Solar- and Sky-Radiation Integrator 1

Transcription

1 VOL. 42, No. 5, MAY, Solar- and Sky-Radiation Integrator 1 D. P. BROWN AND R. A. HARVEY General Electric Company, Richland, Washington (Manuscript received 30 June 1960) ABSRAC An integrator, which automatically and periodically records the weighted average solar radiation from the previous period, has been developed. he instrument is used to obtain a measure of the average solar radiant energy per day. his measurement was previously obtained by manual and often tedious graphical analysis. heoretical considerations involving averaging by passive networks are presented with experimental results which show the accuracy of the instrument. 1. Introduction An integrator was required for averaging the output of a pyrheliometer, an instrument used for measuring the intensity of solar and sky radiation on a horizontal surface. he pyrheliometer output is proportional to the radiation intensity and is calibrated in langleys per minute. he output is 2.49 millivolts per langley with a maximum output of two langleys or 4.58 millivolts. During days in which the sky contains cloud formations, the recorded output of the pyrheliometer is an erratic trace, and, on clear days, the trace approximates the positive half of a sine wave. Since it was desired to obtain a measure of the average radiant energy per day, a procedure was established for obtaining hourly averages from which a daily average was determined. his procedure was completely manual and entailed graphical analyses which were tedious and time consuming, particularly during cloudy days. In order to make the analyses less tedious and time consuming, the integrator described in this report was developed. he instrument utilizes a passive smoothing network which automatically records periodically the weighted average of the previous period's solar radiation. Experimental results indicate that the averages recorded by the instrument compare favorably with those obtained by manual graphical analyses. 2. heory of operation he pyrheliometer has white and black surfaces exposed to the sun and sky light. Since the white surface reflects more energy than the black does, a temperature difference between the two 1 Work performed under Contract No. A (45-1)-1350 for the U. S. Atomic Energy Commission. surfaces exists which is proportional to the amount of incident radiation. his temperature difference is measured by thermocouples, the output of which ranges from zero at night to a maximum of 4.98 millivolts during a bright sunny day. his output is continuously recorded on a Leeds and Northrup Speedomax recorder. Fig. 1 is a typical recording for a sunny day, and fig. 2 is a recording for a typical partly cloudy day. Since the primary objective of the instrument is to perform the task of averaging, the equations governing the response are presented and the response is derived for specific pertinent examples. he response to a time function with a constant slope is considered pertinent because on clear days the slope of the pyrheliometer output changes slowly in comparison to the instrument averaging t'me, and it can be considered almost as a straight line. It is shown that the instrument can be made to indicate exactly the same as the previous manual method of averaging for this case. he second time function considered approaches an impulse. his is assumed pertinent because it is an extreme condition and one that might occur on an overcast day in which' the sun momentarily shines through a hole in the clouds. It is demonstrated that the indicated response for this extreme condition theoretically gives the exact answer over long periods of time, such as several hours, but the incremental values do not coincide with those obtained by the manual method. In this example, it is considered significant that the actual value of the time constant does not affect the final result but only the amount of smoothing and the correspondence with previous manual methods. he expression for the average of a time function at the time tl9 over a particular time interval,, can be expressed by

