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Windows, Baths, and Solar Energy in the Author(s): James W. Ring Source: American Journal of Archaeology, Vol. 100, No. 4 (Oct., 1996), pp. 717-724 Published by: Archaeological Institute of America Stable URL: http://www.jstor.org/stable/506675 . Accessed: 29/05/2014 17:32

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This content downloaded from 128.135.12.127 on Thu, 29 May 2014 17:32:48 PM All use subject to JSTOR Terms and Conditions Windows, Baths, and Solar Energy in the Roman Empire JAMES W. RING

Abstract sea."' One can infer from his writings that Seneca Windows were a prominent feature of Roman archi- regrets this new style and, indeed, it is clear that he tecture and were especially important in the magnifi- looks back with nostalgia on the days of the Roman cent of the Roman A bath buildings Empire. growing when baths were and literature attests to the Romans' use of solar in Republic properly modestly energy dark. heating these large buildings. Edwin Thatcher claimed in 1956 that the windows in such baths did not require of the Early Empire were in the fore- glazing. In this paper I refute this claim, drawing on front of developments both architecturally and tech- modern ideas about solar heat human energy, transfer, nologically and thus make a very interesting study comfort, and the effect of glazed windows to analyze in their own right. This point has been brought out one room in the Baths at Ostia. This analysis and and Nielsen. is compared with that of Thatcher for the same room. ably developed by Yegul2 Vaults, In window size and solar orientation, this room is typ- , and large windows were first found in these ical of Roman baths in many parts of the empire. The baths, where Greek orders were also first combined solar science and of is thus technology today compared with Roman vaults. were developed and with that of the Romans and with that of Thatcher'sdayf* used to heat large rooms and, indeed, to heat the imposing ensembles of large rooms that the great INTRODUCTION imperial baths represented. According to Yegill, The importance of windows in architecture seems Seneca speaks of the recent invention of tubuli, or indisputable. But in its chronological development hollow walls, which maintain an even temperature Western architecture shows striking variation in in the lowest as well as the highest spaces. This in- its treatment of windows. For example, medieval vention also prevents condensation on the walls and churches of the Romanesque style were typically dark increases the area that radiates heat around the bath- and lit only by small windows that pierced massive ers. In these large evenly heated spaces, thousands masonry walls. One of the distinguishing features of bathers could be and often were accommodated. of later Gothic churches was the use of large win- To supply sufficient water, extensive systems dows made possible by outside buttressing. The were developed. Furthermore, as Yegill maintains, Roman Empire prototypes, however, unlike their and is shown in a detailed fashion by D.B. Harden, Romanesque successors, were often lit by magnifi- the Romans by this time had developed glassblow- cently large windows. The Roman public baths of ing and were producing flat panes of window glass.4 the Early Empire are very good examples of this Thus, with all of these elements in hand, it is not anachronism. Seneca, writing in the first century A.D., surprising that the Romans would have utilized the says of these baths: "Nowadays... people regard baths radiant energy of the sun to help heat as well as light as fit only for moths if they have not been so arranged these magnificent buildings. that they receive the sun all day long through the Indeed, so obvious is the Romans' interest in so- of widest windows, if men can not bathe and get a lar heating through their use of large south-facing coat of tan at the same time, and if they can not look windows that in 1956 Edwin Thatcher published a out from their bath-tubs over stretches of land and paper in which he claimed that the large windows

* I wish to thank the American Academy in for a Faculty Fellowship in 1993 during which research lead- help in arranging visits to various Roman sites, including ing to this paper was completed. in the Museum of Glass and Ostia, 1985; Corning Rakow 1 Sen. Ep. 86, trans. R.M. Gummere, 1986, as quoted by for in Library help exploring in 1991; Todd E Yegill, Baths and in (Cambridge, Moore of the Physics Department at Hamilton College for Mass. 1992) 40. his careful reading of the manuscript; Carl Rubino of the 2Yegiil (supra n. 1). ClassicsDepartment at Hamilton for his advice;Fikret Yegiil 3 I. Nielsen, Thermaeand Balnea: The Architectureand Cul- of the Universityof California at Santa Barbaraand Ingrid tural History of Roman Public Baths (Aarhus 1990). E.M. Edlund-Berryof the University of Texas at Austin for 4Yegiil (supra n. 1) 363-65; and D.B. Harden, "Domes- reading an earlier version of this manuscript and advising tic Window Glass:Roman, Saxon and Medieval Studies in about publication; and Richard de Dear of MacquarieUni- Building History,"in E.M.Jope ed., Studiesin BuildingHis- versity,Sydney, for his advice about thermal comfort con- tory. Essays in Recognition of B.H. St.J. O'Neil (London 1961) ditions in these baths. Finally,Hamilton College provided 39-63. 717 American Journal of Archaeology 100 (1996) 717-24

