- Email: [email protected]
A transiently heated needle anemometer
AGRICULTURAL AND FOREST METEOROLOGY ELSEVIER
Agricultural and Forest Meteorology 74 (1995) 227-235
A transiently heated needle anemometer W.L. Bland a'*, J.M. Norman a, G.S. Campbell b, C. Calissendorfff, E.E. M i l l e r a aDepartment of Soil Science, University of Wisconsin-Madison, 1525 Observatory Drive, Madison, WI 53706, USA bDepartment of Agronomy and Soils, Washington State University, Pullman, WA 99163, USA CSoiltronics, Burlington, WA 98233, USA
Received 31 January 1994; revision accepted 5 August 1994
Abstract Convective heat transfer from cylinders warmed above ambient temperature is a standard technique for measuring windspeed. Available devices are typically fragile, expensive, and complex. We developed a heat-transfer anemometer that is robust, inexpensive, and simple. Referred to as a needle anemometer, the sensing portion of the device is a 20 mm length of tubing of 0.56 mm diameter, into which a heater and thermocouple are inserted. Needle temperature measured during alternating 5 s heating and non-heating periods allows estimation of the equilibrium temperature rise above ambient temperature for long heating times. This is inversely related to (windspeed) 1/2 from 0.05 m s -1 to greater than 14 m s -1. In a field trial, the needle anemometer yielded windspeeds of J:0.2 m s -I (Sy.x) of those from a cup anemometer. Precipitation striking the needle caused overestimates of windspeed by up to 20%.
I. Introduction A n e m o m e t e r s for m e a s u r i n g w i n d speed o p e r a t e on several different principles, i n c l u d i n g the d r a g force p r o d u c e d by w i n d flowing p a s t an object (cup a n e m o m e t e r s ) , D o p p l e r shifts in the speed o f s o u n d (sonic a n e m o m e t e r s ) a n d convective h e a t exchange between m o v i n g air a n d a w a r m e r object. E a c h o f these designs has a d v a n t a g e s a n d d i s a d v a n t a g e s . C u p a n e m o m e t e r s are o n l y m o d e r a t e l y expensive, o p e r a t e over a wide r a n g e o f w i n d s p e e d a b o v e 0.5 m s - i , t y p i c a l l y require a small a m o u n t o f electrical p o w e r , m a i n t a i n c a l i b r a t i o n well with a p p r o p r i a t e * Corresponding author. 0168-1923/95/$09.50 © 1995 - Elsevier Science B.V. All rights reserved SSDI 0168-1923(94)02190-2
228
W.L. Bland et al. / Agricultural and Forest Meteorology 74 (1995) 227 235
maintenance, are physically large, and require infrequent maintenance--perhaps yearly. Sonic anemometers are expensive, measure only components of wind and not windspeed directly, maintain calibration very well, operate over a wide range of windspeed above a few centimeters per second, require modest electrical power, are physically large, and may or may not be reliable, depending on the instrument. In this paper we describe a heat transfer anemometer that is inexpensive and reliable, requires little electrical power (0.03 W), is small and robust, and operates from 0.05 to more than 15 m s 1. Transfer of energy from a heated cylinder to a flowing fluid is strongly related to the fluid velocity. This relationship is widely exploited in anemometry, particularly when low velocities must be measured near surfaces and when rapid time response is needed. For air flow normal to a cylinder axis (Monteith and Unsworth, 1990) Nu = ( h D / k ) = A + BRe n
(1)
where Nu is the Nusselt number, h is the convective heat transfer coefficient ( W m 2 °C 1), D is the cylinder diameter (m), k is thermal conductivity of air ( W m -I ° C l), Re = V D / u is the Reynolds number (V is windspeed (m s 1), u is kinematic viscosity (m 2 s-l)), and A, B, and n are empirical factors that vary with Re. The rate of heat transfer, Q ( w m-2), from the cylinder surface is Q = h(Tcylinder - Tair) = ] t / ~ T t.
