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ABSTRACT |
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TOP
ABSTRACT
INTRODUCTION
METHOD
RESULTS
DISCUSSION
REFERENCES
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Pump systems are currently used to test the performance of both
spirometers and peak expiratory flow (PEF) meters, but for certain
flow profiles the input signal (i.e., requested profile) and the output profile can differ. We developed a mathematical model of
wave action within a pump and compared the recorded flow profiles with both the input profiles and the output predicted by the
model. Three American Thoracic Society (ATS) flow profiles and
four artificial flow-versus-time profiles were delivered by a pump,
first to a pneumotachograph (PT) on its own, then to the PT with
a 32-cm upstream extension tube (which would favor wave action), and lastly with the PT in series with and immediately downstream to a mini-Wright peak flow meter. With the PT on its own,
recorded flow for the seven profiles was 2.4 ± 1.9% (mean ± SD)
higher than the pump's input flow, and similarly was 2.3 ± 2.3%
higher than the pump's output flow as predicted by the model.
With the extension tube in place, the recorded flow was 6.6 ± 6.4% higher than the input flow (range: 0.1 to 18.4%), but was
only 1.2 ± 2.5% higher than the output flow predicted by the model (range: 
0.8 to 5.2%). With the mini-Wright meter in series, the flow recorded by the PT was on average 6.1 ± 9.1% below the input flow (range: 23.8 to 2.5%), but was only 0.6 ± 3.3% above the pump's output flow predicted by the model
(range: 5.5 to 3.9%). The mini-Wright meter's reading (corrected for its nonlinearity) was on average 1.3 ± 3.6% below the
model's predicted output flow (range: 9.0 to 1.5%). The mini-Wright meter would be deemed outside ATS limits for accuracy for
three of the seven profiles when compared with the pump's input
PEF, but this would be true for only one profile when compared
with the pump's output PEF as predicted by the model. Our study
shows that the output flow from pump systems can differ from the
input waveform depending on the operating configuration. This
effect can be predicted with reasonable accuracy using a model
based on nonsteady flow analysis that takes account of pressure
wave reflections within pump systems.
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
METHOD
RESULTS
DISCUSSION
REFERENCES
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The use of positive displacement pump systems to test devices used for recording lung function has shown that many hand-held peak expiratory flow (PEF) meters do not record accurately (1, 2), and fail to perform to current specifications (3). These findings depend on the velocity of the piston of the pump being directly proportional to the displaced gas flow. However, it has been recognized that such pump systems may not be able to accurately reproduce certain flow profiles (1), especially when flow is delivered through devices such as a mini-Wright PEF meter (Clement Clarke, Ltd., Harlow, UK) (4), which has a higher impedance than a pneumotachograph. Methods to correct for any inaccuracy have recently been proposed (5, 6), and these use a quasistatic application of Boyle's law, assuming that gas compression is the cause of the errors. However, basic fluid mechanics theory (7) indicates that in an open system, such as the pumps described here, the air within the system will only be dynamically compressed if gas particle velocity divided by the speed of sound (i.e., the Mach number) exceeds 0.3. This criterion is not met with the use of pumps to test pulmonary function equipment.
We analyzed the nonsteady gas flow process of a positive displacement pump, and established a mathematical model with which to predict output flow profiles for a range of inputs and pump geometry. The developed model determines the effect of pressure wave reflections within the pump system, to see whether this can account for the observed differences between the flow that the pump is instructed to deliver (input flow) and the flow that is recorded at the output of the system (output flow). We used a pump with performance characteristics sufficient to produce the most demanding flow-versus-time profiles to test the accuracy of the model's predictions. We also obtained data from normal subjects and patients with airflow limitation to determine whether there is evidence of an effect of these pressure wave reflections in human lungs.
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METHOD |
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TOP
ABSTRACT
INTRODUCTION
METHOD
RESULTS
DISCUSSION
REFERENCES
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Pump
Pump systems in current use have deficiencies that limited their use for the present study, necessitating a new design. The main deficiencies of our original pump design (1) were that the seal was made of Teflon, requiring the pump cylinder to be flooded with silicone oil to effect a good seal, and also that the servo control was not very effective. On the positive side, the pump motor had a very low inertia that allowed rapid acceleration of the piston. With this pump we were able to demonstrate the inaccuracy of PEF meters (1), and this has since been confirmed by others (2). The pump design endorsed by the American Thoracic Society (ATS) (3) has several limitations, including high inertia and limited capability for generating fast response and high flow because of the stepper motor drive's bandwidth.
