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Basic theory of rapid transit-tunnel ventilation
Brown, W. G.
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Basic Theory of Rapid Transit-Tunnel Ventilation
W. G. BROWN
Research Officer, Building Services Section, Division of Building Research, National Research Council, Ottawa, Canada.
An analysis has been carried out for the basic mechanism of rapid-transit tunnel ven- tilation due to train piston action and for the temperatures resulting in the system. Theory i s in general agreement with experimental results, indicating air velocities in tunnels of the order of 1/5 of the train velocity even though trains occupy only 1/4 to 1/3 of the cross-sectional area of the tunnel. Little o f the heat load owing to trains, brakes, passengers and machinery i s conducted through the ground about the tunnel, but the ground serves as a heat reservoir causing the amplitude of the diurnal subway temperature variation to be about half that of the ambient air temperature, and caus- ing a phase shift of several hours. With the help of a partly empirical theory com- bined with experiment i t was possible to indicate approximately the temperatures to b e expected in a tunnel under differing climate conditions. The effect of train scheduling on ventilation rate i s discussed, as i s the possibility o f using models to obtain more precise design data.
Contributed by the Railroad Division of The American Society of Mechanical Engineers for presentation at the ASME-IEEE Railroad Conference, Pittsburgh, Pa., April 7-8, 1965. Manu- script received at ASME Headquarters, December 21, 1964.
Written discussion on this paper will be accepted up to May 10, 1965. Copies will be available until February 1, 1966.
Basic Theory of Rapid Transit-Tunnel Ventilation
W. G.
BROWN
I U n t i l t h e l a t e 1 9 3 0 r s underground r a i l w a y s o r r a p i d - t r a n s i t systems were g e n e r a l l y d e s i g n e d w i t h o u t a t t e n t i o n t o v e n t i l a t i o n , w i t h t h e r e s u l t t h a t h i g h t e m p e r a t u r e s sometimes o c c u r r e d , neces- s i t a t i n g c o s t l y c o n s t r u c t i o n changes t o a l l e v i a t e p a s s e n g e r d i s c o m f o r t . I n t h e London Underground, 1 f o r example, Mount(1)
r e p o r t s t h a t a g r a d u a l i n - c r e a s e over t h e y e a r s i n number of p a s s e n g e r s , t o - g e t h e r w i t h h e a v i e r t r a i n s and g r e a t e r h e a t l o a d s , caused c o n t i n u a l l y r i s i n g t e m p e r a t u r e s u n t i l , f i n - a l l y , a system of f a n s had t o be i n s t a l l e d t o b r i n g t e m p e r a t u r e s t o a r e a s o n a b l e l e v e l f o r t h e comfort of p a s s e n g e r s . A t t h e same t i m e , i n some s t a t i o n s , p a s s e n g e r s and p e r s o n n e l s u f f e r e d d i s - comfort from v e r y h i g h a i r v e l o c i t i e s i n r e s t r i c t - ed p a s s a g e s a t s t a t i o n s and on s t a t i o n p l a t f o r m s .A s a n o t h e r example, Brock
(2)
r e p o r t s over- h e a t i n g i n t h e New York Subway and s u b s e q u e n t abandonment of a h u r r i e d l y i n s t a l l e d r e f r i g e r a t i o n p l a n t . Rasmus( 2 ) ,
i n d e s c r i b i n g t h e v e n t i l a t i o n of t h e Chicago Subway, r e p o r t s t h a t i n e a r l i e r underground r a i l w a y s s t e a d y t e m p e r a t u r e s of over 90 F were n o t uncommon i n Summer, even when t h e o u t s i d e o r s t r e e t l e v e l t e m p e r a t u r e s dropped t o t h e 601s and 7 0 1 s .It c a n be shown e a s i l y t h a t i n t h e t o t a l ab- sence of v e n t i l a t i o n t h e t e m p e r a t u r e i n a subway would r e a c h p e r h a p s 350 F , owing t o a h e a t l o a d from t r a i n s and p a s s e n g e r s a v e r a g i n g about 800 Btu/hr p e r f t of t u n n e l . Obviously no e x i s t i n g r a p i d - t r a n s i t system h a s t e m p e r a t u r e s even ap- p r o a c h i n g t h i s o r d e r of magnitude. On t h e o t h e r hand many, i f n o t most, of t h e e x i s t i n g subways e x p e r i e n c e d i f f i c u l t y a t some l o c a t i o n s i n mid- summer a l t h o u g h a t o t h e r l o c a t i o n s t h e v e n t i l a - t i o n i s more t h a n s u f f i c i e n t t o m a i n t a i n r e a s o n - a b l e t e m p e r a t u r e s .
