INTERNAL REPORT 51
DIRECT SOLAR RADIATION ON VARIOUS SLOPES
FOR 45 AND 47.5 DEGREES NORTH LATITUDE
John Buffo and Leo J. Fritschen
University of Washington
NOTICE: This internal report contains information
of a preliminary nature, prepared primarily for
internal use in the US/IBP Coniferous Forest Biome
program. This Information is not for use prior
to publication unless permission is obtained in
writing from the author.
ABSTRACT
Direct-beam solar radiation is presented
for hourly, daily, and ye?r1y values for
and
47.5 degrees north
latitude.
the calculations are given.
in graphical and tabular form
7 slopes on 16 aspects at 45
Theoretical equations necessary for
Solar altitude and azimuth during the day
and year are also presented for the same latitudes.
INTRODUCTION
Solar energy Is by far the most Important climatic factor; It has many
unrealized uses. For example, it is an important consideration
when
selecting building sites and In designing and landscaping buildings.
In fact, one's personal home comfort is strongly dependent on solar
energy. Solar energy Is a very important parameter in ecological
problems dealing with silviculture, entomology, pathology, and
fire control.
Optimum use of solar energy requires knowledge of the exact location
of the sun and the magnitude of insolation at any time of the
day and
year. Tables and graphs have
been prepared that give the sun's location
(that is, altitude and azimuth) as a function of latitude, time of day,
and year (List 1958, Hutchinson and Cotter 1955). More recently,
computer programs have been written that give not only the
sun's location
but also direct solar radiation on a horizontal surface (Robertson
and Russelo 1968, Furnival et al. 1969). Direct solar radiation
cn
a horizontal surface, however, may not be as useful as radiation on
sloping surfaces or vertical walls. Tables for these values have been
prepared by Frank and Lee (1966) and Fons et al. (1960).
present either daily solar radiation values or coefficientsThe tables
that must
be multiplied by direct solar radiation on a horizontal surface
for
specific aspects, slopes, and latitudes. These
tables are of limited
use because of the range of latitudes and slopes presented
and because
additional calculations are required. Therefore computer programs
were
modified and written to calculate direct solar radiation
on selected
slopes and aspects at 45 and 47.5 degrees north. The results are
presented in tabular form and as graphical plots of hourly and
daily
values
THEORY
Theoretical derivation of the necessary equations have been given by
Byram and Jemlson (1943), Fons et al. (1960), and Frank and Lee (1966),
among others.
The equations are presented here for completeness. A
sample surface is located in the northern hemisphere as shown in Figure
The sun is directly overhead at point p on-meridian Zq with a declinationf,,.
The sample surface is located at latitude ¢, point r, on meridian mq,
which has an hour angle h from the meridian Zq.
The surface ofgi has a
slope a from the horizontal jo, a deviation of a from
the vertical cd,
and an aspect 0 from the north no.
17-7-1.
The sun's rays are striking the surface at' with an altitude angle Sok,
called A. from the horizontal. Thealtitude'A is given by:
sin A - sin
and the
VC,
sin d + cos
;azimuth AZ---f rom-the 'north,
sin AZ -
The'solar Intensity Ion the surface
sin
where
cS
cos h
where AZ = Z + 90
-cos 6 sin
I= iopt/sinA
cos
degrees, is given by:
h/cos A
cfgi is:
sin 8
e = sin A cos a
- cos A sin a sin (Z -
S)
Here Io Is the radiation at the top of the atmosphere on a surface normal
to the sun's rays and p is the atomspheric transmission coefficient.
RESULTS AND DISCUSSION
for slopes in 15-degree increments
wall, 16 aspects in 22.5 degree increments,
on 11 selected days of the year, at latitudes 45 and 47.5 degrees north,
and atmospheric transmission coefficient of 0.9. The possible combinations
are detailed in Table 1.
Other options may have been desirable but were
Hourly solar radiation was computed
ranging from
level to a vertical
eliminated because of costs.
The atmospheric coefficient of 0.9 was selected because it was representative of conditions on the top of a mountain, about 1500 m, on a clear day._
Other values would be useful for some areas where atmospheric pollution
is severe. List (1958) presents seasonal totals of direct solar radiation
with different atmospheric coefficients at many latitudes. These values
may be useful for comparison.
The hourly computations for the 11 days are displayed in three ways as:
(1) isograms of radiation drawn on graphs of time versus slope for specific
days, aspects, and latitudes; (2) daily totals in tabular form for specific
days, aspects, and latitudes; and (3) annual totals in tabular form for
specific slopes, aspects, and latitudes. The details of display are discussed in the following section.
The computed values apply only to surfaces on level terrain where sunrise
and sunset are not restricted by topographic features. Many applications
of the radiation values are in city or mountainous areas, which have
limited day length. Other computer programs can be utilized for these
special
applications.1
Knowledge of the solar altitude and azimuth as a
1Ross Lake Solar Program, Forest Meteorology, College of Forest Resources,
University of Washington.
-2-
function of the time of day and day of year is required to compute solar
List (1958) presents in convenient graphical
radiation for these areas.
form solar azimuths and elevations for the same days and latitudes as
listed in Table 1.