The design chosen for the newly designed pump in the study followed the earlier vertical-axis design (1), with the drive motor located beneath the cylinder and with the outlet on top. The design utilized as many commercially available components as possible, rather than requiring the manufacture of individual special components. The preliminary specifications for the new pump were: a volume displacement of 8 L with an accuracy of better than 5 ml over the full stroke of the piston; a maximum flow of 20 L/s being delivered within 20 ms; a maximum cylinder pressure of 8 kPa, with a maximum allowable leakage of 2 L/min up to this pressure; and a possible working temperature range of 15 to 35° C.
Tests with human subjects have indicated that fewer than 0.2% of subjects have a PEF that exceeds 15 L/s (8), and that the lower 5th percentile for a 10-to-90% rise time (RT) to PEF is 30 ms (9). On this basis, the specifications given earlier should easily exceed those needed to mimic human flow performance.
The pump has an aluminum chamber of 250-mm diameter and 10-mm wall thickness. The piston travel distance was 200 mm, which optimized the geometry for the choice of the pump motor and drive. The large piston diameter was selected because it made it easy to achieve both the force to displace the piston and the accuracy in piston position necessary to meet the stated volume resolution. If the piston diameter had been smaller, the required piston stroke would have been too large, and to generate a peak flow of 20 L/s the necessary piston velocity would have been too high. The dimension of the outlet from the pump was designed to accept standard mouthpieces for spirometers and PEF meters, with a 26-mm I.D. The radius of curvature of the inside edge from the endplate of the pump cylinder chamber to the outlet was more than 0.14 times the outlet tube diameter, in order to help prevent flow eddies and to minimize entry losses (10). The piston motion was provided by a rotating-ball leadscrew connected by a torsion coupling to a brushless direct current servo motor (Electrocraft 3633-3Y motor with AM30 drive; Rockwell Automation Ltd., Crewe, UK) that allowed the possibility of slippage if an extreme load was presented to the motor. The motor was part of a commercial system (Series 1 Motion Coordinator; Trio Motion Technology Ltd, Gloucester, UK) with integral amplifiers, an optical shaft encoder, and a servocontroller that could store 16 separate profiles in erasable, programmable, read-only memory (EPROM), for repetitive use without the need for a computer.
The pump cylinder endplate had two sets of pressure-release valves, with one set for flow in each direction (Figure 1), to prevent any undue wear on the motor if the pump were accidentally discharged with the outlet occluded. An independent pressure port was made in the endplate to allow recording of the cylinder pressure. The piston was fitted with two end-stop limit sensors to initiate an emergency shutdown of the motor in the event of a malfunction. A further "home position" sensor marked the usual start position of the piston. The optimum position of these sensors resulted in the swept volume of the pump between the limiter switches being reduced to 7.45 L.
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The proportional gain and the "stiffness" function of the motor
motion controller were varied to tune the system to achieve the optimal response to a square-wave input of 12 L/s in terms of no overshoot and fastest rise time (RT) from 10 to 90% of the PEF. An RT of
17 ms was achieved, with the piston developing a PEF of 12.1 L/s. Velocity feed-forward gain is normally used in position servo systems to
minimize the dynamic position error when a constant-velocity input is
applied. This was not critical in this application, since the only effect
of velocity feed-forward action would be to modify the delay between
input and output, and it would not change the waveform shape. A series of experiments was undertaken with two of the four artificial
flow-versus-time profiles shown in Figure 2, which spanned the 5th to
the 95th percentiles for RTs and dwell times (DTs) for the PEF found
in humans (9). These profiles were scaled to give a PEF of 12 L/s or
8 L/s. The profiles were used repeatedly, and the proportional software gain was adjusted until the best agreement was achieved between the input signal to the motor and the output piston motion measured by the optical shaft encoder. Regression analysis of the results gave the solution for the required proportional gain as: gain = 140 0.5 × DT90 + 0.125 × RT, where RT is the rise time and DT90
is the dwell time of flow above 90% of the peak. Using this algorithm,
we obtained the results for the two profiles with the extremes of RT
and DT that are shown in the Table 1.
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These results show that very precise flows and profiles can be generated by the piston displacement. The resolution of flow delivered depends directly on both the resolution of the the optical shaft encoder and on the sampling interval of 4 ms. The encoder resolution is to within 4 pulses, which in our configuration equates to a travel distance of 0.00127 cm and a delivered volume of 0.623 cm3. A volume difference of this magnitude when sampling at 4 ms equates to a possible variation in recorded flow of 0.156 L/s, owing to the resolution of the optical shaft encoder. The delivered piston flows were well within this limit, and the variation in delivered flow was always in the second decimal place of flow (L/s), which was more than adequate for the purposes of this pump. The formula given earlier for defining the relevant proportional gain was applicable only for human or similarly shaped waveform profiles. For other profiles, such as square-wave flow input and constant flow (ramp type) profiles, a different proportional gain setting was appropriate.