This content downloaded from 128.135.12.127 on Thu, 29 May 2014 17:32:48 PM All use subject to JSTOR Terms and Conditions 718 JAMES W. RING [AJA 100 of the second-century Forum Baths at Ostia, which ing shortages and expense, the Romans, like the he took to be unglazed, provide "a striking demon- Greeks before them, turned to solar heat."8 stration of the potentialities of the Roman heating method and, in extension, of the principles of ra- DID THE ROMANS USE GLAZING? diant It was this method that made the heating. open The Forum Baths in Ostia were constructed early rooms to we have not matched possible and, date, in the second century A.D. There is no controversy them in a modern It is evident that the building. about the existence of large windows in several of Roman engineers had a greater confidence in ra- the rooms of these baths. These windows faced the diant than we have and a heating greater knowledge south and hence would intercept the sun's beam ra- of what it could accomplish." diation most of the day,particularly from early after- This sets out to this claim. Is paper investigate noon to near sunset, which were the most popular Thatcher's confidence in radiant heating justified? hours for Romans to bathe. These rooms indeed are Could there be the sun enough heating provided by typical of baths built during this period in many parts and the to allow nude bathers to be com- of the empire. fortable even the windows of the baths though large Thatcher gives attention to the whole set of large- were and What do the modern open unglazed? prin- windowed rooms but for our purposes let us con- ciples of passive solar heating and the physics and centrate on one, room 4, which seems to have been of heat transfer tell us about the Forum physiology a warm room, or .Figures 1 and 2 show Baths and Thatcher's claim? the southern elevation and north-south section, re- I take a raised He In addition, up point by Yegiul. spectively, of this room and its window. The dimen- that Thatcher has too far in his claims suggests gone sions are those given by Thatcher. for radiant heating: The question is whether or not the walls, vault, In full admiration of the system'spotential, I still and floor surrounding the nude bather on all but doubt if the implications of radiantheating should the window side can be maintained at a high-enough be stretchedthat far.Not only is the evidencefor win- temperatureto ensure comfort.Thatcher approaches dow glass and window frames (both in wood and the issue from the of the nude bather from the heatedrooms of Romanbaths across standpoint metal) both to the radiant of the sun and the Mediterraneanoverwhelming, but Thatcher'sthe- exposed energy sis, despite its theoretical possibility,seems to refute that given off by the surrounding heated room sur- the preceptsof simple economic logic.It maybe that faces, holding that radiant energy, if the walls are by heating the floor, the walls,and the vaultto a high maintained at close to skin temperature, can by it- degree, sufficient radiant energy could be released self establish a comfortable to offset the effects of low air on a cold temperature. My ap- temperature on the other hand, is to assume that com- winterday, but whyshould fuel and energybe wasted proach, in order to makean open-airhot bath possible when fort will be determined by conditions in the room, the same degree of warmth and comfort could be both the air temperature and the radiant temper- achieved with much lower furnace activity and fuel ature being considered along with air currents, or in a and well-insulatedroom?6 consumption glazed convective flows, and the relative humidity of the This question is also raised byJordan and Perlin in air as prescribed by the American Society of Heat- an article about the use of solar energy in ancient ing, Refrigerating, and Air Conditioning Engineers times.7 They claim that by the first century B.C., (ASHRAE).9Comfortable conditions for nude sub- Rome had to import timber from the fringes of its jects have been studied carefully in climate cham- domains, such as the Alpine regions, in part because bers. An example of one such study,which also shows of the Roman love of bathhouses- there were 800 the importance of heat transfer at the skin surface, baths in Rome alone in the third century A.D.- but is that of de Dear, Ring, and Fanger."1Also of im- also because of the growth of industry and manu- portance is the air flow over the skin. This too has facture. As they point out, "pricesof wood, charcoal, been studied and reported on by,for example,Fanger and small firewood rose steeply. To avoid the grow- and his colleagues."1

