Windows, Baths, And Solar Energy in the Roman Empire

8/8/2019 Windows, Baths, And Solar Energy in the Roman Empire

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Windows, Baths, and Solar Energy in the Roman Empire

James W. Ring

American Journal of Archaeology , Vol. 100, No. 4. (Oct., 1996), pp. 717-724.

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Windows, Baths, and Solar Energy in the Roman Empire JAMES W. RING

Abstract

Windows were a prom inent feature of Roman archi-tecture and were especially important in the magnifi-cent bath buildings of the Roman Empire. A growingliterature attests to the Romans' use of solar energy inheating these large buildings. Edwin That che r claimedin 1956 tha t the windows in such baths did not requireglazing. In this paper 1 refute this claim, drawing onrnodern ideas abo ut solar energy, heat transfer, humancomfort, and the effect of glazed windows to analyzeone room in the Forum Baths at Ostia. This analysisis compar ed with that of Thatcher for the same room.In window size and solar orie ntation, this room is typ-ical of Roman baths in many parts of the empire. Thesolar science and technology of today is thus comp aredwith that of the Romans an d with that ofTh atcher's day.*

INTRODUCTION

c .

I he importance ofwindows in archite cture seemsindisputable. But in its chronological developmentWestern architecture shows striking variation inits treatment of windows. For example, medievalchurches of the Romanesque style were typically darkand lit only by small windows that pierced massivemasonry walls. One of the distinguishing featuresof later Gothic churches was the use of large win-dows made possible by ou tside buttressing. TheRoman Empire prototypes, however, unlike their

Romanesque successors, were often lit by magnifi-cently large windows. The Roman public baths ofthe Early Empire are very good examples of thisanachron ism. Seneca, writing in the first century A.D.,says of these baths : "Nowadays.. . people regard bathsas fit only for moths ifthey have not been so arrangedthat they receive the sun all day long through the

widest of windows, if men can not bathe and get acoat of t an at the same time, and if they can not lookout from thei r bath -tubs over stretches of land a nd

* I wish to thank the American Academy in Rome forhelp in ar ranging visits to various Roman sites, includingOstia, in 1985; the Corning Museum of Glass and RakowLibrary fo r help in explorin g Roman glass in 1991; ToddMoore of th e Physics Departm ent at Hamilton College forhis careful reading of the manusc ript; Carl Rubino of theClassics Dep artm ent at H amilton for his advice; Fikret Yegiilof the University of C alifornia at Santa Barbara and IngridE.M. Edlund-Berry of t he University of Texas at Austin fo rreading an ea rlier version of this manuscript a nd advisingabout publication; and Richard de Dear of Macquarie Uni-versity, Sydney, for his advice abo ut the rmal comfort con -ditions in these baths. Finally, Hamilton College provided

71 7Amer ican Journal of Archaeology 100 (1996) 717-24

sea."' One can infer fiom his writings that Senecaregrets this new style an d, inde ed, it is clear that he

looks back with nostalgia on t he days of t h e RomanRepublic when baths were properly and modestlydark.

Roman baths of the Early Emp ire were in the fore-front of developments both architecturally and tech-nologically an d thu s make a very interes ting studyin their own right. This point has been brought outan d ably develo ped by Yegiil' an d Nielse n." Vaults,domes, and large windows were first found in thesebaths, where Greek or ders were also first combin edwith Roman vaults. Hypocau sts were developed andused to heat large rooms and, indeed, to heat the

imposing ensembles of large rooms that the greatimperial baths represented. According to Yegiil,Seneca speaks of the recent invention of t ubu l i , o rhollow walls, which maintain an even temperaturein the lowest as well as the highest spaces. This in-

vention also prevents condens ation o n the walls andincreases the area that radiates hea t arou nd the bath -ers. In these large evenly heated spaces, thousand sof bathers could be and often were accommodated.

c .

Io supply sufficient water, extensive aqueduc t systemswere developed. Furth erm ore, as Yegiil maintain s,and is shown in a detailed fashion by D.B. Harden,

the Romans by this time had developed glassblow-ing and were pr oduc ing flat panes of window glass.-'Thus, with all of these elements in hand, it is notsurpris ing that the Romans would have utilized t herad iant energy of the s un to hel p heat as well as lightthese magnificent buildings.

