Modeling and comparison of passive solar heating systems
by Robert Eugene Stotts
A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE
in Mechanical Engineering
Montana State University
© Copyright by Robert Eugene Stotts (1980)
Abstract:
Computer models were developed to simulate and compare the thermal responses of direct gain,
indirect gain, isolated gain, and natural passive solar heating systems. The performances of the direct
and indirect gain models were verified using data from the passive test cells at the National Center for
Appropriate Technology in Butte, Montana. A 139 m^2 (1500 ft2) home with standard insulation levels
was used to compare the different types of passive solar heating systems. The collector area, thermal
storage, and heating load were identical for the direct, indirect, and isolated gain systems. The natural
passive system, which uses only the building and its contents for thermal storage and only south-facing
windows for solar radiation collection, was simulated for several different collector areas. The largest
collector area was approximately one half the area used in the other three systems. Comparisons of air
temperature, auxiliary heating requirements, and solar fraction were performed for several different
weather patterns, each of which had the same total insolation and the same ambient temperature
variation. Each of the formal passive systems was found to supply more than half the heating needed
by the home, with the direct gain system outperforming the Trombe wall (indirect gain) system for all
of the weather patterns tested. The natural passive systems also provided significant solar heating,
indicating that locating the normal window area present in a well insulated house on the south wall of
the house results in a substantial solar fraction. STATEMENT OF PERMISSION TO COPY
In p re s e n tin g t h is th e s is in p a r t i a l f u l f i l l m e n t o f th e requirem ents
f o r an advanced degree a t Montana S ta te U n iv e r s it y , I agree t h a t the
L ib r a r y s h a ll make i t
f r e e l y a v a ila b le f o r in s p e c tio n .
I fu rth e r
ag ree t h a t perm ission f o r e x te n s iv e copying o f t h is th e s is f o r
s c h o la r ly purposes may be g ra n te d by my m ajor p ro fe s s o r, o r , in his
absence, by th e D ir e c t o r o f L ib r a r ie s .
It
is understood t h a t any
copying o r p u b lic a tio n o f t h is th e s is f o r f in a n c ia l g a in s h a ll not be
a llo w e d w ith o u t my w r it t e n p e rm issio n .
S ig n a tu re
Date
Feb. 2.2.J9K0
MODELING AND COMPARISON OF PASSIVE SOLAR HEATING SYSTEMS
by
ROBERT EUGENE STOTTS
A th e s is sub m itted in p a r t i a l f u l f i l l m e n t
o f th e requirem ents f o r th e degree
of
MASTER OF SCIENCE
in
Mechanical E n g in ee rin g
Approved:
C h a irp e rs o n , G raduate Committee
G fid u a te Deayi
MONTANA STATE UNIVERSITY
Bozeman, Montana
F e b ru ary , 1980.
ACKNOWLEDGMENTS
The a u th o r wishes to thank D r. R. 0 . W arrington and D r. R. L.
Mussulman f o r t h e i r h e lp and guidance in the perform ance o f th is study
The a u th o r would a ls o l i k e to thank D r. L. P a lm ite r o f th e N a tio n a l
C e n te r f o r A p p ro p ria te Technology f o r h is v a lu a b le a s s is ta n c e in the
v a lid a t io n o f th e computer models.
T h is study was supported by th e E n g in ee rin g Experim ent S ta tio n a t
Montana S ta te U n iv e r s it y , Bozeman, Montana.
TABLE OF CONTENTS
Page
V I T A ..........................................................................................................................................
Ir
ACKNOWLEDGMENTS.................................................................................................................... i i i
LIST OF TABLES ............................................................................................
LIST OF FIGURES
NOMENCLATURE .
. . . . .
v
.....................................................................................................
vi
. .......................................................................................................... , . v i i i .
ABSTRACT.........................................
CHAPTER I .
x ii
INTRODUCTION . % ...................................................................................
■
CHAPTER I I .
CHAPTER I I I .
CHAPTER IV .
I
•'
ANALYTICAL MODELS.............................................. ... ................... .
7
D ir e c t Gain M o d e l ......................................................................
.
7
Is o la te d Gain M o d e l .................................................................
.
23
Trombe W all M o d e l ..........................................................................
27
Model V a l i d a t i o n ..........................................................................
36
RESULTS
..................................... .... .................................... ....
CONCLUSIONS ....................................
. . .
42
. . . . .
64
. . . .
67
APPENDICES:
A.
SELECTION CRITERIA FOR LOCATION OF PROJECTEDPOINTS
B.
DETERMINATION OF THE SUNLIT AREACONFIGURATION..........................
C.
DIFFERENTIAL ELEMENT VIEW FACTORS
D.
NUMERICAL DOUBLE INTEGRATION .
E.
INSOLATION AND AMBIENT TEMPERATURE DATA GENERATION ; . . V
.
.......................................................
..................................................
LITERATURE CITED ..........................................................................................................
70
73
77
.
79
. 8 5
V
LIST OF TABLES
T a b le
Page
2.1
Equations f o r th e C o ordinates o f P o in t Q ■
............................ ....
.
10
2 .2
C o o rd in ates o f th e Window Corner P o i n t s ......................................... ....
10
3 .1
P h ys ic al Param eters o f th e P assive S o la r Systems
46
3 .2
A u x ilia r y H e a tin g and S o la r F ra c tio n s f o r the T e s t
Case S i m u l a t i o n s ..................................... .......................................................
.......................
.
B .l
P o s s ib le C o n fig u ra tio n s o f th e S u n lit Areas . .................................71
E .I
Degree D a y /N ig h t Averages f o r Bozeman, Montana (1 9 6 8 -1 9 7 7 )
.
62
83
vi
LIST OF FIGURES
F ig u re
2 .1
Page
C o n fig u ra tio n and Nom enclature o f th e D ir e c t
Gain System ............................................................................................ ....
2 .2
One P o s s ib le Shape f o r th e S u n lit A r e a s ............................
2 .3
A r b it r a r y P la n a r S urfaces and C o o rd in ate System
f o r View F a c to r D e te rm in a tio n ..........................................
. . . . .
8
12
15
2 .4
Flow C h art f o r th e D ir e c t Gain M o d e l .............................................
22
2 .5
The Is o la te d Gain S y s t e m ................................................................
23
2 .6
Trombe W all System and P r in c ip a l Modes o f Heat
T r a n s fe r
........................................................................................
. . . . .
2 .7
Node C o n fig u ra tio n f o r th e S torage W all S o lu tio n
...................
34
2 .8
D ir e c t Gain V a l i d a t i o n ..............................................................................
38
2 .9
Trombe W all V a l i d a t i o n ..............................................................................
40
3.1
D i r e c t - I n d i r e c t Gain Comparison W ith o u t A u x ilia r y
H e a tin g ,
Case I .
48
D i r e c t - I n d i r e c t Gain Comparison W ith o u t A u x ilia r y
H e a tin g ,
CaseI I .................................................................................................
49
N a tu ra l P assive A ir Tem perature W ith o u t A u x ilia r y
H e a tin g ,
CaseI .
............................................................................................
50
N a tu ra l P assive A ir Tem perature W ith o u t A u x ilia r y
H e a tin g ,
CaseI I .................................................................................................
51
D ir e c t - In d ir e c t Gain Comparison w ith A u x ilia r y
H e a tin g , Case I ....................................................................................
53
D i r e c t - I n d i r e c t Gain Comparison w ith A u x ilia r y
H e a tin g ,
CaseI I .....................................................................
54
N a tu ra l P assive A i r Tem perature w ith A u x ilia r y
H e a tin g ,
Case 1 ...................................................................................................
56
3 .2
3 .3
3 .4
3 .5
3 .6
3 .7
29
vi i
F ig u re
3 .8
3 .9
3 .1 0
Page
N a tu ra l P assive A i r Tem perature w ith A u x ilia r y
H e a tin g , Case I I .....................................................................................................
57
Is o la te d Gain B u ild in g A ir Tem perature w ith
A u x ilia r y H e a tin g , Case I .
. .....................................................................
58
Is o la te d Gain B u ild in g A ir Tem perature w ith
A u x ilia r y H e a tin g , Case I I .......................................................................
C .l
D iffe r e n tia l
Element f o r P e rp e n d ic u la r Surfaces
C.2
D iffe r e n tia l
Elem ent1 f o r P a r a lle l S urfaces
E .l
The Assumed In s o la tio n D is t r ib u t io n
E .2
The Assumed Ambient Tem perature D is t r ib u t io n
. . . .
59
.........................
75
..................................
76
..................................
80
..............................
82
NOMENCLATURE
Description
d is ta n c e from Z a x is to window edge (see F ig u re 2 .2 )
Area
v e n t a re a per u n it w id th o f s to ra g e w a ll
( i n d i r e c t g a in )
d is ta n c e from Y a x is to window edge (see F ig u re 2 .2 )
s to ra g e w a ll th ic k n e s s ( i n d i r e c t g a in )
s p e c if ic h e a t
s p e c if ic h e a t a t c o n s ta n t p ressu re ( a i r )
Y -c o o rd in a te o f a r b i t r a r y p o in t Q (see F ig u re 2 .1 )
Z -c o o rd in a te o f a r b i t r a r y p o in t Q (see F ig u re 2 .1 )
v e n t disch arg e, c o e f f i c i e n t
h e a tin g degree days, as d e fin e d by Equation ( E .5 )
day le n g th (h o u rs )
h e a tin g degree n ig h ts , as d e fin e d by E quation (E .7 )
in itia l
s to ra g e w a ll te m p e ratu re d is t r ib u t io n
d if f u s e r a d ia t io n view f a c t o r f o r r a d ia tio n from s u rfa c e A
to s u rfa c e B
a c c e le r a tio n due to g r a v it y
G rashof number = gB(Tw - Too) x ^ /v 2
window h e ig h t (see F ig u re 2 .2 )
heat tra n s fe r c o e ffic ie n t
average h e a t t r a n s f e r c o e f f i c i e n t
ix
Symbol
hO
hb
D e scriptio n
h e a t t r a n s f e r c o e f f i c i e n t on window s id e o f s to ra g e w a ll
h e a t t r a n s f e r c o e f f i c i e n t on i n t e r i o r s id e o f s to ra g e w a ll
H
room h e ig h t (see F ig u re 2 .1 )
H
s to ra g e w a ll h e ig h t ( i n d i r e c t g a in )
HS
d a ily t o t a l
J
r a d io s it y ( t o t a l r a d ia t io n which leaves a s u rfa c e per u n it
tim e p e r u n it a re a )
k
therm al c o n d u c tiv ity o f s to ra g e w a ll
I
window le n g th
L
room le n g th (see F ig u re 2 .1 )
m
mass
m
mass flo w r a te o f a i r
n
number o f nodes in s to ra g e w a ll
Nu
average N u s s e lt number = 4 r
Pr
P ra n d tl number =
cUve
%
Qr
Qs
in s o la tio n
(see F ig u re 2 .2 )
.
V
average (n u m e ric a l) s o la r f l u x , as d e fin e d by Equation ( E .3 )
s o la r f l u x , as d e fin e d by Equation (E .4 )
net to ta l
s u rfa c e
lo n g -w avelen g th r a d ia n t h e at t r a n s f e r r a te to a
t o t a l absorbed in s o la tio n
r
in filtr a tio n
r a te
R
therm al re s is ta n c e fa c to r
SR
tim e o f s u n ris e = 12 -
(ra te )
( a i r changes per hour)
x
Symbol
Description
SS
tim e o f sunset = 12 +
t
s o la r tim e (h o u rs )
T
te m p e ratu re (a b s o lu te )
^amb
T
max
am bient tem p eratu re
d a i ly maximum am bient te m p e ratu re (see F ig u re E .2 )
Tmean
average (n u m e ric a l) am bient te m p e ra tu re , as d e fin e d by
Equation (E.10)
T
v en t o u t l e t te m p e ratu re (see F ig u re 2 .6 )
out
space
TV
average a i r space te m p e ratu re (see F ig u re 2 .6 )
s to ra g e w a ll te m p e ratu re d is t r i b u t i o n
U
o v e r -a ll heat tra n s fe r c o e ffic ie n t
UA
to t
h e a tin g load
room w id th (see F ig u re 2 .1 )
X
c o o rd in a te
y
c o o rd in a te
Z
c o o rd in a te
a
s u rfa c e azim uth an g le o f in s o la tio n
a
k
therm al d i f f u s i v i t y = —
6
e le v a tio n angle o f in s o la tio n
3
therm al expansion c o e f f i c i e n t
V
e le v a t io n an g le o f i n s o la t io n , as p ro je c te d onto X-Z
plane (see F ig u re 2 .2 )
At
tim e s te p
(see F ig u re 2 .1 )
(see F ig u re 2 .1 )
Xl
Symbol
D e s c rip tio n
Ax
node spacing (see F ig u re 2 .7 )
E
lo n g -w a ve len g th r a d ia t io n e m itta n c e
X
dim en sionless p a ra m e te r, as d e fin e d by Equation ( 2 .3 4 )
U
dynamic v is c o s it y
V
k in e m a tic v is c o s it y = ^
P
d e n s ity
O
S te fa n -B o ltzm a n n c o n s ta n t = 5 .6 6 9 x 10"^ -- aI r 0K4
S u b s c rip ts
a ir
d i r e c t gain system a i r
I
s u rfa c e ( d i r e c t g a in )
I
s to ra g e w a ll node
in t
a ll
P
p r o je c tio n
room
room a i r ( i n d i r e c t and is o la te d g a in )
w a ll
s to ra g e w a ll
win
window
y
Y -c o o rd in a te
i n t e r i o r s u rfa c e s e xc e p t s to ra g e w a ll
( i n th e in s o la tio n d ir e c t io n )
S u p e rs c rip ts
a t th e n e xt tim e
(lum ped)
x ii
ABSTRACT
Computer models were developed to s im u la te and compare th e therm al
responses o f d i r e c t g a in , i n d i r e c t g a in , is o la te d g a in , and n a tu ra l pas­
s iv e s o la r h e a tin g system s.
The perform ances o f th e d i r e c t and in d i r e c t
g ain models were v e r i f i e d using d a ta from th e passive t e s t c e lls a t the
N a tio n a l C e n te r f o r A p p ro p ria te Technology in B u tte , Montana.
A 139 m2
(1500 f t 2 ) home w ith s ta n d ard in s u la tio n le v e ls was used to compare the
d i f f e r e n t types o f pa ss iv e s o la r h e a tin g systems.
The c o lle c t o r a re a ,
therm al s to ra g e , and h e a tin g load were id e n t ic a l f o r th e d i r e c t , i n ­
d i r e c t , and is o la te d gain system s. The n a tu ra l passive system , which
uses o n ly the b u ild in g and i t s c o n te n ts f o r therm al s to ra g e and only
s o u th -fa c in g windows f o r s o la r r a d ia t io n c o l le c t i o n , was s im u la te d f o r
s e v e ra l d i f f e r e n t c o l le c t o r a r e a s . The la r g e s t c o lle c t o r a re a was
a p p ro x im a te ly one h a l f th e a re a used in th e o th e r th re e systems.
Com­
p ariso n s o f a i r te m p e ra tu re , a u x i l i a r y h e a tin g re q u ire m e n ts , and s o la r
f r a c t io n were perform ed f o r s e v e ra l d i f f e r e n t w eather p a tte r n s , each o f
which had th e same t o t a l in s o la tio n and th e same am bient tem peratu re
v a r ia tio n .
Each o f th e form al pa ss iv e systems was found to supply
more than h a l f th e h e a tin g needed by th e home, w ith th e d i r e c t gain
system o u tp e rfo rm in g th e Trombe w a ll ( i n d i r e c t g a in ) system f o r a l l o f
th e w eath er p a tte rn s te s te d .
The n a tu ra l passive systems a ls o provided
s ig n if ic a n t s o la r h e a tin g , in d ic a tin g t h a t lo c a tin g th e normal window
area p re s e n t in a w e ll in s u la te d house on th e south w a ll o f the house
r e s u lts in a s u b s ta n tia l s o la r f r a c t i o n .
CHAPTER I
.