The Chicago Subway a p p e a r s t o have been t h e f i r s t f o r which a r a t i o n a l a n a l y s i s was a t t e m p t e d , b e f o r e c o n s t r u c t i o n , of b o t h t h e v e n t i l a t i n g e f f e c t of t r a i n p i s t o n a c t i o n and t h e h e a t l o a d from motors, p a s s e n g e r s and l i g h t i n g t o be d i s s i p a t e d . This a n a l y s i s was c a r r i e d o u t by Brock
( Z ) ,
who developed e q u a t i o n s f o r t h e p i s t o n e f f e c t based on s t e a d y - s t a t e assumptions and c o n s i d e r e d t h e h e a t - s t o r a g e c a p a c i t y of t h e ground i n d i s s i p a t i n g t h e h e a t l o a d . It w i l l be shown i n t h e p r e s e n t paper t h a t s t e a d y - s t a t e assumptions a r e i n a d e q u a t e f o r 1 Underlined numbers i n p a r e n t h e s e s d e s i g n a t e R e f e r e n c e s a t t h e end of t h e p a p e r . t h e d e s c r i p t i o n of underground r a i l w a y v e n t i l a t i o n . ( A S f a r a s i s known t h e n o n s t e a d y - s t a t e procedure t o be used h e r e has not been a p p l i e d p r e v i o u s l y t o underground r a i l w a y s . ) N e v e r t h e l e s s , t h e work done i n c o n n e c t i o n w i t h t h e Chicago Subway r e s u l t e d i n r e c o g n i t i o n of most of t h e f a c t o r s i n v o l v e d . The i n f o r m a t i o n o b t a i n e d from t h i s s t u d y , combined w i t h e x p e r i e n c e g a i n e d i n e a r l i e r subway systems l e d t o a g e n e r a l l y a d e q u a t e v e n t i l a t i o n system f o r t h e Chicago Subway, a s was e v i d e n t i n f o l l o w - u p meas- urements of a c t u a l a i r f l o w r a t e s and t e m p e r a t u r e s , Rasmus and Brock( 3 ) .
Experience w i t h t h e Chicago Subway was t h e n r e l i e d upon h e a v i l y , some t e n y e a r s l a t e r , and i n t h e d e s i g n of t h e Yonge S t r e e t s e c t i o n of t h e Toronto Rapid T r a n s i t System.Although r e c e n t subways can g e n e r a l l y be de- s c r i b e d a s " a d e q u a t e " from a v e n t i l a t i o n and tem- p e r a t u r e s t a n d p o i n t , t h e y a r e s o few i n number a s t o g i v e e s s e n t i a l l y no i n d i c a t i o n of whether i m -
proved d e s i g n i s p o s s i b l e . It i s n o t c l e a r , f o r example, whether c o n s t r u c t i o n economies may be p o s s i b l e o r whether t e m p e r a t u r e and comfort condi- t i o n s c a n be improved perhaps by s i m p l e changes i n t h e aerodynamic c h a r a c t e r i s t i c s of t r a i n s , t u n n e l s , and v e n t i l a t i o n openings from t u n n e l s t o t h e
s t r e e t , o r even by s m a l l changes i n t r a i n s c h e d u l - i n g .
By way of g e n e r a l o r i e n t a t i o n , F i g . 1 shows a t y p i c a l d o u b l e - t r a c k underground r a i l w a y s e c t i o n drawn a p p r o x i m a t e l y t o s c a l e . I n t r a v e r s i n g t h e t u n n e l , t r a i n s f o r c e a i r ahead of themselves and o u t v e n t i l a t i o n s h a f t s and p a s s e n g e r e x i t s , drawing a i r i n t o t h e t u n n e l t h r o u g h s i m i l a r openings be- h i n d them. Vent ( o r b l a s t ) s h a f t s a t t h e ends of s t a t i o n s s e r v e t h e two-fold purpose of p r o v i d i n g v e n t i l a t i o n and r e d u c i n g a i r v e l o c i t i e s o v e r t h e p l a t f o r m a r e a i n o r d e r t o m a i n t a i n r e a s o n a b l e com- f o r t . F a n s , p r i m a r i l y f o r emergency u s e , a r e usu- a l l y i n s t a l l e d i n f a n s h a f t s between s t a t i o n s . With f a n s i n o p e r a t i v e t h e s h a f t s s e r v e s i m p l y a s v e n t i l a t i o n o u t l e t s . The h e a t g e n e r a t e d by t r a i n d r i v e s , passen- g e r s and l i g h t i n g i s removed by v e n t i l a t i o n a i r ; t h e r e s u l t i s a n average subway t e m p e r a t u r e h i g h e r t h a n t h e average ambient t e m p e r a t u r e by a n amount depending on t h e v e n t i l a t i o n r a t e . Only a s m a l l f r a c t i o n of t h e h e a t g e n e r a t e d i s conducted t h r o u g h t h e ground above t h e subway, b u t t h e ground does s e r v e a s a h e a t r e s e r v o i r and h a s t h e e f f e c t of p a r t i a l l y damping t e m p e r a t u r e f l u c t u a t i o n s gener- a t e d by t h e d i u r n a l and a n n u a l ambient a i r c y c l e s .