(2)
Combining Eqs. (1) and (2) yields = -~
= a + A-~f
(3)
Heat transfer anemometers may either be continuously or transiently heated. Continuously heated anemometers usually require large power inputs (of the order of watts for the unit described by Kanemasu and Tanner (1968)), to elevate cylinder temperature enough that radiant heating by sunshine is negligible. Transient heating offers power savings relative to continuous heating because of a lower duty cycle and lower heating current. A smaller current is possible because radiation errors are eliminated by measuring anemometer temperature before each heating cycle; it need then only be assumed that radiant heating remains constant over a heating or cooling period (of the order of 10 s). The time required to reach equilibrium temperature for a transiently heated anemometer is dependent on windspeed. For a thermally isolated (no axial heat loss) cylinder in a steady flow and in equilibrium with air temperature at the start of heating, the cylinder warms by a first-order process ( d T / d t = Ct, where t is time and C is a negative constant), e.g. Tcylinder changes exponentially in time. Complications may arise if the heated cylinder is not thermally isolated, and if there is insufficient time between heating cycles to allow the cylinder to return to air temperature. Proper design can eliminate significant axial heat conduction. The time constant of a needle heated at rate Q ( W m 1) is found from
pc TVD2d T
dt - Q - hTrD(T - Ti)
(4a)
where p C is volumetric heat capacity of the needle material, T is time-varying needle
W.L. Bland et al. / Agricultural and Forest Meteorology 74 (1995) 227-235
229
temperature, and Ti is the initial needle temperature. Solution of this differential equation yields h T 4h 7rD~( - T0 = 1 - e x p ( - p - - ~ t ) At
(4b)
large
t, T is equal to the equilibrium needle temperature Tf, so = (Tf - Ti) and the time constant 7- = p C D / 4 h . Substitution for h from Eq. (1) with A = 0 , B = 0 . 6 2 , and n = 0 . 4 7 ( 4 0 < R e < 4 0 0 0 ; Monteith and Unsworth, 1990) yields Q/(TrDh)
_ pCD2-nlj n 7-
4Bk V n
(2.37 × 105)D 153 V°'47
(4c)
for stainless steel ( p C = 3.88 x 106 j m - 3 K -1) in 20°C air and lengths in meters. Thus a stainless steel cylinder of 1 mm diameter in a 2 m s i wind has 7- = 6.1 s. A needle anemometer of this diameter at a wind speed of 2 m s -1 thus requires about 24 s to reach within 1% of its final value after a step change in temperature. The transiently heated needle anemometer offers low cost, ruggedness, and small size; traits that make it a desirable instrument for environmental and biophysical studies. We investigated the behavior of this instrument to learn if routine application was feasible, and developed a computational approach for analysis of the signal in terms of windspeed.
2. Materials and methods 2.1. N e e d l e a n e m o m e t e r s
The heated needle anemometers were manufactured by Soiltronics (Burlington, WA). (Mention of a company and/or product does not imply endorsement by the University of Wisconsin-Madison.) The heated length was the end 20 mm of a stainless steel tube that projected 60 mm from the end of a polyvinyl chloride plastic base of 9 mm diameter by 40 mm length (Fig. 1). Shielded heater and thermocouple
~
re
Plastic base
/
0.89 mm dia. tube
Heated zone ~ 1 20 mm long
0.56 mm dia. tube
Fig. 1. Outline view of a needle anemometer. Ruggedness and simplicity are major advantages of the device.
230
W.L. Bland et al. /' Agricultural and Forest Meteorology 74 (1995) 227 235
leadwires entered the base at the end opposite the needle. Fine heater leadwires and thermocouple wires were joined to the leadwires inside the base and extended through the hypodermic tubing. The thermocouple junction was in the middle of the heated section. Finally, space around the wires in the tubing was filled with high thermal conductivity epoxy cement. Early anemometers were fabricated from tubing of 0.89 m m diameter. We believed that conduction of heat down the tubing significantly affected warming of these needles, so later models were fabricated with tubing of 0.56 m m diameter at the end 20 ram, and this was soldered to tubing of 0.89 m m diameter inserted into the base, for an overall exposed tubing length of 56 ram. 2.2. Operation and analysis
When in use, needles were continuously cycled through alternating 5 s heating periods and 5 s cooling (unheated) periods. During heating, 2 V d.c. were applied to the nominally 100 [~ heater, thus requiring a current of 20 m A one-half of the time. After each 5 s period (heating or cooling), an estimate was made of the temperature that the needle would have attained had the heating or cooling continued for a long time (Fig. 2). For a heating cycle, equilibrium temperature was dependent on windspeed, air temperature, and radiant heating of the needle. Equilibrium for a cooling cycle was (possibly time-variant) air temperature and the radiation increment. Thus the difference in equilibrium temperatures of successive heating and cooling cycles was the windspeed-determined temperature elevation above air temperature (ATr) of a needle heated continuously (or for many time constants). A heated cycle temperature record alone is not practical for providing ATr, because at low windspeed long times are required for the needle to return to air temperature during cooling. Continuous heating cannot yield A T r , because air temperature and radiation loads vary. Equilibrium temperatures were estimated using a technique described by Miller 29.0#
v
L ~27.0 ~
~
I
~
~ i
'
~
~-
~ Heating equilibrium
--I _:
25,0
"~ ¢~ Z~
23.0
-1 j Cooling ~- W i n d s p e e d = 0.5 m s ~ equilibrium _~ F j 21.0 ' ! L.~ I , j 0 10 20 30 40 50 60 70 0.5-second intervals
Fig. 2. Example temperature record for a transiently heated needle anemometer in a 0.5 m s f wind. Also shown are equilibrium temperatures that would obtain if heating or cooling continued for a long time.
W.L. Bland et al. / Agricultural and Forest Meteorology 74 (1995) 227-235
231
29-
Tavg
(=c) (3.
", .(~eating cycle
27
(3 25
©
-~ 5 m
Cooling cycle © % .
J
"° ,© ©
21 2 r
-0.8
'
i
-0.6
'
J
-0.4
'
r
-0.2
'
'
0.0
r
0.2
'
J
0.4
'
r
0.6
Ti_1- T o(°C) Fig. 3. Diagrammatic representation of the reduction of needle temperature curves in Fig. 2 to ATf. This method of finding equilibrium values of exponential processes is described by Miller in the Appendix.