INTRODUCTION
S o la r h e a tin g systems a re r e c e iv in g e v e r -in c r e a s in g a t t e n t io n as
a lt e r n a t e means o f space h e a tin g .
c o n v e n tio n a l
This is due to th e s p i r a l i n g costs o f
( e l e c t r i c , gas, and o i l )
h e a tin g .
Two b a s ic types o f s o la r
h e a tin g system s, a c t iv e and p a s s iv e , a re a v a ila b le .
A c tiv e systems are
c h a r a c te r iz e d by c o lle c t o r s , pumps o r fa n s , and e la b o ra te h e a t d e liv e r y
system s, such as plumbing o r v e n ts .
They r e q u ire a la rg e i n i t i a l
t a l in v es tm e n t and may r e q u ire e x te n s iv e m aintenance.
c a p i­
C o n tro l systems
to r e g u la te h e a t d e liv e r y to th e b u ild in g add to the c o m p le x ity o f these
systems.
A p assive s o la r h e a tin g system , on th e o th e r hand, i s , by i t s
n a tu r e , an in t e g r a l p a r t o f the b u ild in g i t h e a ts .
very
P assive systems con­
s i s t o f a la rg e window (c o n ta in e d in th e b u ild in g 's w a lls ) f o r s o la r
r a d ia t io n c o lle c t io n and some means o f therm al s to ra g e .
The storage is
u s u a lly a masonry m a t e r ia l, such as b ric k s o r c o n c re te , o r c o n s is ts o f
w a t e r - f i l l e d c o n ta in e r s .
I f th e s to ra g e is d is t r ib u t e d about th e room
c o n ta in in g the window, th e system is r e fe r r e d to as a d i r e c t gain sys­
tem.
The therm al s to ra g e o f i n d i r e c t gain systems is c o n c e n tra te d in to
a w a ll a s h o rt d is ta n c e from th e window.
This w a ll p re ve n ts most o r a l l
o f th e s u n lig h t from e n te rin g th e l i v i n g space.
Is o la te d gain systems
a re p a ss iv e systems in which th e c o lle c t io n and s to ra g e components are
th e r m a lly is o la te d from th e l i v i n g space.
A means o f t r a n s f e r r in g heat
from th e s to ra g e to th e l i v i n g space, such as a fa n , is necessary.
One
2
example o f t h is ty p e o f system is a s o la r greenhouse.
N a tu ra l passive
systems a re d i r e c t gain systems w ith no added s to ra g e , o th e r than th e
normal c o n te n ts o f th e b u ild in g .
i t s e l f and in i t s
c o n te n ts .
Thermal s to ra g e is o n ly in th e house
P assive systems t y p i c a l l y do no t e f f i c i e n t ­
l y r e g u la te h e a t d e liv e r y to t h e i r b u ild in g s .
A ll the aforem entioned
p a ss iv e system s, e xc e p t th e is o la te d gain ty p e s , r e ly on th e therm al i n ­
te r a c tio n s between th e v a rio u s components o f the system (window, sto ra g e
and th e b u ild in g i t s e l f )
to c o n tro l th e a i r te m p e ra tu re .
l i g h t h o u rs, th e la r g e amounts o f in c id e n t s o la r r a d ia tio n
During day­
( in s o la t io n )
which pass through th e window a re m ostly absorbed by th e s to ra g e .
moderates peak daytim e te m p e ra tu re s .
A t n ig h t , convection and r a d ia tio n
from th e s to ra g e supply heat to th e b u ild in g .
tio n s a re fu n c tio n s o f many v a r ia b le s
This
Since th ese in t e r a c ­
(in c lu d in g the b u ild in g h e a tin g
c h a r a c t e r is t i c s , amount and type o f s to ra g e , window a rea and ty p e , and
th e am bient c o n d it io n s ) , la rg e a i r te m p e ratu re flu c tu a tio n s may o ccu r.
I n d i r e c t g ain systems t y p i c a l l y e x h i b i t lo w er a i r te m p e ratu re f lu c t u a ­
tio n s than do d i r e c t gain o r n a tu ra l passive systems.
N a tu ra l passive
systems e x p e rie n c e th e la r g e s t f lu c t u a t io n s , due to t h e i r r e l a t i v e l y
sm all therm al s to ra g e .
Is o la te d gain systems c o n tro l th e b u ild in g a i r
te m p e ratu re much b e t t e r than o th e r types o f system s, b u t high tem pera­
tu re s may occur w ith in th e c o lle c t o r .
S ince pa ss iv e systems a re in t e g r a l p a rts o f the b u ild in g s they
h e a t, th e y a re not n o rm a lly r e t r o f i t t e d .
In s te a d , the s to ra g e and
3
window a re in c o rp o ra te d in t o the b u ild in g d u rin g o r ig in a l c o n s tru c tio n ,
o fte n using th e s to ra g e f o r s tr u c tu r a l s u p p o rt.
In o rd e r to p ro p e rly
s iz e th e window a re a and s to ra g e f o r th e system , some means o f p r e d ic t ­
in g i t s
lo n g -te rm p e rfo rm a n ce , b e fo re i t
is b u i l t , is d e s ir a b le .
R u le -
of-thum b methods a re a v a ila b le to s iz e th e window a rea and th e therm al
s to ra g e [ I ] .
However, these methods w i l l n o t a c c u ra te ly (w ith in 10%)
p r e d ic t th e a c tu a l perform ance o f th e system .
They a r e , however, use­
f u l f o r making a f i r s t guess.
Two o th e r methods a re a ls o a v a ila b le f o r making an a c c u ra te p re d ic ­
t io n o f th e perform ance o f passive system s.
The f i r s t method is the
use o f computer models o f th e passive system s.
The models a re developed
using b a s ic thermodynamic and h e a t t r a n s f e r equations to approxim ate th e
therm al response o f th e system s.
F. Trombe [ 2 ] perform ed th e f i r s t com­
p re h en s iv e m odeling o f th e passive s o la r system commonly known as the
Trombe w a ll system .
It
is an i n d i r e c t gain system which uses a masonry
w a l l , w ith vents a t th e top and bottom , f o r s to ra g e .
A ir flo w from th e
l i v i n g space through th e vents tr a n s fe r s h e a t, by n a tu ra l c o n v e c tio n , to
th e l i v i n g space.
[3 ,4 ,5 ].
E x te n s iv e m odeling o f t h is system has been perform ed
S everal o f these models were used to o p tim iz e v a rio u s compo­
nents o f th e Trombe w a ll system .
c a b le to w a te r w a ll system s.
Many o f these modpls a re a ls o a p p l i ­
They a re i n d i r e c t gain systems w hich,use
w a t e r - f i l l e d c o n ta in e rs f o r th e s to ra g e w a l l .
Models have a ls o been
developed to s im u la te th e perform ance o f d i r e c t gain systems [ 6 , 7 ] .
The
4
f i r s t o f th ese is a sim ple g eneral model which is a p p lic a b le to any type
o f p a ss iv e system .
The o th e r is a model whose c o m p lex ity r i v a l s
th e d i r e c t gain model developed f o r t h i s s tu d y .
study was d e riv e d from b a s ic p r in c ip le s .
th a t o f
The model f o r th is
L i t t l e m odeling has been done
f o r n a tu ra l pa ss iv e and is o la te d gain systems.
When a pa ss iv e s o la r system model is developed, some means o f t e s t ­
in g i t s
perform ance is needed.
This is u s u a lly done by using the model
to s im u la te th e e x p e rim e n ta lly measured perform ance o f a system o f the
modeled ty p e .
P assive t e s t c e lls a re c u r r e n t ly in use in s e v e ra l p a rts
o f th e c o u n try to o b ta in th e necessary d a ta [ 8 , 9 ] .
Some o f th is data
was used to v e r i f y th e perform ance o f two o f th e models which were
developed f o r t h is s tu d y .
Models were developed to s im u la te the p e r­
formance o f d i r e c t , i n d i r e c t ( Trombe w a l l ) and is o la te d gain systems.
The d i r e c t gain model was a ls o used to s im u la te the perform ance o f n a tu ­
ra l
p a ss iv e s o la r system s.
The o th e r method which is c u r r e n t ly a v a ila b le to p r e d ic t the p e r­
formance o f pa ss iv e systems is a s im p lif ie d method developed by Balcomb
e t al
[ 1 0 ] , a t th e Los Alamos S c i e n t i f i c L a b o ra to rie s
(L A S L ).
The
method is a p p lic a b le to Trombe w a l l , w a te r w a ll and d i r e c t gain systems.
I t e s tim a te s th e perform ance o f th e systems as a fu n c tio n o n ly o f the
b u ild in g h e a tin g lo a d , th e m onthly absorbed in s o la tio n and th e t o t a l
m onthly h e a tin g degree days f o r th e lo c a tio n o f i n t e r e s t .
a l c lim a t ic fa c to r s could be used to improve the method.
Two a d d itio n ­
The f i r s t
5
f a c t o r is th e m o d ific a tio n o f degree days in to degree days f o r daytim e
am bient tem p eratu res and degree n ig h ts f o r n ig h ttim e am bient tem pera­
tu re s (see Appendix E ) .
This m o d ific a tio n has a lre a d y been shown to
in c re a s e th e accuracy o f s im u la tio n s [ 1 1 ] .
c lo u d y /c le a r day f a c t o r (CCDF).
The second f a c t o r is a
I t would account f o r th e d iffe re n c e s in
c le a r and cloudy day groupings f o r d i f f e r e n t g eograp hical lo c a tio n s .
The perform ance o f a l l
pa ss iv e systems is dependent on th e grouping o f
c le a r and clo udy days which o c cu rs .
The dependency on th e CCDF is not
y e t known, b u t th e s im u la tio n s which were perform ed f o r t h is study
re p re s e n t th e f i r s t s te p in d e te rm in in g t h is dependence.
S im u la tio n s
were perform ed f o r th e v a rio u s modeled systems ( d i r e c t g a in , i n d ir e c t
g a in , is o la te d g a in , and n a tu ra l p a s s iv e ) f o r two d i f f e r e n t c le a r and
clo udy day groupings.
The n a tu ra l pa ss iv e system was s im u la te d f o r two
d i f f e r e n t window a re a s .
The f i r s t approxim ated the normal window area
p re s e n t in a re s id e n c e .
The windows were assumed to a l l
south s id e o f th e b u ild in g so t h a t th e y could a l l
be on the
p ro v id e s o la r h e a tin g .
The second window a rea was a p p ro x im a te ly one h a l f th e window area pres-i
e n t in th e th re e form al passive system s.
h e a tin g lo ad were id e n t ic a l
f o r the th r e e .
The window a re a ; s to ra g e and
The s im u la tio n s formed a
b a sis f o r a comparison o f th e perform ance o f th e fo u r system s, f o r the
two t e s t cases.
Comparisons o f d i r e c t gain and Trombe w a ll systems have
a lre a d y been perform ed [ 1 2 ,1 3 ] .
D ir e c t gain systems w ith o u t n ig h t w in ­
dow in s u la tio n were found to perform b e t t e r (e c o n o m ic a lly ) than Trombe
6.
w a ll systems w ith n ig h t window in s u la tio n in southern U .S . c lim a te s .
However, Trombe w a ll systems w ith n ig h t in s u la tio n were found to be
more economical t h a t d i r e c t gain systems w ith n ig h t in s u la t io n in north
e'rn U .S . c lim a te s .
None o f these s tu d ie s in c lu d e s is o la te d gain o r
n a tu ra l p a ss iv e system s.
S ince d a ta f o r v e r i f i c a t i o n o f these two
models were not a v a i l a b l e , th e r e s u lts o f t h e i r s im u la tio n s a re only
p r e lim in a r y and should be t r e a te d a c c o rd in g ly .
CHAPTER I I
ANALYTICAL MODELS
The d i f f e r e n t i a l
eq u atio n s which were used to develop the passive
s o la r system models were o b ta in e d from b a s ic p r in c ip le s .
B asic thermo­
dynamic laws were combined w ith t h e o r e t ic a l and e m p iric a l h e a t t r a n s f e r
e q u a tio n s .
A few s im p lify in g assumptions were used to reduce the com­
p l e x i t y o f th e problem s.
D ir e c t Gain Model
T h is model was developed f o r a s in g le room w ith a s in g le window.
I t c o u ld , however, be e a s ily expanded to model a more complex b u ild in g ,
such as an e n t i r e d w e llin g .
To o b ta in high accuracy and g e n e r a lit y , a l l
modes o f h e a t t r a n s f e r w ith in th e room were assumed to be im p o rta n t.
R a d ia tiv e exchange among th e w a l l s , f l o o r , c e i l i n g , and window was i n ­
c o rp o ra te d in to the model.
Convection from each s u rfa c e in th e room to
th e room a i r and conduction from th ese s u rfa c e s , through th e w a lls , to
th e am bient were a ls o computed.
The d e s ire d degree o f accuracy n e c e s s i­
ta te d knowing the s iz e , lo c a tio n and shape o f the s u n l i t areas w ith in
th e room.
This r e s u lte d in a complex geometry problem .
F ig u re 2.1
tem .
illu s tr a te s
the geometry o f th e modeled d i r e c t gain s ys ­
S u n lig h t e n te rs th e room through th e window a t an e le v a tio n angle
B and a s u rfa c e azim uth a n g le a .
e n te r in g s u n lig h t may f a l l ,
Morning sun has a p o s it iv e a .
The
in p a r t o r in f u l l , on th e f l o o r o r on any
o f th e n o r th , w e s t, and e a s t w a lls .
For t h is model, th e s u n l i t p o rtio n
F ig u re 2.1
C o n fig u ra tio n and Nomenclature o f
th e D ir e c t Gain System
9
o f each o f these s u rfa ce s was t r e a te d as an in d iv id u a l s u rfa c e .
During th e course o f a normal day, th e s iz e , lo c a tio n and shape o f
th e s u n l i t areas w ith in th e room change c o n tin u o u s ly .
These are fu n c ­
tio n s o n ly o f th e room geometry and th e d ir e c tio n o f th e in c id e n t s o la r
r a d ia t io n
(in s o la t io n ).
The d ir e c t io n o f th e in s o la tio n
(a and 3) was
c a lc u la te d using r e la t io n s from D u f fie and Beckman [ 1 4 ] .
These angles
a re fu n c tio n s o n ly o f th e window o r ie n t a t io n
(azim uth an g le and s lo p e ),
th e l a t i t u d e , th e tim e o f day and th e day o f th e y e a r .
P ro v is io n was
made in th e model f o r a n o n - v e r t i c a l , n o n -s o u th -fa c in g window,
Once th e in s o la tio n d ir e c t io n was known, the s i z e , lo c a tio n and
shape o f th e s u n l i t areas were found.
lo c a te th e p ro je c tio n s
The f i r s t s te p in t h is was to
( i n th e d ir e c t io n o f the in s o la t io n ) o f the c o r­
n er p o in ts o f th e window.
For exam ple, c o n s id e r any p o in t Q on the
south w a ll o f th e room, as shown in F ig u re 2 .1 .
p o in t Q a re ( 0 * ^ , 0 ^ ) .
The c o o rd in a te s o f
Expressions f o r th e c o o rd in ates o f i t s
p ro je c ­
t i o n , Qp, a re given in T a b le 2 . 1 , as a fu n c tio n o f where th e p ro je c te d
p o in t f a l l s .
In these e q u a tio n s , the a n g le y (see F ig u re 2 .2 ) is de­
fin e d as
I
Y = ta n " (ta n 3/cos a ) .
(2 .1 )
O b v io u s ly , w ith fo u r d i f f e r e n t sets o f eq u atio n s f o r th e c o o rd in ates o f
Qp, some means o f d e te rm in in g which s u rfa c e the p o in t in te r s e c te d was
needed.
One method o f accom plishing t h is d e te rm in a tio n would be to
Table 2.1
C oordinate
flo o r
Equations f o r the Coordinates o f P o int Qp
W
X
Cb c o ty
Y
Ca + Cb ta n a c o ty
C3 + Wtana
. a
Z
O
Cb - Wtany
T a b le 2 .2
fa lls
%
n o rth w a ll
on th e
L im its
w est w a ll
e a s t w a ll
( I - Ca ) c o ta
-C 3Cota
a
O < X < W
O
O < Y < L
Cb + Cq ta n y c o ta
O < z <cb
L
Cb -
(L -C a )ta n y c o ta
C oordinates o f the Window Corner P oints
C orner P o in t
A
.
