Transient Gas Flow in a Porous Column Frank A. Morrison, Jr.l Lawrence Livermore Laboratory, University of California, Livermore, Calif. 94560
Transient flow of an ideal gas in a porous cylinder is investigated analytically. The results may b e applied to the flow of gas in the backfill following an underground nuclear explosion. Two models of gas flow are considered, isothermal flow and adiabatic flow. The isothermal model is the more realistic for a flow in the long column above a nuclear detonation, but the adiabatic model serves to bound the effects caused by heat transfer. Calculations are made for a column of infinite length, a finite open column, and a finite plugged column. The calculations for the infinite column are valid for flow in a finite column for a well defined period of time, significant compared to the duration time of high cavity pressure. Pressure distributions and mass flow rates are obtained.
T h e purpose of this paper is to present the results of a n analytical study of unsteady gas flow in porous media. This work resulted from an interest in the transient behavior of gases entering the stemming material during a n underground nuclear test. Pressure distributions and mass flow rates have been found as a function of time. These results permit calculation of the feasibility of venting through the stemming material. Further calculations were done to predict the transient pressure distribution when gas a t high pressure is suddenly introduced a t one end of a closed cylinder containing porous material. This flow corresponds to flow in backfill containing a plug and may be used to evaluate the effectiveness of the plug and t o determine the strength necessary for such a plug. Flow in a column of infinite length is also treated. The results may be applied to a finite column for a period of time that may be accurately estimated. Simple expressions are obtained for the mass flow into the column and for the position of the interface between the cavity gas and the gas originally in the column. Formulation of the Problem
I n formulating a model for the gas flow in the porous medium, it is desired that the model be sufficiently detailed t h a t it will predict the salient features of the flow, yet be sufficiently simple t h a t the results will have a broad range of application. T h e basic relations governing the flow of gas in porous media are the continuity equation, the momentum equation, the equation of state, and the first law of thermodynamics. The continuity equation for the unsteady one-dimensional flow of a compressible fluid in a porous media is easily obtained by performing a mass balance on an infinitesimal element. The relation reduces t o
b dX
(pu)
+ e btb-P = 0
where p is the gas density, u is the “apparent” speed (= volume flow rate/unit area), B is the porosity, t is the time, and x is the position coordinate in the direction of flow. Present address, Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, Ill. 61801.
The momentum equation mill be replaced by a constitutive equation, Darcy’s law
where k is the permeability of the medium, p is the fluid viscosity, and p is the fluid pressure. Darcy’s law is valid for a low Reynolds number flow. A number of empirical extensions for higher velocity flows have been proposed. Because of the simpler formulation resulting from the use of Darcy’s law, the absence of good data in estimating coefficients for higher order approximations, and the agreement between experiment and calculations based on Darcy’s law, Darcy’s law will be used here. The permeability and viscosity ail1 be assumed constant. The gas is assumed to obey the ideal gas equation of state
p
=
pRT
(3)
where T is the absolute temperature and R is the gas constant. The first law of thermodynamics (energy equation), as applied t o the one-dimensional flow, may also be obtained by consideration of an infinitesimal element. For an ideal gas, this reduces to Qe =
b b €C” - (PT) CP & . bt
+
(PUT)
(4)
q is the rate of heat transfer per unit volume into the gas. I n the transient flow, the dominant heat transfer to the gas is the local transfer of heat from the solid rather than spatially directed heat transfer through the bed. S o effort \ d l be made to distinguish among the mechanisms for this heat transfer or t o propose phenomenological relations goveriiing it as this is unnecessary for the models considered here. C , and C p are the specific heats at constant volume and constant pressure of the gas, respectively. They have been assumed constant. Any changes in kinetic energy are neglected. This is consistent with the degree of approximation used in the momentum equation. In general, one \Tould also wish t o include a n energy equation for the porous medium and relations governing the heat transfer. Rather than attempt a complete solution, including a model for heat transfer to the gas, two simpler models will be analyzed. These correspond t o the limits of very rapid heat transfer or no heat transfer. The first model is clearly Ind. Eng. Chem. Fundam.,
Vol. 11, No. 2, 1972
191
more realistic for the flow of interest. The second model, however, yields remarkably similar results for the pressure as a function of time and position and may be more realistic in other cases. Considered together, the results of the two analyses bound the actual pressure distribution quite closely.
