Ser
I
no.859
ei t >,, /,a
i
I#
National Research
Council Canada
Conseil national
de recherches Canada
ENCORECANADA: COMPUTER PROGRAM FOR
THE STUDY OF ENERGY CONSUMPTION OF
RESIDENTIAL BUILDINGS IN CANADA
by A. Konrad and B.T. Larsen
Appeared in
Proceedings, Third International Symposium on
The Use of Computers for Environmental Engineering
Related to Buildings
held in Banff, Alberta, 10 12 May 1978
p. 439 450
-
DBR Paper No. 859
Division of Building Research
Price $1.25
OTTAWA
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ENCORE-CANADA:
COMPUTER PROGRAM FOR THE STUDY OF
ENERGY CONSUMPTION OF RESIDENTIAL BUILDINGS I N CANADA
A . ~ o n r a d land
B .T. ~ a r s e n ~
l ~ i v i s i o nof Building Research, National Research Council o f Canada
Ottawa, Canada
2 ~ o r w e g i a nBuilding Research I n s t i t u t e , Oslo, Norway
(Guest worker with DBR/NRC, 1975)
-
The paper d e s c r i b e s t h e mathematical methods employed in t h e
ABSTRACT
ENCORE-CANADA computer program which p r e d i c t s t h e hourly a s we1 1 a s the
annual h e a t i n g requi rements of smal l r e s i d e n t i a l - t y p e bui l d i n g s .
ENCORE-CANADA i s p r i m a r i l y intended f o r r e s e a r c h e r s , d e s i g n e r s and cons u l t i n g engineers i n t e r e s t e d in energy conservation measures. The model
includes t h e e f f e c t s o f thermal s t o r a g e , i n t e r n a l heat g a i n s , basement
and a i r i n f i l t r a t i o n l o s s e s , transmission h e a t l o s s e s and s o l a r heat
gains. The h e a t i n g system is a t h e r m o s t a t i c a l l y c o n t r o l l e d o i l - f i r e d
furnace with warm a i r d i s t r i b u t i o n . Hourly s o l a r r a d i a t i o n and weather
d a t a f o r various Canadian c i t i e s a r e used t o simulate outdoor c o n d i t i o n s .
R ~ S U M E - Les a u t e u r s d e c r i v e n t l e s d t h o d e s mathematiques ut i 1 i s b e s dans
l e programme informatique ENCORE-CANADA qui predi t l e s besoins de chauffage
h o r a i r e s e t annuels des p e t i t s immeubles r C s i d e n t i e l s . ENCORE-CANADA
s ' a d r e s s e s u r t o u t aux chercheurs, aux concepteurs e t aux ingenieurs-conseils
qui s 1 i n t 6 r e s s e n t aux mesures d18conomie de I 1 b n e r g i e . Le modele englobe
l e s e f f e t s de I ' i n e r t i e thermique, l e s gains de c h a l e u r i n t e r n e s , l e s p e r t e s
de c h a l e u r par l e s fondations e t l 1 i n f i l t r a t i o n d ' a i r , l e s p e r t e s par t r a n s mission e t l e s gains de c h a l e u r s o l a i r e . L ' i n s t a l l a t i o n de chauffage 3 a i r
chaud pulsd u t i l i s e I ' h u i l e comme combustible e t e s t rCgl8e par thermostat.
Des donnees m 6 t ~ o r o l o g i q u e se t l e rayonnement s o l a i r e h o r a i r e en d i f f e r e n t s
c e n t r e s c a n a d i e n s . s e r v e n t 3 simuler l e s conditions e x t e r i e u r e s .
INTRODUCTION
The determination o f t h e h e a t energy r e q u i r e d t o
maintain p r e s c r i b e d indoor c o n d i t i o n s i n a b u i l d i n g
i s one o f t h e main concerns o f b u i l d i n g d e s i g n e r s .
(Ayres 1977). For many y e a r s t h e degree-day method
(ASHRAE 1976) was widely used b u t , some of t h e
e s s e n t i a l elements o f energy c a l c u l a t i o n s a r e
a b s e n t from such simple methods (Mitalas 1976).
Owing t o t h e r e l a t i v e l y high c o s t o f energy t h e r e
i s a need t o r e f i n e t h e accuracy o f b u i l d i n g energy
requirement c a l c u l a t i o n s . The u s e o f t h e computer
i s i n e v i t a b l e due t o t h e l e n g t h and complexity of
computations f o r hour -b y-hour a n a l y s i s . Furthermore, i t i s important t o p r e d i c t a c c u r a t e l y n o t
o n l y t h e t o t a l energy consumption of a b u i l d i n g ,
b u t a l s o t h e savings a s a r e s u l t o f energy conserv a t i o n measures. Thus t h e mathematical model
i n c l u d e s many p r e v i o u s l y ignored f a c t o r s .
This paper d e s c r i b e s a mathematical model f o r
hour-by-hour s i m u l a t i o n of energy consumption o f
small r e s i d e n t i a l - t y p e b u i l d i n g s u s i n g r e a l weather
d a t a . The a s s o c i a t e d computer program, ENCORECANADA (Energy Consumption o f Residences i n Canada)
i s t h e Canadian v e r s i o n o f t h e Norwegian ENCORE
program (Larsen 1976 and 1977)
I t a l l o w s one t o
.
study, f o r a given b u i l d i n g design, t h e r e l a t i o n
between h e a t i n g demand and
-
c l i m a t o l o g i c a l f a c t o r s ( e . g. temperature, wind,
solar radiation),
-
b u i l d i n g l o c a t i o n and o r i e n t a t i o n ,
- i n s u l a t i o n l e v e l s i n c e i l i n g / r o o f system, w a l l s
and basement,
-
thermal s t o r a g e i n t h e b u i l d i n g s t r u c t u r e and
contents,
-
t y p e s of f e n e s t r a t i o n ,
exterior surface solar absorptivity,
-
a i r infiltration,
-
thermostat s e t t i n g s , and
-
h e a t i n g system c h a r a c t e r i s t i c s .
OVERVIEW OF MODEL
ENCORE-CANADA r e q u i r e s two i n p u t d a t a s e t s :
the building model, and the other
c o n t a i n i n g hourly weather and s o l a r r a d i a t i o n
d a t a . The b u i l d i n g model h a s t h e following
components:
one describing
-
basement w i t h below- and abcve-grade p o r t i o n s ,
-
thermostatically controlled w a n a i r distribution
h e a t i n g system consis,ting o f o i l - f i r e d furnace,,
.smoke p i p e w i t h barometric damper, and chimney,
-
two-element e l e c t r i c h o t water h e a t e r and s t o r a g e
tank from which water i s drawn according t o a
24 h schedufe,
-
-
l i g h t s which a r e on o r o f f according t o a 24 h
schedule,
- tangent of d e c l i n a t i o n angle, t a n ( & ) ,
-
v a r i o u s h e a t - g e n e r a t i n g e l e c t r i c a l a p p l i a n c e s and
equipment t h a t a r e turned on o r o f f according t o
a 24 h schedule,
-
-
occupants who a r e p r e s e n t i n t h e b u i l d i n g i n
varying numbers according t o a 24 h schedule.
For each day o f t h e year t h e d a t a s e t c o n t a i n s 24
hourly values o f
n o n - p a r t i t i o n e d b u i l d i n g . e n c l o s u r e t h a t includes
c e i l i n g / r o o f system, w a l l s , doors and windows,
Schedules f o r h o l i d a y s may be d i f f e r e n t from those
f o r working days. Using t h e schedules and t h e
s p e c i f i c a t i o n s o f model components, ENCORE-CANADA
computes t h e i n t e r n a l h e a t g a i n due t o
'-
I
1
II
-
occupants,
!
.
