Modelling the thermal pattern of a statue or column exposed to periodic lighting conditions

IL NUOVO CIMENTO VOL. 11 C, N. 5-6 Settembre-Dicembre 1988

Modelling the Thermal Pattern of a Statue or Column

Exposed to Periodic Lighting Conditions (*).

A. BERNARDI and D. CAMUFFO

C N R I C T R - Corso Stat i Unit i 4, 1-35020 Padova

S. VINCENZI

C N R I S D G M - S. Polo 1364, 1-30125 Venezia

(ricevuto il 13 Ottobre 1987)

Summary. - - In the case of monuments, the thermal effects due to periodic lighting conditions in the long run may be very dangerous. In this article the thermal response of a column or a statue cylindrically shaped and subjected to periodic lighting conditions is studied. Two sources are considered: in the first the intensity is time dependent according to a sinusoid (i.e. diffuse solar radiation from a skylight or window or partially closed outdoor environments); in the second according to a square-wave law (i.e. spot-light switched on and off). The model shows some characteristic features as a function of a dimensionless parameter ~, expressed as the ratio between the radius of the cylinder and its penetration depth, which takes into account the thermal properties of the substance and the lighting period. For high values of ~, the amplitude is poorly dependent on azimuth. The shadowed part is practically insensitive to the azimuth variation. For small values of ~ the amplitudes of the thermal wave are very large, causing enhanced thermo- hygrometric cycles on the bodies. The phase lags between the different point depths increase with ~. The phase lags behave differently in three zones of the cylinder: i) the part facing the light, ii) the two transition zones, iii) the shadowed part. For small lighting durations and small ~ the switching on or off of artificial light causes sudden and dangerous variations of temperature. Some hours after the switching off, all the surface of a cylinder with small reaches a homogeneous temperature. The mathematical analysis of the thermal behaviour of a cylinder shows how a physical simulation can be made, by changing radius, substance and lighting period, separately or together.

PACS 92.60 - Meteorology.

(*) Paper presented at the IV Congresso del Gruppo Nazionale per la Fisica dell'Atmo- sfera e dell'Oceano, June 22-24, 1987, Rome.

31 - l l Nuovo Cimento C. 453

454 A. BERNARDI, D. CAMUFFO and s. VINCENZI

1. - I n t r o d u c t i o n .

In the case of precious statues or ancient columns the thermal effects due to periodic lighting conditions in the long run may be very dangerous. In fact, they result in mechanical stresses, evaporation condensation cycles in the micropores and thermal and hygrometric gradients in the surrounding atmosphere which are responsible for phoretic processes associated with the deposition of airborne pollutants (1-5). To understand these processes it is very important to know the thermal properties of the surface and of the inner layers. In works of art it is pratically impossible to measure the behaviour of the parameters inside, unless destructive methods are used. Therefore, models of the heat transfer are the necessary input for additional models of the above-mentioned induced effects. The problem becomes more and more complex as a greater number of conditions regarding the source (i.e. movement and variable intensity) or the body (i.e. shape and composition) are taken into account. A step-by-step procedure has been, therefore, considered.

In previous papers (8,7) we considered simple geometries representative of real cases, i.e. half-spaces, cylinders, spheres. Experimental campaings of building facades (San Francesco's Basilica in Assisi, Orvieto Cathedral, Scrovegni's Chapel in Padua), columns (Trajan and Aurelian in Rome), and statues (San Marco's horses in Venice) gave information on the surface behaviour and only partially (i.e. when possible) on the inner layers. The complete variability of the solar input has been possible only for flat monuments. For different geometries, i.e. cylinder and sphere, the first step was to describe a steady-state lighting corresponding to exhibits in a show room after the transient. The second step, described in this article, is to consider the complete behaviour under periodic lighting conditions. In this step we model the effects of two fixed sources: the first has an intensity time dependent according to a sinusoid (i.e. diffuse solar radiation from a skylight or window or partially closed outdoor environments); the second according to a square-wave law (i.e. spot-light switched on and off). Outdoor monuments heated by solar radiation constitute the more interesting but the more complicated case, due to the variability of the solar input. The next step will be to model an outdoor column or statue exposed to sunlight.