Coordi nates
(0 ,a ,b )
B
(0 ,a ,b + h )
C
( 0 ,a + l,b + h )
D
(O sa + l ,b )
11
c a lc u la t e th e c o o rd in a te s o f th e p ro je c te d p o in t w ith each o f th e sets
o f eq u atio n s and compare th e r e s u lts o f each s e t w ith th e a llo w a b le
l i m it s o f each c o o rd in a te .
The l i m it s a re l i s t e d in T a b le 2 .1 .
The
s e le c tio n c r i t e r i a were based on t h is method, b u t were condensed in to a
s m a lle r s e t o f i n e q u a lit ie s w h ich , when s a t i s f i e d , determ ined the lo c a ­
t io n o f th e p ro je c te d p o in t.
pendix A.
These s e le c tio n c r i t e r i a a re given in Ap­
The p ro je c tio n s o f th e window c o rn e r p o in ts were found using
t h is method.
The c o o rd in a te s o f th e c o rn e r p o in ts
( t o be used in the
e q u atio n s in T a b le 2 .1 ) a re given in T a b le 2 .2 .
A f t e r lo c a tin g Ap, B^, C^, and D^, the s iz e , shape and lo c a tio n o f
th e s u n l i t areas were c a lc u la t e d .
f o r th e s u n l i t a re a s .
There were 19 p o s s ib le c o n fig u ra tio n s
These p o s s ib ilit ie s .w e r e d esig n ated by the s e t o f
lo c a tio n s o f th e p ro je c te d window c o rn e r p o in ts .
s e t is
{A
P
For exam ple, one such
Bvn, and Dvn on th e f l o o r and Cvn on the w est w a l l } .
P
P
P
f i g u r a t io n is i l l u s t r a t e d
in F ig u re 2 .2 .
s u n l i t p o rtio n s o f th e room.
This con-
The shaded areas re p re s e n t th e
Appendix B c o n ta in s a l i s t o f a l l
s ib le c o n fig u ra tio n s t h a t th e s u n l i t areas may assume.
the pos­
A lso in c lu d e d is
a b r i e f d e s c r ip tio n o f th e method which was used to d eterm in e which con­
f ig u r a t io n occurred f o r any in s o la tio n d ir e c t io n .
It
is ap p are n t from F ig u re 2 .2 t h a t the s u n l it areas in the room
have complex g e o m e tric a l shapes which change c o n tin u o u s ly throughout th e
day.
For t h is re as o n , and s in ce d if f u s e r a d ia tio n view fa c to r s are d i f ­
f i c u l t to c a lc u la t e f o r odd shapes, an e q u iv a le n t re c ta n g le was
12
/^ " E q u iv a le n t
I
re c ta n g le
F ig u re 2 .2
One P o ss ib le Shape f o r the S u n lit Areas
13
g en erated f o r each s u n l i t area w ith in th e room.
Each, re c ta n g le had the
same area and a p p ro x im a te ly the same lo c a tio n as the s u n l i t area i t
re p re s e n te d .
As many edges as p o s s ib le o f th e re c ta n g le c o in c id e d w ith
th e co rresponding edges o f th e s u n l i t a re a .
F ig u re 2 .2 i l l u s t r a t e s
the
e q u iv a le n t re c ta n g le s f o r one p o s s ib le ,c o n fig u ra tio n .
The f r a c t io n o f th e in s o la tio n passing through the window in c id e n t
on each s u n l i t s u rfa c e was a ls o d e te rm in e d .
T h is was c a lc u la te d by d i ­
v id in g th e a c tu a l s u n l i t a rea by th e s u n l i t area which would occur i f
th e w a ll
tio n .
(o r th e f l o o r ) was la rg e enough to accomodate a l l
T h is c a lc u la t io n , and a l l
th e in s o la ­
the preced in g ones, were performed
using o n ly th e room dim en sions , a , 3 and th e c o o rd in ates o f A , B , Cn ,
P
h k
and Dp.
The c o o rd in a te s o f th e p ro je c te d window c o rn e r p o in ts were used
whenever p o s s ib le in o rd e r to s im p lify th e d e r iv a tio n o f th e e q u a tio n s .
Each o f th e 19 p o s s ib le c o n fig u ra tio n s has i t s
own s e t o f e q u a tio n s , but
due to space r e s t r i c t i o n s , the e q u atio n s a re not p resented h e re.
The t r a n s m is s iv ity o f th e g la z in g (window) was c a lc u la t e d , as a
fu n c tio n o f th e in c id e n c e angle and th e number o f window panes, using
th e method suggested by D u f fie and Beckman f o r f l a t p la te s o la r c o lle c ­
to rs [ 1 4 ] .
Once th e amount and lo c a tio n o f th e in s o la tio n e n te rin g th e
room was known, the s o la r r a d ia tio n absorbed by each s u r fa c e , Qc .., was
bI
e s tim a te d .
its
Each s u n l i t a re a was assumed to absorb a f r a c t i o n , equal to
s o la r a b s o r p t iv it y , o f th e in s o la t io n
in c id e n t on i t .
The r e f le c t e d
s u n lig h t was assumed to be t o t a l l y d if f u s e and to be d is t r ib u t e d over
14
th e e n t i r e in s id e s u rfa c e o f th e room.
In t h is m odel, r a d ia t i v e heat exchange among th e s u n l i t a re a s ,
w a lls , f l o o r , c e i l i n g , and window was assumed to c o n tr ib u te s i g n i f i c a n t ­
l y to th e t o t a l
therm al response o f th e room.
In o rd e r to d eterm ine the
n e t r a d ia t i v e t r a n s f e r among the v a rio u s s u rfa c e s , the d if f u s e r a d ia tio n
view fa c to r s must be known f o r each p a i r o f heat t r a n s f e r s u rfa c e s .
C onsider any two f l a t s u rfa c e s , such as those shown in F ig u re 2 .3 .
If
th e view f a c t o r from dA-j to A2 , F ^ _ 2 , is known as a fu n c tio n o f x and
y , then F ^
is given, by
b
a
(2. 2)
0
0
Appendix C c o n ta in s th e v a rio u s expressions which were, used f o r F ^ 2
in t h is m odel.
The view f a c t o r F2
coul d be determ ined e x a c tly by p e r­
fo rm in g th e in t e g r a l shown in Equation ( 2 . 2 ) .
However,.-, th e expressions
f o r FfJ-] 2 a re o fte n v e ry complex (see Appendix C) and a re no t e a s ily
in te g r a t e d .
For t h is re a s o n , a num erical method was used to perform
th e necessary double in t e g r a t io n .
The num erical double in te g r a t io n
scheme which was used in t h is model appears in Appendix D.
Using an approach s im ila r to t h a t p resented by Holman [ 1 5 ] , the
e q u atio n s governing steady r a d ia t iv e h e a t tr a n s fe r among th e surfaces
in th e room were found to be
15
F ig u re 2 .3
A r b it r a r y P la n a r S urfaces and C oordinate
System f o r View F a c to r D e te rm in a tio n
c T .4 - J ,
I - E.
fo r i = I , 2 , 3 , . . . , 1 1 .
t io n
11
e i Ai + J 1 (J j
= 0
(2 .3 )
The f i r s t term re p re s e n ts the n e t therm al r a d ia ­
le a v in g s u rfa c e i .
Each term in th e summation re p re s e n ts the net
r a d ia t iv e h e at t r a n s f e r from s u rfa c e j
t i o n , th e sum o f a l l
" J i ) Aj Fj i
to s u rfa c e i .
For steady r a d ia ­
these terms (in c lu d in g the f i r s t term ) must equal
z e ro .
This s e t o f e q u atio n s was solved f o r ( J ) using Gaussian e lim in a ­
tio n .
Knowing ( J ) ,
the n e t (lo n g w av e len g th ) r a d ia t iv e h e at tr a n s fe r
r a te to each s u r fa c e , Qr i , was found, using
16
&.A.
a
■ ( J i - aTi
> T ^ ir
•
(z -*)
In t h is m odel, th e room a i r was assumed to be tra n s p a re n t to s o la r
and therm al r a d ia t io n [ 1 6 ] .
Because o f t h i s assum ption, h e a t may only
be tr a n s fe r r e d to th e room a i r by conduction a n d /o r c o n v e c tio n .
account f o r both modes, h e a t t r a n s f e r c o e f f ic ie n t s were used.
To
Some
means o f d e te rm in in g a h e a t t r a n s f e r c o e f f i c i e n t f o r each s u rfa c e in th e
room was thus needed.
In any s u n l i t room, th e p o rtio n o f th e w a lls and th e f l o o r on which
s u n lig h t f a l l s
d i r e c t l y was expected to h e a t up to a te m p e ratu re h ig h e r
than th e te m p e ra tu re o f e i t h e r th e a i r o r o f th e n o n -s u n !i t
th e room.
This te m p e ratu re d i f f e r e n t i a l
could cause s ig n if ic a n t a i r
movements in th e room, e s p e c ia lly i f th e s u n l i t p o rtio n is
a i r movement could a f f e c t a l l
c re a s in g th e t o t a l
s u rfaces in
la r g e .
This
the a i r in th e room, s i g n i f i c a n t l y i n ­
c o n v e c tiv e h e a t t r a n s f e r to the room a i r .
However,
s in c e a l l methods a v a ila b le f o r acco u n tin g f o r the a i r movement are very
c o m p lic a te d , and s in ce no sim ple e m p iric a l method is a v a ila b le to e s t i ­
mate t h is e f f e c t ,
i t was n e g le c te d .
In s te a d , the a i r movements were
assumed to r e s u l t in com plete ( p e r f e c t ) m ixing o f th e room a i r .
Much work has been done d eve lo p in g e m p iric a l r e la t io n s f o r d e te r ­
m in a tio n o f h e a t t r a n s f e r c o e f f ic ie n t s f o r n a tu ra l c o n ve c tio n from s u r­
faces to s e m i - i n f i n i t e f lu id s
in th e d im en sionless form
[1 5 ].
These r e la tio n s a r e .u s u a lly w r it t e n
17
Nu = ^
= C (G rP r)m,
(2 .5 )
and account f o r both conduction and c o n v e c tio n , s in ce th e y a re e x p e r i- .
m e n ta lly d e te rm in e d .
C and m a re dim en sionless param eters dependent on
th e s u rfa c e shape and o r ie n t a t io n and on th e degree o f tu rb u le n c e p re s ­
e n t in th e flo w , which i s ,
in tu r n , dependent on th e m agnitude o f the
G ra s h o f-P ra n d tl number product (R a y le ig h num ber).
The f l u i d p ro p e rtie s
used in Equation ( 2 . 5 ) a re n o rm a lly determ ined a t th e f i l m
The f i l m
te m p e ratu re .
te m p e ratu re is th e average o f th e f l u i d and th e h e a t tr a n s fe r
s u rfa c e te m p e ra tu re s .
Equation ( 2 . 5 ) was used in th is model to e s tim a te
th e h e a t t r a n s f e r c o e f f i c i e n t , h^, f o r each s u rfa c e .
Each was assumed
to i n t e r a c t w ith the room a i r independent o f a l l o th e r s u rfa c e s in the
room.
The valu es f o r C and m were o b ta in e d from Holman [ 1 5 ] .
In a d d itio n to co n ve c tio n to th e room a i r , each s u rfa c e in th e room
was a ls o assumed to i n t e r a c t w ith th e am bient and to s to re therm al e n e r­
gy.
The h e a t loss from each s u rfa c e to th e am bient was w r it t e n in terms
o f th e e f f e c t i v e therm al re s is ta n c e between the s u rfa c e and the am bient.
The therm al s to ra g e o f each s u rfa c e was assumed to be c o n c e n tra te d in
th e f i r s t p h y s ic a l la y e rs o f th e w a l l , b e fo re the in s u la t io n .
For exam­
p le , c o n s id e r a w a ll c o n s tru c te d o f 2" x 4" wooden studs w ith fib e r g la s s
in s u la tio n between th e studs and w ith an in s id e s u rfa c e o f 1 /2 " gypsum
board (s h e e t r o c k ).
The c o n s tru c tio n o f th e o u ts id e s u rfa c e o f the w a ll
is u n im p o rtan t f o r s to ra g e c o n s id e ra tio n s .
The therm al s to ra g e would be
18
c o n c e n tra te d in th e gypsum board and the sh eet rock would be tr e a te d
(e x c lu d in g th e s u n l i t p o rtio n o f th e w a l l ) as an is o th e rm a l body.
In t h is m odel, th e room's i n t e r i o r s u rfa c e was d iv id e d in to 11 s u r­
fa c e s .
S u rfaces one through fo u r w e re , r e s p e c tiv e ly , th e n o n -s u n lit
p o rtio n s o f th e f l o o r and o f th e n o rth , w e s t, and e a s t w a lls .
' f i v e was th e south w a ll
S urface
(le s s th e window) and s ix was th e c e i l i n g .
S ur­
faces seven through te n w ere , r e s p e c t iv e ly , th e s u n l i t p o rtio n s o f th e
f l o o r and th e n o rth , w e s t, and e a s t w a lls .
s id e pane o f the window.
d iv id u a l
S u rface e le v e n was the i n ­
Each o f th ese s u rfa ce s was t r e a te d as an i n ­
lu m p e d -c a p a c ity system.
The d i f f e r e n t i a l eq u atio n s governing the therm al response o f the
v a rio u s s u rfa c e s in th e room were d e riv e d by p e rform ing a h e a t balance
on each s u rfa c e .
The heat balance was o f th e form
[S o la r r a d ia t io n absorbed] + [n e t therm al r a d ia tio n from o th e r
s u rfa c e s ] - [c o n v e c tio n to the room] - [lo s s to th e am b ien t]
= [s to ra g e ].
S u b s titu tin g th e a p p ro p ria te m athem atical expressions f o r each term r e ­
s u lte d in th e s e t o f (1 1 ) d i f f e r e n t i a l equations
dT.
Qs i + Qr i
- O1A1 Cr1 - Ta 1 r ) - A1 U 1 - TaiJ Z R 1 = (m e),
The c o rresp onding i n i t i a l
c o n d itio n s f o r these equations were
. .
(2 .6 )
19
(2 .7 )
t= 0
The. two r a d ia t io n components ( s o la r and th e rm a l) were t r e a te d indepen­
d e n tly because most common c o n s tru c tio n m a te ria ls have v e ry d i f f e r e n t
absorptances f o r the two ty p e s .
For exam ple, a normal w h ite p a in t has a
s o la r absorptance between 0 .3 and 0 .5 .
The same p a in t , however, has an
e m itta n c e ( a t 25°C ) between 0 .8 5 and 0 .9 5 [ 1 7 ] .
The d i f f e r e n t i a l e q u a tio n f o r th e room a i r was d e riv e d using a sim ­
i l a r ap p ro ach.
The h e a t balance was o f th e form
[c o n v e c tio n from s u rfa c e s ] -
[ i n f i l t r a t i o n ] = [s to ra g e ].
The corresponding m athem atical expressions were s u b s titu te d in to th is
b a la n c e , and th e r e s u lt a n t d i f f e r e n t i a l
The i n i t i a l
e q u atio n was
c o n d itio n f o r th e a i r was
(2 .9 )
In Equation ( 2 . 8 ) , r is th e i n f i l t r a t i o n
Equations ( 2 . 6 )
r a te in a i r changes per hour.
through ( 2 . 9 ) were s o lved n u m e ric a lly using a f o r ­
w a rd -s te p p in g , e x p l i c i t f i n i t e
d iffe r e n c e method.
The a pproxim ation
( 2.
10)
20
was s u b s titu te d in t o Equations ( 2 . 6 ) and ( 2 . 7 ) and th e r e s u lt a n t equa­
tio n s were s o lved f o r I ' .
The s e t o f e x p l i c i t equations to determ ine
th e new s u rfa c e tem peratu res was found to be
Y
■ CY +
+ W a ir + Vamt/Y
AtA.
+ [T - W
T
(h Y
1/ R 1I Y - .
