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A novel gas divider using nonlinear laminar flow
Flow Measurement and Instrumentation xx (xxxx) xxxx–xxxx
Contents lists available at ScienceDirect
Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
A novel gas divider using nonlinear laminar flow ⁎
J. Nuszkowski , J. Schwamb, J. Esposito School of Engineering, University of North Florida, Jacksonville, FL, USA
A R T I C L E I N F O
A BS T RAC T
Keywords: Gas divider Laminar flow Flow measurement Compressible fluid Capillary tube Gas dilution
Gas dividers are important in emissions measurement since they continuously and accurately mix two gases to create a known gas concentration that is needed in the multi-point calibration of gas analyzers. A novel gas divider was designed using nonlinear laminar flow induced from the density change along the capillary channels due to the high-pressure drop (relative to the inlet gas pressure). The minor losses from entrance and exit effects can be ignored due to the high pressure loss from Hagen-Poiseuille's law relative to the minor losses. Small diameter wires inside of a tube were used to create capillary channels through which gas could flow. The gas divider, using nonlinear laminar flow, showed lower measurement uncertainty at high (90%) dilution levels than using linear laminar flow due to the higher-pressure drop at the same volumetric flow rates. Experiments showed the expected gas concentration from using the gas divider to be within 2% of the measured gas concentrations.
1. Introduction Gas dividers are widely used in environmental pollution control, such as automotive and exhaust plumes for the calibrating of gas analyzers used in the measurement of emissions from different pollution emitters. The gas analyzers require multi-point calibration via a gas divider that accurately and continuously mixes two gases at known concentrations. Gas dividers require accurate regulation and measurement of the flow rate from each gas stream. Some of the commonly used techniques for flow measurement with gas dividers are volumetric pumps, critical orifices, thermal mass-flow controllers, and capillary tubes [1,2]. A study by Wright and Murdoch [3] investigated four different commercially available gas dividers. The four gas dividers used were thermal mass flow controllers, capillaries, positive-displacement pumps (varying stroke frequency), and positivedisplacement pumps (varying the number of pumps), respectively. Sherman et al. [4] developed a gas divider based on mass flow technology that achieved residual errors of less than 1% of point for most of the tested analyzer ranges. When using capillary tubes, the small diameter of the capillary generates the laminar flow inside the gas divider. The internal laminar flow follows the Hagen-Poiseuille law and the flow rate increases linearly as the pressure drop across the capillary channel increases. For larger flow rates, multiple capillary tubes of the same diameter are arranged in parallel. Therefore, to have a higher flow rate while still maintaining laminar flow, the gas divider requires more capillary tubes [5]. These laminar flow meters are common in industry to measure the
⁎
flow rate of clean gases. Authors [6,7] have studied the use of a concentric annular duct instead of capillary tube bundles to generate linear laminar flow. Priestman and Boucher [8] showed that the range of other flow meter types could be extended by using a laminar by-pass resistance. The objective of this study was to develop a novel gas divider using a compressible fluid with a high-pressure drop (with respect to the inlet gas pressure) that created nonlinear laminar flow. A gas divider was designed, constructed, calibrated using a reference flow meter, and experimentally verified using an emissions analyzer. In addition, equations were developed that demonstrate the behavior of the nonlinear laminar flow and propagation of error analysis was performed on the gas divider when using nonlinear or linear laminar flow channels. 2. Theory For a gas divider, diluting one gas concentration (by volume) to lower gas concentrations (by volume) requires the mixing of two gas streams. The first gas stream has a gas concentration (by volume) for a specific gas species, called the component gas, and the remainder of that gas stream is another gas species, called the balance gas. The second gas stream consists of only the balance gas. The exit concentration (by volume) of the component gas is dependent on the volumetric flow rates of the gas streams and the inlet concentration (by volume) of the component gas species, as shown in Eq. (1). When diluting a gas with a low concentration ( < 1%), the variation in the densities between the two gas streams and the outlet can be neglected as shown in Eq. (2).
Corresponding author.
http://dx.doi.org/10.1016/j.flowmeasinst.2016.10.016 Received 27 April 2016; Received in revised form 4 October 2016; Accepted 12 October 2016 Available online xxxx 0955-5986/ © 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: Nuszkowski, J., Flow Measurement and Instrumentation (2016), http://dx.doi.org/10.1016/j.flowmeasinst.2016.10.016