For completeness, the graphs are duplicated here
(Figures 24 and 25).2
EXPLANATION OF FIGURES AND TABLES
Isograms of direct solar radiation (hourly values drawn for every 10
and labeled for every 20 cal cm-2) were drawn on graphs of time (hours)
versus slope (degrees) for each of the 16 aspects and each of the 11
days at both latitudes. For convenience, all of the graphs for a
specific day and latitude have been reduced and displayed together
(Figures 2-23).
Some subfigures are labeled double because the east
and west aspects are mirror images with respect to solar radiation.
The degree of slope is denoted across the bottom of each graph. The
reader should note that: (1) the left axis radiation values (0 slope)
for each graph for a given day are the same for the same hours of each
day, (2) the south-facing walls (right axis of subfigure south) are in
sunlight when the north-facing wails (right axis of subfigure north)
are not, (3) the east-facing slopes are in sunlight while the comparable
west-facing slopes are not, (4) the hours of daylight decrease from
summer (Figures 2 and 13) to winter (Figures 12 and 23), and (5) the
degree of slope receiving the most radiation increases from summer to
winter as the sun's altitude decreases.
The second method of presentation of the radiation data is in tabular
form.
The hourly
values were accumulated to form daily totals.
These
are presented in Tables 2 and 3 as a function of degree of slope and
aspect for each of the 11 days and for each latitude.
Annual values of solar radiation for each slope and aspect are given
in Tables 4 and 5. Small errors exist In these totals because it was
assumed that the period from 22 December to 22 June was symmetrical
with the period from 22 June to 22 December. This error is small
compared with variations in atmospheric transmission coefficients and
in determining the slope and aspect of a particular surface.
Solar altitude and azimuth for the two latitudes are given In Figures 24
and 25. The user should refer to Table i for the approximate date of
the various declinations.
For example, the declination of -5 degrees
occurs on 8 March and 6 October.
Using Figure 24, the solar azimuth
at 1000 hours on either of the above dates would be 144 degrees and the
elevation would be 34 degrees above the horizon.
2Courtesy of the Smithsonian Institution Press, Washington, D.C.
REFERENCES
Solar radiation and forest fuel
BYRAM, G. M., and G. M. JEMISON.
1943.
moisture. J. Agric. Res. 67(4):149-176.
1960. Tables for estimating
EONS, W. L., H. D. BRUCE, and A. McMASTERS.
direct beaten solar Irradiation on slopes at 30° to 46° latitude.
USDA Forest Service, Pacific Southwest Forest and Range Experiment
298 p.
Station (Berkeley, California).
Potential solar beam Irradiation on
FRANK, E. C., and R. LEE.
1966.
slopes: Tables for 30° to 50° latitude. USDA Forest Service Research
Paper RM-i8. Rocky Mountain Forest and Range Experiment Station
166 p.
(Fort Collins, Colorado).
Solar Irradiation on south1955.
HUTCHINSON, F. W., and M. 0. COTTER.
facing inclined surfaces. Heat. Piping Air Cond. 27(8):.118-120.
Smithsonian meteorological tables.
1958.
The Smithsonian Institution. 527 p.
LIST, R. J.
ROBERTSON, G. W., and D. A. RUSSELO.
1968.
6th ed., revised.
Astrometeorological
estimator for estimating time when sun is at any elevation, elapsed
time between elevations in the morning and afternoon, and hourly
and daily values of solar energy. Q. Can. Dep. Agric., Plant Res.
Inst., Ottawa, Ont., Agric. Meteorol. Tech. Bull. 14. 22 p.
FURNIVAL, G. M., W. E. REIFSNYDER, E.-WYLER, and T. G. SICCAMA.
Symposium, p. 181-187.
1969.
Proc. Third Forest Microclimate
Kanaskis Forest Experimental Station, Seebee,
Solar climate calculator.
Alberta, 23-26 September 1969.
-4-
Table 1.
Input data from which hourly values, and daily and yearly
totals of direct solar radiation were computed using the atmospheric
transmission of 0.9.
Approximate
date
Declination
(deg)
(min)
22 Dec.
-23
27
21 Jan. & 22 Nov.
-20
9 Feb. 6
Latitude
(deg)
45
47.5
Slope
(deg)
0
Aspect
N
15
NNE-NNW
3 Nov.
-15
30
NE-NW
23 Feb. F, 20 Oct.
-10
45
ENE-WNW
6 Oct.
-5
60
E-W
21 Mar. & 23 Sep.
0
75
ESE-WSW
3 Apr. & 10 Sep.
+5
90
SE-SW
16 Apr. & 28 Aug.
+10
SSE-SSW
& 12 Aug.