In this paper the case of a cylinder has been considered by using a

(1) D. CAMUFFO: Atmos. Environ., 15, 1767 (1981). (2) D. CAMUFFO: Atmos. Environ., 17, 1803 (1983). (8) D. CAMUFFO: Acidification and its Policy Implications, edited by T. SCHNEIDER (Elsevier, Amsterdam, 1986), p. 189. (4) D. CAMUFFO and A. BERNARDI: Fattori microclimatici e conservazione dei beni artistici (Edizioni del Laboratorio, Brescia, 1985). (5) D. CAMUFFO and S. VINCENZI: Sci. Total Environ., 44, 147 (1985). (~) D. CAMUFFO, S. VINCENZI and L. PILAN: Water, Air, Soil Pollut., 23, 441 (1984). (7) D. CAMUFFO and S. VINCENZI: Sci. Total Environ., 53, 179 (1986).

MODELLING THE THERMAL PATTERN OF A STATUE OR COLUMN EXPOSED ETC. 455

mathematical treatement based on the Bessel functions. An extension to the case of a sphere has been already considered for stationary conditions in a previous paper (7) by using the Legendre polynomials. Similarly, the following discussion could be extended to the case of spheres with variable input. Domes characterized by a thin conductive layer of metal (i.e. lead or copper) over an insulating support (i.e. wood or brickwork) require a simpler mathematical treatment, as the radial component of the flux is negligible.

2. - Mathemat i ca l analys is and discussion.

The equation of the heat transfer was solved under the hypotheses that: i) the outgoing flux is constant and ii) the temperature is dependent only on the radial and azimuth distance from the axis of the cylinder, i .e. it is insensitive to disturbances due to the truncation of the cylinder. A more general treatment with outgoing flux dependent on the surface temperature can be made in a first approximation by dividing the flux into two parts: the first constant and the second proportional to the temperature. However, the temperature dependence causes only small departures (6) but a more complex mathematical treatment. The latter condition nearly holds for small radius-to-height ratios or when the cylinder ends with a hemisphere (7).

The solution are presented in terms of the adimensional temperature function Y(~, z, t; ~), dependent on: the azimuth ~ computed starting from the direction of the incoming radiation; the adimensional radial distance z espressed in terms of radius unit, i .e. z = 1 on the surface and z = 0 on the axis, the time t, the adimensional ratio ~ between tha radius a of the cylinder and the penetration depth. For each material and lighting period, the parameter ~ is related to the radius of the cylinder; for each radius and lighting conditions, it is representative of the substance. The actual temperature T is obtained by multiplying Y by the characteristic temperature T*, defmed as follows:

(1) T * = C-Sa/k ,

where C is the absorption coefficient, S the mean value of the incoming flux, k the thermal conductivity.

As the minimum of a periodic lighting is physically nonnegative, from the mathematical point of view, the incoming light is expressed in terms of a variable component plus a stationary component which, when added, transforms the negative minimum into a zero of lighting.

In the case of illumination from a skylight, the source is expressed by a sinusoidal wave plus a stationary component and the solution Y is Y = Y ' + Y" ,

where Y' is the oscillating component and Y" the stationary one. As Y " has been already analysed in a previous paper (7) we will discuss here only the oscillating

456 A. BERNARDI, D. CAMUFFO and S. VINCENZI

0.6

y/

- - 0 . 6

~ ~ 1 8 0 ~ 180 ~

I I I I I -12 - 9 - 6 --3 0 3 6 9 12

t

0.10 yt 0.0~

-0 .0 !

-0 .10

~= 90 ~ \ \ I I 1 I I I L I ~ I I I

t I l I I I --12 - 9 --6 --3 0 3 6 1'2

t

Fig. 1. - Time variation of the oscillating component Y' of the adimensional temperature , computed at the surface (z = 1), for the different azimuth ~: a) ~ = 2; b) ~ = 10.

1.2

A

0.8

0.4

# = 0 . 8

#=1

~ #=2 I I I l ~ l J I ' I i l

30 60 90 120 150 180 ~(c legrees)

Fig. 2. - Ampli tude A of the adimensional tempera ture Y against the azimuth ~, computed at the surface (z = 1) for different values of t,.


Y'. The time variation of Y' on the surface ( i . e . z = 1) is plotted against t in fig. 1. The time evolution of the surface temperature at different azimuths varies sinusoidally and their amplitude decreases for increasing azimuths. We did note that the amplitudes could be only slightly reduced if an outgoing flux dependent on the surface temperature is considered (6) despite our first hypothesis.