decrease at the end (Kidder, 1957) and a slight pressure change a t the end (Jain, 1967). Before proceeding to the similar solution, it is desirable to write eq 5 in dimensionless form. Dimensionless pressure P is defined
P = - -P- - Po
Isothermal Flow
(5) This relation is well known (Muskat, 1946; Scheidegger, 1960). One-dimensional transient isothermal flow governed by this equation has been studied by a number of people. A paper by Xronofsky and Jenkins (1952) is particularly noteworthy. They treat many of the problems considered here for a step application of the pressure a t one end of the cylinder. They did not, however, treat cases of very high pressure ratio (applied pressure/ambient pressure) which interest us here. The character of the solution changes significantly due t o the nonlinearity of the governing equation. Use of a different time scale becomes necessary. ,4 discontinuity appears in the pressure gradient. I n order that the nature of the transient flow in a cylinder of porous material should be clearly seen, the first case to be considered will be flow in a cylinder of infinite extent. The case of primary interest is that of a step application of a constant high pressure a t the end. The resulting solution will be valid for flow in a finite cylinder for short times. The time interval for which the solution is valid for a finite cylinder can be very accurately estimated. This time is significant compared to the duration of high pressure in the cavity following a nuclear explosion. The great advantage in considering the flow in a n infinite cylinder is that a similar solution exists. The gas pressure may be written as a function of a single variable, rather than both time and position variables. Xronofsky and Jenkins (1952) present their results in a manner that recognizes this fact, although they apparently did not take advantage of this simplification in their numerical solution. Analytical approximations of the solution for the similar flow have been presented for the cases of a sudden pressure 192 Ind. Eng. Chem. Fundam., Vol. 11, No. 2, 1972
- Po
PI
Consider a cylinder of porous material of uniform properties initially filled with air a t ambient temperature and pressure. At one end of t h e cylinder, a pressure is applied as a function of time. The other end may be either open to the atmosphere or sealed. Both cases will be treated. If the heat transfer rate is sufficiently large and the heat capacity of the porous medium sufficiently large, the flow will be essentially isothermal over the major portion of the cylinder. The response time for heat transfer between gas and solid is much shorter than the response time for gas flow in a long column. The shorter response time may be safely neglected and the gas and solid temperatures assumed equal. Furthermore, our interest is confined t o times less than the time necessary for a mass of gas equal to the original pore mass to leave the column. The heat capacity of the solid consequently iyill greatly exceed the heat capacity of entering gas. The column temperature will not be significantly affected by the entering gas temperature. The flow is then governed by eq 1, 2, and 3 with heat transfer as necessary to satisfy eq 4 with a constant temperature. Holding the temperature constant and eliminating the apparent speed and density by substituting eq 2 and 3 into eq 1, an equation with pressure as the sole dependent variable results.
where po is the initial gas pressure in the porous media and p , is the applied pressure a t the entrance. A dimensionless position X is defined
X = -
2
L
where L is the length of the cylinder. For flow in the infinite cylinder, the quantity L will cancel out and present no difficulty. A dimensionless time 7 is defined k(p1
- P0)t
€/.LL2 This dimensionless time differs from that of hronofsky and Jenkins (1952), who used k p , t / ~ p P .Use of the scale employed here serves to minimize the effect of the pressure ratio in presenting results for high pressure ratios. I n terms of this pressure ratio A;
N E PI -
Po
the governing equation becomes
bP 1 b2 37 - - - --( P2 23 + x 2& ) Similar Solution
For the flow in an infinite cylinder, due to a step change in the pressure a t the end, the governing differential equation reduces to an ordinary differential equation with the independent variable
The equation becomes (7) The appropriate boundary conditions are
P=latO=O
P+OasO-.