(Appendix). Our application of the method is depicted in Fig. 3. Equilibrium temperatures are intercepts estimated by linear regression of the mean of two successive Tneedle measurements on their difference, for either a cooling or heating cycle. The inverse of the difference of two successive intercepts is (1/AT 0. Our use of the mean of two successive Tneedle measurements is an inconsequential and theoretically sound departure from the procedure of Miller (Appendix). A Campbell Scientific, Inc. (Logan, UT) Model 21x data acquisition and control system was programmed to read needle temperature, turn the heater current on and off, estimate (1/ATf), and convert this to velocity. Execution interval was 0.5 s, for a total of 11 temperature readings in each equilibrium temperature calculation. The program required about 75 steps; this could be reduced by about 20 steps if the correlationcovariance instruction was used to perform the linear regression. A 21x could operate at least five anemometers, if heating current was supplied externally.
2.3. Calibration The relationship between I/ATf and v/V (Eq. (3), with n = 0.5 for simplicity), was determined in two wind tunnels. Velocities from 0.05 to 2.0 m s-~ were calibrated with a small tunnel developed by the late C.B. Tanner, The tunnel cross-section was circular, 70 mm in diameter. Prior to entering the working section, the air stream flowed through a square mesh wire screen (1 wire mm -~, 0.4 mm wire), an array of plastic straws (6.4 mm diameter, 27 mm length), another screen (2 wires mm -1, 0.23 mm wire), then through two screens (4 wires mm -1, 0.11 mm wires), separated from one another and the previous screen by 27 ram. The working section (55 mm length) extended from the fourth screen to the open end. Air was supplied from the laboratory high-pressure air system (620 kPa), and volume flow rate was measured with rotameters (Ratemaster, Dwyer Instruments, Inc., Michigan City, IN) of various
232
W.L. Bland et al. / Agricultural and Forest Meteorology 74 (1995) 227-235
4.0 dspeed1/2=6.43 * (ATf)"1- 0.17 • "7
l
j
3.0
lI
E
,0
"o 2.0
J 4
.s
,Q
o.. 13 {.- 1.0 °~ L
o
,©
I I
0.0
0.1
0.2
- - _ L ~
0.3
0.4
-1 (0C-1) (ATf)
I
0.5
0.6
Fig. 4. Calibration results of the transiently heated anemometer. © , Data from the Department of Soil Science wind tunnel; 0 , data from the Department of Mechanical Engineering tunnel. The theoretically predicted linear relationship (dashed line) from Eq. (3), with n = 0.5 obtained from 0.05 to 14 m s -~ .
capacities, yielding windspeeds accurate to better than + 5 % as calibrated by a pitot tube and Datametrics (Wilmington, MA) Barocel pressure sensor. Calibration of the needle anemometers (and cup anemometer described below) at windspeeds of 0.5-14 m s -~ was performed in a large wind tunnel (exit port 0.4 m x 0.8 m) at the Department of Mechanical Engineering, University of Wisconsin-Madison. Tunnel windspeed was measured with a totalizing windrun fan anemometer (A2 4 inch Ball Bearing Anemometer, Davis Instrument, Baltimore, M D ) that had earlier been checked against a pitot tube. Fig. 4 shows calibration results of the needle anemometer from both wind tunnels. Linearity of the relationship supports the choice of n = 0.5 for this instrument. Results from the small tunnel were inadequate to predict the calibration at 14 m s -1.
2.4. Field comparison An outdoor comparison was made between the heated needle and a cup anemometer (Model 014A, MetOne Instruments, Grants Pass, OR). The cup anemometer calibration was checked in the large wind tunnel; the resulting relationship was: true windspeed (ms - l ) = 0.03 4- 1.070(cup windspeed). All subsequent observations were corrected using this equation. In the field test, both anemometers were mounted at 3 m, above a field of mowed alfalfa and short grass. The needle was mounted with the heated tip below the base, and with a plastic horizontal precipitation shield of 150 m m diameter glued to the base. Windspeed was measured every 5 s from both instruments, and averaged for 5 min intervals.
233
W.L. Bland et al. / Agricultural and Forest Meteorology 74 (1995) 227-235
3. Results and discussion Experimentally determined time constants (data not shown) were about 15% smaller than those predicted by Eq. (4c), probably owing to approximation of the heated element as a solid stainless steel rod. Measurements made during a 30 h precipitation-free period (about 360 5-rain averages) demonstrated excellent agreement between the two anemometers (Fig. 5(A)). Standard error of the estimate of the regression of the cup anemometer on the needle was +0.2 m s -~. Systematic departures from the 1:1 line are evident at the lower and upper ranges of the data, however. Further comparisons with other anemometers designs are needed to understand the sources of these discrepancies. The effect of turbulent intensity was not investigated, but the time constant of the instrument (e.g. 1.8 s at 1.6 m s -~ windspeed) would help integrate fluctuations at frequencies above the several hertz range. Precipitation in the form of wind-driven heavy mist caused error in the relationship between the anemometers (Fig. 5(B)), presumably by altering the needle energy balance. Water on the needle increased heat capacity of the instrument and cooled by evaporation, thereby reducing warming and increasing (1/AT0. We cannot envision a rain shield design that would eliminate the error under these precipitation conditions. Sensitivity of needle anemometers to precipitation limits their utility for routine windspeed monitoring. Effects of solar radiation were not assessed, but we believe that they will be insignificant under most conditions. The transiently heated needle anemometer described offers low cost, sensitivity to windspeeds as low as 0.05 m s -l, ruggedness, maintenance-free operation, and small size. It compared favorably with a widely used cup anemometer during precipitationfree periods. Computational demands of the needle anemometer are high with respect 12.0 V ~ - ~ ~ ~ - - F ~ T - V ~'10.0 f
m~/~
(A)
-b~?" 4
6.0) • 4.0) ~
i
•~ -
Z
2.0 ~J'l"
0.0 0.0
Needle windspeed = 0 98*(Cup windspeed) - 0.03 Syx = 0,2 / , ,', I ,
~
_
~
.