For poorly diffusive media or cylinders with large radius, i . e . with large ~, the amplitudes and azimuthal dependence are dramatically reduced. In the case of greater than 3, the part of the cylinder with ~ greater than 90 ~ is practically insensitive to the forcing. This can be easily observed in fig. 2. In the same figure one can observe that, in any case, the amplitude is practically independent of the azimuth in the shadowed part of the cylinder ( i . e . I~1 > 900) �9 However, the ratio between the amplitudes of the temperatures at ~ = 0 ~ and ~ = 180 ~ is close to unity for small ~ and becomes greater and greater for large values of ~. It can be observed that the small azimuthal dependence of the shadowed part makes it behave similarly to a flat surface, whose temperature depends only on the depth.

The phase lag of the thermal waves v s . the radius a for r = 0 is shown in fig. 3 for some values of ~. The curves are quasi straight lines and their slope is an increasing function of ~. For small values of t~ the thermal waves at different depths tend to be in phase between each other.

A~

24

18

12

6

- - ] ~ = 5

- - / 1 = 3

~=2

i i i i I [ i I i 0.5 z 0

Fig. 3. - Phase lag At of the thermal wave when it penetrates inside the cylinder, for different values of ~. The unit in ordinate is 1/2 of the period.

The phase lag v s . the azimuth is shown in fig. 4 for some values of~. For small values of t~, the phase lags are slightly increasing functions of the azimuth. The greater ~ is, the lesser are the phase lags on the lighted part of the cylinder. On the contrary, they are greater on the shaded part. This means that the thermal surface pattern of poorly diffusive cylinder can be roughly divided into three parts: i) the lighted part where the temperature waves are quite syncronous, i i )

a sudden transition zone, i i i ) the shaded part where the thermal pattern is homogeneous and time independent.


24 _ /J = 10/ / / J = 5 A t _

18_ /~=3

- / z = 2

i

6

I I I I I 0 60 120 180

(p (oLegrees)

Fig. 4. - Phase lag At of the thermal wave on the surface against the azimuth ~, for different values of tL. The unit in ordinate is 1/24 of the period.

In fig. 5 the solution Y' on the surface is computed for some rounded values of t (expressed in unit of 1/24 of the period) and ~ = 2. The largest amplitudes are in the zone facing the light source; the shaded zone is quite insensitive to the azimuth. F o r t = 0 the radiant energy is at maximum. At t = 0 and t = + 6 and similarly at t = + 12 and t = +18 = - 6 the surface t empera tu res in the zone facing the light source are respect ively equal, but not in the other par ts of the cylinder. Note tha t in the par t of the cylinder facing source, the curves at t = + 3 and t = - 9, are close to the maximum and minimum of surface tempera ture . Among the repor ted curves, a double oscillation with two minima is well distinguishable for t = - 3.

0.6

yr

--0.6

t=3

I I I I I I I I --180 --135 --90 --45 0 45 90 135 180

~(degrees)

Fig. 5. - Adimensional temperature Y' on the surface plotted against the azimuth ~ at different times t expressed in 1/24 of the period, z = 1.


4

Y

--2

~=0~ t ~__

I

~o =180 ~

I I I I I I I I --12 --9 - 6 --3 0 3 6 9 12 t

~ . = 1 8 0 o ~ - - - - - 9 0 o

--2 I I I I I I I I -12 -9 - 6 --3 0 3 6 9 12

t

14

I ~ 1 I I I

0 o

~=180 ~

-I~ i i l i i i i I -12 -9 - 6 3 0 3 6 9 12 t

Fig. 6. - Surface the rmal w a v e Y repor t ed for some azimuths against the t ime t, expressed in 1/24 of the period: a) l ight ing duration: 1/6 of the period and ~ = 1; b) as in a) bu t ~ = 4; c) l ight ing duration: 1/2 period and ~ = 1, z = 1.

460 A. BERNARDI~ D. CAMUFFO and S. VINCENZI

The mathematical analysis shows (see appendix) that points at equal distance from the axis and aligned with the incoming beam have temperature differences which can be expressed in a relatively simple form having a sinusoidal time dependence, with amplitude and phase depending only on t~. Therefore, for each internal cylindrical layer the mean flux between each couple of points aligned with the incoming beam is equal to the tangential flux at ~ = 90 ~

In the case of artificial lighting, especially for direct spot-light, the forcing is a square wave, mathematically expressed as a Fourier expansion (see appendix). The overall thermal wave Y has been computed for two cases: lighting duration LD = 4h and LD = 12h per day.

