Modeling of a gas concentration measurement system.

Abstract

Energy expenditure can be calculated via measurement of oxygen consumption and carbon dioxide production. Precise measurement of expired gas concentrations and volume is required for this determination. For a given gas concentration measurement system, the establishment of a model is a good way to effectively use the equipments and achieve more accurate energy expenditure calculations. This paper proposes a simple but effective approach for the modeling of a gas concentration measurement system.

Full text

Page 1

? Modeling of a Gas Concentration Measurement System
L. Wang, S.W. Su, B.G. Celler, and A.V. Savkin
Human Performance Lab, School of Electrical Engineering & Telecommunications
University of New South Wales, UNSW Sydney N.S.W. 2052 Australia
Abstract--Energy
measurement of oxygen consumption and carbon dioxide
production. Precise measurement of expired gas concentrations
and volume is required for this determination. For a given gas
concentration measurement system, the establishment of a
model is a good way to effectively use the equipments and
achieve more accurate energy expenditure calculations. This
paper proposes a simple but effective approach for the
modeling of a gas concentration measurement system.
expenditure can be calculated via
I. INTRODUCTION
Indirect calorimetry is the calculation of energy expenditure
via measurement of oxygen consumption (
dioxide production (
2
VCO )[1][2]. However, the complexity
and expense ofindirect
implementation of this idea technically challenging [3][4].
The most difficult part perhaps is the dynamic measurement
of the concentrations of O2 and CO2 with the desired level of
precision.
2
VO ) and carbon
calorimeters made the
The majority of commercially available indirect calorimeters
use mixing chambers [2] for the collection of expired gases.
Mixing chambers use baffles to interrupt gas flow and thus
prevent streaming of gases and uneven gas concentrations.
This is a necessary step to ensure desired calculation
accuracy of energy expenditure. However, gas samples
collected from a mixing chamber are averaged gas fractions
over time. The sensitivity to changes in metabolism is
therefore reduced. Patents for improved mixing chamber
have been reported. However, for the user of commercial
indirect calorimeters, an acceptable strategy is to set up a
model for the system for the individual user’s setting and
environment. This may lead to more efficient and accurate
use of these systems.
This paper investigates the modeling of the gas (O2 and
CO2) concentration measurement systems. These systems
are typically used to estimate energy expenditure during
exercise based on expired air analysis. For breath-by-breath
analysis, dynamic characteristics, such as time delays and
time constant, are very important for dynamically estimating
energy expenditure.
There is no report on the systematic modeling of
concentration measurement systems. This paper proposes a
simple but effective approach for the measurement system
modeling. Firstly, the model structure is obtained based on
physical principles. Then, model parameters are determined
by using least square method and step change response
analysis [5][6]. Finally, the overall system is modeled by a
two order linear system with time delay. It is found that both
the time constant and time delay are functions of breath
flow-rate and are therefore not time invariant. In the next
section, a brief description of the experimental setup is
given. Model identification is presented in section III, and
experimental results and analysis in section IV.
II. EXPERIMENTAL DESCRIPTION
~
240V
Douglas bag
Measurement unit
Calibration gas
Ambient air
Pump
Turbine flow meter
Mixing chamber
CO2sensor
O2sensor
Mini-pump
CO2 analyser
O2 analyser
Data collection system
Supply tubing
Fig. 1. Experimental settings.
The experimental settings are described in Fig.1. The
concentration measurement system has the following parts:
supply tubing, mixing chamber, CO2 and O2 sensors and
CO2 and O2 analyzers.
First, a Douglas bag is filled with calibration gas (16.00%
O2 and 4.00% CO2), which is pumped to the mixing
chamber through the supply tubing with a constant flow rate.
Inlet flow rate is measured by a turbine flow meter. The
outputs of the analyzers under different inlet flow rate are all
automatically recorded by a computer-based data collection
system.
Proceedings of the 2005 IEEE
Engineering in Medicine and Biology 27th Annual Conference
Shanghai, China, September 1-4, 2005
0-7803-8740-6/05/$20.00 ©2005 IEEE.
6695
Authorized licensed use limited to: IEEE Xplore. Downloaded on March 1, 2009 at 19:29 from IEEE Xplore. Restrictions apply.