Flow Measurement and Instrumentation xx (xxxx) xxxx–xxxx
J. Nuszkowski et al.
Qo,1 Qo,2 R ReD T Vi Vo x Yo Y1 α β ΔP
Nomenclature A d dh dt dw Γ Ki Ke L Le ṁ n P Pi Po Po,cal Q Qo Qo,corr
Yo=Y1
Yo=Y1
Cross sectional area Diameter of capillary channels Hydraulic diameter of the capillary channels Inside diameter of the tube Wire diameter Wetted perimeter Entrance loss coefficient Exit loss coefficient Length of the capillary channels Entrance length Mass flow rate Number of capillary channels Pressure in the capillary channels Inlet pressure of the capillary channels Outlet pressure of the capillary channels Outlet pressure during calibration of the capillary channels Volumetric flow rate Outlet volumetric flowrate Corrected outlet volumetric flowrate
Qo,1 ⎛ ρo,1Qo,1 + ρo,2 Qo,2 ⎞ ⎟ ⎜ ρo ⎠ ⎝
(1)
Qo,1 Qo,1 + Qo,2
(2)
ΔPloss ΔPm μ μcal ρo,1 ρo,2
Q o=
nπd 4 dP 128μ dx
(4)
ṁ =−
P nπd 4 dP RT 128μ dx
(5)
ṁ =−
nπd 4 Po2 − Pi2 128μRTL 2
(6)
nπd 4 ⎛ ∆P ⎞ ⎛ Pi ⎞ ⎟ ⎜ +1⎟ ⎜ 128μL ⎝ 2 ⎠ ⎝ Po ⎠
(7)
Pi=∆P+Po Q o=
⎞ nπd 4 ⎛ (∆P )2 +∆P⎟ ⎜ ⎠ 128μL ⎝ 2Po
(10)
The gas divider system (Fig. 1) consisted of three inlet gas flow measurement sections that allowed three gases to be mixed. Each flow measurement section consisted of a needle valve (to regulate the inlet pressure), an inlet pressure gauge, and the laminar flow tube. The outlet of each flow measurement section connected to a manifold, with a pressure gauge, and then the flow traveled through a flow meter. The pressure gauges measured gage pressure and had a range of 0–400 kPa with an accuracy of 0.5% of full scale. The flow meter was a variable area style flow meter with a range of 1.0×10−5 – 8.3×10−5 m3/s and an accuracy of 5% of full scale. Three different gas bottle mixtures of air (20.5% O2/79.5% N2), NO (1533 ppm NO/balance N2), and N2 (99.999% pure) were used for the experiments. The air and NO gas bottle concentrations were accurate to within 0.1% and 1%, respectively. Common methods in industry of creating laminar flow channels
(3)
Q=−
Q o=
nπd 4 ∆P 128μL
3. Experimental setup
The mass flow rate (Eq. (3)) through a tube is a function of density and the volumetric flow rate. Hagen-Poiseuille's law, Eq. (4), applies to laminar flow for a constant cross sectional area and relates the pressure drop over a small length to the volumetric flow rate. Eq. (5) applies the ideal gas law and Eqs. (4) to (3). The mass flow rate is constant through the tube, and by assuming isothermal conditions throughout the length of the tube, Eq. (5), can be integrated from the inlet to the outlet of the tube and results in Eq. (6). Eq. (7) shows the volumetric flow rate at the outlet was developed using Eqs. (3) and (6). By substituting Eqs. (8) and (9) gives an expression for the outlet volumetric flow as a second-order polynomial of the pressure drop. Neglecting the second order term in Eq. (9) due to the small pressure drop compared to the outlet pressure, results in Eq. (10), which is the form of the Hagen-Poiseuille's law used in other studies with capillaries [5–8]. Calculations in this study use Eq. (9) due to the high-pressure drops relative to the outlet pressure.
ṁ =ρQ
Outlet volumetric flowrate of tube 1 Outlet volumetric flowrate of tube 2 Gas constant Reynolds number based on the hydraulic diameter Temperature of the gas in the capillary channels Inlet gas velocity Outlet gas velocity Length along the capillary channels Gas divider outlet gas concentration, by volume Gas divider inlet gas concentration, by volume Calibration constant for the capillary channels Calibration constant for the capillary channels Pressure drop from the inlet to the outlet of the capillary channels Pressure loss associated with the entrance and exit effects Maximum pressure drop from the inlet to the outlet of the capillary channels Gas viscosity Gas viscosity of the calibration gas Outlet gas density of tube 1 Outlet gas density of tube 2
(8)
(9)
Fig. 1. Gas divider system.
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Fig. 5. Influences of the pressure drop measurement accuracy and gas bottle accuracy on the diluted gas concentration (Assumed an accuracy of 0.5% of full scale for the pressure sensors and 1.0% of full scale for the gas bottle concentration).
Fig. 2. Stainless steel wire bundle creating capillary channels (35 wires are shown, 45 wires were used).
stainless steel wires was an inexpensive method for corrosive gases. Based on the diameters of the wire and the tube, theoretically 35 wires should fit inside the tube, but 45 were inserted. Maximizing the number of wires inside the tubes also limited flow between the channels. 3.1. Calibration of the laminar flow meters The laminar flow meters were each calibrated using the air gas bottle. Using a ten-point calibration with the variable area flow meter, the calibration coefficients were determined for each of the laminar flow meters. The resulting calibration coefficients were based on air requiring them to be corrected for the actual gas flowing through the gas divider. 3.2. Gas divider verification Fig. 3. Influence of the pressure drop range (relative to the outlet pressure) on the nonlinearity of the volumetric flow rate.
To verify the gas divider operation, the calculated gas concentration from the gas divider was compared to the measured gas concentration from the gas analyzer. The gas analyzer used a zirconia oxide sensor that measures O2 and NO concentrations in a gas sample. The gas analyzer had a range of 1800 ppm for NOx and 28% for O2 with an accuracy of ± 3% of full scale. Two experiments were conducted during the study. The first experiment used a varied O2 concentration by mixing the O2 bottle with the N2 bottle using the gas divider. Another experiment varied the NO concentration by mixing the NO bottle with the N2 bottle. As shown by Eq. (9), the volumetric flow rate is influenced by the flowing gas viscosity. These experiments allowed for the variation of the flowing gas viscosity to be tested since the viscosity of air and nitrogen is approximately 14% different at atmospheric conditions. 4. Calculation When the pressure drop approaches the order of the magnitude of the outlet pressure from Eq. (9), nonlinearity plays a role in outlet volumetric flow rate (Fig. 3). Because of the additional term in the volumetric flow equation, the measurement accuracy of the pressure drop across the capillary plays a larger role (Fig. 4). From Fig. 4, linear laminar flow has a relatively constant 0.5% propagation of error while the nonlinear laminar flow ranges from below 0.1% at low flow rates to 1.0% at high flow rates. The impact of the nonlinearity flow through the tubes and the bottle concentration accuracy on the propagation error for the gas divider from Eq. (2) is shown in Fig. 5. When using the gas
Fig. 4. Influence of the pressure drop measurement accuracy on the volumetric flow rate (Assumed an accuracy of 0.5% of full scale for the pressure sensors).