~
2.0 4.0 6.0 8.0 10.0 12.0 Cup windspeed (m/s)
7
0.0 ~ 0.0
~
2.0
-
-
4.0
~
6.0
~
l~
8.0 10.0 12.0
Cup windspeed (m/s)
Fig. 5. Results of outdoor comparisons of the transiently heated needle anemometer with a standard cup anemometer; data points are 5 rain averages. (A) Precipitation-free period, during which agreement was excellent; (B) period of wind-driven, heavy mist and rain. Water on the needle decreased warming, resulting in overestimation of windspeed.
234
W.L. Bland et al. / Agricultural and Forest Meteorology 74 (1995) 227 235
to capacity of current-generation environmental data acquisition equipment, but this will inevitably change in the coming years. Transiently heated needle anemometers should find application in micrometeorological research, particularly within plant canopies.
Appendix: Extrapolation of exponential approaches to equilibrium--E.E. Miller In many experimental situations, after one makes some input change to a system, an output quantity U is seen to approach a final equilibrium value Ul- exponentially with time. This situation requires either a long wait for satisfactory settling down on Uf, or use of some computationally intensive non-linear curve-fitting algorithm. Years ago I happened upon a beautifully simple scheme for extrapolating to Uf directly. It requires only that U be recorded at equal time intervals, whereupon the difference in successive values of U plotted against U becomes a straight line for any exponential approach process. For an exponential final approach of Ui to its end-point value, Uf (where the subscript i simply counts the number of data points after the first point), the residual ( U i - U0 follows the same exponential approach to zero as does the increment AiU = Ui - Ui_ I. The ratio of the residual to the increment is thus a constant, as shown in detail below. The final exponential stage of approach to equilibrium can be written Ui - Uf = ( g 0 - Uf)exp(-ti/7-)
(A1)
where U0 is the initial value of U and r is the time constant for the final approach to equilibrium. The increment of U is A i U = U i - Ui I = (Uo - U f ) [ e x p ( - t i / r ) -
exp(-ti j/r)]
(A2)
so that the ratio of residual to increment is a constant, K, given by K = ( U i - U f ) / A i U = 1/[1 - e x p ( - A t / 7 - ) ]
(A3)
It should be noted that the size of K is controlled by the experimental design choice of the time increment, At. Solving for the final value gives Uf = Ui - K z 2 x i U = Ui - [ d U i / d / k i U l ( / k i U )
(A4)
Thus, a plot of successive values of Ui vs. A iU will extrapolate to U = Uf as A i U approaches zero (Fig. A1). In many experimental situations, the approach to Uf may not be perfectly exponential in the earliest stages, as suggested by the early curved portion of Fig. A1. When the plot has straightened out for a distance safely longer than the final required distance of extrapolation, the remaining plot may reasonably assumed to be straight. Although an anemometer application of this scheme employing computer analysis is used in the main portion of this paper (which stimulated preparation of this muchoverdue note), this method has long been used in a graphical form for our soil physics
W.L. Bland et al. / Agricultural and Forest Meteorology 74 (1995) 227-235
\
\', \\\
~U -
235
J. ~U i
Curvedearly ~,
oo,on \, y ~ ..!I •[U \,
/I
\'x~ //~,Ui+2 Straight-line\~" finalportion~/~ Ui+3 Extrapolation ~ f Uf
,I
zxU
Fig. A1. Graphical version of the scheme for extrapolation. O, Observations made at equal time intervals; O, the set of points used to draw a straight line which extrapolates to the equilibrium value (U0 of the observed parameter.
teaching laboratory at the University of Wisconsin-Madison. In this graphical version, a vertical line is drawn on ordinary graph paper to represent the locus of A i U = 0, and is laid offwith a scale for plotting successive values of Ui. The values of A i U in the horizontal direction may then be obtained by drawing toward the left both horizontal and 45 ° lines through each Ui point, the intersections of these sets of lines for adjacent points being marked as points for developing the extrapolation line. Extrapolation beyond the last data intersections will cross the original vertical line at the final value of Uf, as shown in Fig. A1. The illustrated choice of a line with a slope of 45 ° was entirely arbitrary; more accuracy can usually be obtained in practice by choosing to use a slant closer to horizontal, thereby stretching the scale of AiU as appropriate. This graphical method directly displays early departures from an exponential approach to equilibrium as small, early-time curvatures of the line of experimental extrapolation. The inevitable small erratic errors of the experimental and plotting processes can be 'best fitted' by eye in the placement of the extrapolation line (best judged by using for parallax-free extrapolation a thin pigment-filled scratch on the bottom surface of a clear plastic sheet).
References Kanemasu, E.T. and Tanner, C.B., 1968. A note on heat tranport. BioScience, 18: 327-329. Monteith, J.L. and Unsworth, M.H., 1990. Principles of Environmental Physics, 2nd edn. Edward Arnold, New York, 292 pp.
