in macropores. Currie in 196016
measured the rate of diffusion of hydrogen through a number of materials

in which the void fraction € varied from 0.18 to 0.98 and recommended

the relation:
12

n
De=7€:7_<_1:n21: (5)

where ¥ and n are characteristic éf a given material.

Petersen in 1958LL6 considered the effect of periodic pore con-
strictions (such as in pelleted or extruded porous solids) on the effec-
tive diffusivity. The pore model assumed was a hyperbola of revolution
giving a pore constriction at the vertex of the hyperbola. The solution
to the steady-state diffusion equation for a pore of this shape was
found at different values of £, the ratio of the maximum to minimum
cross section of the pore. Comparison of the rate of diffusion in this
fype pore and in an equivalent cylindrical pore showed that the normal
diffusivity was reduced by a factor of three when B had a value of

twenty-five.

Simultaneous Diffusion and Irreversible Reaction

The process in which a substance diffuses into a rigid medium
with which it reacts irreversibly is encountered in many fields and has
consequently received considerable study. In the most general form of
the problem, the rate of sorption of the reacting substance depends on
both the rate of diffusion of the substance in the medium (or in the
p;oduct of reaction between the reacting substance and the medium) and
the rate of reaction between the diffusing substance and the medium.
One may also have the added complication that the point values of
'diffusivity and the rate of reaction are dependent on the quantity of
the substance that has reacted at the point. To date, an analytical

solution yielding the rate of sorption and associated information has
13

not been found for the general problem. A number of solutions, both
analytical and numerical,.have been obtained for special cases of the
general problem and will be discussed'below. Most of these solutions
are of a steady-state nature; that is, the rate of sorption does not
include the time rate of change of the total quantity of unreacted
substance in the medium. These solutions result from solution of the
appropriate form of the general diffusion equation and can be grouped
according to reaction rate and diffusivity in the following manner:

1. Instantaneous reaction, constant diffusivity.

2. Instantaneous reaction, variable diffusivity.

5. Variable reaction rate, constant diffusivity.

In the first type of solution, distifiguished by an instantaneous
reaction rate and constant diffusivity, the rate of sorption of the
reacting substance is controlled solely by the rate of diffusion of
reactant through the reacted portion of the medium to the reaction
interface, which is of infinitesimal thickness and separates the region
in which complete reaction with the medium has occurred from the region
in which no reaétion has occurred. This type of solution was first
investigated by Hill in 192929 in the study of the diffusion of oxygen
and lactic acid in muscle tissue. Other investigations of this type
include the work of fiermans-inf19h728 on the diffusion of sulphide ions
into a gel containing heavy metal ions, and of the work of Booth in
19&85 who derived the condition for the existenge of a steady—gtate
type of solution for slab geometry. Crank in 195715 showed that the

agreement between the actual solution and the steady state approximation
1L

was dependent on the ratio S/C, where S is the capacity of the medium
for the reacting substance and C is the concentration of the reacting
substance in the fluid adjacent to the medium. For values of s/c
greater than ten, results calculated from the steady-state solution
agree to wichin one per cent of results from the actual solution for
plane, cylindrical, and spherical geometries. Kawasaki et al. in
1962,5u and Scott in 1962,51 encountered this type solution in the
reduction of pellets of ircn oxide and copper oxide, respectively, by
hydrogen and carbon monoxide. 1In these two studies, the diffusional
procesc was that of councerdiffusion of reactant and product gases;
however, this does not change the characteristics of the solution.

The second type of soiution, distinguished by instantaneous
feaction and variable diffusivity, was studied by Olofsson in 195645

and in 196OuuAin the study of uptake of periodate.ions by cellulose

fibers. The change of diffusivity with time was attributed to swelling -

of the fiber, and a sharp reaction interface was observed, as in the
first type of solution.

The third type of solution, distinguished by variable reaction
rate and constant diffusivity, is used widely in the field of hetero-~
geneous. catalysis for the pfediction of catalyst activity, selectivity
and other properties. In this type of solution, one or more reactants
and products penetrate the catalfst particle by diffusion through the
'pofe space in the particle to a depth dictated by the relative races of
diffusion and reaction. A constant value of the diffusivity of

reactants and products is observed since the gross structure of the
15

catalyst is unchanged by the occurrence of a reaction on the pore walls.
The vériability of reaction rate in the catalyst particle is caused by
the dependence of the rate upon the concentration of the reactants.
Hence, in the third type of solution, the reaction zone may be
restricted to a narrow region near the surface of the particle if the
reaction rate is high compared with the diffusion rate, or, the reaction
may occur throughout the entire particle if the reaction rate is low
compared with the diffuéion rate.

The third type of solution was inveétigated by Thiele in 1959.55
He obtained an analytical solution for slab geometry with first- and
second-order reactions, and for spherical geometry with first~order
reaction. Danckwerts in 195717 obtained the general solution for first-
order reaction in various simple geometries, of which the Thiele treat-
ment is a special case. The Thiele treatment has been used widely in
catalysis by Wheeler in 195160 and in 1955,61 by Bokhoven et al. in
195'+,L-L by Barnett et al. in 19615 and by numerous other investigators.
A similar type of solution was also obtained by Ausman et al. in 1962l
for the burning of carbon frém the internal surfaces of a porous cata-
lyst pellet where the burning rate was assumed to be first order with
respect to oxygen concentration. The Thiele.concept has been extended
to nonisothermal conditions by Tinkler et al. in 196156 and by Carberry

in 1961.10

Mass Transfer to the External Surface of the Pellet
The resistance to mass transfer of a component from a fluid

stream to the surface of a solid particle in a fixed bed is normally
16

considered to be the resist;nce to diffusion across a hypothesized
stagnant film of fluid surrounding the particle. The rate of transfer
per unit.area is assumed to be given by the product of a mass transfer
coefficient and the concentration difference across the film. Many
investigations have been made for determining the dependence of the mass
transfer coefficient on factors characterizing the fluid, the soiidJ and
the flow conditions.

Almost all mass transfer data are correlated on the basis of the
j factor originated by Chilton and Colburn in 1931+1LL which is defined
in terms of the mass transfer coefficient and physical properties of
the system. The correlation of j factor as a function of Reynolds
number is observed to consist of two straight line portions that inter-
sect at values of the Reynolds number from 50 to 150. Most investi-
gators are in agreement for the line reprgsenting Reynolds number
greater than 150, and since the values of the Reynolds number in this

r

study are considerably lower than fifty, only investigations pertinent
to the low Reynolds number range will be discussed. Obtaining accurate
mass transfer data in this range is difficult, particularily for
Reynolds numbers lower than five, owing to effects such as backmixing,
free convection, and axial diffusion. These problems and others
specific to the systems used for obtaining mass transfer data have
resulted in wide discrepancies betweefi the results of individual inves-
tigators and disagreement as to the factors on which the mass transfer
coefficient is dependent. For example, Hurt in 19&3,31 Resnick et al.

49

in 1949, and Bar-Ilan et al. in 19572 noted an effect of particle

oy

¥
L7

size greater than the dependence afforded by inclusion of. the particle
diameter in the Reynolds number, while other investigators show only the
variation accounted for by the Reynolds number. Examination of these
studies suggests an effect of particle size for particles less than four
millimeters in diameter; this effect becomes increasingly important as
the particle size is reduced.

Another area of inconsisteficy is\in the definition of the proper

Reynolds number for use in fixed-bed studies. Most investigators use

D G
N =-{%— ; (6)
where
Dp = particle diameter,
G = superficial mass flow rate,

i = viscosity of fluid.

2 Dryden et al. in 1953,20 and Carberry

36

However, Gaffney et al. in 1950,2

in 19609 defined Reynolds number as DpG/pe, while Lynch et al. in 1959

" used DpG/u(l-e). ‘Other differences are observed in the recommended

power dependence of the Reynolds number and the Schmidt number.

Carberry in 19609 derived an expression for mass transfer in
fixed beds on the basis of boundary layer considerations which agrees
well with the results of a number of investigations. This relation
shows a direct dependence of the transfer coefficient on fluid velocity,
which is not present in the Chilton-Colburn correlation.

7

Gupta and Thodos in 19622 examined existing data on mass

transfer in fixed beds for both gases and liquids as the fluid phase
18

and recommended the following relation:

 

¢ ip = 0010 + ——52p2 | ()
NRe - Ool"85
D G , _
for NRe = —Ef-greater than one, with the mass transfer factor being
defined as
k P _M
. _ g gf 2/3

This relation was used in the present study, although resistance to
mass transfer across the stagnant film is important only during the

early stages of sorption.
19

CHAPTER III

MATHEMATICAL MODEL

The over-all process of removal of uranium hexafluoride from a

flowing stream of uranium hexafluoride in nitrogen by a pellet of sodium

fluoride is believed to involve some or all of the following processes:

1.

The movement of uranium hexafluoride from the gas stream to
the external surface of the pellet,lwhich is commonly
depicted as the transfer of uranium hexafluoride across a
stagnant nitrogen film surrounding the pellet.

The diffusion of gaseous uranium hexafluoride through
nitrogen in the pores of the pellet.

The diffusion of gaseous uranium hexafluoride through a
cracked or broken layer of complex on the internal surfaces
of the pellet.

The adsorption of gaseofis uranium hexafluoride on a thin
layer of complex adhering to the underlying sodium fluoride.
The diffusion of adsorbed uranium hexafluoride across the
adherent layer to underlying sodium fluoride.

The reaction of uranium hexafluoride with sodium fluoride

below the complex layer.

Any model proposed to represent the over-all sorption process

must take into account those steps whose rates are of the same order of

magnitude as the rate of the over-all process. The model must also

account for effects resulting from deposition of complex in the pores
20
of the pellet.

.The most striking characteristic of the system under étudy is the
effect of temperature on the rate and on the extent of reaction between
uranium hexafluoride and sodium fluoride. As noted in the previous
section, for reaction between uranium hexafluoride and powdered sodium
fluoride, an increase in temperature over the range 29 to 100°C results
in an increase in the rate of reaction and the extent of reaction at a

59

specific time. However, as noted by Massoth et al., and from the
results of the present study, there is an inverse effect of temperature
on the maximum extent of the reaction of the sodium fluoride pellets
with uranium hexafluoride.

An explanation of this anomalous effect of temperature on the
rate and extent of sorption for finely divided sodium fluoride and for
sodium fluoride pellets is based on the combination of two ideas. The
first idea is that the temperature dependence of most chemical reactions

and of solid-phase diffusion is of the form e_E/RT

, whereas the tempera-
| : : . /2

ture dependence of bulk diffusion of gases is of the form . At some
temperature, a given increase in temperature will increase the point

reaction rate by a greater amount than the point rate of bulk diffusion.
The second idea is that, based on measurement of the crystalline density
of the complex, the pores of a pellet will be closed with complex before
complete reaction of the sodium fluoride can occur. (A maximum reaction

of thirty-three per cent of the sodium fluoride was calculated for the

present pellets.)
.

21

In order to clearly depict the condition outlined above, consider
a sphere of sodium fluoride at a high temperature where the local rate
of reaction of gaseous uranium hexafluoride is high compared with the
rate of diffusion of gaseous uranium hexafluoride through the pores of
the pellet. Under these conditions, the concentration profile of
gaseous uranium hexafluoride would extend into the sphere only a short
distance, and when the pores at the surface have been closed by the
formation of complex, there will be only a thin layer of complex at the
outer surface of the sphere. On the other hand, at a low temperature,

where the rate of reaction is low compared with the rate of diffusion,

‘the radial concentration profile of gaseous uranium hexafluoride is

practically constant, and when the pores are filled at the surface of
the sphere, the pofes at the center of the pellet will also be filled.
In the latter case, a larger loading will result due to the formation
of complex within the interior of the pellet in addition to that at the
surface.

In order to test the hypotheses discussed above it is necessary
to derive the general equation for diffusion with chemical reaction
allowing for variéble diffusivity and variable reaction rate.

The following assumptions will be made in order to make the

" problem more tractable:

1. The pellet is a sphere having a volume equal to the one-
eighth-inch right circular cylindrical pellets used in the
study.

2. The pellet is homogeneous.
22
Radial symmetry.
The temperature variation in the pellet is negligible.
Radiél transfer of uranium hexafluoride in the pellet occurs
as diffusion of gaseous uranium hexafluoride through an inert
gas in the pores of the pellet.
The decomposition pressure of the complex is negligible.
The local rate of reaction between gaseous uranium hexa-

fluoride and unreacted sodium fluoride is of the form:

2 - B(q, T) C. | (9)

The local effective diffusivity for diffusion in the pores of
the pellet is of the form:

D, = D (q, T)- : (10)

Consider a spherical shell of thickness dr which has the condi-

tions noted at the two surfaces of the shell in Figure 1

where

r

concentration of gaseous uranium hexafluoride in the pores
of the pellet,

reaction rate constant,

effective diffusivity of gaseous uranium hexafluoride in the
pellet,

volume void fraction of pellet,

radial distance in pellet.