Ind eed, so obvious is the Romans' interest in so-

lar heating through their use of large south-facingwindows that in 1956 Edwin Tha tcher published apaper in which he claimed that the large windows

a Faculty Fellowship in 1993 du rin g which re search lead-ing to this paper was completed.

1 Sen. Ep. 86, trans. R.M. Gumm ere, 1986, as qu ote d byF Yegiil, Baths and B athi ngin Classical Antiquity(Cambridge,Mass. 1992) 40.

2 Yegiil (supra n. 1).:3 I. Nielsen, Thermae and Balnea: T he Architecture and Cul-

tural History of Roman Public Baths (Aarhus 1990).' Yegiil (supra n. 1) 363-65; an d D.B. Har den , "Domes-

tic Window Glass: Roman , Saxon an d Medieval Studies inBuilding History," in E.M. Jo pe ed., Studies in Building His-tory. Essays in Recognition 0fB.H. St.J. O'Neil (London 1961)39-63.


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718 JAMES W . RING [AJA 100

of t he second-century Forum Baths at Ostia, which

he took to b e unglazed, provide "a striking demo n-

stration of the potentialit ies of the Roman heatingmethod and , in extension, of the principles of ra-

dian t heating. It was this method that made the ope n

rooms possible a nd, to date, we have not ma tched

them in a modern building. It is evident that the

Roman engineers had a greater confidence in ra-

dian t heating tha n we have a nd a greater knowledge

of what it could accomplish."'c .

I his paper sets out to investigate this claim. IsThatcher 's confidence in radiant heatingjustified?

Could there be e noug h heating provided by the sun

an d the hypocaust to allow nude bathers to be com-

fortable even thou gh th e large windows of the baths

were open and unglazed? What d o the modern pr in-

ciples of passive solar heating a nd the physics a nd

physiology of heat transfe r tell us about t he Forum

Baths an d Thatcher 's claim?

In a ddit ion, I take u p a po int raised by Yegiil. H e

suggests that That cher has gone too far in his claims

for radiant heating:

In full admiration of the system's potential, I stilldoubt if the implications of radiant heating shouldbe stretched that far. Not only is the evidence for win-dow glass and window frames (both in wood andmetal ) from the heated rooms of Roman baths acrossthe Mediterranean overwhelming, but Thatcher's the-sis, despite its theoretic al possibility, seems to refutethe precept s of simpl e economic logic. It may be that

by he ating the floor, the walls, an d th e vault to a highdegree, sufficient radiant energy could be releasedto offset the effects of low air t emp era ture on a coldwinter day, but why shou ld fuel and energy be wastedin order to make an open-air hot bath possible whenthe same degree of warmth and comfort could beachieved with much lower furnace activity and fuelconsumpt ion in a glazed and well-insulated room?"

This question is also raised by Jord an and Perlin in

an article ab out the use of solar energy in ancient

times.' They claim th at by the first century B.C.,Rome had to i mpor t t imber fr om the fringes of i ts

domains, such as the Alpine regions, in p art because

of the Roman love of bathhouse s- ther e were 800baths in Rome alone in the thir d century A.D.- but

also because of the growth of industry and manu-

facture. As they point out , "prices ofw ood , charco al,

and small firewood rose steeply. To avoid the grow-

'

E.D. Thatcher, "The Open Rooms of the Terme delForo a t Ostia," MAAR 24 (1956) 169-264.

"egul (supra n. 1) 383.B.Jordan and J. Perlin, "Solar Energy Use and Litiga-

tion in Ancient Times,"SolarLaw Reporter 1.3 (1979) 583-94.Jordan and Perlin (supra n. 7) 587.

"

ASHRAE Handbook, 1977 Fundamentals (New York 1977)

ing shortages and expense, the Romans, l ike the

Greeks before them , turn ed t o solar heat."*

Th e Forum Baths in Ostia were constructed early

in t he se cond cen tury A.D. Th er e is no controversy

about the existence of large windows in several of

the r ooms of the se baths. These windows faced the

south and hence would intercept the sun's beam ra-

diation most of the day, particularly from early after-

noon to near sunset, which were the most popular

hours for Romans to bathe. These rooms indeed are

typical of baths built durin g this peri od in many parts

of the empire.