" E.D. Thatcher, "The Open Rooms of the Terme del ch. 8. 1( Foro at Ostia,"MAAR 24 (1956) 169-264. R.J. de Dear, J.W.Ring, and P.O.Fanger, "Thermal Sen- 6 Yegfil (supra n. 1) 383. sations Resulting from Sudden Ambient Temperature 7 B.Jordan and J. Perlin, "SolarEnergy Use and Litiga- Changes," Indoor Air 3 (1993) 181-92. tion in Ancient Times," Solar Law Reporter1.3 (1979) 583-94. 11P.O. Fanger, A.K. Melikov,H. Hanzawa,andJ.W. Ring, 8Jordan and Perlin (supra n. 7) 587. "AirTurbulence and Sensation of Draught,"Energy and Build- 9 ASHRAE Handbook,1977 Fundamentals (New York 1977) ings 12 (1988) 21-39.

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grade level

10 ft I I I I I I I

Fig. 1. Southern elevation of room 4 at the Forum Baths of Ostia

The problem then becomes one of the transfer neutral environment. On the contrary, they expect of heat from the hypocaust and the sun to the room to feel hot. Thus, strictly speaking, we are not deal- and from the surfaces of the room to the outside ing with the usual comfort scale but rather with mainly through the large window. These processes one biased toward the hot end. For example, the determine the temperatures of the surfaces and thus seven-point ASHRAE scale (cold, cool, slightly cool, the convective drafts and air temperature. Relative neutral, slightly warm, warm, hot) would allow using humidity, of course, depends on the vapor pressure of only the upper point in the warm rooms of the water in the space. In room 4, which apparently had baths. According to Thatcher, the temperatures of no pools or baths, the relative humidity would not the walls and floors of these rooms were -400 C have been particularly high, perhaps about 50%. The (or - 1000 F).12In fact, this is in the range of evapo- temperature and convective flows were thus the pri- rative regulation (sweating) that is adjacent to but not mary determinants of human comfort in this room. part of the comfort zone, i.e., these are, according Naked bathers do not expect or want a thermally to ASHRAE, "uncomfortable" conditions.

12Thatcher (supra n. 5) 190-94.

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Yault

suspensura

tubuli hypocaust praefurnium 10ft ILIIIIIIII

Fig. 2. North-south section of room 4 at the Forum Baths of Ostia

We take conditions to be like those assumed by that temperatures of 700 F for these surfaces will Thatcher: outside temperature is just below freezing, cause the nude bather to radiate heat to them as 300F,'1 the interior surfaces of the room are at well as losing heat to them by convection (and con- 1000 F, and the sun is shining in the window during duction if in contact with them). These will not be December at 250 British thermal units (BTU)/ft2l/hr, warm conditions for him but rather ones somewhat assuming a clear sky. Further, we take the temper- on the cool side. The thermal properties of the ma- ature of the hot gases at the top of the hypocaust terials, i.e., conductivity and coefficients of heat trans- below the floor to be 4000 F, which is consistent with fer for radiative or convective flow, can be found in the experiments of Rookl4 with a small hypocaust the appropriate part of the ASHRAE Handbook.16 in Welwyn, , although somewhat above the With these parameters known, calculations can temperature (-3000 F) found by Kretzschmer15 in be made for the heat flows as shown below in table experiments in Saalburg, Germany. For the purposes 1. The heat flows without and with glass are shown of the argument here, any temperature up to 4000 F diagrammatically in figures 3 and 4, with figure 3 can be posited. The crucial heat flow is that out showing inflows and figure 4 outflows. Here we are through the open window, and if we assume 4000 F concerned with the comparison of flows in versus we are estimating the maximum heat flow in and flows out, for if they are not equal, the temperature thus giving Thatcher the best chance of being cor- of the walls and the room will not be constant at, rect. As does Thatcher, I too assume that the floor and or close to, the desired 1000 E inside wall surfaces are held at - 1000 F, including Certain caveats about these calculations ought to the inside of the vault, which, although unheated by be made clear. None of these heat flows can be said tubuli, by convection and radiation will be at nearly to be precisely defined. The problems in calculation 1000 F if the walls are also at this temperature. Note are:

14 1'This is, according to Thatcher (supra n. 5) 182-83, T. Rook, "The Development and Operation of Roman a quite possible temperature in Ostia in December orJan- Hypocausts,"JAS 5 (1978) 269-82. 15 uary, and in my own experience a low but not outlandish F. Kretzschmer, "Hypokausten," SaalbJb 12 (1953) 7-41. one. 16ASHRAE Handbook (supra n. 9) ch. 11.

