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ENCORE-CANADA : Computer program for the study of energy
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Konrad, A.; Larsen, B. T.
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Ser
I
no.859
National Research
Conseil national
e ,
a
i t > , /,
I
#
Council Canada
de recherches Canada
i
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
Thie publication is being dietributed by the Divieion of Building R e s e a r c h of the National R e s e a r c h Council of Canada. I t should not be reproduced in whole o r in p a r t without permiesion of the original publisher. The Di- vieion would be glad to be of aeoietance in obtaining such permiseion.
Publicatione of the Divieion may be obtained by m a i l - ing the appropriate remittance ( a Bank, Exprero, o r b e t Office Money Order, o r a cheque, made payable to the Receiver General of Canada, c r e d i t NRC) to the National Research Council-of Canada, Ottawa. KIA OR6. Stamps a r e not acceptable.
A l i s t of a l l publications of the Division i s available and may be obtained f r o m the Publications Section, Diviaion of Building Research. National R e s e a r c h Council of Canada, Ottawa. KIA OR6.
ENCORE-CANADA: COMPUTER PROGRAM FOR THE STUDY OF ENERGY CONSUMPTION OF RESIDENTIAL BUILDINGS I N CANADA
A . ~ o n r a d l and B .T. ~ a r s e n ~
l ~ i v i s i o n of Building Research, National Research Council o f Canada
Ottawa, Canada
2 ~ o r w e g i a n Building Research I n s t i t u t e , Oslo, Norway (Guest worker w i t h DBR/NRC, 1975)
ABSTRACT
-
The paper d e s c r i b e s t h e mathematical methods employed in t h e ENCORE-CANADA computer program which p r e d i c t s t h e hourly a s we1 1 a s t h e 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 con- s u l t i n g e n g i n e e r s i n t e r e s t e d in energy c o n s e r v a t i o n 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 h e a t 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 , t r a n s m i s s i o n h e a t l o s s e s and s o l a r h e a t g a i n s . 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 v a r i o u s Canadian c i t i e s a r e used t o s i m u l a t e 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 c h a u f f a g e 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-CANADAs ' a d r e s s e s u r t o u t aux c h e r c h e u r s , aux concepteurs e t aux i n g e n i e u r s - c o n s e i l s 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 g a i n s 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 f o n d a t i o n s 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 g a i n s 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 t h e r m o s t a t . Des donnees m 6 t ~ o r o l o g i q u e s e 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 s i m u l e r l e s c o n d i t i o n s e x t e r i e u r e s .
INTRODUCTION s t u d y , f o r a given b u i l d i n g d e s i g n , t h e r e l a t i o n The d e t e r m i n a t i o n o f t h e h e a t energy r e q u i r e d t o
m a i n t a i n p r e s c r i b e d i n d o o r 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 ( M i t a l a s 1976). Owing t o t h e r e l a t i v e l y h i g h 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 . F u r t h e r - more, 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 s a v i n g s a s a r e s u l t o f energy conser- v 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 .