Page 2

III. SYSTEM MODELING
The typical outputs of CO2 analyzer and O2 analyzer are
given in Fig 2.
0510 15202530 35 40
0
0.5
1
1.5
2
2.5
A typical CO2 analyser output
Time (Seconds)
CO2 Analyser output (Voltage)
Fig 2.a. Typical outputs of CO2 analyzer.
01020304050607080
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
A typical O2 analyser output
Time (Seconds)
O2 Analyser output (Voltage)
Fig. 2.b. Typical outputs of O2 analyzer.
From Fig 2, it can be seen that the outputs of the system
look like the output of a low order stable linear system with
time delay. The time delay is summarized in Table I.
Table I
Time delay(s)
CO2
4.53
3.745
3.5
2.97
Test
No
1
2
3
4
Flow
rate(Liter/s)
1.54
3.40
7.24
11.8
O2
5.53
4.885
4.4
3.89
Basis physical principles together with the above experiment
results leads to the development of a suitable model
structure.
As the calibration gas takes time to pass through the supply
tubing, a time delay caused by this is approximately
= Volume of supply tubing (2.7 liters) / inlet flow rate.
Based on the data in Table I, a linear relationship has been
found between time delay (?) and reciprocal of inlet flow
rate. Parameters were calculated as follows (least square):
O2:
9 . 3
?
/ 6 . 2
2
?
inO
f
?
(1)
CO2:
0 . 3
?
/ 5 . 2
2
?
in CO
f
?
(2)
The estimation results about time delay are summarized in
Fig. 3 and Fig. 4.
024681012
3.5
4
4.5
5
5.5
6
Time delay for O2
Flow rate (Liter/second)
Time delay (Seconds)
Measured time delay
Estimated time delay (2.6/fin +3.9)
Fig. 3.a. Estimation results of time delay for O2.
00.10.20.30.4 0.50.60.7
3.5
4
4.5
5
5.5
6
Time delay for O2
Reciprocal of flow rate (Second/liter)
Time delay (Seconds)
Measured time delay
Estimated time delay (2.6/fin +3.9)
Fig. 3.b. Estimation results of time delay for O2.
024681012
3
3.5
4
4.5
Time delay for CO2
Flow rate (Liter/second)
Time delay (Seconds)
Measured time delay
Estimated time delay (2.5/fin +3.0)
Fig. 4.a. Estimation results about time delay for CO2.
00.1 0.2 0.30.40.50.60.7
3
3.5
4
4.5
Time delay for CO2
Reciprocal of flow rate (Second/liter)
Time delay (Seconds)
Measured time delay
Estimated time delay (2.5/fin +3.0)
Fig. 4.b. Estimation results for time delay for CO2.
Now, we consider the model structure of the mixing
chamber (see Fig. 5.)
Inlet
Outlet
Mixing fan
Cin: inlet gas concentration
fin: inlet flowrate
Cout: outlet gas concentration
fout: outlet flowrate
C: gas concentration in mixing chamber
V: volume of mixing chamber
Mixing Chamber
Fig. 5. A simple physical model of the mixing chamber.
6696
Authorized licensed use limited to: IEEE Xplore. Downloaded on March 1, 2009 at 19:29 from IEEE Xplore. Restrictions apply.

Page 3

Based on physical principles, it can be seen that
VfCfCC
outoutinin
/ )(
??
. Because the gas in the mixing
chamber is well mixed, it is reasonable to assume that
CC ?
. Thus:
?
?
out
VCffC
(
C
outin in
/ )
??
(3)
Generally, equation (3) is a nonlinear differential equation.
However, if we assume that inlet gas concentration (Cin) and
outlet flow rate (fout) is constant or with a very slow variant
rate1, equation (3) can be simplified as a first order linear
differential equation:
k
sC
m
?
)(
1
)(sf
sT
in
m
?
(4)
Where
out
in
m
f
C
k ?
and
out
m
f
V
T
?
with the volume of mixing chamber V approximately 4.2L.
Thus, the mixing chamber can be described by a first order
system. However, its DC gain km and time constant are the
functions of outlet flow rate fout and concentration Cin.
Now, let us consider model structure for the measurement
unit (includes sensors and analysers). Normally, these can be
modeled by a first order system with time delay. However,
in order to suppress high frequency noise, we have added
RC filters to the output of the analysers. It would seem
reasonable to treat the measurement units as second order
systems, however the time constant for the RC filters is less
than 0.015 seconds, which is much less than the time
constant of the measurement unit. From the following results
obtained from singular perturbation analysis [7], it can be
seen that the first order model is a rational selection:
When two or more first order stable system are serially
connected, if one of the system has a very big time constant
Tmax, then the overall system can be simplified as a first
order system with time constant Tmax.
Therefore, two serially connected first order systems with
time delay can be selected as the model structure of the
overall measurement system.
Now, let us determine the parameters of the second order
system for the overall measurement system by using the
“limit theorems of the Laplace transform” method (see [8]).
Specifically, by applying equation (36) of [8], we
approximately calculate the coefficients of the second order
systems based on the data of the second experiment (flow
rate 3.4). The second order model for O2 measurement unit
is then given by:
1
7 . 3
?
s
and for the CO2measurement unit:
) 1
?
53. 0 )(1(
4 . 3
s
1 Normally, this condition can be satisfied when doing mild strength
exercises.
) 1
?
48 . 0)( 1(
1
4 . 3
9 . 3
?
ss
.
Finally, the following models are achieved for the overall
measurement systems under different flow rates:
O2:
)(
1 53. 0
1
) 9 . 3
?
/ 6 . 2 (
?
sG
s
e
mixing
sfin
?

