[2005] IEEE. Reprinted, with permission, from [Wang, S.W. Su, B.G. Celler, and A.V. Savkin, Modeling of a Gas Concentration Measurement System,

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1 [2005] IEEE. Reprted, with perission, ro [Wang, S.W. Su, B.G. Celler, and A.V. Savk, Modelg o a Gas Concentration Measureent Syste, Engeerg Medice and Biology Society, IEEE-EMBS th Annual International Conerence o the, 2005 ]. This aterial is posted here with perission o the IEEE. Such perission o the IEEE does not any way iply IEEE endorseent o any o the University o Technology, Sydney's products or services. Internal or personal use o this aterial is peritted. However, perission to reprt/republish this aterial or advertisg or prootional purposes or or creatg new collective works or resale or redistribution ust be obtaed ro the IEEE by writg to pubs-perissions@ieee.org. By choosg to view this docuent, you agree to all provisions o the copyright laws protectg it

2 MODELING OF A GAS CONCENTRATION MEASUREMENT SYSTEM L. Wang, S.W. Su, B.G. Celler, and A.V. Savk Huan Perorance Lab, School o Electrical Engeerg & Telecounications University o New Sh Wales, UNSW Sydney N.S.W Australia ABSTRACT- Energy expenditure can be calculated via easureent o oxygen consuption and carbon dioxide production. Precise easureent o expired gas concentrations and volue is required or this deteration. For a given gas concentration easureent syste, the establishent o a odel is a good way to eectively use the equipents and achieve ore accurate energy expenditure calculations. This paper proposes a siple but eective approach or the odelg o a gas concentration easureent syste. I. INTRODUCTION Indirect calorietry is the calculation o energy expenditure via easureent o oxygen consuption ( VO 2 ) and carbon dioxide production ( VCO 2 )[][2]. However, the coplexity and expense o direct calorieters ade the ipleentation o this idea technically challengg [3][4]. The ost diicult part perhaps is the dynaic easureent o the concentrations o O 2 and CO 2 with the desired level o precision. The ajority o coercially available direct calorieters use ixg chabers [2] or the collection o expired gases. Mixg chabers use bales to terrupt gas low and thus prevent streag o gases and uneven gas concentrations. This is a necessary step to ensure desired calculation accuracy o energy expenditure. However, gas saples collected ro a ixg chaber are averaged gas ractions over tie. The sensitivity to changes etabolis is thereore reduced. Patents or iproved ixg chaber have been reported. However, or the user o coercial direct calorieters, an acceptable strategy is to set up a odel or the syste or the dividual user s settg and environent. This ay lead to ore eicient and accurate use o these systes. This paper vestigates the odelg o the gas (O 2 and CO 2 ) concentration easureent systes. These systes are typically used to estiate energy expenditure durg exercise based on expired air analysis. For breath-by-breath analysis, dynaic characteristics, such as tie delays and tie constant, are very iportant or dynaically estiatg energy expenditure. There is no report on the systeatic odelg o concentration easureent systes. This paper proposes a siple but eective approach or the easureent syste odelg. Firstly, the odel structure is obtaed based on physical prciples. Then, odel paraeters are detered by usg least square ethod and step change response analysis [5][6]. Fally, the overall syste is odeled by a two order lear syste with tie delay. It is ound that both the tie constant and tie delay are unctions o breath low-rate and are thereore not tie variant. In the next section, a brie description o the experiental setup is given. Model identiication is presented section III, and experiental results and analysis section IV. Douglas bag II. EXPERIMENTAL DESCRIPTION Calibration gas Abient air CO 2 sensor O 2 sensor Data collection syste Fig.. Experiental settgs. 240V ~ Pup Mixg chaber Turbe low eter CO 2 analyser O 2 analyser Supply tubg Mi-pup Measureent unit The experiental settgs are described Fig.. The concentration easureent syste has the ollowg parts: supply tubg, ixg chaber, CO 2 and O 2 sensors and CO 2 and O 2 analyzers. First, a Douglas bag is illed with calibration gas (6.3% O 2 and 4% CO 2 ), which is puped to the ixg chaber through the supply tubg with a constant low rate. Inlet low rate is easured by a turbe low eter. The puts o the analyzers under dierent let low rate are all

3 autoatically recorded by a coputer-based data collection syste. III. SYSTEM MODELING The typical puts o CO 2 analyzer and O 2 analyzer are given Fig 2. Fig. 3.a. Estiation results o tie delay or O 2. Fig 2.a. Typical puts o CO 2 analyzer. Fig. 3.b. Estiation results o tie delay or O 2. Fig. 2.b. Typical puts o O 2 analyzer. Fro Fig 2, it can be seen that the puts o the syste look like the put o a low order stable lear syste with tie delay. The tie delay is suarized Table I. Table I Test Flow Tie delay(s) No rate(liter/s) CO2 O Basis physical prciples together with the above experient results leads to the developent o a suitable odel structure. As the calibration gas takes tie to pass through the supply tubg, a tie delay caused by this is approxiately = Volue o supply tubg (2.7 liters) / let low rate. Based on the data Table I, a lear relationship has been ound between tie delay (τ) and reciprocal o let low rate. Paraeters were calculated as ollows (least square): O 2: τ 2.6 / 3. 9 () O = 2 CO 2: τ CO = 2.5/ 3. 0 (2) 2 The estiation results ab tie delay are suarized Fig. 3 and Fig. 4. Fig. 4.a. Estiation results ab tie delay or CO 2. Fig. 4.b. Estiation results or tie delay or CO 2. Now, we consider the odel structure o the ixg chaber (see Fig. 5.) Inlet C : let gas concentration : let lowrate Mixg Chaber Mixg an C: gas concentration ixg chaber V: volue o ixg chaber Fig. 5. A siple physical odel o the ixg chaber. Outlet C : let gas concentration : let lowrate