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, s o l a r r a d i a t i o n ) ,-
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 c o n t e n t s ,-
t y p e s of f e n e s t r a t i o n ,-
e x t e r i o r s u r f a c e s o l a r a b s o r p t i v i t y ,-
a i r i n f i l t r a t i o n ,-
t h e r m o s t a t 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 .T h i s p a p e r d e s c r i b e s a mathematical model f o r OVERVIEW OF MODEL
hour-by-hour s i m u l a t i o n of energy consumption o f ENCORE-CANADA r e q u i r e s two i n p u t d a t a s e t s : 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 one describing the building model, and the other d a t a . The a s s o c i a t e d computer program, ENCORE- c o n t a i n i n g h o u r l y weather and s o l a r r a d i a t i o n CANADA (Energy Consumption o f Residences i n Canada) d a t a . The b u i l d i n g model h a s t h e f o l l o w i n g i s t h e Canadian v e r s i o n o f t h e Norwegian ENCORE components:
-
basement w i t h below- and abcve-grade p o r t i o n s ,-
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 i n c l u d e sc e i l i n g / r o o f system, w a l l s , doors and windows,
-
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 w a n a i r d i s t r i b u t i o nh e a t i n g system c o n s i s , t i n g o f o i l - f i r e d furnace,, .smoke p i p e w i t h b a r o m e t r i c damper, and chimney,
-
two-element e l e c t r i c h o t w a t e r h e a t e r and s t o r a g etank from which w a t e r i s drawn a c c o r d i n g t o a 24 h schedufe,
-
l i g h t s which a r e on o r o f f a c c o r d i n g t o a 24 h s c h e d u l e ,-
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 t u r n e d on o r o f f a c c o r d i n g 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 v a r y i n g numbers a c c o r d i n g t o a 24 h s c h e d u l e . Schedules f o r h o l i d a y s may b e d i f f e r e n t from t h o s e 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 '- occupants,-
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 w a t e r consumption.I
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 compo- n e n t s t o compute h e a t g a i n and l o s s through - below-grade p o r t i o n o f basement,
-
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,1
-
d o o r s ,I
-
windows, andI
-
a i r i n f i l t r a t i o n .These g a i n s o r l o s s e s , w i t h t h e e x c e p t i o n 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 t e m p e r a t u r e . 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
WEATHER AND SOLAR RADIATION DATAENCORE-CANADA may u s e 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 . Theycorrespond t o t h r e e c o n s e c u t i v e y e a r s o f d a t a f o r t h e f o l l o w i n g 13 Canadian c i t i e s . Vancouver, B .C
.
Edmonton, A l t a . Winnipeg, Man. Ottawa, Ont. F r e d e r i c t o n , N . B.
Toronto, Ont.
Montreal, Que. H a l i f a x , N.S.
S t . J o h n ' s , Nfld. Summerland, B .C.
S u f f i e l d , A l t a . (1970-72) Swift C u r r e n t , Sask. (1970-72) Charlottetown, P .E.I.
(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),-
h o l i d a y i n d i c a t o r (used f o r s e l e c t i n g s c h e d u l e s ) ,-
hour a n g l e when s o l a r a l t i t u d e i s zero ( s u n r i s ea n g l e , ho),
- t a n g e n t of d e c l i n a t i o n a n g l e , t a n ( & ) ,
-
ground r e f l e c t i v i t y , R (0.4 w i t h snowcover, 0.2 w i t h o u t ).
For each day o f t h e y e a r t h e d a t a s e t c o n t a i n s 24 hourly v a l u e s o f
a ) dry-bulb t e m p e r a t u r e of ambient a i r
(O,),
b) t o t a l c l o u d amount i n t e n t h s (C), c) wind speed (V), 1 0 m above ground, a t
m e t e o r o l o g i c a l s i t e , d) wind d i r e c t i o n ,
e) atmospheric p r e s s u r e ( P ) ,
f ) hour a n g l e ( h ) , (sun i s above h o r i z o n i f I h l < l h O O ,
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,), 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) ,
i ) conversion f a c t o r f o r i t e m s 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 a p p l i e s t h e f o l l o w i n g c o n v e r s i o nformula t o items g) and h)
where
I ( 0 ) = average s o l a r r a d i a t i o n d u r i n g 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 d u r i n g 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 p r e v i o u s hour.
The f a c t o r 100/[100-e(O)] i n Eq. (1) is a cloud cover m o d i f i e r d e r i v e d b y 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 s u r f a c e .
