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Flow of gases through channels with reference to porous materials
Flow of gases through channels with reference to porous materials J D Mellor, CSIRO Division D A Love&
of Food Preservation,
Ryde, N.S.W. Australia
CSIRO Meat Research Laboratory, Cannon Hill, Qld., Australia
Knowledge of the gas flow in pumping lines, in internal channels, and in porous materials undergoing vacuum processes is basic to the design and operation of a vacuum system. The gas flow equations for viscous and molecular flow in cylindrical channels are ideally simple, in slip flow they are not so easy, but in porous materials in all the flow regions they are more complicated. The permeability of a porous material has been obtained directly by measuring with a flowmeter the amount of gas flowing through it under an absolute pressure gradient, and the same has been done with model materials. Correlation of the permeability and mean absolute pressure provides basic information on structural factors of the material such as porosity, tortuosity factor, and equivalent pore diameter which could, for example, be applied to vacuum and freeze-drying processes.
1. Introduction
The flow of gases in vacuum systems and in porous materials undergoing vacuum processes may be divided into three regions, which can be specified by limiting values of Knudsen’s number l/d, where 3,is the mean free path, and d is the characteristic channel dimension. Thus, if I/d < 0.01, the flow is viscous, if L/d > 1, the flow is molecular, and if 0.01 < I/d < 1, the flow is slip. F At all pressures the flow of gas is impeded by the channel in which it flows. The channel conductance F is defined as the rate of gas flow Q (micron I/set) in a particular channel per unit pressure difference AP (micron) between two points. The viscous-flow conductance F, and molecular-flow conductance F,,, are readily obtained from kinetic theory for channels, having a simple geometry. In such cases F, varies as the mean absolute pressure P, and F,,, varies independently of P with a single value at a pressure of zero as shown in Figure 1. The derivation of slip-flow conductance is difficult. Generally, an equation for the viscous conductance corrected for slip is derived, namely :
P
+3z 2-*F
F=F
”
16’
where f=fraction
f
Figure 1. Conductance-pressure
m’ of molecules colliding
produce diffuse scattering,;6
=coefficient,
with the walls that assuming
rough
walls. Equation 1 does not predict the pressure dependence that has been observed; it is only a linear line for viscous flow displaced upwards by an amount Lpm
as
equal to the slip term g
shown in Figure 1. Therefore, it is only applicable
from high pressures down to intermediate
pressures.
1 may be generalized
Vacuum/volume
Is/number
12.
for long straight
(Barrer 1963)‘, such as those in Table 1, by replacing the tube diameter d by the hydraulic diameter dh area of cross-section / \ =2x circumference of cross-section \ 1 and by‘introducing two shape factors kb and k,, and becomes
F=A-ddL+~. llqk,
2. Gasflow in Channels Equation
relationships.
channels
Pergamon Press Ltdlfrinted
16
2-,f j-‘3k,
4,fi
1 ’
where A =area of cross-section, l=tube and v =mean molecular velocity. in Great Britain
(2) length, r]==gas viscosity,
625
J. D. Me//or
and D. A. Lovett:
Rectangle:
breadth
-2
Flow of gases
width
3 I .05
through
channels
0.87
The conductance of a channel consisting of a combination of such cross-sections could be treated as a set of circuit elements each with simple conductances, however, there arc dificulties in having to determine experimental parameters for the equations. The molecular-flow conductances of channels having a more complicated geometry, such as orifices and short tubes, has been given by Dushman (I 962)y in which he uses Clausing’s factor, defined by the ratio of the net volume flow rate through the cross-section to the volume flow rate of gas striking an orifice of the same cross-section. Levenson et al (I961 ).’ gives a similar method for more complicated channel systems.