+15
S
24 Jul.
+20
8 Mar. &
1 May
21 May
22 Jun.
&
+23
27
LATITUDE 4ti DEGREES NJRTH, DEC. 22
ASPECT
SLOPE
N
(DEGREES)
NNE
NE
ENE
E
ESE
SE
SSE
NNW
NW
WNW
W
WSW
SW
SSW
S
0
166
160
166
166
166
166
166
166
166
15
36
47
77
117
164
210
253
281
291
30
0
0
24
85
162
245
322
377
396
45
0
0
8
67
157
264
369
447
474
60
0
0
4
55
151
269
392
486
519
75
0
0
3
46
135
258
387
493
530
90
0
0
1
35
118
228
358
455
504
LATITUDE 45 DEGREES NJRTH, JAN. 21
ASPECT
SLOPE
N
NNE
NE
ENE
E
ESE
S
(Oc5RELS)
NNW
NW
WNW
SE
SSE
W
WSW
SW
SSP
0
213
213
213
213
213
213
213
213
213
15
09
82
114
158
208
259
304
334
345
30
0
0
47
117
203
292
374
432
453
45
0
0
20
94
194
308
419
501
530
60
0
0
10
76
183
308
435
536
571
75
0
0
6
63
162
290
423
534
573
90
0
0
4
47
139
253
383
495
536
Table 2.
slopes,
Daily values of direct solar rad'ation computed for selected
aspects, and days at 45 degrees north latitude.
LATITUDE 45 DEGREES NORTH, FEB. 9
ASPECT
SLOPE
N
NNE
NE
ENE
SE
SSE
NW
WNW
E
W
ESE
NNW
WSW
SW
SSW
287
287
2b7
287
287
287
287
287
287
15
130
143
179
227
282
338
384
416
427
30
0
27
96
182
275
371
456
516
538
45
0
0
53
150
265
386
498
581
612
60
0
0
35
127
249
380
506
60?
645
75
0
0
24
107
220
354
483
591
633
90
0
0
19
85
189
306
428
535
579
(UE REtS)
S
LATITUDE 45 OEGRcES NORTH, FEB. 23
ASPECT
SLUP.
(DtGREtS)
360
NrNE
NNW
NE
ENE
E
SSE
WNW
W
ESE
WSW
SE
NW
SW
SSW
360
360
360
360
360
360
360
360
11
15
197
211
247
297
354
409
455
488
499
30
32
74
152
244
343
437
522
582
604
45
0
11
94
205
328
449
556
637
668
3
64
174
303
434
552
'648
686
1
47
145
267
398
517
615
658
0
35
116
226
340
448
540
584
60
75
0
90
Table 2. (continued)
LATITUDE 45 DEGREES NJRTH
MAR. 8
ASPECT
NNE
NE
ENE
E
ESE
SE
SSE
NNW
NW
WNW
W
WSW
SW
SSW
434
434
434
434
434
434
434
434
434
15
274
285
319
368
423
475
522
553
564
30
96
135
212
306
405
497
576
634
655
45
0
40
138
257
381
498
598
673
702
60
0
14
98
216
346
471
579
665
702
75
0
7
71
176
302
424
531
612
653
90
0
3
50
142
250
356
446
520
559
E
ESE
W
WSW
SE
SW
SSE
SSW
SLOPE
(DEGREES)
N
LATITUDE 45 DEGREES NORTH, MAR. 21
ASPECT
SLOPE
(DEGREES)
N
0
NNE
NE
ENE
NNW
NW
512
512
512
512
512
512
*512
512
512
15
360
372
404
449
501
552
593
619
629
30
183
215
293
387
481
569
637
687
704
45
0
92
208
331
452
561
648
707
730
60
0
44
152
282
410
526
616
680
70T
75
0
26
116
236
360
469
553
609
635
90
0
19
88
191
298
390
456
500
520
Table
2. (continued)
LATITUDE 45 DEGREES NJRTH APR*
3
ASPECT
NNE
NE
ENE
E
ESE
SE
SSE
NNW
NW
WNW
W
WSW
SW
SSW
590
590
590
590
590
590
590
590
590
15
451
460
488
530
576
620
055
679
688
30
281
303
372
459
547
626
685
725
740
45
93
157
274
397
509
606
679
725
741
60
5
81
206
338
459
558
631
675
692
75
1
51
157
282
398
490
552
585
595
90
0
34
120
226
325
401
444
458
458
LATITUDE -45 DEGREES NORTH, APR. 16
ASPECT
SLOPE
N
NNE
NE ENE
E
ESE
SE
SLOPE
(DEGREES)
N
(DEGREES)
S
NW
WNW
W
WSW
SW
SSE
SSW
S
NNW
0
659
b59
659
659
659
659
659
659
659
15
536
544
566
601
641
678
708
729
736
30
378
393
449
527
605
671
721
751
764
45
193
229
339
453
556
638
696
729
739
b0
35
124
256
385
496
577
631
657
664
75
14
78
196
319
424
498
538
550
544
90
6
52
147
252
341
399
420
408
391
Table 2. (continued)
LATITUDE 45 DESRcES NJRTH, MAY 1
ASPECT
SLOPE
N
NNE
NE
ENc
E
ESE
SE
SSE
NNW
NW
WNW
W
WSW
SW
SSW
0
730
730
730
730
730
730
730
730
730
15
630
63b
o53
680
711
739
762
776
779
30
488
499
542
607
671