ENCORE-CANADA u s e s t h e i n f o r m a t i o n 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 compo- n 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 -
e = e
1 1 + c o s ( ) Cf a c e o r i e n t a t i o n t h e s u b r o u t i n e DIRDIF c a l c u l a t e s o
+ y
['It - E D (1 - = ) I ( 6 )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 where
a = s u r f a c e s o l a r a b s o r p t i v i t y
Idir = Idn c o s ( i ) E = r a d i a n t i n t e r c h a n g e 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
= Idn{ [ s i n ( & ) sin(R) +cos(G)cos(~)cos(h)]cos(q) b u i l d i n g s i t e s , r e s p e c t i v e l y )
D = d i f f e r e n c e between i n c i d e n t long-wave sky
-
[ s i n ( 6 ) c o s ( R )-
cos(6) sin(R)cos(h)]sin(q)cos(a) r a d i a t i o n and r a d i a t i o n e m i t t e d by 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- c o s ( & ) s i n ( h ) s i n ( q ) s i n ( a )
1
(2) (63.1 w/m2 (20 Btu/hr f t 2 ) was used) where i = i n c i d e n c e a n g l e a = s u r f a c e azimuth a n g l e (measured c o u n t e r - clockwise from s o u t h ) q = s u r f a c e t i l t a n g l e (measured from t h e h o r i zonta 1) L = l a t i t u d e 6 = d e c l i n a t i o n a n g l e 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 H = 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 e q u a l t o 17 w/m2 K,3 Btu/hr f t 2 R).
I n a d d i t i o n t o t h e t h i r t y - n i n e d a t a s e t s mentioned above, t h e mean ( e m ) , a m p l i t u d e (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 func- t 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 c o n s e c u t i v e y e a r s ) a r e s t o r e d i n t h e b l o c k d a t a s u b r o u t i n e 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 o b t a i n e d from a F o u r i e r a n a l y s i s of h o u r l y v a l u e s of ground s u r f a c e s o l - a i r tempera- t 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 u s i n g 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)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 o b t a i n e d from T h r e l k e l d (1962) Y = 0.55 + 0 . 4 3 7 c o s ( i ) + 0 . 313cos2 ( i ) c o s ( 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 + c o s (6) c o s ( t ) c o s ( 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 t e m p e r a t u r e from t h e following e q u a t i o n (ASHRAE 1977) :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 below- grade p o r t i o n of basements. The f o l l o w i n g f a c t o r s were found t o b e important:
- amount of w a l l below grade,
-
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 w a t e r 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.
F a c t o r s such a s f r e e z i n g and thawing, tempera- t 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 p a r a m e t e r s
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 a l g o r i t h m Gfw = s t e a d y - s t a t e h e a t conductance between was s e l e c t e d r a t h e r t h a n i n c o r v o r a t i n g a f i n i t e .., basement f l o o r and w a t e r t a b l 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 G f g c = c o s i n e component of dynamic h e a t conductance between basement f l o o r and
-
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 temper- a 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 c o n s t a n t 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 h o u r l y 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 w i t h t h e
s u b r o u t i n e BTLOSS according t o t h e f o l l o w i n g formulae
ground s u r f a c e
G = s i n e component of dynamic h e a t conduc- f g s t a n c e between basement f l o o r and ground
s u r f a c e .
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 n e g l e c t e d . The conductances Gwg, Gww, Gwgc, G w g s , 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 s u b r o u t i n e BASMNT. I n o r d e r t o a r r i v e a t t h e s e e i g h t conductances, a v e r s i o n of t h e method d e s c r i b e d by Boileau and L a t t a (1968) i s used.2lT 2lT
+ [ G w g c ~ ~ ~ ( n t + p) + G s i n ( ? t +p)]Ba (8) AIR INFILTRATION CALCULATIONS wgs
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 d e s c r i b e d 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 l e n g t h - t o - w i d t h
Qf
= Gfg(om-
gin) + G f w (ewt-
gin)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
2 l ~ 2.rr r e p r e s e n t e d b y s e v e r a l holes e q u a l l y spaced along+ [Gfgccos (T t + p) + G sin(f;- t + p ) ] e a
f gs (9) pressure a t t h e ground l e v e l (P1) i s known, t h e the h e i g h t o f the component. I f t h e indoor
Where mass E l m r a t e o f a i r {Gj) through t h e j - t h h o l e
t = hour of t h e y e a r ( 0
5
t5
n) can be computed f r o m n = number o f hours i n a y e a r (n = 8760 f o ro r d i n a r y y e a r , n = 8784 f o r l e a p year)
J
Owt = water t a b l e temperature -n .
