Health Physics Pergamon Press 1975. Vol. 28 (June), pp. 717-725. Printed in Northern Ireland

FUNDAMENTAL STUDY ON LONGITUDINAL DISPERSION OF TRITIATED WATER THROUGH SATURATED POROUS MEDIA M.FUKUI and K. KATSURAYAMA Research Reactor Institute, Kyoto University, Kumatori-ch6, Sennan-gun, Osaka, Japan

(Received 18 December 1973; accepted 13 June 1974)

Abstract-Dispersion

phenomena of ions or molecules through porous media are found not

only in engineering fields such as the process of ion exchange and filtration, but in natural hydro-systems. Recently the pollution of underground water due to industrial wastes has been discussed, and it is important to forecast the behavior of pollutants. The purpose of this paper is to investigate the effect of the average axial velocity of flow, the effect of size and the shape of porous media and flow length on the dispersion coefficient in more detail, and to examine the application of diffusion type equations to dispersion phenomena. Moreover, numerical analysis of the diffusion equation was tried under a boundary condition of variable concentration, which was not solved analytically in general, and the solutions were compared with the

breakthrough curves under such conditions. INTRODUCTION

IN GENERAL, it is believed that dispersion phenomena are due basically to two main factors. One is molecular diffusion, caused by the concentration gradient of a diffusing substance, and another is the dispersion induced by the microscopic velocity variation of tracer particles through porous media. Until now, two mathematical models were proposed to explain the phenomena. I n the first modcl, porous media were reduced to bundles of capillary tubes and the longitudinal dispersion in one tube was studied. I n another model, it was assumed that the statistical characteristics of porous media were in disorder and the distribution of dispersing substance would be a normal probability distribution after a long distance of flow. But in both models the fundamental equation which express dispersion phenomena quantitatively is finally the same as Fick’s second law of diffusion. DANCKWERTS (1 953) considered a packed column of length L, through which fluid flows with a mean axial velocity V , and he denoted this imaginary plane by x = 0. At time t, the plane x = 0 is at a distance Vt from the entry, and the x-coordinates of the ends of the column are (-Vt) and ( L - Vt), respectively. If packing is randomly arranged, each element of fluid

will travel according to ordinary “random walk” theory. That is, if C is the mean concentration of the tracer at a plane x a t time t,

aciat = II) - azc/ax=, where II) is a “dispersion coefficient,” which must be determined empirically. The solution to equation ( 1 ) is given under the following boundary and initial conditions:

c=o, c=c,, c=o, c=c,,

x>o,

t=0;

xO;

x=m, x =

-m,

t>0;

LAPIDUS and AMUNDSON (1952) stated that, for small rates of flow, the effect of molecular diffusion is pronounced; and they solved the partial differential equation,

aciat = D .

a2c/ax2

- v . aqax

(3)

for the column of infinite length. T h e solution of equation (3) subject to the conditions t = O CEO, x>o, x=o, t>O c-0, x=m, t>O

717

c=c,,

718 FUNDAMENTAL STUDY ON LONGITUDINAL DISPERSION OF TRITIATED WATER

Pore volume of effluent 7

(=Vt/L)

FIG. 1. Solution of the dispersion equation by DANCKWERTS (1953).

By these means, solutions (2) and (4) become

is found to be

c = -1 erfc(+)+ cQ

21 e x p (vx z).

erfc

dDt

x

+ vt

+

c =1 - erfc (k) -1 exp ( 4 ~* erfc ~ )

d7/p,

Q '

the solutions described above can be put in As is evident from equation (7), C/C, is 112 dimensionless form a t the position x = L by introducing the following dimensionless vari- whenever r = 1, i.e. whenever one pore volume has been displaced. But equation (8) does not ables: pass through C/CQ= 0.5 and one pore volume. P, = VL/4D or VL/4D; ( 5 ) The results of calculations with equations (7) and (8) are shown in Figs. 1 and 2. The expansion for erfc(x) should be used, since the errorfunction complement is small while the exponenT h e dimensionless time, T, corresponds physi- tial is large: cally to the number of pore displacements introduced into the medium since the start of the experiment; that is, it is equal to the ratio of 1.3-5 total fluid volume introduced to the free volume of the column.

