Error analysis of the SORP4 sorption microcalorimeter
Wadsö, Lars
Published: 1997-01-01
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Citation for published version (APA):
Wadsö, L. (1997). Error analysis of the SORP4 sorption microcalorimeter. (Report TVBM (Intern 7000-rapport);
Vol. 7121). Division of Building Materials, LTH, Lund University.
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L
UNDUNI
VERS
I
TY
PO Box117
22100L
und
+46462220000
LINIVERSITY OF LUND
LUND INSTITUTE OF TECHNOLOGY
Building Materials
Error analysis of the SORP4 sorption
microcalorimeter
Lars Wadsö
Report TYBM-7121
October 1997
Error analysis of the SORP4 sorption
microcalorimeter
Lars \Madsö
Building Materials, Lund University
Box 118, 22I 00 Lund, Sweden
email: Lars. Wadso@byggtek. lth.se
October, 20 IggT
1 fntroduction
It
is rather difficult to intuitively estimate the influence of different types
of errors on measurements with the soRp4 and our other sorption microcalorimeters. Therefore I have made computer simulations of soRp4
measwements with difierent types of errors. This report gives the result
of these simulations. The sorption microcalorimetric instruments and the
method has been described by wadsö and wadsö (1996, LggT).
We have a measwement from which we would like to evaiuate the true
sample properties" These properties are totaliy independ.ent of our way of
measuring them. From our measurement we get results that may coniain
rneasL|rernent ervors. Mrhen we evaluate the measurements we may have
got coefficients etc. ïvTong, so we make eaaluation errors. In this report I
assess the influence of these two types of errors on the result from soRp
measurements.
2 Analytical treatment
The following nomenclature is used:
1
a
c
ea
ec
€ps
€po
palpa
vapor activity (:plp*r)
vapor content
Slg
palpa
error in vapor activity
error in vapor content
glg
added error in p,
W
added error in p.'
W
€¿.sh error in A,ä
Jlg
M
dry sample weight
g
thermal power of sorption*
n
1ry
P"
thermal pov/eï of vaporization* W
P^u* maximal p.,,
W
L"h sorption enthalpyt
J le
Luh vaporization enthalpy*
t l,
ór,, thermal cross-talk from vap. to sorp. 1
ó"" thermal cross-talk from sorp. to vap 1
Kt) error factor multiplied to p.
1
Kv error factor multiplied to p"
1
Kpmax error factor multiplied to p*"*
1
The following sign conventions are used here: the pa,rameters marked
with a star (*) are always equal to or greater than zero, and the parameter
marked with a dagger (t) is
less than zero (A"h, is the enthalpy of the
'sualy
transition from liquid to vapor,
and a,å is the enthalpy of transitión from
liquid to sorbed phase, cf. Eq. 3).
If we neglect certain small corrections ('initial' and ,sorption time.lag,)
and assume that we have a dry sample at the start of the measurement tie
following equatiors are used for evaiuating SoRp measurements:
T)
1,,
A:I
":
(1)
IDmax
Irt
M - t¡ Jo '"ft)dr
a,h: L"h.fr-*l
,
P,'
Note that we here work with a, c
of each other.
and.
A,h
2
as functions of
(2)
(3)
time; not fgnctions
Equation 1 may be written as follows when an erïor is added to the
thermal power of vaporization:
aJ-eo-1- Pu+e'o
P^*
(4)
This results in the following expression for the error in the activity:
ú^--
€Po
r
(5)
max
This error at time ú is only dependant on the values of epu at time ú.
Equation 2 may be written as follows when an error is added to the
thermal power of vaporization:
c-L
¡t
e-
M . L,h
Jre"?) *
ep,(r))dr
(6)
The resulting expression for the vapor content error is:
lrt
"': M4E Jo er"(r)dr
Q)
The error is dependent on the whole error-history from the start of the
measurement. Errors that change sign (e.g. noise)
cancel
out
themselves.
-uy
Equation 3 may be written as folrows when erïors are added:
A"ä
+
€a"h: A,ä(1
Ï: | "'" )
- P,
* epu'
By expanding, using Eq. 3 and neglecting an e2_termwe get:
ep,(L'"h - L'"h) - ep"a,uh
€A"å
It
È
P,
(S)
(9)
is interesting to note that e6"¡, is inversely proportionar to p,,, but not
dependent on Pr.
3
The material data used in the simulations
Table 1 gives the data used for all computer simulations. The output interval
of 5 s is used in our normal SORP4 measuïements and the other values are
also within the ranges commonly used in measurements with soRp4. I
have used data for th¡ee different types of isotherms: 'linear' (an ideal case),
'sigmoid' (e.g. wood) and 'hydrate' (a stepwise salt hydrate isotherm). Table
2 and Figs 2, 3 and 4 show these three cases.
3
Table 1: The data used in all simu-lations
sample
ty
sample
h
sample weight
0.2 gf cm3
2mm
52 mg
ratio
0.5
simulation ceils in sample ( n) 5
length of simulation
48h
output interval
5s
6 is the ratio of the diffusion coefficients
for the sample and for air, respectively.
4
The computer prograrns
The following MATLAB 4 computer progïams were used in the present study
(the program codes are given in Appendix B):
s4s.m simulates measurements with SoRp4. Input are the sample proper_
ties; output are the thermal pov/ers, the vapor activity and the vâpor
content (all as functions of time).
evalsc.m; er¡alscf.m calculate sorption isotherms and heats of sorption from
the measured (or simulated) thermal power vectors (the program end_
ing with 'f is a function used in the present work; eva,lsc is the normal
program).
master4e.m contain data needed in the evaiuation (a special version of the
normally used master4).
errxx.m
is a program that compares evaluated curves from input data with
and without errors (xx is the number of the error simulation).
inps4s.m inputs the data files.
errplot.m plots the differences between the output from evaluatiors with
and without added errors.
4
Table 2: The knick-points of the sorption isotherms and sorption enthalpies
used in the present paper. The vapor content is in units of
sls arrd ihe
sorption enthalpy has units of Jlg.
linear
sigmoid
0,
c
L"h
a
0.00
1.00
0.0
0.3
-1000
0.00 0.0
0.05 0.02
0.10 0.035
0.40 0.085
0.70 0.15
0.80 0.18
0.90 0.23
0.95 0.27
1.00 0.37
0
hydrate
arh
c
a,
-IIO7 0.00
-836
-669
-376
-188
-177
-54
-27
0.029
0.031
0.229
0.231
0.95
1.00
c
0.0
0.0001
0.0538
0.0539
0.135
0.136
0.200
L"h
0
-1222
-1222
-444
-444
389
444
0
Figure 1 shows in what order the programs mn.
5
The calculations
Table 3 gives an overview of all error calculations made. The procedures
are
described in the following sections. The result is given in a large number
of
figures in Appendix A. Too frnd the result of a cerbain caiculation, look for
the
err-labels (these are actually the names of the MATLAB m-files generating
the results).
NOTE 1: All
errors introduced (except noise) increase the absolute values
of P" and P" (in the present paper).
NOTE 2: rn Appendix A the error is given
as the difference between the
evaluated parameters without any error and the eveluated errors with
error added. As an example, let a,,o"(ú) be the activity calculated with_
out any error added to the simuiated thermar por,Ã/ers and ø",r(ú) the
activity calculated with an error added. Then the plots give ø,,o"(t)
¿*'.(ú) as a function of ø"."(ú). This means that if the thermui pà*",
of vaporization is increased by the error the signs of the result will be
5
ALdA
(*r^t)
(1
hû5k. Tc
L
PnPs
lìar¿S
6cAtv
¿v a[s c
I
( noc)
inp
s
innr
€
1s
Ccrle
Lv t\5c
{
^cbl,
oC sr.)
(¡
err¡lo(
.
r,l^oß
of ñt'þcl
Figure 1: The use of the prograrns in the present study. The program ,err1a,
and the data files 'linres'and 'linnoe' axe examples oi aata Àtes (MATLAB
mat-files); those for the sigmoid and the hydratã cases axe called
'sigres, and
'hydres', and 'signoe' and 'hydnoe', respectively). The circles are ãata, the
rectangles are computer programs, and the rounded rectangles are
data flles.