4 Based on physical prciples, it can be seen that C& = ( C C ) / V. Because the gas the ixg chaber is well ixed, it is reasonable to assue that C = C. Thus: C& = ( C C ) / V (3) Generally, equation () is a nonlear dierential equation. However, i we assue that let gas concentration (C ) and let low rate ( ) is constant or with a very slow variant rate, equation (3) can be sipliied as a irst order lear dierential equation: k (4) C = T s Where C k = and T = with the volue o ixg chaber V approxiately 4.2L. Thus, the ixg chaber can be described by a irst order syste. However, its DC ga k and tie constant are the unctions o let low rate and concentration C. Now, let us consider odel structure or the easureent unit (cludes sensors and analysers). Norally, these can be odeled by a irst order syste with tie delay. However, order to suppress high requency noise, we have added RC ilters to the put o the analysers. It would see reasonable to treat the easureent units as second order systes, however the tie constant or the RC ilters is less than 0.05 seconds, which is uch less than the tie constant o the easureent unit. Fro the ollowg results obtaed ro sgular perturbation analysis [7], it can be seen that the irst order odel is a rational selection: V.7 ( 3.4 s )(0.53s and or the CO 2 easureent unit: ( s )(0.48s ). ) Fally, the ollowg odels are achieved or the overall easureent systes under dierent low rates: O 2: CO 2: e 0.53s (2.6 / 3.9) s e 0.48s (2.5 / 3.0) s G G ixg ixg k Where G = as given equation (4). ixg 3.8 s IV. ESTIMATION RESULTS The ollowg igures (Figures 6-3) show the estiation errors o odels (5) and (6) 2. It can be observed that the estiation error is relatively sall when let low rate is low (.5 L/S, 3.4 L/S), and the estiation error is relatively high when let low rate is high (7.24 L/S,.8 L/S). The reason or the dierence ay be that the pup speed ay vary when the low rate is high. However, because the breath low rate or oderate exercisg levels is low, odels (5) and (6) are good odels or use energy expenditure easureents. (5) (6 ) When two or ore irst order stable syste are serially connected, i one o the syste has a very big tie constant T ax, then the overall syste can be sipliied as a irst order syste with tie constant T ax. Thereore, two serially connected irst order systes with tie delay can be selected as the odel structure o the overall easureent syste. Now, let us detere the paraeters o the second order syste or the overall easureent syste by usg the liit theores o the Laplace transor ethod (see [8]). Speciically, by applyg equation (36) o [8], we approxiately calculate the coeicients o the second order systes based on the data o the third experient (low rate 3.4). The second order odel or O 2 easureent unit is then given by: Fig. 6. Fig. 7. Norally, this condition can be satisied when dog ild strength exercises. 2 Only the dynaic part o the step response is copared the ollowg igures. The dierence o tie delay is shown igure 4 and 5.

5 Fig. 8. Fig. 3 V. CONCLUSION Fig. 9. We have presented this paper siple but practical odels or gas concentration easureent systes. Based on the odels derived, soe useul results can be achieved: The odel gives us soe hts to decrease the delay tie: a. Shorten the length o supply tubg as possible as we can. b. Increase low rate o the i-pup and shorten the length o the th pipe. The odel iplies that the ixg chaber is a low pass ilter. Soe high requency signal cannot be easured. Thus, the volue o the ixg chaber should be careully selected order to ensure terested requency band signal beg captured. Fig. 0 VI. ACKNOWLEDGMENT The authors grateully acknowledge the ancial support o the Australian Research Council (Grant DP045286). Fig. REFERENCES [] Elia M, Livesey G. Theory and validity o direct calorietry durg net lipid synthesis. A J Cl Nutr 988;47(4): [2] Ferranni E. The theoretical bases o direct calorietry: a review. Metabolis 988;37(3): [3] Branson RD, Johannigha JA. The easureent o energy expenditure. Nutr Cl Pract. 2004;9: [4] Keper MA. Indirect calorietry equipent and practical considerations o easureent. In: Weissan C, editor. Probles respiratory care: nutrition and respiratory disease. Philadelphia: JB Lippcott: 989;2: [5] Goodw, G. C., and R. L. Payne. Dynaic Syste Identiication, Experiental Design and Data Analysis. Acadeic Press, New York,977. [6]Ljung L. Syste identiication: theory or the user. Prentice-Hall, Inc., New Jersey, 987. [7] Kokotovic, P. V., Khalil, H. K. & O Reilly, J. Sgular Perturbation Methods Control: Analysis and Design., Acadeic Press, London etc.,986. [8] Rake. V. Step response and requency response ethods Autoatica, Volue 6, Issue 5, Septeber980,Pages: Fig. 2