Time-dependence of the Room Temperature during
intermittent Heating for various Types of Wall
(Dipendenza daI tempo della temperatura di stanza durante riscaldamento
intermittente per diversi tipi di parete)
S. FAGGIANI, P. GALEIATI, G. TUONI
Istituto di Fisica Generale ed Applicata, Università di Pisa, Italia_
Abstract - The influence of intermittent heating on room temperature is
studied for three types of wall and several values of internaI heat
capacity; also the effect of the solar radiation is taken into account.
Moreover, the problem of energy saving is analyzed in dependence of the
heating power and the type of wall.
Sommario - Si studia l'influenza congiunta deI riscaldamento intermitte~
te e dell'irraggiamento solare sulla temperatura di stanza, per diversi
tipi di parete e valori della capacità termica interna. Viene, inoltre,
analizzato il problema deI risparmio energetico in funzione delle caratteristiche termiche della parete, e della potenza riscaldante.
1.
INTRODUCTION
The problem of thermal performance of walls was widely investigated in the
last twenty years. The influence of the room thermal capacity assumed as uniformly distributed was pointed out by ~agrini [1) for the case of periodic st~
te, single-layer opaque wall and sinusoidal externaI temperature variations;
also the air changes in the room and the transient state were then considered
[2,3). The extension of the above problems to the case of multi-layer walls is
due to Warsi and Choudhury [4J, Colurnba and Lo Giudice [5, 6), Boffa, Ferro and
Sacchi [7 ,8J. Particular attention to the influence of the solar radiation and
to the responce-time in the transient state was paid by Faggiani and Fantini [9 J
and by Sovrano and Zorzini [lO,llJ. The transfer function method, firstly introduced by ~titalas and Stephenson [12,13J, has been recently employed for the
typical italian buildings [14] in relation to CNR's Progetto Finalizzato Energetica. Also a nurnber of experimental works were made about these problems
[15- l7J.
Today the increasing cost of fuel andthe consequent necessity of a more
limited time of heating in buildings suggest to extend these studies, in order
to analyze the thermal behaviour of the couple wall-room subjected to solar r adiation and intermittent heating. Therefore, purpose of the present work is to
investigate the time-dependence of the room temperature in relation to three
types of wall and three expositions during intermittent heating. Specifically,
the considered walls are shown in Fig. 1: the first (light) is formed by two
thin layers of asbestos-cement and one of polystyrene foam; the second (medium)
by light-weight bricks, polystyrene foam and plaster; the third (heavy) by two
layers of common bricks and one of polystyrene foam. The three walls are characteri zed .bythe same value of the thermal transmittance (Table 1):
( 2- + 2.t + ~ + ~ + 2- ) -1 = 0.47 W/mZ °c
h.
À1
À2
À3
h,
1227
.R- asbestos
~
polystyrene
loam
cement
I
.....",plaster
~*== polystyrene
loam
.,
polystyrene
.lII:
toam
()
x
o
Fig. 1 - Light , medium and heavy walls.
The influence of the solar radiation is taken into account by means of the
well - known sol-air temperature, te' "hich is shm'il1 in Fig. 2 for the averaged
Thicknesses
~Ia terial
.À
Q'
Light "a11
~'Iedium
Asbestos-cement
0.63
0 . 48
Common bricks
0.70
0.48
Li ght-we íght bricks
0.62
0.48
12
PIaster
0.70
0 . 44
2.0
Po l ystyrene foam
0 .031
1. 25
--
wa11
Heavy "all
0.60
12
6 .0
5.5
5.0
6
Table 1 - Thermal conductivity, W/m 2°C; thermal diffusivity x 10 , m2/s;
thicknesses of the Iayers x 100, m.
1228
condi tions of Pisa and for East, Sout h and West expositions. Concerning the
ratio be t\~en adsor bed and i ncident radiat ion, it has been fixed as 0.85.