Three different sets of material data are used: 'true' (the data entered
into
the simulation), 'noe' (the data evaluated withoru any e.ror), ,"a ,J
firr"
data calculated with an error added).
6
0.3
0
o,
trD
ct¡
L
(D
.o
0.2
CL
40 0
o
L
aD
o
()
o
-200
L
o
0
-600
(¡l
1
o
CL
cl
-c
-800
5-
=
Ì,
0
-1000
0.5
1
0
vapor activity
0.5
vapor activity
2: The sorption isotherm and corresponding sorption enthalpy for
Iis*.
the
hypothetical
linear case used in the present study.
0.4
0
ctt
-200
ct¡
è
o.g
c
o
c
c 0.2
o
(J
CL
(D
o
qt
o
o
a.
(¡l
ct
o
0
-4 00
-600
-800
!
d
=
E
0
0
0.5
-l
000
-1200
1
vapor activity
0
0.5
vapor activity
lis,o" 3: The sigmoid isotherm
and corresponding sorption enthaply (data
for the wood Eucalyptus rcgnans taken from chriseosen a.rd Kelsey
rois¡
I
1
500
0.2
ct)
E'
CD
E
0.15
o
CL
É
o
L
o
o
ø -500
o
0.1
C'
(tl
o
o.
(ú
0.05
!
o
-1 000
d
=
ît
0
0.5
0
-1 500
1
0
0.05 0.1
0.15
vapor content
vapor activity
Figrue 4: The saJt hydrate isotherm and sorption enthalpy (morphine sulphate, taken from our own measurements). Note that the enthalpy plot has
vapor content on the x-axis for claritv.
as follows
- aerr >
?
unoe-? uerr <
0
- a"rrä"r, <
0
anoe
arärro"
0
When a positive error acts on the thermal pov/er of sorption the following result will be obtained:
anoe
- ae,
î-^
unoe
arâoo"
-
uerr
:0
:0
a".rh"r.
NOTE 3:
Figrues 7, 8 and 9 a,re of the same type as the diagrams in Appendix A, but they give the difference between true values (i.e. the values
entered into the simulation shown in Figs. 2, 3 and 4) and the values
8
0.2
Table 3: An overview over the error calcuations
made. The different calculations are in this report called 'errxy', where x
is the number to the left and
y is the letter in the table head.
#
1
2
3
4
5
6
nI
8
Tl Bt
erlor
short term noise
T+8.
disturbance
distur bance
B
20
20
T
20
20
B
1
shift
baseline shifb
T
baseline
baseline slope
calibr ation coefficient B
a
b
0.1
0.5 mea. ¡lW
c
unit
p,W early
p.W ea,I
t-tw
1
B
T
0.1
1
0.1 1
1
10
I calibration
T
1
10
10 P,r,u*
1
10
11 cross-talk
T&B 0.1 1
irop( sorption side) or Bottom (vap orization side)
p.W
p,W
h
24h
too
too high
Yo too
To
cross-talk
* Random noise independently
added to both T and B
calculated without any added error. For the activity
they give plots of
a(t)
. .a"".(t) as a function of ø(ú), where a(t) are the true values and
an""(t) are the values evaruated without u"y áaa"a
error.
NOTE 4z rn the hydrate
case the siopes of the isotherm are either near
zero (rapid change of activity without much change
in vapor content)
or very high (increase in vapor content at nearly constant
activity).
When the iatter is the case the vapor content increase will
move like
a front through the sample. In the sim'lations in
which the sample is
divided into five parts, each vertical step on the isotherm
will be alrnost
completed for one cell before the next (inner) cell
may start to increase
its vapor content. As a result of this five sáar steps or peaks
may be
seen in some of the hydrate result (cf. Fig. 6, g,
err6a and errgb).
5.1
The case with no error
A set of sample data is entered (called 'true') are entered into
the simulation.
When the output from the simulation is evaiuated without
any error
added
I
(called 'noe') we will get a result that differs from the input data. The main
reason for this difference is that the sample has a thickness and the parts
at
different depths of the sample will be at siightty different activities (otherwise
there would be no flow into the sample) *d the evaluation procedure
cannot
take this into account.
Figures 5 and 6 gives the 'true' and ,noe'sorption isotherms and heats
of
sorption. The differences seen in the high slopes of the hydrate isotherm will
probably be corrected by the 'sorption time-iag' correction procedure
beilg
developed. The spikes seen in the hydrate enthalpies are artefacts from
the
simulation. Figures 7-9 gives the difference between the 'true' values and the
'noe' (no error) values for the three simuiated cases when the assumption is
made that the activities are correctl. Note that it is the d,,ifferences that
is
given on the y-axes. The plot is of the sample type as the main result
that
is given in Appendix A. some of the steps and kinks seen in Figs. g and
9 are artefacts from the simulation (changes in the slopes of isotherms and
enthalpies).
5.2
Short term noise
Figures 10 and 11 show typical short term noise from two good baselines.
It
is seen that the short term noise is quite low (it may be compared with
the
maximal thermal power P-.* that is g7b ¡zW for SORP4).
Calculations have been made with normatly distributed white noise with
a standard deviation of 0.1 pw (err1a) and 0.b
¡;w (errlb). The measured
noise from simultaneously measured good baselines (Figs. 10 and 11)
were
also repeated to generate a more reatistic(?) short termìoise (errrc).
lValues of a, c and
A"h are evaluated from the simulations as functions of time. The
true values are just given as a-c-L,"h-ftiplets. If there is a difference between
the true and
the evaluated isotherms it is not possible to state that this is caused by an o-difierence
or
by a c-difference. To make a comparison between two sets of data one has to choose
one
parameter as cottect, and in Figs. 7-9 I have here chosen the activity.
To get the whole
picture one must look at the plots of the sorption isotherms (fig. S) urr|th"
sorption
enthalpies (Fig. 6). In Appendix A I make comparisons between data evaluated
witú and
without added errors- As both these data sets have the same origin (the same simulation)
the differences between any of them (a, c or a.å.) may be ploteJaginst time or (as
I have
done) against the values of one of the variables (I used ¿ eìaluated without added
error).
10
0.4
0.35
0.3
0.25
o
o
0.2
C)
E
o
o-
cl
0
1
5
I
I
Í
0.1
I
0.05
¿
I
0
0
0.1
0.2
0.3
0.4
0.5
vapor activity
0.6
0.7
0.8
0.9
Figure 5: Sorption isotherms for the three cases investigated. Solid lines are
the input data for the simulatiors ('true') ,rrd dashed lines are the er¡aluated
result with no error added ('noe,).
11
1
400
200
0
o
o-
o
ID
-200
o
(É
o
a
-400
G
c
o
o
-600
=
E
-80
0
-1 000
-1200
0
0.05
0.1
0.15
0.2
0.25
vapor content
Figure 6: Differential heats of sorption for the three cases investigated. Solid
lines are the input data for the simulations ('true') and dashed lines are the
evaluated result with no error added ('noe,).
72
x
10
-1
true values minus values eval. with
¡s êrror (ì inear)
1
o
o
-.¡
o
0
(ú
ci
(É
1
0
ct,
2
ctt
o
o
x
10
_.0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
0.2
0.9
0.4
0.5
0.6
1
0
c
o
(,
ci
ñt
2
0
ct
o
o
õ
o
=
ît
-8
-9
1
0
0
vapor actÍvity (no error added)
0.7
0,8
0.9
Figure 7: The difference between the true sample data (entered into the
simulation) and the evaluated sample data with no error added for the linear
case. Note that the differences between the values are given on the three
y-axes. The activity is taken as being the same in both cases (cf. footnote
on page 10).
13
1
X
1
0
true values minus values eval. with no error (sigmoid)
-4
1
o
o
(,
0
(ú
ci
(s
-1
_.0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0,9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
o)
?
o
o
.;
c
o
()
0
10
2
1
0
ó
.ú
ctt
o
o
1
1
20
0
(tt
o
d
=
E
-20
vapor activity (no error added)
Figure 8: The difierence between the true sample data (entered into the simulation) and the evaluated sample data with no error added for the sigmoid
case. Note that the differences between the values are given on the three
y-axes. The activity is taken as being the same in both cases (cf. footnote
on page 10).