That che r gives attention to the whole set of large-

windowed rooms but f or our pur poses let us con-

centrate on one, room 3, which seems to have been

a warm room, or tepidarium. Figures 1 a nd 2 showthe southe rn elevation an d north-south section, re-

spectively, of this ro om an d its window. The d im en -

sions are those given by Thatcher.r .

I he question is whether or not the walls, vault,and f loor surrounding the nud e bather on a ll but

the window side can be maintained at a high-enough

temper ature to ensure comfort. Thatche r approaches

the issue f rom the s tandpoint of the nude bather

exposed both to the radiant energy of th e sun a nd

tha t given off by the s urrounding heated r oom sur-

faces, holding that radiant energy, if the walls are

mainta ined a t close to skin tempe rature, can by it-self establish a comfortable temperature. My ap-

proach, on the other hand, is to assume that com-

fort will be det ermine d by conditions in th e room,both the a i r temperature a nd the radiant temper-

atur e being considered along with a ir currents, or

convective flows, and the relative humidity of the

ai r as prescri bed by the Americ an Society of Heat-

ing, Refrigerating, and Air C ondi tion ing Engineers

(ASHRAE).Womfortable conditions for nud e sub-

jects have been studied carefully in climate cham-

bers. An exam ple of o ne such s tudy, which also shows

the im portance of heat transfer at th e skin surface,is tha t of de Dear, Ring, a nd Fanger.Io Also of im-

portance is the air flow over the skin. This too has

been studied an d rep orte d on by, for example, Fanger

and his col leagues .~~

ch. 8.R.J. de Dear,J.W. Ring, and P.O. Fanger, "Thermal Sen-

sations Resulting from Sudden Ambient TemperatureChanges," Indoor Air 3 (1993) 181-92.

11 P.O. Fanger, A.K. Melikov, H. Hanzawa, a nd J.W. Ring,"Air Turbulence and Sensation of Draught," Energy and Build-

ings 12 (1988) 21-39.


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WINDOWS, BATHS, AND SOLAR ENERGY IN THE ROMAN EMPIRE

grade

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

The problem then becomes one of the transferof heat from the hypocaust and the sun to the roomand from the surfaces of the room to the outside

mainly th roug h the large window. These processesdete rmine the temperatures of the surfaces and thu sthe convective drafts and air tem perat ure. Relativehumidity, of course, depends on the vapor pressure ofwater in t he space. In room 4, which apparently hadno pools or baths, the relative humidity would nothave been particularly high, perhaps abou t 50 %. Thetemp eratu re and convective flows were thus th e pri.mary determinants of human c omfort in this room .

Naked bathers do not expect o r want a thermally

neutral environment. On the contrary, they expectto feel hot. Thus , strictly speak ing, we are not deal -ing with the usual comfort scale but rather with

one biased toward the hot end. For example, theseven-point ASHRAE scale (cold, cool, slightly co ol,neut ral, slightly warm, warm, hot ) would allow usingonly the upper point in the warm rooms of thebaths. According to Thatcher, the temperatures ofthe walls and floors of these rooms were -40' C(o r - 100' F).12 In fact, this is in t he ra nge of evapo-rative regulation (sweating) hat is adjacent to but notpart of th e com fort zone, i.e., these are, accordingto ASHRAE, "uncomfortable" conditions.

l y Thatcher (supra n . 5 ) 190-94.