32
'c
~
24
/ '5(""I
lL\c/1= ~~est
cp'vj"f"'",
. . . *\
,j
18
East,,;
8
o
O
/
"
/.
i!.?V
12
8
4
-
V
20
18
24
day hour
Fig. 2 - Sol-air t emper ature versus time fo r East, South and lVest e:>:posi
tions; January .
EQUATIONS
Under t he hypot hesis of cons t ant properties and perfect thermal
between the l ayers , t he governing equations are :
2.
J2 t,
x2
( 1)
~= a ,
a,
( 2)
(--ª--!.L)
=~
d X x=o
A,
(3 )
t,(x=x, .,)= t 2 ( X = X ,
(4 )
( d t, )
d X x=x,
(5)
at 2
~-
fr
contac~
a
2
=
l
d
[t, (x= o "
h
A,
) - te (
• T
, )]
)
( dt2)
d X X =x,
t2
x2
1229
(6 )
t 2 (X=X,+ X 2 ,T) = t 3 ( X=X ,+ X 2 , T)
(8 )
J t3 )
(;J-x-
(10)
t , (X
x=x,+X 2+X =
3
,T= O) =
E --º-1L
d
t
-À
T
3
h i . [t i (T ) .,1.3
t3(
=t 3 (X
, T
2 ( X, T = O )
(fu)
dX
XX1 • X2 • X 3
-
X = X, • X2' X 3 ' T
= O) = t
+
)1J
a
Q
Equations (1) , (5) and (8) are Fourier's equations for the three Iayers ; (3)
and (6) exp ress the temperature equaIity at the contact interfaces, (4) and
(7) the thermal flux equaIity; (2) and (9) express the externaI and internaI
heat exchanges respectively. About the last two equations , the first one says
that the layer temperHtures have the same value, to. at the init i al instant;
the second ono that the time var iations af the roam temperature depend both on
the heat exchange at the internaI side of the wal~ (x=x,+ x 2+ x 3 ) and on the
heating power. ] f this is eIectrically supplied, Q can be taken as a functio n
of time which assumes the value <i = constant when the heating goes, and the va
lue O when i t does noto In the case of heating by hot water , <i must be expre~
sed as:
<i
=(
SI/Sw )h [( tI - ti )
being Swano S t tho W;Jll and heat exchanger areas, h 'i the heat transfer coefficient bct\;een thc hat water and the room , finally tI the temperature of the
f irst onc . ~!O\:revcr, taking into account that: i) the time - dependence af t t is
like that a[ (i ; i i J during the periodic state the oscillations of t i influence the dif[c r encc t t -ti less than :: 4% and only in the transient state thi s
infl ue ncc can attain '=10% at 'most; ib) the uncertainties af À, a , E and in
particula r h e , h i anel h'i are- often larger than above 1 imi ts ; i t can be can cluded that ror this kind of problem a reasonable accuracy is obtained by si lll
r 1y assumi ng Q = <ia and Q= O 31so in thc case of hot water heating .
Thc abovc equations hovc heen solved fallowing a resolution method repo!.
teu by Ozisik [I RI, \;ith thc aid of a I'liOli6 Olivetti mini - computer(')
(1) For sakc or brcvity furthcr details on the resolution
omi tted.
1230
method must be
3.
RESULTS ANO CONCLUSIONS
Part of the result s of numerical calculations are reported in figures 3,4
and 5. In a11 t hese graphs the zero-time is fixed ",hen the temperature s Df room
and ",alI periodically oscillate as te does : in such a ",ay the influence Df the
arbitrary initial condition (10) vanishe s at al I.
From Fig. 3 it appears that for a given value Df the internaI heat capaci
t y, E, the amplitudes of the room temperature oscillations are lnarkedly larger
in the case of light ",alI, both during the transient and the periodic state.
This observati on i s yet valid ",hen heating is performed t",ice a day instead of
once (Fig. 4) . 1.1oreover, changes in solar exposi tion do not modify this trend.