74
1
x
1
0
true values minus values eval. with no error (hydrate)
-4
1
o
o
y
(,
(ú
0
ci
(tl
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
I0.1
g)
o
o
e
Eo
(J
ci
(tt
g¡
2000
o
o
0
(ú
o -2000
d
=
î, -4 000
vapor activity (no error added)
Figure 9: The difference between the true sample data (entered into the simulation) and the evaluated sample data with no error added for the hydrate
case- Note that the differences between the values are given on the three
y-axes. The activity is taken as being the same in both cases (cf. footnote
on page 10).
15
ntestb.058
o.2
0.1
==
L
0
(u
o
=
È -0.1
rú
E
!
-0.2
(¡J
F
-0.3
0
5oo looo 1500 2000 2500 3000
3500
4000
time / s
Figure 10: An example of a good basline from the bottom (vaporization)
calorimeter in SORP4. This is the baseline used in the errlc calculations.
ntestt.058
0.2
=
0.1
r-
(u
o
=
o
rú
0
-0.1
L
(u
-c
l-
-o.2
-0. 3
0
500
1000
1500
2000
time / s
2500 3000 3500
Figure 11: An example of a good basline from the top (sorption) calorimeter
in SORP4. This is the baseline used in the errlc calculations.
16
4000
5.3
Disturbance
The effect of a disturbâ,nce \¡/as tested by adding a half-period sinusoidal
curve
to the sim'lated thermal por,¡/ers. This disturbance is shown in Fig. 12. A
disturbances is characterized by its starting time (ú1), its length
1nt, t am a
harmonic period) and its amplitude (p2). Tabre 4 gives these parameters
for
the four calculatiorrs made. The disturbances only act on one part (top or
bottom) a time.
All the disturbances increase the absolute values of the thermal porvers.
Mlhen a higher thermal power of vaporization is measured for some time this
will have three effects:
1. The vapor activity will not be correct during the disturbance (afterwards it will be correct again).
2- To the vapor content will be added a small apparent vapor content increase caused by the integration ofthe disturbance. This vapor content
shift seen in the c-result may be calculated by the following equation:
A,c:
Pz
Lt
r'1(
;J,
(10)
Here the nomenclature is as follows:
L,c
P2
Lt
M
L"h
For err2b
vapor content shift
max thermal power of disturbance
time of disturbance
dry mass of sample
heat of vaporization
I calculated 6.0.10-5 and
sls
w
S
C
b
rle
measured 6.0.10-5.
3. The heat of sorption will not be correct during the disturbance, but
afber the disturbance it will be correct again.
Mlhen a higher thermal poluer of sorption is measured only the evaluated
enthalpy of sorption will be affected, and onry as long as the disturbance is
active.
T7
P2
o
o
=
o-
o
E
o
-c
Dt
fl
time
Figure 12: The sinousoidal disturbance added to the thermal powfl curves
in err2 and err3. The disturbance sta¡ts att1, has a duration of Aú, and
has a maximum of P2.
Table 4: The parameters for the disturbances.
BIT t,
err2a
err2b
err3a
err3b
lh Ltls Pzlpw
B
B
0.5
20
600
600
T
T
0.5
20
600
600
18
20
20
20
20
6.4
Baseline shift
Baseline shift is the same as an error in the determination
of the baseline,
i'e' epu and ep" are constant. Equations 5 and 7 may be used without
any
changes. For a the error at time ú is only dependent
ån the values of epu at
time ú' For c the error is dependent on the integral of the
errors auringirre
whole measurement. For a"å the situation is mãre
complex, but Eq. g may
be rewritten as:
€,,"h
ñ
ePo\'uh
- epo\,"h - epr\,uh
P"
l,
^
Here A and B are corstants (becawe A,uh is constant).
In the SORP sorption microcalorimeter we first determine the baseline
with only the dry sample in the calorimetric vessel. To start the
measurement
we then introduce liquid water into the other part
of the vessel. Therefore we
use the baseiine measr¡red without water to correct the
measurement with
water' I believe that this procedure is correct, but if the base[ne
was changed
by the introduction of water we would have a baseline shifb.
only one calculation with 1 pw shift was made for each of the two parts
ofthe vessel: err4a for the bottom (sorption) and errsa for the top (våpor_
ization). Results from other baseline shift may be found by the
,rr" àr Ëqr.
5, 7 and 9.
5.5
Baseline slope
Baseline siope is the same as baseline drift and its comes
from the drift of
some part of the instrument (e.g. the electronics). Baseline
slopes may be
written:
€po: cut
(12)
€P": ort
(13)
Here a" and oo are the slopes (in w/s) of the baserines.
Equation b, 7 and
9 may then be written as follows whe: the errors are included:
lL
""çt¡: P^^*
e"(t):"hfo'o,rdr:#*¡
19
(r4)
(1b)
^ t+\ ^, or\uh e*nlt)È-.ú
or(Lrh - A-fr.)
(16)
The Thermometric thermostated bath we a^re using is very stable, but we have
sometimes seen quite large slopes in the baseiines before the measurements,
especially for the top (sorption) calorimeter. The cause of this is not yet
known, but we guess that the drifis we see are caused by some process in
the sorption vessei (e.g. sorption) or non-matching time-constants in the
measurement and reference vessels.
Two calculations were made for each of the two parts of the vessel: err6a
and err6b for the bottom (vaporization) and err7a and errTb for the top
(sorption). The baseline-slopes are given in Table 3. A baseline slope greater
that zero increases the absolute value of a thermal power.
5.6
Calibration coefficients
For this case Eqs. 5, 7 and t have to be modified
as the error is not added
to, but multiplied with the thermaÌ pou/ers. This can be done by rewriting
the errors into the form previously used by first writing:
KrP,
K"P,
+ epo:
P" + er" :
Pu
that
(17)
(18)
gives:
- 1)&
€p": (". - 1)P,
epu:
(o"
Equation 5, 7 and 9 may be then written as follows:
(o" - 1)P"
e"(t) :
t)
I
ec
(,'"
-
1
(19)
(20)
(21)
max
¡t
M L,h Jn
r"0)a'
(22)
(23)
The calibration coefficients are determined by electrical calibrations. There
âre some problems associated with this; mainly the introduction of heat
conducting copper wires into the vessel. Two error levels were investigated
for each of the two parts of the vessel: 1 and 10%. For each calculation the
calibration coefficient is this much higher than it should be. The results a¡e
called errSb - errSc and errgb - err9c.
20
5.7 Max thermal
po\{/er (p,or*)
The maximal thermal poweï of vaporization (p,''"*) is an important parameter in the evaluation of our measurements. It is a measure of the resistance
to vapor diffusion from the vaporizing liquid to the sorbing sample. In theory
at least, it may be calculated as:
p-"*
: Dr.p,^r!A,"h
L
(24)
Here Do @lslm/Pa) is the diffusion coefficient in air with vapor pressure
as potentiâI, p,ot (Pa) is the saturation vapor pïessure, A (^r) is the cross
sectional area of the tube between the chambers, L
@) is the effective length2
of the diffusion tube, and L"h (Jlg) is the enthalpy of vaporization.
Equation 1 may be written as follows when p-u* is modified by an error
factor:
&
aleo-1-
KP''ax.P-.*
(25)
This may be reduced to the following expression for the resulting error:
P",-
'": P^u*(r -
1
K."-r,.)
In practice, the maximal thermal power is determined from experiments
with water in the vaporization chamber and a drying agent or a saturated salt
solution in the sorption chamber. Different methods yield slightly different
values. The cause of this is not known. The P-..-calculations are done with
two error levels: 1 and 10%. For each calcuration p,,'u* is this much higher
than the true values. The results are called err1Ob - errloc
5.8
Cross-talk
Our double calorimeter is not perfect; one weakness is that a when a thermal
signal is generated in one pa.rt of the calorimeter a small fraction of it wili
be measrue in the other calorimeter. we calr this cross-talk and it may be
formallvdescribedas:
Þ
r-v
D
:_erv _s
- usvrs
(26)
2The effective length difiers
from the true length as it must also take into acount the
diffusive resistance in the top chamber of the calorimetric vessel.
