Chemical Engineering Science, 1972, Vol. 27, pp. 1157- 1169.
Pergamon
Press.
Printed
in Great
Britain
Coupled equilibrium beat and single adsorbate transfer in fluid flow
through a porous medium - II
Predictions for a silica-gel air-drier using characteristic charts*
D. J. CLOSEt and P. J. BANKS
Commonwealth Scientific and Industrial Research Organization, Division of Mechanical Engineering,
Highett, Victoria, Australia
(Firstreceived
16July 1971; accepted
9September 1971)
Abstract-For a sorbent bed through which a single-phase fluid mixture is flowing and transferring
energy and an adsorbate between bed and fluid, and where thermal and sorption equilibrium exists
between bed and fluid at every location, the analysis in Part I has shown that the process can be
described in terms of characteristic potentials and specific capacity ratios. Following transformation of
the energy and adsorbate conservation equations into a pair of equations, each having the form of the
equation for heat transfer alone, these new properties are seen to be analogous to temperature and
bed/fluid specific heat ratio.
The silica-gel-water-air system has been chosen for further study and several data sources were
examined to obtain a compatible set of thermal and sorption properties, for reversible sorption. The
characteristic potentials and specific capacity ratios are presented as plots on psychrometric charts,
and the specific capacity ratios are shown to be sensitive to changes in the curves used to fit the data.
An example is performed which illustrates the use of the charts for predicting the behaviour of a silicagel air-drier operating under equilibrium conditions. Comparisons are made with published measurements taken from a real system, and the equilibrium predictions are seen to provide a guide to the
behaviour of the system.
1. INTRODUCTION
industrial processes including air drying,
humidity control and the drying or cooling of
agricultural products involve simultaneous heat
and mass exchange between a solid sorbent, and
a fluid mixture containing the sorbate and a nontransferable
component.
The term sorbate
denotes the transferable component which may
be either adsorbed or absorbed in the sorbent.
Commonly air-water-vapour
mixtures are passed
through porous beds of the sorbent which is to be
cooled or dried, or through desiccant beds which
dry the air.
A theoretical model frequently employed to
describe such a process assumes that the process
is adiabatic, sorption hysteresis is negligible, the
fluid mixture is at constant pressure, no diffusion
processes occur in the fluid flow direction, the
concentration
in the fluid mixture of the nonMANY
transferable component, such as air (mass per
unit volume of fluid mixture) is a constant, and
there is plug flow of the fluid mixture. Because of
uncertainties
in film transfer coefficients and
diffisivities, transfer of heat and sorbate between
the sorbent and fluid is usually assumed to be
governed by film transfer coefficients, their values
being adjusted to give reasonable agreement
between model predictions and the observed
behaviour of real systems. In single-blow operations the fluid and bed are assumed to be initially
in equilibrium and a step change in the fluid state
then to occur.
In spite of the obvious simplifications contained
in this model, its employment for system analysis
and design has been hampered by the non-linear
and coupled equations derived from it. Consequently a variety of approaches to the analysis
and design problem has been reported in the
*Part I, Gem. Engng Sci. 1972 27 1143.
tNow at Department of Engineering, James Cook University of North Queensland, Townsville, Australia.
1157
‘D. J. CLOSE
and P. J. BANKS
literature. Ross and McLaughlin[l]
and Getty
and Armstrong[21
have adopted
empirical
methods based on the results of a large number of
experiments.
Myklestad [3] has simplified the
system model so enabling the equations to be
solved more readily, while Lee and Cummings
[4] have empirically related the measured behaviour of beds operated under adiabatic conditions to the predicted behaviour of beds operated
under linear isothermal mass transfer conditions.
Hougen and Marshall[Sl, Van Arsdell[61 and
Acrivos [7] have proposed numerical solutions
to the model equations, and with the advent of
large digital computers,
complex models of
systems with air-water-vapour
mixtures passing
through beds of wool fibres [8], silica gel [9] and
activated alumina[lO, 111 have been analysed.
The good agreement achieved between numerical solutions and experimental results[9,10. 111
proves the usefulness of the models employed.
A useful extreme case to study involves the
assumption of thermal and sorption equilibrium
existing between the fluid and adjacent bed. This
situation will be approached in practice when the
number of transfer units or NTU describing both
the heat and sorbate transfer are high and when
diffusion within the sorbent can be neglected.
Cassie [ 121 has solved this case for a bed of wool
fibres with through flow of an air-water-vapour
mixture by applying the well known approach of
Henry [ 131 to coupled diffision, in which constant
thermal and sorption properties are assumed.
The uncoupled
linear equations derived by
Cassie showed that when a step change occurred
in the state of the entering air, two sharp and
separated waves passed through the bed. The
application of the method of characteristics to
the equilibrium case [ 14, 151 provides a refinement of this approach since non-linear effects are
included. Wave fronts of finite and unsteady
width as well as sharp fronts are then predicted.