J c .v2 1 1
G . = a p ( ~ P + ; [++g hj (---)I - P (101 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 temper-
I n I
11
o i' ' oI
a t u r e
0 m = mean annual ground s u r f a c e s o l - a i r temperature
0 = amplitude o f fundamental F o u r i e r 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 o f fundamental F o u r i e r s i n e component o f ground s u r f a c e s o l - a i r temperature f u n c t i o n
G
= s t e a d y - s t a t e h e a t conductance between base- Wg ment w a l l s and ground s u r f a c e= 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 water t a b l e
G = c o s i n e component of dynamic h e a t conductance Wgc between basement w a l l s and ground s u r f a c e G = s i n e component o f dynamic h e a t conductance
Wgs between basement w a l l s and ground s u r f a c e
where s = + 1 f o r a i r i n f i l t r a t i o n , and =
-
1 f o r a i r e x f i l t r a t i o n p = d e n s i t y o f indoor a i r i f s =-
1, and = d e n s i t y o f outdoor a i r i f s = + 1 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 1n: = flow exponent f o r a i r flow through j - t h h o l e
P = outdoor atmospheric p r e s s u r e a t ground l e v e l
R = g a s c o n s t a n t f o r a i r
c = wind p r e s s u r e c o e f f i c i e n t where j - t h h o l e j i s l o c a t e d (depends on wind d i r e c t i o n and
b u i l d i n g o r i e n t a t i o n ) G = s t e a d y - s t a t e h e a t conductance between base-
g = g r a v i t a t i o n a l a c c e l e r a t i o n h . = h e i g h t of j - t h h o l e above ground I 0 = outdoor a i r t e m p e r a t u r e 0 0 . = i n d o o r a i r temperature. 1
Using t h e mathematical model o f t h e f u r n a c e , smoke p i p e , b a r o m e t r i c 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 v a l u e 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 b a l a n c e e q u a t i o n 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 f u r n a c e o p e r a t i o n . I t i s computed by m u l t i p l y i n g t h e sum o f a l l p o s i t i v e Gj I S b y ( 0 i - 0,) and b y 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.
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 b y indoor a i r from occupants, l i g h t s , domestic h o t w a t e r 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 i n c l u d i n g e f f e c t s of t h e r m o s t a t 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 c o n t e x t 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)
.
When a c o n t i n u o u s 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 f o l l o w s : I f b o t h i n p u t , i ( t ) , and o u t p u t , o ( t ) , of a l i n e a r , i n v a r i a b l e system a r e e x p r e s s e d 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) i s known, t h e response t o any g i v e n i n p u t 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 e x p r e s s e d a s a polynomial i n z, a s f o l l o w s :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 r e s p o n s e 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 b e a s s i g n e d a f i n i t e v a l u e b y 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 i s sometimes impossible t o f i n d t h e impulse r e s p o n s e 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 i n p u t 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 b e transformed i n t o t h e r a t i o of two poly- nomials (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 . ,
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
where wl i s c a l l e d Common R a t i o and i s d e f i n e d by The z-transform of i ( t ) i s o b t a i n e d from t h e
Laplace t r a n s f o r m 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 byw = l i m . k (nT)
7
444
Whether given i n t h e form o f Eqs. (14), (15) o r immediate e f f e c t on t h e indoor a i r t e m p e r a t u r e . ( 1 6 ) , i n g e n e r a l , K(z) can always b e thought of a s The d e l a y depends on t h e t y p e of i n t e r i o r f i n i s h , t h e r a t i o of two polynomials, i . e . , K(z)=N(z)/D(z)
.
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 Given t h e c o e f f i c i e n t s Ni and Di of t h e numerator 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 and denominator, r e s p e c t i v e l y , t o g e t h e r w i t h a t h e form given by Eq. (16) and a s a f u n c t i o n o f s e r i e s of sampled v a l u e s i(nT) and o(nT) o f an 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 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 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 nt h e c u r r e n t v a l u e of t h e r e s p o n s e o(qT), t h e l a t t e r t h e s u b r o u t i n e WALLGL f o r f o u r d i f f e r e n t v a l u e s
-
can b e s o l v e d by e q u a t i n g t h e c o e f f i c i e n t s of t h e o f mass p e r u n i t f l o o r a r e a : h i g h e s t power of z on both s i d e s of t h e e q u a t i o n : 146 kg/m2 342 kg/m2 635 kg/m2 976 kg/m2 C d (30 l b / f t 2 ) (70 l b / f t 2 ) (130 l b / f t 2 ) (200 l b / f t 2 ) O(z)D(z) = I(z)N(z) (18) 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 v 0 0.703 0.681 0.676 0.676 c o e f f i c i e n t s , one o b t a i n s v -0.523 1 -0.551 -0.606 -0.646 1 o(qT) = - { i ( 0 ) N p + i ( T ) N +...