with
reference
to porous
materials
porous material, one in each arm between the manometer and the tube. It is used by running the pump and adjusting the needle valve to cause the soap film to move the gas over a measured volume in a given time. Both the vacuum gauge and oil manometer, after compressing the gas in each arm by the McLeod gauge, are then read. The permeability coefficient K for different mean pressures I’ is then calculated. When K is plotted against P a transition flow curve is obtained which approaches asymptotically a viscous-slip flow straight line in the medium pressure range. The slope and the asymptote intercept on the K-axis yield the equivalent pore radius r and the ratio: porosity c to the tortuosity factor (L,/L)” of the porous material. These quantities can be deduced by developing an equation for the asymptote in terms of an equivalent cylindrical model for the porous material along similar lines to those proposed by Barrer (1963)’ to give:
3. Porous materials Transition flow behaviour was INI recognised before Knudsen (1909)6 observed a minimum in the flow rate for a gas at low pressures flowing in a long straight capillary. Knudsen considered the minimum to be due to the transition from viscous to molecular flow and not simply to viscous flow with slip, and he proposed an empirical equation to account for it by combining the viscous and molecular flow equations and an impedance function of the pressure, temperature, gas viscosity, and the diameter of the capillary. The equation (XC Figure I ) is: F = F,, + ZF,,,. where Z
1
(3)
intercept
i)
2’
= 3/c’ +
where I< = 2.5, 1~’= I ci, = 37~1I6 (rough
pores)
or (i, = n/4 (smooth
L,, = actual flow path, L = sample
pores)
thickness
2SO?tl/i.
is the siniplesl expression. I 3.095(//1. This transition flow equation is commonly used today. After Knudsen’s discovery, minima were observed for flow through long narrow slits and. later, for some porous materials (Barrel 1963)‘. Various theories for transition flow with low pressure minima have been published; one theory by Pollard and Present (I Y48)“, offers a rigorous solution to the problem. In connection with the development of drying rate equations for vapour transfer for a new freeze-drying process in which the vacuum pressure is cycled in a prescribed manner (Mellor, 1967)“, the permeability of a gas through a number of freezedried products was measured at IOLI pressures, and found to have a minimum. In some products, however, practically no minimum in the permeability-mean pressure curves occurs, and in a number of others, the curves are concave downward at low pressures. Minimum permeability curves have also been found for a model porous material, and for a sintered glass which was one of few materials previously reported to give a curve concave downward at low pressures. The permeability apparatus (Mellor, 1966): consists of a glass tube with a restriction in the middle for holding a plug of porous material of known thickness and cross-sectional area. A needle valve, and a sensitive flowmeter comprising a soap film moving in a volumetric tube connected to a gas supply at atmospheric pressure are connected in series at one end of this tube. A vacuum gauge and two-stage vacuum pump are connected to the other end, and a differential oil manometer is connected across theglass tube. The sensitivity of the manometer has been increased by interposing two identical McLeod gauges between the manometer and the glass tube holding the
626
(4)
11 =gas ~ =
viscosity
8RT
1
i TM 1 By transposing equations 4 and 5 the equivalent can therefore be found from:
166,%/k I’ =_31;’ And the ratio c/
pore radius
siope intercept’ can be obtained
from either the slope or
the intercept. The porosity c is determined directly (Mellor, 1966)’ by enclosing a sample of the porous material in a glass vessel in which the volume and pressure can be altered and measured simultaneously by movement of a mercury piston. According to Boyle’s law c can be calculated, and the tortuosity factor (&,/I!,) obtained independently of t, either from the slope or intercept equations. Permeability-pressure curves for a bundle of 61 wires 0.06 cm dia by 1.19 cm long, sintered glass (grade I), and freezedried beef and prawn are shown in Figure 2. The minima occur in the range 0.2 to 0.4 torr, and the calculated equivalent pore radii for the wire and beef are approximately equal to half the mean free path at the minimum. This can be expected from kinetic theory. It follows that the asymptotes derived from the latter curves, involve molecular collisions between successive wall collisions, whereas under the conditions where the capillary
J. D. Mel/or and D. A. Lovetf:
Flow of gases through
channels
materials
,
I
0
with reference to porous
I
2
4
3 P,
Figure 2. Permeability-pressure
5
6
7
6
torr
plots showing minima.
diameter equals the mean free path, a considerable number of molecules do not behave in this way. Instead, if the fraction
of the molecules that, on the average, do collide with other molecules between two wall collisions in a viscous-slip flow term and the fraction that do not collide in an additional molecular flow term are considered, an equation, similar to one by Pollard and Present (194Q6, can be derived from kinetic theory for the permeability K in terms of the mean free path 1, giving:
Whole egg IOOcarrot
I
0
I
I
I
2
3
4 p.
