Permeability relationships using Darcy Permeability, Vacuum Decay, Pressure Decay, and Pore Size Distribution methods on graphitic materials

Curba

1966, Vol. 4, pp. 107-114.

Pergamon

PERMEABILITY PERMEABILITY, AND PORE

Press Ltd.

Printed in Great Britain

RELATIONSHIPS VACUUM

DECAY,

USING

PRESSURE

SIZE DISTRIBUTION

ON GRAPHITIC

DARCY DECAY,

METHODS

MATERIALS

W. A. ANDERSON Great Lakes Carbon Corporation,

Niagara Falls, New York

(Received 13 July 1965)

Abstract-A

study has been carried out in an effort to more fully explore the relationships which exist between different methods of permeability measurement on graphitic materials. Much of this work serves to support that which was previously completed by HUTCHEON,LANGSTAFF, WARNERand WICCS. Linear relationships have been confirmed to exist in the laminar flow region between the Darcy Permeability Value and (1) the Vacuum Decay Permeability Value, (2) the Viscous Flow Coefficient term from the Pressure Decay Method, as well as (3) the Slip Flow Term from the Pressure Decav Method when considering results from various grades of graphite having markedly different Pore S&e Distribution Spectra. It was also confirmed that (4) the cumulative Pore Size Distribution Spectra may be analysed to calculate an equivalent Darcy Permeability Value. The Pore Distribution analysis included molded and extruded samples, two of which having a permeability value twofold greater than any tested by WICGS. It is significant that an agreement in permeability values may be obtained using the various methods of measurement. An analysis of the subject relationships has been made in an attempt to evaluate these data using theoretical concepts as applied to the flow of gas through porous media.

Nitrogen gas flow was measured using four rotameters having overlapping scale ranges which permitted flow rate measurements between 5 and 7500 cm3/min. This flow rate measurement was used in Darcy Permeability calculations with gas pressure as measured using a mercury manometer. Helium gas decay was used in vacuum and pressure decay tests.

1. INTRODUCTION

THE OBJECT of this work was to study the various

permeability techniques using several different carbon products to give a wide range in permeability values. A comparison of the results using Vacuum Decay, Pressure Decay, Steady State, and Pore Size Distribution methods was made to present the relationship between permeability values obtained using the subject techniques. Mathematical derivations have been included to clarify any uncertainty which may exist in the variety of available test methods and to present the inter-relationships of importance. 2. MEASUREMENT

OF BI’IY

Each graphite sample was thoroughly dried and then cooled to 25°C in a desiccator prior to performing each test. The sample was then placed in a rubber gasket, compressed in metal cups using a hydraulic jack, and finally tested. This caused compression of the rubber gasket which sealed the sample thus preventing gas from flowing between sample and gasket.

2.7 Darcy permeability Darcy permeability testing requires the use of a constant pressure differential across the sample. The volumetric flow of nitrogen gas through the sample is measured at constant pressure. Darcy’s law may be expressed as K =

pQL/A V’z-Pi)

(1)

which is valid when KA/pL, is a constant in the laminar flow region and where:

107

K =permeability

value in units of Darcys;

p =viscosity of the Auid in centipoise;

W. A. ANDERSON

108

g =volume rate of flow measured in cm3/sec (where the fluid is a gas, 8 is measured in cm3/sec at the mean pressure, F, existing in the specimen during flow); L =length of the specimen in cm ; A =cross-sectional area of the specimen perpendicular to thedirection of gasflow in cm* ; Pa=pressure at the upstream face in atmospheres; Pr=pressure at the downstream face in atmos-’ pheres. When the volume rate of flow is measured at the downstream face of the sample, it is necessary to make the appropriate changes in the above formula to correct for the effect of flow measurement at ressures other than p, the mean pressure. Thus, This results in !% =QIPl where %(Pz+Pr)/2. K = ~J~LQIPI/A(P~*-PI*)