+ i [ ( p - l ) T ] N 1 + w 1 -0.82 -0.87 -0.93 -0.97Do
P- 1Note t h a t v l = 1 + w l
-
vo. Based on Eq. (19),+ i(pT)N 0
-
o(0)D - O ( T ) D ~ - ~-
...
-
9 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 w a l l s u r f a c e d u r i n g 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
-
~ [ ( q - l l ~ l ~ ~ ) (19) according t oHeat t r a n s f e r through w a l l s and c e i l i n g Ls(o) = voQ(0) + v1Q(l) - w1LS(l) (21)
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 i n n e r 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 Heat t r a n s f e r through d o o r s 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 The h e a t flow through doors i s c a l c u l a t e d 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 conduc- assuming s t e a d y - s t a t e c o n d i t i o n s . The s t e a d y - t 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 ) . s t a t e h e a t flow through a door of a r e a Ad and There i s no r e s t r i c t i o n on t h e number of l a y e r s . thermal conductance Ud due t o t h e d i f f e r e n c e The s o l - a i r t e m p e r a t u r e , B ( t ) , a t t h e between e x t e r i o r s o l - a i r temperature, 8 ( 0 ) , and 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 i n d o o r r e f e r e n c e temperature, Or, i s g i v e n by: 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 Qd(0) = UdAd[e(0) - or] (22) 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 d e s c r i b e d by Stephenson and M i t a l a s (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 - Note t h a t Ud must b e g i v e n f o r t h e H used i n c i e n t s , a i and b i , (ASHRAE 1977; M i t a l a s and Eq. ( 6 ) . The h e a t t r a n s f e r r e d t o o r removed from A r s e n e a u l t 1972) t o compute t h e h e a t flow during t h e indoor a i r by a door s u r f a c e d u r i n g t h e t h e p r e s e n t hour, Q(O), g i v e n t h e h e a t flows during 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 p r e v i o u s h o u r s (Q(i)
,
i2
0) and t h e outdoor tem- from an e q u a t i o n s i m i l a r t o (21)p e r a t u r e d u r i n g p r e s e n t and previous hours ( 8 ( i ) ,
i 0 ) The number of p r e v i o u s 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 Ld(0) = vOQd(0) + v l Q d ( l )
-
w1Ld(l) (23)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 ENCORE-
CANADA). n i s l a r g e f o r massive w a l l s , small f o r 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 l i g h t walls. On t h e b a s i s of Eq. (19), Q(0) i s 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 .
computed from .
.
Heat t r a n s f e r through windows
n n
Heat t r a n s f e r through windows o c c u r s b y t h e Q ( o ) =
1
a i A ( @ ( i )-
8,)-
1
bi Q ( i ) (20) combined mechanism of long-wave r a d i a t i o n , conduc-i = O i=l t i o n , convection and b y t r a n s m i s s i o n and a b s o r p -
t 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 w i t h o r without i n t e r i o r shading. where A i s t h e n e t w a l l a r e a . The c o e f f i c i e n t s bi The s o l a r h e a t g a i n through such a window d u r i n g 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 t h e p r e s e n t hour can b e computed from t h e s o l a r p e r s q u a r e metre degree Kelvin ( B r i t i s h thermal h e a t g a i n f a c t o r , SHGF(0)
,
u n i t s p e r hour s q u a r e f o o t degree Rankine). The a i
must b e computed under t h e assumption t h a t H = l / R 1 Qs(0) = AwSwSHGF(0) = AwSw{Idif (0) [tdif +adif F l +
= 17 w/m2 K ( 3 Btu/hr f t 2 R).