The analysis in Part I by Banks [ 151 is general for
sorption without hysteresis, so that measured
sorption and thermal data can be used. Thus an
advantage
normally
accorded
to numerical
techniques can be retained.
This paper examines a system comprising a
silica-gel, water and air and using Banks’ analysis
employs property data reported in the literature.
The intention is to illustrate this approach to
analysing such systems, to examine some of the
effects of properties on system behaviour, and by
comparing predictions from this work with experimental results reported in the literature, to
assess its relevance to real, non-equilibrium
cases.
2. CHARACTERISTIC
EQUATIONS
The equations describing the conservation
energy and water are, from [ 151.
of
P(l--E)$$+peg+pvrg=O,(1)
P(l--r)$+pE$+pvr$=O.
(2)
Since an equilibrium process without hysteresis
is being considered, h, H and W are all functions
of t and w only. Hence Eqs. (1) and (2) may be
transformed to the characteristic form 1151,
$+A$)$ = 0,
i= 1,2,
in which F, and F, are termed
potentials and satisfy the relations
z/%=-($),.i=%y
(3)
characteristic
i= 1,2,
(4)
where
20Zi= (hCrh+(+v+(YW)
- (- l)i[(XOlh+uV+0lW)2_4~wXolh]“*,
i= 1,2.
(5)
The velocity of propagation
in Fi is Aivy where
Ai = (I+ pyi)-‘,
of a small change
i= 1,2,
(6)
and
$=l_-(ai~yh%)=
%V
i= 1,2.
(7)
(WV-%)
Each quantity yi is termed a characteristic specific
capacity ratio since it bears the same relationship
to Fi as the specific heat ratio u does to temperature.
1158
Coupled equilibrium heat and single adsorbate transfer - II
From the definitions of symbols in the notation
list, it is apparent that enthalpy-concentrationtemperature and vapour pressure-concentrationtemperature data are required for both the silicagel-water and air-water-vapour
combinations.
3. PROPERTIES
OF AIR-WATER-VAPOUR
MIXTURES
The enthalpy of air-water-vapour
mixtures at
a pressure of one atmosphere and over the range
of interest can be expressed as
h=
.24t+w[(t-32)i-h,],
where
h, = h,(t),
so that
(r-32) + h,
(yh= *24+w[(l+dh,/dt)]’
(8)
Also
f=
T
w/Mwater
l/Mair’
WlMwater +
or
(9)
if & is 1 atmosphere,
4. PROPERTIES
4. I Compatible
as assumed subsequently.
OF MOIST
SILICA-GEL
data
For sorbents in which the sorption process is
thermodynamically
reversible,
the
sorbate
vapour can be considered a perfect gas, and the
volume occupied by sorbate in the liquid phase is
negligible compared to that when in the vapour
phase, the Clausius-Clapeyron
equation can be
used to link the heat of sorption to the vapour
pressure data. The resulting relation is
(10)
An equilibrium process without hysteresis is
thermodynamically
reversible. Thus from equilibrium vapour pressure-concentration-temperature data, heats of sorption and hence heats of
wetting can be obtained, if sorption hysteresis is
assumed negligible. Clearly the converse is not
true in that some vapour pressure data is required
to obtain a complete set of such data from heat of
wetting results. The Clausius-Clapeyron
equation is also useful in determining whether data
from different sources was actually taken from
the same sorbent material.
Othmer Charts [ 161, which are graphs of In 6
vs. ln & at constant W, are useful tools for determining hJh,, which is the gradient of the curve at
a given point.
4.2 Sources of data
Three sources of data for silica-gel-water-air
were examined. They are
(i) Calorimetrically determined integral heats
of wetting at a single constant temperature,
together
with some vapour pressure
data,
presented by Ewing and Bauer [ 171.
(ii) Vapour pressure-concentration-temperature data from Hubard [ 181.
(iii) An Othmer Chart from Hougen, Watson
and Ragatz [ 191.
The data of Ewing and Bauer appears to be the
most complete. The vapour pressure data is
plotted on an Othmer Chart, Fig. 1, and the
points at constant W are seen to be closely fitted
by straight lines. Hence h,/h, can be assumed
independent
of temperature.
Bullock
and
Threlkeld [9] have expressed the integral heat of
wetting data as two polynomials in terms of W.
