Date: Thu, 27 Jan 94 23:08:42 -0800 From: daveb@ingres.com (David Brower) To: rusin@math.niu.edu (Dave Rusin) Subject: Re: Heat Loss in House (Math model) (Long) Newsgroups: misc.consumers.house I've come across a book that has some relevant math explained, from a practical perspective. Unfortunately, it does not discuss how one adds up losses possible from various mixed types of construction in much detail. The book is "From the Walls In", by Charles Wing, Atlantic-Little, 1979, ISBN 0-316-94740-7. Since this is old and probably hard to get, here are a few relevant parts, all paraphrased or quoted. YMMV. (1) The rate at which a body radiates energy is proportional to A * emissivity * S-B-K * T^4 where A = area of surface emissivity = emissivity of surface S-B-K = Stefan-Boltzman Constant T^4 = absolute temp of the body to the 4th power. The emissivity of a dark hole drilled in a cube of carbon at the MBS is the blackest thing around, and has a value of 1.00. Flat black paint has an emmisivity of around .95, while that of polished aluminum foil (brighter than Reynolds wrap) is only 0.05. Polished foils act as insulators when facing air. If they actually touch another material, the benefit is lost because heat flows by conduction instead of by radiation. (2) The exact equation for the conduction of heat throuch a slab (the heat flow equation) is: H-hr = A * delta(T) / R where H-hr is Btu's lost per hour, and Btu = heat to raise 1 lb. water 1 degree F. A = face area of slave in sqare feet deltat(T) difference in F temp between sides. R = thermal resistance of the slab. (3) The total thermal resistance of any construction is obtained by adding the thermal resistance of each of the component parts. Emissivity of surfaces ------------------------------------------------ boards, one surface 0.90 boards, two facing surfaces 0.82 galvanized steel, white or aluminum paper, one or two surfaces, average 0.20 builder's foil, one surface 0.05 builder's foil, two facing surfaces 0.03 R value of surfaces type direction emmisivity 0.05 0.20 0.90 ---------------------------------------------------- inside wall horizontal 1.70 1.35 0.68 inside ceiling upward 1.32 1.10 0.61 inside floor downward 4.55 2.70 0.92 outside surface Any - - 0.17 15 mph wind outside surface Any - - 0.25 7.5 mph wind R value of Air Spaces, @50 deg. F, delta(30 deg F) air space direction air space emissivity location of flow inches 0.03 0.05 0.20 0.82 ---------------------------------------------------------------- horizontal upward 3/4 1.72 1.67 1.37 0.78 ex. ceiling 1-1/2 1.82 1.76 1.43 0.80 4 2.14 2.06 1.62 0.85 horiz. down 3/4 3.80 3.55 2.39 1.02 ex. floor 1-1/2 6.41 5.74 3.21 1.14 4 10.70 8.94 4.02 1.23 vertical horizontal 3/4 2.95 2.80 2.04 0.96 ex. wall 1-1/2 2.90 2.76 2.02 0.96 4 2.74 2.63 1.94 0.94 pitched 45 upward 3/4 2.02 1.95 1.54 0.83 deg roof 1-1/2 2.09 2.01 1.58 0.84 in winter 4 2.32 2.22 1.71 0.88 pitched 45 downward 3/4 3.47 3.27 2.27 1.01 deg roof 1-1/2 3.50 3.30 2.29 1.01 in winter 4 3.61 3.39 2.33 1.02 R value of materials @50 deg F, delta(30 deg F) material R per inch R total ------------------------------------------------ softwoods 1.25 hardwoods 0.91 plywood 1.25 poly vapor barrier 0.00 Sheetrock 0.90 stone or concrete 0.08 plaster or bricks 0.20 8 in concrete block 1.11 1/8 in cork tile 0.28 cork board 3.50 carpet w/rubber pad 1.20 carpet w/fiber pad 2.1 asphalt roll roofing 0.15 asphalt selvage 0.30 asphalt shingles 0.44 standard exterior door 2.00 insulating glass, 1.50 1/4 in space insulating glass, 2.00 1 in space Fiberglass batt 3.15 Fiberglass, loose 2.2 vermiculite, loose 2.2 polyurethane foam, 6.25-7.50 2 lb/cu ft. polystyrene bead board 3.70-4.25 For a full 6-inch fiberglass batt, R= 3.2 * 6 = 19.2. Note however that when insulation is placed between framing memberss, the