Agricultural and Forest Meteorology 74 (1995) 227-235
A transiently heated needle anemometer W.L. Bland a'*, J.M. Norman a, G.S. Campbell b, C. Calissendorfff, E.E. M i l l e r a aDepartment of Soil Science, University of Wisconsin-Madison, 1525 Observatory Drive, Madison, WI 53706, USA bDepartment of Agronomy and Soils, Washington State University, Pullman, WA 99163, USA CSoiltronics, Burlington, WA 98233, USA
Received 31 January 1994; revision accepted 5 August 1994
Abstract Convective heat transfer from cylinders warmed above ambient temperature is a standard technique for measuring windspeed. Available devices are typically fragile, expensive, and complex. We developed a heat-transfer anemometer that is robust, inexpensive, and simple. Referred to as a needle anemometer, the sensing portion of the device is a 20 mm length of tubing of 0.56 mm diameter, into which a heater and thermocouple are inserted. Needle temperature measured during alternating 5 s heating and non-heating periods allows estimation of the equilibrium temperature rise above ambient temperature for long heating times. This is inversely related to (windspeed) 1/2 from 0.05 m s -1 to greater than 14 m s -1. In a field trial, the needle anemometer yielded windspeeds of J:0.2 m s -I (Sy.x) of those from a cup anemometer. Precipitation striking the needle caused overestimates of windspeed by up to 20%.
I. Introduction A n e m o m e t e r s for m e a s u r i n g w i n d speed o p e r a t e on several different principles, i n c l u d i n g the d r a g force p r o d u c e d by w i n d flowing p a s t an object (cup a n e m o m e t e r s ) , D o p p l e r shifts in the speed o f s o u n d (sonic a n e m o m e t e r s ) a n d convective h e a t exchange between m o v i n g air a n d a w a r m e r object. E a c h o f these designs has a d v a n t a g e s a n d d i s a d v a n t a g e s . C u p a n e m o m e t e r s are o n l y m o d e r a t e l y expensive, o p e r a t e over a wide r a n g e o f w i n d s p e e d a b o v e 0.5 m s - i , t y p i c a l l y require a small a m o u n t o f electrical p o w e r , m a i n t a i n c a l i b r a t i o n well with a p p r o p r i a t e * Corresponding author. 0168-1923/95/$09.50 © 1995 - Elsevier Science B.V. All rights reserved SSDI 0168-1923(94)02190-2
228
W.L. Bland et al. / Agricultural and Forest Meteorology 74 (1995) 227 235
maintenance, are physically large, and require infrequent maintenance--perhaps yearly. Sonic anemometers are expensive, measure only components of wind and not windspeed directly, maintain calibration very well, operate over a wide range of windspeed above a few centimeters per second, require modest electrical power, are physically large, and may or may not be reliable, depending on the instrument. In this paper we describe a heat transfer anemometer that is inexpensive and reliable, requires little electrical power (0.03 W), is small and robust, and operates from 0.05 to more than 15 m s 1. Transfer of energy from a heated cylinder to a flowing fluid is strongly related to the fluid velocity. This relationship is widely exploited in anemometry, particularly when low velocities must be measured near surfaces and when rapid time response is needed. For air flow normal to a cylinder axis (Monteith and Unsworth, 1990) Nu = ( h D / k ) = A + BRe n
(1)
where Nu is the Nusselt number, h is the convective heat transfer coefficient ( W m 2 °C 1), D is the cylinder diameter (m), k is thermal conductivity of air ( W m -I ° C l), Re = V D / u is the Reynolds number (V is windspeed (m s 1), u is kinematic viscosity (m 2 s-l)), and A, B, and n are empirical factors that vary with Re. The rate of heat transfer, Q ( w m-2), from the cylinder surface is Q = h(Tcylinder - Tair) = ] t / ~ T t.
(2)
Combining Eqs. (1) and (2) yields = -~
= a + A-~f
(3)
Heat transfer anemometers may either be continuously or transiently heated. Continuously heated anemometers usually require large power inputs (of the order of watts for the unit described by Kanemasu and Tanner (1968)), to elevate cylinder temperature enough that radiant heating by sunshine is negligible. Transient heating offers power savings relative to continuous heating because of a lower duty cycle and lower heating current. A smaller current is possible because radiation errors are eliminated by measuring anemometer temperature before each heating cycle; it need then only be assumed that radiant heating remains constant over a heating or cooling period (of the order of 10 s). The time required to reach equilibrium temperature for a transiently heated anemometer is dependent on windspeed. For a thermally isolated (no axial heat loss) cylinder in a steady flow and in equilibrium with air temperature at the start of heating, the cylinder warms by a first-order process ( d T / d t = Ct, where t is time and C is a negative constant), e.g. Tcylinder changes exponentially in time. Complications may arise if the heated cylinder is not thermally isolated, and if there is insufficient time between heating cycles to allow the cylinder to return to air temperature. Proper design can eliminate significant axial heat conduction. The time constant of a needle heated at rate Q ( W m 1) is found from
pc TVD2d T
dt - Q - hTrD(T - Ti)
(4a)
where p C is volumetric heat capacity of the needle material, T is time-varying needle
W.L. Bland et al. / Agricultural and Forest Meteorology 74 (1995) 227-235
229
temperature, and Ti is the initial needle temperature. Solution of this differential equation yields h T 4h 7rD~( - T0 = 1 - e x p ( - p - - ~ t ) At
(4b)
large
t, T is equal to the equilibrium needle temperature Tf, so = (Tf - Ti) and the time constant 7- = p C D / 4 h . Substitution for h from Eq. (1) with A = 0 , B = 0 . 6 2 , and n = 0 . 4 7 ( 4 0 < R e < 4 0 0 0 ; Monteith and Unsworth, 1990) yields Q/(TrDh)
_ pCD2-nlj n 7-
4Bk V n
(2.37 × 105)D 153 V°'47
(4c)
for stainless steel ( p C = 3.88 x 106 j m - 3 K -1) in 20°C air and lengths in meters. Thus a stainless steel cylinder of 1 mm diameter in a 2 m s i wind has 7- = 6.1 s. A needle anemometer of this diameter at a wind speed of 2 m s -1 thus requires about 24 s to reach within 1% of its final value after a step change in temperature. The transiently heated needle anemometer offers low cost, ruggedness, and small size; traits that make it a desirable instrument for environmental and biophysical studies. We investigated the behavior of this instrument to learn if routine application was feasible, and developed a computational approach for analysis of the signal in terms of windspeed.