From these polynomials hJh, can be determined
and its variation with W is shown in Fig. 2. The
large discontinuity at a gel water content of 5 percent is caused by a discontinuity in gradient
between the polynomials which meet at that
point. The slopes of the lines fitted to the data in
Fig. 1 were taken from these values of h,/h, and
the good agreement shows the compatibility of
the heat of wetting and vapour pressure data. The
greatest discrepancy is at a gel water content of
11 percent where the disparity is about seven
percent in vapour pressure or three percent in
heat of sorption. Unfortunately, the vapour pressure data is presented in such a manner that
values below a gel water content of about 10
1159
D. J. CLOSE
and P. J. BANKS
70”F, has been plotted in the form of an Othmer
Chart in Fig. 3. Allowing for some scatter, the
points follow straight lines as in Fig. 1. It follows
that h,/h, is independent of temperature so that
Eq. (10) yields
I”
6-
6
p
.c
4-
=U
0
3
s
._
.Z
VI
2-
o
W-O.26
l
w-021
+ W=O.16
x W=O.ll
ln$=+ln$,+C,
1:
2
E
6
’
o-e-
:
CM-
z
04-
?!
0
kl
0.3-
i.z
02 -
(11)
where h,lh, and C are functions of W. The vapour
pressure Q,was regressed upon & to obtain these
functions.
102
Symbd
I”
5
0
c
I
02
0.1
II
0.3 0.4
Saturation
,,I
06 0.6 I
I
2
II
3
4
vapour pressure of water,
II
6
6
IO
in.Hg
2
&
3
.z
IO'
_
+
.
.
.
x
0
0
W
001
002
0.03
0.04
005
006
010
0.20
Fig. 1. Othmer Chart of silica-gel data from Ewing and Batter
[ 171. Points are taken from vapour pressure data, and lines
are drawn at slopes corresponding to values of h,/h, derived
from integral heat of wetting data.
2 d
I
Fig. 3. Othmer Chart showing vapour pressure data from
Hubard [ 181: Points are taken directly from curves in [ 181.
Lines are drawn from Eqs. (12) and (14).
1.4 i
i\
,y
;
I2 +&'
---_
---_
-+x...
-+-.
IO
------_____
-.-.-+.\.
.-+
\.
1%
061
0
\.+__AH
:
--;
I
I
O-05 0.1
W,
I
OIS
lb water/lb
1
0.2
silica
I
0.25
I
0.3
o
I
0.35
I
The plot of h,/h, is shown in Fig. 2. Three
polynomials have been used to fit the calculated
values of h,/h, as a function of W.
I
0.4
o-01 s
w s
0.04,
gel
Fig. 2. h,/h, as a function of W from three sources: 0 Points
from regression fits to Hubard[ 181 vapour pressure data,
from Eq. (11); Polynomial fit to these points from Eqs.
(12). ---Polynomial fit derived from Ewing and Bauer[l7]
integral heat of wetting data, [9]. + Points derived from Hougen, Watson and Ragatz [ 191 vapour pressure data, connected
by a smooth curve.
percent could not be read from the published
graph. Thus a full set of data could not be
obtained.
Hubard’s data [ 181, for temperatures
above
+=
U
8533W3-542W’2+
11.79W+ 1.1557
0.04 S w S 0.1,
+= 45.4W2-8*97W+
21
1.5922
0.1 S w G 0.35,
+=-9*1553W”+7-048W2-1.8966W+1.2789,
2,
(12)
1160
Coupled equilibrium heat and single adsorbate transfer- II
The unbroken line in Fig. 2 is plotted from these
three polynomials.
Taking one atmosphere as 29.91 in. of mercury,
Eq. (11) can be written as
4 = &
[29W(
WI $81‘@u,
(13)
and the function f(W) can be derived from the
values of C. A plot off(W) as a function of W is
shown in Fig. 4. Two methods have been employed to fit this function; a straight line and a set
of polynomials
which provide a continuous
gradient. The reasons for using both these
methods are discussed in the next section. The
straight line function is
f(W) = 2*009W,
(14)
while the polynomials are
Plots from these functions are shown in Fig. 4.
Vapour pressures for moist silica-gel have
been calculated using Eqs. (12), (13) and (14) and
are shown as the lines in Fig. 3. The maximum
deviation in I#Jbetween these and the regression
lines is about 0.2 in. of mercury at 150”F, the
temperature limit used in these calculations. The
corresponding gel water content is 20 per cent.
For that water content, the discrepancy in + is
less than the scatter of points about the regression line. At low water contents the discrepancies are much greater than the scatter of points,
most noticeably at 1 and 2 per cent water
content, and a correct definition of the gel
properties at low gel water contents can only be
obtained by using the more complex equations
off( W) as a function of W given by Eqs. (15).