There fore, a first simple conclusion can be drawn : ",ith the same energy cons~
ption the heavy wall assures smaller 9scillations of ti than those corresponding
to the light "all .
HOl<ever , I<hen E increases the j ust pointed out difference t ends to decrea
se (Fig . 5) : this -means that the characteristic of more constant temperature a~
sured by the heavy wal l i s particularly evi dent for residential buildings and
offices, in "hich E is only due to furniture, internaI "alls and air , but it
could become less evident in the case of warehouses.
Another interesting aspect which distinguishes the heavy "alI from the II
ght one is tha-t the choice of the heatin.z beginning is slightly important for
the fo rmer, while it is very critic 'for the latter. If the beginning instant
changes from Z a.m. t o 8 a .m. (Fig. 3) the amplitude of the r oom temperature
oscillations remains nearly unchanged in the case of heavy "alI, but it varies
from ZOC t o 4°C in that of light wall. In other words , the beginning instant
must be careful l y chosen in relation to the external temperature, te , in the
last case; obviously such a choice becomes more important for the South and We
st expositions because of the marked peak "hich characterizes the time - depende~
ce of their temperature s (Fig. Z) .
Let' s nm; consi der t he problem from a different point of vie", (Fig . 6) . In
order to obtain these graphs the heating is made to start so that the room :em
perature , ti ', reaches the va lue of ZO°C at 9 a.m . Successively the temperature
is controlled every 450 5 until 8 p.m .: if it results larger than ZO°C the hea
ting is stopped (namely Qis put equal O), i f ti results les s than ZO°C the hea
ting is made to start aga i n (Q = (0)' In this ",ay a fixed value of the room tem
perature i s as sured in the time interval f rom 9 a.m. to 8 p.m . Obviously such a
calculation must be repeated for several val ues of Qo in order to evaluate the
energy consumption, because the heat ing time which is necessary to obtain ti
ZO°C at 9 a .m. decreases if Oo is increased [191.
Some time-dependences of ti obta ined by means of these calculations are
reported in Fig . 6; some va1ues of the energy consumption in Table Z.
Qo= 7. 301(/m 2
~o= 2:) . S\\I/ m 2
Light wull
106
98
Itcuvy \;u11
llJ4
101
Table Z- Energy consumption, J per day; ti
constant from 9 a.m. to 8 p.m.
1231
days
22
o
2
-----_ ..
Ii
18 I-
I
medi um wall
---
heavy wall
/~
/
,," ',
;:••,
v· __
......
.l~._~2
/
.
.
,
tA
.< - 'v
SOU) h
,
10
II
' • ...t /'
",=J:=-1'_VI'----="'--.!'./_
I------jl---e =
n
---===j
84 K J Im' 'C
ã.=
....-:: .- -
10
11
~;,':\~::-~'.
lighl wall
14
9
.-----,-----.-----~
_._"-
.............:;-:=:"
3 8
11.6 Wlm'
In
10
2
10
2
18
/ 0_',
!r -::'
."'
.
l
! """ . 0----
S O uI h
14
._."---'.-'---'
....'-.'
:
-r- I~''';
ã."
-'./
n ,.
10
1------1--1- e_ 84 K
"
.....
18
,
...
'--.
/I
!/'--/i~,;t
I'·':.:;·. . .
.....-;--~
10
o
~
ll- --
j"'.",
: --:::',>
14
"~o
11.6 Wlm'
n
6
6
So uI h
J Im"C
"h
8
o
ã,= 11.6
.,
Wlm 2
ri
16
8
o
16
8
o
O
16
8
8
18
18
O
O
O
day hour
'.
Fig. 3
Internal temperature ,
ti,
.,
versus time for heavy , medium and
light wall at three beginning istants of heating .