The possibility of leakage from the pressure-release valves was checked by sealing the outlet and using a constant, steady flow to approximate isothermal conditions. As the piston moved, the chamber pressure was recorded to determine whether there was any deviation in pressure and at what pressure the release valves would respond. The pressure when release occurred was 8 kPa, which was comfortably in excess of the 6 kPa expected from the maximum recommended resistance of 0.4 kPa/L/s for PEF meters (11) when tested at 15 L/s. The command to generate a displaced volume of 7 L within 4 ms resulted in a maximum flow of 34 L/s based on piston velocity. This defines the system slewing rate.
The Model
The positive displacement of a piston-based pump, with air as the working fluid, initiates a pressure wave disturbance in the piston chamber that is propagated through to the output tube. The system comprises a series of coupled elements. The piston in the pump is the wave generator, with the pump chamber being coupled to an outlet tube where entrance losses have been minimized by the design. This tube is then coupled to the complex impedance of the various flowmeters (pneumotachograph and/or mini-Wright meter) attached to the end of the tube, and the system ultimately exhausts to the atmosphere. Analysis of the system is based on deriving nonsteady flow energy equations for the various elements of this coupled system. A full explanation of this is given in the APPENDIX.
Under the normal working conditions for the system, with moderate pressure and temperature, the flow behaves like that of a perfect isentropic gas, and the disturbance can be considered as a plane wave. The maximum cylinder pressure during usual test conditions is below 2 kPa, and the maximum desired volumetric flow of the system is 20 L/s. A flow of 20 L/s implies a particle velocity of about 3.8 m/s through the pump's outlet, giving a Mach number of only 0.01. Consequently, the compressibility effects of the system are negligible (7).
Under these circumstances, a much simpler incompressible model
can be built. Although density does change within the thin layer next
to the pressure wavefront, the change in density in the bulk of air in
the chamber is very small and thus can be ignored. In other words, the
air behaves like a higher-density fluid such as water. The wave speed,
a, is constant, and is given as: 
, where K is the ratio of specific heats, and 
is the density of air.
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where A is the area, p is the pressure, u is the flow velocity, t is time, and x is a displacement in the direction of flow.
The nonsteady flow equation is:
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where D is the pump cylinder diameter, and f is the friction force at the wall per unit mass of fluid, which in this instance is air. A solution to these equations was obtained with the method of characteristics (12), and a detailed analysis of wave effects is outlined in the APPENDIX.
In simple terms, the analysis predicts the flow motion as the piston starts to accelerate and creates a pressure wave within the gas that moves at the speed of sound within the cylinder. A complex system of pressure wave reflections is set up, which continuously change as the piston moves through its stroke length. The main factors that contribute to the process are: the pump cylinder dimensions; the outlet configuration (length and diameter of the tube); the outlet coefficient of discharge (Cd), which relates to the total outlet impedance; the local speed of sound, which depends on gas density and temperature; and the shape of the flow profile to be delivered.
A computer program of the solution was derived that allows the output waveform to be predicted for any input waveform and set of pump parameters. The Cd for the PT and the mini-Wright meter were determined as 0.68 and 0.38, respectively. These values were held constant, and gave the best fit for all experimental data.
Experimental Procedure
The four artificial flow-versus-time profiles (Figure 2) used in the study were of a similar shape to that for human subjects, and could be scaled to give any PEF. In addition, three of the ATS flow-versus-time profiles (3) recommended for testing PEF meters were used (Profiles 10, 15, and 26), and had the extremes for RT and DT among these profiles. Flows having these profiles were delivered to an optimized pneumotachograph (PT) when it was mounted directly on the pump and when a 32-cm extension tube was inserted upstream of the PT. This was repeated with a mini-Wright meter placed between the pump and the PT, with the mini-Wright meter enclosed in a Perspex cylinder so that all the exhausted air from the meter was collected to be passed through the PT (13). The effect of changing the start volume of gas within the pump was also checked, using one of the artificial profiles.
To test whether wave action was present in human lungs during a maximum forced expiratory maneuver, 15 healthy subjects and 15 patients with airflow limitation performed three PEF maneuvers into a PT with and without a 32-cm extension tube added upstream of the PT, doing this in random order. The subjects were naive about how this might influence the recordings. The highest achieved PEF was recorded for each configuration.