The h e a t Q(0) which a p p e a r s on t h e i n s i d e w a l l
+ Idir(0)
itdir
+ Udir ~ 1 )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 (24)
where
A = window s u r f a c e a r e a
W
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
o b t a i n e d by t h e t r a n s f e r f u n c t i o n method a sS = window shading c o e f f i c i e n t ( r a t i o of
W
s o l a r h e a t g a i n through g l a z i n g system t o s o l a r h e a t g a i n through double- s 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 c o n d i t i o n s ) Idif = i n t e n s i t y o f 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 ( M i t a l a s and Stephenson 1962; Stephenson 1965) 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 = 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
dir s i n g l e window pane
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 s u r f a c e 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 ; e s t i m a t e d 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 c o s i n e of t h e i n c i d e n c e a n g l e (ASHRAE 1977; Stephenson 1965) ( i f tdir < 0.0, s e t t = 0.0) d i r (25) + 8.57881 c o s 3 ( i ) -8.38135 cos4 ( i )
The h e a t t r a n s f e r r e d through a window o f thermal
-
conductance Uw due t o indoor/outdoor temperatured i f f e r e n c e i s given by a n e q u a t i o n s i m i l a r t o (22) f o r doors
Lw(0) i s computed i n t h e s u b r o u t i n e 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 t h o s e f o r wall s u r f a c e s . The s o 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 s u b r o u t i n e WNDWGL f o r f o u r 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
When t h e window has i n t e r i o r shading ( b l i n d s , c u r t a i n s o r d r a p e s ) ,
so
= vo and sl = v l , 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 c l o s e d 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 s u r f a c e s o l - a i r temperature, 0 (0),
i scomputed from Eq. (6) a s f o r w a l l s u r f a c e s ( i . e . w i t h a I t i n c l u d e d ) .
I n t e r n a l h e a t g a i n due t o l i g h t s The h e a t 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 o f t h e e l e c t r i c a l power s u p p l i e d 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; M i t a l a s 1973; and 1973-74). The t r a n s f e r f u n c t i o n method y i e l d s t h e following formula f o r t h e computation of c o n v e c t i v e 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 i n p u t t o l i g h t s one and two hours ago, r e s p e c - t i v e l y . The v a l u e s f o r Qu a r e given i n a schedule i n t h e ENCORE-CANADA i n p u t 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 i n p u t t o l i g h t s d u r i n g t h e p r e s e n t 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 s u b r o u t i n e 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 u n i t f l o o r a r e a
I
where t h e window e x t e r i o r s u r f a c e temperature, 0(0), 146 kg/m2 342 kg/m2 635 kg/m2 976 kg/m2 i s computed from Eq. (6) with a I t s e t t o z e r o
( 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. (30 l b / f t 2 ) (70 l b / f t 2 ) (130 l b / f t 2 ) (200 l b / f t 2 ) ( 2 4 ) ) . Note t h a t Uw must b e g i v e n f o r t h e H used
i n Eq. (6)
.
y1 0.5 0.5 0 . 5 0 . 5
Given t h e v a l u e s o f Qs and Qc f o r t h e p r e s e n t
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 water i n a f u l l tank can s t o r e i s Qs and i s given h e a t t r a n s f e r through w a l l s , doors and windows. by
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 e q u a l 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 t u r n e d on o r o f f a c c o r d i n g 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 h e a t g a i n due t o occupancy and a p p l i a n c e s - The h e a t g e n e r a t e d 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 h e a t t r a n s f e r r e d by convection produces an almost i n s t a n t a n e o u s 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
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
of water a t temperature (BoU: + Bin)/2. I f t h e
-
volume of h o t 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
-
B i n ) . The make-up h e a t r e q u i r e d i s-
given bywhere q S ( l ) i s 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).