(5)
CO2:
)(
148. 0
1
) 0 . 3
?
/ 5 . 2 (
?
sG
s
e
mixing
sfin
?
(6)
Where
1
8 . 3
f
)(
?
?
s
k
sG
in
m
mixing
as given in equation (4).
IV.ESTIMATION RESULTS
The following figures (Figures 6-13) show the estimation
errors of models (5) and (6)2. It can be observed that the
estimation error is relatively small when inlet flow rate is
low (1.5 L/S, 3.4 L/S), and the estimation error is relatively
high when inlet flow rate is high (7.24 L/S, 11.8 L/S). The
reason for the difference may be that the pump speed may
vary when the flow rate is high. However, because the
breath flow rate for moderate exercising levels is low,
models (5) and (6) are good models for use in energy
expenditure measurements.
0123456789
?0.5
0
0.5
1
1.5
2
2.5
Analyser output of CO2 (flow rate is around 3.4 L/S)
Time (Second)
Analyser output (Voltage)
Measured output
Estimated output
Fig. 6.
01234567
0
0.05
0.1
0.15
0.2
0.25
Analyser output of O2 (flow rate is around 3.4 L/S)
Time (Second)
Analyser output (Voltage)
Measured output
Estimated output
Fig. 7.
2 Only the dynamic part of the step response is compared in the following
figures. The difference of time delay is shown in figure 4 and 5.
6697
Authorized licensed use limited to: IEEE Xplore. Downloaded on March 1, 2009 at 19:29 from IEEE Xplore. Restrictions apply.

Page 4

02468101214 1618
?0.5
0
0.5
1
1.5
2
2.5
Analyser output of CO2 (flow rate is around 1.54 L/S)
Time (Second)
Analyser output (Voltage)
Measured output
Estimated output
Fig. 8.
0246810 1214161820
?0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
Analyser output of O2 (flow rate is around 1.5 L/S)
Time (Second)
Analyser output (Voltage)
Measured output
Estimated output
Fig. 9.
01234567
0
0.5
1
1.5
2
2.5
Analyser output of CO2 (flow rate is around 7.2 L/S)
Time (Second)
Analyser output (Voltage)
Measured output
Estimated output
Fig. 10
0123456
?0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
Analyser output of O2 (flow rate is around 7.2 L/S)
Time (Second)
Analyser output (Voltage)
Measured output
Estimated output
Fig. 11
00.51 1.522.5 33.5 44.5
0
0.5
1
1.5
2
2.5
Analyser output of CO2 (flow rate is around 11.8 L/S)
Time (Second)
Analyser output (Voltage)
Measured output
Estimated output
Fig. 12
00.5 11.5 22.533.5 4
?0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
Analyser output of O2 (flow rate is around 11.8 L/S)
Time (Second)
Analyser output (Voltage)
Measured output
Estimated output
Fig. 13
V. CONCLUSION
In this paper we have presented the simple but practical
models for gas concentration measurement systems. Based
on the models derived, some useful results can be
achieved:
The model gives us some hints to decrease the delay
time:
a. Shorten the length of supply tubing as possible as
we can.
b. Increase flow rate of the mini-pump and shorten the
length of the thin pipe.
??
??
The model implies that the mixing chamber is a low
pass filter. Some high frequency signal cannot be
measured. Thus, the volume of the mixing chamber
should be carefully selected in order to ensure interested
frequency band signal being captured.
VI. ACKNOWLEDGMENT
The authors gratefully acknowledge the financial support of
the Australian Research Council (Grant DP0452186).
REFERENCES
[1] Elia M, Livesey G. Theory and validity of indirect calorimetry during
net lipid synthesis. Am J Clin Nutr 1988;47(4):591-607.
[2] Ferrannini E. The theoretical bases of indirect calorimetry: a review.
Metabolism 1988;37(3):287-301.
[3] Branson R.D, Johannigham J.A. The measurement of energy
expenditure. Nutr Clin Pract. 2004;19:622–636.
[4] Kemper M.A. Indirect calorimetry equipment and practical
considerations of measurement. In: Weissman C, editor. Problems in
respiratory care: nutrition and respiratory disease. Philadelphia: JB
Lippincott: 1989;2:479- 490.
[5] Goodwin, G. C., and R. L. Payne. Dynamic System Identification,
Experimental Design and Data Analysis. Academic Press, New York,1977.
[6]Ljung L. System identification: theory for the user. Prentice-Hall, Inc.,
New Jersey, 1987.
[7] Kokotovic, P. V., Khalil, H. K. & O’Reilly, J. Singular Perturbation
Methods in Control: Analysis and Design., Academic Press, London
etc.,1986.
[8] Rake. V. Step response and frequency response methods Automatica,
Volume 16, Issue 5, September1980,Pages:519-526
6698
Authorized licensed use limited to: IEEE Xplore. Downloaded on March 1, 2009 at 19:29 from IEEE Xplore. Restrictions apply.