1232
The fo1101áng conclllsions can be dr",m : i ) the iI1crease of 00 leads t o a saving
of energy both for hemry and light "a11, bllt it also implies a smaU levei of
comfort dllri ng the night ; ii) t hi s 1055 of comfort is sma U in the case of heavy " ali, more marked in. that of light wa ll; i ii)wi thin t S ~ the ene rgy saving r,,aches its maximum when Qo is 4 times the minimum which is necessary to steadi l y
"Ill intain ti= 20'C being the externai temperatllre eqllal to te ( i.e. 9.1,12 and
9 .2' C for East, Sollth "nd Wes t exposition respectively).
According to the above conclllsions in the case of light wa ll a marked increase of Qoseems llseful only for buildings destincd to office , where no people
live during the night .
Finally in Fig . 7 the initial (heating) and the final (coo ling) transients
and "Iso the intermediary interval in "hich t i i5 maintained constant wi thin ±
O. 7'C are reported: for the heavy "ali the internai tempera ture decrease i, 51'!.
"e r than that eorresponding to the light "ali , as the increase does during the
heating. ~loreover, during the initial transient with E = 84 kJ/m 2 'C the energy
consumption decreases of about 25% both for heavy and light "aU if 00 ehanges
(rom 7.7 t o 17 lV/ m 2 ; on the eontrary thi s saving beeomes negligible with E =
.12 kJ/ m2 'C.
Conc luding this ,;ork, authors r eJ1)3rk tha! the problem of energy ,aving , "hi
ch is briefly analized here, mus t be considered in dependence of the following
quantitie5:i nternal heat capaeity, type of "a 11 , solar exposition, heating power
and mainly leveI of required comfort. Such a study wil l be made in a future p~
:ler.
day.
o
2
3 8
22r------;-------r----~
9
10
11
r------T-------r----~
lighl wall
medlum wall
18
heavy wall
+----1
Soulh
14r--------r--~.
1 - - - - - j - - E - 8 4 K J Im"C
Õ"11 .6 W/m'
10r-----~~7_~~-L~~~
4 8
16 20 4 8 16 20
O
O
O
00
4 8
16
20
O
.. 8
16 20
4 8
16 20
O day hour O
Fig. 4 - Internai temperature, ti, versus time for heavy, medi um and light
wall; heating twice a day .
1233
days
o
22 t- -------- IIgh I wall
_._.-
,;
2
I
I
3 8
I
!
- - heavy wall
14
~
'\\
ri
M
/;>./
'n
•
00=
•
...
.
i~--
'/""""
~c-~
1------1-- e • 84
,. . "L!
0 0,
..-.... .l
10
n
• '2
11.6 Wlm'
,,~,
I
Soulh
.....' "
Im"C
12
/,
"
K J
ri
'2
18
14
I - - - - - I - - - e = 42
J
",",./
-'-;=
10
.--
/·\T.i"" -~.
\ t/
"
i ~ '/1\:
So ui h
11
10
r-----~----~-----
medium wall
18
9
n
•
K J
11.8
Im' 'C
W
Im'
12
18
So ui h
.;:.::~:.:.:..
....,,,,+- ",. --=
14
E • 16 é K J I m"C
~-- , -'.,~~
.......
10
O
......•
..-.'"
ri
12
0 0.,1.6
Wlm'
ri
O.
'2
oO•
'2
O•
O•
'2
day hour
Fig. 5 - Interna! temperature, ti ,versus time for heavy , medium and
light \;all and for three values of E.
1234
O
days
o
2
21
ti
-
r
~
t9
17
h
r
LIGHT
\
- - -
E
'\
E as t
=
42
WALL
K
J 1m 2
0 0 =23 .5
·c
Wlm 2
15
9
20
9
20
LIGHT
21
.....