STATION EXIT -ENTRANCE
T
FAN SHAFTS VENT SHAFTS7
VENT SHAFTSTFAN
SHAFTSla) L . . . ~ . ~ , ~ . -0 100 2 0 0 3 0 0 4 W 5 0 0 6 0 0 7 0 0 1 0 0 9 0 0 1 W O FT SCALE I A P P R O X I STREET LEVEL
,
P
(b) (c) 0 I 0 2 0 3 0 4 0 3 0 FT SCALE (APPROXIFig. 1 Typical ullderground rapid-transit construction (double track). (a) plan view; (b) tunnel section, including ventilation shafts;
(c) station section, showi~lg platforin
L
1
J
Fig. 2
(a)
Schematic representatio~l of a train ina
single length of tunnel--(u-V) pb
)
Pa
-
(U-V)TRAIN --U
Fig.2 (b) Air velocities as observed from the train
The s u b j e c t m a t t e r of underground r a i l w a y v e n t i l a t i o n d i v i d e s i t s e l f n a t u r a l l y i n t o t h e two s e p a r a t e components of a i r - f l o w r a t e s and t h e tem- p e r a t u r e s t h a t r e s u l t . It i s t h e p u r p o s e of t h i s p a p e r t o carr: development of t h e t h e o r y t o i t s p r a c t i c a l l i m i t and t o i n d i c a t e t h e a p p l i c a b i l i t y of o t h e r methods f o r o b t a i n i n g f i n a l d e s i g n i n f o r - m a t i o n .
MECHANISM OF TRAIN PISTON ACTION
A l t h o u g h t h e s i t u a t i o n w i l l seldom a r i s e i n a c t u a l p r a c t i c e , t h e a n a l y s i s w i l l f i r s t be con- f i n e d t o t h e s i m p l e c a s e of a t u n n e l w i t h o u t open- i n g s , b u t d i s c h a r g i n g f r e e l y i n t o t h e atmosphere a t b o t h e n d s , F i g . 2 ( a ) . A t r a i n of l e n g t h t r a v e l s a t v e l o c i t y
U
i n a t u n n e l o f l e n g t h L . With a i r f l o w i n g t o t h e r i g h t w i t h v e l o c i t y v , a momentum b a l a n c e g i v e s and where p a n d p a r e t h e s t a t i c p r e s s u r e s ahead of a b and b e h i n d t h e t r a i n , f i s D a r c y l s f r i c t i o n f a c t o r , d i s t h e h y d r a u l i c d i a m e t e r of t h e t u n n e l a n d Kei s t h e p r e s s u r e l o s s c o e f f i c i e n t a t t u n n e l en- t r a n c e . s l i s t h e s i g n of v . It has been i n c l u d - e d i n e q u a t i o n ( 3 ) b e c a u s e v and t h e f r i c t i o n may b o t h be n e g a t i v e when t r a i n s t r a v e l i n b o t h d i r e c - t i o n s . C o n s i d e r i n g , now, t h e t r a i n t o be a t r e s t and t h e t u n n e l w a l l s and a i r t o be i n motion t o t h e l e f t , a s i n F i g . 2 ( b ) , we have E q u a t i o n ( 9 ) c a n be r e a r r a n g e d i n t o t h e dimension- l e s s f o r m dv* = -dt* a t bv* t cv* where v* = v/U,
t*
=t/r (r
i s a c h a r a c t e r i s t i c t i m e , e . g . , t h e t r a i n headway), and Here v i s t h e a p p a r e n t v e l o c i t y of a i r b e s i d e t h e it r a i n , a s viewed from t h e t r a i n , and Kc and Kx a r e c o n t r a c t i o n and e x p a n s i o n p r e s s u r e l o s s c o e f f i - c i e n t s . The s u b s c r i p t i r e f e r s t o c o n d i t i o n s be- tween t h e t r a i n and t u n n e l . The f r i c t i o n p r e s s u r e l o s s Fi
i s
p o s i t i v e o r n e g a t i v e d e p e n d i n g upon t h e r e l a t i v e magnitudes of vi and U. A s i m p l e e x p r e s - s i o n f o r Fi i s o b t a i n e d w i t h t h e a p p r o x i m a t i o n Here, t h e f r i c t i o n a l d r a g a l o n g t h e w a l l , c e i l i n g and f l o o r ( s u b s c r i p tw )
i s s u b t r a c t e d f r o m t h e f r i c t i o n a l o n g t h e s i d e of t h e t r a i n ( s u b s c r i p tt ) .