-3ao

500

100.0

1'6

10 1; ; .1 1'8 Pore volume of effluent 7 (=Vt/L)

20

FIG.2. Solution of the diffusion equation by LAPIDUS and AMUNDSON (1952).

M. FUKUI and K. KATSURAYAMA

Here, there are two ways to determine the dispersion coefficient. One way is to obtain the number P,,which is calculated by minimizing the standard deviation between the analytical solution and experimental data of C/Co, and to use equation (5). Another method, which is given by RIFAI(1956), is differentiating equation (7) or (8) with respect to 7 and setting T =: 1, which gives the result

719

Small samples of the effluent from the columns were taken at appropriate intervals and the volume was measured gravimetrically and transformed on pore volume basis. Triated water was selected for the tracer substance because of its small adsorption by the medium. The solutions were identified by using a concentration of tritium of 2.0 x pCi/ml and were analyzed by using liquid scintillation techniques. A 0.5-ml aliquot of sample was transferred to a vial containing 10 ml of scintillation solution, and the sample was counted for 10 min. The scintillation solution was made by mixing 104 g of naphthalene, 0.13 g of POPOP(2,2-p-phenylene-bis-5-phenyloxazole), 6.5 g of PPO(2,5-diphenyloxazole), 700 ml of mcthanol, 500 ml of toluene and 500 ml of dioxane. Other conditions of the experiment, such as column length, flow velocity, and the kind of packed material, will be explained in each run.

(9) Let be replaced temporally by J , which is the slope of the brcakthrough curve at one pore volume. Here the coefficient of dispersion is equal to L * V/4nJ2. Though it seems to be easy to determine the coefficient of dispersion by the latter way, it is difficult to obtain the exact value J empirically, since the slope of the breakthrough curve a t one pore volume is very steep. Thus, the former way is used to determine the coefficient of dispersion in the following arguments.

RESULTS AND DISCUSSION

Prior to examination of the effects of pii;4cal factors on the coefficients of dispersim, it is EXPERIMENTAL PROCEDURE necessary to choose either equation (7) or (3) to Experimental work was conducted with glass determine them. We used equation (8) because beads, having diameters of 0.1, 0.6, 1.O, 3.0 and significant differences between the two equa5.0 mm, and with Toyoura standard sand. tions were not recognized for the various d o w Fragments of crushed glass plates that were velocities shown in ‘Table 1. Moreover, the relascreened through a 5 m m sieve were also tested. tive concentration of the breakthrough curves Colirmns were prepared by carefully packing did not pass through 0.5 a t one pore volume, but known weights of sample into 2 cm i.d. lucite passed between 0.5 and 0.6 in the experiments. tubes, having a stainless steel screen of 200 mesh in the bottom. An outlet tube with stopcock (1) Effect o f the averageflow velocity, the size o f tamping and column length and Maliotte reservoir was connected with a n I n Fig. 3, the breakthrough curves are shown inlet tube for controlling flow. A filter paper was placed above the sample to prevent mixing for various average flow velocities (2.2 x lop3 at the surface boundary. Prior to the applica- to 1.9 x lo-* cmlsec), the column being packed tion of tracer solution, the columns were pre- with uniform Toyoura standard sand. T h e saturated with distilled water and one pore column length was 32 cm and one pore volume was 42.3 ml. I n this region of flow, a significant volume was determined.