uses multiple individual capillary tubes, wrapping stainless steel foil, parallel plates, and concentric annular ducts. The method chosen for this study was inserting 45 0.2 mm-diameter stainless wires inside a 1.4 mm (inner diameter) stainless steel tube (Fig. 2) to create the flow channels inside the tube around the wires. This method of using 3
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linear fit without an offset. The slope for the linear fit with an offset was 1.014 and 1.018. Without the offset, the slope was 0.993 and 1.003. These show residual errors within 0.7–1.8% since a slope of one is a perfect fit. The offsets of 0.36% and 0.17% O2 were less than two percent error at zero concentration, which is within the accuracy of the zirconia analyzer. The accuracy of the gas divider (Fig. 6) was greater than accuracy of the individual flow tubes (Fig. 5) due to the gas divider using the ratio of flow rates through the flow tubes (Eqs. (1) and (2)). Therefore, precision of the calibration flow meter was more important than the accuracy for the accuracy of gas divider. Since air was used to develop the calibration coefficients, a gas mixture with a nitrogen concentration greater than 99% with the remainder being NO was used to verify the viscosity correction given in Eq. (13). Fig. 8 shows the comparison of the gas divider NO concentration to the measured NO concentration for both of the tests, similar to Fig. 6. Test 1 shows the effects of having an increasing NO concentration and test 2 the decreasing NO concentration. The slopes for the linear fits, in the figure, with and without an offset, range from 0.968 to 0.986. The linear fit with an offset was 0.986–0.987. This shows residual errors between the analyzer and the gas divider of 1.3– 1.4%. The offsets of 7 and 20 ppm of NO were less than one percent error at zero concentration, which is within the accuracy of the zirconia analyzer.
divider for low gas concentrations (above 90% dilution), the propagation of error is lower for the nonlinear laminar flow situations since the measured pressure drop will be higher at the same volumetric flow rate. The linear laminar flow had slightly lower propagation of errors from 75% to 20% dilution, but slightly higher propagation of error below 20% dilution. It should be noted that the trends for the propagation of error assumed an accuracy of 0.5% of full scale for the pressure sensors and 1% of full scale for the bottle concentration. It should be noted that pressure sensors with lower and higher accuracy are available in industry and using those accuracies would change the scale of the propagation of error analysis, but not the trends shown. Eq. (9) can be represented as a second-order polynomial (Eq. (11)) with two calibration coefficients, shown as Eqs. (12) and (13). The calibration coefficients are dependent on the viscosity and outlet pressure of the gas used for calibration. With a gas divider application, the viscosity of the fluid changes as the gas mixture changes. Using the relationships developed by Carr et al. [9], the viscosity was calculated for the gas mixture. Eq. (14) corrected the calibration coefficients for the flowing fluid viscosity and outlet pressure. In addition, Eq. (15) determined the required entrance length to develop the laminar flow velocity profile [10]. A Reynolds number below 2300 (Eq. (16)) [10] is needed for laminar flow and at a Reynolds number of 2300, the entrance length is 138 times the hydraulic diameter. The experimental setup that was used (Fig. 1) had a hydraulic diameter (Eq. (17)) and capillary length of 0.009 mm and 305 mm, respectively. At a Reynolds number of 2300, this corresponded to a tube length over 30,000 times the hydraulic diameter and an entrance length of only 1 mm. Therefore, effects due to the entrance length were neglected.
Qo=α (∆P )2 +β∆P nπd 4 128μL
(12)
α=
β 2Po
(13)
∆Ploss=Ki (14)
Le = 0. 06 ReD dh
(15)
ReD < 2300
(16)
dh=
dt2
for laminar flow
Eq. (18) shows the minor losses associated with the entrance and exit effects of the capillaries. With sharp corners, the entrance and exit K-values are 0.5 and 1.0, respectively [10]. For linear laminar flow meters with small pressure drops (relative to the outlet pressure, ΔPm/ Po ≈0.02), these pressure losses can be as much as 10% of the measured pressure with high density gases; negligible, with low density gases (such as hydrogen); and is approximately 4% of the total pressure measurement for air [7]. These high entrance and exit effects are due to the small pressure drops from Hagen-Poiseuille's law, so the entrance and exit effects are on the same order of magnitude.
(11)
β=
⎛ μ ⎞ ⎛ Po, cal ⎞ Qo, corr =Qo ⎜ cal ⎟ ⎜ ⎟ ⎝ μ ⎠ ⎝ Po ⎠
6. Entrance and exit losses
(18)
To understand whether the second order term from the experiments and using Eq. (11) was due to density changes or entrance and exit effects, the volumetric flow rate versus pressure drop data was fitted with four different equations in Fig. 9. The four equations are Eq. (19) (fitting β), Eq. (20) (fitting β and using sharp corner values for the K-values), Eq. (11) (fitting β and α), and Eq. (20) (fitting β and a Kvalue). The curve fit, used through this study (fitting β and α, Eq. (11)), and the curve fit of β and a K-value (Eq. (20)) showed better agreement to the collected data than fitting only β (Eq. (19)) and fitting β while using K-values of 0.5 and 1.0 for the entrance and exit effects (Eq.