Wi s
t h e w i d t h of o p e n i n g between t h e t r a i n and t h e w a l l and Hi s
t u n n e l h e i g h t .s 2 i s
t h e s i g n of( U
-
v i ) . The h y d r a u l i c d i a m e t e r s of t h e t r a i n s i d e and r e m a i n i n g s u r f a c e s a r e 4WH/H and 4WH/(2W+
H),
r e s p e c t i v e l y . Assuming f o r s i m p l i c i t y t h a t f t s=l f w = f , e q u a t i o n( 5 )
becomes I t w i l l a l s o be n o t e d t h a t where Ai s
t h e t u n n e l c r o s s - s e c t i o n a l a r e a , and S u b s t i t u t i n g e q u a t i o n s ( 6 ) and( 8 )
i n t o e q u a t i o n ( 4 ) and e q u a t i n g ( 4 ) t o ( 3 ) now y i e l d s A K t K-m{.
x Here P i s t h e t u n n e l p e r i m e t e r . There a r e two s o l u t i o n s t o e q u a t i o n( l o ) ,
d e p e n d i n g upon w h e t h e r q = 4ac-
b2i s
p o s i t i v e o r n e g a t i v e . These a r e 2 tan-
1 (2cv*+
b) =-
t*r
r
( 1 2 ) A f t e r t h e t r a i n l e a v e s t h e t u n n e l t h e v e l o c i t y c o n d i t i o n s a r e d e f i n e d by e q u a t i o n ( 3 ) w i t h p-
a Pb = 0 and w i t h ( L- A )
r e p l a c e d b y L; t h u s dv* =-
dt* T U 2(13)
From h y d r a u l i c e x p e r i m e n t si t
i s known t h a tTIME, MINUTES
Fig. 3 Tunnel air velocities for a single traverse of trains of different lengths in tunnels of different lengths; train velocity= 3090 fpm (35 mph) ; L = tunnel length;
l
= train lengthK = 0 . 5 ( f o r s q u a r e - e d g e d e n t r y i n t o t h e tunnel) K c
-
0 . 5-
T ]
( c o n t r a c t i o n o v e r f r o n t of t r a i n ) ( e x p a n s i o n o v e r r e a r of t r a i n ) f ~ 0 . 0 3 The v e n t i l a t i o n i n d u c e d by t r a i n s f o r v a r i o u s v a l u e s of L, 1,
W , H and A c a n now be e v a l u a t e d . G e n e r a l l y , t h e c r o s s - s e c t i o n a l a r e a of t r a i n s w i l l be c o n s t a n t a t a b o u t 100 s q f t , henceWH
= (A-100) s q f t . A s t a n d a r d t r a i n s t a n d s 1 2 f t h i g h and r e - q u i r e s 11 f t w i d t h f o r c l e a r a n c e ; c o n s e q u e n t l y t h e minimum p o s s i b l e a r e a of a d o u b l e - t r a c k t u n n e l i s a b o u t 300 s q f t .or
t h e Toronto Rapid T r a n s i t System, A = 358 s q f t .) The l e n g t h of newer s u b - way c a r s i s 75 f t , and u s u a l l y a minimum of3
o r 4 c a r s ( t r a i n l e n g t h = 225 t o 300 f t ) c o n s t i t u t e s a t r a i n . The maximum t r a i n l e n g t h i s governed by t h e l e n g t h of s t a t i o n p l a t f o r m s ( a b o u t 500 f t ).