721
757
773
779
313
329
*25
528
617
677
715
727
728
60
116
192
329
452
5+8
607
635
634
627
75
54
126
255
376
457
516
526
508
487
90
35
90
196
299
375
409
396
355
322
LATITUDE 45 DEESRLES NORTH,
SLOPE
N
(DEGREES)
MAY
21
ASPECT
NNE
NE
ENE
E
NW
WNW
W
ESE
WSW
SE
SW
SSE
SSW
S
NNW
0
799
799
199
799
199
799
799
799
799
15
725
728
738
75b
777
796
810
817
819
30
601
608
X32
680
726
762
783
788
787
436
445
505
594
063
703
119
715
708
60
241
267
394
504
563
619
622
599
582
75
106
175
306
417
491
516
498
454
426
90
b4
123
232
329
389
399
351
292
247
Table 2. (continued)
LATITUDE 45 DEGRcES NJRTH,
NNE
NE
JUN. 22
ASPECT
E
ENE
ESE
SE
SSA
NNW
NW
WNW
W
WSW
SW
SSW
838
838
638
838
838
838
838
833
838
15
781
783
789
799
814
826
834
837
839
30
670
674
388
723
758
783
792
791
785
45
514
520
558
631
686
713
716
701
691
60
323
330
436
53b
600
621
608
572
549
75
151
212
339
439
501
510
476
416
386
90
94
145
256
345
394
388
335
248
205
SLOPE
(OEGRtES)
N
0
Table 2. (continued)
S
LATITUDE 47.5 DEGREES NORTH, JEC. 22
ASPECT
SLOPE
N
(J EGREES)
.4N
Nt
LNE
L
ESE
SS
SSL
iNNW
NW
WNW
W
WSW
SW
SSW
S
0
135
136
136
136
136
136
136
136
136
15
13
29
56
92
135
178
217
243
252
30
0
13
66
135
212
283
333
351
4
52
133
231
33)
4C1
426
45
60
0
2
43
126
239
354
442
472
75
3
1
36
117
231
354
452
466
1
28
133
2J7
331
431
467
S
90
0
,-A T ITUDE 47.5 uEi,kEEz:, NO TH, J4N. 21
4SPcCT
SLOPE
N
( ) Ev REES)
0
16?
15
47
30
0
45
3
AN
NE
14
E
-SL
SE
SSL
1 NW
NW
WVW
W
WSW
Sw
SSW
1o2
182
102
162
132
162
182
1t2
89
131
179
227
270
299
309
33
J8
177
261
339
395
414
13
79
171
279
385
464
492
6
64
164
2z4
405
5G2
536
0
75
3
5
55
147
270
399
5C5
543
90
3
3
42
126
238
366
474
513
Table 3.
slopes,
Daily values of direct solar radiation computed for selected
aspects, and days at 47.5 degrees north latitude.
LATITUDE 47.5 DEGREES NORTH,
F.B. 9
ASPECT
SLOPE
(J EGRESS)
N
0
NNNE
NE
ENE
E
ES E
SE
SSE
N NW
NW
WNW
W
WSW
SW
SSW
25+
254
254
254
254
254
254
254
254
15
.103
116
150
197
249
303
349
380
391
30
0
13
75
155
244
337
421
479
500
0
41
127
237
354
464
546
576
45
°
S
60
0
0
27
108
224
352
476
576
613
75
0
0
18
92
199
330
459
566
607
90
0
0
14
73
172
288
411
518
561
S
LATITUDE
LOPE
47.5 DEGREES NORTH, FEB,. 23
ASPECT
,JNE
NE
ENE
E
ESE
SE
SSE
4NW
NW
WNW
W
WSW
SW
SSW
323
328
328
326
328
328
328
32b
328
15
165
181
217
266
323
377
423
455
467
30
12
53
127
216
315
409
493
552
574
45
0
6
76
164
303
423
531
611
642
60
0
2
54
158
284
413
532
628
667
75
0
u
41
133
252
333
504
663
645
90
0
0
31
108 _216
330
442
536
580
S
N
t 9 E GRESS)
Table 3. (continued)
LATITUDE -+7.S 3EGREES NO TNT MAR. 8
ASPECT
EKE
tN
NL
N
'1
NE
SLOPE
t!)=GREES)
St
SSE
iNW
NW
WNW
W
WSW
SW
SSW
S
0
403
408
408
408
408
408
406
408
4C8
15
245
257
293
34+2
398
+51
498
530
542
30
63
Iii
190
262
383
477
556
617
639
45
0
30
124
240
363
432
585
662
692
60
0
11
88
234
334
4+81
572
b61
699
75
5
65
169
293
419
530
616
657
90
2
47
136
246
355
451
530
571
E
_SE
SE
SSE
S
SW
SSW
LATITUDE 47.5 DEGREES NORTH, MAR. 21
ASPECT
SLOPE
(DEGREES)
NNE
N
NE
LN_
NNW
NW
WNW
W
WOW
483
489
489
489
469
439
489
489
489
15
333
345
378
425
479
531
574
601
611
30
154
186
269
366
463
552
623
674
692
45
0
77
191
314
438
550
640
702
726
60
0
37
143
27u
402
520
615
683
710
75
0
23
111
229
355
468
558
618
646
90
0
18
84
187
297
393
464
515
538
Table 3. (continued)
3
LATITUDE 47#5 DEGREES NORTH, APR.