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JAMES W. RING [AJA 100

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

We take conditions to be like those assumed byThatcher : outside tem perature is just below freezing,:30 F, ' ' the interior surfaces of the room are at100 F, and t he s un is shining in t he window du rin gDecember a t 250 British t hermal units (BTU)lft'lhr,assuming a clear sky. Further, we take the temper-ature of the hot gases at the top of the hypocaustbelow the fl oor to be 400 F, which is consisten t withthe experiments of RookL4 with a small hypocaustin Welwyn, England , although somewhat above th e

temperature ( - 300 F) found by KretzschmerlVnexperiments in Saalburg, Germany. For t he pu rposesof the argum ent here, any temperature u p to 400 Fcan be posited. The crucial heat flow is that out

thro ugh the o pen window, and if we assume 400 Fwe are estimating the maximum heat flow in andthus giving Thatcher the best chance of being cor-rect. As does Thatcher, I too assume that the floor andinside wall surfaces are held a t - 100 F, includ ingthe inside of the vault, which, although unheated bytubuli, by convection an d radiatio n will be at nearly

100' F if the walls are also at this temper ature. Note

that tem pera tur es of 70' F for these surfaces willcause the nude bather to radiate heat to them aswell as losing heat to them by convection (and con-duction if in con tact with the m). These will not bewarm conditions for him b ut r ather ones somewhaton the cool side. Th e thermal properties of the ma-terials, i.e., conduct ivity an d coefficients of heat tr ans -fer for radiative o r convective flow, can be fou nd inthe appropriate part of the ASHRAE Handbook.1"

With these parameters known, calculations can

be made for the heat flows as shown below in table1. Th e heat flows without and with glass ar e shown

diagrammatically in figures 3 and 4, with figure 3showing inflows an d figure 4 outflows. He re we are

concerned with the comparison of flows in versusflows out, for if they ar e not equal, the tem pera tur eof the walls and the room will not be constant at,or close to, the desired 100 F.

Cer tain caveats about these calculations ought tobe made clear. None of these heat flows can be saidto be precisely defined. Th e problem s in calculationare:

This is, according to Thatcher (supra n. 5) 182-83, I-' T Rook, "The Development and Opera tion of Romana quite possible temperature in Ostia in December orJan- Hypocausts," JAS 5 (1978) 269-82.uary, and in my own experience a low but not outlan dish li F. Kretzschmer, "Hypokausten, " SaalbJb 12 (1953) 7-41.

one. " ' A S H R A E Handbook (supra n. 9) ch. 11.


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19961 WINDOWS, BATHS, AND SOLAR ENERGY IN T H E R O M A N E M P IR E

radiation sun's beam radiation

Fig. 3. H ea t flows into ro om 4 with o pen windows (left) and glazed windows (right)

1) Th e dinlension s are not always precisely known.I use those given by Thatcher.

2) In some cases the materials are only guessesas the upper p art of the room has disappeared. Gen-erally I follow 'Thatcher's suggestions.

3) I assume, as does Thatcher, that t he vault is notheated.

4) Th e optical quality of Roman glass varies widelyand t hat used at Ostia in thes e baths is not known.A transmission o f 50% , assumed her e, is probablya quite conservative estimate.

conduction

t hough vault

5) I estimate the temperatures of the lower sus -p e n s u r a (floor) surEace an d the inner surfaces of thetubuli from Rook's and Kretzschmer's experiments.

6) The solar beam radiation is calculated herein th e same way as i t was by Thatche r, agreeing alsowith the met hod used by King an d Hamilton" totest the perfor mance of a solar classroom at Hamil-ton College at close to the same latitude as Ostia.

7) Convective flows are notoriously hard to cal-culate, but since such flows do occ ur in so lar houseswe can expect to achieve ord er of magnit ude res ults

Tconduction through vau lt

f lovs though glass

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

17 J.W. Ring and A. Hamilton, "The Solar Classroom at Research Ins tit ute TP-245-430 , Washingto n, D.C. 1979)Hamil ton College," Proceedings, Conference on Solar Heati ng 107-11.and Cooling Systems, Colorado Spri ngs , Colorado (Solar Energy


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7 2 2 JAMES W. RING [AJA 100

Table 1. Heat Flows for Room 4 with Glazed and

Unglazed Windows

Direction andSource of Flow

Units of100,000B'TUlhr

Into roomHypocaust (full heat)Sun (clear day at winter solstice)

Unglazed windowGlazed window

4 .O

1.30.7

TOTALSUnglazed windowGlazed window

5.34.7

Out of room (outside temperature = 30'ConductionNatural convection and radiation

Unglazed windowGlazed window

'TOTALSUnglazed windowGlazed window

Net Flow ( + = into room)Unglazed windowGlazed window

using th e solar designer's formulas.lx Natural rat her

than forced convection is assume d, i.e., there is nowind blowing. With wind the convective flow isgreater, and could be much greater at high-windvelocities.