___

Table 1. The coegicients of diffusion and dispersionfor variousfiow velocities

Flow velocity V (cmlsec) DiKusioncoefficientD ( cm2/sec) DispersioncoefficientD

2.3 x 10-3

3.9 x 10-3

7.4 x 10-3

9.4 x 10-3

1.9 x I O - - ~ 7.7 x

1.9 x lo4

2.7 x 10-4

6.0 x

7.1 x

1.4 x l W 3

3.9 x

1.8 x

2.8 x

6.4 x lo4

7.4 x lop4

1.5 x

4.1 x lop3

(cm2/sec) -.

Column length L 5

__ __

.~

.

. .

=

32 cm.

-__-. ~

_

..

_

_

- - ..

.. -.

10-2

--__

Toyoura standard sand, uniform glass beads, 1 mm and crushed glass plate sieved from 5.0 mm to 2.0 mm. Above the region of average flow velocity, K x 10-4 cmlsec ( K = I ,- 2), the results of the experiment were formularized by the method of least squares as follows. Toyoura standard sand; D = 0.076 x VoJ'ee Uniformglassbeads, 1 mm; D = 0.178 x Vo.es5. RIFAIet al. (1956) found the following relation for Ottawa sand

1.0

- 1 0

-

.E

-

>

.-

-

0

-

d

-

.

v

V ( W 4 2.2~10-~

a,

3.9~16~

7.4~10-~ o 9.0~16~ o 9.4x 1.9~ lbz X

%

c

-

::

9 1

s 5 0.5a l

&'q

Column length 32(cm) One porn volume 423(ml)

e o v&

I

_

2

-

u

Id-

0

Toyoura sand Glass beads, 5 m m Column length 32cm

O--O-

0-9

o.o

. 10'

I

! I 1 8

I

*

0

. 0

I

I

I I t 1

.

I

I

I

I l l , ,

FIG.4. The relation of P, to average flow velocity.

M. FUKUI and K. KATSURAYAMA

72 1

Average velocity V (Cm/sec)

FIG.5. The dispersion coefficients of three kinds of packed materials calculated from various flow velocities.

1'

0

d2

,o

0.4

0

I

a

d.6

1.b

16

"d8 ; .1 ; .1 h e volume of effluent 7

; .1

(=Vt/L)

FIG.6. Effect of molecular diffusion on the breakthrough curve.

time period the displacement continued at the original flow velocity. I n Fig. 6 the differences between these three conditions are shown. The coefficient of dispersion must be compared under the same condition of mean fluid velocity, on which it depends. In Fig. 7, the coefficients of dispersion are shown. These results were obtained under the same flow velocity (1 1.2 x 10-2cmlsec) ,by using the glass beads of different diameters, as mentioned previously. Because the values of dispersion coefficient should increase with the root of the average diameter of

-

packed material, more examination is necessary. And in Table 2, it is shown that the larger the average diameter of tamping, the greater the ratio, D/V. The reason for this is that the traveling length of microscopic particle of tracer is longer when the size of packed material is increased. Thus, the degree of dispersion could be expressed by the ratio, DIV, which is affected by the shape, the size and the porosity of porous material. I n order to study the effect of column length, columns of three different lengths were used for

722 FUNDAMENTAL STUDY ON LONGITUDINAL DISPERSION OF TKITIATED W A T E R -

1 1

L-

. 1-

Awerage velocity 10~16~(Fm/r3~

o

I

0

L

30 cm

I

d m 8' 10

';A+

I I

,

1

1

0

1

I

1

FIG.8. Concentration history of various column lengths through the packing ofglass beads, 1 mm.

I I I

10-1 Averoge size of glass b e d s (Cm)

(2) Breakthrough curves withf r e e and discontinuous concentration at the boundary

FIG. 7. Effect of the average size of glass beads on the dispersion coefficient.