The Othmer Chart of Hougen, Watson and
I
0 <
0.03 s
0.06 s
0.2 s
W s O.O3,f( W)
W s 0.O&f(W)
W s 0.2,f(W)
W s 0.35,~(W)
= W(24W+ 1.23)
=-10.444W2+3.2967W-O-031
3
=-32.5W4+43.234W
-13.152W 2 +3*1827W-0.02333
=-18.35W3+
14.495W2-1.528W+O.2556
Ragatz [ 191 gives data nearly as complete as that
obtained from Hubard. Values of h,/h, dependent on W alone are indicated and these have been
plotted in Fig. 2. A comparison of the three sets
of sorption data shows the gel of Hougen et al. to
have somewhat different properties from the
other two in that the heat of sorption h, is sometimes less than the latent heat of vaporisation of
pure water h,, at the same temperature. Since it
appears that the gel described by Hubard may be
more typical, its properties have been employed
in the remaining calculations.
4.3 Properties of moist silica-gelfrom
The enthalpy of moist silica-gel is
H = C,t+C,W(t-32)
0
0.05
0.10
W.
lb
0.15
water/lb
0.20
0.25
silica
gel
030
Hubard
+AH,,
(16)
where AH, is the integral heat of wetting, the
expression for which may be written as
0.35
Fig. 4. f(W) as a function of W. Points are derived from intercepts of straight line regression fits to data in Fig. 3. ---(15).
Eq. (14), -Eqs.
1161
AH, = h,
(17)
0
D. J. CLOSE
and P. J. BANKS
and
From (16) and (17), and noting that h,/h, has
been assumed independent of temperature
(l-4)$8
W(d&/dt)
W
=C,+C,B’+!$~(l-k)dW,
(18)
0
and
=C,(t-32)+h,
Hubard’s
(19)
data does not extend
below a gel
water content of 1 percent so that 7 (1 - h,/h,) d W
0
cannot be completely specified from the data.
For this paper it has been assumed that the value
of h,/h, for dry gel is the same as that obtained by
Ewing and Bauer. The integral is then obtained
using the straight line interpolation shown in
Fig. 2 between gel water contents of zero and
one percent.
0.01
The effect of the integral _f (1 - h,/h,) d W is:
The
follows
Cnl
+s (t)
h,
remaining
h,
’ h,’
data and its sources
(23)
are as
0.22 Btu lb-’ “F-l [9]
in polynomial form, atm. [20]
represented as ( 1095.4 - 0.5877t) Btu lb-’
from data given [20].
In the model employed by Banks, the concentration in the fluid mixture of the non-transferable
component p is assumed constant whereas it is
actually a function of both t and w. In using Eq.
(3) to predict the performance of a silica-gel
air-drier it is necessary to choose a suitable mean
value of p. Since the yi do not depend on p they
have been preferred to the Ai in the presentation
of the characteristic properties. For silica-gelwater-air systems pyi % 1, so that from Eq. (6)
there is a simple conversion from yi to Ai,
0
when
W = O-35, approximately
1 percent
of
dH
at
(
>w;
when
W = O-01, approximately
3 percent
of
aH
( at > w’
This term affects only (T. It may be deduced from
Eqs. (5) and (7) that both ai and yt are insensitive
to changes in (+, when 1~~1 6 Iail which applies
for silica-gel-water-air
systems at states of
interest.
From Eqs. (9) and (13)
5= r(z),= (InW%f(wb#d&(~)
+h,.d/dwLf(w)l
ho
&f(w) 1'
aw
= l(l-r$)w-1,
(-)
aw t
h, 1 dA
v--l = <h,.,-,
It may be deduced from Eqs. (5) and (7) that
the yi are sensitive to changes in v and the ai
provided
that ((TV\Q (ail. This
insensitive,
provision applies to a silica-gel-water-air
system
at the states of interest. Sincef( W) and df( W)/
dW influence Y, both methods of fittingf( W), i.e.
Eqs. (14) and (15), have been employed, so that
their relative effect on the yi may be observed.
5. CHARACTERISTIC
PROPERTIES
5.1 Fi and yi charts
With all components of (Y~determined, Eqs. (4)
and (5) can be employed to determine states
corresponding to constant Fiy and values of yi at
these states can then be calculated using Eq. (7).
The differential equations in t and w, (4), were
solved using an integrating routine UNIP [211
(20) which employs a fifth order predictor-corrector
iteration method. Psychrometric
charts have
been
used
to
present
the
characteristic
properties
(21)
and a set of initial values for Eq. (4) have been
chosen to give a uniform coverage of the tem(22) perature and absolute humidity ranges chosen.
1162
Coupled equilibrium heat and single adsorbate transfer - II
The ranges of 70-150°F and O-O-035 lb water/lb
dry air are within the scope of Hubard’s data.
Charts of FI and yi calculated using the straight
line approximation to f( W) are shown in Figs. 5
and 6, while yi calculated using the polynomial fit
are shown in Figs. 7 and 8. Differences between
the FI calculated using the two forms off(W)
could not be detected when plotted on a psychrometric chart. Mollier type charts are used.
Temperature.
*F
Fig. 7. Chart of yl, forf( W) fitted by Eqs. (15).
Temperature.
OF
Fig. 5. Chart showing lines of constant F, and F2.