\---
19
V
~
WALL
m2
E = 42
K J
HEAVY
WALL
I
'C
E as t
17
o
9
20
o
9
20
day hour
o
O
2
days
tI
2 O +--- ,"",,,-,,v-..../""'I - . f - - - - vv"""""r--.,
E=42 KJlm 2 'C
00=
East
18+-____~------~~------~------~~
20
9
9
20
23.5
HEAVY
E
= 42
W
I m~
WALL
K J
1m 2 ~C
0=730
Wlm 2
o .
East
18;-----~------~~-----r------~~
9
20
9
20
O
O
day
hou r
O
Fig. 6 - Cyclic time -dependence of t i for heavy and light ",all and for
two vaIues of 00' Constant internaI temperature from 9 a.m.
to 8 p .m.
1235
days
4
2
3
5
6
22r-------,-------,------,-------,-------,-------r----
o
t: 8 t-------+--:-::../''-,/-,.-''-''-+,.-.-•.-.,.-/:::'-~" " =1"'-"-"-'-/-"-"'-'_\f-L,,_ _ _-l-_ _ _-+\'_.__"
/:/"'r
,
t4
light wall
50uth
heavy wall
E · 84KJ/m 2
°
0
.7.66
0C
W/m
2
tO+-------+-------~----~~----_4-------+------_+----
18
+--l"'-/",:. . /
------~------+------+------+-----__+---
!
14t--! -----+-------~----~~------+-------+------_+--~
K J
Im 2 0C
0 0 .17.4
W/m 2
E . e4
50 ut h
.i
10+-------+-------~----~------_4-------+------_+--~
o
o
o
o
O
O
o
day hour
day.
8
7
9
lO
12
II
13
22.--.----r----.----~---,----_r---~
I
t i"".
N
18
e - S4KJ/m 1
~~
..,~._.~/~/'~
...=..~
.I~~--t-----f----~--
0C
I
a,- O' W/m'
-
_:/.-.=c
..J ===:=bbJJ
.,.....,..../ .......!." .. ,'~
..
14
"" .. __ ,/ ....................---"'.1....
50 ut h
.
-
.
/ ......1.
.....:'.
-'.
10~+1 ___~______~______~___-4______~______~
O
O
O
O
O
O
O
da y hour
Fig. 7 - Tnitial hea ting, constant temperature intenTa] anel final co -
oling for heavy and light \ValI . During the constant telllperat~
re intcrvnl the heating stnrts when t" 19OC and stops when
'
t i ' 20 °C,
1236
L1 ST OF SYi'IBOLS
coo.rdinate, m
X
x 1 ,x 2 ,x 3 th i cknesses of the wa11 layers , m
;'1,i' 2 '},3 thermal conductivities of the waU layers , IV/m °c
u l'u 2 ,u 3 thermal diffusivities of the waU layers, mYs
sol-air externaI temperature, °c
te
roam temperature, °c
ti
t l' t i:,t 3 temperatures of the' wall layers, °c
2
room thermal capacity per unit area , J/m °C
E
heating power per unit area, lV/m 2
ri
externaI heat transfer coefficient, lV/m 2 °C
he
internaI heat transfer coefficient, lV/m 2 °C
hi
uniform "a11 temperature at T=O, °c
to
RE FERECiCES
[I J U . ~1~GRlCiI:
[ 2]
[S]
[o]
Influenza deUa capacità tennica deU 'ambiente sul regime perio
di co stabilizzato di temperatura - La Tennotecnica, vol. 16° (1962)
U.H~GRI~I : lnfluema deI ricambio d' ar ia sulle osci11azioni di temperatura
alI 'interno di un l ocale di riferimento - Atti deI 3° Convegno deI Riscal damento e de11a Ventilaz ione, Padova (1963)
Ci .K.D. CIIOUDHURY, Z.U .A. IVARSI : lVcighting function and transient thennal
response af buil dings . Pi1 rt 1 - Ilomogeneous struc turc - Int. J. I-Iea t Mass
Transfer, vol . 7 (1964)