2. Materials and methods 2.1. N e e d l e a n e m o m e t e r s
The heated needle anemometers were manufactured by Soiltronics (Burlington, WA). (Mention of a company and/or product does not imply endorsement by the University of Wisconsin-Madison.) The heated length was the end 20 mm of a stainless steel tube that projected 60 mm from the end of a polyvinyl chloride plastic base of 9 mm diameter by 40 mm length (Fig. 1). Shielded heater and thermocouple
~
re
Plastic base
/
0.89 mm dia. tube
Heated zone ~ 1 20 mm long
0.56 mm dia. tube
Fig. 1. Outline view of a needle anemometer. Ruggedness and simplicity are major advantages of the device.
230
W.L. Bland et al. /' Agricultural and Forest Meteorology 74 (1995) 227 235
leadwires entered the base at the end opposite the needle. Fine heater leadwires and thermocouple wires were joined to the leadwires inside the base and extended through the hypodermic tubing. The thermocouple junction was in the middle of the heated section. Finally, space around the wires in the tubing was filled with high thermal conductivity epoxy cement. Early anemometers were fabricated from tubing of 0.89 m m diameter. We believed that conduction of heat down the tubing significantly affected warming of these needles, so later models were fabricated with tubing of 0.56 m m diameter at the end 20 ram, and this was soldered to tubing of 0.89 m m diameter inserted into the base, for an overall exposed tubing length of 56 ram. 2.2. Operation and analysis
When in use, needles were continuously cycled through alternating 5 s heating periods and 5 s cooling (unheated) periods. During heating, 2 V d.c. were applied to the nominally 100 [~ heater, thus requiring a current of 20 m A one-half of the time. After each 5 s period (heating or cooling), an estimate was made of the temperature that the needle would have attained had the heating or cooling continued for a long time (Fig. 2). For a heating cycle, equilibrium temperature was dependent on windspeed, air temperature, and radiant heating of the needle. Equilibrium for a cooling cycle was (possibly time-variant) air temperature and the radiation increment. Thus the difference in equilibrium temperatures of successive heating and cooling cycles was the windspeed-determined temperature elevation above air temperature (ATr) of a needle heated continuously (or for many time constants). A heated cycle temperature record alone is not practical for providing ATr, because at low windspeed long times are required for the needle to return to air temperature during cooling. Continuous heating cannot yield A T r , because air temperature and radiation loads vary. Equilibrium temperatures were estimated using a technique described by Miller 29.0#
v
L ~27.0 ~
~
I
~
~ i
'
~
~-
~ Heating equilibrium
--I _:
25,0
"~ ¢~ Z~
23.0
-1 j Cooling ~- W i n d s p e e d = 0.5 m s ~ equilibrium _~ F j 21.0 ' ! L.~ I , j 0 10 20 30 40 50 60 70 0.5-second intervals
Fig. 2. Example temperature record for a transiently heated needle anemometer in a 0.5 m s f wind. Also shown are equilibrium temperatures that would obtain if heating or cooling continued for a long time.
W.L. Bland et al. / Agricultural and Forest Meteorology 74 (1995) 227-235
231
29-
Tavg
(=c) (3.
", .(~eating cycle
27
(3 25
©
-~ 5 m
Cooling cycle © % .
J
"° ,© ©
21 2 r
-0.8
'
i
-0.6
'
J
-0.4
'
r
-0.2
'
'
0.0
r
0.2
'
J
0.4
'
r
0.6
Ti_1- T o(°C) Fig. 3. Diagrammatic representation of the reduction of needle temperature curves in Fig. 2 to ATf. This method of finding equilibrium values of exponential processes is described by Miller in the Appendix.
