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Simple calibrator for low range differential pressure transducers
ARTICLE IN PRESS
Journal of Wind Engineering and Industrial Aerodynamics 92 (2004) 1167–1170 www.elsevier.com/locate/jweia
Short Note
Simple calibrator for low range differential pressure transducers M. Poreh Department of Civil Engineering, Technion – Israel Institute of technology, Technion City Haifa 32000, Israel Received 20 April 2004; received in revised form 30 July 2004; accepted 2 August 2004 Available online 11 September 2004
Differential pressure transducers used for Wind Engineering applications are usually linear. Namely, the output electrical signal from their controller, E, is given by E=E0+S*P, where P is the differential pressure. Their linearity within the designed range of operation is usually high; namely, for given setting of their controller and environmental conditions, the ‘‘slope’’ S is practically a constant within this range. In addition, most of the controllers enable the user to ‘‘zero’’ the electrical output (at zero differential pressure) and some of them make it possible to adjust the value of S. In many wind tunnel studies, where all measurements are made with the same transducer, the exact value of S is not important, as one is interested in dimensionless values, such as u/U and P/rU2, where U is a reference velocity. Thus, relatively inexpensive linear transducers may be used for such cases. In some applications, however, the value of S should be determined. Transducers with a controlled value of S or calibrators which determine its value are usually very expensive and may cost as much as $16,000 [See, for example, [1]]. Moreover, the user does not always have the capacity of checking whether his transducer has maintained its original performance. To overcome this problem, a simple approximate calibrator for differential transducers, which can be easily built at any laboratory, has been
Tel.: +97-248-251-011; fax: +1-509-356-3844.
E-mail address: [email protected] (M. Poreh). 0167-6105/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jweia.2004.08.001
ARTICLE IN PRESS 1168
M. Poreh / J. Wind Eng. Ind. Aerodyn. 92 (2004) 1167–1170
developed by the author. Its design, operation and estimated accuracy are presented in this note. The calibrator is made of two communicating containers, see Fig. 1. Container [1] is open to the atmosphere and container [2] is closed. It is desired that their diameters (D1 and D2) be large, that h2 be small and that D2oD1. With the venting needlevalve [4] of container [2] open, one fills the containers with water up to a desired height, preferably close to the top of the closed container. The needle valve is then closed and a small, precisely measured volume of water Vw is added to the open container. The pressure Dp above the water surface in the closed container, which is measured by the transducer will be Dp ¼ gw Dh1 ;
(1)
where gw is the specific weight of the water and Dh1 is the difference between the levels of the water surfaces in the two containers. Since air is compressible, the water level in the closed container will rise slightly. This rise will be denoted by -Dh2, where h2 is the height of the air column in that container. Its value may be estimated if the volume of the air column V2 is much larger than the volume in the tube [6] and within the positive side of the transducer. This assumption implies that when Dh2/h2 is small, it may be estimated by Dh2 =h2 ¼ DV 2 =V 2 ¼ ð1=bÞDp=Pa ;
(2)
where b is 1.4 for reversibly adiabatic compression and 1 for isothermal compression, and Pa is the atmospheric pressure. It is also clear that, V w ¼ Dh1 A1 Dh2 A2 Dh2 A1 ;
(3)
Fig. 1. Schematic description of the calibrator. (1) A large, open, constant-area container with vertical walls, (2) a closed container, (3) a tube connecting the two containers, (4) a venting needle valve, (5) differential pressure transducer and digital controller, (6) a small diameter tube connecting the closed container to the positive port of the transducer. (The negative port is open to the atmosphere).