Statistics
All data were analyzed with the SPSS software system for Windows,
version 6.0 (SPSS Inc., Chicago, IL), and a probability of < 5% was
taken as significant. Null hypotheses for differences between input
flow and recorded PEF from the pump, and for the PEF for human
subjects with and without the extension tube, were tested with a paired
t test. Null hypotheses for differences in RT for the subjects with and
without the extension tube were tested with Wilcoxon's signed ranks
test. All lung function data from subjects were related to their predicted values, using the method of standardized residuals (14, 15),
which are determined as: Standardized residual = (Recorded value Predicted value)/RSD, where RSD is the residual standard deviation
from the regression equation used.
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RESULTS |
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METHOD
RESULTS
DISCUSSION
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The RTs and DTs for the four artificial profiles and three ATS
profiles are shown in Table 2. The results for recorded PEF, input PEF, and PEF predicted from the model for these profiles with and without the extension tube are shown in Table 3.
With the extension tube in place, the flow recorded by the PT
was 6.6 ± 6.4% (mean ± SD) above the input flow (range: 0.1 to 18.4%), but remained at just 1.2 ± 2.5% above the output
flow predicted by the model (range: 0.8 to 5.2%). The PT recordings of PEF without the extension tube were all within
ATS limits for accuracy (3) when compared with input PEF.
With the extension tube, PEFs for Profiles 1, 2, and 4 were
outside these limits as compared with input PEF, but for only
Profile 1 was PEF above the limit when compared with the
model's predicted PEF.
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When the mini-Wright meter was added in its Perspex
holder, the results in L/s given by the PT placed downstream
from the mini-Wright meter (corrected for the inherent error
of the meter [1]) are shown in Table 4. The flow recorded by
the PT was now on average 6.1 ± 9.1% below the input flow
(range: 23.8 to 0.1%), but was only 0.6 ± 3.3% above the
predicted output flow (range: 5.5 to 3.9%). The corrected
mini-Wright meter reading was 1.3 ± 3.6% below the model's
predicted output flow (range: 9.0 to 1.5%). The profiles with
the shortest DTs, Profiles 1 and 3, gave the biggest discrepancies between the input flow and the predicted output flow, the
flow recorded by the PT, and the recorded flow with the mini-Wright meter. For Profiles 1, 3, and ATS26, the PEF with the
mini-Wright meter was outside ATS accuracy limits as compared with input PEF, but only for Profile 1 was PEF outside the limits when compared with the model's predicted PEF.
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Figure 3 shows the input, recorded, and predicted flow profiles for ATS Profile 26 and for artificial Profiles 1 and 3. It can be seen that the model also predicted the presence of wave phenomena in the buildup to PEF for Profile 3, which was not on the input signal, and also in the drop-down from PEF for Profiles 1 and 3. Wave effects were also evident on the pressure recordings from the pump cylinder. However, the model did not perfectly match the phase timing of these waves, which may have occurred because of servo following error.
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The effect of changing the pump starting volume, using a profile with an RT of 28 ms and DT90 of 34 ms, is shown in Figure 4 and Table 5. The model and the recorded findings were in close agreement, in that when the piston started from the bottom of the pump, the recorded PEF was lower than when it started nearer the top of the chamber, owing to the greater chamber length in which wave effects could occur.
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The data for the short tube configuration for the healthy subjects and those with airflow limitation (AL) are given in Table 6, with the spirometric data also expressed as standardized residuals (SRs). The results for PEF and RT are shown in Table 7 for the configuration with and without the 32-cm extension tube. There were no significant differences between the PEF recordings (p > 0.7, paired t test) and the RT readings (p > 0.48, Wilcoxon's signed ranks test). There was no significant correlation between any change in PEF and the absolute RT for either group of subjects.
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DISCUSSION |
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ABSTRACT
INTRODUCTION
METHOD
RESULTS
DISCUSSION
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Data have been presented indicating that the use of pump systems to test PEF meters is a complex undertaking, and that both the impedance characteristics of the device being tested and the pump configuration have an important influence on the result. Other workers have presented data confirming discrepancies between input and output when using pump systems in this way (5, 6), but these workers have assumed that the problem was due to gas compression within the pump chamber, and that this dynamic compression led to a reduction in the generated flow. We contend that these discrepancies are due to wave effects within pump systems. Both the experimental results presented here and the analysis done with our model indicate that changes in outlet geometry and impedance can lead either to a reduction or to an increase in transient flow output relative to the input waveform.