The rate of diffusion of uranium hexafluoride into the shell is
23

UNCLASSIFIED
ORNL-DWG 63-2024

SODIUM FLUORIDE €
PELLET

SPHERICAL
SHELL

 

 

Figure 1., Spherical Shell Used in Derivation of Diffusion
Equation for Variable Reaction Rate, Diffusivity, and Volume Void
Fraction,
5 oD, d(c + %g-dr)
Rate = Ly(r + dr) D, + 57 dr) ———— - (11)

and the rate of diffusion out of the shell is

Rate = &wre D, %% . (12)

The rate of reaction of uranium hexafiuoride in the shell is
2 18 1L
Rate = lmr dr {a+25r dr} {C +28rdr} , (13)

and the rate of accumulation of gaseous uranium hexafluoride in the

pores of the shell is

Rate = lnrfédr 'g? [e + %-g—i— dr] [C + é— %S— dr]} . (11L_)

From a material balance on the shell,
Rate of Accumulation = Rate of Diffusion In
| - Rate of Diffusion OQut (15)
- Rate of Reaction.
Expanding the above terms and taking the limit as the shell thickness
épproaches zero [neglecting terms of order (dr)2 or higher] yields the

desired relation, which is:

@?g+(2+1_a]’e

e ar2 r De or

 

& (ec) =D Ebo-BC . (16)

In order to complete the definition of the mathematical model, one must
specify relations for ¢, De’ and B.
A linear relation was assumed between ¢ and q which was

e =¢, (1 - q/qma%) , (17)
25

where

m
il

o void fraction of unreacted pellet,

Imax

maximum loading of uranium hexafluoride based on density of

complex and initial void fraction of pellet.

In order to choose the form of De’ one must first characterize
the type of diffusion occurring in the pores of the pellet which could
be bulk, Knudsen, or a combination of the two. The median pore radius
for the unreacted pellets is 6780.angstroms (Appendix E) and the mean
free path (Appendix D) of a uranium hexafluoride molecule in a mixture
of uranium hexafluoride and nitrogen in the present study. is approxi-
mately 300 angstroms. Based on cylindrical pores, the pore radius will
be three times the mean free path when the loading is ninety-eight
per cent of the maximum loading. On this basis, diffusion through the

pores of the pellet will be assumed to be of the bulk type, with De of

the form
a
D =D.. . 7Ye , (18)

where

D, = diffusivity of uranium hexafluoride in nitrogen,

UF6-N2

€ = void fraction,
Y, n = constants.

The form of the point rate of reaction of uranium hexafluoride
remains to be specified. The uranium hexafluoride-sodium fluoride
complex occupies a volume three and one-half times as great as the

volume of the sodium fluoride consumed in its production. Thus, when
26

freshly formed, a film of complex will be in a state of lateral compres-
sion. Although the film adjacent to the sodium fluoride may adhere to
and be braced by the underlying crystalline material, where the film is
not braced by this contact it will yield to the compressional stresses
by cracking and buckling. This yielding will almost certainly result
in flaw.paths or zones of loose structure suited for easy transfer of
a diffusing gas. One might thus consider the sodium fluoride to be
covered by two types of filmé: a thin, tightly packed layer in contact
with the sodium fluoride, which will be of constant thickness, covered
by a broken layer that increases in thickness as sorption proceeds.
Transfer of uranium hexafluoride across the inner layer will be by
diffusion through a crystalline structure. This transfer is slow,
compared with diffusion through gases, and, as pointed out by Jost,32
is normally an activated type of diffusion. The rate of transfer of
uranium hexafluoride through a break in the outer layer may thus be
independent of the thickness of the outer film and dependent only on
the rate of transfer across the inner layer. If all of the cracks in
the outer layer traversed the entire thickness of the layer, the rate
of sorption would be independent of the film thickness, once the thick-
ness exceeded the thickness of the inner layer. Practically, however,
some paths will be obstructed and no transfer can occur along these.
The rate of transport will be determined by the number of paths that
remain unobstructed at any given thickness.

If one defines as pdx the probability that an obstruction will

occur in the portion of a path contained in a film thickness dx, then

X 4
27

the probability that a given path be unobstructed will be e *~. The
number of unobstructed paths per unit area of film will thus be in
proportion to e P*, As shown in Figure 2, if C ié the uranium hexa-
fluoride concentration in the gas adjacent tb the outer surface, the
average concentration at the plane dividing the two films will be
ke-pxC, where k is a constant. The concentration of uranium hexafluo-
ride will be zero where the inner film contacts the sodium fluoride.
The rate of transfer of uranium hexafluoride per unit volume of pellet

is then

dq _ dCc
at = PSPyar ax ° (19)

The form of D for activated diffusion is

D =D e-E/RT
0

If the thickness of the inner layer is small compared with that of the

outer layer,

Accordingly, Equation (19) becomes:

 

dg _ ;¢ cE/RT ke e (20)

where

g = quantity of uranium hexafluoride which has reacted per unit

volume of pellet,

ct
i

time,
 

   
 
 

C,P .
S T 2
R .
o
o8 ¥ &
ll_ ADl_l W
[
mu 23 S
A . Q
S o g
Z 4 e O £
- Z m 3
o
© s
A _ | °
A
>
™
-
o
@
_ kv,
o wc
YX . .
EDI

\\\\\\\\\\\\\\\

/fl\\\\\\/ fl

 
29

wn
H

initial surface area of pellet,
PyaF = density of pellet,
p_ = density of complex,
M.C = molecular weight of complex,

£ = thickness of inner layer of complex,

R = gas constant,

T = temperature in degrees Kelvin, -

C = uranium hexafluoride concentration in the gas phase at the

point being considered.
Several constants in Equation (20) may be combined by defining that
DoSpNaF b
£

, (a1)

PP

b = -

MS
c

and Equation (20) can be written as

%%-: a e-E/RT e ¢ = BC . (22)

A second derivation of Equation (22) is possible based on
slightly different ideas. 1In this derivation, the unreacted sodium
fluoride is considered to be covered by only the broken layer of complex
of the previous-case. As before, the average concentration of uranium
hexafluoride at the sodium fluoride surface is k e P°C. The rate of
reaction per unit area will be assumed to be of the form

-E/RT

Rate = k_ e c,
o .

where ko is a constant. The rate of reaction per unit volume of pellet
30

is then
cl(l:k e-E/RTS ke P*¢c .
dt o
With the definition that
'_ .
a’ =k, Spy,p ks
and the previous definition for b, one obtains the rate equation

¢ o~E/RT .24 ¢

=I5

= a 3

which is of the same form as Equation (22).‘ In the first derivation,

a resistance to reaction is afforded by the adherenthlayer of complex,
whereas, in the second case a resistance to reaction is afforded by the
rate of chemical reaction. While only two interpretations of Equation -
(22) have been given, it is probable that others exist.

Since Equation (16) involves variable coefficients of a rather
complex form, the possibility of achieving an analytical solution is
limited. In such a case, the normal procedure is that of performing
the desired integration by a simple, explicit finite-difference tech-
nique. However, as noted by DuFort and Frankel,21 if the range in time
for which a solution is required is of the order of the time required
for effective diffusion through the distance NAx, the stability crite-
rion requires that the number of time intervals be of the order N
squared. Cases requiring a moderately fine spatial division lead to
impractically large amounts of computer time, even for large computers
comparable to the IBM-7090 computer. More complicated explicit differ-
encing techniques using two or more time rows or various implicit

'techniques have been developed for relaxing the stringent stability
351
. 23 . .
requirements. Since it was not clear that use of one of these methods
would result in a practical time increment (approximately ten seconds
or greater ), an alternative method of solution was sought. Desirable
characteristics for such a system include complete stability and freedom
to determine the degree of convergence in order to minimize the amount

of computer time required for solution of the differential equation.
32
CHAPTER IV

| METHOD OF SOLUTION OF SIMULTANEOUS DIFFUSION
AND REACTION EQUATION

As Crank pointed out, in a medium in which a substance is under-
going simultaneous free diffusion and instantaneous irreversible reac-
tion, the concentration profile of unreacted material closely resembles
a steady-state profile when S/C is greater than ten, where S is the
capacity of the medium for the reacting material, and C is the concen-
tration of the unreacted material in the fluid adjacent to the medium.
In the present study, values of S/C range from 1000 to 14,000; however,
the condition of instantaneous rate of reaction is not completely
satisfied.

Another statement of the same idea is that tfie capacity of the
pores for the accumulation of uranium hexafluoride in gaseous form is
negligible when compared with their capacity for uranium hexafluoride
in the form of the.complex. In the present case, the maximum capacity
of the initial void space is 0.03 per cent of that of the solid material
in the pellet. Since the term on the left of the left of equation (16)
represents the accumulation of gaseous uranium hexafluoride in the
pores of the pellet, one is justified in neglecting this term so that
equation (16) becomes the ordinary differential equation:

2 dD
.“1.._.(.:_.*_ g.;.l__e. ig_é_c=0, (25)
2 r - ‘

dr e

which defines a steady-state profile dependent ofily on values of B and

De' As was the case for equation (16), this equation cannot be solved
53

easily by analytical methods. A method will be developed for effecting
the solution to Equation (23).

Consider a sphere that has been divided into N spherical shells
in each of which 8 and De have constant values. Since B and De can
vary from shell to shell, one is approximating the true radial variation
of these functions by a series of step changes. Within shell m,

Equation (23) reduces to:

 

B
m_ 2 _m m
tra&E "D ¢ =0, (2LL)'

which can be solved analytically to yield the solution

A B
C_=-2cosh kr + == sinh k r , (25)
m r m r m
where
km - vlam.:Dm ?
A , B = constants.
m’ m

-Denote by Rm'the outer radius of the mth spherical shell. 1If one
specifies the 3N wvalues of Dm, km and R.m for the original sphere (which
consists of N shells) one has specified the rate of reaction in the
sphere.. It is desired to calculate this rate of reaction.
If one denotes the center shell, which is actually a sphere, as
shell one and specifies the boundary conditions,'
C1 is finite at r = O,

C1 = C51 at r = Rl’

one obtains the concentration profile of unreacted uranium hexafluoride

in the pores of the shell as:
R. sinh k, r

¢y = Co1 £ IR (26)
171
where
C1 = uranium hexafluoride concentration in the pores of shell
one,
CSl = uranium hexafluoride concentration in the Pores at the

surface of shell one,

r = radius variable in shell one,

Ri = outer radius of shell one,
ky =/By/D;

w
"

reaction rate constant in shell one,

effective diffusivity of uranium hexafluoride in the pores

o
u

of shell one.
The remaining shells have common boundary conditions which are,
for shell m,

C =C_atr =R,

m sm m
Cm = Csm-l at r = Rm-l’

which yields the concentration profile relation:

 

 

 

cosh kmr sinh kmr
R .C tanh k R 0ok K R .  cosh K R
c = m-1 “sm-1 m m-1 m m-1
m r tanh k R - tanh k R '
m m m m-1
(27)
sinh k r cosh kmr
R C cosh kK R tanh k'mRm-l-sinh k R
L o sm m m m m
r tanh k R - tanh k R ’
m m m m-1

By noting that
55

cosh Kk R cosh k R
m m m

1 _ ! m-1 : (28)
tanh k R - tanh k R sinh [k (R_-R_ .)|"’
m m m m-l m ' m m-1

 

one can write the expression for the concentration of unreacted uranium
. . th |
hexafluoride in the pores of the m  shell as:

R ;€. _; sinb [km (R - r)]

 

 

 

f C — - i
m r sinh [km (Rm - Rm-lq.
(29)
R C__ sinh [k (r - R fl
4 ;M _sm m m-1
T sinh [k (R - R fl ’
m ' m m-1
Continuing the development for shell two, from above,
. L R, G, sinh [k2 (R2 -Hrfl
2 r sinh [k2 (R2 - Rlfl
(30)

. R, C_, sinh [kg (r - Rl)]
r sinh [k2 (R2 - qu

An equal flux of uranium hexafluoride across the surface r = Rl

 

between shells one and two requires that

D, — = D, = . (31)

r=R1 r=Rl

This condition results in a relation between the goncentrations at the

surfaces of shells one and two, which is:

 

 

 

 

 

Eg. - k2R2
oy . sl Dl sinh _k2 (R.2 - Rl), (32)
2 ¢, kR D k R - 5
s2 11,210, 2% |
tanh kR, D, tanh [k2 (R2 - Rl)

 

It should be noted that Oé is dependent only.on the known quantities

k;, ky, D, Dy, Ry, and R,.
36

 

 

 

 

Substituting the quantity O% CS2 for CSl in Equation (30) and
requiring equal fluxes at the surface r = R2 yields O% as
D k
> 5"
. D, sinh [k3 (R5 - RE)] (53)
% = A s |
where
. k2R2 - _ _Rloékg
tanh[k2 (R2 - le] sinh [k2 (R2 - R, )]
-1 + Eé 1 + oh : REI% 3 .
D, ta [kB (R3 - R2)

 

The quantity 0% may be noted to depend only on known quantities.
In 2 similar manner, one can show that for the mth shell,
D R k
m m

D . sinh [k (R -R _)I
g - D 1 m ' m m-1 ‘] , (54)

 

 

 

 

where

 

and that

(35)

 
57

The rate of diffusion of uranium hexafluoride into shell m is

 

 

 

 

 

 

 

 

given by:
o dC
Rate of Diffusion = 4R © D —f = =
m m m dr
r=R
m
(36)
Rmkm Rm 1Ofikm
= 4R D C -1+ — - o — -
m m sm tanh [km (R.m - Rm-lj] sinh [km (Rm - %n-lq
The rate of reaction in the mth shell is also obtained by:
R
m 2
Rate of Reaction = 4B u/\ C r dr =
m m m
Rm-l
(37)
2 2 km
- LHTDmCsm [Rm % Rm—-lil [tanh[ k (R -R )] ]
m ' m m-1
] R Roo1 (1+a)k koo R
sinh [k (R -R )] m m m-1l
m ‘' m m-1
If one denotes by 6m and gm the follofiing relations;
km
%y = tanh [k R_-R )] ) (38)
‘ m ' m m-1
km
€n © 5inh [k (R -R )] ? (39)
m * m m-1
the results of the above derivation can be summarized as:
Dm
Dm_1 ngm ‘ .
am = D . ()-I-O)
R .5 -R Ot -1+ —"-(1+R ,5)
m-1"m~1 m-2 m-1°m-1 D m-1"m

m~1
38

Rate of Reaction = 4yD C [kRe r QR ) 8
m m sm m m m-1 m
(1)
-RR . (L+@)E -R + O%Rm_l} .
Rate of Diffusion into shell N =
(k2)

- AFRNDNCSN (-1 f‘RNBN - RN—loth) = ¢CSN ?
where ¢ = &WRNDN (- 1+ Redy - RN_long).