Tempcroture, ‘F
Fig. 8. Chart of y2, forf( W) fitted by Eqs. (15).
Temperature,
OF
Fig. 6. Chart of y1 and y2, forf(W) fitted by Eq. (14).
It is apparent from Figs. 6-8 that the correlations off( W) and h,/h, had a significant effect on
the yi, especially yz. The variation in (l/f(W)) x
d(f( W))/dW around W = O-20, Fig, 4, caused
irregularities in both y1 and yz, Figs. 7 and 8; the
position on a psychrometric
chart of the line
W = 0.20 is shown on Fig. 9. From the discussion in Subsection 4.2, it follows that the difference in correlations causing these irregularities
is less than the scatter of the experimental data.
Hence these irregularities may be an unwarranted
complication.
The discontinuity in y2 at W = O-04 may mask
any irregularities due tof( W) in that region, but
some waviness in yl, Fig. 7, can be detected. The
discontinuity in y2 is caused by a discontinuity in
d(h,/h,)/dW at W = 044. It is possible that the
1163
D. J. CLOSE and P. J. BANKS
Temperature,
OF
Fig. 9. Chart comparing F, and h, and F, and W.
effect is a spurious one due solely to the method
of fitting h,/h, as a function of W. It has been
included as an illustration of the sensitivity of y2
to the gradient d (h,/h,) /d W. It is noteworthy that
this discontinuity has so small an effect on y1 and
the aI that discontinuities in these curves are not
noticeable. The sharp change in slope of y2 lines
near their discontinuity is caused by the increase
in the gradient of h,/h, near its discontinuity.
Figures 5 and 6, excluding the region of low gel
water contents where irregularities in the yi
occur, are charts showing the characteristic
properties
of a simplified silica-gel-water-air
system. The behaviour of this system is discussed
in Subsection 6.1 to illustrate the use of such
characteristic charts.
5.2 Comparison
with predicted trends
Banks [ 151 has predicted
the behaviour
of the
Fi and yi as the extreme cases of small and large
sorbability, defined by (~3Wlaw),, are approached.
For large sorbability, the F1 and F, tend to
near constant fluid mixture enthalpy h, and constant sorbate content of the sorbent, W. For
small sorbability, the limits are constant sorbate
content of the fluid w and constant temperature t.
Selected F1 are shown in Fig. 9 together with the
relevant extremes. It is seen that the calculated
characteristics
lie between the extremes, so
confirming the predictions. In fact the behaviour
of the system approaches quite closely the case
of large sorbability.
The limits of the yi are: for small sorbability,
y1 approaches zero while y2 approaches the
specific heat ratio (T; for large sorbability, y1
tends to a limit less than (T and y2 approaches
infinity. For the system under study c varies
from 0.92 to 2.5 for the system states covered by
the calculations. The ranges between these limits
are given in Fig. 10, and compared with the
ranges of values of y1 and y2 taken from Fig. 6.
As the diagram shows, both the yt lie within the
limits predicted by Banks.
6. SYSTEM
BEHAVIOUR
FOLLOWING
A STEP
CHANGE IN THE INLET AIR STATE
Prediction from Fi and yi charts
Amundson et af.[14] discussed the formation
of sharp and widening fronts which occur when
benzene vapour is sorbed by carbon. In the terms
of the parameters introduced by Banks, the form
of the front is determined by the variation of the
yI with the change. If the appropriate yi increases
as the bed changes from its initial to final state,
6.1
Predicted range
of Y,
O-
Gredicted
range of r2
Range of Calculated
I
‘O“
I
I
kid
calculated
CT
I
kii&xGd
y,
_co
lo2
72
Fig. 10. Comparison of the rang&of the calculated yr with the ranges predicted by Banks [151.
1164
Coupled equilibrium heat and single adsorbate transfer - II
then a widening front occurs. A sharp front forms
when the appropriate yi decreases with the
change from its initial to final state. For a simplified silica-gel-water-air
system, Figs. 5 and 6
show that widening fronts will tend to form when
a change in F, of the bed can be described as
from a high equilibrium air enthalpy to a low, or
when a change in F2 results in an increase in gel
water content. Changes in the opposite sense
will produce sharp fronts.
To illustrate the use of the characteristic
charts, an example will be performed. A bed of
gel, 6 in. deep, is in equilibrium with air passing
through it at state B, Figs. 5 and 6. The dry
density of the bed P( 1 - E) is 45 lb ft3 and the
void fraction E is 0.496. The entering air state
then undergoes a step change to state R, Figs. 5
and 6. The approach velocity of the air is 47.5 ft
min.-‘.
As the changes in the state of the air and bed
involve changes in both the Fi, two change fronts
will pass through the bed. Assuming that the
equilibrium model is a valid one, then Figs. 5 and
6 can be used to predict the system behaviour.