ARTICLE IN PRESS M. Poreh / J. Wind Eng. Ind. Aerodyn. 92 (2004) 1167–1170
1169
where A1 and A2 are the areas of the containers and Vw is the added volume of water. It follows from the above equations that Dp ¼ ðgw V w =A1 Þ=½1 þ C;
(4)
where Cð1= ¼ bÞðgw h2 =Pa Þ1 þ ðA2 =A1 Þ:
(5)
In many cases, the correction factor C is small, particularly if both A2/A1 and h2 are small. For example, for h2=50 mm and A2/A1=0.25, and Pa=10,000 mm H2O, C=1 50/10000 1.25=0.00625 for b=1 and 0.00446 for b=1.4. As the value of b is not known, using an intermediate value of b=1.2, the error in the calculation of Dp in this example would be less than 0.1%. It follows from Eq. (4) that when such accuracy is not essential or when C is much smaller than 1, the induced pressure Dp may be estimated by Dp ¼ gw V w =A1 :
(6)
Clearly, the accuracy of the above equations for calculating Dp is affected by the accuracy of measuring the volume Vw and the area A1. Standard Pyrex pipettes usually discharge a known volume of water with a very high accuracy. One may also raise the water level in the open container by inserting into the water a heavier-thanwater object of a known volume. For example, Teflon sphere hanging on a thin wire. The calculation of the area A1, on the other hand, is not as simple. If the container has a circular cross section, A1 can be determined by measuring its diameter at the water level at various orientations and using their average value. If the water level is close to the edge of the container, the average diameter of the open container can be measured with an accuracy of 0.15 mm. Thus, for D=150 mm, the area A1 may be determined with an accuracy of 0.2%. We have also assumed in the above calculations that the open container is practically vertical. Note that a 4 1 inclination, which is rather large and can be visually detected, will produce a 0.25% increase in the effective area of A. For ideal conditions, mainly clean container walls and pure water, there should not be any effect of surface tension on Dp, as the shape of the meniscus near the walls of the containers in the two containers before and after the addition of Vw should be the same. However, it is possible; that when the inner surface of the wall of the open container along the water-air interface is not perfectly clean, smooth or uniform, the circular meniscus might not be uniform and its shape before and after the addition of Vw would not be exactly the same. The effect of such non-uniformity is expected to be small for large diameter containers (D470 mm) and no effect is expected in the closed container, where the movement of the water surface is very small. In spite of this conclusion, following Pope and Harper [2], it is recommended to clean the walls of the inner cylinder and to add a wetting agent to the water before calibrating very low-range pressure transducers. In view with the above discussion, one may conclude that the overall accuracy of such calibrators would usually be of the order of 1% and that an accuracy of the
ARTICLE IN PRESS 1170
M. Poreh / J. Wind Eng. Ind. Aerodyn. 92 (2004) 1167–1170
order of 0.5% can be achieved for carefully designed built and maintained calibrators, which is sufficient for most Wind Engineering applications.
Acknowledgements Thanks are due to Prof. A. Shavit of the Department of Mechanical Engineering at the Technion-Israel Institute of Technology for reviewing the draft of this note and for his constructive comments and advice. References [1] M. Girard, R. Haines, An automated controller for low differential pressure around atmosphere, Symposio de Metrologia, Queretaro-Mexico May 29, 2002, 2000. Available from: /[email protected]. [2] A. Pope, J.J. Harper, Low-Speed Wind Tunnel Testing, Wiley, New York, 1996.
Journal of Wind Engineering and Industrial Aerodynamics 92 (2004) 1167–1170 www.elsevier.com/locate/jweia
Short Note
Simple calibrator for low range differential pressure transducers M. Poreh Department of Civil Engineering, Technion – Israel Institute of technology, Technion City Haifa 32000, Israel Received 20 April 2004; received in revised form 30 July 2004; accepted 2 August 2004 Available online 11 September 2004
Differential pressure transducers used for Wind Engineering applications are usually linear. Namely, the output electrical signal from their controller, E, is given by E=E0+S*P, where P is the differential pressure. Their linearity within the designed range of operation is usually high; namely, for given setting of their controller and environmental conditions, the ‘‘slope’’ S is practically a constant within this range. In addition, most of the controllers enable the user to ‘‘zero’’ the electrical output (at zero differential pressure) and some of them make it possible to adjust the value of S. In many wind tunnel studies, where all measurements are made with the same transducer, the exact value of S is not important, as one is interested in dimensionless values, such as u/U and P/rU2, where U is a reference velocity. Thus, relatively inexpensive linear transducers may be used for such cases. In some applications, however, the value of S should be determined. Transducers with a controlled value of S or calibrators which determine its value are usually very expensive and may cost as much as $16,000 [See, for example, [1]]. Moreover, the user does not always have the capacity of checking whether his transducer has maintained its original performance. To overcome this problem, a simple approximate calibrator for differential transducers, which can be easily built at any laboratory, has been
Tel.: +97-248-251-011; fax: +1-509-356-3844.