(1970-72)
(1970-72)
(1972-74)
Each s e t c o n t a i n s t h e c i t y ' s l a t i t u d e (R) md t h e
f o l l a w i n g d a t a f o r each day o f t h e y e a r :
d a t e (day, month),
holiday i n d i c a t o r (used f o r s e l e c t i n g schedules),
hour a n g l e when s o l a r a l t i t u d e is zero ( s u n r i s e
a n g l e , ho),
ground r e f l e c t i v i t y , R (0.4 w i t h snowcover, 0.2
without) .
a ) dry-bulb temperature of ambient a i r (O,)
b) t o t a l cloud amount i n t e n t h s (C),
c) wind speed (V), 10 m above ground, a t
meteorological s i t e ,
d) wind d i r e c t i o n ,
e l e c t r i c a l a p p l i a n c e s and equipment,
l i g h t s , and
h o t water consumption.
e) atmospheric p r e s s u r e (P),
f ) hour a n g l e ( h ) , (sun i s above horizon i f
Ihl<lhOO,
g) i n t e n s i t y o f d i f f u s e sky r a d i a t i o n (M,),
S i m i l a r l y , t h e program u s e s weather and s o l a r r a d i a t i o n d a t a and t h e s p e c i f i c a t i o n s o f model compon e n t s t o compute h e a t gain and l o s s through
h) i n t e n s i t y of d i r e c t s o l a r r a d i a t i o n p e r u n i t
a r e a normal t o s u n ' s r a y s (Mdn) ,
- below-grade p o r t i o n o f basement,
i ) conversion f a c t o r f o r items
-
c e i l i n g / , r o o f system and e x t e r i o r w a l l s i n c l u d i n g
above grade p o r t i o n o f basement,
-
doors,
windows, and
-
g) and h ) ,
(e)
.
Of t h e s e , a ) through e) a r e measured o r observed
q u a n t i t i e s ; f ) through i ) a r e computed v a l u e s .
ENCORE-CANADA appl i e s t h e following conversion
formula t o items g) and h)
a i r infiltration.
These g a i n s o r l o s s e s , w i t h t h e exception o f t h e
below-grade p o r t i o n of t h e basement a r e computed
w i t h r e s p e c t t o an a r b i t r a r i l y chosen indoor
r e f e r e n c e temperature. The sum o f v a r i o u s h e a t
g a i n s and l o s s e s computed w i t h r e s p e c t t o t h e
r e f e r e n c e temperature a r e transformed i n t o h e a t i n g
demand based on indoor a i r temperature c o n t r o l l e d
by t h e r m o s t a t .
I
Suffield, Alta.
Swift Current, Sask.
Charlottetown, P .E.I
WEATHER AND SOLAR RADIATION DATA
ENCORE-CANADA may use any one o f 39 preprocessed
weather and s o l a r r a d i a t i o n d a t a s e t s . They
correspond t o t h r e e consecutive y e a r s o f d a t a f o r
t h e following 13 Canadian c i t i e s .
Vancouver, B .C .
Edmonton, A l t a .
Winnipeg, Man.
Ottawa, Ont.
Fredericton, N.B .
Toronto, Ont .
Montreal, Que.
H a l i f a x , N.S
S t . John's, Nfld.
Summerland, B .C .
.
where
I ( 0 ) = average s o l a r r a d i a t i o n during p r e s e n t
hour,
e(0) = p e r c e n t d i f f e r e n c e between computed
and measured average t o t a l s o l a r r a d i a t i o n ( d i r e c t p l u s d i f f u s e ) on a h o r i z o n t a l s u r f a c e during p r e s e n t hour,
M(0) = c a l c u l a t e d i n s t a n t a n e o u s s o l a r r a d i a t i o n a t p r e s e n t hour,
M(l) = c a l c u l a t e d i n s t a n t a n e o u s s o l a r r a d i a t i o n a t previous hour.
The f a c t o r 100/[100-e(O)] i n Eq. (1) is a cloud
cover modifier d e r i v e d by comparing computed and
measured t o t a l s o l a r r a d i a t i o n on a h o r i z o n t a l
surface.
ENCORE-CANADA u s e s t h e information i n t h e
weather and s o l a r r a d i a t i o n d a t a s e t s t o compute
t h e s o l a r r a d i a t i o n i n t e r c e p t e d by b u i l d i n g compon e n t s . The number o f d i s t i n c t s u r f a c e o r i e n t a t i o n s f o r a given b u i l d i n g envelope a r e determined
i n t h e s u b r o u t i n e SOLRAD. For every d i s t i n c t s u r f a c e o r i e n t a t i o n t h e subroutine DIRDIF c a l c u l a t e s
t h e d i r e c t and d i f f u s e components o f s o l a r r a d i a t i o n p e r u n i t a r e a according t o ASHRAE procedure
(ASHRAE 1975). The d i r e c t component i s given by
e = eo + 1y
= Idn{ [ s i n ( & )sin(R) +cos(G)cos(~)cos(h)]cos(q)
- [sin(6)cos(R) - cos(6) sin(R)cos(h)]sin(q)cos(a)
(2)
H
i = incidence a n g l e
a = s u r f a c e azimuth a n g l e (measured counterclockwise from south)
q = s u r f a c e tilt angle (measured from t h e
hor i zonta 1)
=
latitude
6 = d e c l i n a t i o n angle
h = hour a n g l e .
The d i f f u s e component of s o l a r r a d i a t i o n i s
given by
where t h e r a t i o of d i f f u s e v e r t i c a l sky r a d i a t i o n
t o d i f f u s e h o r i z o n t a l r a d i a t i o n , Y, i s obtained from
Threlkeld (1962)
Y = 0.55 + 0.437cos(i)
+ 0. 313cos2 ( i )
C
(1 - = ) I
(6)
= surface solar absorptivity
D = d i f f e r e n c e between i n c i d e n t long-wave sky
r a d i a t i o n and r a d i a t i o n emitted by a h o r i zontal s u r f a c e f a c i n g t h e sky
(63.1 w/m2 (20 Btu/hr f t 2 ) was used)
where
L
1 + cos( )
E = r a d i a n t interchange f a c t o r f o r a h o r i z o n t a l
s u r f a c e f a c i n g t h e sky (assumed t o be 0.9,
0.5 and 0.1 f o r f l a t , suburban and urban
building s i t e s , respectively)
= Idn c o s ( i )
- c o s ( & )s i n ( h ) s i n ( q ) s i n ( a ) 1
- E D
where
a
Idir
['It
=
c o e f f i c i e n t of h e a t t r a n s f e r by long-wave
r a d i a t i o n and convection (assumed t o b e
c o n s t a n t and equal t o 17 w/m2 K,
3 Btu/hr f t 2 R).
In addition t o t h e thirty-nine data s e t s
mentioned above, t h e mean (em), amplitude (8,)
and phase (p) of t h e main s i n u s o i d a l component
f o r t h e ground s u r f a c e s o l - a i r temperature funct i o n ( f o r t h e same t h i r t e e n c i t i e s and t h r e e
consecutive years) a r e s t o r e d i n t h e block d a t a
subroutine SOLBLK. em, 0, and p a r e used a s i n p u t
t o t h e below grade basement and h e a t l o s s model.