nd w2
− 4A = P dt + nd w
ρi 2 ρ Vi +Ke o Vo 2 2 2
(17)
5. Results and discussion The three tubes were calibrated for the two calibration coefficients with the variable area style flow meter (Fig. 6) using Eq. (11). On the figure, the error bars shown for the volumetric flow rate and pressure drop are the accuracy of the measuring device. It should be noted that the accuracy of the pressure sensors ( ± 0.5% full scale) was much greater than the accuracy of the variable area style flow meter (5% full scale). For tubes 2 and 3, the calibration curves were similar while the coefficients varied for all three tubes. Any variability in the coefficients was due to the variability in the wire diameter and location of the 45 wires inside each tube. The concentration of O2 in the gas mixture was varied by using tubes 1 and 2 of the gas divider and measured using the zirconia oxide sensor (Fig. 7). Test 1 was increasing in gas concentration, from 0% to 20.5%, and test 2 was decreasing in concentration, from 20.5% to 0%. Fig. 7 shows two sets of trend lines: a linear fit with an offset and a
Fig. 6. Calibration curves for the three flow tubes.
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J. Nuszkowski et al.
Fig. 9. Non-linear volumetric flow curve fits based on four different equations.
7. Conclusions A gas divider was developed based on nonlinear laminar flow. The nonlinearity was created from the high-pressure drop (relative to the outlet pressure), which resulted in significant gas density change along the channels. In order to create laminar flow channels, small diameter wires were bundled together inside of a tube. The pressure loss from entrance and exit effects can be neglected due to the high-pressure drop from Hagen-Poiseuille's law relative to the minor losses from entrance and effects. The measurement uncertainty for a gas divider with nonlinear laminar flow compared to linear laminar flow was lower (1.5%) at 90% dilution levels, slightly higher (0.1%) at 50% dilution, and approximately the same at no dilution. This was due to the higher-pressure drop of nonlinear laminar flow for the same volumetric flow rate as linear laminar flow as a result of the second order term in the volumetric flow rate equation. The gas divider showed residual errors within 2% of the measured gas concentrations.
Fig. 7. Comparison of the O2 concentration from the gas divider to the measured concentration.
Acknowledgements The authors would like to acknowledge the financial support from the University of North Florida Foundation Board. References [1] R.S. Barratt, The preparation of standard gas mixtures. A review, Analyst 106 (1265) (1981) 817–849. [2] Gas analysis – Preparation of Calibration Gas Mixtures Using Dynamic Volumetric Methods – Part 5: Capillary Calibration Devices, ISO 6145-5:2009, International Organization for Standardization, Geneva, Switzerland [3] R.S. Wright, R.W. Murdoch, Laboratory evaluation of gas dilution systems for analyzer calibration and calibration gas analysis, Air Waste 44 (4) (1994) 428–430. [4] Sherman, M., Mauti, A., Rauker, Z., Dageforde, A., Evaluation of Mass Flow Controller Gas Divider For Linearizing Emission Analytical Equipment, SAE Technical Paper 1999-01-0148. 〈http//doi.org/10.4271/1999-01-0148〉 1999 [5] D. Rimberg, Pressure drop across sharp-end capillary tubes, Ind. Eng. Chem. Fundam. 6 (4) (1967) 599–603. [6] J. Wojtkowiak, C.O. Popiel, Inherently linear annular-duct-type laminar flowmeter, J. Fluids Eng. 128 (1) (2006) 196–198. [7] Alvesteffer, W.J. and Lawrence, W.E., A Laminar Flow Element with A Linear Pressure Drop Versus Volumetric Flow Asme Fluids Engineering Division Summer Meeting, Proceedings of FEDSM’98, 1982 [8] G.H. Priestman, R.F. Boucher, The biased laminar by-pass fluidic flowmeter, J. Fluids Eng. 127 (6) (2005) 1199–1204. [9] N.L. Carr, R. Kobayashi, D.B. Burrows, Viscosity of hydrocarbon gases under pressure, J. Pet. Technol. 6 (10) (1954) 47–55. [10] R.W. Fox, A.T. McDonald, Introduction to Fluid Mechanics, 5th Edition, John Wiley & Sons, New York, 1985.
Fig. 8. Comparison of the no concentration from the gas divider to the measured concentration.
(20)). The resulting K-value from curve fitting was 62.8, which was much higher than the K-value of 0.5 and 1.0 for sharp corners. This high Kvalue and the high pressure drop (relative to outlet pressure, ΔPm/Po ≈4) led the authors to believe the differences between the two sets of curves was due to the accuracy of the variable area style flow meter (5% of full scale) and not entrance and exit effects. Therefore, the entrance and exit effects can be ignored due to the high-pressure drop (relative to the outlet pressure) from Hagen-Poiseuille's law.
Q o=
β [(∆P )2 +∆P ] 2Po
(19)
Q o=
β [(∆P − ∆Ploss )2 +(∆P−∆Ploss )] 2Po
(20)
Dr. John Nuszkowski is an Assistant Professor at the University of North Florida. His research goals are the improvement of air quality by reducing the exhaust emissions from
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Jeffrey Schwamb is a research and development mechanical engineer at Medtronic. Jeffrey Schwamb may be reached at schwamb.jeff@gmail.com.
engines and vehicles through improved combustion of petroleum-based fuels and the use of alternative fuels. Dr. Nuszkowski's research includes in-use vehicle emissions, advanced combustion, alternative fuels, vehicle powertrain optimization, and large bore diesel emissions reduction. Dr. Nuszkowski may be reached at john.nuszkowski@unf. edu.