(8)
m
Equation 7 \vas solved numerically on a high-speed digital computer using a Runge-Kutta technique. The results are shown in Figure 1. Note that use of the similarity variable 8, as defined here, serves to compress the results into a narrow band, the effect of the pressure ratio becoming less important as the pressure ratio increases. S o t e also the discontinuity in the pressure gradient for the case of infinite pressure ratio. The pressure steadily decreases, going t o zero abruptly a t 8 equal to 0.81. The pressure here has wavelike behavior. no effect of the applied pressure occurring for large values of 0. It is this behavior that permits an accurate calculation of the time for which the similar solution holds for the finite cylinder.
For a n infinite pressure ratio, eq 6 and, consequently, eq 7 become identical with the equation governing nonlinear diff usion with the diffusivity proportional to the concentration. This case was studied b y Wagner (1950), who first obtained this result. T h e numerical solutions were generated b y obtaining successively better approximations for d P l d 8 a t e equal t o zero until a solution was obtained satisfying the boundary condition a t infinity. For the infinite pressure ratio, the initial derivative is approximately -0.8875. The nature of the discontinuity in the first derivative may be seen more clearly b y examining the behavior of eq 7 for small values of P . For any finite value of S and for sufficiently small P, the term involving the square of the pressure may be neglected and eq 7Yeduces to
_1_ d2P _ +,20- d P
N
- I de2
do
=
8
Figure 1.
Pressure distribution in infinite column
Expanding the derivatives in eq 7 , we see that the position of the interface also corresponds t o the position where
0
dZP -=o do
This is a well known linear differential equation. The same equation governs transient heat conduction, or diffusion with a constant diffusivity, corresponding t o a step change in t h e dependent Yariable a t the surface of a semiinfinite region. The solution is
Similarly, the mass flow into the infinite column may be determined as a function of time. The mass that has entered the column is ncc
m
erfc is the complementar> error function. The position of the interface betweeii the gas that has entered the infinite column and the gas initially in the column may be determined. ;it any time, the amount of gas lying between the interface and infinity exceeds the mass originally in that portio11 of the column by the mass of gas originally lying between the position of the interface and the origin
s,'
(p
- po)EAds
= poeA~
PdO
=
G0 T
but
or
where limo,m PO = 0 has been used. This is a stronger condition than limoie P = 0, the boundary condition applied to the differential equation for P. It is satisfied, nonetheless, as eq 9 shows. Replacing the integrand on the right-hand side using eq 7 and then integrating
Combining eq 11 aiid 12 and rearranging, we find that the interface is located a t the position where
aP -
de
+2e=0
Jn
(p
- po)eAdz
(14)
Again, the density is proportional to the pressure. Using the definitions of P and 0
The integral is given in eq 12. The dimensionless pressure is unity when 0 vanishes. Consequently, using the definition of 7 , there results
(10)
A is the area of the column normal t o the direction of floa. Using the fact that the density is proportional t o the pressure 111 the isothermal flow and using the definitions of P a i d 8,this relation becomes
1'
=
(13)
The derivative dPld8, evaluated at e equal to zero, is known, because this is the initial condition that must be determined in order that the numerical solution of eq 7 satisfy the boundary condition as 0 goes to infinity. The solution, thus far, is valid for the pressure distribution in an infinite cylinder resulting from a sudden application of a constant pressure at the end. This solution is also valid for the flow in a finite cylinder for short times, being exact in the case of infinite pressure ratio. We wish now to examine the f l o ~in a finite cylinder open to the atmosphere a t one end and subject to a sudden pressure increase a t the other end. The final steady flow solution will first be examined as the resulting pressure distribution restricts the choice of numerical techniques that may be applied to the transient problem. Steady Flow in a Finite Column
In the steady state, the pressure distribution is given by eq 5 or 6 with the temporal derivative equal to zero. K i t h the boundary conditions P
=
1a t X = 0
P
=
OatX
=
(16)
1
the pressure distribution is
P =
d N Z - (NZ - 1)X - 1 N-1
Ind. Eng. Chem. Fundam., Vol. 11, No. 2, 1972
(17) 193
Transient Flow in a Finite Column
The unsteady flow in a finite column due to a sudden constant pressure applied a t X equal t o 0 and open to the atmosphere a t X equal to 1 is given by eq 6 with boundary conditions 16 and the initial condition