(2) with Qr=volume rate of flow measured at the downstream face in cm3/sec. 2.2 Vacuum decay permeability Vacuum decay permeability measurements are performed under conditions of transient pressure differential across the sample. A vacuum is drawn on the system adjacent to one side of the sample. Helium gas at atmospheric pressure is maintained on the other side of the sample. The vacuum system is isolated from the sample when optimum vacuum has been attained at time t=O. The increase in pressure on the vacuum side is measured as a function of time as helium gas diffuses through the sample. Again, Darcy’s law may be used to derive the appropriate formula which applies for such apparatus. The Permeation Constant (k) may be defined as k=K/p or k = Qd/A (P2 -PI) (3) This formula may be used to derive the permeability coefficient for use with a vacuum decay system, which is (4) *The transient nature of this test requires that Q1 be expressed in differential form. Thus, Q,=V?~ may be substituted in (3). A simple integration of this differential equation leads to (4).

where L=length

of the specimen

in cm;

.4=cross-sectional area of the specimen perpendicular to the direction of gas flowincm’; Vo=static volume amounting to the evacuated volume into which the gas decays during testing in cm3 ; P2 =constant pressure at the upstream face in mm Hg; Pticinitial pressure at the downstream face before decay begins in mm Hg; Py=pressure at the downstream face when decay ends in mm Hg; At=time required for the pressure to increase from Prr to Pra in set; Q r=volume rate of flow measured at the downstream face in cm3 atm/sec. 2.3 Pressure decay permeability Pressure decay permeability measurements are performed under conditions of transient pressure differential across the sample. A positive pressure of helium gas is introduced on one side of the sample. The other side of the sample is held at atmospheric pressure. Time-pressure data are recorded as helium gas pressure decays from the static volume chamber through the test specimen after isolation from the pressure supply. Again, the Permeation Constant (k) may be defined as k=KIp or k =

QzLIA(P2 - PI)

(5)

Formula (5) may be used in deriving the permeability coefficient for use with a pressure decay system”, which is

where L=length of the specimen in cm; Vo=static volume amounting to the pressurized volume from which the gas decays during testing in cm3 ; PI =constant pressure at the downstream face in psig; *The transient nature of this test requires that Qs be expressed in differential form: dP Q1= Vo-$ Substitution of this in (5) produces a differential equation which leads to (6) upon integration.

PERMEABILITY

RELATIONSHIPS

P:i=pressure at the upstream face measured at some time 1: during the decay period in psig ; Pzf=preasure at the upstream face measured at some time 12) tr during the decay period in psig; t=t2- tl in set; QZ = volume rate of flow measured at the upstream face in cm3 atmjsec. This pressure decay value is normally expressed as a function of the mean pressure across the sample. Pm =(pzi+p2’)‘2+ 29.4 The logarithmic

1 atm , for P, =I atm

series In(x)=

for X> 0 may be used to approximate where x =-.

For

p2i

-pl

y2,

-Pl

p2i

-pl

p2,

-PI

the above

USING DARCY PERMEABILITY

109

3. RESULTS

3.f Pressure decay Each sample was tested using the corresponding apparatus. The Vacuum Decay and Darcy Permeability tests each provided one point for each sample. It was necessary to calculate a number of data points for various values of mean pressure in the Pressure Decay Analysis. Figure 1 is a typical pressure decay curve obtained during the pressure decay test in which the upstream pressure decayed from 70 psig to 5 psig with a downstream pressure of 0 psig. A table, similar to that shown beneath in Fig. 1 was prepared from each pressure decay curve. The Permeability Coefficient, &I, was then plotted on another graph as a function of Mean Pressure (Pm) across the sample. This variation of Permeability Coefficient with hlean Pressure is presented in Figs. 2 and 3 for a number of the samples tested. The linearity (indicating Iaminar flow) of each curve was of interest as was the fact that the Permeability Coefficient value from vacuum decay testing fell on the pressure decay curve at a Mean Pressure of 3 atm abs, The close correlation of data from pressure decay and vacuum decay tests served as a

thus ignoring terms of an order >3 in the power series produces an error of < 10”: Since data obtamed

in this test involved

1<

13,

approximate solution was used to simplify lations thus resulting in

an

calcu-

Time,

ccc

FIG. 1. Pressure decay raw data curve. Graphite sample 4.

p*z

PCl

t

(psi)

(psi)

(se4

(cm*/sec)

65 55

55 45

24 34 47 71 122 360

2.91 2.46 2.23 1.96 1.72 1.17

35

Where Vo = 940 cm8

25 15

25 1.5 5

KP

3.04 2.70 2.36 2.02 1.68 1.34

W. A. ANDERSON

110

MWn OTMplre @X088 lhe 80mpb.

ohn abt.