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sun's beam radiation sun's beam radiation conduction conduction through glass from tubuli from tubuli

conduction conduction fromhypocaust fromhypocaust

10 ft 10oft IU1111111111 I111111111

Fig. 3. Heat flows into room 4 with open windows (left) and glazed windows (right)

1) The dimensions are not always precisely known. 5) I estimate the temperatures of the lower sus- I use those given by Thatcher. pensura (floor) surface and the inner surfaces of the 2) In some cases the materials are only guesses tubuli from Rook's and Kretzschmer's experiments. as the upper part of the room has disappeared. Gen- 6) The solar beam radiation is calculated here erally I follow Thatcher's suggestions. in the same way as it was by Thatcher, agreeing also 3) I assume, as does Thatcher, that the vault is not with the method used by Ring and Hamilton"7 to heated. test the performance of a solar classroom at Hamil- 4) The optical quality of Roman glass varies widely ton College at close to the same latitude as Ostia. and that used at Ostia in these baths is not known. 7) Convective flows are notoriously hard to cal- A transmission of 50%, assumed here, is probably culate, but since such flows do occur in solar houses a quite conservative estimate. we can expect to achieve order of magnitude results

conduction though vault conduction through vault

radiation

flows through glass convection-loops

radiation

10 ft 10 ft II!11111111 LLIIIIJlllI

Fig. 4. Heat flows out of room 4 with open windows (left) and glazed windows (right)

17J.W.Ring and A. Hamilton, "The Solar Classroom at Research Institute TP-245-430, Washington, D.C. 1979) Hamilton College," Proceedings, Conferenceon Solar Heating 107-11. and Cooling Systems,Colorado Springs, Colorado(Solar Energy

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Table 1. Heat Flows for Room 4 with Glazed and much more than adequate to provide the outward Unglazed Windows flow (80,000 BTU/hr) in radiation, conduction, and convection the window and in con- Units of through glazed Direction and 100,000 duction through the vault. The sun alone on sunny Source of Flow BTU/hr days could provide most of the energy to maintain 100' F temperatures. Indeed, even with the fires re- Into room duced on sunny there would be some (full heat) 4.0 days, probably Hypocaust thermal stored in the floors and walls that Sun (clear day at winter solstice) energy Unglazed window 1.3 would maintain the temperature as the sun goes Glazed window 0.7 down. On days when the sun is obscured by clouds, TOTALS the hypocaust with a reduced fire, or being on only Unglazed window 5.3 part of the time, could by itself easily maintain the Glazed window 4.7 temperature even with the outside temperature at Out of room (outside temperature = 300 F) its coldest point of the season, i.e., the design tem- Conduction 0.4 perature of 300 E Natural convection and radiation The stored thermal energy, which may come from window 52.5 Unglazed either the sun or the or both, can be Glazed window 0.4 hypocaust handled quite easily by the heavy masonry walls, TOTALS vault, and floor of this room. With such surround- Unglazed window 52.9 Glazed window 0.8 ings extra heat in the room can pass readily by con- duction into the masonry without heating the air Net Flow (+ = into room) in the room excessively, i.e., much above 100 .E At Unglazed window - 47.6 Glazed window + 3.9 night, when the sun is down (even with the fire out), this stored heat will flow back into the room to offset the cooling that inevitably will occur. Note that using the solar designer's formulas.'" Natural rather wooden shutters closing the window area at night than forced convection is assumed, i.e., there is no would enhance this storage considerably. Such heat wind blowing. With wind the convective flow is storage is an important element in solar house de- greater, and could be much greater at high-wind sign and the Romans seem to have incorporated this velocities. element into their designs as well. The thickness of Under such circumstances we can expect only to the floor, or suspensura, is especially interesting in estimate these flows. Even with these rough estimates, this regard and it may be that the thickness chosen, however, some important conclusions can be i.e., 15 in, is not necessary structurally but aids in reached. The estimates for glazed and unglazed win- long-term heat retention. dows are given in table 1 above. The details of the Thus, we see that Yeguil is correct in his claim that calculations follow in an appendix. the Romans would have been wise to use glass in their bath windows. Furthermore, we see that RESULTS AND DISCUSSION Thatcher is probably overly enthusiastic in claiming The results of the calculations summarized in table such efficiency for radiant heating. Indeed, with I and in the appendix indicate that with an open the windows open, the rooms in these baths could window the input is - 530,000 BTU/hr while the not have been maintained at 1000 F and a nude outflow is - 5,290,000 BTU/hr. In such a case it is ob- bather would soon have become very chilly. Thatcher vious that equilibrium is not possible and that the neglected to consider fully the very great heat flow 100' F surfaces will rapidly cool toward 300 E The out of the room due to the convective flow through same would be true for a nude bather whose skin the open window. He does not completely ignore the temperature normally should be - 930 E Indeed the possibility of an air current but claims that such a outflow would equilibrate with the inflow only when current would only exist with no wind, and normally the wall and floor surfaces are within 100 F or less there would be some wind. He then says, "A wind of the outside temperature, i.e., at 400 E pressure of any but the lowest magnitude would nul- On the other hand, if the window_ is glazed the lify the action and set in motion the various air cur- heat from the hypocaust (400,000 BTU/hr) would be rents already described."'' These currents, Thatcher