The effect of a stagnant gas film at the surface of the sphere
éan be taken into account by equating the rate of reaction in the sphere
to the rate of transfer across the film, which yields:

0C e =k, a (Cp - C o), | (43)
where

k = mass transfer coefficieqt across film,

a = surface area of sphere having volume equal to actual pellet,

CB = uranium hexafluoride concentration in the bulk stream.

Solving for CsN yields:

Cp

CsN 1+ ¢/k a ° (44 )
g
The total rate of sorption by the sphere is then:

k a ¢

-8
total rate = kg <+ °p° (45)

The relations given by Equations (40), (41), (44), and (45) may
be used in a finite-difference manner for calculating the fate of
reaction, the total quantity which has reacted at a given time, and
other similar quantities. The time increment for any given time inter-

val can be chosen such that the maximum change in the value of B or De

 
59

is some prescribed fréction of its present valfie. In this manner, the
degree of convergence from # time increment basis can be chosen at will.
Similarily, since the method is cofipletely stable, one can choose the
degree of convergence on a distance increment basis. In this manner,
one can calculate approximate values of parameters with the expenditure
of a small amount of computer time and improve the degree of convergence
when the vélues for the parameters are approximately equal to the
required values.

.The convergence of the numerical method was examined (Appendix

F) for the case of B and D, of the form:

 

 

2
R
B = p* ; 2
2
e r

for which an analytical solution was obtained. It was concluded that

the method was convergent.
Lo
CHAPTER V

. MATERTALS AND EQUIPMENT

Materials

The sodium fluoride used in these tests was in the form of com-
pacted right circular cylinders and was part of a shipment of commercial-
grade sodium fluoride from the Harshaw Chemical_Company. The material
was prepared by a two-step process consisting of the compaction of
powdered sodium bifluoride followed by heating to approximately 300°C
in air for removing the hydrogen fluoride. A material having a surface
area of about one square meter per gram and a void fraction of 0.35 to
0.50 is,produced. Typically, this material has the composition shown
in Table XII (Appendix E) and consists of a mixture of whole and broken
pellets (Figure 3).

Repeated sorption of uranium hexafluoride followed by désorption
at 4L00°C in fluorine results in an initial reduction in capacity for
uranium hexafluoride of about thirty-five per cent. During continued
cyclical operation, the capacity increases slowly and approaches that
of pellets treated at 400°C in fluorine for one hour, which is about
five per cent less than the capacity of "as-received" pellets. The
initial variation in pellet capacity due to cyclical operation is within
that observed between different lots of commercial pellets.

The pellets used in this study were fluorinated at LOO°C for one
hour prior to use. The material fiad a surface area (determined by

nitrogen adsorption) of 0.86 square meter per gram, and a void fraction
41

UNCL ASSIFIED
PHOTO &1968

 

Figure 3. Typical One-eighth-inch Right Circular Cylindrical
Sodium Fluoride Pellets.

 
Lo

of 0.45. The mean pére size was 6780 angstroms. A porogram for this
material, together with other pertinent information is presented in
Appendix E.

The nitrogen used in the study had been 0il pumped and was dried
in a Drierite column prior to use. )

The uranium hexafluoride used in this study was obtainea from
the Oak Ridge Gaseous Diffuéion Plant and contained iess than 200 part;
per million of volatile impurities the most of which was hydrogen
fluoride.

The fluorine used for conditioning the sodium fluoride and
various parts of the experimentél apparatus was obtained in gaseous
form from the Oak Ridge Gaseous Diffusion Plant. Prior to use, the gas
was passed through a bed of sodium fluoride at room temperature for

removal of hydrogen fluoride.

EguiEment

A flow diagram for the equipment used in this study is shown in
Figure 4. Basically, the equipment provided means for preparing gas
mixtures of the desired composition at a controlled flow rate, means
for controlling the temperature of the gas mixture and fhesorption
vessel, and means for a second determination of the flow rate of the
two gases used.

The flow rate of uranium hexafluoride was set by maintaining a
controlled pressure drop across a calibrated capillary. The nitrogen

t

was metered through a rotameter from a constant-pressure nitrogen
 

ROTAMETER
CONSTANT
NPRESSUS;JPiE, CALIBRATED
2 CAPILLARIES

 

 

PRESSURE
CONTROL VALVEi

 

 

 

 

 

UF SUPPLY

OFF-GAS .__@

WET TEST

 

PRESSURE
GAUGE

<

    
 

PREHEATER

COIL

UNCLASSIFIED
ORNL-DWG 63-2026

SORPTION
VESSEL

[y

 

 

 

 

 

 

 

 

 

 

UFg
TRAP

 

UF4
TRAP

 

 

 

 

METER

Figure 4, Flow Diagram for Equipment Used in the Study of Sorption of Uranium Hexafluoride

by Sodium Fluoride,

en
Ll

supply. The two gases were introduced into a common line that led to
the gas preheater which consisted of a coil of three-eighthsfinch tubing
fifty feet in length. The p;eheated gas then flowed through the sorp-
tion vessel and through a sodium fluoride trap for removal of uranium
hexafluoride which passed through the sorption vessel. Both the gas
preheater and the sorption vessel were immersed in an agitated oil bath
that was controlled to within 0.1°C of the desired operating tempera-
ture. After removal of the uranium hexafluoride, the nitrogen was
metered by a wet test meter.

A bypass around the sorption vessel and the sodium fluoride trap
was provided so that the gases did not flow through the sorption vessel

during startup and shutdown of the metering system.
5

CHAPTER VI
EXPERIMENTAL PROCEDURE

The experimental phase of the study consisted mainly of the
determination of the loading of uranium hexafluoride on single layers
of sodium fluoride pellets during a prescribed time interval under a
given set of conditions. Some work was necessary for preparation of
samples of the uranium hexafluoride-sodium fluoride complex for deter-

mination of crystalline density.

Differential -Bed Studies

 

The differential-bed runs were made by using a single layer of
sodium fluoride pellets placed between two four-and-one-fourth-inéh
sections of three-millimeter glass beads used as entrance and exit
sections. As shown in Figure 5, the bed was constructed from one-and-
one-half-inch diameter schedule~forty nickel pipe. The glass beads
were conditioned before use by exposure to fluorine at L4L00°C for one
hour.

To make a run, the sorption vessel was loaded and placed in the
oil bath. At least one hour was allowed for the bed temperature to
reach that of the oil bath. As shown in Appendix G, after 0.45 hour,
the difference between the center-line temperature of the bed and the
bath temperature will be less than two per cent of the original tempera-
ture difference. Prior to starting a run, the uranium hexafluoride and

nitrogen flow rates were set, and the stream was allowed to flow through
46

UNCLASSIFIED
ORNL-LR-DWG 66094

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1,61
A
—
Inlet — 7
7
3/8~inch [/ 1 1/2-inch Schedule 40
7 f Nickel Pipe
/] /]
] /]
/] /]
2 )
: 7
] 2
]
/ /
% 7
/2 — 3 mm glass beads
’
41/4"
L/
9 3/ 0 2
7 |
B 000000000000 e Single layer of 1/8-inch
; 2 ? Sodium Fluoride
4 /
] /]
4 "
L]
] /]
] /]
7 /
g - a 3 mm glass beads
4 1/4" /) %
" /]
] 4
¢ ]
¢ 4
g A 3/8uinch
] % =inc
/] )
g_____étzsz"_____fi/ J/ -
e t
7 \\\\‘\\\\\\\\\\\ia !

Figure 5. Sorption Vessel Used in Differential-Bed Studies.
b

the preheater coil for at least five minutes in order to establish a
constant concentration in the coil. During this period, the stream was
byfassed around the sorption vessel and the sodium fluoride bed. A run
was started by diverting the stream through the differential-bed. Gas
was passed through the bed for a predetermined length of time, after
which the stream was diverted into the bypass. About ten seconds were
required for effecting the valving changes necessary for diverting the
gaé stream. The uranium hexafluoride flow was stopped and the nitrogen
flow was continued through the preheater coil for two minutes to free
the coil'of uranium hexafluoride. The nitrogen flow was then diverted
through the sorption vessel for one minute to free the bed of unreacted
uranium hexafluoride. The bed was then sealed, removed from the oil
bath, and allowed to cool in a dry box containing dry air. The sodium
fluoride was removed from the reaction vessel in the dry box and placed
in a sealed container for weighing on an analytical balance having a
capacity of 200 grams. The sodium fluoride trap was treated similarly

and weighed on an analytical balance having a two-kilogram capacity.

Determination of Density of Complex

For determination of the crystalline density of the uranium
hexafluoride-sodium fluoride complex, pellets of sodium fluoride that
had been treated with fluorine for one hour at 400°C were ground with a
mortar and pestle in a dry box. A half-inch layer of the powder in a
nickel dish was exposed to a stream of dilute gaseous uranium hexafluo-

ride (five to ten mole per cent uranium hexafluoride in nitrogen) at
48
100°C, after which the resultant material was again-ground with a mortar
and pestle in a dry box. The material was repeatedly exposed to uranium
hexafluoride an& ground until the desired weight gain had occurred.
Samples of the material and of the original sodium fluoride were then

submitted for crystalline density determination by toluene immersion.
\
k9

CHAPTER VII
EXPERIMENTAL RESULTS

Fifty usable differential -bed runs were made in order to provide
data on the rate of sorption of uranium hexafluoride by one-eighth-inch
pellefs of sodium fluoride. The concentration profiles of sorbed
uranium hexafluoride on pellets from a number of runs were examined
both photographicall& and by a reflected X-ray technique. Two samples
of the uranium hexafluoride~sodium fluoride complex were prepared for
determination of the crystalline density, which is required for use of

the mathematical model.

Differential -Bed Studies

 

Differential-bed runs were made with a single layer of sodium
fluoride pellets in the following range of operating conditions:

Uranium hexafiuoride concentration O0.57 to 10.9 mole per cent

Temperature 29 to 100°C
All runs were made with a nitrogen flow rate of 0.129 gram mole per
minute at atmospheric pressure. Appro;imately 4.5 grams (about 130
pellets) of one-eighth-inch right circular cylindrical sodium fluoride
pellets were used in each run. The results, plus operating conditions,
are presented in Tables I through VI. Data from some runs was not
usable due to experimental difficulties such as poor control of the

flow of the nitrogen or uranium hexafluoride, or other malfunctions of

the apparatus.
50
TABLE I

EXPERIMENTAL RESULTS FROM DIFFERENTIAL=-BED RUNS AT
29°C WITH 2.62 MOLE PER CENT URANIUM HEXAFLUORIDE

 

 

 

Time, Weight Gain,
Run Number min gms UF6/gm NaF
60 3 0.1204
59 5 0.1781
58 10 0.2388
61 12 0.3260
57 20 0.4590
54 45 0.6200

25 60 0.6980

 

 
51
TABLE II

EXPERIMENTAL RESULTS FROM DIFFERENTIAL-BED RUNS AT
50°C WITH 2.35 MOLE PER CENT URANIUM HEXAFLUORIDE

 

 

 

Time, Weight Gain,
Run Number min gms UF6/gm NaF
39 3 - 0.1459
W7 3 0.1604
37 5 0.2367
46 5 0.2290
35 10 0.3821
45 10 0.3820
34 15 0.3900
38 15 0.4161
41 15 0.4020
36 20 0.4580
43 20 0.4331
Lo 25 0.4740
by 30 0.5250
48 45 0.6110
50 60 0.6160

51 60 0.60%6

 

 
52
TABLE III

EXPERIMENTAL RESULTS FROM DIFFERENTIAL-BED RUNS AT
100°C WITH O.57 MOLE PER CENT URANIUM HEXAFLUORIDE

 

 

 

Time; Weight Gain,

Run Number min gms UF6/gm NaF
20 3 0.0937
19 5 0.1161
14 - 10 0.1622
13 15 0.2333%
23 15 0.2188
21 20 0.2403
22 25 0.2611
15 30 0.2849
69 45 0.3140

 

 
25

TABLE IV

EXPERIMENTAL RESULTS FROM DIFFERENTIAL-BED RUNS AT
100°C WITH 2.45 MOLE PER CENT URANIUM HEXAFLUORIDE

 

 

 

Time, Weight Gain,
Run Number min gms UF6/ gm NaF
28 3 0.1825
26 5 . 0.2575
27 8 0.3009
25 10 0.3189
2L 15 0.3640
29 20 0.k237
3] 25 0.4310
33 30 0.4480

52 60 o.5ouo

 

 
5L
TABLE V

EXPERIMENTAL RESULTS FROM DIFFERENTIAL-BED RUNS AT
100°C WITH 8.51 MOLE PER CENT URANIUM HEXAFLUORIDE

 