I t is dificult to obtain an analytical solution to equation (3) under general boundary condi-

glass beads, of 1 and 5 m m diameter. T h e typical breakthrough curves of tritium tagged water for this experiment are shown in Figs. 8 and 9. The gradient of breakthrough curves decrease with longer column length. Generally, the gradient of tracer front changes with flow length by the equation

__ -,=a

ax

I=&

tions and much effort will be needed, even if we use the concept of superposition of the solutions (HOUGHTON, 1963) under thc condition of pulse-wise inflow, instead of that of free and discontinuous concentration. A finite difference I O y - - - -

- '0

2 t

0

3

2

t=LIv

[Glass

beads,

D

The proportionality of the dispersion coefficient to the average flow velocity is also shown in Fig. 10, even though the column length changes. So the ratio DlV is also independent of the column length in laminar flow, and the coefficients of dispersion are nearly equal in various column lengths, when they are compared by using the same average velocity (Table 3).

t

0

LL* o~ 10

4'0

L

50

U

60

-

1

70

(cm)

3.6 x

FIG.9. Concentration history of various column lengths through the packing of glass beads, 5 mm.

Crushed glass plate 1.6 x 10-1 cm Toyoura sand 7.4 x 1 W c m Column length L = 20 cm.

_

1.8 x 10-l

1.5 x 10-1

8.8 x ___

_

80

Effluent volume I m l )

Table 2. The relation of the ratio D/V to the size of glass beads ~_ ~- . Glass beads (mm) 0.1 0.6 1.o 3.0 5.0

D/y

1

_

- _ -_

.

~

~~

__

3.3 x 10-1 ____.

90

M. FUKUI and K. KATSURAYAMA

72 3

dary condition, are shown in Fig. 12.

I.C. c/co=o O l Z l l 7 = 0 B.C. C/C, = s i n ( 2 n ~ )Z = 0 0 < T 5 0.5 c/co= 0 Z=O 0.5 ( 7 . I n the experiment, the column of40 cm length was packed with glass beads of 1 mm diameter and one pore volume of the column was approximately 50 mI. The average velocity of flow was about 1 x cmlsec and P, obtained under these conditions was nearly equal to 90. As it is difficult to measure the concentration of inflow continuously, due to the low energy ,%ray emission of the tritium radionuclide, the concentration of inflow was observed as a slug of known volume and concentration. T h e slug was subsequently displaced by using distilled water. I n Fig. 13 breakthrough curves are shown, where the pore volumes of the slugs are 0.1, 0.3

Glass beads 5mm

Column len$h

15cm 32cm e 48cm 0

a

I

I

I

1 I I L

~~

3

Average velocity V ( C W s e d

FIG.10. The dispersion coefficients of three kinds of column lengths obtained from various flow velocities.

method is useful for solving a diffusion type equation with general boundary conditions. I n this work the finite difference approximation was used according to the Crank-Nicolson method (LEE, 1968). I n Fig. 11 a comparison of the solution by finite difference method with that by analytical method to equation (3) is shown. Here the condition of constant inflow was used and there are no significant differences between the two methods. The solutions by the finite difference method of equation (3) under the following boundary conditions as an example of the general boun-

FIG. 1 1 . Comparison of breakthrough curve by the analytical method and the finite difference approximation.

Table 3 . Dispersion coe>cients of various jacked materials for an averageflow velocity

Packed material Toyoura sand 20 32 Column length (cm) Dispersion coefficient 8.2 x lo-‘ 7.6 x 10-4 (crn21sec) Average flow velocity Y

=

1

x

Glass beads (1 mm) 20 40

1.5 x l F 3 1.1

10-2cm/sec.

x

15 3.0 x

Glass beads (5 mm) 32 48

4.1 x

4.0 x

724 FUNDAMENTAL STUDY ON LONGITUDINAL DISPERSION OF TRITIATED WATER .o 1.01

"

0.2

a4

06

1.0

08

1.2

1.4

1.6

1.8

Pore volume of effluent 7 (-1

FIG.12. The breakthrough curve calculated by finite difference approximation with the boundary condition of sinusoidal wave inflow.