In fig. 6 the surface thermal wave Y ( t ) is reported for some azimuths and = 1 and ~ = 4. Y ( t ) refers to the average value of the axis. In fig. 6a) (LD = 4h, = 1), for all azimuths, the temperature is sensitive to the forcing input. The

switching on causes a sudden warming on the lighted part: the shadowed one follows with some delay. The switching off causes a sudden drop in temperature in the lighted part; on the shadowed part the rise in temperature continues again and the cooling occurs gradually with some delay. We note that the limit between the lighted and shaded zones, i .e . ~ = 90 ~ behaves closely to the average value of the temperature of the whole cylinder. A few hours after the switch off, all the curves are pratically undistinguishable showing a quite homogeneous temperature on the whole surface. In fig. 6b) (LD--4h, ~ = 4) the increase of t~ causes a reduction of the amplitude and sensibility to the forcing. In fact, for

= 90 ~ the time evolution is very small. For most of the period, the surface temperature is not homogeneous. For a longer external forcing (see fig. 6c) with LD = 12h and t~ = 1) all surface temperatures at various azimuths are distinct, except at the very end of the cooling period. The curve of the warming period is symmetrical with reference to the cooling period. The temperature wave at

= 180 ~ is close to sinusoid. The temperature dependence on ~ is reported in fig. 7a) for LD = 4, z = 1 and

= 0. The attenuation of the amplitude is greater for increasing values of ~. For > 1, some hours after the switch off the surface temperature seems to be

pratically independent of~. On the axis of the cylinder (fig. 7b)) very conductive media or thin cylinders behave like straight lines, similarly to the mean temperature of the body. The increse of ~ causes a more smoothed curve. For

> 3 the curves are barely distinguishable in the graph. From the physical point of view, it appears that the thermal behaviour of a

cylinder can be simulated by means of other cylinders with different radius a, or substance, or lighted with different period P, if one or two of these parameters is chosen accordingly. In fact, t~ is expressed by

( 2 ) , = a

where p is the density and c the specific heat, so that a suitable choice of these


9.~

4.0

--4.0

-9 .4

a)

-12 - 9 - 6 -3 o

!% I I I I 3 6 9 12

t

--3

--6

b)

i I I I --12 --9 --6 --3 0

I I I I 3 6 9 12

t

Fig. 7. - Thermal wave Y plotted against the time t (expressed in 1/24 of the period) for different ~'s. Lighting duration: 1/6 of the period: a) on surface, i.e. z = 1; b) at the centre, i.e. z=O.

parameters leaving t* unchanged is related to the same thermal behaviour of the cylinder. That is, in two cylinders the temperature departures AT (referred to the mean temperature of the axis of the cylinders) of corresponding points and at corresponding times are proportional, i.e.

(3) AT1/AT2 = T~/ T ~ = C1 S1 al k2/C~S--~ a~ k l .

This can be easily shown in fig. 8 where the relationship between the radius a

462 A. BERNARDI, D. CAMUFFO and S. VINCENZI

u

160

140

p = 5

=2

p= l

120

100:

80

60

4(

�9 I I i I

Cu(24)

p=0.5

I I I

I

Ag(24)

Fe(24.) B(6),

I

iAg(6) I i

I B(24) I

Cu(6)

I "~ ' I i I I I i I I I I J I J I 0 20 4.0 60 80 100 120 14.0 160

6(cm)

Fig. 8. - Relationship between the radius a and the penetration depth ~ for some values of~.

and penetration depth ~ is plotted for some values of ~. The straight lines correspond to physical situations characterized by the same value of t~. The points marked correspond to different substances and three different periods. A vertical displacement shows the different thermal behaviour when only the radius varies. A horizontal displacement shows the different thermal behaviour when the substance or the period or both are changed.

3 . - C o n c l u s i o n s .

The mathematical model of the thermal wave inside a cylinder exposed to periodic lighting can be used as an input to compute the mechanical stresses as well as its thermo-hygrometric behaviour. The model of the surface temperature pattern and evolution is used to compute the phoretic deposition of airborne pollutants. An indepth knowledge of these processes is very important in the


science of conservation, especially for the case of delicate exhibits lighted in show rooms with spot-light or sky-light or outdoor statues in niches or other partially closed environments.