(Appendix). Our application of the method is depicted in Fig. 3. Equilibrium temperatures are intercepts estimated by linear regression of the mean of two successive Tneedle measurements on their difference, for either a cooling or heating cycle. The inverse of the difference of two successive intercepts is (1/AT 0. Our use of the mean of two successive Tneedle measurements is an inconsequential and theoretically sound departure from the procedure of Miller (Appendix). A Campbell Scientific, Inc. (Logan, UT) Model 21x data acquisition and control system was programmed to read needle temperature, turn the heater current on and off, estimate (1/ATf), and convert this to velocity. Execution interval was 0.5 s, for a total of 11 temperature readings in each equilibrium temperature calculation. The program required about 75 steps; this could be reduced by about 20 steps if the correlationcovariance instruction was used to perform the linear regression. A 21x could operate at least five anemometers, if heating current was supplied externally.
2.3. Calibration The relationship between I/ATf and v/V (Eq. (3), with n = 0.5 for simplicity), was determined in two wind tunnels. Velocities from 0.05 to 2.0 m s-~ were calibrated with a small tunnel developed by the late C.B. Tanner, The tunnel cross-section was circular, 70 mm in diameter. Prior to entering the working section, the air stream flowed through a square mesh wire screen (1 wire mm -~, 0.4 mm wire), an array of plastic straws (6.4 mm diameter, 27 mm length), another screen (2 wires mm -1, 0.23 mm wire), then through two screens (4 wires mm -1, 0.11 mm wires), separated from one another and the previous screen by 27 ram. The working section (55 mm length) extended from the fourth screen to the open end. Air was supplied from the laboratory high-pressure air system (620 kPa), and volume flow rate was measured with rotameters (Ratemaster, Dwyer Instruments, Inc., Michigan City, IN) of various
232
W.L. Bland et al. / Agricultural and Forest Meteorology 74 (1995) 227-235
4.0 dspeed1/2=6.43 * (ATf)"1- 0.17 • "7
l
j
3.0
lI
E
,0
"o 2.0
J 4
.s
,Q
o.. 13 {.- 1.0 °~ L
o
,©
I I
0.0
0.1
0.2
- - _ L ~
0.3
0.4
-1 (0C-1) (ATf)
I
0.5
0.6
Fig. 4. Calibration results of the transiently heated anemometer. © , Data from the Department of Soil Science wind tunnel; 0 , data from the Department of Mechanical Engineering tunnel. The theoretically predicted linear relationship (dashed line) from Eq. (3), with n = 0.5 obtained from 0.05 to 14 m s -~ .
capacities, yielding windspeeds accurate to better than + 5 % as calibrated by a pitot tube and Datametrics (Wilmington, MA) Barocel pressure sensor. Calibration of the needle anemometers (and cup anemometer described below) at windspeeds of 0.5-14 m s -~ was performed in a large wind tunnel (exit port 0.4 m x 0.8 m) at the Department of Mechanical Engineering, University of Wisconsin-Madison. Tunnel windspeed was measured with a totalizing windrun fan anemometer (A2 4 inch Ball Bearing Anemometer, Davis Instrument, Baltimore, M D ) that had earlier been checked against a pitot tube. Fig. 4 shows calibration results of the needle anemometer from both wind tunnels. Linearity of the relationship supports the choice of n = 0.5 for this instrument. Results from the small tunnel were inadequate to predict the calibration at 14 m s -1.
2.4. Field comparison An outdoor comparison was made between the heated needle and a cup anemometer (Model 014A, MetOne Instruments, Grants Pass, OR). The cup anemometer calibration was checked in the large wind tunnel; the resulting relationship was: true windspeed (ms - l ) = 0.03 4- 1.070(cup windspeed). All subsequent observations were corrected using this equation. In the field test, both anemometers were mounted at 3 m, above a field of mowed alfalfa and short grass. The needle was mounted with the heated tip below the base, and with a plastic horizontal precipitation shield of 150 m m diameter glued to the base. Windspeed was measured every 5 s from both instruments, and averaged for 5 min intervals.
233
W.L. Bland et al. / Agricultural and Forest Meteorology 74 (1995) 227-235
3. Results and discussion Experimentally determined time constants (data not shown) were about 15% smaller than those predicted by Eq. (4c), probably owing to approximation of the heated element as a solid stainless steel rod. Measurements made during a 30 h precipitation-free period (about 360 5-rain averages) demonstrated excellent agreement between the two anemometers (Fig. 5(A)). Standard error of the estimate of the regression of the cup anemometer on the needle was +0.2 m s -~. Systematic departures from the 1:1 line are evident at the lower and upper ranges of the data, however. Further comparisons with other anemometers designs are needed to understand the sources of these discrepancies. The effect of turbulent intensity was not investigated, but the time constant of the instrument (e.g. 1.8 s at 1.6 m s -~ windspeed) would help integrate fluctuations at frequencies above the several hertz range. Precipitation in the form of wind-driven heavy mist caused error in the relationship between the anemometers (Fig. 5(B)), presumably by altering the needle energy balance. Water on the needle increased heat capacity of the instrument and cooled by evaporation, thereby reducing warming and increasing (1/AT0. We cannot envision a rain shield design that would eliminate the error under these precipitation conditions. Sensitivity of needle anemometers to precipitation limits their utility for routine windspeed monitoring. Effects of solar radiation were not assessed, but we believe that they will be insignificant under most conditions. The transiently heated needle anemometer described offers low cost, sensitivity to windspeeds as low as 0.05 m s -l, ruggedness, maintenance-free operation, and small size. It compared favorably with a widely used cup anemometer during precipitationfree periods. Computational demands of the needle anemometer are high with respect 12.0 V ~ - ~ ~ ~ - - F ~ T - V ~'10.0 f
m~/~
(A)
-b~?" 4
6.0) • 4.0) ~
i
•~ -
Z
2.0 ~J'l"
0.0 0.0
Needle windspeed = 0 98*(Cup windspeed) - 0.03 Syx = 0,2 / , ,', I ,
~
_
~
.