 

 

Time, Weight Gain,
Run Number min gms UF6/ gm NaF
67 5 0.423
66 10 0.510
65 15 0.584
63 25 0.601
62 30 . 0.672
6k L5 » 0.645

68 60 0.641

 

 
55

TABIE VI

EXPERIMENTAL RESULTS FROM DIFFERENTIAL-BED RUNS SHOWING VARTATION
OF EFFECTIVE CAPACITY FOR URANIUM HEXAFLUORIDE WITH TEMPERATURE

 

 

 

Run Temperature, Time, UFg Conc., Weight Gain,
Number °C min Mole % gms UF6/gm NaF
78 30 300 10.9 1.130
80 50 300 10.9 0.930
64 100 45 8.51 0.645

68 100 60 8.51 0.641

 

 
56

Examination of Partially Reacted Pellets

 

Partially reacted pellets from a number of runs were sectioned
for determination of the distribution of sorbed uranium hexafluoride.
The contrast between the pale yellow‘color of the complex and the white
color of sodium fluoride allowed qualitative determination of the dis-
tribution of the uranium hexafluoride in a pellet by microscopic and
photographic means. Partially reacted pellets, which have been sec-
tioned axially, are shown in Figures 6 and 7. By use of the proper
filter, the pale yellow of the complex was made to appear grey. The
pellets in Figure 6 are from a run at 50°C and éontain the maximum
quantity of uranium hexafluoride which will be sorbed at that tempera-
ture. A considerable variation in penetration of uranium hexafluoride
is observed not only between individual pellets but also between dif-
ferent areas of the same pellet. Of particular interest is the slight
penétrafiion near the corners Qf some pellets since this type of profile
is not éfiserved in the case of diffusion of a substance into a finite
cylinder of constant properties. It is felt that the differences in
penetfation are due to variafiion in the density of different areas of
the pellet. The method of manufacture of the pellets, compression of
a powder, would be expected to produce variations in density, with high-
‘dengity areas in the finreacted corners of the sectioned pellets.
Sectioned pellets containing the maximum quantity of uranium hexafluo-
ride which will be sorbed at 100°C are shown in Figure 7. The profiles
are similar to those at 50°C, except that the penetration is not so

great. Most of the pellets that have sorbed the maximum amount of
59

uranium hexafluoride at 30°C show no unreacted areas.

The concentration profiles of sorbed uranium hexafluoride on a
number of pellets were measured by use of a geflected X-ray technique.
In this method, the target area is.bombarded with monoenergetic X rays
having an energy sufficiently high to excite the element of interest;
excitedratoms then emit X rays having an energy characteristic of the
excited element.; The count rate of emitted X rays is assumed propotr-
tional to concentration of the element of interest. Use of this method
allowed the measurement of the uranium concentration inAa circular area
100 microns in diameter. The method is somewhat time-consuming, and,
in view of the irregularities in the profile within a pellet as well as
betWeen‘pellets, only enough profiles were measured to establish_tfie:
type of profile, that is, whether the profile is sharp or e#tends oVef.
a region as predicted by the mathematical model. A typical experimental
profile resulting from sorption at 100°C is shown in Figure 8 and ié
observed to be diffuse, in agreement with the mathematical model. |
Profiles in some pellets from the same run exténded farther into the -
pellet, with about five per cent of the pellets being reacted to the
center. The profile extended farther into the pellet at lower tempera-
tures, and a larger percentage of the pellets were reacted to ther

center.

Density of'Complex

 

Two samples of the uranium hexafluoride-sodium fluoride complex

were prepared by repeated grinding and.exposure of powdered sodium
60

UNCLASSIFIED
ORNL-DWG 63-2716

\ |/

1.0

 

 

 

 

 

 

 

0.5

 

 

 

 

 

 

 

 

RELATIVE SOLID PHASE CONCENTRATION

 

 

 

 

 

 

 

 

 

 

 

 

 

\_t /
-1.0 -0.4 0 +0.4 +1.0
RELATIVE RADIAL POSITION

 

Figure 8. Typical Solid-Phase Uranium Hexafluoride Concentration
Profile for Pellets Reacted at 100°C,
61
fluoride to gaseous uranium hexafluoride. Conditions for preparing
the samples are given in Table VII. Measurements of the sample density
were made by a toluene pycnometric method at 26°C.

Duplicate measurements on sample one, which contained 2.58 grams
of uranium hexéfluoride per gram of sodium éluoride, yielded sample
densities of 3.935 grams per cubic centimeter and 3.885 grams per cubic
centimeter. Sample two, which contained 3.54 grams of uranium hexafluo-
ride per gram qf sodium fluoride, had a density of 4.128 grams per cubic
centimeter.

For calculation of the crystalline density of the uranium hexa-
fluoride-sodium fluoride complex, the formula UF6'2NaF waé assumed. It
was also assumed that in a mixture of complex and sodium fluoride each
material exhibited the density of the pure material. On this basis,
63.5 per cent of the sodium fluoride in sample one was complexed, and
the resultant complex densities are L.13 grams per cubic centimeter and
4.07 grams per cubic centimeter, based on the two reported sample
densities. In sample two, 8L.5 per cent of the sodium fluoride was
complexed, and the resultant complex density was 4.20 grams per cubic
centimeter. The average of these figures, 1.13 grams per cubic centi-
meter, was used for determination of qmax" the maximum quantity of
uranium hexafluoride which can react per unit volume at a point in the
pellet. ‘The density of a sample of the initial sodium fluoride was
determined to be 2.78 grams per cubic centimeter, which compares quite
59

favorably with the reported value of 2.79 grams per cubic centimeter.

The samples were analyzed before and after exposure to toluene to detect
62
TABLE VII

WEIGHT -GAIN AND EXPOSURE DATA FOR PREPARATION OF URANIUM
HEXAFLUORIDE-SODIUM FLUORIDE COMPLEX AT 100°C

Weight Gains are Per Unit Weight of Sodium Fluoride

 

 

1st Exposure  2nd Exposure 5rd Exposure . Total
Weight Weight

 

Sample Time, Weight Time, Weight Time,
Number hr Gain hr Gain hr Gain Gain
1 18.0 2,138 18.0 0.385 6.0 0.054 2.5TT
1.155 5.2 0.382 3, 5Lk

2 6.2 2.007 5.3

 

 
63
interaction with possible leaching or extraction of the uranium hexa-
fluoride. As shown in Table VIII, no change was noted in the uranium
content of sample two, and a change of less thafi four per cent was noted
for sample one. The uranium content of the toluene'after exposure to
the complex was below the limit of detection (less than 0.003 micrograms
per milliliter ). It was concluded that interactionm, if present, céuld

be safely neglected.
6L
TABLE VIII

URANIUM CONTENT OF SAMPLES OF THE COMPLEX
BEFORE AND AFTER EXPOSURE TO TOLUENE

 

 

Weight Per Cent Uranium

 

 

Sample Before Toluene After Toluene
Number Calculated Exposure Exposure
1 48.79 48.75 | 46.92

2 52.70 51.14 51.13

 

 
65
CHAPTER VIII

ANALYSIS AND DISCUSSION OF RESULTS

The final evaluation of the mathematical model rests on a com-
parison of experimental and model-predicted data. This comparison and
a discussion of the application of the results of the study to fixed-bed

sorber design are given in the following sections. A discussion of

experimental error is also presented.

Differential-Bed Data

 

Several relations [Equations (40), (41), (L&), and (45)] were
defived for use in the solution of the diffefential equation describing
sorption by a single pellet with variable reaction rate constant and
variable diffusivity. In addition to these equations, one has the three
relatidns for the variation of volume void'fraction; effective diffu-
sifiity, and reaction rate constaht,‘all of which contain constants whose
values must be determined from experifien£a1 data or from a parameter
search. The constants to be detéfmined in the relation for.volume void
fraction,

e=¢, (L -4qa/q_ ) , (17)
are ¢, the initial void fraction (0.L45 for_the pellets'in this séudy),
and 9 ase? the maximum quantity of uranium hexafluoride which can accu-
mulate per unit volume of'pellet. This quantity was calculated from
the initial volfime Qoid fraction and the cryétaliine densities of sodium

fluoride and the complex UF6'2NaF and was found to be 0.00595 gram mole
66
of uranium hexafluoride per cubic centimeter of pellet or 1.37 grams of
uranium hexafluoride per gram of sodium fluoride.

The relation for the effective diffusivity in the pellet,

n .
P = Pup w7 € (18)

contains two constants, 7 and n, which must be determined in the param-

eter search. Values of DUF6-N2’

in nitrogen, were calculated as shown in Appendix C.

the diffusivity of uranium hexafluoride

The relation for the reaction rate constant B,

B = a e-E/RT oD

2
contains three constants (a, E, and b) which must be determined in the
parameter search.

Without additional information, the parameter search would
involve five constants whose values must be determined. Omne can,
however, derive relationships between the constants which will markedly
decrease the amount of computer work necessary for determination of the
values of the constants. The first relation is based on Danckwerts'!
analytical solutioan for the steady~state rate of sorption with irre-
versible reaction for constant reaction rate constant and constant
diffusivity, which is:

dq _ EgaCB#W [- 1 + kR coth kR] D,
dt kga + 4R [- 1 + kR coth kR] D,

 

Knowing the initial rate of sorption, this expression sets a relation-

ship between the initial values of B and De'
67
The second relation between constants is that afforded by appli-
cation of the point rate equation to the surface of the pellet. The

point rate of reaction at the surface is

dq _ , e-E/RT e-bq c,

dt N’

which integrates to
t
1+abe-E/RTfC e | .
sN

O

O
il
o~
—
B

From experimental data, one knows the time at which sorption ceases, and
hence the value of q which will be 9ax” One does not know the time

variation of CS however, a close approximation is the assumption that

N;

Then, 1if t is the time at which q has the value ¢ , one has the
max : max
relation

In [}1 + abe_E/RT C_t ] 3
. B max

which sets a relation between the value of b and the initial value of B.

o=

qmax -
With the two relations that have been developed, at a given
temperature one has only two parameters, ¥ and_n, whose values are
independent. Hence; at a fixed temperature, the parameter search
ififiolves only two parameters. Based on ififormation from the literature
on the Qbserved dependence of De on ¢, it was decided to try 6n1y two
values for n; these were.1.5 and 2.0. It was apparent very early that

n should have the value of 2.0. Hence the major part of the parameter
68
search involved only two independent parameters, y and E.

In order to accomplish the parameter search, the relations
discussed above were coded in FORTRAN for a finite-difference solution
using the IBM-7090 computer at the Central Data Proéessing Facility of
the Oak Ridge Gaseous Diffusion Plant, Oak Ridge, Tennessee. The
FORTRAN statements for the computer code are given in Appendix H. For
most of the computer calculations, the spherical pellet upon which the
model is based was divided into forty shells of equal volume; the length
of each time increment was chosen such that the maximum'change’in the
point reaction rate constant or the point diffusivity during the time
increment would be five per cent of its current value or less. In this
method of solution, the time increments were short initially (about 0.5
second) and increased continuously as the solution progressed so that
after about one hour of computed time had been accumulated, time incre-
ments of approximately 100 seconds were observed. For calculations
covering the first five hours of sorption, approximately 0.006 hour of
computer time was required.

As shown in Appendix F, the calculational method was observed to
converge to within one per cent of the analytical solution for B and De
of a functional form similar to the experimental case when forty equal-
volume shells were used. As a further check on cbnvergence, a number
of the calculations were repeated using eighty shells and a maximum
changé in the current value of B or De of 2.5 per cent; a difference of
less than one per cent in calculated values for uranium hexafluoride

loading was observed for the two cases.
69

The relations that resulted from the parameter search are:

2
D =0.369 D, . €
and
‘ _TOOO
B =6.25 x 10%¢ RI e71590q

A comparison of the experimental data_on variation of pellet loading
with time with model-predicted values for the loading using these rate
relations is shown in Figures 9 through 13. Experimental and calcu-
lated values for pellet capacity as a function of temperature are shown
in Figure 1l4. The root-mean-square error for all points-in the study
was 9.5 per cent; the largest error for a single point was 31 per cent.

The values of the parafieters should be examined to determine
whether they are consistent with the proposed model. The labyrinth
factor for the pellets, ye, has an initial vaiue of 0.166, which is
typical for porous materials. It should be noted that this factor is
independent of temperature and the gases taking part in the diffusion
process.