They were obtained from a F o u r i e r a n a l y s i s of
hourly v a l u e s of ground s u r f a c e s o l - a i r temperat u r e computed from Eq. (6) by s e t t i n g a = 1 - R,
cos(q) = 1, E = 0.9, by using measured v a l u e s f o r
t h e t o t a l s o l a r r a d i a t i o n i n c i d e n t on t h e ground
s u r f a c e , and by assuming a l i n e a r v a r i a t i o n o f H
with wind speed (V)
CALCULATIONS OF BELOW-GRADE BASEMENT HEAT LOSS
The f i n i t e d i f f e r e n c e program TRUMP (Edwards
1968) was used t o i n v e s t i g a t e t h e importance of
v a r i a b l e s a f f e c t i n g h e a t flow from t h e belowgrade p o r t i o n of basements. The following
f a c t o r s were found t o be important:
- amount of wall below grade,
cos(i)
2
-0.2
and t h e i n t e n s i t y of d i f f u s e ground r a d i a t i o n i s
given by
-
mean thermal p r o p e r t i e s of c o n c r e t e and s o i l ,
- mean and fundamental s i n u s o i d a l component of
ground s u r f a c e s o l - a i r temperature,
-
depth and temperature of water t a b l e ,
e x t e n t and thermal r e s i s t a n c e of i n s u l a t i o n ,
- p e r i m e t e r and shape of basement.
+ cos (6) cos ( t ) cos(h) ]
The t o t a l r a d i a t i o n I t = ( I d i r + I d i f ) i s
e s s e n t i a l t o t h e computation of s u r f a c e s o l - a i r
temperature from t h e following equation (ASHRAE
1977) :
Factors such a s f r e e z i n g and thawing, temperat u r e dependence of thermal p r o p e r t i e s , snow cover,
p o s i t i o n of i n s u l a t i o n , and d i u r n a l and weekly
t r a n s i e n t s were found t o have a second o r d e r
e f f e c t on c a l c u l a t e d h e a t l o s s f o r a h e a t i n g
season. Since most of t h e important parameters
a r e seldom known a c c u r a t e l y , a s i m p l i f i e d algorithm
was s e l e c t e d r a t h e r t h a n incorvorating,. a f i n i t e
d i f f e r e n c e o r f i n i t e element approach. The h e a t
flow from t h e basement was assumed t o b e t h e sum of
-
s t e a d y - s t a t e h e a t flow t o ground s u r f a c e a t mean
annual s o l - a i r temperature,
- p e r i o d i c heat flow t o ground s u r f a c e with tempera t u r e f l u c t u a t i o n equal t o fundamental s i n u s o i d a l
component of s o l - a i r temperature, and
-
s t e a d y - s t a t e h e a t flow t o water t a b l e a t mean
annual deep ground temperature.
The basement i n t e r i o r s u r f a c e temperature was
assumed constant throughout t h e h e a t i n g season.
The components f o r w a l l and f l o o r were t r e a t e d
s e p a r a t e l y t o s i m p l i f y comparison with t h e TRUMP
f i n i t e d i f f e r e n c e r e s u l t s . Thus, i n ENCORE-CANADA
t h e hourly v a l u e s of h e a t flow through basement
w a l l s (Qw) and f l o o r (Qf) a r e computed with t h e
subroutine BTLOSS according t o t h e following
formulae
+
Qf
2lT
[ G w g c ~ ~ ~t( n+ p)
= Gfg(om
-
gin)
2l~
+ [Gfgccos (T t + p)
+
+
G
wgs
2lT
sin(? t +p)]Ba
-
G f w (ewt
+G
Where
t = hour of t h e year (0
sin(f;- t + p)]ea
5
t
G
G
fgc
fgs
= c o s i n e component of dynamic h e a t
conductance between basement f l o o r and
ground s u r f a c e
= s i n e component of dynamic h e a t conduc-
tance between basement f l o o r and ground
surface.
The f u n c t i o n 0 ( t ) = 0, + e a [ s i n ( 2 a / n + p ) ]
d e s c r i b e s t h e p e r i o d i c boundary c o n d i t i o n f o r
temperature a t t h e ground s u r f a c e . The f i r s t two
terms on t h e r i g h t hand s i d e s of Eqs. (8) and (9)
r e p r e s e n t s t e a d y - s t a t e h e a t flow from t h e basement
t o the ground s u r f a c e and t o t h e water t a b l e ,
r e s p e c t i v e l y . The remaining terms represent the
dynamic h e a t flow between basement and pound
s u r f a c e . The dynamic heat flow between basement
and water t a b l e i s neglected. The conductances
Gwg, Gww, Gwgc, Gwgs, Gfg9 Gfw, Gfgc and Gfgs a r e
computed once f o r a p a r t i c u l a r basement i n t h e
subroutine BASMNT. In order t o a r r i v e a t t h e s e
e i g h t conductances, a version of t h e method
described by Boileau and L a t t a (1968) i s used.
(8)
gin)
2.rr
f gs
Gfw = steady - s t a t e h e a t conductance between
basement f l o o r and water t a b l e
(9)
5 n)
AIR INFILTRATION CALCULATIONS
The a i r i n f i l t r a t i o n c a l c u l a t i o n s a r e based on
t h e model described i n d e t a i l by Konrad e t a 1
(1978). This model i s l i m i t e d t o e i g h t d i f f e r e n t
b u i l d i n g height-to-width and length-to-width
r a t i o s . The leakage opening of each b u i l d i n g
component ( c e i l i n g , walls, windows and doors) i s
represented b y s e v e r a l holes e q u a l l y spaced along
the height of the component. I f t h e indoor
pressure a t t h e ground l e v e l (P1) i s known, t h e
mass E l m r a t e o f a i r {Gj) through t h e j - t h hole
can be computed f r o m
n = number of hours i n a y e a r (n = 8760 f o r
o r d i n a r y year, n = 8784 f o r l e a p year)
Owt
= water t a b l e temperature
0. = c o n s t a n t basement i n t e r i o r s u r f a c e temperIn
ature
0
m
= mean annual ground s u r f a c e s o l - a i r
temperature
-n .
J
G . = a p ( ~ P+;
I
11
G
Wg
= s t e a d y - s t a t e h e a t conductance between base-
ment w a l l s and ground s u r f a c e
=
G
-
(101
-
1, and
+
1
1
n:
= flow exponent f o r a i r flow through j - t h
hole
P = outdoor atmospheric p r e s s u r e a t ground
level
R = gas c o n s t a n t f o r a i r
between basement w a l l s and ground s u r f a c e
ment f l o o r and ground s u r f a c e
I
R. = r e s i s t a n c e t o a i r leakage of j - t h h o l e
c
j
= wind p r e s s u r e c o e f f i c i e n t where j - t h hole
i s located (depends on wind d i r e c t i o n and
building orientation)
= s t e a d y - s t a t e heat conductance between base-
fg
-P
1 for a i r exfiltration
= d e n s i t y of outdoor a i r i f s =
= c o s i n e component of dynamic h e a t conductance
G
= s i n e component of dynamic h e a t conductance
Wgs between basement walls and ground s u r f a c e
1
'o
p = d e n s i t y of indoor a i r i f s =
ment w a l l s and water t a b l e
Wgc
1
hj (---)I
i'
s = + 1 f o r a i r i n f i l t r a t i o n , and
= s t e a d y - s t a t e h e a t conductance between base-
G
[++g
o
where
0 = amplitude of fundamental Fourier s i n e
a
component of ground s u r f a c e s o l - a i r
temperature f u n c t i o n
p = phase of fundamental Fourier s i n e component
o f ground s u r f a c e s o l - a i r temperature
function
J
c .v2
V = wind speed
g = gravitational acceleration
h . = h e i g h t of j - t h h o l e above ground
I
0 = outdoor a i r temperature
0
0 . = indoor a i r temperature.
