With radiation loss, a roof with an absolute Rankine temperature T would have 0.1714E-8(T^4-Ts^4)+(T-(35.5+460)2 = 150 Btu/h, where Ts is the effective sky temperature in Rankine degrees. Duffie and Beckman's 2006 Solar Engineering of Thermal Processes book has equation (3.9.2) for sky temperature Ts = Ta[0.711+0.0056Tdp+0.000073Tdp^2+0.013cos(15t)]^(1/4), where Ts and Ta are in degrees Kelvin and Tdp is the dew point temp in degrees Celsius and t is the number of hours from midnight.
Ta = 35.5+460 = 495.5 degrees R, ie 495.5/1.8 = 275.3 degrees K. NREL says the humidity ratio w = 0.0028 pounds of water per pound of dry air on an average December day in Allentown, which makes the partial pressure of water in air Pa = 29.921/(0.62198/w-1) = 0.135 "Hg, which makes the dew point Tdp = 9621/(17.863-ln(Pa)) = 484 R, ie 484-460 = 24 F, ie -4.2 C. With t = 12 hours (noon), cos(15t) = -1, so Ts = 275.3[0.711-0.0056x4.2+0.000073x4.2^2-0.013)]^(1/4) = 249.6 K, ie 1.8x249.6 = 449.3 R, ie -10.7 F, 46 F degrees less than the air temperature.
And 0.1714E-8(T^4-449.3^4)+(T-495.5)2 = 150 makes T = (1141-(T^4-449.3^4)0.1714E-8)/2. Plugging in T = 530 on the right makes T = 537.8 on the left. Repeating this makes T = 533.7, 535.9, 534.8, 535.3, 535.0, and 535.2 R, ie 75.2 F, which is not much greater than 70 F, so the roof can't provide much space heating in December, even at noon in direct beam sun.
http://rredc.nrel.gov/solar/calculators/PVWATTS/version1/US/code/pvwattsv1.cgi says 2.62 kWh/m^2 (2.62x317 = 831 Btu/ft^2) of sun falls on a south roof with an 18.4 degree slope on an average 26.6 F January day with a 34.3 high and a 0.0022 humidity ratio in Allentown. February, March, April, and May bring 3.36, 4.43, 5.32, and 5.65 kWh/m^2 with 29.3, 37.7, 0.0024, 39.4, 48.8, 0.0032, 49.7, 60.4, 0.0046, 60.3, 71.3, and 0.0074 average and high temps and humidity ratios, and so on. If 80% of the sun falls on the roof in all but 3 hours of daylight, opaque attic roof heating does not look at all promising:
10 PI=4*ATN(1)
20 S=1.714E-09'Stephan-Boltzman constant
30 LAT=40.5'north latitude (degrees)
40 DATA 26.6,34.3,2.62,.0022,-20.9
50 DATA 29.3,37.7,3.36,.0024,-13.0
60 DATA 39.4,48.8,4.43,.0032,-2.4
70 DATA 49.7,60.4,5.32,.0046,9.4
80 DATA 60.2,71.3,5.65,.0074,18.8
90 DATA 69.4,80.0,5.80,.0104,23.1
100 DATA 74.1,84.5,6.06,.0122,21.2
110 DATA 72.2,82.3,5.43,.0120,13.5
120 DATA 64.7,75.1,4.70,.0097,2.2
130 DATA 53.2,63.8,3.87,.0064,-9.6
140 DATA 43.1,51.8,2.48,.0044,-18.9
150 DATA 31.8,39.2,2.25,.0028,-23.0
160 FOR MONTH=1 TO 12
170 READ TA,TM,SUN,W,DEC
180 TDF=(TA+TM)/2'average daytime temp (F)
185 TDFP=INT(10*TDF+.5)/10'rounded tdf
190 TDR=TDF+460'daytime temp (R)
200 TD=(TDF+460)/1.8'daytime temp (K)
210 PA=29.921/(.62198/W-1)'vapor pressure ("Hg)
220 TDPF=9621/(17.863-LOG(PA))-460'dew point temp (F)
230 TDP=(TDPF-32)/1.8'dew point temp (C)
240 TSKY=TD*(.698+.0056*TDP+.000073*TDP^2)^.25'sky temp (K)
250 TSKYF=1.8*TSKY-460'sky temp (F)
260 X=-TAN(PI*LAT/180)*TAN(PI*DEC/180)'find day length
270 ICOS=-ATN(X/SQR(-X*X+1))+PI/2'inverse cosine (radians)
280 DAYL=2/15*ICOS*180/PI'day length (hours)
290 SINT=.8*317*SUN/(DAYL-3)'solar intensity (Btu/h-ft^2)
300 TR=530'initial roof temp (R)
305 FOR I=1 TO 10
310 TR=(2*TDR+SINT-S*(TR^4-TSKYR^4))/2'roof temp (R)
315 NEXT I
320 TRF=TR-460'roof temp (F)
330 PRINT 500+MONTH;"'";TDFP,TSKYF,DAYL,SINT,TRF
340 NEXT MONTH
01 30.5 -17.9104 9.462032 102.8209 31.75281
02 33.5 -14.2666 10.48372 113.86 37.80908
03 44.1 -1.409027 11.72648 128.7402 50.34619
04 55.1 13.38834 13.08379 133.7942 59.52539
05 65.8 30.77084 14.25374 127.3213 64.54016
06 74.7 45.41767 14.84857 124.1399 69.43659
07 79.3 52.98224 14.5795 132.7187 75.28638
08 77.3 50.6933 13.57763 130.1849 73.11206
09 69.9 39.54871 12.2507 128.8465 67.81403
10 58.5 21.59882 10.89256 124.3491 58.66553
11 47.4 5.783173 9.73297 93.41019 40.42139
12 35.5 -10.78049 9.165865 92.54176 31.68912
The average roof shingle temps are less than 70 F except for the months of July and August, when the outdoor temperature is warm enough that the house doesn't need heat.
This could be refined with a simulation using hourly TMY2 weather data, but so far it looks doomed...
Nick
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