The value of E, TOOO calories per mole, may be slightly low for
the activation energy for diffusion in crystalline materials, however
few data are available on systems comparable to the present one. Using
Equation (21), one can assumé a typical value for the diffusivity of
uranium hexafluoride in the crystalline layer of complex and calculate
its thickness. A diffusivity of 0.6 x 10-7 square centimeters per

second corresponds to a thickness of 16 angstroms, or about three
UNCLASSIFIED
ORNL-DWG 63-2609

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5

0.4
: /
on
\‘00-3 //
Y o
= o””"
0 .
< Q/
2 @
3 02 /
- -
w / EXPERIMENTAL CONDITIONS
z ° T 100°C
a / Cg 0.57 mole % UFR

_3 2
o’ G 4.99x1073 g/sec-cm
/
oX ~
o |
0 400 800 1200 1600 2000 2400 2800 3200 - 3600 4000
TIME (sec)

Figure 9., Comparison of Experimental and Model-Predicted Data Showing Variation of Pellet
Loading with Time at 100°C and 0.57 Mole Per Cent Uranium Hexafluoride.

oL
T1

UNCL ASSIFIED
ORNL-DWG 63-2610

 

 

 

 

 

 

 

PELLET LOADING (g UFg /g NaF}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Q.7
0.6
//
0.5 //
/ ®
°
®
0.4 ////!/’
®
03 &
/ EXPERIMENTAL CONDITIONS
® T 100°C
Cg 2.45 mole % UFg
3 6 642 x10™> g/sec-cm?
0.2 7
@
04
0
0 400 800 1200 1600 . 2000 2400 2800 3200 3600 4000

TIME (sec}

. Figure 10. Comparison of Experimental and Model-Predicted Data
Showing Variation of Pellet Loading with Time at 100°C and 2.45 Mole
Per Cent Uranium Hexafluoride,
12

UNCLASSIFIED
ORNL-DWG 63-26{1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.7
®
/———-——— o ¥
06 //-
/://///

0.5 /
w
o
= ./
o
< 04
5 EXPERIMENTAL CONDITIONS
= 7 100°C
g ¢, 8.51 mole % UF,
O ~2 2
< ¢ 1.01 x10°° g/sec-cm
O
303
'_
wl
-
-
Wl
o

0.2

X

0
0 400 800 1200 1600 2000 2400 2800 3200 3600 4000

TIME {sec)

Figure 11, Comparison of Experimental and Model-Predicted Data
Showing Variation of Pellet Loading with Time at 100°C and 8.51 Mole
Per Cent Uranium Hexafluoride,
5

UNCLASSIFIED
ORNL-DWG 63-2608

 

0.7

1

 

0.6 //
0.5 /

 

Ne

 

0.4

X

 

 

 

 

/ EXPERIMENTAL CONDITIONS
0.3
/ T 50°C
G, 2.35 mole % UFg

}/ G (:‘:.0(:?:(10'3’g/sec-cm2
0.2 /

04

 

 

PELLET LOADING ( gUFg /g Naf )

 

 

.

 

 

 

 

 

 

 

 

 

 

 

 

o 400 800 200 1600 2000 2400 2800 3200 3600 4000
TIME (sec)

Figure 12. Comparison of Experimental and Model-Predicted Data
Showing Variation of Pellet Loading with Time at 50°C and 2.35 Mole
Per Cent Uranium Hexafluoride.
Th

UNCLASSIFIED
ORNL-DWG 63-2607

]

 

0.7

0.6 ° /

 

05 ' A~

 

 

0.4 /
A
0.3 EXPERIMENTAL CONDITIONS

T 29°C
Cg 2.62 mole % UFg

 

 

 

 

 

PELLET LOADING (gqUFg /g NoF )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

®
/ 6 6.23x107%g/sec-cm?
0.2
®
.
041
o
0 400 800 1200 1600 2000 2400 2800 3200 3600 4000
TIME (sec)

Figure 13. Comparison of Experimental and Model-Predicted Data
Showing Variation of Pellet Loading with Time at 29°C and 2.62 Mole
Per Cent Uranium Hexafluoride.
>

UNCLASSIFIED
ORNL-DWG 63-2717

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.8
w
O
Zo).
™~
1.2
~O oy
5 :‘*~,~
9 \\
O \
3‘. L
< N
O \
—
w 0.6
w =
= ‘
o \\
0 .
20 50 80 110 130

TEMPERATURE (°C)

Figure 14, Comparison of Experimentally Determined and Model-
Predicted Results on Variation of Effective Pellet Capacity with
Temperature.
76
molecular layers; both values are felt to be realistic for the system
at hand.

The values of E and a are likewise felt to be of the proper
order of magnitude for the second interpretation of Equation (22) in
which the rate of the chemical reaction was considered to contribute
a resistance to the rate of sorption.

The effect of a number of pellet properties on sorption rate
and capacity have been considered. Thus far, however, the effect of
surface area has not been discussed. A dependence of sorption charac-
teristics on surface area is predicted through the dependence of the
constants a and b on the surface area, as shown in Equation (21). A
direct dependence of a on S is predicted; and an inverse dependence of
b on S. The net result will be a decrease in effective capacity as the
surface area is increased. Calculations were made using the computer
code for pellets having the same properties as those of the study, with
the exception of surface area. Results are shown in Figure 15 for the
pellets of this study, which had a surface area of 0.86 square meter

per gram, and for pellets having surface areas of 1.0 and 1.2 square

meters per gram. The effective capacity of pellets with a surface area
of 1.1 square meters per gram is about sixty-five per cent that of
pellets having a surface area of 0.86 square meters per gram. This
result is in agreement with data resulting from repeated use of sodium
fluoride pellets. After the first sorption cycle pellets from this
study had a surface area of approximately 1.1 square meters per gram
and an effective capacity of about sixty-five per cent of the initial

capacity.
[

UNCLASSIFIED

1 8 ORNL-DWG 63-2745

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

w

o

5

™~ .

0 1.2

z — 0.86 m 2/gm

< ,

U \ \

o ‘S \ 1.0 m2/gm \\

= 0.6 S —~— —
\\* ——

———
0.0
20 50 80 ' 110 | 130

- TEMPERATURE (°C)

, Figure 15, Calculated Values of Effective Pellet Capacity for
Pellets having an Initial Void Fraction of 0.45 Showlng Effect of
Pellet Surface Area.
T8
Appiication of Data to Sorber Design

The results from the differential-bed studies enable one to
predict the rate of removal of uranium heiafluoride from a gas stream by
a pellet of sodium fluoride at conditions which may be time dépendent;
theoreticaliy, one can also use the results to predict the performance
of sorber systems such as fixed or moving beds. The calculations neces-
sary for treatment of the general case are somewhat involved and require
a finite-difference integration in both time and distance throughout the
system. For this reason, the general rgsults will not be included in
this report.

One can consider the results for a specific cése which is fre--
quently encountered and which will serve to exemplify the effects of two
system parameters; the temperature, and the diameter of the pellet. The
case to be considered is that of a sorber system operating under condi-
tions such that the pellet loading reaches the effective capacity
throughout most of the sorber system. Such conditions include sorber
systems having a low gas velocity (0.5 ce;timeter per second in a bed
ten centimeters deep) or systems in which the bed occupies an extended
length (150 centimeters at a gas velocity of 5.0 centimeters per second).
These results, shown in Figure 16, also indicate the minimum quantity
of sodium fluoride of the type used in this study that can be expected
to sorb a given quantity of uranium hexafluoride. The results will be
useful until more detailed information is available from the general

solution.
PELLET CAPACITY (gUF,/gNaF)

1.8

9

ORNL-DWG 63-2714

- UNCLASSIFIED

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

14 MAXIMUM CAPACITY
v T T AT
.o o\_\ 1/16-IN. DIAMETER
| \\\\\\ -~.~5-‘~.. |
].0—._ \ ,
N s T _
. \ \\1/8-|N. DIAMETER )
~ N L
Ty \
\\\\\\“-~\‘\\\\"‘O 3/1éj;jt‘h~\"--o
DIAMETER ~—
® . - \\.
S~ T —
0.4 1/4-IN. i -
DIAMETER T |
\-‘-
0
20 40 60 80 100 130
TEMPERATURE (°C)
Figure 16. Calculated Values of Effective Pellet Capacity for

Pellets having an Initial Void Fraction of 0.45 and a Surface Area of
0.86 Square Meter per Gram.
80

Discussion of Error

A number of possible sources of error exist in the experimental
techniques used in this studyf The major contribution to error probably
arose from the variation of pellet characteristics between individual
pellets. A second source of error is in the flow rates of the two gases,
nitrogen and uranium hexafluoride, and hence the concentration of the
gas stream entering the sorption vessel. It is felt that the concentra-
tion was known to within five per cent of the concentration value.’ A
third source of error is in the temperature control of the pellets dur-
ing sorption; the pellet temperature is felt to be known to within 1.0°C
during the initial stageé of sorption, and to a much closer degree dur-

ing the subsequent period.
81
CHAPTER IX

CONCLUSIONS AND RECOMMENDATIONS

Conclusions

fhe following‘conclusions can be drawn from the results of this
study:

1. Experimental data on the rate and maximum extent of sorption
of uranium hexafluoride by pelleted sodium fluoride has been correlated
with a root-mean-square error of 9.5 per cent.

2. The apparent mechanisms controlling the rate of sorption of
uranium hexafluoride by sodium fluoride pellets are transfer of firanium
hexafluoride across a stagnant gas film surrounding the pellet, diffu-
sion of gaseous uranium hexafluoride in the pore space of the pellet,
and diffusion of uranium hexafluoride through a layer of uranium hexa-
fluoride~sodium fiuoride complex covering unreacted sodium fluoride.

3. Sorption of uranium hexafluoride at a point in a sodium
fluoride pellet results in a decrease in the volume void fraction, the
effective diffusivity of gaseous uranium hexafluoride, and the reaction
rate at the point.

4. The maximum quantity of uranium hexafluoride that can be
sorbed at a point in a pellet of sodium fluoride depends only on the
-initial void fraction of the pellet at the point. For void fraction

values less than 0.807, incomplete reaction of the sodium fluoride will

occur.
-y

82

5. Cessation of sorption of uranium hexafluoride by pelleted
sodium fluoride occurs when the pores at the external pellét surface
have been filled with complex.

6. The effective capacity of pelleted sodium fluoride for uranium
hexafluoride is inversely dependent on the temperature.

7. The density of the uranium hexafluoride-sodium fluoride com-
plex UFg'2NaF is 4.13 grams per cubic centimeter at 26°C.

8. Considerable variation in characteristics éxist in individual\
pellets as well as between pellets of commercial sodium fluoride.

9. A useful, general calculational method has-been derived for
systems involving variéble reaction rate constants and/or variable
diffusivity which is applicable when the steady-state approximation is

valid.

Recommendations

 

The ultimate objective of the study of sorption of uranium hexa-
fluoride by sodium fluoride bellets is the prediction of the performance
of sorption systems such as fixed-bed or moving-bed sorbers. The com-
puter code resulting from this study can be used to generate data for
pellets of’specified characteristics which can be used in calculations
on specific sorber systems.

Until such calculations are made, the data 6n variation of ef-

fective pellet capacity with system parameters can be used for design

of sorber systems.

The experimental data on sorption rate should be extended to

approximately 200°C; the diluent gas for this work should be fluorine
&3
in order to avoid decomposition reactions which the complex can undergo

at higher temperatures.
10.
11.

12.

13.

1h.

84
LIST OF REFERENCES

Ausman, J. M., and C. C. Watson, "Mass Transfer in a Catalyst Pellet
During Regeneration," Chem. Engr. Sci., 17, 323 (1962).

Bar-Ilan, M., and W. Resnick, "Gas Phase Mass Transfer in Fixed Beds
at Low Reynolds Numbers," Ind. Eng. Chem., 49, 313 (1957).

Barnett, L. G., R. E. C. Weaver, and M. M. Gilkeson, "Effect of Mass
Transfer on Solid-Catalyzed Reactions: The Dehydrogeneration of

Bokhoven, C., and W. van Raayen, "Diffusion and Reaction Rate in
Porous Synthetic Ammonia Catalysts,” Jour. Physical Chem., 5@, 471

(1954 ).

Booth, F., "Note on the Theory of Surface Diffusion Reactions,"
Trans. Faraday Soc., 4L, 796 (1948).

Bruggeman, D. A. G., "Berechnung verschiedener physikalischen
Konstanten von heterogenen Substanzen," Ann. Phys. (Leipzig), 2,

636 (1935).

. Buckingham, E., U. 5. Dept. Argic. Bureau of Soils Bull. No. 25

(1904 ).

- Cabrera, N., and N. F. Mott, "Theory of the Oxidation of Metals,"

Report on Progress in Phys., 12, 163 (1949).

Carberry, J. J., "A Boundary-Layer Model of Fluid-Particle Mass

Carslaw, H. S., and J. C. Jaeger, Conduction of Heat in Solids, 2nd
Ed., Oxford University Press, ‘London (1959).

Cathers, G. I., M. R. Bennett, and R. L. Jolley, "Formation and
Decomposition Reactions of the Complex UF6.3NaF," ORNL-CF-5T7-4-25

(1957).

Cohen, A. F., "Thermal Conductivity of Sodium Fluoride Crystal at
Low Temperatures,'" J. Appl. Physics, 29, 870 (1958).

Colburn, A. P., "A Method of Correlating Forced Convection Heat
Transfer Data and a Comparison with Fluid PFriction," Trans. Am.
Inst. Chem. Engrs., 29, 174 (1933).

 
15.

16.

17.

18.

19.

20.

2l.

22.

03,

2k.

25.

26.

27

28.

85

Crank, J., "Diffusion With Rapid Irreversible Immobilizatiom,"
Tréans. Faraday Soc., 53, 1083 (1957).

Currie, J. A., "Gaseous Diffusion in Porous Media. Part 2. Dry
Granular Materials," Brit. J. Appl. Phys., 11, 318 (1960).

Danckwerts, P. V., "Absorption by Simultaneous Diffusion and
Chemical Reaction in Particles of Various Shapes and into Falling
Drops," Trans. Faraday Soc., 47, 101k (1957).

DeMarcus, W. C., and M. P. Starnes, "The Intermolecular Interaction
of UF, Molecules, " K-1114 (1954 ).

DeWitt, R., "Uranium Hexafluoride: A Study of the Physico-Chemical
Properties," GAT-280 (1960).