1
Using t h e mathematical model o f t h e furnace,
smoke pipe, barometric damper and chimney o u t l i n e d
by Konrad e t a 1 (1978), t h e program ( s u b r o u t i n e s
FRNACE, INFILT, FLOWS, PCOEF, WIND, CHMNEY, FFACT
and PRESSR) computes t h e mass flow r a t e of g a s e s i n
t h e chimney (Gc) a s a f u n c t i o n of f u r n a c e load
f a c t o r f o r any assumed value o f t h e indoor p r e s s u r e
a t ground l e v e l . The t r u e p r e s s u r e i s computed by
an i t e r a t i v e procedure from t h e mass balance
equation f o r a i r flow through m h o l e s and t h e gas
flow i n t h e chimney, i . e . ,
The a i r i n f i l t r a t i o n load, Li, v a r i e s w i t h o u t door a i r temperature, indoor a i r temperature, wind
speed, wind d i r e c t i o n and furnace o p e r a t i o n . I t i s
computed by multiplying t h e sum o f a l l p o s i t i v e
Gj I S by ( 0 i - 0,) and by t h e s p e c i f i c h e a t of a i r
a t temperature ( 0 i + 0,)/2.
I f both i n p u t , i ( t ) , and output, o ( t ) , of a
l i n e a r , i n v a r i a b l e system a r e expressed i n terms
of t h e i r z-transforms, I (z) and O(z) r e s p e c t i v e l y ,
then t h e r a t i o K(z) = O ( z ) / I ( z ) i s a z-transform
f u n c t i o n f o r t h e system. I f K(z) is known, t h e
response t o any given input is o b t a i n e d by forming
t h e product K(z) I (z) . To determine K(z) , one
f i n d s t h e impulse response of t h e system expressed
a s a polynomial i n z, a s follows:
where t h e c o e f f i c i e n t s , k(nT), a r e c a l l e d response
f a c t o r s and p = m. I f t h e response decays t o z e r o
with time, p can be assigned a f i n i t e v a l u e by
t r u n c a t i n g t h e polynomial. I n p r a c t i c e , it is
sometimes impossible t o f i n d t h e impulse response
s i n c e i t i s a s s o c i a t e d with i n f i n i t e h e a t f l u x .
I n such c a s e s a t r i a n g u l a r p u l s e of u n i t a r e a i s
used a s input t o f i n d t h e response f a c t o r s (Kimura
1977; Stephenson and M i t a l a s 1967). K(z) can
always be transformed i n t o t h e r a t i o of two polynomials (Mitalas 1978) simply by m u l t i p l y i n g
Eq. (14) by a polynomial q u o t i e n t which e v a l u a t e s
t o 1, i . e . ,
HEAT TRANSFER CALCULATIONS
The z-transform method
ENCORE-CANADA employs t h e z-transform method t o
compute:
-
h e a t t r a n s f e r r e d through c e i l i n g , w a l l s , windows
and doors,
-
h e a t gained by indoor a i r from occupants, l i g h t s ,
domestic h o t water and e l e c t r i c a l a p p l i a n c e s , and
- h e a t i n g demand and room a i r temperature including
e f f e c t s of thermostat c o n t r o l system.
The z-transform method i s c l o s e l y r e l a t e d t o t h e
c l a s s i c a l Laplace t r a n s f o r m a t i o n . The method i s
perhaps e a s i e s t t o understand i n t h e context of
sampled d a t a feedback c o n t r o l systems f o r which i t
was o r i g i n a l l y developed (Truxal 1955). I t s a p p l i c a t i o n t o h e a t t r a n s f e r problems i n b u i l d i n g s i s
& s c r i b e d by M i t a l a s (1978) .
I t i s o f t e n convenient f o r simple computations t o
s e t j = 1 and w r i t e K(z) i n t h e form
When a continuous f u n c t i o n of time, i ( t ) , i s
sampled a t r e g u l a r time i n t e r v a l s , T, t h e r e s u l t i n g
f u n c t i o n , i l ( t ) , can b e d e s c r i b e d i n terms of t h e
u n i t impulse f u n c t i o n , u o ( t ) , a s follows:
where wl i s c a l l e d Common Ratio and i s defined by
The z-transform of i ( t ) i s obtained from t h e
Laplace transform o f i t ( t ) by s u b s t i t u t i n g z f o r
exp(Ts) . I t i s given by
w
=
lim .
k (nT)
n - t m k [ ( n - 1)T]
7
444
Whether given i n t h e form o f Eqs. (14), (15) o r
(16), i n g e n e r a l , K(z) can always be thought of a s
t h e r a t i o of two polynomials, i . e . , K(z)=N(z)/D(z)
Given t h e c o e f f i c i e n t s Ni and Di of t h e numerator
and denominator, r e s p e c t i v e l y , t o g e t h e r with a
s e r i e s of sampled values i(nT) and o(nT) o f an
e x c i t a t i o n and a response r e s p e c t i v e l y , except f o r
t h e c u r r e n t v a l u e of t h e response o(qT), t h e l a t t e r
can be solved by equating t h e c o e f f i c i e n t s of t h e
h i g h e s t power of z on both s i d e s of t h e equation:
.
immediate e f f e c t on t h e indoor a i r temperature.
The d e l a y depends on t h e type of i n t e r i o r f i n i s h ,
f u r n i s h i n g s and c o n s t r u c t i o n . I t is p o s s i b l e t o
d e r i v e z - t r a n s f e r f u n c t i o n s f o r t h i s system i n
t h e form given by Eq. (16) and a s a f u n c t i o n o f
mass p e r u n i t f l o o r a r e a (ASHRAE 1977; M i t a l a s
1972). The following c o e f f i c i e n t s a r e s t o r e d i n
t h e subroutine WALLGL f o r f o u r d i f f e r e n t v a l u e s
o f mass p e r u n i t f l o o r a r e a :
146 kg/m2
342 kg/m2
635 kg/m2
C
976 kg/m2
d
O(z)D(z) = I(z)N(z)
v
I f N(z) has p + l c o e f f i c i e n t s and D(z) h a s q + l
c o e f f i c i e n t s , one o b t a i n s
1
o(qT) = - { i ( 0 ) N p + i ( T ) N
Do
0
(30 l b / f t 2 ) (70 l b / f t 2 ) (130 l b / f t 2 ) (200 l b / f t 2 )
(18)
+
v
...
+i[(p-l)T]N1 +
...
-
w
0
1
1
0.703
0.681
0.676
0.676
-0.523
-0.551
-0.606
-0.646
-0.82
-0.87
-0.93
-0.97
P- 1
o(0)D
- O(T)D~-~
+
i(pT)N
-
~ [ ( q - l l ~ l ~ ~ )
9
(19)
Heat t r a n s f e r through w a l l s and c e i l i n g
Note t h a t vl = 1 + w l - vo. Based on Eq. (19),
t h e amount of h e a t t r a n s f e r r e d t o o r removed from
t h e indoor a i r by a wall s u r f a c e during t h e
p r e s e n t hour i s computed i n t h e s u b r o u t i n e WALLGL
according t o
Ls(o) = voQ(0)
A t y p i c a l wall c o n s t r u c t i o n i s shown i n F i g . 1.
The inner and o u t e r l a y e r s of a i r and t h e a i r s p a c e
a r e assumed t o have a thermal r e s i s t a n c e (R1 and
R6) b u t no thermal c a p a c i t y . The o t h e r l a y e r s a r e
s p e c i f i e d by t h e i r t h i c k n e s s (T), thermal conduct i v i t y (K), d e n s i t y (D) and s p e c i f i c h e a t (S).
There i s no r e s t r i c t i o n on t h e number of layers.