The t r a i n headway, o r f r e q u e n c y of t r a i n s i n e a c h d i r e c t i o n , v a r i e s from a b o u t z1/4 min d u r i n g r u s h h o u r s t o 6 min f o r n i g h t - t i m e o p e r a t i o n . Normal daytime o p e r a t i o n g e n e r a l l y r e q u i r e s a headway of about j1/2 t o 4 min. T r a i n s u s u a l l y move a t 30 t o 40 mph i n t r a v e r s i n g a t u n n e l s e c t i o n .or
t h eTIME. MINUTES
net air flow to left no net air flow Fig. 4 Two examples of two-way traffic in a 2500- ft tunnel; (train length = 500 ft, train headway = 3. 67 min). (a) Train enters tunnel from left; (b) train leaves tunnel a t right; (c) train enters tunnel froin right; (d) train leaves tunnel a t left
t h e a i r v e l o c i t y i n t u n n e l s of d i f f e r e n t l e n g t h s f o r two c o n d i t i o n s of t r a i n l e n g t h , t h e t r a i n mak- i n g o n l y a s i n g l e p a s s t h r o u g h t h e t u n n e l . It w i l l be s e e n t h a t t h e maximum a i r v e l o c i t y a t t a i n e d a s t h e t r a i n l e a v e s t h e t u n n e l i s a p p r o x i m a t e l y t h e same, i r r e s p e c t i v e of t u n n e l l e n g t h . F o r t r a i n s of 250 f t and 500 f t l e n g t h t h e maximum v e l o c i t i e s o b t a i n e d a r e about 0.16 and 0 . 2 0 of t h e t r a i n ve- l o c i t y . The t o t a l v e n t i l a t i o n o b t a i n e d d o e s . n o t v a r y g r e a t l y w i t h t u n n e l l e n g t h .
The f o r e g o i n g r e s u l t s c o r r e s p o n d t o r a i l w a y t u n n e l s r a t h e r t h a n t o r a p i d - t r a n s i t s y s t e m s be- c a u s e of t h e l a r g e d i f f e r e n c e s i n t r a i n f r e q u e n c y between t h e two s y s t e m s . E a r l i e r methods of e v a l u a t i n g t h e v e n t i l a t i o n r a t e s f o r b o t h r a i l w a y and r a p i d - t r a n s i t s y s t e m s c o n t a i n s e r i o u s e r r o r s by n e g l e c t i n g t i m e dependence
( 2 ) ;
t h i s i s e q u i v a - l e n t t o h a v i n g t r a i n s f o l l o w one a n o t h e r i n t o t h e t u n n e l i n d e f i n i t e l y a t s h o r t i n t e r v a l s of t i m e . Although t h i s c o n c e p t i s i n a p p l i c a b l e f o r t r a i n s ,i t
i s a p p r o p r i a t e f o r h e a v i l y t r a v e l l e d a u t o m o b i l e t u n n e l s( 2 ) .
VENTILATION PRODUCED BY TRAINS MOVING I N BOTH DIRECTIONS AT UNIFORM TIME INTERVALS
f o l l o w i n g examples a t r a i n s p e e d of 35 mph i s a s -
sumed. ) With t r a i n s moving i n b o t h d i r e c t i o n s i n t h e t u n n e l , b o t h t h e f r e q u e n c y (headway) and p o s i t i o n VENTILATION PRODUCED BY SINGLE TRAINS when p a s s i n g o b v i o u s l y w i l l a f f e c t t h e n e t v e n t i l a -
t i o n r a t e . F o r example, l i t t l e v e n t i l a t i o n i s p r o - Using t h e . f o r e g o i n g e q u a t i o n s and d a t a and duced i f t r a i n s e n t e r b o t h ends of t h e t u n n e l s i - w i t h A = 358 s q f t , examples a r e g i v e n i n F i g . 3 of m u l t a n e o u s l y , whereas maximum v e n t i l a t i o n o c c u r s
tem of t u n n e l and openings i n t h e a c t u a l subway.
-800
1
I I t I,
I I II
0 1 2 3 4 5 6 7 8 9TIME, MINUTES
Fig.5 Measured air velocities in a vent shaft of a rapid- transit system (north-east vent shaft, College Street Sta-
tion, Toronto Rapid-Transit System)
i f t h e y p a s s one a n o t h e r o u t s i d e t h e t u n n e l . Be- tween t h e s e two extremes t h e v e n t i l a t i o n may have any value depending on t h e t r a i n s c h e d u l i n g . Two examples of v e l o c i t y d i s t r i b u t i o n , a s dependent on t r a i n s c h e d u l i n g , a r e g i v e n i n F i g . 4 . These have been c a l c u l a t e d w i t h t h e h e l p of t h e p r e v i o u s e q u a t i o n s , based on a t r a i n l e n g t h of 500 f t , tun- n e l l e n g t h of 2500 f t and w i t h t r a i n headway of 3.67 min (normal daytime headway i n t h e Toronto Rapid T r a n s i t System). Account was t a k e n of t h e p e r f o r a t e d c e n t e r w a l l by assuming i t s o n l y e f f e c t t o be a n i n c r e a s e i n p e r i m e t e r P by t h e amount 2H i n parameter c and i n e q u a t i o n s ( l l ) , ( 1 2 ) and ( 1 4 ) . I n one c a s e a t r a i n e n t e r s t h e t u n n e l j u s t a s t h e t r a i n from t h e o p p o s i t e d i r e c t i o n l e a v e s and r e s u l t s i n a n e t through-flow of a i r i n one d i r e c t i o n a l o n g t h e t u n n e l . I n t h e second example t h e r e i s no n e t a i r f l o w through t h e t u n n e l , t h e t r a i n s being scheduled t o e n t e r t h e t u n n e l a t time i n t e r v a l s of one h a l f of t h e headway.