The curves in Fig. 11 show distributions of
outlet air temperature and absolute humidity as
functions of time. Time zero corresponds to the
arrival of the step change in the fluid state at the
entrance to the bed. Initially, air leaves the bed at
state B and continues to leave at this condition
until a time is reached corresponding
to the
propagation velocity u ( 1 + 117~)-l of F, at state
B. The outlet air state remains at the F, corresponding to B until the change in F1 is complete.
This is necessarily so since all the y2 lying along
the F1 characteristic between B and C (the intersection point of the F, characteristic through B
and the F, characteristic through R) are greater
than all the y1 along the F2 characteristic between
C and R.
The change in F, appears as the first and
narrower front in Fig. 11, the times corresponding to the emergence of the air states from the
bed being derived from the y1 of the states. On
reaching C, the change in F, is complete and air
continues to leave the bed at this state until the
F, front arrives at the end of the bed. The time
elapsed corresponds to the value of yZ at C. Since
the change in F, has been completed, the change
in F2 occurs along an F, characteristic. Again the
time taken for a state to emerge corresponds to
the value of y2 at that state. When the change in
F2 is completed the air and gel are in equilibrium
at state R.
150
0020
- 0016
4L
,
Scale break
- 0.016
\e
f!
\
State
l~-
so-
C
‘i
\
Stute R - 0.010
\
------\\
\
- 0008
L_-
60
0
I
I
I
2
I
3
--J
4
2
from step
6
6
0006
hr
min
Time
4
change
in air inlet
state
Fig. 11. Time distribution of outlet air state following step change in air inlet state, predicted from characteristic charts, Figs. 5 and 6.
1165
D. J. CLOSE and P. J. BANKS
The curves illustrate the large difference in
speed of the two fronts, and also the more variable nature of y2 which changes by a factor of
nearly 10 between states C and R, compared with
a factor of less than 3 for y1 between B and C.
This is illustrated by the wider form of the second
or change in F2 front as compared to the first or
change in F, front.
6.2 Comparison
of predictions
behaviour of a silica-gel bed
with measured
Bullock and Threlkeld [9] have presented measured time distributions of outlet air temperature
and absolute humidity from beds of silica-gel
which had been subjected to a step change in the
state of the entering air. The change was in
absolute humidity only.
To compare the measurements with predictions from the equilibrium model it is necessary
to deal with sharp fronted or discontinuous
waves [15], as for both experimental situations
the change in F, wave is of this form. In each
experiment, the sharp front changes the bed state
from its initial value to a point on the F, characteristic passing through the new air inlet state.
Across this front, conservation
considerations
require that,
Aw
-=AW
including finite film transfer coefficients. The
discrepancy was attributed to ‘an error in the
experimental curve because of inaccuracy of the
humidity sensor at extremely low humidities’.
For the first change, the measured front is
dispersed while the predicted one is sharp. Both
second fronts are dispersed, but the measured
one more so. The measurements for the second
front are incomplete, but it may be inferred that
the two distributions must intersect. This follows
from the fact that under equilibrium conditions
the bed and outlet air states reach the air inlet
state after a finite time. This situation is reached
after infinite time in the experimental case, where
finite transfer coefficients
govern exchanges
between the fluid and bed. Extension of the
predictions to much longer times than shown
would involve crossing the line of discontinuity
in y2 shown on Fig. 6. At the time when this line
is reached a discontinuity in state would occur,
160
.-._
---__
u"
2
0
&
E"
r-"
loo-
Ah
AH’
ao/
and the front speed to be given by
F-1
(
--------.
5
I
I
:
(o)
_L,Scale
break
’
0.010-
=$.!K.
P
2
>
Using these relations together with those in
section 2, predicted time distributions of outlet
air temperature
and absolute humidity for an
experiment [91 have been determined, assuming
that the silica-gel has the properties represented
by Eqs. (12)-( 14), and compared with the experimental distributions in Fig. 12. The predictions
are seen to indicate the main features of the
experimental results.
The large discrepancy in humidity at the start
was found also by Bullock and Threlkeld, who
compared the experimental results with those of
the numerical solution for a theoretical model
O~Oce
.'
0006-
min
Time from
step change
hr
in inlet
c’ *’
_.=
state
Fig. 12. Comparison of time distribution of outlet air state
in an experiment from Bullock and Threlkeld[9] with that
predicted from the equilibrium model, using the silica-gel
properties represented
by Eqs. (12)-( 14); from
experiment, ----from
equilibrium model.
1166
Coupled equilibrium heat and single adsorbate transfer - II
displacing the predicted time distributions towards those measured.
The more highly dispersed nature of the fronts
from the experimental measurements
is to be
expected considering that irreversible processes,
namely mixing and convective
and diifusive
transport, occur and are not accounted for in the
equilibrium model.