P = Oatr = O
The adequacy of the constant source pressure boundary condition for actual tests is discussed below. For short times, the solution is given b y the similar solution obtained for the infinite column. For long times, the solution is given by the steady-state solution. I n general, the flow is governed by the partial differential equation, eq 6, with two independent variables. An explicit self-starting program was written to solve this problem. Fourth-order finite difference approximations were used for derivatives in the X direction. The results of the calculations for a pressure ratio of 45 are shown in Figure 2. Note the propagation of the pressure distribution in agreement with the similar solutions. The pressure ratio is chosen to agree with the pressure ratio measured by Olsen (1967). The mass flow rate at the surface, X equal to 1, and the total efflux a t the surface were calculated for several pressure ratios. The dimensionless mass flow rate is determined from eq 2 and 18 and the definitions. At the surface
X
Figure 2.
Pressure distribution in open column
d_ Mdr
r
Figure 3.
Mass flow rate out of column
+
dm - -- PUA dt
2
M=--
-
2
m
+)
dX
'sr" odX
2P (P2+ -)dr
N-1
The results of these calculations are shown in Figures 3 and 4. The dotted line in Figure 4 corresponds t o the time at which all of the gas originally in the column has exited. h'ote how rapidly the steady-state flow rate is reached once significant outflow commences. In order to determine the actual times following a n underground nuclear explosion corresponding to the dimensionless times used here, let us consider a fictitious event based upon Olsen's (1967) measurement for a n actual test. The measured pressure ratio for the event described by Olsen is approximately 45. The depth of burst was nearly 240 m and the resulting cavity radius was 28.5 m. The path length to the surface is then about 210 m. For the sake of example, let us choose a gas viscosity of 1O-IO lbr-hr/ft2 and a sand backfill of porosity one-third and a permeability of 20 darcies. For a pressure ratio in this range, the time for which the similar solution is valid is the time corresponding approximately to 0 equal to 0.81 and X equal t o 1. From the definition of 0
, 1
and is independent of position in the steady state. A dimensionless mass is defined as
r =
~
4(0.81)2
=
0.38
From the definition of r and assuming that the backfill communicates with the cavity, the similar solution is valid for
mo
where 1710 is the initial mass of air contained in the column. From eq 2 and 17 and the definitions, the dimensionless mass flow rate is dM dr
-N - l d- ( p * +
The total efflux is
Because of the behavior of this solution near X equal to 1, the solution of transient flow problems by applying certain standard techniques to eq 5 or 6 could result in significant error. Such difficulties could be encountered in programs designed to solve problems of heat transfer with variable conductivity or diffusion with variable diffusivity. Any finite difference approximation to the derivative of a function involves neglecting terms of order (Ax)"-' d"f/dx" and higher in a Taylor series expansion of the function. Az is the step size. At X equal to 1, the n t h derivative of P is, from eq 17, of order NZn-l. Consequently, the error resulting from a finite difference approximation of d P / d X a t X equal to 1 is of the order (AX)n-1N2n-1.I n order that this error can be reduced within reasonable limits, AX must be smaller than N - 2 . As values of N exceeding 40 are common in nuclear tests, this represents a serious restriction. The difficulty is avoided by treating P 2 2 P / ( N - 1) as the dependent variable for all differentiation in the X direction. This is a well behaved function in the steady state. The mass flow rate of gas is given b y
M=-
(20)
-N+l 2
194 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 2, 1972
(19)
in this case. This time is significant compared to the duration time of the high cavity pressure, which was about 10 min in the event described b y Olsen. The effect of using different stemming
materials may be readily determined from their porosities and permeabilities. The adequacy of the constant pressure boundary condition in any case can be determined by comparing the calculated time to the duration time of high pressure. The calculated time will depend upon the choice of the medium, the length of the path, and the applied pressure. Olsen's measurements &howthat the carity pressure decays slowly until the cavity collapses and then falls off suddenly. X time varying cavity pressure is easily included in the finite column analysis. The similar solution is valid only for a constant cavity pressure. r
Transient Flow in a Sealed Column
Figure 4.