For purposes of comparison a n example of ve- l o c i t i e s measured i n a v e n t s h a f t of t h e Toronto Rapid T r a n s i t System i s g i v e n i n F i g . 5 . It i s n o t t o be implied t h a t t h e c a l c u l a t e d r e s u l t s i n F i g . 4 a r e i n agreement w i t h t h o s e of F i g . 5 , but only t h a t t h e g e n e r a l behavior and magnitufies a r e s i m i - l a r . The d i f f e r e n c e s i n shape and xlagnitude of t h e v e l o c i t y p r o f i l e s a r e due t o rne complex s y s -
6
F i g . 4 i n d i c a t e s average a i r v e l o c i t i e s i n t h e t u n n e l of about 200 t o 300 fpm, corresponding t o flow r a t e s i n t h e t u n n e l of 70,000 t o 100,000 cfm. The average v e l o c i t y range f o r t h e vent s h a f t i n F i g . 5 i s about t h e same a s t h a t i n F i g . 4 , but t h e s h a f t c r o s s - s e c t i o n a l a r e a i s 217 s q f t
(compared w i t h t h e t u n n e l c r o s s s e c t i o n of 384 s q f t ) .
DESIGN METHODS FOR TUNNELS WITH VENTILATION SHAFTS AND OPENINGS
The f o r e g o i n g c o n s i d e r a t i o n s have shown t h e g e n e r a l magnitude of t h e v e n t i l a t i o n t h a t can be expected i n t u n n e l s without openings. The a n a l y - s i s cannot be c o n s i d e r e d t o be e x a c t because f r i c - t i o n and p r e s s u r e - l o s s c o e f f i c i e n t s were e s t i m a t e d from s i m i l a r h y d r a u l i c s i t u a t i o n s . Considerable d i f f i c u l t y a r i s e s even i n a t t e m p t i n g t o account f o r t h e p e r f o r a t e d c e n t e r w a l l of a double-track system. I n a c t u a l f a c t , t h e a i r moving ahead of and behind t h e t r a i n would r e q u i r e a d i s t a n c e of s e v e r a l hundred f e e t b e f o r e i t s v e l o c i t y would be uniform on both s i d e s of t h e c e n t e r w a l l .
I n a n a c t u a l subway t h e r e a r e passenger open- i n g s a t s t a t i o n s and v e n t i l a t i o n s h a f t s a t i n t e r - v a l s a l o n g t h e t u n n e l . These openings i n no way can be c o n s i d e r e d a s demarking s e p a r a t e l e n g t h s of t u n n e l r e q u i r i n g only t h e p r e v i o u s method of a n a l y - s i s . R a t h e r , t h e openings s e p a r a t e t h e a i r f l o w , some of t h e a i r l e a v i n g through t h e opening and t h e remainder c o n t i n u i n g i n t h e t u n n e l . The r e - s i s t a n c e t o a i r flow through b o t h branches a t such a j u n c t u r e depends n o t only on t h e geometry of t h e c o n s t r u c t i o n , b u t a l s o on t h e r a t i o of t h e v e l o c i - t y i n t h e branch t o t h a t upstream of t h e branch.
It must be emphasized t h a t t h e s e r e s i s t a n c e s can o n l y be o b t a i n e d by experiment e i t h e r i n t h e f i e l d o r on s c a l e models.