. ( 2 . 11)
The r e s u lt in g e x p l i c i t exp re s sio n f o r th e new a i r te m p e ratu re was
Ta i r "" (Inc)a i r
|
-("lc I a i r
At
*
[-
Ai hi Ti + r ^mc^ a ir Tamb +
11
T 1 Ai hi - r ("1cl a i r l Ta i r
( 2 . 12 )
In o rd e r to guarantee s t a b i l i t y o f th e num erical method, th e c o e f f i ­
c ie n ts o f Ti in Equations ( 2 .1 1 ) and o f Tf l i r in Equation ( 2 .1 2 ) must be
n o n -n e g a tiv e [ 1 8 ] .
T h is r e s t r i c t i o n fo rc e d th e fo llo w in g r e s t r ic t io n s
on th e tim e s te p , A t:
(m e )./A i
At -
(h i + V R i )
(2 ,1 3 )
and
At <
("lcIair
( 2 .1 4 )
T1 hiA
i + ("lcIairr
A ll o f th e p reced in g equations were in c o rp o ra te d in t o a computer
21
code to s im u la te th e thermal response o f the d i r e c t gain system.
2 .4 i l lu s t r a t e s
Figure
the programming l o g i c which was used f o r t h i s model.
This f l o w c h a r t , though g r e a t l y s i m p l i f i e d , shows the e s s e n t i a l fe a tu r e s
o f th e l o g i c .
The most c om plicated and th e longe st p o r tio n on the model
was the s u b ro u tin e which c a l c u l a t e d the geom etric parameters ( i n c l u d in g
the r a d i a t i o n view f a c t o r s ) .
Each tim e the view f a c t o r s were r e c a lc u - .
l a t e d , many (up to 55) numerical double i n t e g r a t i o n s were performed.
It
was e s tim a te d t h a t t h i s p a r t o f the model r e q u ire d more than 75% o f the
to tal
cpu tim e needed f o r each run.
In t h i s model, the r a d i a t i v e h e a t t r a n s f e r to each s u rfa c e ( s o l a r
and th e r m a l) was c a l c u l a t e d only once f o r each s e t o f d a ta .
This proce­
dure r e s u l t e d in l i m i t a t i o n s on the tim e s te p between successive data
s ets ( D T ) .
.
These l i m i t a t i o n s were found, during the model v e r i f i c a t i o n
proc edu re , to be 0 .2 5 hours du rin g high i n s o l a t i o n periods
the day) and to be 0 .5 0 hours
at
n ig h t.
(i.e .
during
The model could be m o d ifie d
to accept h o u r ly data s im p ly by a l t e r i n g the computer code to r e c a lc u ­
l a t e the r a d i a t i v e h e at t r a n s f e r r a te s s e v e ra l
data s te p .
(four, o r more) times per
The maximum time step (D E LT), which was used to c a l c u l a t e
the new te m p e ra tu re s , was found using Equations ( 2 . 1 3 ) and ( 2 . 1 4 ) .
A
tim e step equal to 99% o f the maximum a llo w a b le was used to avoid i n ­
s t a b i l i t y due to computer r o u n d - o f f e r r o r , y e t s t i l l
p o s s ib le computation tim e .
o b ta in th e f a s t e s t
22
Figure 2 , 4
Flow Chart f o r the D i r e c t Gain Model
23
I n c id e n t
s o la r
ra d ia tio n
D ire c t
gain
system
(c o lle c to r)
room
Figure 2 .5
The I s o l a t e d Gain System
P r o v is io n was a ls o made in t h i s model f o r n ig h ttim e c o l l e c t o r w in ­
dow i n s u l a t i o n .
Whenever the i n s o l a t i o n dropped to z e r o , the thermal
r e s is ta n c e o f the window, R ^ , was inc re a se d by the R - f a c t o r o f the w in ­
dow i n s u l a t i o n .
This R - f a c t o r may be any d e s ire d v a lu e .
l a t i o n again became n o n -z e ro , R ^ was reduced to i t s
When the in s o ­
o rig in a l
v a lu e .
I s o l a t e d Gain Model
An i s o l a t e d gain passive s o l a r system is any passive system in
which the c o l l e c t i o n and sto ra g e areas a re th e r m a lI y i s o l a t e d from the
b u i l d i n g t h a t the system is intended to h eat ( f o r example, a s o l a r green
house).
A model f o r i s o l a t e d gain was developed using a d i r e c t gain sys
tern a tta c h e d to a b u i l d i n g , as i l l u s t r a t e d
in Figure 2 . 5 .
The b u i l d i n g
24
and the c o l l e c t o r (t h e a tta c h e d d i r e c t gain system) were assumed to have
one common w a l l , the north w a ll o f the c o l l e c t o r .
A fan provided a i r
exchange between the b u i l d i n g and the c o l l e c t o r .
I t was obvious t h a t any model f o r t h i s type o f i s o l a t e d gain system
should be based on the d i r e c t gain model.
Because o f th e s i m i l a r i t y o f
the systems, th e d i r e c t gain model was used, w ith a few m o d i f i c a t io n s ,
to model th e thermal response o f an i s o l a t e d gain system o f the type
shown in Fig u re 2 . 5 .
It
i s ap p are n t tha t, the geometry o f, the atta c h ed
d i r e c t gain system would be unchanged.
The r a d i a t i v e h eat exchange
equations f o r the c o l l e c t o r were a ls o not changed by i t s
b u ild in g .
a d d i t i o n to the
The c o n v e c tiv e h e a t t r a n s f e r from the v a rio u s surfa ce s in th e
c o l l e c t o r to the c o l l e c t o r a i r was a f f e c t e d by the la r g e a i r movements
caused by th e fa n .
However, l i k e the thermal a i r c u r r e n t s , i t
c u l t to account f o r t h i s e f f e c t .
Because o f t h i s d i f f i c u l t y ,
is d i f f i ­
the a i r
movements caused by the fan were assumed to r e s u l t in p e r f e c t mixing o f
th e c o l l e c t o r a i r .
Conduction from the surfa ce s in th e room,through
t h e i r corresponding w a l l s was the same f o r a l l
two and e i g h t (n o r th w a ll s u r f a c e s ) .
was to th e b u i l d i n g a i r ,
surfaces e xcept numbers
Heat t r a n s f e r from these surfaces
r a t h e r than to the ambient.
To account f o r
t h i s d i f f e r e n c e . Equations ( 2 . 6 ) and ( 2 . 1 1 ) were m o d ifie d .
For i equal
to 2 or 8 , Tamb was re p la c e d by the b u i l d i n g a i r te m p e r a tu re , Troom.
O th e rw is e , the equations were s t i l l
a p p lic a b le .
25
A nother term was added to the h e a t balance e q u atio n d e s c r ib in g the
c o l l e c t o r a i r te m p e ra tu re .
This term rep res e n te d the h e a t t r a n s f e r , due
to the f a n , from the c o l l e c t o r a i r to the room a i r .
mCp(Ta1- r - Troom) .
a ir,
Since i t
I t was o f the form
re p res e n te d a heat loss from th e c o l l e c t o r
i t was i n s e r t e d w it h a n e g a tiv e s ig n .
Applying t h i s m o d ific a t io n
to Equation ( 2 . 8 ) r e s u l t e d in
11
I
.
A-jhj(T1- - Ta_jr ) - 'f'(mc)ai y ^ a i r
^amb^ " mcp ^ a i r ~ ^room^
C T a ir
'= ^m
c^air dt- •
(2-15)
The te m peratu re o f the b u i l d i n g a i r , Tyioom, was th e most im p o rta n t
r e s u l t d e s ir e d from the i s o l a t e d gain model.
tu re , a d iffe r e n tia l
To determ ine t h i s tempera­
e q u a tio n d e s c r ib in g i t was needed.
The equation,
was d e riv e d in the same manner as f o r the c o l l e c t o r a i r te m p e ra tu re .
A
h e at b a la n c e , e q u a tin g n e t h eat gains to s to r a g e , r e s u l t e d in
™cp ^ a ir ~ "*Voom^ ~ ^ to t^ ro o m ~ ^amb^
+ V
T2 - Troom VR2 + A8 (T a - Tro J Z R g = (me) room
room
• (2 .1 6 )
dt
The f i r s t term in t h i s e q u a tio n re p re s e n ts the heat pro v id ed to the
b u i l d i n g a i r from the c o l l e c t o r a i r
(due to the f a n ) .
re p re s e n ts th e h e a t loss to the am bient.
h e a tin g lo a d , i n c lu d in g i n f i l t r a t i o n .
The second term
UAto1. is th e t o t a l
b u ild in g
The o th e r two terms on the l e f t -
hand s id e o f the e q u a tio n r e p r e s e n t c o n d u c tio n , through th e w a l l , from
26
the c o l l e c t o r sto ra g e to the room a i r .
( 2 . 1 6 ) re p re s e n ts the s to r a g e .
The r ig h t-h a n d s id e o f Equation
I f t h i s s to r a g e , and hence the time con­
s t a n t o f th e a i r , was n e g le c te d , the room a i r tem perature f l u c t u a t e d
w i l d l y and the r e s u l t s were unusable.
The d i f f e r e n t i a l
equations f o r t h i s model were solved by the same
method as was used f o r th e d i r e c t gain model.
A f i r s t - o r d e r approxima­
t i o n was made f o r the tim e d e r i v a t i v e i n each d i f f e r e n t i a l
th e equations were solved f o r the new tem p e ratu re s .
a t r u n c a t i o n e r r o r on th e o r d e r o f At .
e q u a tio n , and
This procedure had
The r e s u l t a n t e xpression f o r
th e new c o l l e c t o r a i r te m peratu re was found to be
I
air " (Hic)a1r
(me)
A1M1T1 + I-(Hic)a 1 rTalnb + me Troom
I= I
a ir
- I-(Hic)a i r - mep -
A1M1
'a ir
‘
(2 .1 7 )
The r e s u l t a n t expre s sion f o r the new room a i r tem peratu re was
T1
room
(mct^oom J mcPl a i r + UAt o t Tamb + A2T2/ R 2 + A8T8/ R 8 +
(me)
L
room
■ mcP " UAt o t " A2/ R 2 ' A8/R 8
At
ro o m .
J
:
(2 .1 8 )
J
There w ere , or course, d i f f e r e n t l i m i t a t i o n s on A t , which were d e t e r ­
mined from Equations ( 2 . 1 1 )
( 2 . 1 7 ) , and ( 2 . 1 8 ) .
(w ith the p r e v io u s ly discussed c h ang es),
These l i m i t s were
27
(me)./A
(2 .1 8 )
At <
K m c I a 1 r + mCp +
(2 .1 9 )
A1H1
and
( 2 . 20 )
At <
The computer model f o r the i s o l a t e d gain system was v i r t u a l l y id e n ­
tic a l
to th e d i r e c t gain model, except f o r the m o d ific a t io n s discussed
in t h i s s e c t i o n .
The same r e s t r i c t i o n s on the data tim e s te p were
a p p lic a b le .
Trombe Wall Model
A Trombe w a ll pass iv e s o l a r system i s one type o f i n d i r e c t gain
system.
I t c o n s is ts o f a l a r g e window w ith a masonry w a ll
sto ra g e lo c a te d a s h o r t d is ta n c e behind the window.
f o r thermal
Vents in the top
and bottom o f the w a ll a llo w room a i r to be heated by n a tu r a l convection
o v er the sun s id e o f the w a l l .
B a c k - d r a f t dampers a re necessary to p r e ­
v en t re v e rs e c i r c u l a t i o n o f the a i r a t n i g h t .
I f re v e rs e c i r c u l a t i o n is
not p re v e n te d , the vents a r e , in th e absence o f n ig h t ( c o l l e c t o r ) window
i n s u l a t i o n , a n e t thermal disadvantage to the system [ 3 ] .
28
Fig u re 2 .6 i l l u s t r a t e s
a Trombe Wall system and the modes o f heat
t r a n s f e r used to develop t h i s model.
s id e s u rfa c e o f the w a l l .
S u n lig h t is absorbed by the sun
This h eat is convected to the a i r space and
conducted through the s to ra g e w a l l .
Flow o f a i r through th e v e n ts , con­
v e c tio n from the room s id e s u rfa c e o f th e sto ra g e w a ll and convection
from the room's i n t e r i o r surfaces heat th e room a i r .
The room a i r was
assumed to be tr a n s p a r e n t to thermal r a d i a t i o n from th e s to ra g e w a l l ,
and o n ly r a d i a t i v e i n t e r a c t i o n s w ith th e i n t e r i o r surfa ce s were con­
s id e r e d .
A h e a t b a la n c e , e q u a tin g n e t h eat gains to s to r a g e , was used to
d e r iv e th e d i f f e r e n t i a l
room a i r .
e q u atio n d e s c r ib in g the thermal response o f the
This e q u atio n was found to be
(U b ,t) - T
) + (UA)-JT., - T
huA.
b e w a i l v V us w
'room' ' xvriyI n t x 1i n t
'room)
+ r(m c) room ( T am^ - Troom) + mcp (T 0Ut ~ Tfroom) - ( mc)y»oom
room
dt
( 2 . 21 )
The f i r s t term re p re s e n ts convection from the i n t e r i o r s u r fa c e o f the
sto ra g e w a l l .
The second term accounts f o r convection from the room's
in t e r i o r s u rfa ce s .
The t h i r d term re p re s e n ts the h e at loss to the am­
b i e n t due to i n f i l t r a t i o n , and the f o u r t h term re p res e n ts h e at gain by
a i r flo w through the v e n ts .
The r i g h t - h a n d s id e o f the e quation r e p r e ­
sents thermal sto ra g e i n the room a i r .
In t h i s model, thermal i n t e r a c t i o n s between the i n t e r i o r surfaces
o f the room and th e r e s t o f the room were considered to be im p o rta n t.
29
In c id e n t
s o la r
ra d ia tio n
in filt
conduction
convection
room
co n ve c ti
convection
s[ ace
Fig u re 2 .6
heatin g
load
ra d ia tio n
Trombe Wall System and P r in c ip a l Modes
o f Heat T r a n s fe r
amb
30
The e n t i r e i n s id e s u r fa c e o f the room (e x c lu d in g the s to ra g e w a l l ) was
t r e a t e d as one is o t h e r m a l , lum pe d-c apa c ity body.
The d i f f e r e n t i a l
equa­
t i o n d e s c r ib in g th e thermal response o f th e room's i n t e r i o r s u rfa c e was
o b ta in e d using a h e a t balance s i m i l a r to t h a t used f o r th e room a i r .
The r e s u l t a n t e quation was
dT.
^r + ^ in t^ ro o m
” ^ in t^ + ^ t o t ^ a m b
.
” "^inV “ ^mc^ i n t ~ d t
(2 .2 2 )
where Qyi is th e r a d i a t i v e h e at t r a n s f e r r a t e from the s to ra g e w a l l ,
given by
a W3 1 I
^ ( M
)
- T fn t)
(2 .2 3 )
r
1 ~ eW a l l +
1
eW a l l
V ll 1
~ £in t
^ in t
eIn t
The second term in Equation ( 2 . 2 2 ) re p re s e n ts convection from the i n ­
t e r i o r s u r fa c e to the room a i r .
The t h i r d term re p re s e n ts h e a t loss (by
conduction through the w a l l s ) to the am bient.
The r i g h t - h a n d s id e o f
th e e q u atio n accounts f o r thermal s to ra g e in the room's i n t e r i o r s u r ­
face.
This sto ra g e was assumed to be s i g n i f i c a n t .
In o rd e r to s im u la te the t o t a l
thermal response o f th e system, the
■
te m p e ratu re d i s t r i b u t i o n i n the s to ra g e w a l l , as a f u n c tio n o f tim e , was
needed.
The w a ll was approximated as a o n e -d im e n s io n a l, t r a n s i e n t heat
flo w problem.
The general d i f f e r e n t i a l
d u c tio n problem is
e quation f o r t h i s ty p e o f con­
31
32T.
The w a ll a ls o had the i n i t i a l
I _V
a 9t
( 2 . 24)
c o n d itio n
Tw( X 9O) = f ( x )
(2 .2 5 )
and th e boundary c o n d itio n s
x=0 " Qs + ^O ^w all^ s p a c e "
( 2 . 26 )
and
x-b " % + hbAWanCTw(b- t ’ - Trool,] -
(2 -27>
The average a i r space te m p e ra tu re , Tspace, was found by equating
th e h eat d e l iv e r e d to th e room ( v i a th e v e n ts ) to the sum o f the con­
v e c tio n from the s to ra g e w a ll
to the a i r space and from the a i r space
(through the window) to th e am bient.