27
P":
P"
- 6u"P,
(27)
Here the c-coeflîcients a.re positive as
the absolute values of both p" and p,
will decrase (the endothermic and exothermic
heats will be partly decreased
the top to the bottom need only be
d vice-versa); we need not count it
as that is taken care of by the normal
Equation 5, T
and' g
may be then written as folows:
(28)
e"(t):
€^"h N
*h
P
fra,"{n"n
/o'
*e)dr
- a,å.) _ ó',auå
(2s)
(30)
Here svmmetric
cross-talt<s of 0.170 andTZo aretested
(err11a and err11b)
In practice the cross-tark does not seem to
be symmetric, but the cross_tark
will probably not be much different from the
cases tested.
6
Discussion and conclusions
when making these conclusions I have defined
the following limits of small
allowable erïoïs:
ø < 0.005 Pa/Pa
c < 0.0015 g/g
lA.ål < 10 J/s
Errors less than this are allowable with the
soRp4 sorption microcaiorime.
ter.
This report shows that
Short-term noise will only disturb the measurements
at the end of the
measurements (at vapor activities approaching
1.00). ftr" *rptiå"
enthalpy is most critical, but it is not
difficultio see when the result
starts to get wrong as the noise wil start to
make the curve funner_
shaped.
22
Disturbances (of u¡known shape, size and time) will of cource be impossible to correct for. It is, however, known that almost all SoRp4 mea_
surement curves are perfectly smoth (except for the noise), so distur_
bances are rare, and as all measurements are repeated this shou-ld
not
be a problem.
Base-line shifts is a problem (if they exist). It is very important to have a
good base-line before the measurement starts. The 1 pW base-line
shifb
used in the calculations is a quite high base-line shift if compared
with
the noise and slope of the best base-lines seen during our measurements.
Base-line slope is probably not a problem in the way calculated, as it is
difficult to see why one shourd have a constant slope during 4g h. The
errors are quite small, except for the sorption enthalpy at the end of
the measurement.
calibration coefficients must of cource be well-known. The l%-case
is
probably quite realistic for the SoRp4 today, as we have difficulties
in assessing the influences from the calibration copper wires. For this
case \¡/e get a higher ¿-error than the above limit druing the first part
of the measurement.
Error in P-.* will only
affect the calculation of the activity. The L%o-case
is probably realistic and will give higher errors than the limit. Better
measurements of P-.* are neccessary.
Cross-talk is not a problem in the present instrument (the cross-talks a,re
less than 1%).
7
Acknowledgements
I thank Nils and Dorthi
study.
8
Troëdsson Research Fundation for supporting this
References
Christensen, G.N. and Kelsey, K. (1959), Holz Roh- werkstoff 17(b) 1gg-208
23
Wadsö,
I. and Wadsö, L. (1996), Thermochim. Acta,
Wadsö,
I. and Wadsö, L. (IggT), J. Thermal Anal., 49
24
2TB 2TT_
I045_L052
Appendix A: the result
x
5
10
-4
errl a linear
o
o
y
(J
(It
0
ci
(ú
5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
x
olr
-r
1 0
,l
o,
9n
(¡)
o-5
o
ci
S
-ro
ct
o
o
0
G'
(D
!
-50
=
E
vapor activity (no error added)
x
5
10
-¿
errl a sigmoid
o
o
-.;
o
0
(ú
ci
(It
5
0
CD
0
trD
x
10
-a
0.1
0.2
0.3
0.4
0,5
0.6
0.7
0.8
0.6
0.7
0.8
0.7
0.8
0.9
1
L
o
L
o
5
5o
()
ci
(tt
>-5
0
0.1
0.2
0.3
0.4
0.5
0.9
1
ct)
o
o
(5
20
0
o
-20
ît
0
0.1
0.2
0.3
0.4
0.5
0.6
vapor activity (no error added)
0.9
1
x
5
10
er¡1a hydrate
-¿,
o
(¡)
(J
(õ
0
Ci
cl
5
0
o,
g)
x
10
_.0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
o
ot-
r05
C
o
o
á
So 0
g)
50
o
o
0
(tl
(l)
d
=
Ìt
-50
0
vapor activity (no error added)
1
err
x 10-3
o
o
2
C)
0
(s
lb linear
o_
9-z
0
g,
?
o
o
4
x
10
_.0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.'1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2
¿
E
o
(J
0
o-
(tt
2
0
qi
200
o
o
0
(s
o
,i
=
E
-200
0
vapor activity (no error added)
1
2
x
10
-3
errl b sigmoid
o
(¡)
C)
(ú
0
ci
(ú
-2
o,
?
_.0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
x
5
10
1
!
o
o
-;
c
o
()
0
-5
ci
GI
-10
'1
q¡
100
o
o
0
(\l
o
-'
=
E
-1 00
vapor activity (no error added)
1
x
10
-3
errl b hydrate
o¿
o
-.;
(J0
(s
ci
9-z
0
x
CD
'10
_r
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.3
0.4
0.5
0.7
0.8
1
cr¡
Lb
o
õ4
-)
c2
o
:0
o_
S-e
0
o)
o
o
0.1
1
200
0
(ú
o
d
E -200
0
0.1
0.2
0.6
vapor activity (no error added)
0.9
1
errl c linear
x 104
2
o
o
.;t¡n
aõ
ci
ct
-2
0
x10
9r
u)-
, 0.1 0.2 0.3 o.4 0.5
0.6
o.7 0.8 0.9
1
-o
o
-:0
c
o
ct.
ci
(Il
>-¿
0
0.1 0.2 0.3 0.4
0.5
0.6 0.7 0.8 0.9
1
ct)
-
izo
o
(ú
U
o
E -20
d
=
E -40
0
0.1
0.2
0.3 0.4 0.5 0.6
vapor activity (no error added)
0.7
0.8 0.9
1
x
2
10
errl c sigmoid
-4
o
L
(¡)
-(tt()
0
ci
(l
2
0.1
0.2
0.3
0.4
0.5
0,6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
x 10
etJ ^
\¿
ot
_r
o
o
10
o
C)
ci
ct^
>-¿
o)
10
o
o
0
I
(ú
o
.=
E
1
0
vapor activity (no error added)
2
x
10
-4
errl c hydrate
o
o
(,
0
cl
ci
(ú
2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
.8
0.9
0
(t
q)
2
x
10
_r
1
o
o
c
o
0
C)
ci
(tl
2
CD
1
20
o
(l)
0
(ú
(D
d
=
It
-20
vapor activity (no error added)
1
x
o
o
¿
o
(ú
10
efizo,linear
-3
20
1
0
ci
(ú
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
x
g)
10
-5
?0
o
6-¿
E-4
C'
ó-6
G'
('t
ì_0
2 -20
o
ão -40
-
-60
o -go
vapor activity (no error added)
x
10
x
10
-3
en2a sigmoid
20
o
o
õG'
10
o-
(ú
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
CD
_r
1
?o
o
(D
t;
c.
o
_^
C'
ci
ct
-6
1
CD
0
o -20
o
(tt
o
-40
-60
=
1r
-90
vapor act¡vity (no error added)
1
x
o
o
()
ct
10
err2o. hydrate
-3
20
0
ci
ct
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
x
cn
'10 -5
1
?o
o
6-¿
.)