Dryden, C. E., D. A. Strang, and A. E. Withrow, "Mass Transfer in
Packed Beds at Low Reynolds Number," Chem. Eng. Progr., 49, 191
(1953). - -

DuFort, E. C., and S. P. Frankel, "Stability Conditions in the
Numerical Treatment of Parabolic Differential Equations," Math.

Tables and Aids to Computation, 7T, 135 (1953). °

Evans, U. R., "Laws Governing the Growth of Films on Metals,"” Trans.

Electrochem. Soc., 83, 335 (1943).

Forsythe, G. E., and W. R. Wasow, Finite-Difference Methods For
Partial Differential Equations, J. Wiley and Sons, Inc., New York

(1960). |

Freund, T., "Diffusion and Gas Sorption Rates Obeying the Elovich
Equation," J. Chem. Phys., 26, 713 (1957).

Gaffney, B. J., and T. B. Drew, "Mass Transfer from Packing to
Organic Solvents in Single Phase Flow Through a Column, " Ind. Eng.
Chem., 42, 1120 (1950).

Grosse, A. V., "A Method For Handling and Purifying UFg in Glass
Vessels By Means of Alkali Fluoride Getters," MDDC-1083 (1945).

Gupta, A. S., and G. Thodos, "Mass and Heat Transfer in the Flow of
Fluids Through Fixed and Fluidized Beds of Spherical Particles,”

Hermans, J. J., "Diffusion with Discontinuous Boundary," J. Colloid

Sci., 2, 387 (1947).
29.

30.

31.

32.

535

3k,

55

36.

57

38.

59.

Lo.

L1,

Lo,

86

Hill, A. V., "The Diffusion of Oxygen and Lactic Acid Through
Tissues, " Roy. Soc. of London, 104B, 39 (1929).

Hodgman, C. D., R. C. Weast, and C. W. Wallace, Handbook of
Chemistry and Physics, 35th Ed., Chemical Rubber Publlshlng
Company, Cleveland (1953).

 

Hurt, D. M., "Principles of Reactor Design, Gas-Solid Interface
Reactions,” Ind. Eng. Chem., 35, 522 (1943).

 

 

Jost, W., Diffusion in Solids, Liquids, Gases, Academic Press, Inc.,
New York (1960 ). :

Katz, S., "Apparatus for the Gasometric Study of Solid-Gas Reactionms,
Sodium Fluoride with Hydrogen Fluoride and Uranium Hexafluoride, "

ORNL-349T (1963 ).

Kawasaki, E., J. Sanscrainte, and T. J. Walsh, "Kinetics of
Reduction of Iron Oxide with Carbon Monoxide and Hydrogen,"

Landsberg, P. T., "On the logarithmic Rate Law in Chemisorption and
Oxidation," J. Chem. Phys., 23, 1079 (1955).

Lynch, E. J., and C. R. Wilke, "A New Correlation for Mass Transfer
in the Flow of Gases Through Packed Beds and for the Psychrometric
Ratio," UCRL-8602 (1959).

Martin, H., A. Albers, and H. P. Dust, "Double Fluorides of Uranium
Hexafluorlde," Z. Anorg. Allg. Chemie, 265, 128 (1951).

 

Masamune, S., and J. M. Smith, "Pore Diffusion in Silver Catalysts,"
A.I.Ch.E.J., 8, 217 (1962).

Massoth, F. E., and W. E. Hensel, Jr., "Kinetics of the Reaction of
Uranium Hexafluoride with Sodium Fluoride Powder, Pellets, and
Crushed Pellets," GAT-230 (1958).

Maxwell, C., Electricity and Magnetism, Clarendon Press, Oxford
(1873).

 

McBain, J. W., The Sorption of Gases and Vapors By Solids, G.
Routledge and Soms, Ltd., London (1932).

 

Milford, R. P., S. Mann, J. B. Ruch, W. H. Carr, "Recovering
Uranium Submarine Reactor Fuels," Ind. En ng. Chem., 53, 357 (1961).
L3,

L,

L5,

L6.

LT,

L8.

Ll-9.

50.

51.

52.

23

54,

D5

56.

2T

58.

87

Olofsson, B., "A Method of Calculating Diffusion in Fibres Coupled
with Irreversible Adsorption or Rapid Reaction,” J. Textile Inst.,

L7, TuEL (1956).

Olofsson, B., "Diffusion with Rapid Irfeversible.Immobiliiation,"
Swedish Inst. Textile Research, 64, 371 (1960).

 

Perry, J. H., Chemical Engineers Handbook, 3rd Ed., McGraw-Hlll Book
Company, Inc., New York (1950).

 

Petersen, E. E., "Diffusion in a Pore of Varying Cross Section,"

A.L.Ch.E.J., 4, 343 (1958).

Prater, C. D., "The Temperature Produced by Heat of Reaction in the
Interior of Porous Particles," Chem. Engr. Sci., 8, 284 (1958).

Reid, R. C., and T. K. Sherwood, The Propertles of Gases and Liquids,
McGraw-Hill Book Company, Inc., New York (1958).

Resnick, W., and R. R. White, "Mass Transfer in Systems of Gas and
Fluidized Solids," Chem. Eng. Progr., L5, 377 (1949).

 

Ruff, 0., and A. Heinzelmann, "Uranium Hexafluoride," Z. Anorg.
Allgem. Chem., 72, 63 (1911).

Scott, C. D., "The Rate of Reaction of Hydrogen from Hydrogen-Helium
Streams with Fixed Beds of Copper Oxide," ORNL-3292 (1962).

Sutherland, K. L., and M. E. Winfield, "Transient Rates of Gas
Sorption," Australian J. Chem., 6, 234 (1953).

 

Taylor, A. H., and N. Thon, "Kinetics of Chemisorption," J. Am.
Chem. Soc., Th, k169 (1952).

Taylor H. S., and S. Glasstone, A Treatise on Physical Chemistry,
Vol. II, Van Nostrand, New York (192

Thiele, E. W., "Relation Between Catalytic Activity and Size of
Particle," Ind. Eng. Chem., 31, 916 (1939).

Tinkler, J. D., and A. B. Metzner, "Reaction Rates in Nonisothermal
Catalysts,"” Ind. Eng. Chem., 53, 663 (1961).

Trapnell, B. M. W., ChemlsorEtlon, Academic Press Inc., New York
(1955).

Wakao, N., and J. M. Smith, "Diffusion in Catalyst Pellets," Chem.
Eng. Sci., 17, 825 (1962). |
88

 

59. Washburn, E. W., International Critical Tables, Vol. I, McGraw-Hill
Book Company, Inc., New York (1926). .

60. Wheeler, A., "Reaction Rates and Selectivity in Catalyst Pores,"
Advances in Catalysis, Vol. III, 249, Academic Press Inc., New York

(1951).

 

61. Wheeler, A., "Reaction Rates and Selectivity in Catalyst Pores,"
Catalysis, Vol. III, 105, Reinhold Publishing Corp., New York
(19555-

62. Wilke, C. R., "A Viscosity Equation for Gas Mixtures," J. Chem.
Phys., 18, 517 (1950).

63. Worthington, R. E., "The Reactions of Sodium Fluoride with Hex and
Hydrogen Fluoride," IGR-R/CA-200 (1957).
APPENDICES
o1
APPENDIX A

TEMPERATURE OF PELLET DURING SORPTION

The temperature difference between the interior and the external
surface of a sodium fluoride pellet during sorption is of interest.
Also of interest is the temperature difference between the external

surface of the pellet and that of the gas sfream.

Temperature Difference in Pellet
Heat of sorption will cause a temperature difference between the
interior and the surface of a porous pellet during the sorption of

b7

uranium hexafluoride. Prater in 1957, from an analytic solution of
the steady-state equations for heat and mass transfer in a porous solid,
derived an equation relating the temperature difference to the reactant
concentration difference and showed it to be independent of the kinetics
of the reaction and of the particle geometry. The relation obtained
was: |
o,
T -T =—¢— (¢, -c) (46)

where

T = temperature at any point within the particle which has a

concentration C, °C,

T = temperature at the surface of the particle, °C,

C = concentration at any point within the particle, moles/cmi,

AH = heat of reaction, cal/mole,

D = effective diffusivity of the particle, cm?/sec,
92
K = thermal conductivity of the particlé, cal/sec+cm*°C.
For the system at hand, the concentration in the interior of the
pellet is zero, so that the maximum temperature difference is then

- AH D
T-T =———C_. (4T)

s K s
The thermal conductivity of sodium fluoride is reported to be 0.036
cal/sec*cm*°C at 100°C.13 The thermal conductivity of the pellet was
taken as half this value. The initial wvalue of De for the pellet is
0.0089 cm?/sec. The maximum value of CS in the study is 0.45 x 10-5
g mole/cmj. The value of (- AH) is 23,200 cal/mole. Substitution of

these values into equation (47) yields the temperature difference as:

 

_ (23,200)(0.0089)(0.45 x 107°)

T-1T 0.018 |

S
T - TS = 00052000

Hence, radial variation in pellet témperature will be negligible.

Temperature Difference Between Pellet and Gas Stream

 

A conservative estimate of the temperature difference between
the pellet surface and the gas stream will be obtained if one neglects
the effect of heat capacity of the pellet. A heat balance on the pellet

then yields the relation:

ha (T, -1,) = 2 (- m), (48)
where
h = heat transfer coefficient,
a = external surface area,
T = pellet surface temperature, ,
93

gas temperature,

sorption rate,

2 &5«

heat of sorption.

Values of h were calculated from the j factor correlation for heat
transfef in fixed beds.27

A plot of temperature difference between the pellet surface and
the gas stream for the series at 100°C is shown in Figure 17. As would
be expected, the maximum temperature difference was observed for the
series with 8.5 mole per cent uranium hexafluoride. The initial differ-
ence of 9.0°C decreased to less than 1.0°C after 8.0 minutes. As
discussed in Chapter VIII, the difference between pellet temperature and

gas temperature was taken into account during the finite-difference

calculations.
TEMPERATURE DIFFERENCE (°C)

ol

UNCLASSIFIED

10 ORNL-DWG 63-2746

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

4 ~———8.5 MOLE % UF,
2 I\ ___———2.45 MOLE % UF,
P~
0.57 MOLE % UF,
l

 

 

0 200 400 600 800 1100
TIME (sec)

Figure 17. Time Variation of Temperature Difference Between
Pellet Surface and Gas Stream,
95
APPENDIX B

VISCOSITY OF URANIUM HEXAFLUORIDE-NITROGEN MIXTURES

The viscosity of uranium hexafluoride-nitrogen mixtures in the
range of interest is needed. Wilke62 showed that the viscosity of a

binary mixture of nonpolar gases at low pressure may be represented as:

 

 

 

 

Ky Mo (49)
K. = + 3 9
mix © 1+ (y,/yy) ¢, 1+ (y,/y,) 05
where
Boix = viscosity of mixture at low pressure,
Hys Bp = viscosity of pure components,
Y10 Yo = mole fractions of components,
\1/2 1/42
[1+(u/u)/ (M/M)/]
o = 172 2 1
- 2
2 2/5 (1 +u /u)H/°
172
1/2 1/&]2
. . [1 + (/i )7 (/M)

- 21

2/2 (1 + wym )2

molecular weights of components.

il

‘ My M
 Values of the viscosity of the pure components were calculated

L5

from reported correlations of experimental data. For nitrogen, Perry

gives the viscosity in centipoise as:

/2

L = 0.00144 T + 118

in the temperature range 15 to 100°C, where T is the temperature in

19

degrees Kelvin. For uranium hexafluoride, DeWitt ~ gives the viscosity
96

in micropoise as:

u = 0.6163 TO'955,
where T is the temperature in degrees Kelvin.

Values for the viscosity of the mixture were calculated using
Equation (49). It was observed that within the temperature and concen-
tration range covered, the viscosity for the mixture could be repre-
sented to within less than one per cent of values calculated from

Equation (49) by the simpler linear relatiom:

M

ix -~ ”1y1 + l-‘-2 (1 - yl): : (50)
which was used in the code for calculation of p . . Values of p .

mix mix
calculated from the two relations in the temperature range 29 to 100°C

and the concentration range 0.57 to 8.5 mole per cent uranium hexafluo-

ride are given in Table IX.
97
TABLE IX

VISCOSITY OF URANIUM HEXAFLUORIDE-NITROGEN MIXTURES IN THE
TEMPERATURE RANGE 29 TO 100°C AND URANIUM HEXAFLUORIDE
CONCENTRATION RANGE 0.57 TO 8.5 MOLE PER CENT
AT ONE ATMOSPHERE

Viscosity Values in Centipoise

 

 

Mole % Uranium Hexafluoride

 

 

 

Temperature, 0.57% 245% 8.5%
°C Eq. 49 Eq. 50 Eq. 49 Eq. 50 Eq. 49 Eq. 50
29 0.01802 0.01798 0.01796 0.01787 0.01762  0.0l755
50 0.01896 0.01893 0.01892 0.01882 0.01860 0.01850

100 - 0.02113 0.02109 0.02114 0.02099 0.02083 0.02066

-

 

 
One

98

APPENDIX C
BULK DIFFUSIVITY OF URANIUM HEXAFLUORIDE

of the best correlations for predicting the bulk diffusivity

in a binary mixture of nonpolar gases is a result of modern kinetic

is reported by Reid and Sherwood,+8 as:

 

 

 

theory and
0.001858 /2 [(M + M )/M M ]1/2
1 12
D = ’ (51)
12 P o 2
e Sp
where
Dy, = diffusivity of component one in a mixture of one and two,
square centimeters per second,
T = temperature, degrees Kelvin,
Ml’ M.2 = molecular weights of components one and two,
P = total pressure, atmospheres,
1
015 = 5 (91 *+ 9);
015 0y = Lennard-Jones force constants for components one and two,
QD = collision integral presented in tabular form in reference
48 which is a function of kT/elE’
k ~ _k 1
€ - - 2
12 ‘/6162 El. ig
k k
€, €5
R Lennard-Jones force constants for components one and two.
For nitrogen, Reid and Sherwood48 give force constant values of:

o = 3.681 angstroms,
99

%-: 0l1.5 degrees Kelvin.
For uranium hexafluoride, DeMarcus and Starne‘s18 give force constant

values of:

Q
Il

5.2232 angstroms,

i

~|m

429 degrees Kelvin.