John Esposito is a Graduate Assistant for the Mechanical and Aerospace Department at the University of Florida. John Esposito may be reached at jce316@ufl.edu.
6
Contents lists available at ScienceDirect
Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
A novel gas divider using nonlinear laminar flow ⁎
J. Nuszkowski , J. Schwamb, J. Esposito School of Engineering, University of North Florida, Jacksonville, FL, USA
A R T I C L E I N F O
A BS T RAC T
Keywords: Gas divider Laminar flow Flow measurement Compressible fluid Capillary tube Gas dilution
Gas dividers are important in emissions measurement since they continuously and accurately mix two gases to create a known gas concentration that is needed in the multi-point calibration of gas analyzers. A novel gas divider was designed using nonlinear laminar flow induced from the density change along the capillary channels due to the high-pressure drop (relative to the inlet gas pressure). The minor losses from entrance and exit effects can be ignored due to the high pressure loss from Hagen-Poiseuille's law relative to the minor losses. Small diameter wires inside of a tube were used to create capillary channels through which gas could flow. The gas divider, using nonlinear laminar flow, showed lower measurement uncertainty at high (90%) dilution levels than using linear laminar flow due to the higher-pressure drop at the same volumetric flow rates. Experiments showed the expected gas concentration from using the gas divider to be within 2% of the measured gas concentrations.
1. Introduction Gas dividers are widely used in environmental pollution control, such as automotive and exhaust plumes for the calibrating of gas analyzers used in the measurement of emissions from different pollution emitters. The gas analyzers require multi-point calibration via a gas divider that accurately and continuously mixes two gases at known concentrations. Gas dividers require accurate regulation and measurement of the flow rate from each gas stream. Some of the commonly used techniques for flow measurement with gas dividers are volumetric pumps, critical orifices, thermal mass-flow controllers, and capillary tubes [1,2]. A study by Wright and Murdoch [3] investigated four different commercially available gas dividers. The four gas dividers used were thermal mass flow controllers, capillaries, positive-displacement pumps (varying stroke frequency), and positivedisplacement pumps (varying the number of pumps), respectively. Sherman et al. [4] developed a gas divider based on mass flow technology that achieved residual errors of less than 1% of point for most of the tested analyzer ranges. When using capillary tubes, the small diameter of the capillary generates the laminar flow inside the gas divider. The internal laminar flow follows the Hagen-Poiseuille law and the flow rate increases linearly as the pressure drop across the capillary channel increases. For larger flow rates, multiple capillary tubes of the same diameter are arranged in parallel. Therefore, to have a higher flow rate while still maintaining laminar flow, the gas divider requires more capillary tubes [5]. These laminar flow meters are common in industry to measure the
⁎
flow rate of clean gases. Authors [6,7] have studied the use of a concentric annular duct instead of capillary tube bundles to generate linear laminar flow. Priestman and Boucher [8] showed that the range of other flow meter types could be extended by using a laminar by-pass resistance. The objective of this study was to develop a novel gas divider using a compressible fluid with a high-pressure drop (with respect to the inlet gas pressure) that created nonlinear laminar flow. A gas divider was designed, constructed, calibrated using a reference flow meter, and experimentally verified using an emissions analyzer. In addition, equations were developed that demonstrate the behavior of the nonlinear laminar flow and propagation of error analysis was performed on the gas divider when using nonlinear or linear laminar flow channels. 2. Theory For a gas divider, diluting one gas concentration (by volume) to lower gas concentrations (by volume) requires the mixing of two gas streams. The first gas stream has a gas concentration (by volume) for a specific gas species, called the component gas, and the remainder of that gas stream is another gas species, called the balance gas. The second gas stream consists of only the balance gas. The exit concentration (by volume) of the component gas is dependent on the volumetric flow rates of the gas streams and the inlet concentration (by volume) of the component gas species, as shown in Eq. (1). When diluting a gas with a low concentration ( < 1%), the variation in the densities between the two gas streams and the outlet can be neglected as shown in Eq. (2).
Corresponding author.
http://dx.doi.org/10.1016/j.flowmeasinst.2016.10.016 Received 27 April 2016; Received in revised form 4 October 2016; Accepted 12 October 2016 Available online xxxx 0955-5986/ © 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: Nuszkowski, J., Flow Measurement and Instrumentation (2016), http://dx.doi.org/10.1016/j.flowmeasinst.2016.10.016