Loc(o) = ( 1 - r ) Q O ( 0 ) (30) I f E l
'
qm(0), h e a t e r element No. 1 producese(0) = q,,,(O) amount o f h e a t and t h e tank w i l l b e The g a i n due t o r a d i a n t h e a t t r a n s f e r from completely f i l l e d with h o t water, i . e . , %(O) = Q s . 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 I f E l < q,,,(O) and E2
2
q m ( 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 g e n e r a t i n g e(0) = qm(0) L o r ( 0 ) = r [ u l Q o ( l ) + U2Q0(2)1-
WILor(l) (31) amount of h e a t . I f E2 < q,,,(O), t h e demand cannotbe met f o r t h i s hour and o n l y e ( 0 ) = E 2 amount of Notice t h e one hour delay between e x c i t a t i o n and h e a t i s generated, i . e . , qs(0) = Qs - qm(0) + E2.
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 This procedure i s implemented i n t h e s u b r o u t i n e 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, Lo(0) = ( 1 - r)Qo(0) + r [ u l Q o ( l ) + u2Qo(2)1
-
e(O), consumed by t h e h o t water system d u r i n g t h ep r e s e n t hour. I t i s assumed t h a t
-
w L (1)1 o r (32) Qh(0) = [e(O)
-
q R ( 0 ) 1 / 2 + qg(0) p o r t i o n of e(O)c a u s e s t h e indoor a i r temperature t o r i s e . The h e a t t r a n s f e r mechanism is assumed t o b e i d e n t i c a l The r a d i a n t component of h e a t gain during previous
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 .
hour i s simply Thus, t h e i n t e r n a l h e a t gain, Lh (0) ,
from t h e domestic h o t w a t e r system i s computed from Eq. (34) Lor(l) = Lo(l)
-
Loc(l) = Lo(l)-
( l - r ) Q o ( l ) (33) (Qh r e p l a c e s Qo) by t h e s u b r o u t i n e IGAIN1.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 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 d u r i n g t h e Lo(0) = ( 1
-
r ) Q o ( 0 ) + [wl(l- '1
+ r u l l Q o ( l ) + p r e s e n t hour, Ln(0), i s given byThe 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
+ Lw(0) + LU(O) + LO(O) + Le(0) + Lh(0)
t o S O and u2 i s i d e n t i c a l t o s l f o r h e a t t r a n s f e r (37)
through windows. Lo(0) i s computed by t h e sub-
- .
r o u t i n e IGAIN1. where t h e components a r e
4 Heat g a i n from e l e c t r i c a l a p p l i a n c e s and equip-
ment, Le(0), i s t r e a t e d i n t h e same manner a s h e a t gain from occupants.
Heating from domestic h o t water system 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 a n e l e c t r i c h o t 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 o u t p u t of El and E2, r e s p e c - t i v e l y . The incoming c o l d water i s a t temperature B i n and t h e h o t water l e a v i n g t h e tank i s a t tem- p e r a t u r e Bout. The maximum amount o f h e a t t h e
Qw
: basement f l o o r , Eq. (8) Qf : basement w a l l s , Eq. (9) Li : a i r i n f i l t r a t i o nLS : w a l l s 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 a p p l i a n c e s
Lh : h o t w a t e r system.
The n e t e f f e c t o f indoor temperature v a r i a t i o n d u r i n g 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 n e g l e c t e d i n ENCORE-CANADA. Thecomponents Lu, Lo, Le and Lh can b e c o n s i d e r e d independent o f indoor temperature v a r i a t i o n s . How-
c ever, t h e components L i , Ls, Ld and Lw, which a r e
4 computed w i t h r e s p e c t t o a c o n s t a n t indoor 4 r e f e r e n c e t e m p e r a t u r e , O r , 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 . T h e r e f o r e t h e e f f e c t o f i n d o o r temperature v a r i a t i o n cannot b e n e g l e c t e d . 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 d e v i a - t 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, O r , 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 d e r i v e d from h e a t b a l a n c e 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 e n c l o s u r e (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) w i t h 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. ( 2 1 ) , ( 2 3 ) , (28), (29) and ( 3 4 ) . 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
They were d e r i v e d 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 i n d o o r 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 r e s p o n s e 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 g a i n , L n ( t ) . A p p l i c a t i o n 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 temperatureBy d e n o t i n g
v1 = glA + w [ K + I ( 1 ) ]
1 a s (39) 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 = f l o o r s u r f a c e a r e a
Kas = t o t a l thermal conductance between indoor
a i r and s u r r o u n d i n g s (computed i n t h e sub- r 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 l o a d p e r u n i t indoor/outdoor temperature d i f f e r e n c e , j h o u r s ago.