The s o l - a i r temperature, B ( t ) , a t t h e
e x t e r i o r s u r f a c e i s given by Eq. ( 6 ) . The h e a t
flow, Q ( t ) , a t t h e i n s i d e s u r f a c e of t h e wall i s
r e l a t e d t o t h e temperature d i f f e r e n c e 8 ( t ) - 8, by
a z - t r a n s f e r f u n c t i o n . A method t o o b t a i n t h e
t r a n s f e r f u n c t i o n a s a r a t i o of two polynomials i n
z i s described by Stephenson and Mitalas (1971) and
M i t a l a s (1968). ENCORE-CANADA r e q u i r e s n c o e f f i c i e n t s , a i and b i , (ASHRAE 1977; Mitalas and
Arseneault 1972) t o compute t h e heat flow during
t h e p r e s e n t hour, Q(O), given t h e heat flows during
previous hours (Q(i) , i 2 0) and t h e outdoor temp e r a t u r e during p r e s e n t and previous hours ( 8 ( i ) ,
i
0 ) The number of previous hours f o r which
d a t a a r e r e q u i r e d depends on t h e number of
c o e f f i c i e n t s (24 i s t h e upper l i m i t i n ENCORECANADA). n i s l a r g e f o r massive w a l l s , small f o r
l i g h t walls. On t h e b a s i s of Eq. (19), Q(0) i s
computed from
+
v1Q(l) - w1LS(l)
(21)
Heat t r a n s f e r through doors
The h e a t flow through doors i s c a l c u l a t e d
assuming s t e a d y - s t a t e c o n d i t i o n s . The steadys t a t e h e a t flow through a door of a r e a Ad and
thermal conductance Ud due t o t h e d i f f e r e n c e
between e x t e r i o r s o l - a i r temperature, 8 ( 0 ) , and
indoor r e f e r e n c e temperature, Or, i s given by:
Qd(0) = UdAd[e(0) - or]
(22)
Note t h a t Ud must b e given f o r t h e H used i n
Eq. ( 6 ) . The h e a t t r a n s f e r r e d t o o r removed from
t h e indoor a i r by a door s u r f a c e during t h e
p r e s e n t hour i s computed i n t h e s u b r o u t i n e DOORGL
from an equation s i m i l a r t o (21)
Ld(0) = vOQd(0)
+
vlQd(l)
-
w1Ld(l)
(23)
where t h e t r a n s f e r f u n c t i o n c o e f f i c i e n t s vo, v l
and w l a r e i d e n t i c a l t o t h o s e f o r w a l l s .
.
.
Heat t r a n s f e r through windows
n
Q(o) =
1
i=O
n
ai A ( @ ( i ) - 8,)
-
1
bi Q ( i )
(20)
i=l
where A i s t h e n e t wall a r e a . The c o e f f i c i e n t s bi
a r e dimensionless; t h e c o e f f i c i e n t s a i a r e i n w a t t s
p e r square metre degree Kelvin ( B r i t i s h thermal
u n i t s p e r hour square f o o t degree Rankine). The a i
must be computed under t h e assumption t h a t H = l / R 1
= 17 w/m2 K (3 Btu/hr f t 2 R).
The h e a t Q(0) which appears on t h e i n s i d e wall
s u r f a c e during t h e c u r r e n t hour, does n o t have an
Heat t r a n s f e r through windows occurs by t h e
combined mechanism of long-wave r a d i a t i o n , conduct i o n , convection and by transmission and absorpt i o n of s o l a r r a d i a t i o n . Consider a window
g l a z i n g system with o r without i n t e r i o r shading.
The s o l a r h e a t g a i n through such a window during
t h e p r e s e n t hour can be computed from t h e s o l a r
h e a t g a i n f a c t o r , SHGF(0) ,
Qs(0) = AwSwSHGF(0) = AwSw{Idif (0) [tdif +adif F l +
+
Idir(0)
itdir
+
Udir
~ 1 )
(24)
L
where
and previous hour, t h e h e a t t r a n s f e r r e d t o t h e i n door a i r from a window s u r f a c e is obtained by t h e
t r a n s f e r f u n c t i o n method a s
A
= window s u r f a c e area
S
= window shading c o e f f i c i e n t ( r a t i o of
W
W
Idif
s o l a r h e a t g a i n through glazing system
t o s o l a r h e a t gain through doubles t r e n g t h s i n g l e pane window g l a s s under
i d e n t i c a l c l i m a t i c conditions)
= i n t e n s i t y of d i f f u s e s o l a r r a d i a t i o n
from Eq. (3)
Idir = i n t e n s i t y of d i r e c t s o l a r r a d i a t i o n
from Eq. (2)
tdif = 0.8 = t r a n s m i s s i v i t y f o r d i f f u s e r a d i a t i o n of s i n g l e 0.3175 cm (1/8 i n )
t h i c k o r d i n a r y window pane (Mitalas and
Stephenson 1962; Stephenson 1965)
Lw(0) i s computed i n t h e subroutine WNDWGL. The
t r a n s f e r f u n c t i o n c o e f f i c i e n t s vo, vl and w l a r e
i d e n t i c a l t o those f o r wall s u r f a c e s . The so and
s1 c o e f f i c i e n t s f o r a window without shading ( i . e .
no b l i n d s , c u r t a i n s o r drapes) a r e s t o r e d i n t h e
subroutine WNDWGL f o r f o u r values of mass p e r u n i t
floor area
tdir = t r a n s m i s s i v i t y f o r d i r e c t r a d i a t i o n of
s i n g l e window pane
adif = 0.054 = a b s o r p t i v i t y f o r d i f f u s e r a d i a t i o n of s i n g l e window pane (Mitalas and
Stephenson 1962; Stephenson 1965)
= a b s o r p t i v i t y f o r d i r e c t r a d i a t i o n of
a
s i n g l e window pane
dir
F = inward-flowing f r a c t i o n of absorbed
s o l a r r a d i a t i o n f o r s i n g l e window pane
(given by r a t i o of o u t s i d e surface
thermal r e s i s t a n c e t o t o t a l r e s i s t a n c e ;
estimated t o b e 1/3)
The t r a n s m i s s i v i t y and a b s o r p t i v i t y f o r d i r e c t
s o l a r r a d i a t i o n a r e f u n c t i o n s of t h e cosine of t h e
incidence angle (ASHRAE 1977; Stephenson 1965)
( i f tdir < 0.0,
set t
dir
= 0.0)
(25)
+ 8.57881 cos3 ( i ) -8.38135 cos4 ( i )
-
The h e a t t r a n s f e r r e d through a window of thermal
conductance Uw due t o indoor/outdoor temperature
d i f f e r e n c e i s given by an equation s i m i l a r t o (22)
f o r doors
where t h e window e x t e r i o r s u r f a c e temperature, 0(0),
i s computed from Eq. (6) with a I t s e t t o zero
( s o l a r r a d i a t i o n i s a l r e a d y accounted f o r i n Eq.
( 2 4 ) ) . Note t h a t Uw must b e given f o r t h e H used
i n Eq. (6)
.
Given t h e values o f Qs and Qc f o r t h e present
When t h e window has i n t e r i o r shading (blinds,
c u r t a i n s or drapes), so = vo and sl = vl, and t h e
v a l u e s of S, and Uw a r e changed. When t h e window
has a closed s h u t t e r on t h e e x t e r i o r , S, = s o =
0.0, t h e thermal conductance, Uw, i s changed and
t h e e x t e r i o r surface s o l - a i r temperature, 0 (0) , i s
computed from Eq. (6) a s f o r w a l l s u r f a c e s ( i . e .
with a I t included).
I n t e r n a l h e a t gain due t o l i g h t s
The heat t r a n s f e r r e d t o indoor a i r by l i g h t s
r e p r e s e n t s a s i g n i f i c a n t p o r t i o n of t h e e l e c t r i c a l
power supplied s i n c e e l e c t r i c l i g h t s have f a i r l y
low e f f i c a c y . Some of t h e energy i s d i s s i p a t e d by
convection, and a s i g n i f i c a n t p a r t by r a d i a t i o n
(Kimura 1977; Mitalas 1973; and 1973-74). The
t r a n s f e r function method y i e l d s t h e following
formula f o r t h e computation of convective and
r a d i a n t h e a t g a i n from l i g h t s :
where Qu(l) and Qu(2) a r e t h e e l e c t r i c a l energy
input t o l i g h t s one and two hours ago, respect i v e l y . The values f o r Qu a r e given i n a schedule
i n t h e ENCORE-CANADA input d a t a . Notice t h a t
&(O) does n o t appear on t h e r i g h t hand s i d e of
Eq. (29), i n d i c a t i n g a long d e l a y between t h e
energy input t o l i g h t s during t h e present hour and
t h e consequent r i s e i n indoor a i r temperature.