'8J.D. Balcomb ed., Passive Solar Buildings (Cambridge, '1 Thatcher (supra n. 5) 233-34. Mass. 1992) 149-52.

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believed, flowed from north to south, or vice versa, Thatcher, it seems, was too sanguine about radiant across the room; in the north wall they passed heating but, nevertheless, the Romans deserve high through the door, cracks around the door, or the praise for their use of solar energy. Even Seneca, no lunette at the top of the vault, and in the south wall, admirer of conspicuous consumption and easy liv- through the upper part of the window. ing, might have admired the frugality that the com- This assertion seems incorrect because a wind bination of "the widest of windows" and glass panes blowing in, or eddying through the southern win- in baths demonstrated. dow, will not nullify the convective effect. Rather it will cause a change from natural convection to forced Appendix convection, changing and distorting the geometry Calculations of Heat Flows for Room 4 of the convective loop and, as a result, increasing the mixing of cold and hot air and thus increasing the with and without Glazing heat loss above that caused by natural convection A. Heat flows into the room: alone. The net result would be to set the 52.5 x 105 (i) The heat flow through the suspensura: this is a heat BTU/hr heat loss calculated above as a minimum conduction problem where Q, the heat flow per hour, value and in windy situations to expect this loss to is given by: be even greater with the concomitant effect of an with k the thermal of the even faster lowering of the bath temperature, more Q kAAT conductivity concrete Ax the thickness of the quickly chilling the nude bather. slab, Note that the natural convective flow calculated slab, A the cross-sectional area, and AT the temperature difference between here is only a rough estimate. But it is about 10 times the top and bottom of the slab. the inflow so that even if it is overestimated by a fac- Using British engineering units with tor of two, it still will be times than the many greater k = 11 BTU-in/oF-ft2-hr(value used by Thatcher) inflow. With it will be even than cal- wind, greater A = 1,200 ft2 culated here. AT = 400 - 100 = 3000 F (using maximum hot Finally, to return to Yegiil's point, the estimates gas temperature under suspensura) of heat flows here show not only that nude bathing Ax = 15 in in Roman baths would not have been possible with- Q = 2.6 x 105 BTU/hr out but also that with glazing glazing during sunny (ii) The heat flow through the heated walls: the sun with a little from the days, only help hypo- k = 7.0 BTU-in/oF-ft2-hr(value used by Thatcher) caust and its furnaces, and hence little wood burned, A = 1,760 ft2 (3 vertical walls) could have maintained the temperature of these AT estimated to be 200 - 100 = 1000 F room surfaces at - 100' E Furthermore, on cloudy Ax = 9 in days the hypocaust with only a low or intermittent Q = 1.4 x 105 BTU/hr fire would have been able to sustain this tempera- (iii) The sun's radiant energy in through the window: ture. And even at night the large thermal storage In December in Rome, the sun at noon is only about capacity would have kept temperatures from drop- 240 above the horizon and the sun's beam intensity is ping very fast so that by the next morning the amount approximately 250 BTU/hr/ft2. Note that compared of heat necessary to return to - 1000 F might have with 228 at the winter solstice, the noon value on a south- been relatively small. Thus with a normal mix of facing vertical surface at equinox would be 285 x cos 420, or 212 BTU/hr/ft2 at this latitude of 420 N. At the sunny days, a considerable savings of fuel could be summer solstice this value would be 116. At noon, re- accomplished even in the depths of winter. At other gardless of the season, the sun's beam radiation on this seasons even more savings could be expected. The surface would always be at its maximum for the day. sun would therefore provide a substantial part of Note also that these intensity values agree with those the heat This of required. result, course, is in accord used by Thatcher: with Jordan and Perlin's observations about the in- A = 560 ft2 creasing cost of fuel during this period of rapid Q = 250 A cos 240 = 1.3 x 105 BTUlhr growth of Roman industry, commerce, manufacture, or with Roman glass: and population. Fuel costs would have provided a Q 0.65 x 105 BTU/hr - strong incentive for using glazed windows and the Thus, the total energy input a 5.3 x 105 BTUlhr, i.e., sun's energy. the sum of the above items, or with glass 4.65 x BTUlhr. - In summary, the Romans apparently did display 105 considerable know-how in the design of their baths B. Heat flows out of the room: when judged by the standards and practices of mod- (iv) Conduction through vault to the outside: ern science and technology 2,000 years later. k = 7.0 BTU-in/oF-ft2-hr,as in the walls