E-4
ó-6
C'
G'
o,
ì_o
I
o
-20
ão -40
- -60
õ -go
vapor activity (no error added)
1
enzb linear
o 0.02
o
C)
0.01
(tl
ci
(ú
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
x
ct¡
CD
o
o
C
10
_r
1
2
0
2
o
o 4
o-
cD
0
o
-1 00
o
(ú
(l)
J
=
E
-200
-300
vapor activity (no error added)
1
eï2b sigmoid
o 0.02
o
o 0.01
ci
(ú
(ú
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
x
o,
10
_u
1
E)
-0
o
L
3¡.-1
.)
o-4
(J
ci
ct
-b
1
o,
0
o
o
-t
00
G'
o -200
-c
d
=
Ì, -300
vapor activity (no error added)
1
enzb hydrate
o 0.02
o
(J
0.01
cl
ci
(s
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
x
cr¡
10
_u
1
ct)
-0
o
6-1
-c
o-4
()
ci
(l-b
cr¡
1
0
o -200
o
-4 00
cl
o
d
=
E
-600
-800
vapor activity (no error added)
1
x
err3a linear
104
1
o
0.5
o
o
0
õ
L
o- -0.5
G
-1
0.1 0.2
0.3
0
0.1
0.2
0.3 0.4
0
0.1
0.2
0
x 1o-4
ct)
0.4 0.5
0.7
0.8
0.9
0.6 0.7
0.8
0.9
0.7
0.8
0.9
0.6
1
1
o,
o 0.5
o 0
L
c
o
o
-0.5
cL
(U
-1
0.5
1
o¡60
b40
L
o
õ20
o
E0
E
0.3
0.4 0.5
0.6
vapor activity (no error added)
1
x
err3a sigmoid
1o-4
1
g
o
o.s
ùo
(ú
Ê -o.s
0
1
x
o,
o)
0.1 0.2
0.3
0.4 0.5
0.6
1o-4
0.7 0.8 0.9
1
1
50. 5
o
{i
0
C,
8 -0. 5
ci
(õ
0
0.1 0.2 0.3
0.4
0
0.1 0.2
0.4 0.5
1
0.5 0.6
0.7
0.8 0.9
1
o¡60
-l
b40
L
(¡)
Ë20
(D
-
Ë0
E
0.3
0.6
vapor activity (no error added)
0.7 0.8
0.9
1
err3a hydrate
x 104
1
L
o
L
q)
0.5
+i
(J
0
ñ
-1
trt)
c',
0.1 0.2 0.3
0.4
0
0.1 0.2
0.3
0.4 0.5
0
0.1
0.3
0.4 0.5 0.6
0
x
0.5 0.6 0.7
0.8
1o-4
0.9
1
1
-n 5
ov.
L
o
0
ri
c
8 -0. 5
ci
G
1
0.6
0.7 0.8
0.9
1
or60
ä40
L
L
o
õ20
c)
E0
E
0.2
vapor activity (no error added)
0.7
0.8 0.9
1
err3b linear
x 104
1
L
o
0.5
c)
{..;
o
(E
0
ci -0.5
o
-1
g
o)
0.1 0.2 0.3 0.4 0.5 0.6
0
x
0.7
0.8 0.9
1
104
1
L
o 0.5
0)
¿c
0
o
o
-0.5
ci
(õ
-1
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.8
0.9
1
0
0.1
0.8 0.9
1
o) 400
300
o
o 200
L
L
L
(ú
(¡)
100
õ
0
0.2
0.3 0.4 0.5 0.6
vapor activity (no error added)
0.7
err3b sigmoid
x 104
1
g
o
o.s
tio
(E
Ê-o s
-1
0
x 104
g1
(')
0.1 0.2 0.3
0.4
0.5 0.6 0.7
0.8
0.9
0.1 0.2 0.3
0.4
0.5 0.6 0.7
0.8
0.9
1
b0. 5
L
o
+i
0
c
8 -0. 5
ci
G
1
0
1
g 400
-t
L
300
0)
200
o
(5
o 1 00
-c
E
0
0
0.1
0.2
0.3 0.4 0.5 0.6
vapor activity (no error added)
0.7
0.8 0.9
1
err3b hydrate
x 104
1
os
Þ
o
tio
G
& -o.s
0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.8
0.9
1
0
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1
o) 40
.?
o 30
o
(É 20
o
E
=
10
vapor activity (no error added)
-
x
10
-1
errgc linear
o
L
o
()
ct
0
ci
(ú
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
ot
o)
o
o
e
c
x
_o
1
0
1
1
0
o
o
ci
GI
1
1
q¡ 350
-
o
300
o
(ú
o
250
=
!t
vapor activity (no error added)
1
x
10
-4
errgc sigmoid
1
o
o
o
(ll
0
ci
(s
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
o)
ctt
o
o
c
x
10
_o
1
0
o
()
ci
G'
1
o¡
350
o
300
o
(ú
o
.i
250
=
E
vapor activity (no error added)
E
x
10
errgc hydrate
-4
1
o
o
C'
(ú
0
ci
(ú
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0,3
0.4
0.5
0.6
0.7
0.8
0.9
0
ot
E¡
o
o
c
x
10
-4
1
1
0
o
C'
ci
cr¡
400
o
o 200
G'
o
d
=
E
0
vapor activity (no error added)
1
errl0b linear
0
o
o
.)
o
(ú -0.005
ci
aS
-0.01
0
x
ctI
10
-4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
\l
0.8
0.9
I
?
o
ol
._,
c
0
o
(J
ci
(It
>-l
0
o)
x
10
-4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.9
1
1
o
L
o
0
(s
o
d
E
=
I
0
0.1
0.2
0.3
0.4
0.5
0.6
vapor activity (no error added)
0.7
0.8
err10b sigmoid
0
o
o
(JG'
0.005
o-
(ú
-0.01
x
g,
?
o
o
0
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
_.0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
_o
1
0
C
o
C)
ci
(s
0
o,
o
o
x
10
1
0
cl
o
=
E
1
0
0.1
vapor activity (no error added)
err10b hydrate
0
o
o
-C(tt) -0.005
ci
(ú
-0.01
trt)
q,
o
o
J
0
x
10
-A
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0,9
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
0
o
(J
ci
(Il
1
0
x
o,
10
_o
L
o
o
0
cl
o
d
=
E
1
0
vapor activity (no error added)
errl0c linear
0
o
o
(J
ct
-0.05
ci
õ
-0
1
0
x
CDl
\l
1
0
1
0
_r
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
_o
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
o)
o
.0
o)
o
o
ci
(ú
>-l
0
x
CD
o
o
1
0
(ú
o
d
=
E
1
0
vapor activity (no error added)
1
err10c sigmoid
0
o
o
(J
(ú
5
-0.0
ci
(ú
-0.1
g)
o¡
_o
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
_o
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
x
10
x
10
1
1
L
o
(D
-)
c
o
()
0
ci
(ú
1
0
o)
o
o
1
1
0
(ú
(¡)
d
-c1
0
vapor activity (no error added)
1
r
I
I
I
err10c hydrate
0
o
o
I
C'
(ú
-0.05
d
G'
-0.1
ct
o¡
o
o
0
x
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
_o0.1 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
_o
1
0
E
o
(,
ci
(ú
1
0
x
ct¡
o
o
10
1
0
(ú
o
!
=
E
1
0
0.1
0.2
vapor activ¡ty (no error added)
x
0
10
-3
errl 1a linear
o
o -0.5
-;
C'
ct
ci
1
(s
-1.5
q¡
4
ot
o
o
c
0
x
10
-A
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2
o
C)
o-
(¡t
0
0
1
g3
ot
LÉ
o
(ú
so1
d
=0
E
0
vapor activity (no error added)
1
0
x
10
errl 1a sigmoid
-3
o
0)
-0.5
o
ct
-1
ci
(s
-1.5
g¡
?
o
o
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0
4
x
10
_o
2
L
o
(J
ci
ct
0
g,
a
o
o
2
1
(ú
o
0
=
E
1
vapor activity (no error added)
.8
0.9
x
10
errl 1a hydrate
-3
0
o
o -0.5
()
a\t
ci
(E
1
1
5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
g4
?
x
1
0
-4
1
o
o
¿2
g
o
(¡
ci
9o
gt
2
o
o
0
(U
o
!