Fro. 2. Pressure decay permeability coefficient ns a function of mean pressure across the sample using He gas for graphites as noted.

free molecular path length in the fluid. The linear relationship (Fig. 4) between slope of the permeability coefficient curves and Darcy Permeability is an indication that the latter is linearly dependent on the viscous flow contribution in the sample under test. Experimental errors contribute to the failure of this curve to pass through the origin. The Permeability Coefficient Axis intercepts of the Permeabiiity Coefficient Curves are also of significance in that they represent the cont~bution of molecular flow to the overall permeability coefficient value. An intercept of zero would indicate the absence of molecular flow. Molecular flow occurs when the distance between the walls confining the fluid is less than the free molecular path in the fluid. This molecular flow term is represented by the permeability coefficient value which would be obtained under zero pressure. This means that the gas flow through a capillary is greater than that which would be expected under a given pressure gradient for pure viscous flow. The linear relationship in Fig. 5 signifies that the Darcy Permeability varies linearly with the molecular Aow term. Again, experimental errors were found to contribute to failure of the curve to pass through the origin.

3. Pressure decay permeability coefficient as a function of mean pressure across the sample using He gas for graphites as noted. FIG.

check on the depen~b~i~ of the two methods of measurement. These pressure decay permeability data were further analysed by plotting the Permeability Coefficient slope and the intercept of the Pressure Decay Permeability Coefficient curves as a function of the experimental Darcy Permeability (meaaured with 1
relationship betwe& Darcy slope of the pressure decay coefficient curves.

FXG. 4. The

and

the

permeability permeability

PERMEABILITY

RELATIONSHIPS

USING

DARCY

PERMEABILITY

111

media. The Mean Hydraulic cribed in the formula: m=ko6

Bo

D =4ko6

0

04

0.2 PermeabilOy

0.6

cceffvxnt-0x1s

/ 0.6

intercept.

cm2/sec

FIG. 5. The relationship between Darcp permeability and the permeability coefficient axis intercept for the pressure decay permeability coefficient curves.

Graphite is a porous material composed of pores and pore openings of varying dimension. Thus, the existence of molecular, viscous and slip flow (where the distance between the walls confining the fluid is equal to the mean free path of the fluid) influences the passage of fluids through the pores the degree of which depends on the Mean Free Path of the Auid in use. Turbulent flow can also exist when the distance between the walls confining the fluids is far greater than the free molecular path in the fluid. The analysis of turbulent flow is very complex and has not been attempted. -4 further analysis of the pressure decay data may be obtained by considering the calculation of Permeability Coefficient (B in units of cm’) from the Permeability Coefficient for viscous flow (Bo in cm2) and the Permeability Coefficient for molecular flow (Ko in cm). For consolidated bodies with uniform pore size of circular cross-section, WxcsC2) introduces the formula: B= -$ HUTCHEON et

(D’+SZU)

dc3) describes 4’ (Tortuosity Factor) in terms of Bo and Ko as: q2=(Ro/Ko2)koEor512 where ko=2.5 based on work by Millikan for rough surfaces and &=0.59 by Wyllie and Rose which applies to all pores in all

m, is des-

‘KE

D, is equal to 4m thus result-

The Pore Diameter, ing in

Thus,

Radius,

Bo ‘Ko’

B=Bo+1.36

Koll

(9)

where Bo=the slope of the pressure decay curve times gas viscosity in poise, (also times 0.988 x 10W6to convert cm’/secatm to cm’/poise). Ko=the intercept of the pressure decay curve times 6.03 x 10V6 (a molecular velocity factor to convert cm’/sec to cm) ’ j=3.107 x lo-‘:T . which is the Mean Free

Pm2

Path in cm.