The psychrometric chart in Fig. 13 shows the
paths of the outlet air states from the experiment considered above (termed Experiment l),
and another (Experiment
2)[9], together with
Ff lines from Fig. 5. The measurements for the
second front in Experiment
1 are seen to be
close to the F, characteristic followed by the
relevant predictions from the equilibrium model,
despite the disparity in the time distributions
noted above. A similar comparison for Experiment 2 shows a discrepancy, though the appropriate numerical solution[9] follows the F1 line
very closely.
The predicted state path in each first front is a
jump from the initial bed state to a point on the F,
characteristic
through the air inlet state. This
point is close to the intersection point of this F1
characteristic and the F2 characteristic through
the initial bed state, as shown for Experiment 1.
The measured state paths are seen to follow this
F, characteristic fairly closely.
Thus Fig. 13 shows that the first front (closely
an F1 change front) and the F, change front are
70
80
90
100
110
to a large extent separated in each experiment,
despite dispersion due to irreversible processes.
As a result, the state between the fronts is closely
predicted by the equilibrium model.
Acknowledgements-The authors wish to thank Mr. R. V.
Dunkle and Mr. J. W. Sutherland of CSIRO for helpful discussions. The first author is grateful for the tenure of a
Graduate Scholarship at Monash University during the
completion of this work.
NOTATION
Ai
c
C,
ratio with 21of propagation velocity of
small change in Fi, i = 1,2
intercept defined by Eq. (1 l), in.Hg
specific heat of dry silica-gel, B.t.u.
I,.-‘OF-1
CW specific heat of saturated liquid water,
B .t .u . lb-“F-l
characteristic
potentials, i = 1,2; see
Eq. (4), arbitrary units
f(W) function defined by Eq. (13)
mixture,
h enthalpy of air-water-vapour
B.t.u. (lb dry air)-’
H enthalpy of moist silica-gel, B.t.u. (lb
dry gel)+
h latent heat of vaporisation of water,
B.t.u. lb-’
(ahlaw),(aH/a W),, isothermal differh,
ential heat of sorption of water on
silica-gel, B.t.u. (lb water))’
isothermal
integral heat of wetting of
AH,
water on silica-gel, B.t.u. (lb dry gel)-’
Fi
120
Temperature,
130
140
OF
paths followed by the outlet air state in two experiments from Bullock
Fig. 13. Comparison between: and Threlkeld[9]; ---- Lines of constant Fi from Fig. 5, through air inlet and initial bed states; and 0 the
state between fronts predicted by the equilibrium model for Experiment 1.
1167
D. J. CLOSE
and P. J. BANKS
Mti, molecular weight of air
M water molecular weight of water
P density of dry silica-gel granules not
including intergranular voids, lb ftW3
r 4/&,, relative pressure of water in airwater-vapour mixture
I temperature, “F
v mean velocity of fluid within intergranular voids in bed, ft min-’
V velocity of sharp fronted change in
properties, ft min-’
w absolute humidity of air, lb water (lb dry
air-l
W water content of silica-gel, lb water (lb
dry gel)+
Greek
ah
symbols
-(dt/aw)h,
“F
ai
QW
Yi
E
5
e
- (~/dw),~, i = 1,2. “F
- (atlaw) w, “F
characteristic
specific capacity
i=1,2
void fraction of packed bed
r(aWar),
time, min
ratios,
A I- WUaWh/W&Wt
P PC1 _E)IPE
u -( atlaw),
OF
of air in air-water-vapour
P concentration
mixture, lb dry air (ft3 mixture))’
cr (aHiW,/(Wat),
4 vapour pressure of moist silica-gel, atm
saturation vapour pressure of water, atm
mix:; total pressure of air-water-vapour
ture, atm
REFERENCES
[l] ROSS W. L. and MCLAUGHLIN
E. R., Trans.Am. Sot. Heat.Air-Condit.
Engrs 1955 61321.
[2] GETTY R. J. and ARMSTRONG W. P., Ind. Engng Chem. Proc. Des. Deu. 1964 3 60.
[3] MYKLESTAD O., Int. J. Heat Mass Transfer 1968 11675.
141 LEE H. and CUMMINGS W. P., Chem. Engng Prog. Symp. Ser. 1967 63 (No. 74) 42.
[51 HOUGEN 0. A. and MARSHALL W. R., Chem. Engng Prog. 1947 43 (No. 4) 97.
[61 VAN ARSDEL W. B., Chem. Engng Prog. Symp. Ser. 1955 51 (No:16) 47.
[71 ACRIVOS A., Ind. Engng Chem. 1956 84 703.