Lu(0) i s computed i n t h e subroutine IGAIN2, where
t h e z - t r a n s f e r f u n c t i o n c o e f f i c i e n t s , y l and y2,
a r e s t o r e d f o r f o u r d i f f e r e n t v a l u e s of mass p e r
unit f l o o r area
146 kg/m2
342 kg/m2
635 kg/m2
976 kg/m2
(30 l b / f t 2 ) (70 l b / f t 2 ) (130 l b / f t 2 ) (200 l b / f t 2 )
y1
y2
0.5
0.5
0.5
0.5
-0.32
-0.37
-0.43
-0.47
I
The w l c o e f f i c i e n t s a r e i d e n t i c a l t o t h o s e used f o r
h e a t t r a n s f e r through w a l l s , doors and windows.
The c o e f f i c i e n t s J1 and J 2 i n Eq. (29) a r e
normally s e t equal t o 1 f o r l i g h t s t h a t a r e turned
on o r o f f s t r i c t l y according t o a given schedule.
However, l i g h t s may a l s o be turned on o r o f f
according t o t h e a v a i l a b i l i t y of s u n l i g h t . Thus,
J 2 = 0 i f t h e r e was s u n l i g h t two hours ago; J1 = 0
i f t h e r e was s u n l i g h t one hour ago.
I n t e r n a l heat gain due t o occupancy and appliancesThe h e a t generated by occupants, Qo, i s t r a n s f e r r e d t o t h e surroundings p a r t l y by r a d i a t i o n ,
p a r t l y by convection. The heat t r a n s f e r r e d by
convection produces an almost instantaneous r i s e i n
surrounding a i r temperature. I f t h e f r a c t i o n of
h e a t t r a n s f e r r e d by r a d i a t i o n i s r (0.5 i s a
reasonable v a l u e ) , t h e f r a c t i o n t r a n s f e r r e d by
convection i s
Loc(o) = ( 1 - r)QO(0)
(30)
The gain due t o r a d i a n t h e a t t r a n s f e r from
occupants f o r t h e p r e s e n t hour i s given by an
equation s i m i l a r t o (29) f o r l i g h t s
Lor(0) = r[ulQo(l)
+
U2Q0(2)1 - WILor(l)
(31)
Notice t h e one hour delay between e x c i t a t i o n and
response. By adding Eqs. (30) and (31) one
o b t a i n s t h e t o t a l h e a t g a i n due t o occupancy a s
Lo(0) = ( 1 - r)Qo(0)
-
+
r[ulQo(l)
u2Qo(2)1
+
-
w L (1)
1 or
(32)
The r a d i a n t component of h e a t gain during previous
hour i s simply
Lor(l) = Lo(l)
-
Loc(l) = Lo(l)
-
(l-r)Qo(l)
(33)
water i n a f u l l tank can s t o r e i s Qs and i s given
by
-
r)Qo(0)
+
[wl(l
- 1'
+
rullQo(l)
+
The t r a n s f e r f u n c t i o n c o e f f i c i e n t ul i s i d e n t i c a l
t o S O and u2 i s i d e n t i c a l t o sl f o r heat t r a n s f e r
through windows. Lo(0) i s computed by t h e subr o u t i n e IGAIN1.
Heat gain from e l e c t r i c a l appliances and equipment, Le(0), i s t r e a t e d i n t h e same manner a s heat
gain from occupants.
'
qm(0), h e a t e r element No. 1 produces
I f El
e(0) = q,,,(O) amount of h e a t and t h e tank w i l l b e
completely f i l l e d with hot water, i . e . , %(O) = Q s .
I f E l < q,,,(O) and E2 2 qm(0), h e a t e r No. 1
t u r n s o f f . and h e a t e r No. 2 w i l l meet t h e demand
( i . e . , qs(0) = Qs) by generating e(0) = qm(0)
amount of h e a t . I f E2 < q,,,(O), t h e demand cannot
be met f o r t h i s hour and only e(0) = E 2 amount of
h e a t i s generated, i . e . , qs(0) = Qs - qm(0) + E2.
This procedure i s implemented i n t h e subroutine
HOTWR t o a r r i v e a t t h e t o t a l e l e c t r i c a l energy,
e(O), consumed by t h e h o t water system during t h e
present hour. I t i s assumed t h a t
Qh(0) = [e(O) - qR(0)1/2 + qg(0) p o r t i o n of e(O)
causes t h e indoor a i r temperature t o r i s e . The
heat t r a n s f e r mechanism is assumed t o b e i d e n t i c a l
t o t h a t f o r occupants and e l e c t r i c a l a p p l i a n c e s .
Thus, t h e i n t e r n a l h e a t gain, Lh (0) , from t h e
domestic hot water system i s computed from Eq. (34)
(Qh r e p l a c e s Qo) by t h e subroutine IGAIN1.
THERMOSTAT CONTROL SYSTEM
The n e t h e a t l o s s o r h e a t g a i n during t h e
p r e s e n t hour, Ln(0), i s given by
+
Lw(0)
+
LU(O)
+
LO(O)
+
Le(0)
+
Lh(0)
(37)
where t h e components a r e
Qw
Most small r e s i d e n t i a l - t y p e b u i l d i n g s i n Canada
have an e l e c t r i c hot water h e a t e r which c o n s i s t s of
a tank of volume V and an upper and lower h e a t e r
element with maximum output of El and E2, respect i v e l y . The incoming c o l d water i s a t temperature
B i n and t h e hot water leaving t h e tank i s a t temThe maximum amount of h e a t t h e
p e r a t u r e Bout.
-.
4
: basement f l o o r , Eq. (8)
Qf : basement walls, Eq. (9)
Li : a i r i n f i l t r a t i o n
Heating from domestic hot water system
-
where q S ( l ) is t h e amount o f h e a t l e f t from t h e
previous hour and qg(0) r e p r e s e n t s t h e standby
l o s s from t h e tank (CSA 1973) and t h e d i s t r i b u t i o n
l i n e s (Schultz and Goldschmidt 1978).
and hence, by s u b s t i t u t i n g i n Eq. (32), one o b t a i n s
Lo(0) = ( 1
-
where p i s t h e d e n s i t y and C i s t h e s p e c i f i c h e a t
If the
of water a t temperature (BoU: + Bin)/2.
volume of hot water demand during t h e p r e s e n t hour
i s v(O), t h e amount of h e a t , qd(O), r e q u i r e d t o
r a i s e i t s temperature from Bin t o Bout i s p v ( 0 ) x
Cp (gout - Bin). The make-up h e a t r e q u i r e d i s
given by
LS : walls and c e i l i n g / r o o f system, Eq. (21)
Ld : doors, Eq. (23)
Lw : windows, Eq. (28)
LU : l i g h t s , Eq. (29)
Lo : occupants, Eq. (34)
Le : e l e c t r i c a l appliances
Lh : hot water system.
c
4
4
The n e t e f f e c t o f indoor temperature v a r i a t i o n
during a h e a t i n g season on basement h e a t l o s s e s
(Qw and Qf) i s neglected i n ENCORE-CANADA. The
components Lu, Lo, Le and Lh can be considered
independent o f indoor temperature v a r i a t i o n s . However, t h e components L i , Ls, Ld and Lw, which a r e
computed with r e s p e c t t o a c o n s t a n t indoor
r e f e r e n c e temperature, Or, a r e f u n c t i o n s o f indoor/
outdoor temperature d i f f e r e n c e . Therefore t h e
e f f e c t of indoor temperature v a r i a t i o n cannot b e
neglected.