framing members present alternate heat paths of lower R value. To account for the greater heat loss, we make use of the effective R value, R-eff, wich varies with the situation: R-eff of fiberglass batt in wall framing Fibreglass Batt 2x4 2X6 R=3.2/inch 16" OC 24"OC 16" OC 24" OC ------------------------------------------------------- nominal 11.2 12.8 17.6 19.2 effective 9.2 10.5 15.0 15.8 (4) To find the total thermal resistance of a floor, wall or roof, simply draw a picture of the construction so that you can identify all of the different resistive elements; list each element and it's resistive contribution; add all the contributions to find the total resistance. Example: item R ------------------------------------------------ outside surface, emmis. = 0.9 0.17 1" pine board 1.25 1/2" sheathing plywood 0.63 6" fibreglass in 2x6 studs, 24" OC R-eff 15.80 6 mil poly vapor varrier 0.00 1/2" sheetrock 0.45 inside surface, emmis. = 0.20 0.68 Total R = 19.0 (5) A degree-day is day where the average temperature for that day is one different from 65 deg. F. To compute the heat loss, plug into: H-annual = 24 * A * DD / R where H-annual is the heat loss in BTUs. (6) Infiltration is a special case of heat loss through convection. It is the physical replacement of warm air by cold outside air (or the reverse in summer). Infiltration amounts to a rate of heat loss. It is quantifed in either cu/ft/min (cfm) or total house volume exchanges per hour. The former is useful in tight, new construction, where 10 cfm/person is standard. Total air exchanges per hour are more useful in older buildings where the infiltration unfortunately far exceeds minimum ventilation standards! An old wood frame building with windows rattling in the wind will totally change its air anywhere from two to ten times per hour depending on condition and exposure to wind. New construction of average workmanship will be about one exchange per hour. It is possible, by attention to detail, to build or retrofit to only 1/2 exchange/hr. The hourly heat loss through infiltration is H-hour - 0.0182 * Q * delta(T) where H-hour = BTU/hr Q = cu ft/hr (exchange rate * volume of structure) delta(T) = inside to outside temperature, deg. F. 0.0182 = weight of 1 cu ft. or air * specific heat of air (7) Gains One human occupant adds about 170 BTU/hr. Solar gain through a window is given by H-solar = A * SC * 24 * HD * SF * Y / 1550 where H-solar = BTUs through window during heating season A = window area in sq. ft. SC = shading coeeficient 24 = hours/day HD = heating days/pr SF = solar factor Y = climatic factor (This assume a normal percentage of window area. Areas in excess of 15% of wall area may result in solar overheating and necessary venting of excess heat. A includes sashs, at about 80% glass and 20% wood. SC is the ratio of radiation through the window to a single glzed window exposed to sun from sunrise to sunset. Each layer absorbs about 10% of the radiation, so SC for the case of no sky shading is: single glazed 1.0, double glazed 0.9, triple glazed 0.8. this should be multiplied by the % of the sun's path that is blocked. A south facing window is in the shade 50% of the day and has an SC of about 0.9 * .5 = .45. HD are the number of days where bodies plus other sources (lights, TVs, computers) aren't enough to heat the house. It's about 100 in Miami, 150 in Phoenix, 225 in LA and Richmond VA, 250 in NYC, and 350 in parts of Minnesota and San Francisco (fog, don'tcha know). SF Solar factor window lattitude orientation 24 deg 32 deg 40 deg 48 deg -------------------------------------------------- N 9 8 7 6 NE, NW 16 14 11 9 E, W 37 33 29 25 SE, SW 51 51 50 46 S 55 59 62 59 Y accounts for cloudiness. It's about 1500 in Miami, 1300 in LA, 1000 in San Francisco, 750 in Buffalo and Syracuse, and 550 in Seattle. -dB --