Using these values,

015 = %-(5.681 + 5.22%2) = 4.452 angstroms,
€12
- = 91.5)(439) = 200.1 degrees Kelvin.

These values were used in Equation (51) to calculate values of the bulk
diffusivity of uranium hexafluoride in nitrogen at one atmosphere which

are shown in Table X.
100
TABLE X

DIFFUSIVITY OF URANIUM HEXAFLUORIDE IN
MIXTURES OF URANIUM HEXAFLUORIDE AND
NITROGEN AT ATMOSPHERIC PRESSURE
IN THE TEMPERATURE RANGE

 

 

 

29 TO 100°C
Temperature, | Diffugivity,
°C em®/sec
29 0.0809
50 0.0920

100 0.1200

 

 
101
APPENDIX D

MEAN FREE PATH OF URANIUM HEXAFLUORIDE IN MIXTURES OF
URANIUM HEXAFLUORIDE AND NITROGEN
In order to decide which type-of diffusion occurs in the pores of
a pellet, it is necessary to calculate the mean free path of uranium
hexafluoride in a uranium hexafluoride-nitrogen mixture.
The mean free path of a component in a two-component gas mixture

5k

can be derived by means of the kinetic theory of gases and is given by:

 

» (52)

 

where
A = mean free path of type one molecules,
Nl’ N2 = gas densities of molecules of type one and type two,
dl’ d2 = molecular diameters of molecules of type one and type two,
Ml’ Mé = molecular weights of type one and type two molecules.

19

The molecular diameter of uranium hexafluoride is:
d L.2 -8 i
1 =" 9 x 10 °~ centimeters,
, . . 30
and the molecular diameter of nitrogen is
-8 .
d2 = 3.15 x 10 centimeters.

For the present system, the term involving the molecular weights

becomes:

+
M_LE__ME.= 122_2#=5,683,

2
102

Assuming ideal gas behavior, the densities of type one and type

two molecules are given by:
N, = 6.023 x 107 Py/Rr,
23 ‘
N, = 6.023 x 107 P(1 - y)/RT,

where y is the mole fraction of type one molecules.

The equation for the mean free path is then reduced to:

8

I x 107, (53)

A= 51176 - 0.575 3)

 

which was used to calculate values in Table XI.
103
TABLE XI

MEAN FREE PATH OF URANIUM HEXAFLUORIDE IN URANIUM HEXAFLUORIDE-
NITROGEN MIXTURES IN THE TEMPERATURE RANGE 29 TO 100°C AND
COMPOSITION RANGE 0.5 TO 8.5 MOLE PER CENT URANIUM
HEXAFLUORIDE AT ATMOSPHERIC PRESSURE

 

 

 

 

Temperature, Mean Free Path, angstroms
°C 0.5 mole % 2.5 mole % 8.5 mole %
29 257 260 : 268
50 275 * 278 287

100 | 318 321 331

 

 
104

APPENDIX E
PROPERTIES OF THE SODIUM FLUORIDE PELLETS

The sodium fluoride pellets used in this study were nominal one-
eighth-inch right circular cylinders. They were manufactured by the
compaction of sodium bifluoride followed by heating to about 300°C in
order to volatilize the hydrogen fluoride. Prior to use, the pellets
were contacted with elemental fluorine at 400°C for one hour in order to
fluorinate impurities and remove residual hydrogen fluoride.

The fluorinated pellets had the following properties:

Average wéight of all pellets 0.0349 grams per pellet

Average weight of whole pellets 0.036 grams per pellet

Average length of whole pellets 0.1251 inch

Average diameter of whole pellets 0.1208 inch
The weight loss of the pellets during fluorination was 1.6 per cent of
the original weight. The chemical composition of the fluorinated
pellets is shown in Table XIIL.

The pore volume distribution, porosity, and surface area of the
. pellets were determined by both nitrogen and mercury porosimetry by the
Special Analytical Service Group at the Oak Ridge Gaseous Diffusion
Plant. The following properties were noted:

Surface area by nitrogen adsorption 0.856 square meter per gram

Median pore radius N 0.678 micron

Average pore radius 0.791 micron

Void fraction 0.40
105
_TABLE XII

COMPOSITION OF SODIUM FLUORIDE PELLETS BEFORE AND AFTER
FLUORINATION AT 400°C FOR ONE HOUR

 

 

Element Content of Element, wt %4 Assumed Content of Compound, wt %
or Before After Chemical Before After
" Compound Fluorination Fluorination Compound Fluorination Fluorination

 

Na 53.9 53.7 NaF + 94.8 97.0
HF 1.23 0 NaF*HF 3.81 0

Si 0.34 0.34 NasSiFg 2.28 2.28
SO, 0.065 0.0042 NasSO, 0.096 0.006
Fe 0.0176 0.0173 Fe0s 0.025 0.025
C < 0.1 < 0.1 C < 0.1 <.l

Totals 101.11 99.41

 

 
106

The variation of fractional pore volume with average pore radius is
shown in Figure 18.

The reported void fraction of 0.40, based on the pore volume
contained in pores of 3.7 micron radius or smaller, was lower than the
calculated void fraction of 0.45 based on external dimensions and the
crystalline density of sodium fluoride (2.79 grams per cubic centimeter ).
It is prdbable that the pore volume in pores having radii greater than

‘3.7 microns would account for the discrepancy.
UNCLASSIFIED
ORNL-DWG 63-2715

 

 

o=

 

 

1.0

 

 

 

 

 

 

 

 

 

ol )
]\
e N
o <
\

 

 

 

o
2
AVERAGE PORE RADIUS {u)

 

 

 

 

 

\\-o_—..

 

 

4

 

 

 

 

Porogram of Sodium Fluoride Pellets Treated with Fluorine for One Hour at 400°C,

Figure 18.

Lot
108
APPENDIX F

CONVERGENCE CHARACTERISTICS OF NUMERICAL METHOD

It is desirable to check the convergence characteristics of the
numerical method of calculation. Sifice one does not have the analytical
solution to the general problefi which has been solved numerically, a
direct comparison is not possible. One can, however, examine the degree
of convergence for a fiumber of special cases.

The numerical method was derived fér solution of the relation

¢ 2.1 %Pejdc B

St it T & 5 =0 (54)
dr

for arbitrary dependence of 8 and De on r and with the boundary condi-

tions
C=C atr =R,
dC
ir = Oatr = 0.

It should be noted that for constant values of B and D> the numerical
solution using one shell reduces to the analytical solution.

In the problem at hand, the forms of B and D, are such that after
some reaction has occurred, the values of B and De increase as one moves
along the radius of the sphere from the surface to the center. If one
chooses B and De to be of this generalyform and 'specifically to be:

R 2

r

 

 

 

 
w7

109

Equation (54) is reduced to

d C

L2.Bc-o. (55)

This equation may be integrated analytically to yield the solution

C =A cosh/%*_ir +Bsinh‘/§r. (56)

Applying the boundary conditions which were used for the numerical

procedure, namely,

o
n
o
i)
rt
H
I

&

[
O
o
ct
~
il
O
s

results in the solution:

B - |
cosh ——r
C=Cc, A (57)
cosh/%R

The rate of reaction in the sphere is then:

 

Rate = LrRD* SO
dr
r=R,
which yields the final result,
Rate = &WRECS,/fi* D* tanh . /B%/D¥* R, (58)

25

which is of the same form as the solution obtained by Thiele in 1939
for a reaction in a single pore.

The numerical method resulted in the relations:

Rate = bR D.C (-1 +RB - ohRN_lgN) , (59)
 

 

 

Dm
Dm-l ngm :
Ofm = Dm 2 (60 )
R-1%m-1 " Bao%ne1bmer "1 YD (1 *Rpa® )
m~1
where
k
5 = 3 \
- - 2
m  tanh [km (Rm Rm-lj]
K

m
tn = Sinh [ Ry - Rm_l)] ’

km =“/§m7]¥ ’

If the sphere under consideration is of radius R and is divided
into N shells of equal volume, the outer radius R.m of shell m is given
by:

1/3
R.

 

m
Rm - (N
Choosing the same relations for B and De as used in the analytical

solution, and taking the values of Bm and Dm in shell m to bé’the values

of B and De at the outer radius of shell m, one obtains the relations:

 

 

2/3
D - ox (11 ,
m m
2/3
N
Bm=B*(I; 2
kK = /B -k
m ./ D¥
Kk

 

 

 

 
111

 

 

 

o = 75 7
m o 1/3 o - 1 ?
sinh{ KR [( fi) - ( = 1) T
D'm m - 1)2/5

 

Defining the dimensionless quantity 6 as
& = kR =,./B¥/D¥ R
and substituting the above relations into Equation (60) yields an

expression for Ofi dependent only on m, N, and 6 which. is:

ott/3 m o 1Y
—sinh{e[(fi) '( N ) 5} , | (61)

m B

s fe)”

 

 

where

 

o \1/3
y =L

o[22 - (22

o . o\1/3
( N 2) Ofi-l

sinh{ 6 [(m;I 1)1/3 - (m X 2)1/5]}
5"

tanh{e [(;_1)1/5 (o 1)1/5“

A similar substitition into Equation (59) yields the total rate of

 

 

 

 

 

 

Ly,

 

sorption for the sphere as:
112

 

Rate = LTRD*C -1+ o
sN N -1
tanh{ e l:l - ( N

O (N 1:1 1)'1/3 6

sinh{ 6 [ - (N N 1)1/3 ” |

Denoting the ratio of the two expressions for the sorption rate by T,

 

1/3] .

 

(62)

 

 

where

_ Rate from numerical solution -
~ Rate from analytical solution

 

2

yields an expression for 1 in terms of N and 6. Values of 7 calculated
for several values of N and 6 using equations (58) and (62) are shown

in Figure 19. Values of 6 in the present study are in the range five to
eighty so that an error of less than one per cent will occur when forty
shells are used in the calculation. Although the error encountered in
the solution of the actual case will differ from this value, it is felt
that Figure 19 represents a good estimate of the error which can be

expected.

 
CONVERGENCE RATIO

T

UNCLASSIFIED
ORNL-DWG 63-2010

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

——f—: 7——-— it
%:{:E 0-4—"
0.8 —
-—_-___—-i-_—fi
/ /}j/
/ 6=.1
0.6 /
0.4 l
0.2
0 20 40 - 60 80 100
N, NUMBER OF SHELLS
- Figure 19. Variation of Convergence Ratio of Number with Shells and Dimensionless Parameter 6.

¢t
11k
APPENDIX G

HEAT TRANSFER CHARACTERISTICS OF DIFFERENTIAL-BED AND GAS PREHEATER

In the dififerentialhbed studies, the bed and a gas preheater were
placed in a constant temperature bath. Two calculations related to
heat transfer are of interest. The first of these is the difference in
temperature between that of the center-line of the bed and that of the
bath before a run is started; the second is the inlet gas temperature
to the differential-bed during a run.

A conservative estimate of the time required for the difference
in temperature between the center-line of the bed and the bath to be
some fraction of the initial temperature difference will be obtained if
one treats the bed as an infinite cylindgr. Carslaw and Jaeger11 give
the radial temperature distribution in an infinite cylinder that is
initially at constant temperature and has a constant surface tempera-
ture. In order to use the solution one must know the thermal diffu-
sivity of the differential-bed, which is essentially a bed of three
millimeter diameter glass beads. Assuming a diffusivity of 0.002 square
centimeter per second, after O.45 hour the temperature difference
between the center-line of the bed and the bath will be less than two
per cent of the initial temperature difference. Thus, one hour is
sufficient time to allow the differential-bed to achieve the temperature
of the bath.

The preheater consisted of fifty feet of three-eighths-inch

tubing. If average values for the heat capacity and the transfer
i)

115
coefficient are used, a heat balance yields the relation:
\

d
Fcp a%: hrrd (TW -T), (63)

wheré

F = gas flow rate,

C = heat capacity of gas,
T = temperature of gas,

T = temperature of coil,
x = distance along coil,
h = heat transfer coefficient,
d = inside diameter of coil.

Integration of equation (63) yields:

T - T - b’n__d-x

e p (6%)
T - T. 2 .
w 1 ’ :

where T k£ is the initial gas temperature.

i
If one assumes a heat transfer coefficient of 0.5 Btu/hr-ft2-°F, the
temperature ratio has a value of 10-6 for a typical differential-bed

TUun.
116

APPENDIX H

COMPUTER CODE

The computer code was written in FORTRAN and consisted of the

main program and a number of subroutines which are given below. The

dimension cards and common cards for the subroutines were identical to

those for the main program and are not reproduced below.