A z - t r a n s f e r f u n c t i o n which r e l a t e s t h e deviat i o n of indoor a i r temperature, B i , from t h e
r e f e r e n c e temperature, Or, t o t h e n e t amount of
h e a t added t o o r removed from t h e indoor a i r can be
derived from h e a t balance c o n s i d e r a t i o n s and t h e
thermal c h a r a c t e r i s t i c s of t h e b u i l d i n g enclosure
(ASHRAE 1977; M i t a l a s 1972). The t r a n s f e r f u n c t i o n
has t h e form of Eq. (16) with p = 1, and t h e
c o e f f i c i e n t wl i s i d e n t i c a l t o t h e one used i n
Eqs. (21), (23), (28), (29) and (34). The
c o e f f i c i e n t s vo, v l and v2 a r e c a l c u l a t e d from
v1 = glA
+
w [K
1 as
+
I(1) ]
(39)
They were derived f o r u n i t f l o o r a r e a under t h e
assumption o f n e g l i g i b l e a i r i n f i l t r a t i o n and
conductance between indoor a i r and e x t e r i o r
surroundings. Equations (38) t o (40) a d j u s t t h e
normalized c o e f f i c i e n t s f o r a s p e c i f i c s i t u a t i o n
(ASHRAE 1977; M i t a l a s 1972).
Let t h e e x c i t a t i o n be t h e d e v i a t i o n of indoor
a i r temperature from t h e r e f e r e n c e temperature,
B i ( t ) - Or, and l e t t h e response b e t h e a l g e b r a i c
sum of t h e h e a t demand, E ( t ) , and t h e n e t h e a t
l o s s o r h e a t gain, L n ( t ) . Application of Eq. (19)
y i e l d s a r e l a t i o n s h i p between h e a t demand and i n door a i r temperature
By denoting
one can r e w r i t e Eq. (41) a s
where
g . = normalized room a i r t r a n s f e r f u n c t i o n
1
c o e f f i c i e n t s ( i = 0, 1, 2)
A = floor surface area
Kas
Since both E(0) and e i ( 0 ) a r e unknown, a n o t h e r
independent r e l a t i o n s h i p between h e a t demand and
indoor temperature during t h e p r e s e n t hour i s
r e q u i r e d . T h i s i s provided by t h e mathematical
model of t h e thermostat c o n t r o l system.
= t o t a l thermal conductance between indoor
a i r and surroundings (computed i n t h e subr o u t i n e FCFACT)
I ( j ) = a i r i n f i l t r a t i o n h e a t i n g load p e r u n i t
indoor/outdoor temperature d i f f e r e n c e ,
j hours ago.
The normalized room a i r t r a n s f e r f u n c t i o n coeff i c i e n t s , go, g l and g2, a r e s t o r e d i n t h e s u b r o u t i n e
DEMAND f o r f o u r d i f f e r e n t v a l u e s of mass p e r u n i t
f l o o r area
A simple thermostat, which is s e t a t temperat u r e BS, f u n c t i o n s a s shown i n Fig. 2 where t h e
h e a t o u t p u t , E, of t h e h e a t i n g system i s p l o t t e d
a g a i n s t t h e indoor a i r temperature, Bi.
A band,
Bb, about t h e s e t p o i n t , €Is, determines when t h e
h e a t i n g system i s t u r n e d on o r o f f .
When t h e temperature f a l l s below es - %/2,
t h e h e a t i n g system t u r n s on; when t h e temperature
r i s e s above Os + 8b/2, t h e h e a t i n g system t u r n s
o f f . During a 1 h period t h e thermostat may go
through s e v e r a l on/off c y c l e s . Consequent1 y, t h e
on/off c y c l e s of a thermostat c o n t r o l system cann o t be simulated i n hour- by -hour c a l c u l a t i o n s .
I n s t e a d , t h e s i m p l i f i e d r e p r e s e n t a t i o n shown i n
F i g . 3 i s used. I t i s based on t h e o b s e r v a t i o n
t h a t w i t h i n t h e t h e r m o s t a t band, 8b, t h e t o t a l
amount o f h e a t produced i s p r o p o r t i o n a l t o t h e
indoor a i r temperature.
Mathematically, t h e p r o p o r t i o n a l band thermos t a t c o n t r o l system i s r e p r e s e n t e d a s follows:
E(0) = S ei(0) + Eo(0)
es(o)
- d eb 5
E(0) = 0.0
5
es(0)
+
f
ei(o) > es(o)
+
f eb
ei(o)
9,
(44b)
(44~)
where t h e s l o p e , S, and t h e v e r t i c a l i n t e r c e p t ,
Eo(0), a r e given by ( s e e F i g . 3)
s = - -Emax
(45)
Ob
and
E 0 (0) =
4
Emax
-
S es(0)
(46)
r e s p e c t i v e l y . Equations (43) and (44a, b, c ) can
be solved simultaneously f o r ei(0) and E(0) t o g i v e
t e s t s , however, can o n l y prove t h e r e l a t i v e
v a l i d i t y of t h e program r e s u l t s . Absolute
v a l i d i t y remains t o be determined by comparison
with measurements. C o l l e c t i o n of d a t a from houses
l o c a t e d i n Ottawa i s c u r r e n t l y being undertaken by
t h e Division of Building Research o f t h e National
Research Council of Canada.
The u s e f u l n e s s of t h e r e s u l t s of a computer
s i m u l a t i o n of b u i l d i n g energy consumption is
d i r e c t l y r e l a t e d t o t h e l i m i t a t i o n s bf t h e mathem a t i c a l model and t h e accuracy o f t h e a v a i l a b l e
input d a t a . More o f t e n than n o t , b u i l d i n g s c i e n c e
r e l i e s on crude e s t i m a t e s o f t h e p r o p e r t i e s of
b u i l d i n g m a t e r i a l and environmental c o n d i t i o n s .
I n a d d i t i o n , complicated s t o c h a s t i c processes such
a s weather, s o l a r r a d i a t i o n , s o l - a i r temperature
and i n t e r n a l b u i l d i n g environment, a r e d i f f i c u l t
t o model without s i m p l i f y i n g assumptions. When a n
assumption proves t o be i n a p p r o p r i a t e , a more
d e t a i l e d , p o s s i b l y more complicated, mathematical
model has t o b e adopted.
ENCORE-CANADA was w r i t t e n with t h e i n t e n t t o
study a s many a s p o s s i b l e of t h e f a c t o r s t h a t
i n f l u e n c e energy consumption i n small r e s i d e n t i a l type b u i l d i n g s . This n e c e s s i t a t e d a d e t a i l e d ,
numerically s t a b l e model and, consequently, a
lengthy computer program. The amount of arithmet i c computing time r e q u i r e d p e r s i m u l a t i o n of a
h e a t i n g season i s a t p r e s e n t o f t h e o r d e r of one
hour on an IBM 360/67 computer, with about 5/6th
of t h e time being s p e n t by i n f i l t r a t i o n c a l c u l a t i o n s . I t i s expected t h a t i f more e f f i c i e n t
numerical methods a r e adopted f o r i n f i l t r a t i o n
c a l c u l a t i o n s , t h e computing time f o r one simulat i o n would be 10 t o 15 minutes.