ASPECT
NNE
NE
ENE
E
ESE
SE
SSE
NNW
NW
WNW
W
WSW
SW
SSW
570
570
570
570
570
570
570
570
570
15
.426
435
465
508
557
603
640
E65
675
30
253
276
349
441
531
614
676
718
733
45
63
137
256
382
497
599
676
725
742
60
5
73
195
328
452
557
634
683
701
75
0
48
150
276
395
493
561
599
611
33
116
223
326
407
456
477
480
SLOPE
(DEGREES)
N
0
90
S
LATITUDE 47.5 DcGR-EES NORTH, A?R. 16
ASPECT
SLOPE
N
(DEGREES)
ANE
NE
ENE
E
ESE
SE
SSE
ANW
NW
WNW
W
WSW
SW
SSW
0
642
642
642
542
642
b42
642
642
642
15
515
523
547
584
626
665
697
719
726
30
352
368
430
512
594
664
717
748
762
45
165
207
324
443
550
637
699
734
745
60
31
117
248
380
495
582
640
670
678
75
15
76
193
318
427
506
552
568
564
52
148
254
347
410
436
429
415
90
Table 3. (continued)
LATITUDE 47.- DEuREES NURTN' MAYPICT
AS t
E
ESE
Sc.
SSt
WNW
W
WSW
SW
SSW
718
713
718
718
718
71b
718
638
667
700
732
756
771
775
465
476524
595
664
718
758
776
782
257
366
409
519
614
630
721
737
738
1G1
181
326
+*6
550
615
646
649
645
75
53
125
253
376
472
527
543
529
508
90
35
90
197
302
382
421
417
376
348
tNc
cSL
SE
SSE
S
SW
SSw
.JNc
;NW
Nt
NW
713
716
15
613
619
30
45
N
SLOPE
(0 EG RECS)
EN
S
LP TITUUE 47.5 UEORLES NO (Tri, MAY 21
ASPECT
4NE
N
SLOPE
(DEGREES)
NE
ANW
NW
WNW
W
WSW
792
792
792
792
792
792
792
792
792
0
15
710
715
726
746
773
792
808
616
820
30
580
588
616
669
721
762
786
795
793
45
411
422
489
534
660
706
728
728
723
60
213
25U
3b4
496
564
627
635
61b
663
75
161
167
299
415
494
526
516
477
451
90
64
121
230
329
393
410
379
316
275
Table
.
(continued)
LATITUDE 47.5 O GRcES NORTH,
SLOPE
(DEGREES)
JUN. 22
ASPECT
N N E
NE
ENE
E
ESE
SE
SSE
NNW
NW
WNW
W
WSW
SW
SSW
83+
634
834
.834
834
834
834
834
834
15
772
775
782
794
811
825
834
839
842
30
658
662
678
719
759
787
799
799
794
45
495
i05
548
629
690
723
728
71b
707
60
305
316
432
538
608
634
624
592
572
75
155
213
341
445
512
526
495
440
411
90
101
153
261
35
406
404
353
272
230
Table 3.
N
(continued)
YITUOE 45 DEGREES, NORTH
ASPECTS
0
15
30
SLOPE (DEGREE)
60
45
75
90
N
186389
141420
95228
57442
27803
12543
7704
NNE / NNW
186389
144750
100979
65162
38079
24620
17083
NE / NW
186389
153801
119028
89733
68208
52439
39604
ENE / WNW
186389
166819
144899
124548
105654
87536
69339
E/W
186389
181775
172449
159942
143821
123571
100778
ESE / WSW
186389
196216
198047
191819
177203
155406
126958
SE / SW
186389
208311
218951
217736
203755
179504
146077
S$E / SSW
186389
216248
232510
234433
221001
194328
156193
S
186389
219052
237183
239995
226464
198674
158275
Table 4. Yearly values of direct solar radiation
slopes and aspects at 45 degrees north latitude.
computed for selected
LATITUDE 47.5 DEGREES, NORTH
ASPECTS
0
15
30
SLOPE (DEGREE)
45
60
75
90
N
178802
133783
89539
53210
25132
12365,
7999
NNE / N44
178802
137086
95127
60742
35734
23937
37117
NE / NW
178802
146234
112559
85077
65405
51021
39144
ENE / WVW
178802
159411
138487
119590
102259
85565
68384
W
178802
174520
166234
155112
140410
121512
99849
ESE / WSW
178802
189163
192129
187283
174309
153952
126751
SE / SW
178802
201383
213264
213594
201337
178949
146983
SSE / SS4
178802
209398
227001
230574
219008
194394
158018
S
178802
212285
231604
236215
.224745
198871
160509
E
/.
Table 5. Yearly values of direct solar radiation computed for selected
slopes and aspects at 47.5 degrees north latitude.
Figure I. Illustration of the angles necessary for the theoretical
calculations of solar radiation on a particular surface in the northern
hemisphere.
20/4
I8/6
1618
8/16
6/la
4/20
2/22
18/6
16/a
6/16
4/20
2/22
2:/4
I3/6
10/8
30
93
13/14
8/16
0/16
0'
4/20
d OP T,E101?'i08'd::ST
2/20
15
30
45.
SLOPE wrnj)
6:
75
:0
0
15
5
>
SLLRE
tE.,asi
75
90
0
15
30
45
SLOPE xsf
Figure 2. Isograms of hourly values of direct solar radiation for various
slopes at 45 degrees north latitude on 22 June. For westerly exposures,
the time is read from top to bottom (left time axis); for easterly
exposures, the time is read from bottom to top (right time axis).