This content downloaded from 128.135.12.127 on Thu, 29 May 2014 17:32:48 PM All use subject to JSTOR Terms and Conditions 724 J.W. RING, WINDOWS, BATHS, AND SOLAR ENERGY IN THE ROMAN EMPIRE = A = 1,880 ft2 Using A = 280 ft2, H = 10 ft, and AT 701 F as AT = 100-30 = 700 F above: Ax = 24 in Q = 6.7 x 106 BTU/hr Q = 3.8 x 104 BTU/hr This is 25% more than the estimate above. I use the smaller number in table 1. (v) The natural convective flow through the open window: (vi) The radiant heat flow through the window: Here the convective loop will have hot air exiting Here the Stefan-Boltzman law governs the heat flow through the top of this large window and cold air com- (the same equation used by Thatcher). ing in at the bottom. See figure 4. This flow will be caused Q = oAE(T14 - T24) by the stack effect in which hot, less dense gas is forced with o = 1,730 x 10-12 BTU/hr-ft2-(OF absolute)4 out of the room at the top of the window and, by the A = area of body in question (the window in this same effect, cold air that is more dense will flow in at case) the bottom. Solar house designers use the formula given E = 0.9, a factor accounting for emissivities and sol- below to calculate this air flow.20 id angles subtended by the hot and cold bodies CFM = 9.4Aeff'HAT as seen through the window. Again this is the where Aeff is the effective area (ft2) through which value used by Thatcher. the flow enters and/or leaves the space, and H is the T1 and T2 are 5600 F and 4900 F absolute, respec- effective height of this area. In this case the Aeff is half tively, for the hot inner surfaces at 1000 F and the area of the window while H is half the height of outside surfaces (or air) at 30 F.E the window as half the window is used for outward flow Q = 3.5 x 104 BTU/hr and the other half for inward flow, or Aeff = 260 ft2, If the window is glazed, a combination of convection, = across of air H = 10 ft (mean height), and AT = 100 - 30 700 F radiation, and conduction boundary layers and: both on the inside and outside as well as the glass itself CFM = 7.0 x 104 ft3/min can be treated according to the following equation:22 Then the heat transferred will be: Q = UAAT Q = CFM x 60 x AT x 0.018 where U = 1.10 BTU/hr-ft2-oF,which includes all where 0.018 is the volumetric specific heat of air in three types of heat flow = = BTU/ft3 at these temperatures. A area in ft2 of window 560 Q = 5.2 x 106 BTU/hr AT = 700 F Note that another equation for this convection heat Q = 4.3 x 104 BTU/hr transfer mentioned by Balcomb is that of Weber and Kearney,2'who arrived at it by similitude modeling and full-scale testing: DEPARTMENT OF PHYSICS Q = 4.6 W(dAT)312where W is width and d is height HAMILTON COLLEGE of opening, which when converted to the vari- ables used above becomes CLINTON, NEW YORK 13323 Q = 13 A -H (AT)3'2 [email protected]

21)Balcomb (supra n. 18) 149-52. Amherst, Massachusetts, October 1980 (American Section of 21 D.D. Weber and R.J. Kearny, "Natural Convection the International Solar Energy Society, Newark,Del. 1980) Heat Transferthrough an Aperture in PassiveSolar Heated 1037-41. 22 Buildings," Proceedings,5th National Passive Solar Conference, ASHRAE Handbook (supra n. 9) ch. 11.

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