This r e s u l t e d in
2mc Troom + \n n ^ a m b z^ w i n + ho \ a l l Tw ^ s t ^
space
2m cP + W
( 2 . 28)
rw
I in + ^O^wa11
The v e n t o u t l e t te m p e r a tu re , To u t , was found using
^out ~ 2^space ~ ^room'
( 2 . 29)
32
The mass flo w r a t e o f a i r through the v e n t s , m, was a ls o needed.
I t was found using
(2 .3 0 )
The term w i t h i n the bra ck e ts was developed by Balcomb e t a I
[ 3 ] and
re p re s e n ts the v o lu m e tr ic flo w r a t e pe r u n i t area o f g l a z i n g .
d e n s i t y , Pa ^r > was c a l c u l a t e d using the id e a l
The a i r
gas law .
Equations ( 2 . 2 1 ) and ( 2 . 2 2 ) were solved using the same method as
was used f o r the d i r e c t gain model.
A firs t-o rd e r, f in it e
d iffe re n c e
approxim ation was made f o r each time d e r i v a t i v e , and the equations were
then solved f o r the new te m p e ra tu re s .
This approxim ation r e s u lt e d in a
t r u n c a t i o n e r r o r on the o r d e r o f At .
The e x p l i c i t e xpre s sion f o r the
room's new i n t e r i o r s u rfa c e tem peratu re was found to be
r
The new room te m peratu re was c a l c u l a t e d using
r
in t^ in t +
room
mc^roonJamb
- (UA)1 nt - r(mc')
room
(2 .3 2 )
j
33
The d i f f e r e n t i a l
trib u tio n
e q u a tio n d e s c r ib in g the sto ra g e w a ll
tem peratu re d i s ­
(E q u a tio n ( 2 . 2 4 ) ) was solved using a f i n i t e d i f f e r e n c e te c h ­
nique suggested by Carnahan e t a I
[1 8 ].
The w a ll was d iv id e d in t o n
lum pe d-c apa c ity nodes, e v e n ly spaced Ax a p a r t , as shown in Figure 2 . 7 .
Nodes I and n were 4 r t h i c k , and the o th e r ( i n t e r i o r )
th ic k .
nodes were a l l
Ax
A f i r s t - o r d e r a pproxim ation f o r the tim e d e r i v a t i v e and a second
o rd e r a pproxim ation f o r the space d e r i v a t i v e were s u b s t i t u t e d in t o Equa­
tio n (2 .2 4 ) .
The e q u a tio n was then solved f o r the new te m peratu re o f
each node, r e s u l t i n g in
Si = xSi-I +
- 2xIS1 + xS hi
<2-33>
where
X =
.
(2 .3 4 )
Ax
These equations a re o n ly a p p l ic a b l e to i n t e r i o r nodes (2 £ i £ n - 1 ) , and
r e s u l t e d in a t r u n c a t i o n e r r o r on the o rd e r o f (Ax
3
2
+ At ) .
Two methods were a v a i l a b l e to s a t i s f y the boundary c o n d itio n s o f
the s to ra g e w a ll
(Equations
to l i n e a r l y approxim ate
peratu res.
3Tw
( 2 . 2 6 ) and ( 2 . 2 7 ) ) .
The f i r s t o f these was
and s o lv e e x p l i c i t l y f o r the boundary tem­
This method was not used because i t ignored th e thermal
sto ra g e o f th e boundary nodes.
This s to ra g e was only h a l f t h a t o f each
i n t e r i o r node, but was considered to be s i g n i f i c a n t , e s p e c i a l l y f o r
coarse g r i d spacings.
34
node
n
Figure 2 .7
Node C o n fig u ra tio n f o r the Storage Wall S o lu tio n
The method which was used in v o lv e d perform ing a h e at balance on
each boundary node.
The balance equated n e t thermal gains to s to ra g e.
A f i r s t - o r d e r a pproxim ation f o r each tim e d e r i v a t i v e was then s u b s t i ­
tu te d and the r e s u l t a n t equations were solved f o r the new boundary tem­
peratu res.
The r e s u l t i n g e quation f o r node I was
(2 .3 5 )
The e x p l i c i t e q u atio n f o r the new te m peratu re o f node n was found to be
(2 .3 6 )
35
In o r d e r t o guarantee s t a b i l i t y o f the numerical method, l i m i t s on
At were needed.
They were d e riv e d from the e x p l i c i t Equations ( 2 . 3 1 )
through ( 2 . 3 6 ) , using the n o n -n e g a tiv e c o n d itio n c i t e d e a r l i e r .
The
l i m i t a t i o n s on A t were thus found to be
At <
(me).
in t
(2 .3 7 )
(UA)j n t + UAt o t . •
(me)
At <
VWll
room
(2 .3 8 )
+ (UA)i n t + r (m c )room + k P
2
At <
,
(2 .3 9 )
pcAx
At <
- 2 (h 0 + k/A x)
and
At <
(2 .4 0 )
pcAx
(2 .4 1 )
b + k/Ax) '
Equations ( 2 . 2 3 ) and ( 2 . 2 8 ) -
( 2 . 4 1 ) were used to w r i t e a computer
code to s im u la te the thermal response o f an i n d i r e c t gain passive s o l a r
system o f th e Trombe w a ll
ty p e .
The model i s a ls o a p p l i c a b l e to s o l i d
w a ll systems (systems w it h no v e n ts ) m erely by s e t t i n g
equal to z e r o .
The model used e s s e n t i a l l y the same programming l o g i c as d id the d i r e c t
gain model
(see Fig u re 2 . 4 ) .
The o n ly d i f f e r e n c e s i n th e l o g i c were the
e l i m i n a t i o n o f the geometry s o l u t i o n and o f the complex r a d i a t i o n s o lu ­
tio n .
Since the r a d i a t i v e h e a t t r a n s f e r f o r the Trombe w a l l system is
36
given e x p l i c i t l y by Equation ( 2 . 2 3 ) ,
i t was r e c a l c u l a t e d each time the
new system tem peratures were c a l c u l a t e d .
t u r e d i s t r i b u t i o n was c a l c u l a t e d f i r s t ,
The new sto ra g e w a l l tempera­
then the new a i r te m p e ra tu re .
The new i n t e r i o r s u rfa c e tem peratu re was c a l c u l a t e d l a s t .
c a l c u l a t i o n s were based on the o ld system te m p e ratu re s .
A l l o f these
The new a i r
space and v ent o u t l e t tem peratures were then c a l c u l a t e d , using the o th e r
new te m p e ra tu re s .
This procedure was fo llo w e d f o r each successive time
s te p .
The d i r e c t gain model has t i g h t r e s t r i c t i o n s on the a llo w a b le i n ­
t e r v a l between successive data sets (d a ta tim e s t e p ) .
This is due to
h o ld in g th e r a d i a t i v e h e at t r a n s f e r r a te s constant f o r s e v e ra l succes­
s iv e tim e s te p s .
This model, however, does not have the same r e s t r i c ­
t i o n s , because the r a d i a t i v e heat t r a n s f e r is r e c a l c u l a t e d f o r each
tim e s te p .
During th e model v a l i d a t i o n procedure, th e a llo w a b le data
tim e s te p f o r t h i s model was found to be one hour.
Model V a l i d a t i o n
The d i r e c t gain and Trombe w a ll models were v e r i f i e d using data
from pass iv e t e s t c e l l s a t th e N a tio n a l Center f o r A p p r o p r ia te Technolo­
gy (NCAT), lo c a te d in B u t t e , Montana.
passive systems were not a v a i l a b l e .
Data f o r i s o l a t e d gain and n a tu r a l
However, since t h e i r models are
e s s e n t i a l l y th e same as the d i r e c t gain model, they may be as accurate
as the d i r e c t gain model.
37
The d i r e c t gain model was v a l i d a t e d f o r an e ig h t day p e rio d be g in ­
ning November 14 and ending November 21, 1978.
th e measured and s im u la te d t e s t c e l l
Figure 2 . 8 i l l u s t r a t e s
a i r tem p e ratu re s .
th e ambient te m peratu re and the i n s o l a t i o n .
Also shown are
The t e s t c e l l
a i r tempera­
t u r e was measured using a s h ie ld e d thermocouple [ 1 9 ] lo c a te d near the
c e n t e r o f the t e s t c e l l .
The i n s o l a t i o n was the t o t a l
in c id e n t s o la r
ra d ia tio n
( f o r the p receding hour) measured on a v e r t i c a l , s o u th -f a c in g
s u rface.
The data were o b ta in e d from a r e p o r t by B ickle/C M [ 2 0 ] , which
was subm itted to th e S o la r Energy Research I n s t i t u t e
1979.
(SERI) on June 12,
The r e p o r t c o n ta in ed h o u r ly NCAT i n s o l a t i o n and ambient tempera­
t u r e d a ta f o r fo u r te e n days.
Hourly globe tem peratures f o r a d i r e c t
gain and Trombe w a l l t e s t c e l l were a ls o inclu d e d in the r e p o r t .
The
globe te m p e ratu re was an a pproxim ation o f the mean r a d i a n t tem perature
o f each system.
I t was measured w ith a thermocouple in s id e a t o i l e t
bowl f l o a t ( s p h e r e ) .
The f l o a t was p a in te d b la c k and was suspended i n
th e c e n t e r o f the t e s t c e l l a p p ro x im a te ly 3 0 .5 cm ( I
c e ilin g .
The d i r e c t gain t e s t c e l l
L a r r y P alm it e r a t NCAT.
f o o t ) from the
a i r tem peratures were ob ta in e d from
The globe te m peratu re data were not used f o r
th e d i r e c t gain model v a l i d a t i o n .
I n s t e a d , the a i r te m p e ratu re was
used, because o f th e d i f f i c u l t y o f c a l c u l a t i n g the globe te m p e ratu re .
The d i f f i c u l t y arose because o f the presence o f d i f f u s e s u n l i g h t a t
e v e ry p o in t in the t e s t c e l l .
The d i f f u s e s u n l ig h t caused the measured
globe tem peratures to be h ig h e r than the tem peratures o f th e surfaces i n
>
Experimental air temperature
Simulation air temperature
Ambient
Temperature
i
11/14
11/15
11/16
11/17
i
11/18
/A
11/19
I
11/20 11/21
Day
Fig u re 2 . 8
D i r e c t Gain V a l i d a t i o n
39
the room.
In s p e c tio n o f F ig u re 2 . 8 shows good agreement between th e measured
and s im u la te d a i r te m p e ra tu re s .
The mean d e v ia t io n o f th e sim ulated
te m peratu re from th e measured values was 0 . 1 °C ( 0 . 2 ° F ) f o r the e ig h t-d a y
p e r io d .
The r o o t mean square (rms) d e v i a t i o n was 1 .4 °C ( 2 . 6 ° F ) and th e
maximum d e v i a t i o n was 5 . 4°C ( 9 . 7 ° F ) .
The Trombe w a ll system model performance was v e r i f i e d f o r a s ix -d a y
p e r io d b eginnin g November 19, 1978.
The s im u la te d globe tem perature was
e s tim a te d using a m athem atical a pproxim ation o f a sphere w it h the same
s i z e and l o c a t i o n as th e measuring d e vic e in the t e s t c e l l s .
A sphere
10.1 cm (4 in c h e s ) i n d ia m e te r w ith an e m i s s i v i t y o f 0 .9 5 was lo c a t e d i n
the c e n t e r o f th e room, 3 0 .5 cm ( I
f o o t ) from the c e i l i n g .
R a d ia tiv e
i n t e r a c t i o n s between the sphere and the sto ra g e w a ll and between the
sphere and the room's i n t e r i o r s u rfa c e were considered.
th e sphere to the a i r was a ls o con sid e re d .
sphere was used to approxim ate i t s
Convection from
A steady h e a t balance on the
te m p e ra tu re .
Figure 2 .9 shows the
measured and s im u la te d globe tem peratures f o r the t e s t c e l l .
te m peratu re arid i n s o l a t i o n are a ls o in c lu d e d .
shows good agreement w ith th e measured v a lu e s .
The ambient
The s im u la te d tem perature
During th e f i r s t two
days, the d e v i a t i o n o f the s im u la tio n r e s u l t s from the d a ta is r e l a t i v e ­
ly la rg e .
This i s due to in a c c u ra c ie s i n the i n i t i a l
were used f o r the s im u l a t i o n .
to overcome these i n i t i a l
tem peratures which
However, th e model e x h i b i t s th e a b i l i t y .
in a c c u r a c ie s and becomes more a c c u ra te as the
------- Experimental room temperature
------Simulation room temperature
Ambient
Temperature
80
70
60
50
40 ° F
30
20
10
0
-F*
O
300
200
kw 0.5
1 0 0
11/19 11/20 11/21
11/22 11/23 11/24
Day
F ig u re 2 .9
Trombe Wall V a l i d a t i o n
0
B
hr-Ft2
41
the s im u la tio n proceeds.
This is a good p ro p e rty f o r a model to possess
because th e model is thus s e l f - c o r r e c t i n g .
The mean d e v i a t i o n o f the
s im u la te d te m peratu re from the measured was - I . I °C ( - 1 . 9 ° F ) f o r the
p e r io d .
The rms d e v i a t i o n was 2 . Q0C ( 3 . 6 ° F ) and the maximum d e v ia t io n
was 7 .2 °C ( I 3 ° F ) .
CHAPTER I I I
RESULTS
The purpose o f t h i s study was to perform a comparison o f , f o u r ba sic
types o f passive s o l a r h e a tin g systems.
g a in ,
The types compared were d i r e c t
i n d i r e c t g a in , i s o l a t e d gain and n a tu r a l
passive systems.
A com­
p u te r model was developed to s im u la te th e thermal response o f each type
o f pass iv e system, as d e scribed in Chapter I I .
chosen to
r e p r e s e n t the i n d i r e c t gain systems,
mon type o f passive s o l a r system in use
to d a y .
A Trombe w a ll system was
since i t i s the most com­
N a tu ra l passive systems
were in c lu d e d because i t was suspected t h a t r e l o c a t i o n o f th e normal
window area p re s e n t in a house to the south s id e o f the b u i l d i n g would
r e s u l t in
s i g n i f i c a n t s o l a r h e a t in g .
A
d i r e c t gain ( o r n a tu r a l p a ss iv e)
system is
i l l u s t r a t e d in F ig u re 2 . 1 .
A
Trombe w a ll and a n . i s o l a t e d gain
system a re shown in Figures 2 .6 and 2 . 5 , r e s p e c t i v e l y .
The fo u r passive systems were s im u la te d f o r two s ix - d a y periods o f
d i f f e r e n t c l e a r and cloudy day groupings.
T o ta l i n s o l a t i o n and the am­
b i e n t te m peratu re v a r i a t i o n were i d e n t i c a l
f o r the two p e r io d s .
f i r s t p e r io d , case I ,
cloudy days.
The
c o n s is te d o f th r e e c l e a r days fo llo w e d by th re e
This rep res e n te d a "worst" case s i t u a t i o n , where the t h e r ­
mal s to ra g e o f each system was r e q u ire d to heat i t s b u i l d i n g f o r a r e l a ­
t i v e l y lo n g , l o w - i n s o l a t i o n p e r io d .
The second grouping, case I I ,
began
w ith a c l e a r day which was fo llo w e d by a l t e r n a t i n g cloudy and c l e a r days
This rep res e n te d a "best" case s i t u a t i o n where the sto ra g e o f each sys­
tem was allo w e d one f u l l
c l e a r day to recharge be fo re being c a l l e d upon
43
to h e a t i t s
b u i l d i n g f o r the next (c lo u d y ) day.
For reasons e x p la in e d in
Chapter I I ,
h o u r ly and s u b -h o u rly i n s o l a t i o n and ambient tem perature data
were r e q u ire d in o rd e r to perform the d e s ir e d s im u la tio n s .