75
90
ELST
:0UT .EAST/BEST
EAST/:EST
S 3 5 T I i 1 1 E ST
0-
-4C
60
80
20
-9
F- 0
4/20
.ORT8
2/22
0
CO
80
SLOPE fpasm)
61
75
3C
C
15
33
45
E0
SLOPE !man)
75
90
0
35
3C
85
SLOPE (L3S1
Figure 3. Isograms of hourly values of direct solar radiation for various
slopes at 45 degrees north latitude on 21 May and 24 July. For westerly
exposures, the time is read from top to bottom (left time axis); for
easterly exposures, the time is read from bottom to top (right time
axis).
.'5
90
20
40
60
0
''-5
03
SLOPE
6:
b3
4
'E4
75
30
a
3;
S! APE
45
73
;=3s:
0
15
3_
Y5
SLODE
Figure 4. Isograms of hourly values of
solar radiation for various
slopes at 45 degrees north latitude on l direct
May and 12 August. For westerly
exposures, the time is read from top to bottom (left time axis);
for
easterly exposures, the time is read from bottom to top (right time
axis).
75
9i
0
E0
80
3
7:0 R TF
15
30.
45
SLOPE Cz=f
%
75
30
0
15
O R T F E F S Tf'.1? T!;
45
30
6
SLOPE 'fxesi
10:T UUE
75
0R TH
0
15
43
30
_3
0'E
61
.. rJ
Isograms of hourly values of direct solar radiation for various
For
slopes at 45 degrees north latitude on 16 April and 28 August.
westerly exposures, the time is read from top to bottom (left time axis);
for easterly exposures, the time is read from bottom to top (right time
Figure 5.
axis).
75
30
20/4
18/6
16/8
0-
6/18
4/20
SO:TH SOUTVEIST/SCGTH SOUTIYEST
2/22
EAST SOUTIIE:ST/JEST
EAST/WEST
SOUTH .EST
18/6
16/8
0
60
60
10/14
8/16
6/18
4/20
2/22
0
:S
30
YS
SLOPE (:em)
E7
75
30
0
15
30
-5
0PE
61
(
Seta
75
30
0
45
15
SLOP E
t7--m-0
Figure 6. Isograms of hourly values of direct solar radiation for various
slopes at 45 degrees north latitude on 3 April and 10 September. For
westerly exposures, the time is read from top to bottom (left time axis);
for easterly exposures, the time is read from bottom to top (right time
axis).
550
20/4
10/14
60-
6/18
r0-
4/20
S)!"f
EAST
S0UT-E:ST/S0CTH 50UT HWEST
EAST/WEST
SOUTHEAST/WEST SOUTHWE:T
18/6
166/3
14/10
12/12
10/14
8/16
6/12
OOA'HEAST/'::NTa,EST
0
10
30
45
E3
SLOPE 1ma[a:
OOPTH
15
90
0
15
30
YS
SLOPE figs)
/2
75
90.
3
15
3 0
.
45
E:
SLOPE
Figure 7. Isograms of hourly values of direct solar radiation for various
slopes at 45 degrees north latitude on 21 March and 23 September. For
westerly exposures, the time is read from top to bottom (left time axis);
for easterly exposures, the time is read from bottom to top (right time
axis).
75
90
SO
5
SLOPE W.-=)
6
75
30
45
60
SLOPE CwEs)
75
i0
15
3C
5
E)
SLOPE (y.ffi)
Isograms of hourly values of direct solar radiation for various
For westerly
45
degrees north latitude on 8 March and 6 October.
slopes at
exposures, the time is read from top to bottom (left time axis); for
axis).
easterly exposures, the time is read from bottom to top (right time
Figure 8.
75
50
- -0
i
li
LS
SLOP E
E3
:row.
75
9G
0
90
SLOPE
!fs
s;
i0
:>
su
SLOPE 0vu
Figure 9. Isograms of hourly values of direct solar radiation for various
slopes at 45 degrees north latitude on 23 February and 20 October. For
westerly exposures, the time is read from top to bottom (left time axis);
for easterly exposures, the time is read from bottom to top (right time
axis).
75
SC.` 2CL3
S 0 U 7 tEAS T/SOU T=:EST
EAST/i09T- SO.Tt. EST
8/IE
E A]T
EAST/JEST
S]U7VEA:T/.:EST S0 C'. II UEST
JORT,i :;GR-.nEAST/:.CRT IORTi,wES:
2/220
15
45
EO
SLOPE '
7
9C
C
31
45
SLOPE (gig)
'5
15
0
15
45
G+
SLOPE 'acs'
Isograms of hourly values of direct solar radiation for various
Figure 10.
For
slopes at 45 degrees north latitude on 9 February and 3 November.
westerly exposures, the time is read from top to bottom (left time axis);
for easterly exposures, the time is read from bottom to top (right time
axis).
75
8C
8 5-.