E-mail address: [email protected] (M. Poreh). 0167-6105/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jweia.2004.08.001
ARTICLE IN PRESS 1168
M. Poreh / J. Wind Eng. Ind. Aerodyn. 92 (2004) 1167–1170
developed by the author. Its design, operation and estimated accuracy are presented in this note. The calibrator is made of two communicating containers, see Fig. 1. Container [1] is open to the atmosphere and container [2] is closed. It is desired that their diameters (D1 and D2) be large, that h2 be small and that D2oD1. With the venting needlevalve [4] of container [2] open, one fills the containers with water up to a desired height, preferably close to the top of the closed container. The needle valve is then closed and a small, precisely measured volume of water Vw is added to the open container. The pressure Dp above the water surface in the closed container, which is measured by the transducer will be Dp ¼ gw Dh1 ;
(1)
where gw is the specific weight of the water and Dh1 is the difference between the levels of the water surfaces in the two containers. Since air is compressible, the water level in the closed container will rise slightly. This rise will be denoted by -Dh2, where h2 is the height of the air column in that container. Its value may be estimated if the volume of the air column V2 is much larger than the volume in the tube [6] and within the positive side of the transducer. This assumption implies that when Dh2/h2 is small, it may be estimated by Dh2 =h2 ¼ DV 2 =V 2 ¼ ð1=bÞDp=Pa ;
(2)
where b is 1.4 for reversibly adiabatic compression and 1 for isothermal compression, and Pa is the atmospheric pressure. It is also clear that, V w ¼ Dh1 A1 Dh2 A2 Dh2 A1 ;
(3)
Fig. 1. Schematic description of the calibrator. (1) A large, open, constant-area container with vertical walls, (2) a closed container, (3) a tube connecting the two containers, (4) a venting needle valve, (5) differential pressure transducer and digital controller, (6) a small diameter tube connecting the closed container to the positive port of the transducer. (The negative port is open to the atmosphere).
ARTICLE IN PRESS M. Poreh / J. Wind Eng. Ind. Aerodyn. 92 (2004) 1167–1170
1169
where A1 and A2 are the areas of the containers and Vw is the added volume of water. It follows from the above equations that Dp ¼ ðgw V w =A1 Þ=½1 þ C;
(4)
where Cð1= ¼ bÞðgw h2 =Pa Þ1 þ ðA2 =A1 Þ:
(5)
In many cases, the correction factor C is small, particularly if both A2/A1 and h2 are small. For example, for h2=50 mm and A2/A1=0.25, and Pa=10,000 mm H2O, C=1 50/10000 1.25=0.00625 for b=1 and 0.00446 for b=1.4. As the value of b is not known, using an intermediate value of b=1.2, the error in the calculation of Dp in this example would be less than 0.1%. It follows from Eq. (4) that when such accuracy is not essential or when C is much smaller than 1, the induced pressure Dp may be estimated by Dp ¼ gw V w =A1 :
(6)
Clearly, the accuracy of the above equations for calculating Dp is affected by the accuracy of measuring the volume Vw and the area A1. Standard Pyrex pipettes usually discharge a known volume of water with a very high accuracy. One may also raise the water level in the open container by inserting into the water a heavier-thanwater object of a known volume. For example, Teflon sphere hanging on a thin wire. The calculation of the area A1, on the other hand, is not as simple. If the container has a circular cross section, A1 can be determined by measuring its diameter at the water level at various orientations and using their average value. If the water level is close to the edge of the container, the average diameter of the open container can be measured with an accuracy of 0.15 mm. Thus, for D=150 mm, the area A1 may be determined with an accuracy of 0.2%. We have also assumed in the above calculations that the open container is practically vertical. Note that a 4 1 inclination, which is rather large and can be visually detected, will produce a 0.25% increase in the effective area of A. For ideal conditions, mainly clean container walls and pure water, there should not be any effect of surface tension on Dp, as the shape of the meniscus near the walls of the containers in the two containers before and after the addition of Vw should be the same. However, it is possible; that when the inner surface of the wall of the open container along the water-air interface is not perfectly clean, smooth or uniform, the circular meniscus might not be uniform and its shape before and after the addition of Vw would not be exactly the same. The effect of such non-uniformity is expected to be small for large diameter containers (D470 mm) and no effect is expected in the closed container, where the movement of the water surface is very small. In spite of this conclusion, following Pope and Harper [2], it is recommended to clean the walls of the inner cylinder and to add a wetting agent to the water before calibrating very low-range pressure transducers. In view with the above discussion, one may conclude that the overall accuracy of such calibrators would usually be of the order of 1% and that an accuracy of the
ARTICLE IN PRESS 1170
M. Poreh / J. Wind Eng. Ind. Aerodyn. 92 (2004) 1167–1170
order of 0.5% can be achieved for carefully designed built and maintained calibrators, which is sufficient for most Wind Engineering applications.
Acknowledgements Thanks are due to Prof. A. Shavit of the Department of Mechanical Engineering at the Technion-Israel Institute of Technology for reviewing the draft of this note and for his constructive comments and advice. References [1] M. Girard, R. Haines, An automated controller for low differential pressure around atmosphere, Symposio de Metrologia, Queretaro-Mexico May 29, 2002, 2000. Available from: /[email protected]. [2] A. Pope, J.J. Harper, Low-Speed Wind Tunnel Testing, Wiley, New York, 1996.