On occasion, a solid plug is placed in the stemming material. The flow in this case is analyzed t o determine the pressure on the plug as a function of time. Considering a flow given by eq 6, the boundary conditions appropriate t o a step change in the pressure a t one end of the column are
P
=
1atX
=
dP 0; - = O a t X = 1 dX
Total mass out of column
(21)
The second boundary condition corresponds to no mass flow through the plug. The initial condition is given by eq 20. The program used t o solve the partial differential equation in the previous section was used here with the appropriate boundary conditions. A typical result is shown in Figure 5. The pressure acting on the sealed end is shown in Figure 6 for several pressure ratios.
P
0
Adiabatic Flow
0.2
0.4
0.6
0.8
1.0
X
Figure 5. Pressure distribution in sealed column
In order to obtain a second bound on the actual pressure distribution in the porous column, let us consider a transient adiabat'ic gas flow. Muskat, (1946), among others, describes an ideal gas flow, with p p - Y constant, as an adiabatic flow. Such a flow is isentropic, but, as any gas flow in porous media is irreversible, it cannot be bot'h isentropic and adiabatic. Describing such a flow as adiabatic is not correct' as it would require a most fortuitous distribution of heat transfer. Rat'her, the adiabatic flow is described b y eq 1, 2, and 3 as before and eq 4 with no heat transfer. I n the steady st'ate, the f l o is ~ again isothermal, being a Joule-Thomson flow. I n the t,ransient' f l o ~ ,the pressure distribution is determined by substituting eq 2 and 3 into the first lan-, eq 4.Eliminating p , 26, and T and setting 0 equal t o 0, the pressure is given by T
Figure 6.
Pressure at end of sealed column
y is the ratio of specific heats. Note that, aside from the factor
y,eq 22 is identical n i t h eq 5 .
It follon s, therefore, that all of the pressure distributions calculated for the isothermal floa apply also t o the adiabatic flow if we merelj replace T by T ~ equal , to 77, and a by as, equal to a K i t h this change, Figures 1, 2, and 5 apply equally Jvell to the adiabatic flon . The only change is that all predicted actual times are reduced b y the factor y. The two model., ibothermal flow and adiabatic flow, therefore, bound the actual pressure diitribution in the finite column rather closely. Because the actual temperature a t the exit of the finite open column will be essentially ambient in the actual case of flow through the backfill above a nuclear test, the mass flow rate and the total efflux are similarly bounded. For the sake of completeness, the temperature distribution in the similar flon will also be determined. It is not suggested that the resulting solution nil1 resemble the actual case with
&.
heat transfer, except for exceptionally short times or for shorter columns than t'hose that prompted this investigation. The result is of interest, however, because it demonstrates that the temperature distribution for a compressible flow in porous media, like the pressure distribution, can be markedly different from the distribut,ion for an incompressible flow. It further indicates a possible source of instability in the numerical solution of problems of transient gas f l o in ~ porous media lvith or lvit,hout heat t,ransfer. Define a dimensionless t,emperature
T T z -
To
where To is the ambient temperature. Substituting eq 2 and 3 into eq 1 and using eq 22, the temperature distribution may be found in terms of the pressure distribution. Ind. Eng. Chem. Fundam., Vol. 11,
No. 2, 1972
195
But the integral is identically given by
I\
Substituting for the integrand on the right-hand side using eq 26 and then integrating
Larn(p
0.2
0
0.4
0.8
0.6
8,
Figure 7.