The model has shown some characteristic features, as summarized below. It has been seen that, for high values of~, the amplitude depends very poorly

on the azimuth. In particular, the shadowed part is pratically insensitive to the azimuth variation.

For small values of ~ the amplitudes of the thermal wave are very large, causing enhanced thermo-hygrometric cycles on the bodies.

The phase lags between points at different depths increase with ~. The phase lags behave differently in three zones of the cylinder: i) the part facing the light, ii) the two transition zones, iii) the shadowed part.

For short lighting durations and small ~ the switching on or off of artificial light causes sudden and harmful temperature variations. Some hours after the switching off, the whole cylinder surface with small ~ reaches a homogeneous temperature.

The mathematical analysis ofthe thermal behaviour of a cylinder shows how a physical simulation can be made, by changing radius, substance and lighting period, separately or together.

This study has been supported by the Commission of the European Communities. Research and Development Programme in the Field of Environmental Protection. Contract CEE/CNR EV4V-0051-I (A).

A P P E N D I X

The periodic thermal wave in a cylinder is obtained as a solution of the equation

(A.1) aT 1/ .a2T+ 1 aT . 1 a2T~ at =~

where the symbols are explained in the text. This equations is considered under the boundary conditions:

_K;ar] = (A.2) ~Sr]r=a -CS(t) cos+ ~+B+y(Ts(~,t)-'T~),

where the intensity of the incoming light S(0 is defined by

(A.3)


and the overbar represents the mean value and the period P is 2=/oJ and

(A,4) cos+ 9 = 1 1 ~ cos (nrd2) ~ A~ cos ng, ~ + - ~ c o s g + ~ ~ n----~Z i cosng= n~2 n=O

where cos +9=cos9 if cosg>0; cos9 § if cosg<0; B is a constant representing the mean_ value of the flux leaving the cylinder, averaged both in time and azimuth. T~ is the mean value of the surface temperature Ts(9, t) and y is a constant.

The t e m p e r a t u r e c a n be referred it to its mean value, by putting O(r, 9, t) = T (r, 9, t) - T~ , which still fulfils eq. (1). Then, by imposing that energy balance must be zero for each period, boundary condition (2) becomes

(A.5) ~rr],=, = CS ~ A , cos n9 +

+ CS ~ z~ exp [im~t] ~ A , cos n9 - y0~(9, t). m = l ~ 1

By expressing the solution of eq. (1) O(r, 9, t) in terms of two components, stationary and time-dependent, i.e. O(r, 9, t) = O"(r, 9) + O'(r, 9, t), the boundary condition too can be expressed in two parts, namely

(A.6)

[ K[~r = CS ~ A~ cosn9 - y0"'s[9~," I \~r]r=a n=l

= CS ~ zm exp [im~t] ~ A~ cos n 9 - y 0~ (9). m = l n=o

If we introduce the adimensional parameter q = y~/k already discussed in (~), the stationary 0" becomes

0 ,( zcos ( l + q ~ ~o2 n 2 - n + q ] '

n = 2k with k integer. Let us assume for 0' the following expression:

(A.8) O' -- ~ ~ Dm~ R,~(r) exp [im~t] cos ng, m = l n=O

where D ~ are constants to be determined and Rm~(r) satisfies the Bessel equation of first kind

(A.9) d 2 R + 1 dR i + R = 0 , dy~ Ym dym

where y~ -- (r ~-m)/~. This equation is satisfied by the J~( y~ exp [i 3=/4]), which is nondivergent at


the origin and whose real and imaginary parts are, respectively, expressed by the Kelvin functions BerJym) and Bein(y~), so that

(A.10) R~.(r) = Be rn(Ym) + i Be i j y,~),

which can be expressed in terms of the adimensional variables z and ~z Vr~ instead of r and y~, respectively.

The constants Dm~ are complex and are determined by means of the conditions (6) as a function of the derivatives Ber'(ym) and Bei'~(ym) at z= 1.