Although the idea of compression due to piston motion is intuitively reasonable, air does not necessarily behave as a compressible fluid under these circumstances. Boyle's law can be applied under steady-state conditions in a closed system, but when a pump is used to test PEF meters, the system is open at one end and conditions are never those of the steady state. If Boyle's law is applied with a freeze-frame approach, all dynamic aspects of the system are ignored. This compression approach may give an approximate correction for the errors involved (5, 6), but fails to consider the complex dynamic transients within the pump cylinder. For instance, the compression approach could never predict the finding of a greater output flow than input flow, as demonstrated at the bottom of Figure 3. With the pump under consideration, and with an outlet tube of 26 mm I.D., the local Mach speed will only reach the critical value of 0.3 when the exhaust flow from the pump is about 35 L/s. However, the slewing speed of the system is 34 L/s, and significant dynamic gas compression effects therefore do not ordinarily occur. When a mini-Wright meter is attached, the outlet area is different and so the local particle velocity is different. With a flow of 800 L/min the variable orifice area developed in a mini-Wright meter is such that the local Mach number is well below 0.1, and significant dynamic gas compression effects therefore do not occur.
The gas flow analysis used in our study identifies wave action within pump systems as being significant, and has yielded reasonably accurate predictions of performance. The data from human subjects indicated no evidence of a wave action developing within the bronchial tree. This is intuitively as expected, in that the multiple branches of the bronchial system would not facilitate the propagation of compression or rarefaction pressure waves along the airways. Although the model derived here is a substantial aid in predicting the performance of the system, it suffers from the disadvantage of relying on the use of an empirical use of a discharge coefficient (Cd) for the device applied to the outlet of the pump. It is not possible to accurately measure the exact Cd for an instrument, and for devices offering a complex impedance to flow, the Cd may change under different operating conditions. This limits the usefulness of the approach used in our study, but does not negate its importance in understanding what contributes to the performance of pump systems. For instance, the model predicts that input profiles with very short DTs lead to greater reduction in output PEF from a pump through a mini-Wright meter than do profiles with very short RTs, as shown in Figure 5 and confirmed by our experiments.
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The ATS has recommended the use of 26 arbitrarily chosen flow-versus-time profiles for testing PEF measuring devices (3). A different approach would be to use a more limited test regimen such that one can be confident that the devices being tested can accurately record PEF for at least 95% of the target population. The 5th to 95th centiles for RT and DT for PEF have recently been published for subjects with airflow limitation (9), and both are shorter than that found for healthy subjects. These data were used in this study to derive artificial profiles that spanned the ranges found. The performance characteristics of PEF meters that need to be tested are their linearity and accuracy, frequency response, and resistance. PEF meters should provide linear readings across the desired range, and this property can be tested with pump systems having constant flow profiles, profiles with slow RTs, or cusp profiles (1) all of which are free of wave effects in pump systems. It is easy to measure the PEF meters' resistance at the same time. Meters must also have a sufficiently good frequency response to record PEF when the RT and DT are short. The frequency response of a PT can be verified by a variety of means (16) that are suitable for a PT, which has a continuous analogue output, but which are not suitable for PEF meters. For PEF meters with no analogue output, the frequency response could be tested by using a single profile with an RT and DT chosen to be at the lower 5th centile for subjects with airflow limitation.
The use of an explosive decompression device (19) to deliver this type of profile may avoid these problems, and such a device has been modified to be able to do this (20). The principle of an explosive decompression device is to pressurize a chamber of known volume to a known pressure, and to then release the gas suddenly. The gas can be delivered in such a way as to have a flow profile of a specified shape by controlling the shape of the outlet. In this system there is no piston accelerating the gas, and wave effects are therefore less likely. This sort of device may provide a way to calibrate hand-held meters and check that their performance is adequate for recording flow for human subjects. The only drawback with an explosive decompression device is that it needs independent calibration with a flow meter that is linear, has an adequate frequency response, and has a continuous analogue output, such as a PT. However, this disadvantage is also true for the previously suggested means for correcting a pump's output (5, 6) by iteratively changing the input profile until a PT records the desired output flow.
We conclude that discrepancies between the input and output from pump systems can be explained by the application of fluid mechanics and wave action. Such discrepancies limit the ability of a pump to deliver flow-versus-time profiles with RTs and DTs. The linearity of PEF meters can be successfully tested with a pump, but the frequency response aspects of their performance may be best tested by using a suitable flow profile provided by an explosive decompression device.
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Footnotes |
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Correspondence and requests for reprints should be addressed to Dr. M. R. Miller, Department of Medicine, University Hospital Trust, Selly Oak Hospital, Birmingham B29 6JD, UK. E-mail: m.r.miller{at}bham.ac.uk
(Received in original form May 27, 1998 and in revised form September 20, 1999).
Acknowledgments: Supported by Contract MAT1-CT-930032 from the European Community.
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References |
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TOP
ABSTRACT
INTRODUCTION
METHOD
RESULTS
DISCUSSION
REFERENCES
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1.
Miller, M. R.,
S. A. Dickinson, and
D. J. Hitchings.
1992.
The accuracy
of portable peak flow meters.
Thorax
47:
904-909
2.