Flow Measurement and Instrumentation xx (xxxx) xxxx–xxxx
J. Nuszkowski et al.
Qo,1 Qo,2 R ReD T Vi Vo x Yo Y1 α β ΔP
Nomenclature A d dh dt dw Γ Ki Ke L Le ṁ n P Pi Po Po,cal Q Qo Qo,corr
Yo=Y1
Yo=Y1
Cross sectional area Diameter of capillary channels Hydraulic diameter of the capillary channels Inside diameter of the tube Wire diameter Wetted perimeter Entrance loss coefficient Exit loss coefficient Length of the capillary channels Entrance length Mass flow rate Number of capillary channels Pressure in the capillary channels Inlet pressure of the capillary channels Outlet pressure of the capillary channels Outlet pressure during calibration of the capillary channels Volumetric flow rate Outlet volumetric flowrate Corrected outlet volumetric flowrate
Qo,1 ⎛ ρo,1Qo,1 + ρo,2 Qo,2 ⎞ ⎟ ⎜ ρo ⎠ ⎝
(1)
Qo,1 Qo,1 + Qo,2
(2)
ΔPloss ΔPm μ μcal ρo,1 ρo,2
Q o=
nπd 4 dP 128μ dx
(4)
ṁ =−
P nπd 4 dP RT 128μ dx
(5)
ṁ =−
nπd 4 Po2 − Pi2 128μRTL 2
(6)
nπd 4 ⎛ ∆P ⎞ ⎛ Pi ⎞ ⎟ ⎜ +1⎟ ⎜ 128μL ⎝ 2 ⎠ ⎝ Po ⎠
(7)
Pi=∆P+Po Q o=
⎞ nπd 4 ⎛ (∆P )2 +∆P⎟ ⎜ ⎠ 128μL ⎝ 2Po
(10)
The gas divider system (Fig. 1) consisted of three inlet gas flow measurement sections that allowed three gases to be mixed. Each flow measurement section consisted of a needle valve (to regulate the inlet pressure), an inlet pressure gauge, and the laminar flow tube. The outlet of each flow measurement section connected to a manifold, with a pressure gauge, and then the flow traveled through a flow meter. The pressure gauges measured gage pressure and had a range of 0–400 kPa with an accuracy of 0.5% of full scale. The flow meter was a variable area style flow meter with a range of 1.0×10−5 – 8.3×10−5 m3/s and an accuracy of 5% of full scale. Three different gas bottle mixtures of air (20.5% O2/79.5% N2), NO (1533 ppm NO/balance N2), and N2 (99.999% pure) were used for the experiments. The air and NO gas bottle concentrations were accurate to within 0.1% and 1%, respectively. Common methods in industry of creating laminar flow channels
(3)
Q=−
Q o=
nπd 4 ∆P 128μL
3. Experimental setup
The mass flow rate (Eq. (3)) through a tube is a function of density and the volumetric flow rate. Hagen-Poiseuille's law, Eq. (4), applies to laminar flow for a constant cross sectional area and relates the pressure drop over a small length to the volumetric flow rate. Eq. (5) applies the ideal gas law and Eqs. (4) to (3). The mass flow rate is constant through the tube, and by assuming isothermal conditions throughout the length of the tube, Eq. (5), can be integrated from the inlet to the outlet of the tube and results in Eq. (6). Eq. (7) shows the volumetric flow rate at the outlet was developed using Eqs. (3) and (6). By substituting Eqs. (8) and (9) gives an expression for the outlet volumetric flow as a second-order polynomial of the pressure drop. Neglecting the second order term in Eq. (9) due to the small pressure drop compared to the outlet pressure, results in Eq. (10), which is the form of the Hagen-Poiseuille's law used in other studies with capillaries [5–8]. Calculations in this study use Eq. (9) due to the high-pressure drops relative to the outlet pressure.
ṁ =ρQ
Outlet volumetric flowrate of tube 1 Outlet volumetric flowrate of tube 2 Gas constant Reynolds number based on the hydraulic diameter Temperature of the gas in the capillary channels Inlet gas velocity Outlet gas velocity Length along the capillary channels Gas divider outlet gas concentration, by volume Gas divider inlet gas concentration, by volume Calibration constant for the capillary channels Calibration constant for the capillary channels Pressure drop from the inlet to the outlet of the capillary channels Pressure loss associated with the entrance and exit effects Maximum pressure drop from the inlet to the outlet of the capillary channels Gas viscosity Gas viscosity of the calibration gas Outlet gas density of tube 1 Outlet gas density of tube 2
(8)
(9)
Fig. 1. Gas divider system.
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Fig. 5. Influences of the pressure drop measurement accuracy and gas bottle accuracy on the diluted gas concentration (Assumed an accuracy of 0.5% of full scale for the pressure sensors and 1.0% of full scale for the gas bottle concentration).
Fig. 2. Stainless steel wire bundle creating capillary channels (35 wires are shown, 45 wires were used).
stainless steel wires was an inexpensive method for corrosive gases. Based on the diameters of the wire and the tube, theoretically 35 wires should fit inside the tube, but 45 were inserted. Maximizing the number of wires inside the tubes also limited flow between the channels. 3.1. Calibration of the laminar flow meters The laminar flow meters were each calibrated using the air gas bottle. Using a ten-point calibration with the variable area flow meter, the calibration coefficients were determined for each of the laminar flow meters. The resulting calibration coefficients were based on air requiring them to be corrected for the actual gas flowing through the gas divider. 3.2. Gas divider verification Fig. 3. Influence of the pressure drop range (relative to the outlet pressure) on the nonlinearity of the volumetric flow rate.
To verify the gas divider operation, the calculated gas concentration from the gas divider was compared to the measured gas concentration from the gas analyzer. The gas analyzer used a zirconia oxide sensor that measures O2 and NO concentrations in a gas sample. The gas analyzer had a range of 1800 ppm for NOx and 28% for O2 with an accuracy of ± 3% of full scale. Two experiments were conducted during the study. The first experiment used a varied O2 concentration by mixing the O2 bottle with the N2 bottle using the gas divider. Another experiment varied the NO concentration by mixing the NO bottle with the N2 bottle. As shown by Eq. (9), the volumetric flow rate is influenced by the flowing gas viscosity. These experiments allowed for the variation of the flowing gas viscosity to be tested since the viscosity of air and nitrogen is approximately 14% different at atmospheric conditions. 4. Calculation When the pressure drop approaches the order of the magnitude of the outlet pressure from Eq. (9), nonlinearity plays a role in outlet volumetric flow rate (Fig. 3). Because of the additional term in the volumetric flow equation, the measurement accuracy of the pressure drop across the capillary plays a larger role (Fig. 4). From Fig. 4, linear laminar flow has a relatively constant 0.5% propagation of error while the nonlinear laminar flow ranges from below 0.1% at low flow rates to 1.0% at high flow rates. The impact of the nonlinearity flow through the tubes and the bottle concentration accuracy on the propagation error for the gas divider from Eq. (2) is shown in Fig. 5. When using the gas
Fig. 4. Influence of the pressure drop measurement accuracy on the volumetric flow rate (Assumed an accuracy of 0.5% of full scale for the pressure sensors).