Und er su ch circumstances we can expect only toestima te these flows. Even with these roug h estimates,however, some important conclusions can be

reached. The estimates for glazed and unglazed win-dows are given in table 1 above. The details of thecalculations follow in an appendix.

RESULTS AND DISCUSSION

Th e results of the calculations summar ized in table1 and in the appendix indicate that with an openwindow the input is - 530,000 ETUlhr while theoutflow is - 5,290,000 BTUlhr. In such a case it is ob-vious that equilibrium is not possible and that the100' F surfaces will rapidl y cool toward 30' F. Thesame would be true for a nude bather whose skintempera ture normally should be - 93' F. In de ed theoutflow would equil ibr ate with th e inflow only whenthe wall a nd floor surfaces are within l o 0 F o r lessof th e outsid e temper ature, i.e., at 54 O0 F.

O n the o the r hand, if the window is glazed theheat fro m t he hypocau st (400,000 B'TUlhr) would be

lX .D. Balcomb ed., Passive Solar Buildings (Cambridge,Mass. 1992) 149-52.

much more than adequate to provide the outward

flow (80,000 BTUlhr) in radiation, conduction, and

convection through the glazed window and in con-duction through the vault. Th e sun alone on sunnydays could provide most of the energy to maintain

100 F tempera tures. I ndee d, even with the fires Fe-

duced o n sunny days, there would probably b e som ethermal energy stored in the floors and walls thatwould maintain the temperature as the sun goesdown. O n days when t he s un is obscured by clouds,the hypocaust with a reduced fire, or being on onlypart of the time, could by itself easily maintain the

tempera ture even with the outside temp eratur e atits coldest point of the season, i.e., the design tem-

per atu re of 30 F.'The stor ed ther mal energy, which may com e from

either the sun or the hypocaust or both, can behandled quite easily by the heavy masonry walls,

vault, and floor of this room. With such surround-ings extra heat in the r oom c an pass readily by con -duction into the masonry without heating the airin the room excessively, i.e., much above 100 F. Atnight, when t he s un is down (even with the fire out),

this sto red heat will flow back into t he ro om to offsetthe cooling that inevitably will occur. Note thatwooden shutters closing the window area at nightwould en han ce this storage considerably. Such heatstorage is an import ant element in solar house de-sign and the Romans seem to have incorpo rated thiselement into their designs as well. 'The thickness of

the floor, or s uspens ura, is especially interesting inthis regard a nd it may be that the thickness chosen,

i.e., 15 in, is not necessary structurally but aids inlong-term heat retention.

Thu s, we see that Yegiil is corr ect in h is claim t hatthe Romans would have been wise to use glass intheir bath windows. Furthermore, we see that

Thatche r is probably overly enthusiastic in claimingsuch efficiency for radiant heating. Indeed, withthe windows open, the rooms in these baths couldnot have been maintained at 100' F and a nudebat her would soon have become very chilly. Tha tche rneglected to con sider fully the very great heat flowou t of the ro om d ue to th e convective flow throug hthe o pen window. He does not completely igno re thepossibility of an air current but claims that such acurr ent would only exist with n o wind, and normallyther e would b e so me wind. H e th en says, "A windpressu re of any but the lowest magnit ude would nu l-lify the action and set in motion the various air cur-ren ts alre ady described."""These curr ents , 'Thatcher

I q Thatcher (supra n. 5 ) 233-34.


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19961 WINDOWS, BATHS, AND SOLAR ENERGY IN THE ROMAN EMPIRE 723

believed, flowed from nor th t o sout h, or vice versa,

across the room; in the north wall they passedthrough the door, cracks around the door, or the

lunette at t he top of the vault, and in the s outh wall,

through the upp er part of the window.