Equations can be d e r i v e d f o r t h e s i t u a t i o n of m u l t i p l e openings by t h e same p r o c e s s used i n o b t a i n i n g e q u a t i o n s (11) and ( 1 4 ) . The r e s u l t , however, i s a s e t of simultaneous n o n l i n e a r d i f - f e r e n t i a l e q u a t i o n s w i t h nonconstant c o e f f i c i e n t s . The o n l y p r a c t i c a l way t o s o l v e t h e s e would be w i t h t h e h e l p of a n a n a l o g computer, whose f a c i l i - t i e s c o u l d handle a t most only
3
o r4
t u n n e l open- i n g s . As g e o m e t r i c a l models of t h e v a r i o u s open- i n g s a r e r e q u i r e d t o o b t a i n f r i c t i o n c o e f f i c i e n t s ,i t
i s only a s m a l l s t e p f u r t h e r t o c o n s i d e r a com- p l e t e model of a subway s e c t i o n . Resort t o models i s not n e c e s s a r y , of c o u r s e , i n planning conven- t i o n a l subways i n moderate c l i m a t e s ;i t
i s o n l y n e c e s s a r y t o d e s i g n on t h e b a s i s of p a s t e x p e r i - ence. For s p e c i a l c o n d i t i o n s such a s i n warm c l i - mates o r where novel v e n t i l a t i o n schemes a r e pro-posed, t h e method of models may prove u s e f u l i n e v a l u a t i n g a system p r i o r t o a c t u a l c o n s t r u c t i o n . Models w i l l be d i s c u s s e d l a t e r i n c o n n e c t i o n w i t h subway t e m p e r a t u r e s .
HEAT BALANCE AND SUBWAY A I R TEMPERATURE
I n t h e f o l l o w i n g development a s i m p l i f i e d and p a r t l y e m p i r i c a l procedure w i l l be used t h a t w i l l i n d i c a t e t h e mechanism by which h e a t i s ex- changed and a l l o w approximate c a l c u l a t i o n of sub- way a i r t e m p e r a t u r e . ( I n t h i s c o n n e c t i o n
i t
should be p o i n t e d out t h a t t h e f u l l t h e o r y p r e - s e n t s no e s s e n t i a l mathematical d i f f i c u l t i e s , but i s s u f f i c i e n t l y complex t h a t i t s development i s unwarranted f o r t h e p r e s e n t p u r p o s e s . ) P e r f e c t mixing of t h e a i r i n t h e t u n n e l w i l l be assumed. It t h e n becomes i m m a t e r i a l whether t h e a i r ex- change t a k e s p l a c e by a l t e r n a t e i n j e c t i o n and e j e c t i o n t h r o u g h v e n t i l a t i o n openings o r by con- t i n u o u s i n j e c t i o n a t one l o c a t i o n and e j e c t i o n a t a n o t h e r . Under t h i s c o n d i t i o n t h e t u n n e l a i r tem- p e r a t u r e i s everywhere e q u a l . It w i l l be a p p r e c i a t e d t h a t t h e f o r e g o i n g assumption p l a c e s s e v e r e r e s t r i c t i o n s on r e l a t i n g t h e v e n t i l a t i o n r a t e s t o t h e r e s u l t i n g tempera- t u r e s . Consequently, i n t h e f o l l o w i n g d e r i v a t i o n s e m p i r i c a l c o e f f i c i e n t s w i l l be a t t a c h e d t o t h e s i m p l i f i e d t h e o r y t o account f o r d e v i a t i o n s . To o b t a i n t h e s e c o e f f i c i e n t s t h e t e m p e r a t u r e s measured i n a n a c t u a l s t a t i o n w i l l be r e l a t e d t o average v e n t i l a t i o n r a t e s and h e a t l o a d s f o r b o t h t u n n e l and s t a t i o n . I n a l e n g t h of underground r a i l w a y whose t o t a l volume i s V and whose s u r f a c e a r e a i sA s ,
t h e h e a t e n t e r i n g t h e t u n n e l by v e n t i l a t i o n (when t h e o u t s i d e a i r t e m p e r a t u r e i s h i g h e r t h a n t h e t u n n e l a i r t e m p e r a t u r e ) and from t r a i n s and pas- s e n g e r s i s absorbed i n t o t h e ground s u r r o u n d i n g t h e t u n n e l and by t h e a i r ( i n t h e t u n n e l ) , which t h e n changes t e m p e r a t u r e . T h i s h e a t b a l a n c e may be w r i t t e n a s : d T i P c E l Q(TO
-
T i ) + E 2 qL L = q G A s + P cV
-
P P,,