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4 328 BULLEIN AMERICAN MEEOROLOGICAL SOCIEY ti+{!2) F{t.) = j 1 Jtl-{ f " fit) dt *l-(/2) (1) where F(fa) is the average value. In order to express the average as a continuous time function, the general approach of weighting functions is used. he time function, f(t-r), is multiplied by a weighting function w(t) and the results are integrated to give F(t) = fit- r)w(r)dr. (2) he running average can be obtained by letting w(t) equal l/over the interval /2 to +/2 and by letting w(r) equal zero elsewhere. w(t) = j. = 0 ~ 2 ~ - 2 ~ 2 > > r (3) With this particular weighting function, F(t) becomes 1 CI2 Fif) = ~ f(t- r)dr (4) i J-/2 and, at time fa, i rri2 Fiti) = J, fih - r)dr. (5) 1 J-/2 he answer obtained by this expression is identical to that obtained by eq (1). A delayed running average can be indicated by letting so that w{t) =~ 0 < r< 0 0 > r > Fit) fit ~ r)dr. - ij- J f 0 (6) (7) Eq (7) differs from (4) only in the limits of integration. It is considered significant, however, because this is the answer obtained by the manual method at Hanford. he only difference is that, for any given time interval, the average value is recorded at the end of the interval instead of at the center. his is important from the instrumentation viewpoint because, for coincidence of instrument and manual readings under easy comparison conditions, it is desirable to have the same time delay. he weighting function given by eq (3) can- not be instrumentally obtained exactly since prediction is required, and that of eq (6) requires the use of recording equipment with later playback. A simple practical system employs a low-pass resistance-capacitance (RC) filter. his can be shown to provide correct averages for the daily radiation-incidence period, but it gives a different short-time result. he weighting function for the RC filter is given by w(t) = J[c e ~,RC 0 < r < oo r < 0 (8) so that the expression for the average, from (2), becomes Fit) = J c ^ fh ~ r)e~^ dr. (9) he instrument average as determined by eq (9) is compared with that of the manual method, indicated by eq (7), in the following examples. Example 1: A linearity rising-time function f{t) = At. (10) he true running average is found from (4) to be I /2 Ftit) = (t - r)d7 (11) /2 which results in Ft{t) = At. (12) he delayed running average from (7) is Fait) = which results in r )dl Fait) = A (* - f ) (13) (14) his is shown in fig. 3a. he approximate running average obtained with the RC filter, from (9), is Fnait) = ic r (t ~ r)6 ~,RC d ' his results in If we let Fucit) = A it-rc). RC = j, (15) (16) (17)

5 VOL. 42, No. 5, MAY, Comparison of the filter response with the delayed running average (14) shows that they are the same. he normal situation is not a constant rate of rise for an infinite time as indicated by the integration limit of (15) ; therefore, a transient component is introduced which decays to approximately two per cent in two averaging periods. In four averaging periods, the transient component decays to about 0.03 per cent. Over a complete daily cycle, the integral of the approximate running average can be shown to be equal to the integral of the true or the delayed running averages. FIG. 3a. ime relationship between a linear function and the delayed running average. he dotted line indicates the initial transient. Example 2: An impulse For the purposes of this discussion, an impulse is defined as a square pulse of A units of solar and sky radiation in magnitude and l/a units of time duration so that the total area under the curve is unity. he time function is assumed to be zero at all other times. his can be considered as an impulse if the time duration, l/a, is considerably shorter than the averaging time or the RC time constant. In the case of the delayed running average, the impulse contribution for the pertinent averaging period is of magnitude 1/ for the averaging interval so that the total area under the curve is still unity. If the impulse occurs at time t = 0, Fd(t) = = 0 0 < t < 0 > t>. (19) he total contribution of the impulse at the end of the day is found by multiplying the length of the averaging period by the average value over this period. or G = Fd (t) (20) FIG. 3b. ime relationship between a linear time function and the exponentially weighted running average with RC /2. he dotted line indicates the initial transient. the filter output, shown in fig. 3b, is FRC(t) (18) which gives G = 1; (21) (22) hence, it can be seen that the contribution of the unit impulse is unity by the manual method of averaging. With RC smoothing of the impulse, F(t) becomes FRC(t) =A(l-e~ t ' RC ) 0<K1/A = A(l-e- 1,ARC )e-^t- 1 ' A^RC ~j<t< 00 = 0 *<0. After a long time, this contribution is (23) G = r FRC(t)dt, (24)

6 330 BULLEIN AMERICAN MEEOROLOGICAL SOCIEY FIG. 4a. ime relationship between an impulse and the delayed running average. A is assumed to be very large. which, from (23), becomes ri/a G = A I Jo (1 - e-ti* 0 ) dt + A( 1 - e~ l ' ARC ) which results in X I e-a-im)/^ dtf (25) J l/a G = 1. (26) Fig. 4 shows the time relationships for the impulse. Of interest at this point is the fact that the actual value of the RC time-constant is not a factor in the final result. A longer RC timeconstant results in better smoothing, but would not indicate exactly as previous averaging methods on clear days. An averaging time of one hour was used in the FIG. 4b. ime relationship between an impulse and the exponentially weighted running average with RC = /2. previous method; therefore, an RC time-constant of thirty minutes was indicated. It was decided, however, that an average would be recorded each half-hour; consequently, a time-constant of fifteen minutes was employed. his caused the continuous curve for the approximate average, FRC, to differ from that of the previous manual method, Fd, but the simple process of determining the average of two half-hour periods corresponded with the one-hour average recorded at hourly intervals. 3. Description A block diagram of the solar integrator is shown in fig. 5. he pyrheliometer is located FIG. 5. Solar- and sky-radiation integrator block diagram.