This assertion seems incorrect because a wind

blowing in, or eddying through the s outhern win-dow, will not nullify t he convective effect. Rath er it

will cause a change from natura l convection to forced

convection, changing and distorting t he geometry

of the convective loop and, as a result, increasingthe mixing of cold and hot a ir an d thus increasing the

heat loss above that caused by natural convectionalone. The net result would be to set the 52.5 x 10'BTLJlhr heat loss calculated above as a minimum

value and in windy situa tions to expect this loss to

be even greater with the concomitant effect of an

e\.en faster lowering of th e bath temperat ure, mor e

quickly chilling the nude bather.Note that the natural convective flow calculated

here is only a rough estimate. But it is abo ut 10 times

th e inflow so tha t even if it is overestimated by a fac-

tor of two, it still will be many times gre ater t han the

inflow. With wind, it will be even greater than cal.

culated here.

Finally, to return to Yegiil's point, the estimates

of heat flows here show not only that nud e bathing

in Roman baths would not have been possible with-

out glazing but also that with glazing during sunny

days, the sun with only a little help from the hypo-

caust and its furnaces, and hence li t t le wood burned,could have maintained the temperature of these

room surfaces at - 100 F. Furthermore , on cloudydays the hypocaust with only a low or intermittentfire would have been able to sustain this tempera-

ture. And even at night the large thermal storage

capacity would have kept temperatures from drop-

ping very fast so that by the next morni ng the amou nt

of heat necessarv to return to - 100' F mieht have"been relatively small. Thus with a normal mix of

sunny days, a considerable savings of fuel could b e

accomplis hed even in the de pth s of winter. At othe r

seasons even more savings could be expected. Thesun would therefore provide a substantial part ofthe hea t requi red. This result , of course, is in accord

with Jord an and Perlin's observations about the in-

creasing cost of fuel during this period of rapid

growth of Roman industry, commerce , manufacture,and population. Fuel costs would have provided astro ng incentive for using glaz.ed windows a nd th esun's energy.

In summary, the Romans apparent ly did displayconsiderabl e know-how in the design of their baths

when judged by the standards a nd practices of mod-

ern science and technology 2,000 years later.

Thatcher, i t seems, was too sanguine about radiant

heat ing but, nevertheless, the Romans deserve high

praise f or their use of solar energy. Even Seneca, no

admirer of conspicuous consumpt ion a nd easy liv-

ing, might have admi red t he frugality that th e com-

bin ati on of "the widest of windows" a nd glass panes

in baths demonstrated.

A p p e n d i x

Ca lcu la t ions o f He a t F lows fo r R oom 4wi th an d wi thou t G laz ing

A. Heat flows into th e room:

(i) The he at flow thro ugh the sus pensura: this is a heatconduction problem where Q , the he at flow per hour,is given by:

Q = - kAAT with k the therm al conductivity of theA x concrete slab, Ax the thickness of the

slab, A the cross-sectional area , and ATthe temperature difference betweenthe top and bottom of the slab.

Using British engineering units withk = 11 BTU-inlF-ft2-hr value used by Thatcher)A = 1,200 ft2AT = 400 - 100 = 300 F (using maximum hot

gas temperature under suspensura)Ax = 15 in

Q = 2.6 x lo 5 BTUlhr

(ii) The heat flow through the heated walls:k = 7.0 BTU-inlF-ft2-hr value used by Thatcher)

A=

1,760 ft2 (3 vertical walls)AT estimated to be 200 - 100 = 100 FAx = 9 in

Q = 1.4 x 10QTUlhr

(iii) The sun's radiant energy in through the window:In December in Rome, the sun a t noon is only about

24O above the ho rizon a nd the sun's beam intensity isapproximat ely 250 BTUlhrlft2. Note that comp aredwith 228 at the winter solstice, he noon value on a south-facing vertical surface at equinox would be 285 x cos4Z0, o r 212 BTUlhrfft2 a t this la ti tude of 42' N. At thesummer solstice this value would be 116. At noon, re-gardless of the season, the sun's beam radiatio n o n thissurface would always be a t it s maximum for the day.Note also that these intensity values agree with thoseused bv Thatcher:

A = 560 ft2Q = 250 A cos 24' = 1.3 x lo 5 BTUlhr

or with Roman glass:Q 0.65 x 105 BTUlhr

Thus, the total energy input 5.3 x lOQTUlhr, i.e.,the sum of the above items, or with glass a 4.65x lo 5 BTUlhr.