~
2.0 4.0 6.0 8.0 10.0 12.0 Cup windspeed (m/s)
7
0.0 ~ 0.0
~
2.0
-
-
4.0
~
6.0
~
l~
8.0 10.0 12.0
Cup windspeed (m/s)
Fig. 5. Results of outdoor comparisons of the transiently heated needle anemometer with a standard cup anemometer; data points are 5 rain averages. (A) Precipitation-free period, during which agreement was excellent; (B) period of wind-driven, heavy mist and rain. Water on the needle decreased warming, resulting in overestimation of windspeed.
234
W.L. Bland et al. / Agricultural and Forest Meteorology 74 (1995) 227 235
to capacity of current-generation environmental data acquisition equipment, but this will inevitably change in the coming years. Transiently heated needle anemometers should find application in micrometeorological research, particularly within plant canopies.
Appendix: Extrapolation of exponential approaches to equilibrium--E.E. Miller In many experimental situations, after one makes some input change to a system, an output quantity U is seen to approach a final equilibrium value Ul- exponentially with time. This situation requires either a long wait for satisfactory settling down on Uf, or use of some computationally intensive non-linear curve-fitting algorithm. Years ago I happened upon a beautifully simple scheme for extrapolating to Uf directly. It requires only that U be recorded at equal time intervals, whereupon the difference in successive values of U plotted against U becomes a straight line for any exponential approach process. For an exponential final approach of Ui to its end-point value, Uf (where the subscript i simply counts the number of data points after the first point), the residual ( U i - U0 follows the same exponential approach to zero as does the increment AiU = Ui - Ui_ I. The ratio of the residual to the increment is thus a constant, as shown in detail below. The final exponential stage of approach to equilibrium can be written Ui - Uf = ( g 0 - Uf)exp(-ti/7-)
(A1)
where U0 is the initial value of U and r is the time constant for the final approach to equilibrium. The increment of U is A i U = U i - Ui I = (Uo - U f ) [ e x p ( - t i / r ) -
exp(-ti j/r)]
(A2)
so that the ratio of residual to increment is a constant, K, given by K = ( U i - U f ) / A i U = 1/[1 - e x p ( - A t / 7 - ) ]
(A3)
It should be noted that the size of K is controlled by the experimental design choice of the time increment, At. Solving for the final value gives Uf = Ui - K z 2 x i U = Ui - [ d U i / d / k i U l ( / k i U )
(A4)
Thus, a plot of successive values of Ui vs. A iU will extrapolate to U = Uf as A i U approaches zero (Fig. A1). In many experimental situations, the approach to Uf may not be perfectly exponential in the earliest stages, as suggested by the early curved portion of Fig. A1. When the plot has straightened out for a distance safely longer than the final required distance of extrapolation, the remaining plot may reasonably assumed to be straight. Although an anemometer application of this scheme employing computer analysis is used in the main portion of this paper (which stimulated preparation of this muchoverdue note), this method has long been used in a graphical form for our soil physics
W.L. Bland et al. / Agricultural and Forest Meteorology 74 (1995) 227-235
\
\', \\\
~U -
235
J. ~U i
Curvedearly ~,
oo,on \, y ~ ..!I •[U \,
/I
\'x~ //~,Ui+2 Straight-line\~" finalportion~/~ Ui+3 Extrapolation ~ f Uf
,I
zxU
Fig. A1. Graphical version of the scheme for extrapolation. O, Observations made at equal time intervals; O, the set of points used to draw a straight line which extrapolates to the equilibrium value (U0 of the observed parameter.
teaching laboratory at the University of Wisconsin-Madison. In this graphical version, a vertical line is drawn on ordinary graph paper to represent the locus of A i U = 0, and is laid offwith a scale for plotting successive values of Ui. The values of A i U in the horizontal direction may then be obtained by drawing toward the left both horizontal and 45 ° lines through each Ui point, the intersections of these sets of lines for adjacent points being marked as points for developing the extrapolation line. Extrapolation beyond the last data intersections will cross the original vertical line at the final value of Uf, as shown in Fig. A1. The illustrated choice of a line with a slope of 45 ° was entirely arbitrary; more accuracy can usually be obtained in practice by choosing to use a slant closer to horizontal, thereby stretching the scale of AiU as appropriate. This graphical method directly displays early departures from an exponential approach to equilibrium as small, early-time curvatures of the line of experimental extrapolation. The inevitable small erratic errors of the experimental and plotting processes can be 'best fitted' by eye in the placement of the extrapolation line (best judged by using for parallax-free extrapolation a thin pigment-filled scratch on the bottom surface of a clear plastic sheet).
References Kanemasu, E.T. and Tanner, C.B., 1968. A note on heat tranport. BioScience, 18: 327-329. Monteith, J.L. and Unsworth, M.H., 1990. Principles of Environmental Physics, 2nd edn. Edward Arnold, New York, 292 pp.