The d e r i v a t i o n s given a r e based on a n o i l f i r e d furnace h e a t i n g system w i t h warm a i r d i s t r i bution; t h e b u i l d i n g i s assumed t o have no i n t e r n a l p a r t i t i o n s . ENCORE-CANADA, however, can a l s o
simulate e l e c t r i c a l l y heated, p a r t i t i o n e d b u i l d i n g s . In t h i s c a s e many o f t h e program a l g o r i t h m s
a r e s i m p l i f i e d and computing t i m e s a r e reduced.
ACKNOWLEDGEMENTS
The a u t h o r s wish t o express t h e i r g r a t i t u d e t o
G.P. Mitalas f o r h i s generous a s s i s t a n c e throughout t h i s r e s e a r c h work. The basement h e a t l o s s
model and a s s o c i a t e d s u b r o u t i n e s were o r i g i n a l l y
produced by C . J . S h i r t l i f f e who c o n t r i b u t e d
m a t e r i a l t o t h e s e c t i o n on basement h e a t l o s s
c a l c u l a t i o n s . The TRUMP f i n i t e d i f f e r e n c e comput a t i o n s were c a r r i e d out by W.C. Brown.
The indoor a i r temperature, 8 i ( O ) , and t h e h e a t
supplied by t h e h e a t i n g system during t h e p r e s e n t
hour, E(O), a r e computed according t o Eqs.
(47a, b, c ) i n t h e subroutine DEMAND. Note t h a t
e i , es, Z, Eo and E v a r y with time. The thermostat
s e t t i n g s , es, a r e s p e c i f i e d f o r every hour of t h e
day by a schedule i n t h e ENCORE-CANADA i n p u t d a t a .
CONCLUSIONS
The s e t of mathematical procedures used by t h e
ENCORE-CANADA computer program has been d e s c r i b e d .
T r i a l runs with ENCORE-CANADA produced r e s u l t s t h a t
e x h i b i t expected behaviour
Such computational
.
This paper i s a c o n t r i b u t i o n from t h e Division
o f Building Research, National Research Council of
Canada and i s published with t h e approval of t h e
D i r e c t o r of t h e Division.
REFERENCES
ASHRAE 1975. Procedure f o r determining h e a t i n g
and c o o l i n g loads f o r computerizing energy
c a l c u l a t i o n s ; algorithms f o r b u i l d i n g h e a t
t r a n s f e r s u b r o u t i n e s , compiled and published by
ASHRAE Task Group on Energy Requirements f o r
Heating and Cooling of Buildings, p. 17-32.
ASHRAE 1976. Handbook, 1976 Systems Volume,
American S o c i e t y o f Heating, R e f r i g e r a t i n g and
Air-Conditioning Engineers, p . 43.8.
ASHRAE 1977. Handbook and Product Directory,
Fundamentals, American S o c i e t y of Heating,
R e f r i g e r a t i n g and Air-Conditioning Engineers,
Ch. 25 and 26.
MITALAS, G.P., 1972.
T r a n s f e r f u n c t i o n method of
c a l c u l a t i n g c o o l i n g loads, h e a t e x t r a c t i o n and
space temperature,
ASHRAE J o u r n a l , 14, 12,
p. 54-56.
MITALAS, G .P., 1973.
caused b y l i g h t s ,
p . 37-40.
C a l c u l a t i n g c o o l i n g load
ASHRAE J o u r n a l , 15, 6,
AYRES, J .M. 1977. P r e d i c t i n g b u i l d i n g energy
requirements - t h e computer dilemma, Heating/
Piping/Air c o n d i t i o n i n g , 49, p . 44-50.
MITALAS, G.P., 1973-74. Cooling l o a d caused by
l i g h t s , Trans. Can. Soc. Mech. Eng., 2, 3,
p . 169-174.
BOILEAU, G .G., and LATTA, J .A., 1968."Calculation of
basement h e a t losses," National Research Council
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MITALAS, G.P., 1976. Net annual h e a t l o s s f a c t o r
method f o r e s t i m a t i n g h e a t requirements of
b u i l d i n g s , Building Research Note No. 117,
D i v i s i o n of Building Research, National
Research Council o f Canada, Ottawa.
CSA 1973. Canadian S t a n d a r d s Association, Standard
C191-1973,
Performance requirements f o r
e l e c t r i c s t o r a g e - t a n k water h e a t e r s , Canadian
S t a n d a r d s A s s o c i a t i o n Inc , 178 Rexdale Blvd.,
Rexdale, Ont., M9W 1R3.
.
EDWARDS, A.L. 1968. "TRUMP: A computer program f o r
t r a n s i e n t and s t e a d y s t a t e temperature d i s t r i b u t i o n s i n multidimensional systems,"Lawrence
R a d i a t i o n Laboratory, U n i v e r s i t y of C a l i f o r n i a ,
Livermore, (UCRL-14754-REV. I ) , ( D i s t r i b u t e d by:
Clearinghouse f o r Federal S c i e n t i f i c and
Technical Information, U .S. Department of
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KIMURA, K.-1. 1977. S c i e n t i f i c Basis o f A i r
Conditioning. Applied Science P u b l i s h e r s Ltd.,
London.
KONRAD, A . , LARSEN, B .T., and SHAW, C . Y . 1978.
"Programmed computer model of a i r i n f i l t r a t i o n i n
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t h e Use of Computers f o r Environmental
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A l b e r t a , Canada.
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T r a n s a c t i o n s , 74, P a r t 11, p . 182.
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. .
.
MITALAS, G P , and ARSENEAULT, J .G , 1972.
"FORTRAN IV program t o c a l c u l a t e z - t r a n s f e r
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SCHULTZ, W .W., and GOLDSCHMIDT, V .W.,
E f f e c t of d i s t r i b u t i o n l i n e s on stand-by l o s s
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84, P a r t 1.
STEPHENSON, D.G ., 1965.
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p . 81-86.
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S o l a r Energy, IX, 2,
STEPHENSON, D . G . , and MITALAS, G .P., 1967.
Cooling load c a l c u l a t i o n s by thermal r e s p o n s e
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.
THRELKELD, J .L , 1962. Thermal Environmental
Engineering, Ch. 14, S o l a r Radiation,
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TRUXAL, J . G . , 1955. Automatic Feedback C o n t r o l
System S y n t h e s i s , McGraw-Hill, New York, Ch. 9 .
7
450
1
( INSIDE
I
I
I
I
I
I
I
I
E
I
I
1
1
I
I
1
I
I I
ill
I
OUTSIDE I
SURFACE
OF WALL I
I
1
I
I
8(t)
SOL-AIR
TEMP.
I
1
R3
1
I
1
IK2
I
I
1
~2
I
I
1
AIR
FILM
1
I
7
T~
TI
SURFACE
1 OF WALL
8
1
1
lK4
IK5
I
CONST.
REFERENCE
TEMP.
!'bl
I
4
I Q(r)
I
b
J
I
,
~
I
1
I AIRSPACE I
52
I
I
I
I
I
'
I
4
1
54
~ 5 1I
1
L
1
I I
I AIR
s5
1
FIGURE 1
IF 1 L M
*,
I
*
--------F=--= 2 --=,---Emmy
TYPICAL WALL CONSTRUCTION
E
,
'-
-1
I
L
I
I
I
1
1
IA
I
I
I
I
I
I
I
I
I
I
j
I
====s====2======
--&
;-
o
v
oi
FIGURE 2
I L L U S T R A T I O N O F A SIMPLE THERMOSTAT C O N T R O L
A C T I O N ( N O T T O SCALE)
b
I
E
A
E0
P\
1
Eo
=
-2
Emax
-
s 8,
\
\
\
.' \
\
Z
\
\
-
\
\
\
\
+
\
Emax
FIGURE 3
Emox
-
SLOPE
-__-------I
2
-
5 =
-
P R O P O R T I O N A L B A N D THERMOSTAT C O N T R O L A C T I O N
( N O T TO SCALE)
Emox
-
b
D
'i
>
I