C
SLUT FEAST/SOUT-+EST
SCUT;
20/4
18/6
1618
10/14
8/16
6/18
4/20
EAST :10RThEAST/U EST
:: 70T PUEST
2/22
16/0
3/16
'
0
4/20
;0RT SE AST/;TORT I:::EST
2/22
45
L
75
..
i;.
ti
CO
75
00
PE
Figure 11. Isograms of hourly values of direct solar radiation for various
slopes at 45 degrees north latitude on 21 January and 22 November. For
westerly exposures, the time is read from top to bottom (left time axis);
for easterly exposures, the time is read from bottom to top (right time
axis).
21/4
18/6
16/8
60
8 0-
87
6/16
6/16
4/2
SOJ:
S3uTH
SO OTE0T
OCTii 0ST!5CJT
2/2
20/4
18/6
16/8
T 14110
E 12/12
33
u0
10/14
1-20
6/16
6/13
4/20
.;ST
EAST/.:EST
'll f:EST
ORTHEAST/JEST
2/22
20/,
18/£
16/8
T 14/10
20
N
-
20
20
E 12/12
-33
0 10/14
8/16
6/18
4/20
IOPTnE:ST,'.3PT-EST
:I C9 TH
2/22
J
15
30
5LCPE ce-a)
.v
-
75
50
.5
rTTTHE.=ST/ti C^T!i
45
30
6:
S.CPE 0 - E55
=T IIL_5
7S
00R7
11
4S
SLOPE (x
s'.
Figure 12. Isograms of hourly values of direct solar radiation for various
slopes at 45 degrees north latitude on 22 December. For westerly exposures,
the time is read from top to bottom (left time axis); for easterly exposures,
the time is read from bottom to top (right time axis).
75
40
0EAST/ EST
4/20
U ORThc AST/::OR Tf-E?T
3
1
41
S_O,E 'x M!'
Figure 13.
.
7i
®3
o
is
45
31
S_OPE 5
E11
)
7i
Sc
>O
4>
S.OPE '47='
Isograms of hourly values of direct solar radiation for various
slopes at 47.5 degrees north latitude on 22 June. For westerly exposures,
the time is read from too to bottom (left time axis); for easterly exposures,
the time is read from bottom to top (right time axis).
:7
L
EA 3T
507T.'E'. ST/ti:
EAST /t EST
_JT°IEST
I'-6C
20
-89
974 T4
7
15
it
45
SLIFE (m-av
G
91
0
3L
.
IS
3
4C
S:
SL1PE iL_5r
Figure 14. isograms of hourly values of direct solar radiation for various
slopes at 47.5 degrees north latitude on 21 May and 24 July. For westerly
exposures, the time is read from top to bottom (left time axis); for
easterly exposures, the time is read from bottom to top (right time axis).
75
90
L
37201
'ORTHE+ST'OORTHLEST
'22
i5
50
t5
S:3PI
(cr,-g3)
61
75
3(
0
30
55
SLOPE ( rs)
60
75
i0
0
a;
SLOSE S3s)
Isograms of hourly values of direct solar radiation for various
slopes at 47.5 degrees north latitude on I May and 12 August. For westerly
exposures, the time is read from top to bottom (left time axis); for
easterly exposures, the time is read from bottom to top (right time axis).
Figure 15.
75
30
S OUT IEA ST/SO10
::=_ST
18/6
16/8
14/10
12/12
8/16
4/20
2/22
0
EAST SOUTHEASTIVEST SOU'H0EST
15
39
45
SLOPE
Figure 16.
50
75
EAST/WEST
90
0
91
SLOAE x .i.
15
37
4S
Isograms of hourly values of direct solar radiation for various
slopes at 47.5 degrees north latitude on 16 April and 28 August.
For
westerly exposures, the time is read from top to bottom (left time axis);
for easterly exposures, the time is read from bottom to top (right time
axis).
S:
SLOPE 4xAA
75
3;
S9'' T F
OUTN EST
S O U T L
E A S T JS O U T HfST
A/i9
H, AST/:1'rTH::: ST
% 'PTh
I G R TH E AST/10RTH :: 0RTHJSST
75
7:
PE
Figure 17. Isograms of hourly values of direct solar radiation for various
slopes at 47.5 degrees north latitude on 3 April and 10 September. For
westerly exposures, the time is read from top to bottom (left time axis);
for easterly exposures, the time is read from bottom to top (right time
axis).
99
SGCHEAET;SCUT-ST
2/22
lsograms of hourly values of direct solar radiation for various
Figure 18.
For
slopes at 47.5 degrees north latitude on 21 March and 23 September.
axis);
to
bottom
(left
time
westerly exposures, the time is read from top
for easterly exposures, the time is read from bottom to top (right time
axis).
i
SOV:H
5^_31,'0.ST'-:3Tb.:E cT
4
6
8
0
2
0
50
SL
45
P E ttx:
E0
75
SC
,
41
S L C^ E
C0
75
91
,
15
sum.
39
45-
.5
S; 1 P E
Isograms of hourly values of direct solar radiation for various
Figure 19.
47.5
degrees north latitude on 8 March and 6 October. For
slopes at
cft time axis);
westerly exposures, the time is read from top to bottom
^ (right time
from
bottom
to
the
time
is
read
for easterly exposures,
(
'
axis).
75
49
20/4
18/6
16/a
0
30
20
20
40
4
0
'5 2
8/16
0
6/16
a/20
00534 10R7iEAST/;0311
2/22
0
15
30
45
SLOPE
)i .s)
60
75
9C
0
15
')0?T FOEST
:1 0 PT
SLOPE lx'mI LOPE
33
45
43
75
90
1
15
33
45
42
Figure 20.