Figure 3. Permeability-pressure
where K’=Knudsen’s permeability coefficient. Equation 7 with the structural constant k ~2.5 has been found to give a better fit to the experimental results than Knudsen’s equation 3 for transition flow with a minimum, eg data for freeze-dried beef can be fitted to within a few per cent. Other types of transition flow are represented by the permeability-pressure curves shown in Figure 3. The curves for freeze-dried carrot and strawberry have no curvature down to the limit of pressure measurement (0.005 torr) in the permeability apparatus. Pore size for the two products is about 60 ,U radius. The minimum condition where the capillary diameter equals the mean free path is no criterion on which to judge the anomalous transition flow behaviour in these products. Neither is it in the case of the curves for freeze-dried tomato concentrate and whole egg having calculated pore radii of about 200 p; these curves are concave downward at low pressures. Pore size in the latter products has been found to be highly dependent on rate of freezing before freeze drying. Further permeability measurements are in progress and the kinetic theory is being extended to transition flow behaviour where permeability-pressure curves have a concave downward trend. In addition, a search is being undertaken for a suitable porous material exhibiting this phenomena. It may also be possible to construct a model porous material for this case. Data on the permeability of freeze-dried materials has been found to be very useful for calculating vacuum drying and freeze-drying rates as shown by Mellor (1966)’ where mass transfer and not heat transfer is the controlling mechanism.
Strawberry
*
5
6
7
6
torr
plots showing linear and concave
downward curves.
References
’ R M Barrer, App Mat Res, 2, 1963, 129. B S Dushman, “Scientific Foundations of Vacuum Technique”, J M Lafferty (editor), (John Wiley & Sons, NY). 3 L L Levenson, N Milleron and D H Davis, Tram A VS 7rh Nut Vat Congr, 1961,p 312 (Pergamon Press, Oxford). 4 M Knudsen, Ann Phys, 28, (1909), 75. 5 W G Pollard and R D Present, Phys Rev 73, (1949), 762. B J D Mellor, British Patent No 1,083, (1967), 244. 7 J D Mellor, In “Advances in Freeze Drying”. (Editor L Rey) 1966, pp 75-88 (Hermann Press, Paris).
627
of Food Preservation,
Ryde, N.S.W. Australia
CSIRO Meat Research Laboratory, Cannon Hill, Qld., Australia
Knowledge of the gas flow in pumping lines, in internal channels, and in porous materials undergoing vacuum processes is basic to the design and operation of a vacuum system. The gas flow equations for viscous and molecular flow in cylindrical channels are ideally simple, in slip flow they are not so easy, but in porous materials in all the flow regions they are more complicated. The permeability of a porous material has been obtained directly by measuring with a flowmeter the amount of gas flowing through it under an absolute pressure gradient, and the same has been done with model materials. Correlation of the permeability and mean absolute pressure provides basic information on structural factors of the material such as porosity, tortuosity factor, and equivalent pore diameter which could, for example, be applied to vacuum and freeze-drying processes.
1. Introduction
The flow of gases in vacuum systems and in porous materials undergoing vacuum processes may be divided into three regions, which can be specified by limiting values of Knudsen’s number l/d, where 3,is the mean free path, and d is the characteristic channel dimension. Thus, if I/d < 0.01, the flow is viscous, if L/d > 1, the flow is molecular, and if 0.01 < I/d < 1, the flow is slip. F At all pressures the flow of gas is impeded by the channel in which it flows. The channel conductance F is defined as the rate of gas flow Q (micron I/set) in a particular channel per unit pressure difference AP (micron) between two points. The viscous-flow conductance F, and molecular-flow conductance F,,, are readily obtained from kinetic theory for channels, having a simple geometry. In such cases F, varies as the mean absolute pressure P, and F,,, varies independently of P with a single value at a pressure of zero as shown in Figure 1. The derivation of slip-flow conductance is difficult. Generally, an equation for the viscous conductance corrected for slip is derived, namely :
P
+3z 2-*F
F=F
”
16’
where f=fraction
f
Figure 1. Conductance-pressure
m’ of molecules colliding
produce diffuse scattering,;6
=coefficient,
with the walls that assuming
rough
walls. Equation 1 does not predict the pressure dependence that has been observed; it is only a linear line for viscous flow displaced upwards by an amount Lpm
as
equal to the slip term g
shown in Figure 1. Therefore, it is only applicable
from high pressures down to intermediate
pressures.