Since a c tu a l
measured data o f th e d e s ire d form were not a v a i l a b l e f o r Bozeman, Montana,
the r e q u ir e d data were generated such t h a t they rep res e n te d t y p i c a l mid^
w i n t e r c o n d itio n s in Bozeman.
The ambient tem perature data had 8.55°C
days ( h e a t i n g ) and 18.3 °C n ig h ts each day.
The t o t a l
d a ily in s o la tio n
was r e q u ire d to be 3 .7 6 kw-hr/m
g la z in g (1195 B t u / f t ) f o r c l e a r days
and 10% o f t h i s on cloudy days.
The cloudy day i n s o l a t i o n was determined
from e x p e rim en tal d a ta .
Study o f i n s o l a t i o n data from B u t t e , Montana and.
Los Alamos, New Mexico re v e a le d t h a t th e t o t a l
i n s o l a t i o n on cloudy days
is a p p ro x im a te ly 10-12% o f t h a t o f c l e a r days.
Appendix E contains the
equations (and t h e i r d e r i v a t i o n s ) which were used to g e n e ra te the neces­
s ary data from these average c o n d itio n s .
Each passive s o l a r system was in c o r p o r a te d i n t o a W e l l - i n s u l a t e d
s in g le f l o o r d w e llin g w ith a f l o o r a rea o f 139 m
(1500 f t
).
The house
was c o n s tru c te d such t h a t the e q u i v a l e n t R -f a c to r s o f th e b u i l d i n g com­
ponents were:
Foundation P e rim e te r
Walls
R = 3 .3 5 ,
(1 9 );
C e ilin g
R = 6 .6 9 ,
( 3 8 ) ; and
Windows
R = 0 .3 5 9 ,
(2 .0 4 ).
\
44
A l l windows ( i n c l u d i n g the c o l l e c t o r ) were assumed to be covered w ith
2
2
R 1 .7 6 ~ ~
( 10
— -)
i n s u l a t i o n a t n ig h t because o f th e in c re a se in
thermal e f f i c i e n c y which passive s o l a r h e a tin g systems always e x h i b i t
w ith i t s
use [ 1 0 , 1 1 ] .
change p e r hour.
were i d e n t i c a l
g a in ) systems.
The i n f i l t r a t i o n
r a t e was taken to be 1 /2 a i r
The c o l l e c t o r a r e a , thermal sto ra g e and h e a tin g load
f o r th e formal passive ( d i r e c t , i n d i r e c t , and i s o l a t e d
This was done so t h a t th e systems would be compared un­
der i d e n t i c a l c o n d i t i o n s .
The thermal s to ra g e o f the d i r e c t gain system
was assumed to be d i s t r i b u t e d e v e n ly over the p o r tio n o f the room r e c e i v ­
ing d i r e c t i n s o l a t i o n
( i ; e . the f l o o r , n o r t h , w e s t, and e a s t w a l l s ) .
The thermal sto ra g e o f th e i n d i r e c t gain system was c o n ce n tra te d in a
high d e n s it y ( 2 4 0 0 . kg/m3 ) c oncre te w a ll 4 0 .6 cm (16 i n . )
3 0 .5 cm ( I
ft)
from the window.
t h i c k , lo c a te d
The i s o l a t e d gain system was assumed to
be a d i r e c t gain system 3 .0 5 m (10 f t )
tached to th e south w a l l o f th e house.
deep by 1 5.2 m (50 f t )
long a t ­
The thermal s to ra g e was d i s t r i ­
buted e v e n ly over th e c o l l e c t o r ' s f l o o r and north w a l l , because those
two s u rfa c e s re c e iv e d most o f the i n s o l a t i o n .
O
a p p ro x im a te ly 0 .2 3 6 m / s
A fan which d e liv e r e d
(500 cfm) o f a i r was used to p ro v id e a i r i n t e r ­
change between the c o l l e c t o r and the house.
The n a tu r a l passive system was s im u la te d f o r two d i f f e r e n t window
areas.
The f i r s t a rea was a p p ro xim a te ly the normal window area present
in a house.
Al I windows were assumed to be lo c a te d on th e south w a ll o f
the b u i l d i n g , so a l l
could provide s o l a r h e a t in g .
The second window
45
a rea was a p p ro x im a te ly one h a l f o f t h a t used in the form al passive sys­
tems.
The thermal s to ra g e o f the n a tu r a l
by summing the thermal s to ra g e o f a l l
passive systems was estim a te d
in te rn a l w a lls .
They were con-
.
s t r u c t e d o f 2" x 4" wooden studs w ith 1 /2 " gypsum board on both s id e s .
In a d d i t i o n , the i n s id e s u rfa ce s o f th e e x t e r n a l w a l l s
board) were assumed to s to r e h e a t .
( 1 / 2 " gypsum
The thermal s torage o f th e b u i l d i n g 's
contents was approximated as equal to t h a t o f the house i t s e l f .
This
e s tim a te i s probably c o n s e r v a t iv e , s in c e most houses c o n ta in several
l a r g e , massive a p p lia n c e s , such as a r e f r i g e r a t o r and a s t o v e , . i n a d d i­
t i o n to c a b i n e t s , f u r n i t u r e , and o th e r p r o p e r ty .
The s to ra g e was then
d i s t r i b u t e d e v e n ly o v er the f l o o r , n o r t h , w e s t, and e a s t w a l l s .
The
c o l l e c t o r window a rea and thermal s to ra g e o f each s im u la te d system a re
l i s t e d i n Table 3 . 1 .
The tem peratures o f th e passive system components were i n i t i a l i z e d
such t h a t the thermal s to ra g e o f each was n e a r ly f u l l y charged b e fo re the
f i r s t cloudy day was encountered.
i s o l a t e d gain and n a tu r a l
A ll
components o f the d i r e c t g a in ,
passive systems were i n i t i a l i z e d a t 18.3°C (65
° F ) , sin ce the tim e c o n s ta n t o f t h e i r s to ra g e is s h o rt (one day o r le s s ).
The i n d i r e c t gain system 's s torage w a ll
tem perature d i s t r i b u t i o n was
i n i t i a l i z e d somewhat h ig h e r because i t s
tim e c onstant i s l a r g e r (s e v e r a l
d a y s ).
These i n i t i a l
c o n d itio n s were combined w ith th e p h y sic a l pa ra ­
meters and w eather data c i t e d above and used
to
s im u la te th e passive
systems (e x c e p t the i s o l a t e d gain system) w ith no a u x i l i a r y h e a t in g .
46
Table 3.1
Physical
Parameters o f the Passive S o la r Systems
Window
(C o lle c to r)
Area m2
( f t 2)
Thermal
Storage
k w -h r/°C
(B tu Z 0F)
D i r e c t Gain
3 4 .8
(3 7 5)
6 .8 5
(1 3000)
I n d i r e c t Gain
(Trombe W a ll)
3 4 .8
(3 7 5)
6 .8 5
(1 3000)
I s o l a t e d Gain
3 4 .8
(3 7 5 )
System Type
N a tu ra l
Passive
■
r
-
6 .8 5
(1 3000)
9 .2 6
(1 0 0)
2 .7 4
(5200)
1 7 .7
(1 9 0 )
2 .7 4
(5200)
The room a i r te m peratu re o f each system was allow ed to " f l o a t " w ith no
upper o r low er l i m i t s .
h e a t in g .
The systems were then s im u la te d w it h a u x i l i a r y
Whenever th e a i r te m peratu re was c a l c u l a t e d as being less than
1 5.6 °C (SO0F ) , i t was r e s e t to 1 5 .6 °C .
A te rm , equal to the a u x i l i a r y
h e at s u p p lie d , was then added to the a i r energy balance e q u atio n to main­
t a i n the b a la n c e .
The t o t a l
a u x i l i a r y h e a t s u p p lie d f o r the t e s t pe rio d
was p r i n t e d a t the end o f each s im u l a t i o n .
s im u la te d o n ly w ith a u x i l i a r y h e a t in g .
The i s o l a t e d gain system was
I f the c o l l e c t o r a i r tem perature
was h ig h e r than th e b u i l d i n g a i r te m p e ra tu re , the fan c o n t r o l l i n g a i r
flo w between the c o l l e c t o r and the b u i l d i n g was turned on whenever the
47
b u i l d i n g a i r te m peratu re dropped below 1 5 .6 ° C .
I t was tu rn e d o f f i f the
b u i l d i n g a i r te m peratu re rose above 2 1 . 1°C ( 7 0 oF ) » o r i f th e c o l l e c t o r
a i r te m peratu re was les s t h a t the b u i l d i n g a i r te m p e ra tu re .
The a u x i ­
l i a r y h e a tin g was c a l c u l a t e d i n the same manner as i t was f o r the o th e r
th r e e system s.
The r e s u l t s o f the d i r e c t and i n d i r e c t gain s im u l a t i o n s , w ith o u t
a u x i l i a r y h e a t in g , a re i l l u s t r a t e d ,
and 3 . 2 ,
re s p e c tiv e ly .
f o r cases I and I I ,
in Figures 3.1
Also shown a re the ambient te m p e ratu re and in s o ­
l a t i o n d i s t r i b u t i o n s f o r both cases.
The d i r e c t gain system, as e x p e c t­
ed, e x h i b i t s l a r g e r a i r te m peratu re f l u c t u a t i o n s f o r both cases.
The
i n d i r e c t gain system m a in ta in s a h ig h e r a i r tem perature d u rin g the cloudy
days o f both cases.
For case I I ,
both systems exp e rien ce a h ig h e r f i n a l
a i r te m peratu re ( a t the end o f day 6 ) .
This is due to th e low er peak and
mean a i r tem peratures which o c cu r, on c l e a r days, in case I I .
Lower
tem peratures r e s u l t in low er h e at loss r a t e s , which in tu r n cause the
s to ra g e to lose les s h eat over the p e r io d , r e s u l t i n g in h ig h e r f i n a l
te m p e ra tu re s .
Figures 3 .3 and 3 . 4 i l l u s t r a t e
the r e s u l t s o f the n a tu r a l
passive systems ( f o r both window a r e a s ) f o r cases I and I I ,
Both systems e x h i b i t la r g e a i r te m peratu re f l u c t u a t i o n s .
re s p e c tiv e ly .
For both cases,
th e f l u c t u a t i o n s a r e l a r g e r than those o f the d i r e c t gain system.
is due to th e r e l a t i v e l y small thermal sto ra g e o f the n a t u r a l
systems (see T able 3 . 1 ) .
This
passive
These systems a ls o exp e rien ce h ig h e r f i n a l a i r
tem peratures f o r case I than f o r case I I .
However, the d i f f e r e n c e in
Direct gain
Indirect gain
Ambient tem perature
- 300
hr-Ft
Day
Figure 3.1
D i r e c t - I n d i r e c t Gain Comparison Without A u x ilia r y Heating, Case I .
------ Direct gain
------- Indirect gain
40 0F
Ambient tem perature
Day
Figure 3.2
D i r e c t - I n d i r e c t Gain Comparison Without A u x ilia r y Heating, Case I I .
30
20
10
0
80
70
60
50
40
30
20
10
0
Figure 3 ,3
Natural Passive A i r Temperature Without A u x ilia r y Heating, Case I
Figure 3 .4
N a tu ra l
Passive A i r Temperature W ithout A u x i l i a r y H e a tin g , Case I I .
52
fin a l
tem peratures (between the two cases) is less than t h a t f o r e i t h e r
the d i r e c t o r i n d i r e c t gain systems.
N a t u r a l I y 9 the a i r tem perature is
g r e a t e r f o r th e l a r g e r window area system in both cases.
The a i r tem­
p e r a t u r e o f th e th r e e systems ( d i r e c t g a i n , i n d i r e c t g a in , and n a tu ra l
p a s s iv e ) i s m a in tain e d w e ll above ambient by the s o l a r h e a t in g .
the systems performs w e ll on cloudy days, showing l i t t l e
None o f
o r no a i r tem­
p e r a t u r e in c re a s e due to a b s o rp tio n o f cloudy day i n s o l a t i o n , sin ce l o s ­
ses to th e ambient outweigh these g a in s .
The r e s u l t s o f the s im u la tio n s o f th e d i r e c t and i n d i r e c t gain pas­
s iv e s o l a r h e a tin g systems, w ith a u x i l i a r y h e a t in g , a re i l l u s t r a t e d in
Figures 3 .5 and 3 . 6 .
Both systems m a in ta in the a i r te m p e ratu re a t or
above 15.6 °C f o r the f i r s t th r e e ( c l e a r ) days o f case I .
The d i r e c t gain
system r e q u ire d a small amount o f n ig h tt im e a u x i l i a r y h e a tin g f o r the
f i r s t th r e e days because o f i t s
l a r g e r a i r tem perature f l u c t u a t i o n s .
The
i n d i r e c t gain system m a in ta in s the a i r tem perature above 15 . 6 °C f o r the
th r e e days w it h o u t th e a id o f a u x i l i a r y h e a t in g .
The presence o f a u x i ­
l i a r y h e a tin g is evidenced by a uniform 15 . 6 °C a i r te m p e ra tu re .
Both
systems drop to a steady a i r tem peratu re o f 15.6°C a t about the same
tim e e a r l y in th e f o u r t h day.
Beyond t h i s tim e , th e a i r tem peratu re is
m a in tain e d by a u x i l i a r y h e a t in g .
In case I I
(F ig u r e 3 . 6 ) ,
however, the
d i r e c t gain system appears to outperform the i n d i r e c t gain system.
Both,
systems m a in ta in the a i r te m peratu re above 15.6°C f o r th e f i r s t day, but
both drop to and remain a t 15.6 °C f o r most o f the second.
During each
Direct gain
Indirect gain
80
70
60
50
40
Figure 3.5
D i r e c t - I n d i r e c t Gain Comparison with A u x ilia r y Heating, Case I .
Direct gain
Indirect gain
80
70
60
50
40
Figure 3.6
D i r e c t - I n d i r e c t Gain Comparison with A u x il i a r y Heating, Case I I .
55
succeeding c l e a r day, the d i r e c t gain system a t t a i n s about the same peak
te m p e ra tu re , and the system s u p p lie s a l l
1 /2 o f each c l e a r day.
provides a l l
the r e q u ir e d h e a tin g f o r about
The i n d i r e c t gain system, on the o th e r hand,
■i
the r e q u ir e d h e a tin g o n ly f o r s h o r t e r p erio ds each succeed­
ing c l e a r day.
The d i f f e r e n c e in the performance o f the. two systems is
due to the d i f f e r e n c e in s to ra g e s u rfa c e a r e a .
The d i r e c t gain system
has a much l a r g e r s to ra g e s u rfa c e a r e a , which enables i t
to respond,
q u i c k l y to absorbed i n s o l a t i o n .
Figures 3 .7 and 3 . 8 i l l u s t r a t e , th e r e s u l t s o f th e t e s t case s im u la ­
tio n s o f the n a tu r a l passive systems w i t h a u x i l i a r y h e a t in g .
perform in much the same manner as th e d i r e c t gain system.
The systems
Both systems
e x h i b i t a i r tem peratures i n excess o f 1 5.6 °C during th e sunny p o r tio n o f
each c l e a r day, i n d i c a t i n g t h a t s o l a r gain is p ro v id in g a l l
h e a t f o r t h a t p o r tio n o f the day.
the necessary
In both cases I and I T , the h e ig h t and
le n g th o f the te m p e ratu re peaks a re ( f o r each system) i d e n t i c a l f o r each
c l e a r day a f t e r day I .
This is because the s torage o f each system has a
r e l a t i v e l y s h o r t tim e c o n s ta n t and thus re a c ts r a p i d l y to ab so rp tio n o f
s o la r ra d ia tio n .
Figures 3 .9 and 3 .1 0 i l l u s t r a t e th e h o u rly average b u i l d i n g a i r
te m peratu re o f the i s o l a t e d gain system f o r cases I and I T ,
r e s p e c tiv e ly .
Due to the a c t io n o f the f a n , which was turned on and o f f s e v e ra l times
p e r hour, the b u i l d i n g a i r tem peratu re v a r i e d c o n tin u o u s ly between
1 5 . 6 QC and 2 1 .1 °C .
The f l u c t u a t i o n was g r e a t e s t when the c o l l e c t o r a i r .