T= absolute temperature Pm=Mean Pressure in dynes/cm’ o= I .90 x lo-* cmt4) (molecular diameter of He gas calculated from the gas viscosity). The Darcy Permeability the formula : Kd=lO*

is then calculated

B

from

(10)

The Bo and Ko values were calculated from the curves in Figs. 2 and 3 and tabulated in Table 1. Permeability calculations were made utilizing formulas (9) and (10). Darcy Permeability values calculated from pressure decay data are of the same order of magnitude as the values calculated using direct measurement techniques. A closer correlation would probably exist if the assumptions of uniform pore size and circular crosssection were not used in the derivation of formula (9). Samples 4 and 5, made from an all-flour mix formulation, gave the best correlation as would be expected from the inherent uniform pore structure. Factors such as turbulent flow in samples 1 and 2 and errors in calculation from the graph for samples 7,s and 9 may contribute to the difference

0.23 x lO& 0.15 x10-6 0.14 x 10-s 0.13 x l&6 0.030 x 1w 0.018 x 1W

1.4 x lo-‘0

0.60 x lo-“’

0.43 x lo-‘0

0.35 x lo-‘0

0.42 x 1O-lo

0.084 x lo-‘@

4(E)

S(M)

6(M)

7(M)

8(E)

9(E)

calculated from B=g (Dv’+8M)m).

0.157

0.176

0.213

0.229

0.266

0.209 9.3 x 1w

F$

A+

FROM

2.3 x 1W

Bt

0.97 x IO-10

.23 x lo-“’

1 .o x lo-‘0

3.7 x lo-‘@

4.3 x lo-‘U

4.4 x l&‘@

(cm*)

Bt

PORE-DISTRIBUTION

0.00097

0.0023

0.010

0.037

0.043

0.044

Darcys

04

DATA

0.140 x lo-‘0

0.51 x lo-‘0

0.76 x 1O-1o

0.87 x 10-1s

1.1 x 10-1s

2.1 x lo-‘0

4.6 x lo-‘@

5.1 x lo-‘0

Permeability co&cient (cm’)

DATA

l*Z denotes direction of molding or extrusion. X and Y are perpendicular to each other and to Z.

fKpd= 10sB.

tpermeability

Porosity (Based onEobufkvol.)

M=Molded. E = Extruded. *Mean free path of N, at 20°C and 1 atm abs.

0.56 x lo-’

1.3 x l(r

0.83 x lO-’

1.8 x lo-& 1.0x 10-b

2.7 x lo-’

S(M)

9(E) 13(E)

4.3 x lo-’ 3.9 x 10-o

5.1 x lo-’

5.1 xl&’

11(M)

10(M)

4.6 x lo-‘

5.9 x lo-’

12(M)

Sample identity

M = Molded. E = Extruded. l =He gas. t =At 1 arm.

DECAY

Mean free path (cm)

A*

1. hESURE

PERMEABILITY

0.55 x 102

2.9 x lo-lo

TABLE 2.

0.59 x 104

3.3 x lo-‘0

l(M)

2(M)

Mean pore diameter Dv Dtt# (cm) (cm)

Ko Mole. flow term (cm)

Bo

Viscous flow term (cm’)

Sample

identification

?‘.WLE

0.0020

0.0060

o.op62

0.0072

0.010

0.018

0.043

0.041

0.0024

0.0019

0.0038

0.031

0.061

0.033

0.00066

0.00084

0.0068

0.031

0.068

0.043

0.0013

0.00084

0.0068

0.031

0.065

0.043

Experimental Darcy permeability X Z+* Y Darcys Darcys Darcys

0.0014

0.0051

0.0076

0.0087

0.011

0.020

0.046

0.051

Darcy permeability From press. Decay . Direct meas. (Darcys) (Darcys)

8

z!

:

ii

3

PERMEABILITY

RELATIONSHIPS

in values for the two methods of analysis. The hypothetical nature of the theoretical interpretation leaves much to the imagination in this particular analysis. 3.2 Vacuum decay Data from the vacuum decay permeability test are presented in Fig. 6 as a function of Darcy Permeability. A straight line drawn through the array of data points (vacuum decay and Darcy permeability at mean pressures of 0.5 and 1 atm abs. respectively) has a slope of 0.0243. Thus, the Darcy Permeability may be calculated from the vacuum decay permeability value by simply multiplying by a factor of 0.0243 which holds for the existing test conditions. Samples having a permeability lower than that which can be measured using standard steady-state methods may be tested using vacuum decay techniques providing a value of Darcy Permeability using the aforementioned conversion factor. 3.3 Pore size distribution In WIGGS’ analysis to correlate bility with Pore Size Distribution,

Darcy Permeathe expression

USING

DARCY

B=go

PERMEABILITY

(Dw2+8 Dm)(“) was utilized. An electrical

analogy viscous reduced tion of resultant

was used to derive separate equations for and molecular flow. These equations were to Maxwell’s formula under the assumpa material of uniform pore size. The equations : =o

and

s

E”D-DtndE

0.4

0.6

HOC ot I/,otm

I.2

otn..