Z .11 . A. \,ARS I, \i. K. n. D-IOI/lH IRY : lve -ight::ing flJnction and tnm sient thcrmal
response o[ buildings. r~rt 2 - Comrositc structurc - Int . J . IIcat j''1(lSS
Transfer, vol . 7 (1964)
;'1. CClUf,UlA , G. LO GllJDICL :Trasmissione dei calore in regime periodico Atti deU ' AccaJe-nia di Sc ien zc , Lettere ed Arti di Palenno, vaI. ZS (1964 )
~!.GJLlJHllA , G. UJ GTlPJICL:Trasmissiane dei calare in regime pcriaJico ne 211 edifici - Quadcrn i dcl1 'lstltuto di Fisica Tccnica Jel1'lJnivcrsit~ ui
Palermo, n. 9
(]~J6S)
C. BOFFJ\ , V.fTREO , A.SACOII : I~cgif!lc tcnnico variahilc in cdifici prcfabbrl
cati - La Tcnnotecnj ca , vol . 20 ( I ~)ó6)
C. 110ITA, V. FUmO , A. SACOII : Tahellc numeric hc rer i l calcolo r"rida de] p.1
ramctri carattcristici ui parcti cOfllf1Ostc - 1,(1 TCrTIotccnica , vol. 20 (l~)6Cl)
S.I·'j\r;CIJ\.\jI, A. FJ\.\lT INI: Flusso tcnnico <Jttravcrso paret! sottO] nJste ;JJ irraggiamento solare durante i I perioJo J';:.lVvimnento - 1\tt.! Licl SO Convcgno
de] I(iscaldamento e del]a Venti la z ione , Padava ( 19(,S)
H.SOVIU\.\lO , C.ZOJ{ZINI: Scclta e rnisura delle grandezze atte a carattcrizza
re i I comportamento tennico Je! le rareti in regime v:lr i:l bi le - [,;1 Termotecnicl , vol . 20 (I')(,()J
[r I]
1\1.S(JVHA~() ,
(;.zrIl{lI ,~I :
II cocrr icie nte Ji trasmissione K eJ il tempo di r i
spost;! Tonello stuuio dei cornporLunento tcnnico delle parct i - I.a Termotcc n i ca, vo I. 2() (I ~)()(»)
[12 J
C.I'J1ITAI.i\S: I(oom thennal response ractors - ASllliflE Transactions, vo1. 73
(I %7 J
1237
[13]
[14]
[IS]
[16]
[1 7]
[18]
[19]
G.P.MITALAS, D.G.STEPHENSO:-J : Calculation of heat conduction transfer fun.':.
tion for multi-layer slabs - ASHRAE Transactions, vol . 77 (1971)
B.BON1, R.POZZ1 , G.TREBB1: Un metodo di cal colo delle funzioni di trasferi
mento relative ai comportament o termico in regime variabile di un edificio Atti dei 2° Seminario Informativo sul Risparmio Energetico nel Riscaldamen to degli Edifici, CNR , ~li lano 5- 6 llarzo (1979)
P . BONDI , A. SACCHI : Experiments of t hermal oscillations on large I<a11s Proceedings of the 8th Conference on Thermal Conductivity, Plenum Press ,
N.Y. (1968)
P. DI FILIPPO, G. ZORZIN1 : l-lisura dei coefficiente di trasmissione termica 11
e dei tempo di risposta Todelle pareti - La Termotecnica , vol. 23 (1969)
P.DI F1LIPPO: Te~po di risposta delle pareti : determinazione sperimentale La Termotecnica , vol. 24 (1970)
~I.N . OZ1SIK : Heat Conduction - J . lViley and Sons, N.Y . (1980)
F .DE PONTE, L FORNI\SIERI , G.TRAPANESE , R. ZECCII1N: .llnalisi dei fabbisogno
tennico degli edifici con riscaldamento intenni ttente - Conclizionamento dell 'aria. Riscaldamento . Refrigerazione, n . 9 ( 1976)
1238
r