=
1 27
dP do,
-o,(p - 1) - - p -
(28)
Combining eq 27 and 28, we find that the position of the interface corresponds to the position where
1.2
1.0
l)do,
dP do,
Temperature distribution in infinite column
+ 270, = 0
and
p = o
(23) The temperature distribution in the similar flow is found by substituting the adiabatic equivalent of eq 6 into eq 23 and changing variables 2(7
dT do, - (P
+
dP do,
The density vanishes at the int,erface. The vanishing density and infinite t,emperature at the interface result from the assumed step application of high pressure. The local highpressure gradient produces high gas velocity and a high dissipation rate causing the increase in gas temperahre. -1calculation similar to t'hat' used to locate the interface yields the mass added to the porous column; the mass added to the infinite column is given by eq 14. The integral is evaluated using eq 28, yielding
- 1)T& -
&)(E
+2
(24) 4
Equivalently, we may define a dimensionless density
p E -P PO
Substituting eq 2 into eq 1, the density distribution may be found in terms of the pressure distribution.
(P
Y bp =
where pl is the density of the entering gas. Either the t'eniperature distribution or the density distribution may now be determined. The distribution of one of t'hese variables combined with the pressure distribution and the equation of state yields the other. JYe choose to determine the temperature distribution. The first-order differential equation governing the temperature distribution, eq 24, has two boundary conditions to satisfy.
g)
T
=
TI a t @ ,= O TO
-
T-1 For the similar flow, the density distribution is given by
d 2P dij
-p
de,2
For both the density and the temperature, a singularity in a t least the first derivative exists in the similar flow a t the point where dP do,
+ 270%= 0
as a n inspection of eq 24 and 26 shows. I t may be shown that this point corresponds to the position of the interface between the gas originally in t,he column and the gas that has entered the column. The argument' is similar to that used for isothermal flow. The position of the interface is given by eq 10. I n dimensionless form and changing variables
196 Ind.
Eng.
Chem. Fundam., Vol. 1 1 , No. 2, 1972
asO,-.
(29)
03
The equation can satisfy both boundary conditions because the flow consists of tlvo distinct open domains separated by t'he interface. I n the adiabatic approximation, there is no thermal communication behveen these domains. The behavior of the solution in the neighborhood of the int,erface is best seen by examining t'he solution of eq 24 in this neighborhood. Integrat,ing between t,wo points A and B sufficiently close t'o the interface In
(E)
2(y =
-
Pi +
l)&i
1
(0:i
0
i
- 02) - 0,"
r1
(30)
where the superscript i indicates the interface. If h and B lie on the same side of the interface, this relation may be used t o relate them. If they lie on opposite sides of the interface, one may not correspond to a physical t'ernperature. If T-$ is real and if the interface lies between X and B, then eq 30 iniplies that TB does not correspond to the real teniperature a t position B. The solution of eq 24 has two branches. Each branch is real on one side of the interface and satisfies the boundary condition on that side of the interface.
tion has been found for flow in a n infinit’e column, fiiiit,e open column, and a finite closed colurnn. Mass flow rates Ivere obtained for the flow in infinite and finite open colunins. The time for the effects of such gas flow to appear at the surface above a nuclear test is det’ermined. By usiug backfill of sufficiently small permeability, this time may be made quite long compared to the duration of high pressure after detonation of the device. Two models were used to describe t’he flow. These were adiabatic and isothermal flow. The isothermal flow corresponds more nearly to reality. The adiabatic model serves to bound any deviations of the act’ual pressure distribution from that predicted by t,he isothermal model.