d
=
î,
2
vapor activity (no error added)
1
errl 1b linear
0
o
o -0.005
C)
(E
ci -0.01
(õ
-0.015
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
x
g,
4
o¡
1
0
_r
L
o
o
¿
2
L
o
()
ci
(ú
trt¡
0
30
o 20
L
o
(ú 1
o
=
E
0
0
vapor activity (no error added)
err11b sigmoid
0
o
o -0.005
C'
(ú
ci -0.01
ct
-0.01
5
0
94
cr¡
x
10
_.0.1
0.2
0.3
0.4
0.5
0.6
0.1
0.2
0.3
0.4
0.5
0.6
0.3
0.4
0.5
0.7
0.8
0.9
0.7
0,8
0.9
0.7
0.8
1
o
(D
-:2
g
o
(t
o-
9o
0
930
9.20
o
(l
o
+
d
l0
î=r0
0
0.1
0.2
0.6
vapor activity (no error added)
0.9
1
errl l b hydrate
0
o
(D
¿
C)
-0.005
(tl
ci -0.01
G'
-0.01
ct¡
CD
o
o
c
5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
4
x
10
-3
2
o
(,
ci
(ú
0
CD
o
o
(ú
20
0
o
d
=
!l
-20
vapor activity (no error added)
Appendix B: the computer prograrns used
Program errl a.m
o/ær¡la short term noice as measured
inps4s
errcase:'short term noise'
Pv:Pv+O. 1 e{ *randn(size(Pv))
Ps=Ps+0. I e-6*randn(size(Ps));
;
[t,Pv,Ps,as,cs,Dsh] =evalscf('master4e',t,Pv,Ps)
:
figure(l)
clf
plot(anoe,cnoe,'-r')
hold on
plot(as,cs,'--')
hold off
xlabel ('vapor activity')
ylabel(Vapor content , g/g')
title( ['SORP4 error analysis :',iso,' tested with',errcase])
figure(2)
clf
plot(cnoe,Dshnoe,'-g')
hold on
plot(cs,Dsh,'-J)
hold off
xlabel(Vapor content , g/g')
ylabel('differential heat of sorption , J/g')
title(['SORP4 error analysis:',iso,' tested with',errcase])
Program: inps4s.m
%oinps4s
o/o
linres, sigres, hydres are results from simulations
7o linene, sigene, hydene are the above ñles evaluated
oÂ
with no errors added
clear
type=inpu('Which s4s simulation? (lin:
addpl=input('Add plot? (yes: l)');
l, si52,
hyd=3 )')
;
if t1'pe==l
load linene
tnoe=t; cnoe=cs;anoe=as;Dshnoe=Dsh,
o/oPvnoe=Pv,Psnoe=Ps;
load linres
iso='linear isotherm';
elseif type==2
load sigene
tnoe=t;cnoe=cs,anoe=as,Dshnoe=Dsh; ToPvnoe=Pv;Psnoe:Ps;
load sigres
iso='sigmoid isotherm' I
elseif type==3
load hydene
o/ohmoe=Pv;Psnoe=Ps;
tnoe:t;cnoe=cs;anoe=as;Dshnoe=Dsh;
load hydres
iso='hydrate isotherm' ;
end
disp(iso)
disp('Evaluated case with no eror: tnoe, cnoe. ano€, Dshnoe, Rnoe, Psnoe')
disp(")
disp('AFTERRUN:')
disp('In memory: anoe, cnoe, Dshnoe (noe=no error)')
disp(' as, cs, Dsh (evaluated with error)')
disp('
aiso, ciso, hhhh (start (true) d¿ta)')
Program: err.m
o/æ¡r
o/ølalcr.l for the plots should be in 'namn'
subplot(3 I l)
plot(anoe,anoe-as);
Toxlabel('vapor activity (noe)')
ylabel(Vap. act. error')
title(namn)
subplot(3
l2)
plot(anoe,cnoeæs):
o/odabel('vapor activity (noe)' )
ylabel('vap. cont. error, gig')
subplot(3
l3)
plot(anoe,Dshnoe-Dsh) :
xlabel('vapor activity (no error added))
ylabel(ditr heat error, J/g')
Program: s4s.m
%
%
%
%
%
sorp4sil - a one-dimensional simulation of
the vapor transport and sorption in SORP4
for the testing ofthe influence oferrors
% This MATLAB 4 proglam simulates a measurement with the sorption microcalorimeter
the program:
% SORP4 (not the sorption enthaþes...). The following input is asked for
(mm)
height of the samPle
densitY of the samPle (mg)
ratio of the dif.coeffs in sample and in air
number of calculation layers in sample
%
%
%
%
h
rho
delta
n
þ
%
%
o/o
tend
tout
end of simulation (s)
time interval between outputs (s)
If you give no input the following standard input will be used:
o/oh:2 mrn, rho=O.15 glcm3, delta=0.5 (M=37 mg), n=5, tend=20000, tout:200, linear isotherm.
%
% It is also possible for the user to make other changes in the program:
%
%
%
%
%
%
o/o
vsat, psat,
As
Dvh
Dp,air
properties of vapor in air
surface area of sample (m2)
vaporization enthalpy (J/g)
aiso, ciso, hhhhsorption isotherm
k
a(l)
C,
& enthalpy
capacities
& conductancies of model
activity atvaporization surface
A simulation gives the following numerical ouput (+a number of plots):
%
%
%
%
%
t
a
Da
cv
Pv, Ps
(s) [a vector]
activity in each computational cell
[a matrix]
the activity difference over the sample (1)
apparent concentration in sample (g/g) [a vector]
thermal power of vaporization and sorptin (W) [vectors]
time
(l)
%oThe following points should be noted:
î//o
- The simulation is by simple forward differences (Fick's law and mass
%
balances are calculated in small time increments). The time step dt is
o/
/o
automatically calculated as the maximum time step found in any part of the
model at each time.
%
o/
- During the simulation t, a and c are used and the ouþut is stored in
/o
%
tt. aa and cc. After the simulation the result is transferred to
%
t. a and c. The thermal powers of vaporization and sorption are given
o/
/o
in Pv and Ps.
o/
- Calculations are made with Sl-units: g, m, s, Pa
/o
- Standard temp€rature is 25oC and the vapor forming liquid is water
%
- For each cell only the activity (a) is save4 but as equilibrium is
%
Yo
assumed the vapor concentration can be found with the sorption isotherm
%
o/
/o
o/e
Lars Wadsö 970522 970527 970814 970820 970922 970924 970929
-physical ¡lotqvsat=23: YovaQor content (Úm3)
-----------
psat:3160;
--
---
o/oYàpoÍ pressure
@a)
Dpair:l82e-9; %odifürsion coefficient of watervapor in air (glPalrnls)
As=130eó; Tocross sectional area of sample (m2)
Dvl¡e 2 4 40 : %ioheat of vaporization ( J/g)
Frnax:40Oe- 9 ; o/omaximal vapor fl ow (g/s)
o/e--------------------user input (or standârd case)--------fdt=0.9; o/olime step factor
h:inpu('heiglrt of sample (mm) :');if (h:=0)l(h==n);h=2;end;h=h/1000; %(m)
rho:input('density of sample (g/cm3) :');if (rho:=[])l(rho==0);rho=0.2;end;rho=rho* 1e6; o/oglm3
delta:input('Dp(sample)/Dp(air) :');if (delta==O)l(delta::[]);delta=0.5;end;M:h*As*rho;
disp(") ;disp( ['Sample mass =', num2str(M* I e3 ),' mg] ) ;
n:input('number of calculation layers in sample :r);if (n==0)l(n==[]);n:5;end;
tend=input('end of simulation (s) :');if (tend==0)l(tend==[])fend=24*3600:end
tout=input('time interval between ouþuts (s) :');if (tout==0)l(tout:=[]);tout=5;end
o/v
------
--
-------sample data-----
isoq'pe=input('type of isotherm ? (l=linear, 2=sigmoid, 3=hydrate) :');
Toaiso are the activity lariclçoints on isotherm
%ociso are the concentration knickpoints on isotherm (g/g)
%ohhhh are the differential heâts of sorption (J/g(water))
if (isotyps==0)l(isotypæ= []);isotypæ 1 ;end oástandard case
if isotype== I o/olinear
aiso:[O ll;ciso:[O 0.3];hhhh:[-1000 0l;
elseif isotl,pe:=2 %"sigmoid (wood Eucallptys regnans, Christensen & Kelsey 1959)
aiso:[0 0.05 0.10 0.40 0.70 0.80 0.90 0.95 1.00];
ciso:[0 0.02 0.035 0.085 0.15 0.18 0.23 0.27 0.37];
hhhh:[265 200 160 90 45 28 13 5 0]*(-4.18):
elseif isotype==3 %ohydrate (Morphine sulphate)
aiso=[0 0.029 0.031 0.229 0.231 0.95 1.0];
ciso:[00.0001 0.0538 0.0539 0.135 0.1360.2];
hhhh=[O -22 -22 -8 -8 7 8]*1e3/18;
end
xi:diff(ciso)./diff(aiso); %slope of isotherm
ix:diff(aiso)./diff(ciso): Toinverse xi
o/e -
---- - ----------show
ñgure(l);clf
input-----
subplot(221)
plot(aiso,ciso,'*');hold on;plot(aiso,cise,'J),hold off
xlabel('relative activity');ylabel('vapor content (g/g)')
subplo(223)
plot(aiso,hhhh,t*');hold on:plot(aiso,hlhh,'-');hold off
xlabel('relative activity');ylabel('enthalpy (J/gw)')
subplot(122)
text(O,5,['sample density:',num2str(rho),' kg/m3']);text(O,4,['sample height=',num2str(h*1000),' mm']);
text(0,3,fsample mass='.num2str(M*1e6),' mg']);text(O,2,l'diff.ratio=',num2str(delta)])l
text(O,1,[int2str(n),' simulation cells']);te¡(0.0,['simulation ends at',num2str(tend),' s']);
text(O,-l,['ouput interval: ',num2str(tout),' s']);axis([-I l0 -2 5]);set(gca,'Visible'.'Off)
disp('Press any key to continue (Ctrl-c to abort)');pause
V"--------------------simulation data (cf. report)-nn:l+5+n; Tonumber of last active conductance in sample
Y a:0.7 2e-6;VF I 43 e-6 :
.
