Buy_Sell wrote:

... If the interior of a room was darker then more heat could be absorbed
without the sheet... But if the interior the room was lighter then the dark
sheet in the window would be a better heat source. Do we agree on this?

Yes. Here are 2 scenarios with a 20 ft^2 R2 window admitting 1 kW
(3412 Btu/h) of sun and indoor and outdoor temps of 70 and 30 F:

I. No sheet. The room gains 3412ALeff Btu/h and (70-30)20ft^2R2 = 400 Btu/h
flows out through the window.

II. A 20 ft^2 black sheet at temp Ts (F) absorbs 3412 Btu/h and loses heat
to the room and the Tw (F) window pane.

We can find the room depth that makes the net heatflow into the room equal
in each scenario using equation 4.12.1 on page 198 of the 2006 3rd edition
of Duffie and Beckman's Solar Engineering of Thermal Processes:

ALeff = ALi/(ALi+(1-ALi)Aa/Ai), where

ALeff is the effective absorptance of an cavity opening,
ALi is the absorptance of the inner surface of the cavity,
Aa is the area of the aperture of the cavity, and
Ai is the area of the inner surface.

A very large window in front of a shallow room has an Aa/Ai of about 1,
so ALPHAeff is close to Ai. A small window in front of a deep room
(like a crab trap) has Aa/Ai close to zero, so ALPHAeff is close to 1.

20 TI=70'indoor temp (F)
30 TA=30'ambient temp (F)
40 AW=4*5'window area (ft^2)
50 RW=2'US R-value of window (ft^2-F-h/Btu)
60 GA=1.5'air film R-value
70 TW=TI'initial indoor window pane temp (F)
80 TS=TI'initial sheet window pane temp (F)
90 TM=(TS+TW)/2'mean sheet-pane temp (F)
100 GR=4*1.714E-09*(TM+460)^3'sheet-pane radiation conductance (Btu/h-F-ft^2)
110 RS=RW+1/(GA+GR)'sheet-outdoor R-value
120 TMR=(TS+TI)/2'mean room-pane temp (F)
130 GSR=GA+4*1.714E-09*(TM+460)^3'sheet-room radiation conductance
140 TS=(3412/AW+TI*GSR+TA/RS)/(GSR+1/RS)'sheet temp (F)
150 QWS=(TS-TA)*AW/RS'heatflow through window (Btu/h)
160 TWS=TA+QWS/AW/(RW-1/GA)'indoor window pane temp (F)
170 IF ABS(TWS-TW).1 THEN TW=TWS:GOTO 90
180 SHEETFLOW=3412-QWS'sheet heatflow to room (Btu/h)
190 QW=(TI-TA)*AW/RW'heatflow through window with no sheet (Btu/h)
200 ALEFF=(SHEETFLOW+QW)/3412'cavity absorptance
210 FOR AR=.1 TO .91 STEP .1'room surface absorptance
220 AI=ALEFF*(1-AR)*AW/(AR*(1-ALEFF))'room surface (ft^2)
230 DR=(AI+AW-128)/32'breakeven room depth (feet)
240 PRINT AR,TS,DR
250 NEXT

Ar Ts (F) Dr

..1 120.3131 44.43828
..2 120.3131 17.87535
..3 120.3131 9.021034
..4 120.3131 4.59388
..5 120.3131 1.937587
..6 120.3131 .1667242
..7 120.3131 -1.098178
..8 120.3131 -2.046854
..9 120.3131 -2.784713

A black sheet might keep a room warmer if it had 8'x8' north and south walls
and a 0.1 room surface solar absorptance (including the floor), if the room
depth were less than 44 feet. It wouldn't help much if the room absorptance
were greater than 0.3, and a room with absorptance higher than 0.6 would be
warmer without the sheet. Black greenhouse shadecloth with an open mesh and
80% absorptance would allow more room air to flow near the window...

Appendix 2.4 on page 285 of Bruce Anderson's 1976 Solar Home Book lists
materials and solar absorptances and longwave emittances:

material a e (for selective surfaces)

White plaster 0.07
Fresh snow 0.13
Aluminum foil 0.15 0.05
White paint on aluminum 0.20
Whitewash on galvanized iron 0.22
Ice, with sparse snow cover 0.31
Snow, ice granules 0.33
Aluminum oil base paint 0.45
White powdered sand 0.45
Polished zinc 0.46 0.02
Chromium 0.49 0.08
Green oil base paint 0.50
Bricks, red 0.55
Concrete 0.60
Galvanized iron, clean, new 0.60 0.13
Red oil base paint 0.74
Grey paint 0.75
Desert surface 0.75
Red oil base paint 0.74
Galvanized sheet iron, oxidized 0.80 0.28
Dry sand 0.82
Cupric oxide on sheet aluminum 0.85 0.11
Green roll roofing 0.88
Dark grey slate 0.89
Nickel black on galvanized iron 0.89 0.12
Black gloss paint 0.90
Wet sand 0.91
Black tar paper 0.93
Water 0.94
Black paint on aluminum 0.96 0.88

Nick