The normalized room a i r 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 , go, g l and g 2 , 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 a r e a
S i n c e b o t h 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 d u r i n g 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 t h e r m o s t a t c o n t r o l system.
A simple t h e r m o s t a t , which i s s e t a t tempera- t u r e BS, f u n c t i o n s a s shown i n F i g . 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 t e m p e r a t u r e , Bi. A band, Bb, about t h e s e t p o i n t , €Is, d e t e r m i n e s 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 t e m p e r a t u r e 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 p e r i o d t h e t h e r m o s t a t may go through s e v e r a l on/off c y c l e s . Consequent1 y, t h e o n / o f f c y c l e s of a t h e r m o s t a t c o n t r o l system can- n 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 u s e d . 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 t e m p e r a t u r e .
Mathematically, t h e p r o p o r t i o n a l band thermo- s 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 f o l l o w s :
E(0) = S e i ( 0 ) + Eo(0) 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 e s ( o )
- d
eb
5
ei(o)5
e s ( 0 ) +f
9, (44b) with measurements. l o c a t e d i n Ottawa i s c u r r e n t l y being u n d e r t a k e n by C o l l e c t i o n of d a t a from housest h e D i v i s i o n of Building Research o f t h e N a t i o n a l Research Council of Canada.
E(0) = 0 . 0 e i ( o ) > es(o) +
f
e
b ( 4 4 ~ ) 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 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 , 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 mathe- Eo(0), a r e g i v e n by ( s e e F i g . 3) m 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 i n p u t 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 Emax
s = - -
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 . Ob (45) I n a d d i t i o n , complicated s t o c h a s t i c p r o c e s s e s such a s weather, s o l a r r a d i a t i o n , s o l - a i r t e m p e r a t u r e and 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 tt o model without s i m p l i f y i n g assumptions. When a n assumption proves t o b e i n a p p r o p r i a t e , a more E 0 (0) =
4
Emax-
S e s ( 0 ) (46) d e t a i l e d , p o s s i b l y more complicated, mathematicalmodel h a s t o b e adopted. r e s p e c t i v e l y . Equations (43) and (44a, b, c ) c a n
be s o l v e d s i m u l t a n e o u s l y f o r ei(0) and E(0) t o g i v e
The indoor a i r temperature, 8 i ( O ) , and t h e h e a t s u p p l i e d by t h e h e a t i n g system d u r i n g t h e p r e s e n t hour, E(O), a r e computed a c c o r d i n g t o Eqs.
(47a, b , c ) i n t h e s u b r o u t i n e DEMAND. Note t h a t
e i ,
es,
Z, Eo and E v a r y w i t h t i m e . The t h e r m o s t a t s e t t i n g s ,es,
a r e s p e c i f i e d f o r e v e r y 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 r u n s w i t h ENCORE-CANADA produced r e s u l t s t h a t e x h i b i t expected behaviour
.
Such computationalENCORE-CANADA was w r i t t e n w i t h t h e i n t e n t t o s t u d y 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 - t y p e 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 , n u m e r i c a l l y s t a b l e model and, consequently, a l e n g t h y computer program. The amount of arithme- t 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, w i t h a b o u t 5 / 6 t h 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 simula- t i o n would b e 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 f u r n a c e h e a t i n g system w i t h warm a i r d i s t r i - b u t i o n ; 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, c a n a l s o s i m u l a t e e l e c t r i c a l l y h e a t e d , 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 e x p r e s s t h e i r g r a t i t u d e t o G.P. M i t a l a s f o r h i s generous a s s i s t a n c e through- o u t 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 compu- t a t i o n s were c a r r i e d o u t b y W.C. Brown.
This paper i s a c o n t r i b u t i o n from t h e D i v i s i o n o f Building Research, National Research Council of Canada and i s p u b l i s h e d with t h e a p p r o v a l of t h e D i r e c t o r of t h e D i v i s i o n .
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