7 VOL. 42, No. 5, MAY, FIG. 6. Power supply. about 75 ft from the recorder and is connected to the integrator chassis. he output of the pyrheliometer is connected through the closed contacts of a microswitch to the recorder. A 100-K potentiometer is mechanically attached to the recorder and has 300 v applied across it. he slider of the potentiometer is connected to the input of the RC network. Since the position of the slider is determined by the recorder and since the recorder position is determined by the input from the pyrheliometer, which can vary from 0 to 5 mv, the input to the RC network can vary from 0 to 300 v, an amplification of 60,000. he stability of this amplifier depends on the stability of the 300-v power supply (fig. 6). he RC network which receives this amplified signal consists of one 200-megohm resistor, one 250-megohm resistor and eight one-microfarad capacitors in parallel (see fig. 7). his combination results in an RC time constant of 14.8 min. 2 Because of the long-time constants involved, it was necessary to use large resistances and Mylar 3 capacitors. It was also necessary to keep these components spotlessly clean and well insulated to minimize leakage. o keep these components clean, they were enclosed in a metal box and the box was sealed. 2 he effects on the time constant of the other resistors in the circuit are negligible. 3 rademark of the E. I. DuPont Corporation. FIG. 7. Integrating network for solar- and sky-radiation integrator.

8 332 he output of the integrator was taken from a voltage divider to reduce the voltage of the integrator circuit to that which could be applied directly to the 0- to 5-millivolt Speedomax recorder. In this way, the same recorder was used to record the periodic averages that was used to record the solar intensity. o accomplish this, the recorder is connected directly to the pyrheliometer for 29 min and to the integrator for one minute by a timer operating the microswitch indicated in fig. 5. Readings every thirty minutes were selected because the recorder chart is divided into intervals of thirty minutes. By allowing the integrator to read out for one minute, twenty-nine minutes were left for integrating during each halfhour period. Figs. 1 and 2 illustrate how the integrator readings are recorded on the chart. Since it was desired, in deriving the equation for the function representing the average, F(t), that be twice the RC time constant, the time constant should be 14.5 min. he difference between the manual and instrument readings introduced by making the time constant 14.8 min is negligible. During the one-minute readout, the integrator is in a positive feedback loop which must have a closed loop gain of unity; otherwise, the readout value will drift from the proper value. 4. Experimental results After the instrument was installed and operating satisfactorily, data were obtained to determine the accuracy of the integrator. hese data were obtained from three sources: the integrator output as recorded on the strip chart at half-hour intervals and averaged for each hour, the value obtained by using the manual method, and values obtained by measuring the area of the curve with a planimeter. he values for each hourly average for a ten-day test period covering one day in July, six days in August, and three days in November of 1958 were compared by noting the difference in values obtained. able 1 shows that, for this test period, there were 130 hourly averages taken and that 92 per cent of the integrator readings BULLEIN AMERICAN MEEOROLOGICAL SOCIEY ABLE 1. Number of MSP readings (S) and integrator readings (/) which differ from planimeter values (P) by the values shown in left-hand column Difference Number of Number of (langleys/min) S-P I-P otal were within ± 0.01 langley/min of these obtained by measurement with the planimeter. Also indicated is that 93 per cent of the values obtained by the manual procedure were within ± 0.01 langley/min. By summing the hourly averages for each day and multiplying by sixty, the total number of langleys recorded for that day was obtained. By summing the total number of langleys for the tenday test period, 5000 langleys were obtained by measurement with the planimeter, 5021 langleys by use of the integrator, and 4996 langleys by use of the manual procedure. his is an error of 0.42 per cent for the integrator. 5. Conclusions he results obtained by the experimental instrument are accurate, and considerable time is saved on partly cloudy days. he concept of obtaining an average by the means described in this appears applicable to the particular situation for which the instrument was developed. he same techniques can be applied to any recorder on which precision potentiometer can be mounted. Acknowledgment. he authors wish to thank G. D. Linsey and W. H. Reading, III, for their efforts in the development of the instrument, and D. E. Jenne who compiled the experimental data from the pyrheliometer charts.

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