asim:O.105:
p¿=1 1 -asim)/8Æmax;RF asimJ 2 ß max;Rc=W
2
I nl
delta/Dpair/psat/As;
lezeros([l nn]);
C=zeros(fl nn]);
k(l):1/Ra:
k(2 4):ones(
:
It
3l)I 2 /Ra:.
k(5)=l/(Ra+Rb):
k(6)=l/(Rb+Rc);
k(7:nn-1)=ones([l n- l])/2/Rc:
k(nn):O;
c(1):0;
I 4])*Va*vsat;
C(6)=!6*u..t'
C(7:nn)=ones( [ I n])*N{/n*xi(
C(2:5)=ones([
I
%-------------simulation
a:zeros([
a(l)=l;
I
);
initialization------
nn+ I ]): Toactivities
o/oactivitY of water
c:zeros([l nn+ l]); T0concentrations
q=zeros([I nn]): Zoflows (kgls)
(9s)
Qv:O; 7oQ keeps track of the apparent flow since last output
o/omax time step
dt:fdt*min(C(2:nn)./(k( I :nn-1)+k(2 :nn)));
tdispmin( [t end{ 20 3 600] ) ;
plustdisptdisp;
t=0: Totime in simulation (s)
o/omaximal
possible activity
amax:aiso(length(aiso)) ;
amaxend=0: Towhen amaxend=l the simulation has to be stopped
toutnext=tout; o/osecond output time (first is at t:0)
leSones(n nf);o/oparton isotherm in which each sample part is at each time
o/ocounter for outPuts
nout:ceil(tend/tout)+ I ; o/approx. no of outputs
out:1;
L
tt=zeros([nout l]); o/oin tt the sim. time is saved
aa=zeros([nout nn]); Zoin aa the activities are saved
aa(l,1)=l; "/oactittty of source=l from t=0
cc=zeros([nout nn]); o/oinccthe activities are saved
da=zeros([ I nn]);dc=da;
ll); Zoin
ll); o/oin
rations are saved
e over the sample is saved
vap. are saved
ll);o/oin
I
l);
konst:l.05*dlC(7);o/oto make the simulation run faster
%-_______ ____________simulation___________
tic
disp( ['simulation started with dt=', num2 str(dt),'
while (t<tend)&(amaxend==Q¡
t=t+dt' %oincrement time
q(
1
:
s']
);
nn)=(a( I :nn)-a(2 : nn+ I )). *k( I : nn) ; o/æalculate fl ows
if a(7)+konst*(q(6)-q(7)þamax;amaxend:l;disp('end of isot. reached');end Zætop before going outside
isotherm
da(2:nn¡¿1*(q(1:nn-l)q(2:nn))./c(2:nn); zocalculate differences in activities
ind:findstr(int2str((a(7:nn)+da(7:nn))>aiso(leg+1)),'l'),
isotherm
7ofind index of sample cells that has changed leg on
if ind.*[];
t:t-dt;
dt I
:(aiso(leg(ind)+
I
)-a(ind+6))./(q(ind+S )q(ind+6)). *C(ind+6)
indx:min(fi nd(ones(size(dr
dt:dtl(indx);
1
))
*min(dt
I
)=:dt
;
I )) ;
t=t+dt;
da(2:m¡=¿1*(q(l:nn-l)-q(2:rur)).lc(2:nn);
o/æ,alculate
end
Qv:Qv+q1 t ¡*
new differences in activities
dt "/addfl ow rate from liquid
dc(2 :nn;=¿1*(q( 1 :nn- I )-q(2 :nn));
c(2 :nn):c(2 : nn)+dc(2 :nn),
a(2 6)= c (2 6). / C(2 :6) ;
:
:
nn):aiso(leg)+ix(leg). * (c(7 nn)/(tl4/n)-ciso(leg))
if toutnext<t o/otime for output?
if tdisp<t
a(7
:
:
;
tdisptdisgrplustdispt
disp(['t=',num2str(t/3600),'h with dt:',num2str(dt),' s (',num2str(t/tend*100),'%)']);
end
out=out+l; oZindex in ouþut vectors
tt(out):L Toouþut time
aa(out, I :nn+ l):a( I :nn+l );Da(out)=(a(7)-a(nn))*n/(n-l
);
cc(out, I :nn* I ):c( I :nn+ I ) / M n);
w(out)=cv(out- I )+Qv/lvÍ. Qv:0 ;
Pv(out)=q(1)*Dvh;
Ps(out)=-1r'sum((q(6:nn-t)-q(7:nn)).*(interpl(aiso,hhhh,a(7:nn))-Dvh)');
toutnext=toutnext+tout; o/otime for next ouþut
end
if
ind-[]
o/æneprt entering new
leg
leg(ind(indx))=leg(ind(indx))+l;C(ind(indx)+6):tryfln**1(leg(ind(indx))):
Todisp( ['C(',int2str(ind(indx)*6),')=', num2str(C(ind(indx)+6))]
dFfdt*min(C(2 :nn)./(k( I :nn-l)+k(2:nn))):
end
end
toc
y"- - ---- - - ----------------cleaning
up---
tic
L=length(tt); %ol-=length of vectors
);
if out<L o/ælear the unused pa.rts of aa, cc, Pv, tt
aa(:,out+l:L):[];cc(out+l:L)=[;tt(out+1:L)=[;Pv(out+1,L):[];Ps(out+l:L):[];
end
a:aa,c=cc;ccv:cv;PPv:Pv;PPs=Ps;t=tt; Toouput is in t, a, cv, Pv and Ps
tt=linspace(0,tout* (length(t)- I ),length(t))' ;ttt=t;
for k:l:nn;disp(['Interpolating in a(:,',int2str(k),')']);a(:,k)=interpx(t,a(:,k),tt);end;
for lsl:nn;disp(['Interpolating in c(:,',int2str(k),')']);c(:,k)=interpx(t,c(:,k),tt);end;
disp('Interpolating w, h, Ps, Da');
cv:interpx(t,cv,tt);
Pv:interpx(t,Pv,tt);
Ps:interpx(t,Ps,tt);
Da=interpx(t,Da,tI);
t:tt,
o/æleu aa cc tt o/ælear the simulation var. to save spaæ
toc
-------- -------plot result------figure(2)lclf
subplot(l2l) Toplot of activities as function of time
o/e--
col='rgggg
o/owate=red; gas phasægleen; sample=yellow
';
for z=l:7+n-1
eval ( ['plot(t,
a (:,2),',"
",col(z), "",')' ] )
hold on
end
xlabel('time / s')
ylabel('activity')
subplot(122)
for
z=I:j+n-l
eval( ['plot(!c (:,2),',"",cot(z), "",')'] )
hold on
end
xlabel('time / s')
ylabel('conc.')
