## Thursday, September 5, 2013

### Opaque attic roof heating revisited

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