B. Hea t flows ou t of the room:

(iv) Conduction through vault to the outside:

k = 7.0 BTU-inlF-ft2-hr, s in the walls


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724 J.W. RING, WINDOWS, BATHS, AND SOLAR ENERGY IN THE ROMAN EMPIRE

A = 1,880 ft'AT = 100-30 = 7 0 FAx = 24 in

Q = 3.8 x 104 BTUlhr

(v) Th e natura l convective flow thro ugh the openwindow:

Here the convective loop will have hot air exitingthrou gh t he top of this large window and cold air corn-ing in at the bottom. See figure 4. This flow will be causedby th e stack effect in which hot, less dense gas is forcedout of the room at the top of the window and, by thesame effect, cold ai r that is more dens e will flow in atthe bottom. Solar house designers use t he for mula givenbelow to calculate this air flow.2')

CFM = 9 . 4 ~ ~ ~ -where A,* is the effective are a (ft2) hrough which

the flow enters andlor leaves the space, and-H is theeffective height of this area. In t his case the AeK s halfthe area of the window while

His half the height of

the window as half t he window is used fo r outward flowand the other half for inward flow, or A,E 260 ft2,H 10 ft (mean height), a nd AT = 100 - 30 = 70' Fand:

CFM = 7.0 x 104 fGlminThen the heat transferred will be:Q = CF M x 60 x AT x 0.018where 0.018 is the volumetric specific heat of a ir inBTUlfts at these temperatures.

Q = 5.2 x 10VTUlhrNote that another equation for this convection heat

transfer mentioned by Balcomb is that of Weber andK e a r n e ~ , ~ ' ho arrived at it by simil itude modeling an dfull-scale testing:

Q = 4.6 W(dAT)"I2 where W is width a nd d is heightof opening, which when conve rted to t he vari-ables used above becomes

Q = 13 A- (AT)""

Balcomb (supra n. 18) 149-52.21 D.D. Weber and R.J. Kearny "Natural Convection

Heat Transfer through an Aperture in Passive Solar HeatedBuildings," Proceedings, 5th N ational Passive Solar Conference,

Using A = 280 ff', H = 10 ft, and AT = 70 F asabove:

Q = 6.7 x lo6 FKUlhr This is 25% mor e t han the es timate above. I use

the smaller number in table 1.

(vi) The radiant heat flow through the window:He re the Stefan-Boltzman law governs the h eat flow(the same equation used by Thatcher).

Q = oAE(T14 - Tz4)with o = 1,730 x lo -" BTUIhr-ft'.(OF a b ~ o l u t e ) ~

A = area of body in question (the window in thiscase)

E = 0.9, a factor ac count ing for emissivities an d sol-id angles subt ended by the hot and cold bodiesas seen through the window. Again this is thevalue used by Thatcher.

T I and T2 are 560' F and 490' F absolute, respec-tively, for t he hot inne r surfaces at 100 F andoutside surfaces (o r air) at 30 F.

Q = 3.5 x 104 BTUlhrIf the window is glazed, a combination of convection,

radiation, an d conduction across boundary layers of airboth o n t he inside a nd outside as well as th e glass itselfcan be treated according to the following e q ~ a t i o n : ' ~

Q = UAATwhere U = 1.10 BTUlhr-ft2-OF, which includes all

three types of heat flowA = area in ft2 of window = 560AT = 70' F

Q = 4.3 x lo 4 BTUlhr

DEPARTMENT OF PHYSICS

HAMILTON COLLEGE

CLINTON, NEW YORK 13323

[email protected]

Amherst, Massachusetts, October 1980 (American Section ofthe International Solar Energy Society, Newark, Del. 1980)1037-41.

22ASHRAE Handbook (supra n. 9) ch. 11.
mailto:[email protected]:[email protected]

Windows, Baths, And Solar Energy in the Roman Empire

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