Isograms of hourly values of direct solar radiation for various
slopes at 47.5 degrees north latitude on 23 February and 20 October. For
westerly exposures, the time is read from top to bottom (left time axis);
for easterly exposures, the time is read from bottom to top (right time
axis).
75
30
0
0
6
SOU T?
SC
' ' H 2OUTHEAS T/S0,7H SDLTHUES7
P..E A S1!50516:"EST
D
20
I
0
EA5T SDC'!i E AST/'EST S3UT'2_ EST
a0R T11
35
30
45
53
SLOPE :sr7
75
E-5' :DOT"EAST/'.:EST
EAST/::EST
:0
0
'10PTkEA ST047' % ORT H i E S T
30
45
5L0PE
60
75
tl CTT6:'EST
00RTH
9C
i5
33
45
SLOPE
(
60
s)
Figure 21. Isograms of hourly values of direct solar radiation for various
slopes at 47.5 degrees north latitude on 9 February and 3 November. For
westerly exposures, the time is read from top to bottom (left time axis);
for easterly exposures, the time is read from bottom to top (right time
axis).
75
s0
10/4
18/6
16 ?8
r-- 0
20
4 C_
60
14/10
0
12/12
x-89
10/14
8/16
P
6/18
4/23
SOOT-
SOUTH
2/22
10 UT 4EAST ISOJ TG.'.;ST
20/4
16/8
6/18
4/20
2/22
16/6
16/8
0
20
N
1
8/16
0
6/19
4/20
.ORTREAST/10RTR0_ST
I
(
10
0
gORTN UO8THEAST/".CRT! 00RTNNES?
OORT-
2/22
15
02
i5
SLOPE (z
C:
xs)
75
15
3)
SLOPE
45
5O
--
m)
75
s0
0
:5
TO
45
E)
_; UPE
Figure 22. Isograms of hourly values of direct solar raidation for various
slopes at 47.5 degrees north latitude on 21 January and 22 November. For
westerly exposures, the time is read from top to bottom (left time axis);
for easterly exposures, the time is read from bottom to top (right time
axis).
7,
93
2014
1816
16/8
4/10
Y
go-
2/12
60
'-76
8/16
6/18
4/20
SOOT°
2/22
SOUT- i0U'H!EAST/S uT! SOOT-':!E S'
SO2THEAST/SOOT`.:EST
0
8/6
6/8
/10
20
2/12
28
0/14
8/16
0
6/18
4/20
EAST/AEST
EAST
2/22
C1'!:EAST/A7ST
0/1
8/6
6/8
4/10
20
2/12
20
28
0/14
8/16
6/18
4/20
';OR-HE; ST
/ 20T
BORT0 OORTHEAS7170RTH 10ETHHES-
;T
'IORTH
2/222
0
15
3O
45
6E
SLOPE t.'ss)
75
90
8
15
45
L.
SLOPE 'zexa)
30
75
50
0
10
30
41
cc
I;
SL0°E
lsograms of hourly values of direct solar radiation for various
Figure 23.
slopes at 47.5 degrees north latitude on 22 December. For westerly exposures,
the time is read from top to bottom (left time axis); for easterly exposures,
the time is read from bottom to top (right time axis).
SO
45°
N
S0LA
A L T
R
I
TUDE
AND
AZ
I
MUTH
X 'V\ \ I-T-L / / "-/ r V \
2y0 k\ A
AL A 70
80
280
WEST
EAST
100
260
250
\
X\\ \
t> 7 Cl/X / `'Q1 110
.
120
TRACK
190
SOUTH
170
1
2
3
4
north latitude.
May
May
21, July 24
1, Aug. 12
Apr. 16, Aug. 28
Apr. 3,, Sept. 10
7
- 5°
10
-10°
-15°
-20°
Mar. 8,
Feb.23,
Feb. 9,
Jan 21,
11
-23°27'
Dec. 22
9-
Solar altitude and
June 22
+ 5°
+ 0°
8
45 degrees
+20°
+15°
+10°
6
5
Figure 24.
Decli- Approx. dates
nation
+2327
Mar.
21, Sept.23
Oct. 6.
Oct.20
Nov. 3
Nov.22
azimuth for selected days of the year at
47° 30'N.
S0LAR
ALT ITUDE
350
340 /'T-
290
L,-T Y
NORTH
AND
AZ
I
MUTH
10
7`-...`20
1\1 Y Y\ \ = / /''Z X Y \
H X70
280
WEST
nation
Approx. dates
+23°27' J u ne 22
190
SOUTH
+20°
+15°
+10°
170
+
+
May
May
Apr.
21, July 24
1, Aug. 12
16, Aug. 28
10
23
Apr. 3, ,Sept.
Mar. 21,Sept.
7
- 5°
8
9
10
-10°
-15°
-20°
11
-23°27' Dec. 22
Mar. 8, Oct. 6
Feb. 23, Oct. 20
Feb. 9, Nov. 3
Jan 21, Nov. 22
25.
Solar altitude and azimuth for selected days of the year at
47.5 degrees north latitude.
Figure