Gardner, R. M.,
R. O. Crapo,
B. R. Jackson, and
R. L. Jensen.
1992.
Evaluation of accuracy and reproducibility of peak flowmeters at
1,400 m.
Chest
101:
948-952
3. American Thoracic Society. 1995. Standardization of spirometry: 1995 update. Am. J. Respir. Crit. Care Med. 152: 1107-1136 [Medline].
4. Wright, B. M.. 1978. A miniature Wright peak-flow meter. Br. Med. J. 2: 1627-1628 .
5. Hankinson, J. L., J. S. Reynolds, M. K. Das, and J. O. Viola. 1997. Method to produce American Thoracic Society flow-time waveforms using a mechanical pump. Eur. Respir. J. 10: 690-694 [Abstract].
6. Navajas, D., J. Roca, R. Farré, and M. Rotger. 1997. Gas compression artefacts when testing peak expiratory flow meters with mechanically-driven syringes. Eur. Respir. J. 10: 901-904 [Abstract].
7. Mironer, A. 1979. Engineering Fluid Mechanics. McGraw Hill, New York. 506-564.
8. Miller, M. R., D. M. Grove, and A. C. Pincock. 1985. Time domain spirogram indices: their variability and reference values in non-smokers. Am. Rev. Respir. Dis. 132: 1041-1048 [Medline].
9.
Miller, M. R.,
O. F. Pedersen, and
P. H. Quanjer.
1998.
The rise and
dwell time for peak expiratory flow in patients with and without airflow limitation.
Am. J. Respir. Crit. Care Med.
158:
23-27
10. Massey, B. S. 1970. Mechanics of Fluids. Van Nostrand Reinhold, New York. 185-187.
11. Quanjer, P. H., M. D. Lebowitz, I. Gregg, M. R. Miller, and O. F. Pedersen. 1997. Peak expiratory flow: conclusions and recommendations of a Working Party of the European Respiratory Society. Eur. Respir. J. 10(Suppl. 24):2s-8s.
12. Wylie, E. B., and V. L. Streeter. 1978. Fluid Transients. McGraw Hill, Hew York. 31-65.
13. Pedersen, O. F., T. R. Rasmussen, O. Omland, T. Sigsgaard, P. H. Quanjer, and M. R. Miller. 1996. Peak expiratory flow and the resistance of the mini-Wright peak flow meter. Eur. Respir. J. 9: 828-833 [Abstract].
14.
Miller, M. R., and
A. C. Pincock.
1988.
Predicted values: how should we
use them?
Thorax
43:
265-267
15. Quanjer, P. H., G. J. Tammeling, J. E. Cotes, O. F. Pedersen, R. Peslin, and J.-C. Yernault. 1993. Standardized lung function testing. Eur. Respir. J. 6(Suppl. 16):5-40.
16. Andersen, H. R., and O. Bergsten. 1987. Blood Pressure Measurements and Methods, 2nd ed. S&W Medico Teknik A/S, Albertshund, Denmark. 54-65.
17.
Jackson, A. C., and
A. Vinegar.
1979.
A technique for measuring frequency response of pressure, volume, and flow transducers.
J. Appl.
Physiol.
47:
462-467
18. Zock, J. P.. 1981. Linearity and frequency response of Fleisch type pneumotachometers. Pflügers Arch. 391: 345-352 [Medline].
19. Pedersen, O. F., N. Naeraa, S. Lyager, C. Hilberg, and L. Larsen. 1983. A device for evaluation of flow recording equipment. Bull. Eur. Physiopathol. Respir. 19: 515-520 [Medline].
20. Pedersen, O. F., T. R. Rasmussen, S. K. Kjaergaard, M. R. Miller, and P. H. Quanjer. 1995. Frequency response of variable orifice type peak flow meters: requirements and testing. Eur. Respir. J. 8: 849-855 [Abstract].
21. Bannister, F. K. 1958. Ackroyd Stuart Memorial Lectures, Vol. 2: Pressure Waves in Gases in Pipes. University of Nottingham, Nottingham, UK.
22. Bannister, F. K., and G. P. Mucklow. 1948. Wave action following sudden release of compressed gas from a cylinder. Proc. Inst. Mech. Engrs. 159: 269 .
23. Benson, R. S. 1982. The solution of non-steady flow equations by the method of characteristics. In J. H. Horlock and D. E. Winterbone, editors. The Thermodynamics and Gas Dynamics of Internal Combustion Engines, Vol. 1. Clarendon Press. Oxford, UK. 73-97.