181 NORDON P., Int. J. Heat Mass Transfer 1964 7 639.
[9] BULLOCK C. E. and THRELKELD
J. L., Trans. Am. Sot. Heat. Refrig. Air-Condit. Engrs 1966 72 (Part 1) 301.
1101 CARTER J. W., Trans. Instn Chem. En,vrs 1966 44 T253.
ill] CARTER J. W., Trans. lnstn Chem. En&s 1968 46 T213.
[12] CASSIE A. D. B., Trans. Faraday Sot. 1940 36 453.
[13] HENRY P. S. H.,Proc. R. Sot. 1939A171215.
[14] AMUNDSON N. R., ARIS R. and SWANSON R., Proc. R. Sot. 1965 A286 129.
[15] BANKS P. J., Chem. Engng Sci. 1972 27 1143.
[I61 OTHMER D. F., Znd. Engng Chem. 1940 32 841.
[17] EWING D. J. and BAUER G. T.,J. Am. Chem. Sot., 1937 57 1548.
[18] HUBARD S. S., Ind. Engng Chem. 1954 46 356.
[19] HOUGEN 0. A., WATSON K. M. and RAGATZ R. A., Chemical Process Principles Part 1. Wiley, New York 1943.
[201KEENAN J. H. and KEYES F. G., Thermodynamic Properties ofSteam. Wiley, New York 1937.
[21] TAVERNINI L., Simulation 1966 7 (No. 5) 263.
RhsumC- Si l’on considere un lit adsorbant
a travers lequel s’ecoule un melange liquide 1 phase unique
oti a lieu un transfert d’energie et un adsorbat entre le lit et le Iiquide, oii il existe un Bquilibre thermique
et d’adsorption entre le lit et le liquide en tous points, I’analyse dans la partie I montre que le procede
peut Ptre d&it en termes de potentiels caracdristiques et de rapports de capacitt. Si l’on transforme
les equations de conservation de l’energie et de l’adsorbat en deux equations, chacune ayant la
forme de I’equation du transfert de chaleur, ces nouvelles proprietes sont analogues au rapport de
chaleur specifique de la temperature et du lit-fluide.
Le systeme silice-gel-eau-air
est choisi pour une etude plus poussee et plusieurs sources de
donntes sont examinees pour obtenir un ensemble compatible de proprietes thermiques et d’adsorption, dans le cas de l’adsorption reversible. Sur un graphique psychometrique sont rep&ends
les
potentiels caracteristiques et Ies rapports de capacites specifiques; ceux-ci semblent Ctre sensibles a
des changements dans les courbes, utilises si l’on veut faire correspondre celles-ci aux don&es. Un
exemple illustre l’utilisation des graphes pour la prediction du comportement d’une secheuse silicegel-air, operant darts des conditions d’equilibre. On fait la compataison avec des mesures publiees et
provenant d%n systeme reel; les predictions de l’equilibre sont un guide pour le comportement du
systlrme.
1168
Coupled equilibrium heat and single adsorbate transfer-
II
Zusammenfassung - Die Analyse im l.Teil hat gezeigt, dass fur ein Sorptionsbett durch welches sich
eine einphasige Fliissigkeitsstr6mung
hindruchbewegt und Energie tibertr%gt und fur ein Adsorbat
zwischen Bett und Fliissigkeit, und wo an jedem Ort zwischen Bett und Fltissigkeit Warme-und
Sportionsgleichgewicht
besteht, der Prozess mit Hilfe von charakteristischen Potentialen und spezifischen Kapazitatsverhaltnissen
beschrieben werden kann. Nach Umwandlung der Energie - und Adsorbatserhaltungsgleichungen
in ein Paar von Gleichungen, deren jede die Form einer Gleichung fur
Warmeiibertragung allein besitzt, wird gezeigt, dass diese neuen Eigenschaften analog dem Temperatur und Bett/Fliissigkeit spezifischem W%rmeverh%ltnis sind.
Fur die weitere Untersuchung wurde das Silikagel-Wasser-Luft
System gew%hlt und es werden
verschiedene Datenquellen untersucht un ein en widerspruchsfreien
Satz von Warme-und
Sorptionseigenschaften fiir reversible Sorption zu erhalten. Die charakteristischen Potentiale und spezifisthen Kapazit;itsverh%ltnisse
werden in psychometrischen
D&grammen aufgetragen, und es wird
gezeigt, dass die spezifischen Warmeverhahnisse
gegeniiber Anderungen in den zur Anpassung an
die Daten verwendeten Kurven empfindlich sind. Es wird ein Beispiel angefuhrt, das die Verwendung
der Diagramme zur Voraussage des Verhaltens eines Silikagel Lufttrockners, der unter Gleichgewichtsbedingungen arbeitet, erliiutert. Es werden Vetgleiche mit veriiffentlichten Messungen eines
wirklichen Systems angestellt, un es erweist sich, dass die Gleichgewichtsvoraussagen
einen Leitfaden
zum Verhalten des Systems darstellen.
1169