80
70
60
50
40
Figure 3 .7
Natural passive A i r Temperature with
A u x il i a r y Heating, Case I .
30
Figure 3 .8
17.7 m Z
9 .3 m 2
Natural Passive A i r Temperature w ith A u x ilia r y Heating, Case I I .
Figure 3 .9
I s o l a t e d Gain B u ild in g A i r Temperature w ith A u x i l i a r y H e a tin g , Case I .
30
20
10
0
F ig u re 3 .1 0
80
70
60
50
40
I s o l a t e d Gain B u ild in g A i r Temperature w it h A u x i l i a r y H e a tin g , Case I I
in
kO
60
te m p e ratu re was much g r e a t e r than the minimum tem peratu re needed to heat
i
the b u i l d i n g .
Whenever th e c o l l e c t o r a i r tem perature f e l l
to a le v e l
which a llo w e d the fan to run c o n tin u o u s ly , an ab ru p t r i s e in the average
b u i l d i n g a i r te m p e ratu re o c cu rre d .
This s i t u a t i o n caused the unexpected
r i s e i n b u i l d i n g a i r te m p e ratu re e a r l y i n day 3 o f case I .
This phenom­
enon a ls o caused th e odd shapes o f the tem perature r i s e s on day I o f
case I and on days I and 3 o f case I I .
The i s o l a t e d gain system demon­
s t r a t e s good s o l a r h e a tin g c a p a b i l i t i e s by m a in ta in in g th e b u i l d i n g a i r
te m p e ratu re above 1 5 .6 ° C , w ith o u t the a id o f a u x i l i a r y h e a t in g , w e ll i n t o
the cloudy days o f both cases.
In o r d e r to compare th e performances o f the s im u la te d passive s o l a r
h e a tin g systems, some d e f i n i t i o n o f system e f f e c t i v e n e s s i s necessary.
The most common measure o f system e f f e c t i v e n e s s is th e s o l a r f r a c t i o n .
It
i s d e fin e d as the f r a c t i o n o f the b u i l d i n g ' s h e a tin g load which is
s u p p lie d by s o l a r h e a t in g .
The h e a tin g load was c a l c u l a t e d f o r each
system, using s tandard ASHRAE techniques [ 1 7 ] , as the energy needed to
m a in ta in the b u i l d i n g a i r tem peratu re (above am bient) a t 1 5.6 °C (6 0 °F )
f o r the s ix - d a y p e r io d , w it h o u t s o l a r h e a t in g .
During the s i m u l a t i o n s , .
th e b u i l d i n g a i r was heated o n ly by h e a t from the thermal s to r a g e , the
a u x i l i a r y h e a t source and s o l a r r a d i a t i o n .
The heat s u p p lie d by s o l a r
r a d i a t i o n was thus d e fin e d as
[ s o l a r h e a t in g ] = [ h e a t i n g lo a d ] + [ n e t change i n thermal s to r a g e ]
- [ a u x ilia r y h e a tin g ].
61
The n e t change in thermal s to ra g e was c a l c u l a t e d from th e i n i t i a l
fin a l
s to ra g e te m p e r a tu re s .
tem was les s than the i n i t i a l
s to ra g e was n e g a t iv e .
I f the f i n a l
s to ra g e te m p e ratu re f o r a sys­
te m p e ra tu re , the net change i n thermal
The s o l a r f r a c t i o n was then d e fin e d as the s o l a r
h e a tin g d iv id e d by th e h e a tin g lo a d .
to tal
and
T a b le 3 .2 l i s t s
th e h e a tin g lo a d ,
a u x i l i a r y h e at s u p p lie d , n e t change in thermal s to ra g e and the
s o l a r f r a c t i o n o f each s im u la te d system, f o r both t e s t cases.
As e x p e c te d , the form al passive s o l a r systems achieved h ig h e r s o l a r
f r a c t i o n s f o r the "b est" case (case I I )
than f o r case I .
However, the
d i f f e r e n c e between the two cases is much s m a lle r than was a n t i c i p a t e d
(o n ly a few p e r c e n t ) .
This was probably due to the i n i t i a l
p e ra tu re s a n d /o r th e le n g th o f the t e s t p e r io d s .
system tem­
In s p e c tio n o f Table
3 .2 r e v e a ls t h a t each system e x h i b i t s a d i f f e r e n t magnitude o f p e r f o r ­
mance change between the two t e s t cases.
Each n a tu r a l
performs v i r t u a l l y the same f o r cases I and I I .
passive system
The i s o l a t e d gain sys­
tem, however, e x h i b i t s the l a r g e s t d i f f e r e n c e in perform ance, w h ile the
d i r e c t and i n d i r e c t gain system performances d i f f e r some, b u t not as
much as those o f the i s o l a t e d gain system.
The magnitudes o f the d i f ­
fere n ce s seem to be d i r e c t l y r e l a t e d t o th e le v e l o f h e a t d e l i v e r y con­
t r o l p r e s e n t i n th e system.
The n a tu r a l
passive system w it h the l a r g e r
c o l l e c t o r window a rea produced a s i g n i f i c a n t l y l a r g e r s o l a r f r a c t i o n
than d id the one w ith the s m a lle r window a r e a .
but
This r e s u l t was expected,
in s p e c tio n o f the r e s u l t s f o r these two systems in d i c a t e s t h a t
T a b le 3 . 2
System
A u x i l i a r y H e a tin g and S o la r F r a c tio n s f o r the T e s t Case S im u la tio n s
H e a t i ng A u x i l i a r y
Load
Heating
kW'-hr
kw-hr
( I O 6Btu)
( I O 5Btu)
Case I
Net Change
i n Storage
kw-hr
( I O 5Btu)
S o la r
F r a c t io n
%
Case I I
Net Change
in Storage
kw-hr
( I O 5Btu)
A u x ilia ry
Heating
kw-hr
( I O 5Btu)
S o la r
F r a c t io n
%
D i r e c t Gain
562
(1 .9 2 )
171
(5 .8 5 )
-1 9
(-0 .6 5 )
66.1
166
. (5 .6 8 )
-1 9
(-0 .6 5 )
6 7 .0
I n d i r e c t Gain
562
(1 .9 2 )
96
.(3 .2 7 )
-1 1 2
(-3 .8 2 )
63.1
114
(3 .9 0 )
-8 2
(-2 .8 1 )
65.1
I s o l a t e d Gain
565
(1 .9 3 )
166
(5 .6 8 )
-1 9
(-0 .6 5 )
6 7 .2
168
(5 .7 5 )
+19
(+ 0 .6 5 )
7 3 .6
9.3m2
486
(1 .6 6 )
316
(1 0 .8 )
-2 3
(-0 .7 8 )
3 0 .2
316
(1 0 .8 )
-2 3
(-0 .7 8 )
3 0 .2
17.7m2
512
(1 ,7 5 )
269
(9 .1 9 )
-2 3
(-0 .7 8 )
4 3 .0
269
(9 .1 8 )
-2 3
(-0 .7 8 )
43.1
N a tu ra l Passive
.
63
doubling the c o l l e c t o r , area does not double the s o l a r f r a c t i o n .
This i s .
in p a r t due to the in c re a s e in heat loss which always accompanies i n -
creased window a r e a .
For both t e s t cases, the d i r e c t gain system outperform ed the i n ­
d i r e c t gain ( Trombe w a l l ) system.
This is c o n s is te n t w it h the r e s u lts
o f Wray and Balcomb [ 1 2 ] f o r s to ra g e mass to window area r a t i o s in e x ­
cess o f 830 kg/m2 (170 l b / f t 2 ) .
The r a t i o f o r the form al passive s o l a r
p
h e a tin g systems used in t h i s study was 976 kg/m
p
(200 l b / f t
).
CHAPTER IV
CONCLUSIONS
The r e s u l t s o f t h i s study i n d i c a t e t h a t passive s o l a r h e a tin g sys­
tems a re good means o f space h e a t in g .
With adequate c o l l e c t o r area and
thermal s to ra g e a passive system may h e at a home f o r a f a i r l y
p e rio d (one f u l l
long cloudy
day or l o n g e r ) ; however, some source o f make-up h eat is
o ft e n needed to m a in ta in human com fort l e v e l s du rin g these cloudy
p e r io d s , e s p e c i a l l y in c o ld c lim a t e s .
For the c o n d itio n s o f t h i s s tu d y ,
the d i r e c t gain system is a b e t t e r choice f o r space h e a tin g than is the
Trombe w a l l system.
However, t h i s does not mean t h a t a d i r e c t gain sys­
tem i s a b e t t e r choice than an i n d i r e c t gain system, s in c e w a te r w a ll
systems c o n s i s t e n t l y o u t-p e r fo r m Trombe w a ll systems [ 1 0 ] , and may p e r ­
form b e t t e r than d i r e c t gain systems.
I s o l a t e d gain systems, which are
r e t r o f i t t a b l e , e x h i b i t th e p o t e n t i a l to p rovide space h e a tin g comparable
to d i r e c t and i n d i r e c t gain systems w it h the same c o l l e c t o r area and
thermal s to r a g e .
Although formal passive s o l a r h e a tin g systems may e a s i l y provide
more than 50% o f the space h e a tin g requirem ents o f a d w e llin g under
average m id - w in t e r c o n d itio n s i n Bozeman, Montana, mere r e l o c a t i o n o f
th e normal window area p re s e n t in a house onto the south w a ll o f the
b u i l d i n g r e s u l t s in s u b s t a n t i a l s o l a r h e a t in g .
This r e l o c a t i o n would be
d i f f i c u l t and expensive f o r houses which have a lr e a d y been b u i l t , but
would c o st l i t t l e
o r nothing a t the o r i g i n a l
c o n s tr u c tio n l e v e l .
c o l l e c t o r window i n s u l a t i o n is d e s i r a b l e f o r use w ith a l l
types o f
N ig h t
65
passive s o l a r h e a tin g systems because o f th e in c re a s e i n thermal e f f i ­
c ie n c y which always accompanies i t s
The performance o f a l l
use.
passive s o l a r systems seems to be dependent
on th e d i s t r i b u t i o n o f th e i n c id e n t s o l a r r a d i a t i o n ( i n s o l a t i o n ) over
th e p e rio d i n q u e s tio n .
The performance o f passive systems appears to
be b e s t when the i n s o l a t i o n i s spread e v e n ly over the p e r io d , such as
when c l e a r and cloudy days a l t e r n a t e .
A decrease from t h i s performance
i s e x h i b i t e d by passive s o l a r systems when th e i n s o l a t i o n i s more con­
c e n t r a t e d , such as when c l e a r and cloudy days occur in groups o f th re e
each.
The magnitude o f t h i s decrease is dependent on the type o f system
c o n s id e re d .
N a tu ra l passive systems, which e x h i b i t l a r g e a i r tempera­
t u r e f l u c t u a t i o n s when a u x i l i a r y h e at is not s u p p lie d , e x p e rie n c e a
n e g l i g i b l e decrease in performance from one d i s t r i b u t i o n to the o th e r .
On the o t h e r hand, i s o l a t e d gain systems, which c o n tr o l h e a t d e l i v e r y to
th e b u i l d i n g a i r b e t t e r than the o th e r types o f passive systems s tu d ie d ,
e x h i b i t the g r e a t e s t decrease in performance when c l e a r and cloudy days
occur in groups o f t h r e e , r a t h e r than a l t e r n a t i n g .
T h is seems to show a
d i r e c t r e l a t i o n s h i p between the degree o f heat d e l i v e r y c o n tr o l in h e r e n t
in th e system and the magnitude o f the decrease in system performance
(fro m one i n s o l a t i o n d i s t r i b u t i o n to th e o t h e r ) .
This tr e n d is als o
e x h i b i t e d by d i r e c t and i n d i r e c t gain systems, and o n ly a p p lie s to sys­
tems w it h equal c o l l e c t o r area and thermal s to ra g e .
APPENDICES
APPENDIX A
SELECTION CRITERIA FOR LOCATION OF PROJECTED POINTS
Development o f th e d i r e c t gain model re q u ire d a s e t o f s e l e c t i o n
c rite ria
to determ ine th e l o c a t i o n o f . p r o j e c t e d window c o rn e r p a i n t s .
Table 2.1
lis ts
th e e quations which were used to determ ine th e c o o r d i­
nates o f each p r o je c te d p o i n t .
However, these c o o rd in a te equations ^ r e
dependent on where w i t h i n the room th e p r o je c te d p o in t f a l l s .
The equa­
ti o n s a re d i f f e r e n t f o r each s u r fa c e ( f l o o r , north w a l l , west w a l l , and
e a s t w a l l ) upon which the p o in t may f a l l . , The procedure which was used
to determ ine th e p rope r s e t o f equations to use was as f o l l o w s :
F irs t,
th e p o i n t was checked to see i f
i t fe ll
on the f l o o r .
If
t h i s was t r u e , then the i n s o l a t i o n d i r e c t i o n met
C^boty £ W,
(A. I )
Ctj - Wtany £ 0 ,
(A .2 )
Cfa + C^tanycota £ O
(A .3 j
and one o f e i t h e r
or
'
Cg -
(L - Ca H a n y c o t a £ 0 .
I f these c r i t e r i a were not m et, then th e north w a ll was checked,
p o in t f e l l
on the north w a l l , the f o l l o w i n g c o n d itio n s were met:
(A. 4 )
I f the
68
and one o f e i t h e r
( I - Ca ) c o ta > W
( A . 6)
or
-Ca COta ^ W.
( A . 7)
I f these c r i t e r i a were a ls o not m et, th e west w a ll was t e s t e d .
fe ll
The p o in t
on the west w a l l i f th e i n s o l a t i o n d i r e c t i o n s a t i s f i e d
-C _cota < 0 ,
a
—
(A .8)
+ C^cotytana > I ,
(A .9 )
(L - Ca ) t a n y c o ta _> 0 ,
(A. 1.0)
(L - Cg ) c o ta £ W.
( A . 11)
I f none o f the preceding s e ts o f c r i t e r i a were met, then the p r o je c te d
p o in t f e l l
on the e a s t w a l l .
However, i n o rd e r to d e t e c t any e r r o r s in
t h i s method, the e a s t w a ll was a ls o t e s t e d .
to f a l l
I f the p o in t was found not
on any o f the fo u r a llo w a b le s u r fa c e s , an e r r o r message was
p rin te d .
This e r r o r message remained in the model throughout i t s d e v e l­
opment and use and was never encountered d u rin g any run.
p o in t f e l l
on the e a s t w a ll
The p r o je c te d
i f the f o l l o w i n g c o n d itio n s were met:
(L - Cg ) c o ta £ 0 ,
(A .1 2 )
69
Cb + Cgtanycota >_ 0 ,
( A . 13)
-C^cota < W.
a
—
( A . 14)
and
These c r i t e r i a were d e r iv e d using p h y sic a l reasoning combined w ith the
i n h e r e n t p h y s ic a l l i m i t a t i o n s o f the d i r e c t gain system.
APPENDIX B
DETERMINATION OF THE SUNLIT AREA CONFIGURATION .
The d i r e c t gain model c onta ins a s u b ro u tin e which generates e q u iv a ­
l e n t re c ta n g le s f o r each s u n l i t area i n the room (see Chapter I I ) .
th e r e
a re 19 p o s s ib le c o n f ig u r a t io n s t h a t th e s u n l i t areas could assume.
Table
B .I
lis ts
the lo c a tio n s o f the p r o je c te d window c o rn e r p o in ts (Ap , Bpl
C , and D ) f o r each p o s s ib le c o n f i g u r a t i o n .
H
r
Each o f these cases has i t s
own s e t o f eq u atio n s f o r th e c a l c u l a t i o n o f the s iz e s and lo c a tio n s o f
th e e q u i v a l e n t r e c t a n g l e s .
Each o f the s e ts was in c o r p o r a te d i n t o the
s u b ro u tin e as an i n d i v i d u a l s e c t i o n , n e c e s s i t a t i n g some means o f d e t e r ­
m ining which s e c tio n to execute f o r any i n s o l a t i o n d i r e c t i o n .
The pro­
gramming l o g i c which was used to accomplish the s e l e c t i o n proceeded as
fo llo w s :
A v a r i a b l e was used to i n d i c a t e on which s u rfa c e
( f l o o r , north
w a l l , west w a l l , o r e a s t w a l l ) each p r o je c te d window c o rn e r p o in t
fe ll.