1 may be generalized
Vacuum/volume
Is/number
12.
for long straight
(Barrer 1963)‘, such as those in Table 1, by replacing the tube diameter d by the hydraulic diameter dh area of cross-section / \ =2x circumference of cross-section \ 1 and by‘introducing two shape factors kb and k,, and becomes
F=A-ddL+~. llqk,
2. Gasflow in Channels Equation
relationships.
channels
Pergamon Press Ltdlfrinted
16
2-,f j-‘3k,
4,fi
1 ’
where A =area of cross-section, l=tube and v =mean molecular velocity. in Great Britain
(2) length, r]==gas viscosity,
625
J. D. Me//or
and D. A. Lovett:
Rectangle:
breadth
-2
Flow of gases
width
3 I .05
through
channels
0.87
The conductance of a channel consisting of a combination of such cross-sections could be treated as a set of circuit elements each with simple conductances, however, there arc dificulties in having to determine experimental parameters for the equations. The molecular-flow conductances of channels having a more complicated geometry, such as orifices and short tubes, has been given by Dushman (I 962)y in which he uses Clausing’s factor, defined by the ratio of the net volume flow rate through the cross-section to the volume flow rate of gas striking an orifice of the same cross-section. Levenson et al (I961 ).’ gives a similar method for more complicated channel systems.
with
reference
to porous
materials
porous material, one in each arm between the manometer and the tube. It is used by running the pump and adjusting the needle valve to cause the soap film to move the gas over a measured volume in a given time. Both the vacuum gauge and oil manometer, after compressing the gas in each arm by the McLeod gauge, are then read. The permeability coefficient K for different mean pressures I’ is then calculated. When K is plotted against P a transition flow curve is obtained which approaches asymptotically a viscous-slip flow straight line in the medium pressure range. The slope and the asymptote intercept on the K-axis yield the equivalent pore radius r and the ratio: porosity c to the tortuosity factor (L,/L)” of the porous material. These quantities can be deduced by developing an equation for the asymptote in terms of an equivalent cylindrical model for the porous material along similar lines to those proposed by Barrer (1963)’ to give:
3. Porous materials Transition flow behaviour was INI recognised before Knudsen (1909)6 observed a minimum in the flow rate for a gas at low pressures flowing in a long straight capillary. Knudsen considered the minimum to be due to the transition from viscous to molecular flow and not simply to viscous flow with slip, and he proposed an empirical equation to account for it by combining the viscous and molecular flow equations and an impedance function of the pressure, temperature, gas viscosity, and the diameter of the capillary. The equation (XC Figure I ) is: F = F,, + ZF,,,. where Z
1
(3)
intercept
i)
2’
= 3/c’ +
where I< = 2.5, 1~’= I ci, = 37~1I6 (rough
pores)
or (i, = n/4 (smooth
L,, = actual flow path, L = sample
pores)
thickness
2SO?tl/i.