If we put

(A.11) I Mp(y~) = [ Be r~ (y~) + Be i~ ( y~)]~J2,

l ~b~(ym) = tg -~ [Be in( ym)/Be rn(Ym)],

we obtain

N J ym) =

Ber,2(~ ~ + B e i , 2 ~ o ~1/2 [ n ~ Y m ) n ~ Y m ) J ,

~'(Ym) = = tg -~ [Be i'( ym)/Be r'( y~)],

(A. 12) o' = T___~* ~ ~ Am am M,(~z) exp [i (mo)t + ~(~z) - ~* (~z)] n:O Yn( )

where I = tz V ~ and

(A. 13)

(q)2 M~n (~)+ 2q

~* (~) = tg -~ t g G (a)

MJ~) cos (~b" (i) - ~ (~)),/2 Nn(~)

q Mn(~) sinG(~)

;( N J;() sin ~" (~)

q MJ;() cos~bn(~) 1+

2 Nn(2) COS ~b n (2) ]

When q <<t~, then

(A. 14) Sn = 1, * ' ~n (~) = ~ (~).

In these conditions the functions 0' and r assume a very simple form, as described in the text.

When the input S (t) is described by a square wave, in the general formula (3) we substitute S = So ~ LD/2n and am = 2 sin (m~o LD/2)/(m LD/2), where So is intensity of the incoming light and LD the lighting duration.

When the incoming intensity is sinusoidal, a physical representation of S (t) may be written in the form

(A. 15) S (t) = S (1 + exp [i~ot])

obtained from eq. (A.3) with zl = 1 and Zm = 0 for m > 1. We note that, in order


to avoid negative lighting, o- 1 must not be g rea te r than 1. In this case, the adimensional solution described in the t ex t Y = Y ' + Y" becomes

(A.16) Y(z,~,t)= ~ Anz~cosn~+ l ~ M,(~z) n=l n ~z ~=o An ~ cos(~ot- (~'(/z)-~bn(/ZZ)),

when q = 0. In these conditions the amplitude G (z, ~) and phase 5t(~, z) become, respect ively

G

(A.17)

1 f/| A Mn(t~z) )2 = - ~ ~ - - sinA~n(~z) + ~ l/ .=0 N~(tz) cosn~

/ ~ M ~ ( t ~ z ) )2}1 + (~oA~ ~ cosn~ cos 5~bn(~z) ,

M.(~z)

At(~, Z) = tg -1 ~ ~~ cosn~ sin A~n(~z)

. M~(t~z) .ooEA cosn

where hr = ff~ (~) -- ~bn(~z).

�9 R I A S S U N T O

Viene modellata la propagazione dell'onda termica in una colonna o in una statua, geometricamente ide:~tificabili con un cilindro, nel caso di illuminazione da luce periodica. Sono state considerate due tipi di sorgenti: la prima con un'intensit~ variabile sinusoidal- mente nel tempo (che pub rappresentare un'illuminazione dovuta a luce solare diffusa penetrante, ad esempio, da un lucernaio o da finestre parzialmente chiuse); la seconda con intensiti~ variabile come un'onda quadra (rappresentante una luce artificiale periodi- camente accesa e spenta). I1 modello ha identificato alcuni andamenti caratteristici, espressi in funzione di un parametro adimensionale tL come rapporto t r a i l raggio del cilindro e la profondit~ di attenuazione, la qual.e congloba tutte le caratteristiche termiche della sostanza e del periodo di illuminazione. E stato osservato che per alti valori di ~ la dipendenza delrampiezza dell'azimut diviene molto scarsa. Viceversa, per piccoli valori di t~ le ampiezze delronda termica divengono piuttosto marcate, causando pericolosi cicli termoigrometrici nelle opere d'arte. Gli sfasamenti tra punti a differenti profondit~ crescono con il crescere di ~. In superficie gli sfasamenti presentano una diversa dipendenza azimutale, caratterizzando tre zone: i) la parte direttamente affacciata alla luce; ii) le due zone transizionali; iii) la parte in ombra. Si ~ osservato che per piccoli periodi di illuminazione e per piccoli ~ continue accensioni e spegnimenti diluce artificiale causano improvvise e dannose variazioni di temperatura; sono inoltre necessarie alcune ore affinch~ la superficie raggiunga una temperatura omogenea. L'analisi matematica del comportamento termico di un cilindro fornisce gli elementi necessari per poter eseguire una simulazione in laboratorio, potendo opzionalmente cambiare alcuni parametri anzich~ altri, tra cui raggio del cilindro, tipo di materiale e periodo di illuminazione.

t Pe3lOMe He rIO.rlyqeHo.

Modelling the thermal pattern of a statue or column exposed to periodic lighting conditions

Documents

Transcript of Modelling the thermal pattern of a statue or column exposed to periodic lighting conditions