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APPENDIX |
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The method adopted for the analysis of wave effects in fluids depends upon the magnitude of the changes in pressure and particle velocities. If the changes are very small, then acoustic wave theory can be used, and all points on the wave are considered to be propagated with the same velocity (i.e., the local speed of sound). It has been shown that numerical solutions based on acoustic wave theory have negligible error, provided that the pressure changes are less than 7.5 kPa (21, 22). These conditions pertain in a positive displacement, piston-based device as illustrated in the subsequent discussion.
In the chamber filled with air, the pressure disturbance initiated by piston movement is propagated through the air. The propagation speed is that of sound-wave speed. For disturbances of low amplitude, the resultant change in thermodynamic properties can be neglected. The sound speed in a perfect gas can be calculated from:
![α=&cjs0371;[γRT]](/content/vol161/issue6/fulltext/1887/img004.gif)
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(1) |
where 
is the ratio of specific heat (1.4 for air), R is the gas
constant (287.04 m2/s2 K for air), and T is the thermodynamic
temperature. For normal room conditions, the speed of a
sound wave is about 340 m/s. For systems like that shown in
Figure 6, an airflow output V(t) is the result of an input in the
form of piston movement v(t), or:
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(2) |
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where n1, n2,...nn denote the other relevant variables.
Under normal working conditions for the pump system, the airflow behaves like a perfect isentropic gas flow. The disturbance induced by the piston movement can be considered as a plane wave. Thus, basic flow equations can be derived from the flow analysis based on the element shown in Figure 7. If the side area of an element whose length is dx changes from A to A + dA, then the volume of the element is given by V= (A + dA/2)*dx, and can be simplified as A*dx.
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Continuity Equation
The unsteady continuity equation for the fluid element shown in Figure 7 sets the increase in mass in the control volume equal to the net rate of flow into the control volume.
The increase in mass in the control volume is:
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The net rate of flow into the control volume is:
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Consequently, the continuity equation is:
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Expanding this equation and dividing it through by dx results in:
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Dividing the preceding equation by A and regrouping gives:
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The total time derivative term states that for any variable N = N(t,x(t)), the total time derivative of N is:
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Therefore, the continuity equation can be written as:
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(3) |
Momentum Equation
The momentum equation states that the rate of change of momentum within the control volume equals the sum of forces applied to the surface of the control volume. The surface forces include pressure forces and shear forces on the control volume. The three pressure forces are the pressure forces applied to both ends of the pump cylinder and the pressure force applied to side surface (p is used as the distributed pressure on the side for simplicity), and are given as:
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The shear force, due to friction at the wall, is 
Adx, where is the friction force per unit mass of fluid.
The rate of change of momentum of the control volume consists of the net change of momentum in the control volume and the net efflux of momentum from the control surface. This can be written as:
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where the last three terms inside the bracket are those of the continuity equation (Equation 3):
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Hence the momentum equation is:
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Expanding this and dividing by Adx gives:
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(4) |
The friction term is rewritten through the use of Darcy's formula,
and f is the friction factor. The friction factor is determined from
the Moody diagram (12), and depends on the cylinder material, the inner dimension of the cylinder, and the inner wall roughness. The term 
is introduced to ensure that the wall friction always acts in the opposite direction to the direction of flow.
The Incompressible Gas Model
The volume flow of the system is of the order of 12 L/s, which implies a particle velocity of air much smaller than 1 m/s and a Mach number for this flow that is far less than 0.1. The compressibility of gas can be neglected and a much simpler incompressible model can be built. In using such a model, it is assumed that the density of the air is not changed significantly during the process, and that the air behaves as a fluid with a higher density, such as water. In fact, although the density changes within the thin layer next to the pressure wavefront, the change in density of the bulk of fluid within the chamber is small, and can be neglected.
The definition of the bulk modulus of elasticity of a fluid is:
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where V is volume, and yields:
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(9) |
The wave speed, a, is constant, and can be defined as:
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(10) |
Using flow rate, q = Au, Equation 3 is then obtained as:
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(11) |
and Equation 4 is rewritten as:
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(12) |
The Method of Characteristics
The problem of complex flow can be solved by the method of
characteristics (23). The partial differential Equations 11 and 12 can be manipulated to yield two normal differential equations applied to two sets of lines, the characteristic lines. First, Equation 11 is multiplied by a factor 
, and the result is then added to Equation 12, giving:
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(13) |
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(14) |
Equation 13 can then be written as:
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(15) |
In the t-x plane, the grids formed by the interception of the two sets of lines are not strictly regular rectangles. However, since the particle velocity, u, is far smaller than wave speed, the rectangular grid can be considered to be as regular as the space-time grid shown in Figure 8.
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Equation 14 defines two groups of lines in the t-x plane, which are termed characteristics lines. In other words, the Equations 11 and 12 are converted to a single ordinary differential equation with an independent variable, as in Equation 15.