These v a r i a b l e s were then used to d i r e c t program execution
to th e p roper s e c t i o n ^
Since t h i s model was w r i t t e n i n F o rtra n IV
programming language, computed GO TO statements were used.
This
type o f s ta te m e n t t r a n s f e r s c o n tr o l o f the program to d i f f e r e n t
l o c a t i o n s , depending on the value o f a v a r i a b l e ( i . e .
mentioned v a r i a b l e s ) .
Execution o f th e su b ro u tin e was t r a n s f e r r e d
through a s e t o f 17 computed GO TO s ta te m e n ts .
t r a n s f e r r e d c o n tr o l
the a f o r e ­
The f i r s t statem ent
to v a rio u s o th e r GO TO statem ents as a fu n c tio n
o f the l o c a t i o n o f Brs.
P
These s ta te m e n ts , in t u r n , t r a n s f e r r e d
71
Table B .l
P o s s ib le C o n fig u ra tio n s o f the S u n l i t Areas
Location o f
C o n fig u ra tio n
AP
.
I
flo o r
2
flo o r
3
BP
: ■
S
dp
flo o r
flo o r
flo o r
flo o r
west w a ll
flo o r
flo o r
flo o r
West w a ll
4
north w a ll
north w a ll
north w a ll
5.
flo o r
north w a ll
north w a ll
flo o r
6
flo o r
north w a ll
west w a ll
flo o r
7
flo o r
no rth w a ll
west w a ll
west w a ll
8
north w a ll
north w a l I
west w a l I
west w a ll
9
flo o r
west w a ll
west w a l l
flo o r
10
f l oor
west w a ll
west w a ll
west w a ll
west w a ll
west w a ll
west w a ll
e a s t w a ll
north w a ll
flo o r
Tl
12
.
west w a ll
flo o r
13
e a s t w a ll
e a s t w a ll
north w a ll
14
e a s t w a ll
east w a ll
north w a ll
15 •
16
17
flo o r
e a s t w a ll
flo o r
west w a ll
••'\
•.
north w a ll
.
flo o r
north w a ll
e a s t w a ll
flo o r
flo o r
e a s t w a ll
flo o r
flo o r
e a s t w a ll
east w a lI •
e a s t w a ll
18
e a s t w a ll
e a s t w a ll
19
e a s t w a ll
e a s t w a ll
.
e a s t w a ll
.
;
- flo o r
. f l otir
e a s t w a ll
.
72
c o n tr o l as a f u n c tio n o f ano th e r p r o je c te d p o i n t .
was re pea te d u n t i l
Bp f e l l
the proper case was reached.
on the f l o o r (Ap must a ls o f a l l
This procedure
For example, i f
on the f l o o r ) , th e next
s ta te m e n t checked th e l o c a t i o n o f Cp , and the f o l l o w i n g statements
checked the l o c a t i o n o f D^.
P
As a check on t h i s method, the s u b ro u tin e contained an e r r o r mes­
sage which was p r i n t e d i f the s u n l i t a rea c o n f i g u r a t i o n d id not match
one o f th e 19 a llo w a b le ones.
The e r r o r message was never encountered
d u rin g any ru n , i n d i c a t i n g t h a t a l l
c onta ined in T a b le B . l .
o f the p o s s ib le c o n fig u r a tio n s are
The p o s s i b i l i t i e s were d e riv e d using physical
reasoning and th e i n h e r e n t p h y sic a l l i m i t a t i o n s o f the system.
APPENDIX C
DIFFERENTIAL ELEMENT VIEW FACTORS
The method which was used in the d i r e c t gain model to c a l c u l a t e the
d i f f u s e r a d i a t i o n view f a c t o r s n e c e s s ita te d expressions f o r view fa c to r s
from d i f f e r e n t i a l
elements to p la n a r s u r fa c e s .
Equations were needed
f o r two c o n f i g u r a t i o n s , p e r p e n d ic u la r surfa ce s and p a r a l l e l
s u rfa c e s .
The equations were d e riv e d from expressions obta ine d from S e ig e l a n d .
Howell [ 2 1 ] .
F ig u re 3.1
illu s tra te s
d i c u l a r to s u rfa c e Ag.
along i t s
bottom edge.
a d iffe re n tia l
The element l i e s
e lem ent, dA^, which is perpen­
in a plane which i n t e r s e c t s Ag ,
,Using the nomenclature given in the f i g u r e , the
view f a c t o r from dA-j to Ag, F^^ g, is given by
f
(C .l)
F ig u re C.2 i l l u s t r a t e s
p a ra lle l
to s u rfa c e Ag.
a d iffe re n tia l
element which l i e s
in a plane
The expression f o r F ^ ^ g was found by summing
th e view f a c t o r s from th e element to each s e c tio n o f Ag ( a , b , ; c , and d).
The terms were then regrouped, r e s u l t i n g in
74
I
dl-2
2 tts
7= = = =
VT
+(x/s)2
( tan "1 ,
v
w -
Vl
+ ( ^m
)2
( ta n " 1 V l + ( ^ ) 2
tan "1 , ( " - .W A
Ji/s_
V l +(x/s)2
+
tan
+
tan
-I
-I
(
)
•
(h-,y)/s
)/s
Vi
V i + (y /s )2
+ (y /s )2
f t #
+
V l +(x/s)2
y /s
^ tan "1
Vi
y /s
)
z
X
\2
4 ^ ) 2 /
(w -x )/s
j +
Vi + (y /s )2
a
)
(C .2 )
Equations ( C . l ) and ( C . 2 ) were used in c o n ju n c tio n w it h Equation
(2 .2 )
to c a l c u l a t e th e d i f f u s e r a d i a t i o n view f a c t o r f o r each p a i r o f
h ea t t r a n s f e r su rfa c e s w i t h i n the d i r e c t gain system.
75
Fig u re C .l
D iffe re n tia l
Element f o r P e rp e n d ic u la r Surfaces
76
Fig u re C.2
D iffe re n tia l
Element f o r P a r a l l e l Surfaces
APPENDIX D
NUMERICAL DOUBLE INTEGRATION
In o r d e r to c a l c u l a t e the d i f f u s e r a d i a t i o n view f a c t o r f o r each
p a i r o f h e at t r a n s f e r s u rfa ce s w i t h i n th e d i r e c t gain model, as presen­
te d in Chapter I I ,
was needed.
a method f o r perform ing numerical double i n t e g r a t i o n
Such methods appear in many numerical methods t e x t s .
The
method which was used i s v e ry s i m i l a r t o , but v/as developed independent­
l y from , t h a t presented by Gerald [ 2 2 ] .
a p p l ic a b l e to any numerical
The technique which fo llo w s is
i n t e g r a t i o n method.
used f o r t h i s development because i t
Simpson's method was
i s th e most commonly used numerical
i n t e g r a t i o n method.
Suppose the i n t e g r a l ,
I,
o f the fu n c t io n F ( x , y )
is d e s i r e d , d e fin e d
as
(D .l)
T h is e q u atio n may be r e w r i t t e n as
(D .2 )
where
,x
n
G(y) =
I
x,
0
F (x ,y )d x .
(D .3 )
78
Using a Simpson's r u l e i n t e g r a t i o n . Equation (D .2 ) may be approximated
as
I =
[G (y 0 ) + 4G (y1 ) . + 2G(y2 ) + 4G(y3 ) + 2G(y^) + . . .
+ 4G(y m „ 3 ) * 2 6 ( y m„ 2 ) 4 46 ( y m „ , ) + G(ym) ]
(D .4 )
where
f*n
G(Y1 ) = J
F ( x , y 1 )d x .
( D .5 )
xO
However, G(Yi ) may a ls o be approxim ated, using Simpson's r u l e , as
G(Yi ) =
[F (X Q 1Yi ) + 4 F (x i ,Y i ) + 2 F (x 2 ,Yi ) + . . .
+ 2 F (x ^ _ 2 , y i ) + 4 F (x n_-|,Yi ) + F(XfilYi ) ] .
(D .6 )
Numerical double i n t e g r a t i o n may thus be performed by using Equa­
tio n
( D . 4 ) , w ith each term determined from Equation ( D . 6 ) .
APPENDIX E
INSOLATION AND AMBIENT TEMPERATURE DATA GENERATION
As e x p la in e d in Chapter I I ,
the v a rio u s passive s o l a r system models
used to perform t h i s study could o n ly be run w ith h o u r ly o r sub -h o u rly
i n s o l a t i o n and ambient tem peratu re d a ta .
Actual measured h o u r ly i n s o l a ­
t i o n data were not a v a i l a b l e f o r Bozeman, Montana.
They a r e a v a i l a b l e
f o r G reat F a l l s , Montana, bu t th e e x t r a p o l a t i o n o f t h i s d a ta to Bozeman
could r e s u l t in l a r g e e r r o r s , due to s i g n i f i c a n t e l e v a t i o n and c lim a tic
d i f f e r e n c e s between th e two l o c a t i o n s .
however, a re a v a i l a b l e f o r Bozeman.
Average d a i l y i n s o l a t i o n v a lu e s ,
The necessary s u b -h o u rly i n s o l a t i o n
d a ta were generated such t h a t they s a t i s f i e d the average c o n d itio n s .
The i n s o l a t i o n was assumed to be d i s t r i b u t e d throughout th e day in a
s in u s o id a l
form , as shown i n Fig u re E . l .
The general e q u atio n f o r the
i n s o l a t i o n d i s t r i b u t i o n , qs , o f t h i s form is
( f o r d a y l i g h t hours)
2rr(t-SR)
Qa v e I 1 " c° s
The t o t a l
)•
(E .l)
i n s o l a t i o n f o r th e day, Hg , i s given by
(E .2 )
H
s
Equation ( E . l ) was s u b s t i t u t e d i n t o ( E . 2 ) , the i n t e g r a t i o n was performed
and qave was found to be
qave - H -
(E .3 )
80
12
s o la r tim e
(hours)
-
Figure E . l
The Assumed I n s o l a t i o n D i s t r i b u t i o n
The necessary s u b -h o u rly i n s o l a t i o n data were thus generated using
qS '
If0 -
c ° s 2l,( q' SR) ) ;
SR < t < SS
= 0 a t n ig h t.
(E .4 )
The day l e n g t h , DL, was c a l c u l a t e d using r e l a t i o n s o b ta in e d from D u f f i e
and Beckman [ 1 4 ] .
It
i s a fu n c tio n o n ly o f the day o f the y e a r and o f
the l a t i t u d e o f the t e s t l o c a t i o n .
H ourly ambient tem peratu re data a re a v a i l a b l e f o r Bozeman, as they
a re f o r any U.S. Weather S e rv ic e r e c o rd in g s t a t i o n .
used
These data were not
d i r e c t l y , because a s p e c i f i c number o f h e a tin g degree days and
n ig h ts were d e s ire d f o r the t e s t case ambient tem peratu re d i s t r i b u t i o n .
81
H e a tin g degree days, DD, a re d e fin e d as
DD = h e a tin g degree days ( f o r one day)
<Tr e f ' W
(E .5 )
dt
where Tr e f = r e fe r e n c e tem peratu re = 1 8 . 3°C ( 6 5 ° F ) .
Degree n i g h t s , DN, a re d e fin e d the same, e xcept t h a t th e i n t e g r a t i o n is
performed o n ly f o r n ig h tt im e ambient te m p e r a tu re s .
The h o u r ly ambient
te m peratu re data which were a v a i l a b l e were used to determ ine the average
t o t a l monthly h e a tin g degree days and n ig h ts f o r Bozeman (fro m 1968 to
1 9 7 7 ).
These values a re l i s t e d in Table E . l .
The sum o f degree days
and n ig h ts equals c onve ntiona l degree days, DDcony.
The s im u la tio n s which were performed f o r t h i s study used f i c t i c i o u s
( h y p o t h e t i c a l ) ambient tem peratu re d a ta .
was assumed to be s in u s o id a l
The tem perature d i s t r i b u t i o n
in form , as shown in Fig u re E . 2 .
The
general e q u a tio n f o r t h i s d i s t r i b u t i o n is
mean
(E .6 )
The d i s t r i b u t i o n was a ls o c o n s tra in e d to have a number o f h e a tin g degree
days and n ig h ts equal to the d a i l y average f o r January i n Bozeman, as
determ ined from Table E . l .
Degree days a re d e fin e d by Equation ( E . 5 ) .
T
amb
82
t
O
12
Fig u re E.2
24
s o la r tim e
The Assumed Ambient Temperature D i s t r i b u t i o n
Conventional degree days a re d e fin e d as
DDc o n v = DD + DN
I
Equation ( E . 6 ) was s u b s t i t u t e d in t o
24
(E .7 )
<Tr e f * Tamb>d t -
( E . 5 ) and the i n t e g r a t i o n was p e r ­
formed, r e s u l t i n g in
D0 = Tr= f & + Tmean<? s i " W
' #
' T-max %
Ti
Equations ( E . 6 ) and ( E . 8 ) were then s u b s t i t u t e d in to
^24
•
(E .8 )
( E . 7 ) and the i n t e ­
g r a t io n was perform ed, r e s u l t i n g in
°N = T r e f O
Tmax
T
Bk) + T
24'
s1n W
[—
meanL24
i] +
(E .9 )
T able E . I
Degree D a v /N iq h t Averages f o r Bozeman. Montana ClQRR
Month
Degree Days °C
Degree Nights °C
January
265
567
February
228
413
March
250
361
190
230
127
141
June
66
79
J u ly
31
46
August
32
57
September
78
A p ril
'
May
'
977)
.
147
October
134
277
November
173
405
December
235
522
Equations ( E . 8 ) and ( E . 9 ) were solved s im u lta n e o u s ly f o r T
IIIGdi I
and T
IlldA
,
r e s u l t i n g in
(E.10)
Tmean = Tref ' <DD + DN>
and
T
max
s in #
UL , / -r
r e f 24
v ref
-D D -
d
n
H^1n^
Ok)
24;
(E.n)
- DD
The necessary s u b -h o u rly ambient te m p e ratu re data were then generated
84
using Equation ( E . 6 ) , w it h T
( E . 1 0 ) and ( E . T 1 ) .
IU tS c lii
and T
IliQ A
determined from Equations
The data then had th e r e q u ire d number o f h e atin g
degree days and n i g h t s .
LITERATURE CITED
LITERATURE CITED
1.
Stromberg, R. P. e t a l , Passive S o la r B u ild in g s :
A Com pilation o f
Data and R e s u lt s , Sandia L a b o r a to r ie s p u b l i c a t i o n SAND 7 7 -1 2 0 4 , 1977.
2.
Trombe, F . ,
B -l-7 3 -1 0 0 ,
3.
B a !comb, J. D . , J . C. Hendstrom, a n d -R-. D. McFarland, "Passive S o la r
H e a tin g o f B u i l d i n g s , " Los Alamos S c i e n t i f i c L a b o r a to r ie s p u b l i c a ­
t i o n LA-UR-77-1162, 1977.
4.
Bi I gen, E. and R. J e l d r e s , On the O p tim iz a t io n o f Trombe Wall S o la r
C o l l e c t o r s , American S o c ie ty o f Mechanical Engineers p u b l i c a t i o n
78-WA/S0L-13, 1978.
5.
Ohanessian, P. and W. W. S. C h a r te r s , "Thermal S im u la tio n o f a Pas­
s iv e S o la r House Using a Trombe-Michel Wall S t r u c t u r e , " S o la r EnerSy., v o l . 2 0, pp. 2 7 5 -2 8 1 , 1978.
6.
P a l m i t e r j L . , T. W heeling, and B. C o r b e t t , "Measured and Modeled
P assive Performance in Montana," I n t e r n a t i o n a l S o la r Energy S o c ie ty
Annual M e e tin g , Denver, Colorado, p roceedings, 1978.
7.
Emery, A. F . , c. J . Kippenhan, D. R. Heerwagen, and G. B. V are y ,
The S im u la tio n o f B u ild in g Heat T r a n s f e r f o r Passive S o la r Systems,
American S o c ie ty o f Mechanical Engineers p u b l i c a t i o n 79-WA/S0L-38,
1979.
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