2

\O ©COCN 1 AN

10

11
12

13
14

15

DIMENSI{N C(100),R(100),CDR(100 ), TCDR(100),SCDR(100),
1D(100), DELTA(100),ST(100),RSI(100),RPST(100),RPD(100 ),ALPHA(100),
2¢S(100),RATE(100 ), v(100),Q(100),F(100), BETA(100),

3TE(25 ), RD(100),DR(100)

c¢MM$N C,R,CDR, TCDR, SCDR, D, DELTA, SI,RSI,RPSI,
1RPD,ALPHA,CS, RATE,V,Q, F, BETA,C1,C2,C3,CL, C5,
2¢6,¢7,¢8,c9,c10,C11,C12,C13,V14,C15,C16,CB,C18,C19,
3QTYT, N, TE, C20,PHIL,C21, TRAT, TAB, DTAB, S, CSN,

4s1,s2, 83, sS4, TIME, RD, C22,C23,C2k4, C25, DR, C26, €27, €28, C29, C30
1 READINPUTTAPE10,100,C3,C6,C7,C8,C9,Cl0,C11,Cl2,
1c13,clk4,c15,Cc16,CB,C19,C21, DTAB, S, S1,52,53, Sk, CSN, C25,

2C26,C27,C28
READINPUTTAPE10, 103, N

CALL QUTIN

TAB=DTAB

TE(1 )=N

cS(N )=CSN

cl=Cé*(1.-c7/C8)

C30=C16 ‘
co=(CcT*436./(Co*8L. })-1.+C1
c5=8k4./CT

c23=c21/(c5*c2)

QTHT=0.

C29=C3

TIME=0.

TE(6)=0.

D$20M=1,N

Q(M)=0.

ALPHA(M)=0.
D(M)=C16*(C1**C11)
BETA(M)=C3"
c(M)=SQRTF(BETA(M)/D(M))
TE(2)=F1¢ATF(M;/FL¢ATF(N)
R(M)=C15%(TE(2 )**0.333333333 )
DR(M)=R(M)-TE(6)

CDR(M )=C(M }*DR(M) -

 
117

V(M)=k.1888%C15*C15%C15/TE(1)
20 TE(6 )=R(M)
21 CALL DELSI
22 CALL MULT
23 CALL ALPHAM
2L CALL CON
CL=TRAT*C26
cl8=ck+c27
C3=C29*EXPF(C28*Ck/(1.98T*C2T7%C19))
Cl6=C30%((C18/caT y**1.5)
25 CALL RATEM
26 CALL DELTAT
27 CALL NUCON
28 QTQT=QTPT+TRAT*C21
29 TIME=TIME+C21
30 IF(TAB-TIME )31,31,33%
31 CALL QUTPUT
32 TAM=TAB+DTAB -
33 IF(D(N)-C25)34,3k4,36
36 IF(C19-TIME )3k4,3k,21
34 IF(s)35,35,1
35 CALL EXIT
100 FQRMAT((ET.4))
103 FQRMAT(I3)
END( 1,9,0,9,0,0,0,0,0,0, 0,0,0, O)

SUBRQUTINE QUTIN
(Dimension Cards)
(Common Cards)
1 WRITEQUTPUTTAPE 9,101,C3,C6,CT7,C8,C9,C10,C11,C12,
1c13,clL,cl5,c16,CB,C19,C21,DTAB, S, S1, S2, S3, Sk, CSN, C25,C26,C2T,C28
2 WRITEQUTPUTTAPE 9,102,N
% RETURN
102 FORMAT(1HOI3)
101 F$RMAT(1H0E11.A,9E12.&)
Exn(1,1,0,0,0,0,0,0,0,0,0,0,0,0,0)

SUBRQUTINE DELSI
(Dimension Cards)
(Common Cards )
DG8M=1, N

TE(1 )=EXPF(CRD(M))
TE(2 )=EXPF( -CDR(M))
TE(1 )=(TE(1 )+TE(2 )*0.5
TCDR(M )=TANHF(CDR{M) )
SCDR(M )=TE(1 }*TCDR(M)

WV F\W D
O =1 O WO

\N O =

L

— =
= C\W OO 0W W o

118

DELTA (M )=C(M)/TCDR(M)
SI(M):C(M)}SCDR(M)

RETURN

END(]-) 1,0, O) O) O: O) 0, O,Q, O) 0, O: O) O)

SUBRQUTINE MULT
(Dimension Cards)
(Common Cards)
RSI=R(1 *SI1(1)

RD(1 )=R(1 *DELTA(1)
DOSM=2, N
RST(M)=R(M)}*SI(M)
RD(M )=R(M *DELTA (M)
I=M-1

RPST(M)=R(I }*S1(M)
RPD(M )=R(I }*DELTA(M)
RETURN
ENn(1,1,0,0,0,0,0,0,0,0,0,0,0,0,0)

SUBROUTINE ALPHAM

(Dimension Cards)

(Common Cards)

DP3M=2, N

I=M-1

ALPHA (M )=(D(M *RST(M)/D(M) )/ (RD(I )-RPST (I *ALPHA(I)
1-1.+(p(MM*(1.+RPD(M)))/D(T))

RETURN

enn(1,1,0,0,0,0,0,0,0,0,0,0,0,0,0)

SUBRQUTINE CON

(Dimension Cards)

(Common Cards)
PHI=12.59*%R(N *D(N }*(-1.+RD(N)-ALPHA(N }*RPSI(N))
C20=C12% ((C13+C1L*CS(N) p*0.6666T)
TRAT=C20¥PHI*CB/ (PHI+C20)
cS(N)=CcB/(1.+(PHI/C20))

I=N-1

J=N

CS(I)=CS(J P*ALPHA(J)

I=I-1

J=J-1

IF(1)11,11,7

RETURN
EnD(1,1,0,0,0,0,0,0,0,0,0,0,0,0,0)
W=

On\n

T
8

==
= OV OO0 FW o=

\JTE WO =

119

SUBRGUTINE RATEM

(Dimension Cards)

(Common Cards)

TE(1)=R(1 *R(1)

RATE(1 )=12.59%D(1 }*CS(1 y*(TE(1 *DELTA(1)-R(1))

RATE(1 )=RATE(1)/v(1)

DYTM=2, N

I=M-1

TE(2 )=R(M )*R(M) | :
RATE (M)=12.59%D(M y*CS (M }* ( (TE(2 }+ALPHA(M)*TE(1 ) )*DELTA (M)

1-RPST (M )}*R(M)*(1.+ALPHA(M) )-R(M)+ALPHA(M)*R(I))

RATE (M )=RATE(M)/V(M)

TE(1)=TE(2)

RETURN
enn(1,1,0,0,0,0,0,0,0,0,0,0,0,0,0)

SUBRQUTINE DELTAT

(Dimension Cards)

(Common Cards)

M=N

C21=BETA(M )}*CS(M)

M=M-1

C2L=BETA (M }*CS (M)

IF(ca21-c2k4)3,6,6

M=M+1
C21=C2%%C2/ (BETA (M }*CS(M)*C10)
cel=c23*(c1-(coxF(M)) )/ (BETA(M*CS(M))
IF(c21-c24)11,11,10

c21l=Cc2k

RETURN |
eNn(1,1,0,0,0,0,0,0,0,0,0,0,0,0,0)

SUBRQUTINE NUCN

(Dimension Cards )

(Common Cards )

DOTM=1,N
Q(M)=Q(M)+RATE (M )*C21
F(M)=C5%Q(M)"
D(M)=C16%((C1-C2%¥F (M) J**C11)
BETA (M )=C3*EXPF( (-C10 )*F(M))

6 C(M)=SQRTF(BETA(M)/D(M))
7 CDR(M)=C(M)*DR(M)

8

RETURN
Enn(1,1,0,0,0,0,0,0,0,0,0,0,0,0,0)
120

SUBRQUTINE (QUTPUT
(Dimension Cards)
(Common Cards)
1 IF(S1)3,3,2
2 WRITEQUTPUTTAPE 9,101(CS(M),M=1,N)
3 IF(s2)5,5,k4 |
L WRITEQUTPUTTAPE 9,101, (RATE(M),M~1 N)
5 IF(S3)7,7,6
6 WRITEQUTPUTTAPE 9,101, (Q(M),M=1,N), (F(M),M=1, N)
7 IF(s4)9,9,8
8 WRITEQUTPUTTAPE 9,101, (BETA(M),M=1,N), (D(M),M=1,N)
9 WRITEQUTPUTTAPE 9, 101,QT{T, TIME,CL
10 RETURN
101 FQRMAT(1HOEIl.k4,9E12.4)
END(1,1,90,0,0,0,0,0,0,0,0,0,0,0, o)
121

APPENDIX I
ORIGINAL DATA

The original data are presented in gfaphical and tabular form in
Chapters VII and VIII of this report. The procedures, analyses, and
exfierimental data are recorded in the unclassified notebooks A-2099-B,
pages 120 to 270, and A-3075-G, pages 1 to 150, of Oak Ridge National

Laboratory.
122
LIST OF SYMBOLS

constant in point reaction rate equation, sec

constant in point reaction rate equation, cm5 pellet/g mole
UF6 reacted

concentration of reacting fluid in the fluid phase, g moles/
Cm3

heat capacity, cal/g mole-°C

concentration of reacting fluid in the fluid phase at the
surface of the sheli under consideration, g moles/cmj
molecular diameter, cm

effective diffusivity in shell under consideration, cm?/sec
normal diffusivity, cm?/sec

effective diffusivity in NaF pellet, cm2/sec

diameter of particles in a fixed bed, cm

cm?/sec

diffusivity of UF6 in N2,

activation energy for diffusion or chemical reaction, cal/

mole

gas flow rate, g moles/min

superficial mass flow rate, g/cmgosec

heat transfer coefficient, cal/cm?-sec'°C

mags transfer factor, dimensionless

mass transfer coefficient across external gas film, cm/sec
square root of ratio of reaction fate constant to diffu-
sivity in shell being considered, cm-1

shell number numbered from center shell

 
o

[ S

Re

Sc

123
molecular weight
exponent in relation for effective diffusivity
number of spherical shells into which pellet is divided
molecular density in a gas, molecules/cm;
Reynolds number, DpG/p, dimensionless
Schmidt number, p/Dp, dimensionless

decomposition pressure of UF6—NaF complex, mm Hg

total pressure, atm

- mean partial pressure of nontransferring gas in film, atm

point loading of reacted UF6 in pellet, g moles/cm§
‘maximum point loading of UF6 in pellet, g moleé/cm3
radius variable in pellet, cm

outer radius of shell being considered, cm

gas constant

capacity of a media for a reacting substance, g moles/cm:
surface area of pellet, cme/g

time, éec

temperature, °C or °K

temperature of gas stream in differéntial-béd, °C
temperature of gas entering preheater, °C

temperature at surface of sodium fluoride pellet, °C

wall temperature of preheater, °C

linear distance in pellet, cm’

linear distance along preheater coil, cm

mole fraction of component in fluid phase
Greek Letters

x

124
~
ratio of fluid phasé concentration at the outer radius to
that at the inner radius for the shell being considered,
dimensionless
ratio of maximum to minimum cross section of a pore,
dimensionless
reaction rate constant, sec”!
constant in relation for effective diffusivity
function defined in derivation of differencing equations,
cm"1
volume void fraction in NaF pellet, ratio of void volume in
pellet to total pellet volume, dimensionless
external bed porosity, ra;io of void volume external to
particles to total bed volume, dimensionless
initial volume void fraction in NaF pellet
Lennard-Jones force constant, °K
dimensionless quanfity B/D R
mean free path of uranium hexafluoride in uranium hexafluo-
ride-nitrogen mixture, cm
fluid visCosifiy, g/cm*sec
function defined in derivation of differencing equations,
cm-l,
fluid density, g/ém3

t

Lennard-Jones force constant
125
T . ratio of sorption rate from numerical solution to rate from

analytical solution

¢ function defined in derivation of differencing equations,
em’ /sec
QD "~ collision integral

Subscripts

1, 2 components one and two
B refers to bulk gas stream
f refers to external gas film

m refers to shell m
.

v
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127

INTERNAT, DISTRIBUTION

1. Biology Library
2-4. Central Research Library
5. Reactor Division Library -
6-7. ORNL — Y-12 Technical Library
Document Reference Section
8-42. Laboratory Records Department
43. Laboratory Records, ORNL R.C.
44. J. B. Adams (K-25)

45. Co M.
46. E. G.
47. Re E.
48. W. H.
49. W. L.
50. E. L.
51. F. L.
52. H. W.
53. C. E.

Blood
Bohlmann
Brooksbank
Carr
Carter
Compere
Culler
Godbee
Guthrie

54. R. W. Horton
55. 8. Katz

56. C.
57. R.
58. J.
59. A.
60-74. L.
75. R.
7. J.
77. W
78. W.
79 Je
80. M.
8l. J.
g2. C.
83. A.
84. M.
85. P.
86. J.
g87. T.

E.
B.
T.
P.
E.
P.
F.
S.
W.
B.
Je
A.
W.
M.
E.
H.
J.
H.

ORNL- 3494
UC-4 — Chemistry
TID-4500 (23rd ed.)

Larson

Lindauer

Long

Malinauskas
McNeese

Milford

Murdock

Pappas (K-25)

Pitt, Jr.

Ruch

Skinner

Swartout

Weber (K-25)
Weinberg

Whatley

Frmett (consultant)
Katz (consultant)
Pigford (consultant)

88. C. E. Winters (consultant)

EXTERNAL: DISTRIBUTTION

89. Research and Development Division, AEC, ORO
90-604. Given distribution as shown in TID-4500 (23rd ed.) under
Chemistry category