is the siniplesl expression. I 3.095(//1. This transition flow equation is commonly used today. After Knudsen’s discovery, minima were observed for flow through long narrow slits and. later, for some porous materials (Barrel 1963)‘. Various theories for transition flow with low pressure minima have been published; one theory by Pollard and Present (I Y48)“, offers a rigorous solution to the problem. In connection with the development of drying rate equations for vapour transfer for a new freeze-drying process in which the vacuum pressure is cycled in a prescribed manner (Mellor, 1967)“, the permeability of a gas through a number of freezedried products was measured at IOLI pressures, and found to have a minimum. In some products, however, practically no minimum in the permeability-mean pressure curves occurs, and in a number of others, the curves are concave downward at low pressures. Minimum permeability curves have also been found for a model porous material, and for a sintered glass which was one of few materials previously reported to give a curve concave downward at low pressures. The permeability apparatus (Mellor, 1966): consists of a glass tube with a restriction in the middle for holding a plug of porous material of known thickness and cross-sectional area. A needle valve, and a sensitive flowmeter comprising a soap film moving in a volumetric tube connected to a gas supply at atmospheric pressure are connected in series at one end of this tube. A vacuum gauge and two-stage vacuum pump are connected to the other end, and a differential oil manometer is connected across theglass tube. The sensitivity of the manometer has been increased by interposing two identical McLeod gauges between the manometer and the glass tube holding the
626
(4)
11 =gas ~ =
viscosity
8RT
1
i TM 1 By transposing equations 4 and 5 the equivalent can therefore be found from:
166,%/k I’ =_31;’ And the ratio c/
pore radius
siope intercept’ can be obtained
from either the slope or
the intercept. The porosity c is determined directly (Mellor, 1966)’ by enclosing a sample of the porous material in a glass vessel in which the volume and pressure can be altered and measured simultaneously by movement of a mercury piston. According to Boyle’s law c can be calculated, and the tortuosity factor (&,/I!,) obtained independently of t, either from the slope or intercept equations. Permeability-pressure curves for a bundle of 61 wires 0.06 cm dia by 1.19 cm long, sintered glass (grade I), and freezedried beef and prawn are shown in Figure 2. The minima occur in the range 0.2 to 0.4 torr, and the calculated equivalent pore radii for the wire and beef are approximately equal to half the mean free path at the minimum. This can be expected from kinetic theory. It follows that the asymptotes derived from the latter curves, involve molecular collisions between successive wall collisions, whereas under the conditions where the capillary
J. D. Mel/or and D. A. Lovetf:
Flow of gases through
channels
materials
,
I
0
with reference to porous
I
2
4
3 P,
Figure 2. Permeability-pressure
5
6
7
6
torr
plots showing minima.
diameter equals the mean free path, a considerable number of molecules do not behave in this way. Instead, if the fraction
of the molecules that, on the average, do collide with other molecules between two wall collisions in a viscous-slip flow term and the fraction that do not collide in an additional molecular flow term are considered, an equation, similar to one by Pollard and Present (194Q6, can be derived from kinetic theory for the permeability K in terms of the mean free path 1, giving:
Whole egg IOOcarrot
I
0
I
I
I
2
3
4 p.
Figure 3. Permeability-pressure
where K’=Knudsen’s permeability coefficient. Equation 7 with the structural constant k ~2.5 has been found to give a better fit to the experimental results than Knudsen’s equation 3 for transition flow with a minimum, eg data for freeze-dried beef can be fitted to within a few per cent. Other types of transition flow are represented by the permeability-pressure curves shown in Figure 3. The curves for freeze-dried carrot and strawberry have no curvature down to the limit of pressure measurement (0.005 torr) in the permeability apparatus. Pore size for the two products is about 60 ,U radius. The minimum condition where the capillary diameter equals the mean free path is no criterion on which to judge the anomalous transition flow behaviour in these products. Neither is it in the case of the curves for freeze-dried tomato concentrate and whole egg having calculated pore radii of about 200 p; these curves are concave downward at low pressures. Pore size in the latter products has been found to be highly dependent on rate of freezing before freeze drying. Further permeability measurements are in progress and the kinetic theory is being extended to transition flow behaviour where permeability-pressure curves have a concave downward trend. In addition, a search is being undertaken for a suitable porous material exhibiting this phenomena. It may also be possible to construct a model porous material for this case. Data on the permeability of freeze-dried materials has been found to be very useful for calculating vacuum drying and freeze-drying rates as shown by Mellor (1966)’ where mass transfer and not heat transfer is the controlling mechanism.
Strawberry
*
5
6
7
6
torr
plots showing linear and concave
downward curves.
References
’ R M Barrer, App Mat Res, 2, 1963, 129. B S Dushman, “Scientific Foundations of Vacuum Technique”, J M Lafferty (editor), (John Wiley & Sons, NY). 3 L L Levenson, N Milleron and D H Davis, Tram A VS 7rh Nut Vat Congr, 1961,p 312 (Pergamon Press, Oxford). 4 M Knudsen, Ann Phys, 28, (1909), 75. 5 W G Pollard and R D Present, Phys Rev 73, (1949), 762. B J D Mellor, British Patent No 1,083, (1967), 244. 7 J D Mellor, In “Advances in Freeze Drying”. (Editor L Rey) 1966, pp 75-88 (Hermann Press, Paris).
627