uses multiple individual capillary tubes, wrapping stainless steel foil, parallel plates, and concentric annular ducts. The method chosen for this study was inserting 45 0.2 mm-diameter stainless wires inside a 1.4 mm (inner diameter) stainless steel tube (Fig. 2) to create the flow channels inside the tube around the wires. This method of using 3
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linear fit without an offset. The slope for the linear fit with an offset was 1.014 and 1.018. Without the offset, the slope was 0.993 and 1.003. These show residual errors within 0.7–1.8% since a slope of one is a perfect fit. The offsets of 0.36% and 0.17% O2 were less than two percent error at zero concentration, which is within the accuracy of the zirconia analyzer. The accuracy of the gas divider (Fig. 6) was greater than accuracy of the individual flow tubes (Fig. 5) due to the gas divider using the ratio of flow rates through the flow tubes (Eqs. (1) and (2)). Therefore, precision of the calibration flow meter was more important than the accuracy for the accuracy of gas divider. Since air was used to develop the calibration coefficients, a gas mixture with a nitrogen concentration greater than 99% with the remainder being NO was used to verify the viscosity correction given in Eq. (13). Fig. 8 shows the comparison of the gas divider NO concentration to the measured NO concentration for both of the tests, similar to Fig. 6. Test 1 shows the effects of having an increasing NO concentration and test 2 the decreasing NO concentration. The slopes for the linear fits, in the figure, with and without an offset, range from 0.968 to 0.986. The linear fit with an offset was 0.986–0.987. This shows residual errors between the analyzer and the gas divider of 1.3– 1.4%. The offsets of 7 and 20 ppm of NO were less than one percent error at zero concentration, which is within the accuracy of the zirconia analyzer.
divider for low gas concentrations (above 90% dilution), the propagation of error is lower for the nonlinear laminar flow situations since the measured pressure drop will be higher at the same volumetric flow rate. The linear laminar flow had slightly lower propagation of errors from 75% to 20% dilution, but slightly higher propagation of error below 20% dilution. It should be noted that the trends for the propagation of error assumed an accuracy of 0.5% of full scale for the pressure sensors and 1% of full scale for the bottle concentration. It should be noted that pressure sensors with lower and higher accuracy are available in industry and using those accuracies would change the scale of the propagation of error analysis, but not the trends shown. Eq. (9) can be represented as a second-order polynomial (Eq. (11)) with two calibration coefficients, shown as Eqs. (12) and (13). The calibration coefficients are dependent on the viscosity and outlet pressure of the gas used for calibration. With a gas divider application, the viscosity of the fluid changes as the gas mixture changes. Using the relationships developed by Carr et al. [9], the viscosity was calculated for the gas mixture. Eq. (14) corrected the calibration coefficients for the flowing fluid viscosity and outlet pressure. In addition, Eq. (15) determined the required entrance length to develop the laminar flow velocity profile [10]. A Reynolds number below 2300 (Eq. (16)) [10] is needed for laminar flow and at a Reynolds number of 2300, the entrance length is 138 times the hydraulic diameter. The experimental setup that was used (Fig. 1) had a hydraulic diameter (Eq. (17)) and capillary length of 0.009 mm and 305 mm, respectively. At a Reynolds number of 2300, this corresponded to a tube length over 30,000 times the hydraulic diameter and an entrance length of only 1 mm. Therefore, effects due to the entrance length were neglected.
Qo=α (∆P )2 +β∆P nπd 4 128μL
(12)
α=
β 2Po
(13)
∆Ploss=Ki (14)
Le = 0. 06 ReD dh
(15)
ReD < 2300
(16)
dh=
dt2
for laminar flow
Eq. (18) shows the minor losses associated with the entrance and exit effects of the capillaries. With sharp corners, the entrance and exit K-values are 0.5 and 1.0, respectively [10]. For linear laminar flow meters with small pressure drops (relative to the outlet pressure, ΔPm/ Po ≈0.02), these pressure losses can be as much as 10% of the measured pressure with high density gases; negligible, with low density gases (such as hydrogen); and is approximately 4% of the total pressure measurement for air [7]. These high entrance and exit effects are due to the small pressure drops from Hagen-Poiseuille's law, so the entrance and exit effects are on the same order of magnitude.
(11)
β=
⎛ μ ⎞ ⎛ Po, cal ⎞ Qo, corr =Qo ⎜ cal ⎟ ⎜ ⎟ ⎝ μ ⎠ ⎝ Po ⎠
6. Entrance and exit losses
(18)
To understand whether the second order term from the experiments and using Eq. (11) was due to density changes or entrance and exit effects, the volumetric flow rate versus pressure drop data was fitted with four different equations in Fig. 9. The four equations are Eq. (19) (fitting β), Eq. (20) (fitting β and using sharp corner values for the K-values), Eq. (11) (fitting β and α), and Eq. (20) (fitting β and a Kvalue). The curve fit, used through this study (fitting β and α, Eq. (11)), and the curve fit of β and a K-value (Eq. (20)) showed better agreement to the collected data than fitting only β (Eq. (19)) and fitting β while using K-values of 0.5 and 1.0 for the entrance and exit effects (Eq.