0.2

0.4

0.8 1.0 1.2 1.4 1.6 1.8 Pore volume of effluent T (-)

a6

FIG.13. The forecast and the history of effluent concentration under the boundary condition of pulse-wise inflow of different pore volume.

and 0.5 and the concentrations at the inflow boundary of the slug, C/Co, are all 1.0. Under these conditions all curves of effluent are similar to the numerical solutions. In Fig. 14 the effluent curve is shown, where slugs of three different concentrations inflow continuously as follows.

-Pe 90 6 a5

o

+ -

d

I -

-

a

4 Wove of inflow -02

I

a

,

I

0 I T < 0.1, = 1.0 0.1 5; 7 < 0.2 = 0.173 0.2 I r < 0.3, =o 0.3 < T.

ClC, = 0.594

\ . U2

1.6

Pore volume ofeffluent

7 (-1

FIG.14. The forecast and the history of effluent concentration with different inflow slugs.

I n Fig. 15 the inflow condition of the slug was discontinuous as given by the following

M. FUKUI and K. KATSURAYAMA

11

-0.0

-06 -04

-0.2

725

-a0

Wove of inflow

Fore volume of effluent 7 (-1

FIG. 15. The forecast and the history of effluent concentration under the boundary condition of three slugs of discontinuous and different concentration inflow.

conditions : ClC, = 1.000 = 0.442

=o

0 0.7 0.4 0.2 0.5

57

< 0.2,

I T < 0.8 5T

< 0.5

I T < 0.4,

I T < 0.7, 0.8 IT.

I n all cases shown in Figs. 13, 14 and 15 the solutions obtained by the finite difference approximation give good results. The behavior of a tracer particle is determined and found to be applicable for more complicated boundary conditions. T h e results described above justify the application of the diffusion type equation to dispersion phenomena. CONCLUSIONS

I n this study the following matters were verified. The breakthrough curves of triated water are almost similar under the condition of the same length of column in laminar flow below Reynolds number 10, even when the mean interstitial velocity of the flow is changed. The Peclet numbers (P,) calculated from the breakthrough curves are nearly the same and, inevitably, the coefficients of dispersion D are directly proportional to the velocity V as given by the relation, D = VL/4P,. A linear relationship is also found in the case of using a packed material of fragments of crushed glass plates. Consequently, the coefficient of dispersion itself does not represent the degree of dispersion, because the dispersion coefficient deoends on the mean velocitv of flow.

This relationship is independent of the degree of dispersion when the effluent curve is the same. This suggests that it is not essential to adopt the diffusion type equation to examine the dispersion phenomena. However, one can predict the behavior of a pollutant quantitatively, once the dimensionless parameter P, is known. The prediction can be made because the ratio Dl V,which is independent of travel time and distance, can be considered as a characteristic value of a tamping. Here it was shown that the larger the average diameter of packed material, the greater the ratio. The coefficient of longitudinal dispersion is nearly equal to that of molecular diffusion below the mean velocity of K x cmlsec ( K = 1 2). Molecular diffusion exceeds dispersion if the above velocity is lower than the stated value. The breakthrough curve of inflow with free and discontinuous concentration could be estimated by the finite difference method of the diffusion type equation. N

REFERENCES BRENNERH., 1962 Chem. Engng Sci. 17,229-243. DANCKWERTS P. V., 1953, Chem. Engng Sci. 2, 1-13. HOUGHTON G., 1963, J . phys. Chem. 67,84-88. LAPIDUS L. and AMUNDSON, N. R., 1952, J . fibs. Chem. 56,984-988. LEE P. S., 1968, Quasilinearization and Znuariant Imbedding (New York: Academic Press). RIFAIM. N. E., WARREN J., KATJFMAN and TODD D. K., 1956, Dispersion phenomena in laminar flow through porous media, Sanit. Eng. Research Lab., Rept. 3, p. 157 (University of California, Berkeley).