In a section of pipe shown in Figure 8, x locates a point and
t is the time at which the dependent variables p and q are to be
determined. The interface variables of the pipe are state variables for both ends of the pipe. This section of pipe is divided
into N subparts with N + 1 points. The state variables are sampled at discrete instants of time. The time interval between
two sampling instants is 
t, during which the variables are assumed to be unchanged.
![X=[p<SUB>1,</SUB>p<SUB>2,</SUB>…p<SUB>n+1,</SUB>q<SUB>1,</SUB>q<SUB>2,…</SUB>..q<SUB>n+1</SUB>]](/content/vol161/issue6/fulltext/1887/img034.gif)
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(16) |
First, the equation along the positive characteristic line, noted
as C+ in Figure 8, is considered. Multiplying Equation 15 by
aAdt (with the + sign only), and integrating from A to P,
gives:
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Since adt = dx, the equation can be written in finite-difference form:
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(17) |
The integration of the friction term is an approximation, and is
adequate when x and t are small enough. The corresponding C
equation is integrated in a similar manner, the only difference being that adt = dx. This gives:
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(18) |
If the state variables at points A and B are known, the equations can be solved simultaneously to determine variables qp and pp.
The method can be applied to solve for all of the variables
for the pipe. If state variable set X, at time t = t0 = kt, is known, then the value at t = t0 (k 1)t can be determined from conditions involving interface variables and the following 2N equations:
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(19) |
where CPI and CMI are constants.
Therefore, by collecting the known variables b = a/A, and
substituting:
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then CPI and CMI are given as:
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(20) |
All the 2(N-2) internal variables for the pipe can be calculated by solving the 2(N-2) equations from Equation 19. Variables at points I = 1 and I = N + 1 can be determined with two boundary conditions and the two remaining equations from Equation 19.
Boundary Conditions
The analysis of the pump system is based on the so-called "object-oriented technology." The system is considered to consist of two components (i.e., two component objects in the program) and three interfaces between them, and the environment (three linkers, or interface objects, in the program). The two components are the air in the pump cylinder and the pipe, which interact with each other and with the environment via the three linkers. The distributed state variables are used to describe the system dynamics, and the state variable at one space point at time t0 is therefore decided by the states at its neighboring space points at t0 and at time t0+dt. The three linker functions are defined as follows.
For the pump cylinder, the linker relates piston velocity (piston.v) to the flow developed within the cylinder adjacent to the piston (pq1), and is given as: pq1 + A · piston.v, where A is the piston cross-sectional area.
The chamber-pipe linker is a typical series coupling that relates pressure and flow conditions at the inlet (cham.ppn+1 and cham.pqn+1) to those at the outlet (pipe.pp1, pipe.pq1) for a section of pipe, and is given by: cham.ppn+1 = pipe.pp1, and cham.pqn+1 = pipe.pq1.
The state variables have to be noted with reference to the space distance and time history. For instance, the term cham.ppn+1 refers to the pressure in the pump cylinder at point n + 1 at time t0 + dt. The equation cham.ppn+1 = pipe.pp1 means the pressure in cylinder chamber at point n + 1 is equal to the pressure in the pipe at point 1.
The pipe-meter linker relates the coupled mini-Wright meter to the outlet:
![pipe.pq<SUB>n+1</SUB>=AC<SUB>d</SUB><RAD><RCD>[2(pipe.pp<SUB>n+1</SUB>−P<SUB>0</SUB>)/ρ]</RCD></RAD>](/content/vol161/issue6/fulltext/1887/img041.gif)
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where Cd is the equivalent flow discharge coefficient of the system. The discharge coefficient Cd can be determined from the Bernoulli equation, which gives,
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where A is the cross sectional area of the flow channel in the
meter, is the gas density, and p is the pressure drop across the meter. p and Q can be measured experimentally. This
steady-flow equation is valid, given that the pressure changes
are small relative to the ambient pressure.
The values for Cd and the friction factor f in the model will vary with both the flow and pressure level within the system. As these change dynamically, Cd and f will vary during the delivery of gas from the pump. However, these changes in Cd and f are small, and can be ignored when considering problems of this sort, where the pressure changes are small (12). Cd values of 0.68 and 0.38 have been selected for the PT and mini-Wright meter, respectively, since they best fit all experimental data.
The input parameters for the computer model are: (1) length of pump chamber; (2) diameter of pump chamber; (3) length of outlet tube; (4) diameter of outlet tube; (5) equivalent discharge coefficient Cd; and (6) flow-versus-time input profile of piston movement. The output parameters of the computer model are: (1) the outlet flow profile with respect to time; (2) the chamber pressure profile with respect to time; and (3) the outlet tube pressure profile with respect to time.
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