nd w2
− 4A = P dt + nd w
ρi 2 ρ Vi +Ke o Vo 2 2 2
(17)
5. Results and discussion The three tubes were calibrated for the two calibration coefficients with the variable area style flow meter (Fig. 6) using Eq. (11). On the figure, the error bars shown for the volumetric flow rate and pressure drop are the accuracy of the measuring device. It should be noted that the accuracy of the pressure sensors ( ± 0.5% full scale) was much greater than the accuracy of the variable area style flow meter (5% full scale). For tubes 2 and 3, the calibration curves were similar while the coefficients varied for all three tubes. Any variability in the coefficients was due to the variability in the wire diameter and location of the 45 wires inside each tube. The concentration of O2 in the gas mixture was varied by using tubes 1 and 2 of the gas divider and measured using the zirconia oxide sensor (Fig. 7). Test 1 was increasing in gas concentration, from 0% to 20.5%, and test 2 was decreasing in concentration, from 20.5% to 0%. Fig. 7 shows two sets of trend lines: a linear fit with an offset and a
Fig. 6. Calibration curves for the three flow tubes.
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Fig. 9. Non-linear volumetric flow curve fits based on four different equations.
7. Conclusions A gas divider was developed based on nonlinear laminar flow. The nonlinearity was created from the high-pressure drop (relative to the outlet pressure), which resulted in significant gas density change along the channels. In order to create laminar flow channels, small diameter wires were bundled together inside of a tube. The pressure loss from entrance and exit effects can be neglected due to the high-pressure drop from Hagen-Poiseuille's law relative to the minor losses from entrance and effects. The measurement uncertainty for a gas divider with nonlinear laminar flow compared to linear laminar flow was lower (1.5%) at 90% dilution levels, slightly higher (0.1%) at 50% dilution, and approximately the same at no dilution. This was due to the higher-pressure drop of nonlinear laminar flow for the same volumetric flow rate as linear laminar flow as a result of the second order term in the volumetric flow rate equation. The gas divider showed residual errors within 2% of the measured gas concentrations.
Fig. 7. Comparison of the O2 concentration from the gas divider to the measured concentration.
Acknowledgements The authors would like to acknowledge the financial support from the University of North Florida Foundation Board. References [1] R.S. Barratt, The preparation of standard gas mixtures. A review, Analyst 106 (1265) (1981) 817–849. [2] Gas analysis – Preparation of Calibration Gas Mixtures Using Dynamic Volumetric Methods – Part 5: Capillary Calibration Devices, ISO 6145-5:2009, International Organization for Standardization, Geneva, Switzerland [3] R.S. Wright, R.W. Murdoch, Laboratory evaluation of gas dilution systems for analyzer calibration and calibration gas analysis, Air Waste 44 (4) (1994) 428–430. [4] Sherman, M., Mauti, A., Rauker, Z., Dageforde, A., Evaluation of Mass Flow Controller Gas Divider For Linearizing Emission Analytical Equipment, SAE Technical Paper 1999-01-0148. 〈http//doi.org/10.4271/1999-01-0148〉 1999 [5] D. Rimberg, Pressure drop across sharp-end capillary tubes, Ind. Eng. Chem. Fundam. 6 (4) (1967) 599–603. [6] J. Wojtkowiak, C.O. Popiel, Inherently linear annular-duct-type laminar flowmeter, J. Fluids Eng. 128 (1) (2006) 196–198. [7] Alvesteffer, W.J. and Lawrence, W.E., A Laminar Flow Element with A Linear Pressure Drop Versus Volumetric Flow Asme Fluids Engineering Division Summer Meeting, Proceedings of FEDSM’98, 1982 [8] G.H. Priestman, R.F. Boucher, The biased laminar by-pass fluidic flowmeter, J. Fluids Eng. 127 (6) (2005) 1199–1204. [9] N.L. Carr, R. Kobayashi, D.B. Burrows, Viscosity of hydrocarbon gases under pressure, J. Pet. Technol. 6 (10) (1954) 47–55. [10] R.W. Fox, A.T. McDonald, Introduction to Fluid Mechanics, 5th Edition, John Wiley & Sons, New York, 1985.
Fig. 8. Comparison of the no concentration from the gas divider to the measured concentration.
(20)). The resulting K-value from curve fitting was 62.8, which was much higher than the K-value of 0.5 and 1.0 for sharp corners. This high Kvalue and the high pressure drop (relative to outlet pressure, ΔPm/Po ≈4) led the authors to believe the differences between the two sets of curves was due to the accuracy of the variable area style flow meter (5% of full scale) and not entrance and exit effects. Therefore, the entrance and exit effects can be ignored due to the high-pressure drop (relative to the outlet pressure) from Hagen-Poiseuille's law.
Q o=
β [(∆P )2 +∆P ] 2Po
(19)
Q o=
β [(∆P − ∆Ploss )2 +(∆P−∆Ploss )] 2Po
(20)
Dr. John Nuszkowski is an Assistant Professor at the University of North Florida. His research goals are the improvement of air quality by reducing the exhaust emissions from
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Jeffrey Schwamb is a research and development mechanical engineer at Medtronic. Jeffrey Schwamb may be reached at schwamb.jeff@gmail.com.
engines and vehicles through improved combustion of petroleum-based fuels and the use of alternative fuels. Dr. Nuszkowski's research includes in-use vehicle emissions, advanced combustion, alternative fuels, vehicle powertrain optimization, and large bore diesel emissions reduction. Dr. Nuszkowski may be reached at john.nuszkowski@unf. edu.
John Esposito is a Graduate Assistant for the Mechanical and Aerospace Department at the University of Florida. John Esposito may be reached at jce316@ufl.edu.
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