lo3
-T
Nomenclature
A
cross-sectional area of porous column specific heat a t constant pressure specific heat at’ constant volume permeability length of porous column dimensionless mass, m / m o mass of gas mass of gas originally in column pressure ratio, p,/po P = dimensionless pressure, ( p - p o ) / ( p l - PO) p = pressure p , = initial pressure of gas in column p l = applied pressure y = rate of heat’ transfer per unit volume to gas R = ideal gas constant T = temperature To = initial temperature of gas in column Tl = temperature of gas entering the column T = dimensionless temperature, TITO T = dimensionless temperature; T / T o for gas initially in column, T , T 1 for gas entering column t = time u = apparent speed X = dimensionless position, s / L z = position coordinate in direction of flow
= = C, = k = L = 111 = m = mo = .\- =
C,
I
lo
1
0
Figure 8.
02
0.4
06
C
Temperature distribution in infinite column
The numerical solution of eq 24 then consists of finding two solutions, each valid for one side of the interface and sat,isfyiiig the boundary coiidit’ioii on that side. These solutions are best writ,ten in terms of a dimensionless t,emperature where
T
= - for e, < e,’
T TI
T
E -
T for To
e, > eai
Xunierical solutions were obtained using a Runge-Kutta method on the high-speed digital computer. These are shown for selected values of the pressure ratio in Figures 7 and 8. In a n actual flow with heat transfer, t,he temperature distribution will resemble the adiabatic solution for only very short times. I n addition. changes in kinetic energy, neglected here, would be significant, as the pressure rapidly changes a t the boundary of the column. Certain features of this solution will be retained, however. One would expect to see a high-temperature pulse traveling down the coluniri gradually spreading and dissipat’ing. I n addition, the solutions shown here for the isothermal and adiabatic f l o w demonstrate that, although the heat t’ransfer mechanisms will significantly affect the temperature distribution, they have remarkably litt,le effect on the pressure distribution. Conclusions
Unsteady ga? flow in a porous column, resulting from the sudden application of a constant high pressure a t one end, has been studied analytically. The resulting pressure distribu-
GREEKLETTERS y = ratio of specific heats, C,/C, e = porosity B = similarity variable for isothermal flow, Had/, e, = similarity variable for adiabatic flow, 9:2d< u = viscosity p = density“ PO = density of gas initially in porous column i j = dimensionless density, p po 7 = dinienslonle- time for isothermal flow k ( p l - p o ) t e p L 2 T, = dimensionless time for adiabatic flov y k ( p l - p o ) t / e p L 2 literature Cited
Aronofsky, J. S., Jenkinq, Aronofaky, Jenkins, It., Proc. 1st L-5‘.
.\-at. Congr’. d p p l . J l e c h . , 763 (1952). Jain, SS.. K., Indian J . Pitre A p p l . P h p . 5 , 608 (1967). Kidder, H . E.. Kidder. E., J . A A pmp l . J/wh. 24. 24, 329 119.57). (19.57). I I u s k a ft ,, AI AI.,, “The Flow of Homogeneous Hon1ogeneous Fliiidi Fliiids through Porou,q Inc., Ann Arbor. llich llich.,, 1046. Media,” J. K,Edwards, Inc Olsen, C. X., J . G ~ o p h z l sRes. . 72, 5087 Geophys. ;087 (1967). Scheidegger, il. E., “The Physics of Flow through Porous Media,” llacmillan. Media.” llacmillan, X e w York. York, N . Y., 1930. 1950. Wagner, C., J . Chem. Phys. 18, 1227 (1950) (1950). ~
RECEIVED for review January 28, 1971 ACCI:ITI:DNovember 1.5, 1071
This work n-as performed under the auspices of the .!I F.Atomic Energy Commission.
Ind. Eng. Chem. Fundom., Vol. 1 1 , No. 2, 1972
197