Toplot of thermal power as a function of time
%oplot(t,Pv* 1e6)
Toxlabel('time / s')
Toylabel('Thermal power of vaporization / uW')
hold off
ñgure(3);cH
subplot(l2l)
plo(t,Da)
xlabel('time / s')
ylabel('activity difference over sample')
subplot(122)
plot(t,cv)
xlabel('time / s')
ylabel('mean concentration (kdkg) in sample')
figure(2);subplo(121) ToReady to zoom on sample activities
%-------------------------end
Program evalscf.m
function [t,Pv,Ps,as,cs,Dsh] =evalscf(masterfi lename,t,Pv,Ps,Pmafact)
% EVALSCF An function that evaluates measurements with the Lund
\_
%
SorPtion Calorimeter.
o/
/o
%
function [t,Pv,Ps,as,cs,Dshl=evalscf(masterfilename,t,Pv,Ps,Pmafaxt)
o/
/o
%
Lars Wadsö January 8,1997,970918
global OK C'OON
filen=masterfilename; %oinput('Enter name of m-file with input data (e.g. "indata") : ');
eval(filen); %oinputs neccecary data for the evaluation
%-
-
--
------
-----Pv-evaluation-----
ifPv
source=:2
lt dt U txl:filinaf(filev);
U=subbl(U,l);
if (te<O)l(te==[l)
te=fi ndte(t,u, l,'injection') ;
end
ind=round(teldt);
t:t(ind: length(t))+e;
U:U(ind:length@);
Pv=tian(t,U,epv,tav,TUv,TdUv);
elseif Pv_source=:l o/ot,Pv already in memory
ToPv is the thermal power of vaporization
dr=t(2)-r(l);
end
if mean(k)>O
Pv:-Pv:
o/oit is
assumed
that endo. Pv is less than zero
end
if corrl:= I
eval(incofile) o/oiruøl correction (tci, Pciv. tendic)
tFdt:dt:tendic:
L:length(tt);
P_init_saved=Pv( I :L)
:
ÈPv(L)/Pciv(length(Pciv)) ;
Pv( I :L)=lnlsrpl (tci,Pciv,tt,'spline')*f,
end
if glycerol==0
qv:-Pv/Dvh;
else
lry,av,Dvh]=corrglyc@v,Vgw,cgw);
o/æonection for g-w mix
end
if corr2:=O
as=av-qvlPmax.*Dvh, o/øs is the vapor pressure of the sorbing sample
qs=qv;
else o/æorrection for sorption time lag
[as,qs]=corrstl(k1,k2,k3,
end
if corrl==l
as( I
o/oIf
C
l2,C23,t,qv,av)'.
ainttial correction
I L]):
has been made....
:L)=as(L+ l)*ones([
end
cs= (m0-mdry)/mdry+cumsum(qs) *dUmdry;
Tosubplo(121)
Toplot(as,cs,'.')
Toaxis([0 I 0 1.05*max(cs)l);
o/oxlabel ('relative vapor activity')
Toylabel('vapor concentration (g_vapor/g_dry_sample)')
o/*itle(filev)
var strl='t dt U Pv F as cs';
%-
-
-
-----------------Ps-evaluation -------
ifPs sou¡ce-O
ifPs source=:2
lt dt U txl=filinaf(files);
U:subbl(U,l);
t=t(ind: length(t))-te;
U=U(ind:length(U));
Ps:tian(t,U,epss,tas,TUs,TdUs);
%oPs
is the thermal power of sorption
elseifPs source==l
7ot. Ps already in memory
end
if length@s)>length(Pv)
Ps=Ps(l:length(Pv));
elseif length@s)<length@v)
Pv:Ps(l:length(Ps));
end
t=t(l:length@s));
if mean@s)<O o/olnthe equations it is assumed that
Tothe exothermic Ps is greater than zero
Ps:-Ps;
end
¡s¡=@s./Pv+l)*Dvh; Tdsh is the sorption enthalpy liquid->sorbed phase (<0)
% subplot(I22)
oá plot(as,Dsh,'.')
Vo o/æ,r;s(ÍO1 0 1.05*max(cs)l):
7o xlabel('relative vapor activity')
oZ ylabel('sorption enthalpy liquid->sorbed phase (J/g_vapor)')
% rirle(files)
var_str2='Ps, Dsh':
else
var str2=":
end
disp('--------
------------r)
disp( f Variables',var_strl.var_st¡2.' are left in memory'] );
disp('
Program file: master4e. m
%oMASTER4E
ToMaster file for indâta to EVALSC for ERROR ANALYSIS
%
o/Ã-ars Wadsö January
o/
/o
%
Yo
8, 1997,
v
s
9709 18, 971 0 1 0
vaporization
sorptlon
%----FILE S vaporization
Pv_source=l; ToIf Pv_source=:l : t, Pv are rn memory
o/ollPv source==2 : input from file ("filev")
%ofi
lev:'c:\measurem\cmc\ntestb.06
filev:'error anal.';
o/oif W source=:l "filev" will
I'l
be used as title in plot
%----FILES sorption-----Ps_sou¡ce:l: ToIf Ps source:=0 : no evaluation of sorption mea.
o/olf Ps source:=l : t. Ps are in memory
%olf Ps sou¡ce:=2 : input from file ("files")
%ofi
les:'c:\measurem\cmc\ntestt. 06
l';
files='error anal.';
Toif Ps_sou¡ce==l "files" will be used as title in plot
%----CORRECTIONS--(if corrx=:l conecrion x wilt be made)-
corrl=l; o/oittttial
corr2:0;
o/osorption
time lag
glycerol=O; o/ogJycercl-water mixture
m0=52e-3; yo inttial and dry weight of sample (g)
mdry:m0;
%ù>>TEXTS<<<
TEXT_date='8 jan I 997';
TEXT_temp'25oC';
TEXT_calorim='SORP4'.
TEXT_sample=Fiber board @urch) dried over CaCl2,;
TEXT_vapor'Water (millipore)' ;
TEXT_extra=";
epv:O.096529e4;
tav:144;,
epss:O.124069e-6;
tas:l3l:
TUv:20 TdUv: 20 ;TU s:20 ;TdUs=2 0 ;
Dvh=2440; %dhv is the enthalpy of vaporisation (J/g)
Pmax=975e-6;%oPmaxis the maximal thermal power (W)
if exist('Pmafact')
:
Pmax:Pmax*Prnafact
Ènafact:[];
end
av:l.o/ærctäty of vapor
trO:
Totime of injection (give te<O if not known)
%-----CORRECTION: build-up of inirial gradient---
if corrl==l
this was used for the error analys¡s!
incofile='incos4';
end
%-----CORRECTION: correction for time-lag of sorption--
if corr2== I
not appl¡ed for error analys¡s!
vsat=23:
qma>ePmax/Dvh:
ê-4.54e-5;
V=1.43e-6;
L:64e-3;
act=O.885;
k1=qmax/(acV2);
k2=k1:
k3:qmax/(l-act);
cl2:vsat*314*A*L:
C23:vsat*(V+ I / 4* A*L);
end
%-----CORRECTION: glycerol-water mixture---not applied for error analysis!
if glycerol==l
Vgw=le-6; o/ointtial volume of glycerol-water mixture (m3)
cgrv:O.5; Yointtial concentration of glycerol-water mixture (%glyc.)
end