The average daily max temp is 92.7 F in July in Houston. On a still sunny day,
a square foot of light-colored south shingles might absorb 0.2x250 = 50 Btu/h
and lose heat to outdoor air with a 2 Btu/h-F conductance, which makes the
equivalent shingle temp 92.7+50/2 = 118 F, so we might have something like
this, with attic air temp T (F), viewed in a fixed font:

T
0.5 | R1 | -----
-----www-------www------| -- |---|
| | RATT -----
| 118 | I
--- |
- outside|inside
| |
-

If 1250(118-T)/1.5 Btu/h that flows into the attic equals I that flows out
and I = 16.6Asqrt(H)(T-93)^1.5, with lesser vent area A ft^2 and height diff
H in feet, A = 190/144 = 1.3 ft^2 and H = 8' make T = 118-(T-93)^(2/3)/13.5
= 117.4 F, after 2 iterations. With A = 8 ft^2 (eg 2 2'x2' gable vents),
T = 114.5. Not a big difference, esp since the larger soffit area will add
about 40% to the flow in the first case, which isn't counted above.

A radiant barrier can help by increasing the R1 above.

Not much, altho larger vents help. A thermal chimney would also help
(how much?), by increasing H. It might have a transparent south side
or roof to collect sun and raise the "chimney" air temp and bouyancy
and flow in summertime and collect wintertime heat.

20 ASR=1250'south roof area (ft^2)
30 RSH=.2'shingle reflectivity
40 TA=92.7'outdoor temp (F)
50 GSA=2'shingle-outdoor conductance (Btu/h-F-ft^2)
60 TTS=TA+250*RSH/GSA'Thevenin equiv shingle temp (F)
70 FOR RATT=1 TO 4 STEP 3'R-value of roof (1-4 for radiant barrier)
80 KROOF=ASR/(1/GSA+RATT)'roof coefficient
90 H=8'vent height difference (feet)
100 FOR A=8 TO 32 STEP 8'vent area (ft^2)
110 C=16.6*A*SQR(H)/KROOF
120 T=TTS'initialize attic air temp (F)
130 TL=T'save previous value
140 IF TTA THEN T=2*TA-T'reflect up
150 T=TTS-C*(T-TA)^.666'find new value
160 IF ABS(T-TL).1 GOTO 130'iterate
170 PRINT RATT,A,T
180 NEXT:NEXT

RATT A (ft^2) Tattic (F)

1 8 114.2156
1 16 111.3843
1 24 109.0334
1 32 107.0474

4 8 109.0334
4 16 104.0215
4 24 101.0059
4 32 99.13989

Nick

The average max temp is 92.7 F in July in Houston. On a still sunny day,
a square foot of light-colored south shingles might absorb 0.2x250 = 50 Btu/h
and lose heat to outdoor air with a 2 Btu/h-F conductance, which makes the
equivalent shingle temp 92.7+50/2 = 118 F, so we might have something like
this, with attic air temp T (F), viewed in a fixed font:

T
0.5 | R1 | -----
-----www-------www------| -- |---|
| | RATT -----
| 118 | I
--- |
- outside|inside
| |
-

If 1250(118-T)/1.5 Btu/h that flows into the attic equals I that flows out
and I = 16.6Asqrt(H)(T-93)^1.5, with lesser vent area A ft^2 and height diff
H in feet, A = 190/144 = 1.3 ft^2 and H = 8' make T = 118-(T-93)^(2/3)/13.5
= 117.4 F, after 2 iterations. With A = 8 ft^2 (eg 2 2'x2' gable vents),
T = 114.5....

A radiant barrier can help by increasing the R1 above.

... A thermal chimney would also help (how much?), by increasing H.
It might have a transparent south side or roof to collect sun and
raise the chimney air temp and bouyancy and flow in summertime...

.... and make the average attic temp only 2 F above the outdoor temp.

20 ASR=1250'south roof area (ft^2)
30 RSH=.2'shingle absorptivity
40 TA=92.7'outdoor temp (F)
50 GSA=2'shingle-outdoor conductance (Btu/h-F-ft^2)
60 TTS=TA+250*RSH/GSA'Thevenin equiv shingle temp (F)
70 RATT=4'R-value of roof (1-4 for radiant barrier)
80 KROOF=ASR/(1/GSA+RATT)'roof coefficient
90 H=8'vent height difference (feet)
100 A=8'vent area (ft^2)
110 C=16.6*A*SQR(H)/KROOF
120 T=TTS'initialize attic outlet temp (F)
130 TL=T'save previous temp
140 IF TTA THEN T=2*TA-T'reflect
150 T=TTS-C*(T-TA)^.666'new attic temp (F)
160 IF ABS(T-TL).1 GOTO 130'iterate
170 C=16.6*A*SQR(H*(T-TA))'airflow (cfm)
180 PRINT C,(TA+T)/2'airflow and average attic temp (F)
190 C=1000'airflow estimate (cfm)
200 T=TA'average attic temp estimate (F)
210 TC=TA'average chimney temp estimate (F)
220 DT=KROOF*(TTS-T)/C'attic air temp gain (F)
230 T=TA+DT/2'average attic temp (F)
240 MA=H*.075*530/(460+T)'attic air mass (psf)
250 TAT=TA+DT'temp at top of attic (F)
260 CSUN=.9*250*32'sun into chimney (Btu/h)
270 CGAIN=CSUN-(TC-TA)*32'net chimney gain (Btu/h)
280 DTC=CGAIN/C'chimney air temp gain (F)
290 TC=TAT+DTC/2'average chimney temp (F)
300 MC=H*.075*530/(460+TC)'chimney air mass (psf)
310 MB=MA+MC'total bouyant air mass (psf)
320 MO=2*H*.075*530/(460+TA)'outdoor air mass (psf)
330 DP=MO-MB'bouyancy force (psf)
340 C=1975*A*SQR(DP)'airflow (cfm)
350 IF ABS(C-CL).1 THEN CL=C:GOTO 220'iterate
360 PRINT C,T,TC

airflow attic temp chimney temp
(cfm) (F) (F)

1518.035 100.8667 H = 8' vent height difference

1496.56 94.82298 99.28098 H = 8' + 2'x4'x8' solar chimney

Nick