1.6

2.0

2.4

cmZ/sec

FIG. 6. The relationship between Darcy permeability and helium diffusion coefficient at a mean pressure across the sample of 4 atm abs. These data are based on an array of 8 different grades of graphite samples. H

(11)

=.

(12)

_=,,, D+2D,n

where D=pore

diameter,

Dv=effective Dm=effective flow,

pore diameter for viscous flow, pore diameter for molecular

E=porosity accessible through ings larger than D, Ev=porosity accessible through ings larger than Dv, Em=porosity accessible through ings larger than Dm, Eo=total porosity,

pore openpore openpore open-

The -.‘-I I+22 cumulative pore size distribution curve was plotted with pore diameter on a logarithmic scale and fraction of bulk volume on a linear scale. Next, a transparent template (Z as a function of x) was constructed on the same scale by solving equation (12) to determine the mean pore diameter for molecular flow. The transparent template may now be traversed over the cumulative pore distribution curve until the area bounded by the curve, total porosity level (Eo), and Z for x=0 (D=Dm) is equal to the area bounded by the curve, zero porosity level, and Z for x=0 (D= Dm). The position of the point Dm on the horizontal (Pore Diameter) axis is the mean pore diameter for molecular flow. A similar analysis may be made to determine the mean pore diameter for viscous flow. In this case equation (11) is solved by setting were solved by defining a function

0

113

z*-1 -=xorZ= 1+22* Now, the position

x=

1+x I/* 1-2x ( > of the point Dv (for this new

W. A. ANDERSON

114

template) on the horizontal axis is the mean pore diameter for viscous flow. The cumulative pore size distribution curves for a number of different graphite samples were analyzed in this manner to determine Dv and L&n the results of which are shown in Table 2. Porosimeter data take into account the largest throat leading into a pore whereas Darcy Permeability measurements are influenced by the average throat dieter. This could account for the generally higher values of permeability calculated from pore distribution spectra. It is evident from this analysis that pore distribution data may be used to determine the order of magnitude of the Darcy Permeability especially where a sample having a uniform pore structure is under evaluation. Again, the hypothetical nature of the theoretical relationships reduces the possibility of practical application of this theory. 4. CONCLUSIONS

It has been experimentally confirmed that the Darcy Pe~eab~~ value (Kd) may be calculated from the Helium Diiiusion Coefficient (KvJ) vacuum decay value using the expression : Kd=Q.O243

Ku

The Darcy Permeability value is also linearly related to the Viicous Flow Coefficient term and

the Molecular Flow term from the pressure decay data. This Darcy value may be calculated using data from the pressure decay analysis. It has also been confirmed that a relationship exists between Darcy Permeability and the Pore Distribution Spectra in that the Darcy Permeability value may be approximated through calculations using the mean diameter for molecular flow and the mean diameter for viscous ffow. These relations~ps have been shown to exist for both extruded and molded graphites having a broad range of permeability values. author is indebted to the Great Lakes Carbon Corporation for permission to reksase this paper and to D. M. RYMER for his work in performing the tests.

Acknowledgements-The

RBFERENCBS SCHEIDEGGER A. E., The Physics of Flow Through a Porous Media. Macmillan, New York (1960). Wroos P. K. C., The relation between gas permeability and pore size distribution in consolidated bodies, Proceed&s of the Conference MI Industrial Carbon and Gmpkite, p. 252. Sot. Chem. Ind., London (1958). J. M., iv+ONGSTAFP B. and WAXNER R. K., flow of gases through a fine-pore graphite,

HUTCHEON

The

Proceedings of tke Conference on Imh&rial Carbon and Graphite, p. 259. Sot. Chem. Ind., London (1958). Handbook of Chmtirtry and Physics. Chemical Rubber Publishing Company,

Cleveland (1958).