( 1 5 ) where p cp = s p e c i f i c h e a t p e r u n i t volume of a i r El = r a t i o of a p p a r e n t v e n t i l a t i o n i n a s t a t i o n t o average f o r s t a t i o n p l u st
unne 1 E2 = r a t i o of a p p a r e n t h e a t l o a d i n a s t a - t i o n t o a v e r a g e f o r s t a t i o n p l u st
unne 1 Q = v e n t i l a t i o n r a t e ( a v e r a g e f o r s t a t i o n p l u s t u n n e l ) T , T . = o u t s i d e a i r and t u n n e l a i r tempera- 0 1 , t u r e s , r e s p e c t i v e l y qLL = h e a t l o a d on t h e subway from t r a i n s , p a s s e n g e r s , l i g h t i n g and machinery (L i s t u n n e l l e n g t h ) qGAs = h e a t f l o w i n g i n t o t h e ground dT.1=
r a t e of change of t u n n e l a i r tempera- d t t u r e w i t h time Temperature V a r i a t i o nThe ambient a i r t e m p e r a t u r e i s assumed t o be e x p r e s s e d , a p p r o x i m a t e l y , by
T ~ = T
o
+ C A sin(w t t e ) t c D s i n ( w t t e A A D D ) ( 1 6 ) and t h e subway a i r t e m p e r a t u r e by - T . = T. t BA sin w t t BD sin w t 1 1 A D-
where T
,
Fi
a r e t h e mean a n n u a l v a l u e s of t h e am- b i e n t and t u n n e l a i r t e m p e r a t u r e s , C i s t h e ampli- t u d e of v a r i a t i o n . o f ambient a i r t e m p e r a t u r e , and B i s t h e amplitude of t u n n e l a i r t e m p e r a t u r e v a r i a - t i o n . S u b s c r i p t A d e n o t e s a n n u a l , and D d e n o t e s d a i l y c o n d i t i o n s .w = B / Y , whereY i s the period
t
i s t i m e and e i s t h e phase s h i f t .With t h e subway a i r t e m p e r a t u r e v a r y i n g s i - n u s o i d a l l y , t h e f o l l o w i n g e x p r e s s i o n i s a v a i l a b l e f o r t h e h e a t flow i n t o t h e ground about t h e t u n n e l
(6
pages 65-67):qG=kGpA
(2
( s i n w A t i c o s w A t ) t t c o s w t ) t S(Ti
-
To)](18)
D Here kG i s t h e ground t h e r m a l c o n d u c t i v i t y , a i t s d i f f u s i v i t y , and S t h e shape f a c t o r f o r c o n d u c t i o n from t h e t u n n e l t o t h e ground above. (Tunnel w a l l and a i r t e m p e r a t u r e s a r e assumed e q u a l . )E q u a t i o n
(18)
assumes a l l c y c l i c a l h e a t flow t o be p e r p e n d i c u l a r t o w a l l s , c e i l i n g and f l o o r and i s approximate i n t h a ti t
does not account f o r d i s - t o r t i o n a t c o r n e r s . Furthermore, i f t h e t u n n e l c e i l i n g i s l e s s t h a n about 10 t o 1 5 f t deep, t h e e q u a t i o n must be a l t e r e d t o account f o r t h e f a c t t h a t t h e d e p t h i s f i n i t e ( f o r t h e c a l c u l a t i o n method i n t h i s case ( r e f e r e n c e 6 , page 1 1 0 ) .A f t e r s u b s t i t u t i n g e q u a t i o n s ( 1 6 ) , ( 1 7 ) and
(18)
i n t o e q u a t i o n (151, expanding s i n( w t
+
e ),
e q u a t i n g t h e c o e f f i c i e n t s of s i nw t
and of cosw t ,
and where and m C c o s e - m C s i n e B = l t r n l t n ( 2 0 ' ~ q u a t i o n s ( l 9 ) , ( 2 2 ) , ( 2 3 ) now become e = t a n -1 (&) ( I + rr3
Because t h e geometry of underground t u n n e l s
w i l l n o t u s u a l l y v a r y g r e a t l y , t h e f o r e g o i n g r e l a - t i o n s h i p s may be s i m p l i f i e d by s u b s t i t u t i n g r e p r e - s e n t a t i v e v a l u e s f o r a l l v a r i a b l e s e x c e p t qL,
Q,
El, E2. F u r t h e r m o r e , t h e v a r i a b l e s f o r s t a t i o n s and t u n n e l a r e n e a r l y t h e same; hence, o n l y s t a t i o n c o n d i t i o n s w i l l be c o n s i d e r e d , t h e s e b e i n g t h e more i m p o r t a n t . The f o l l o w i n g v a l u e s a r e assumed:
With t h e f o r e g o i n g s u b s t i t u t i o n s t h e equa- t i o n f o r t h e subway t e m p e r a t u r e becomes 1 . 6 5 x 10-'E Q C s i n ( 0 . 2 6 2 t ) t 2 D
( 2 4 )
2 ( 1 . 1 3 4 ) t ( 1 t 1 . 6 5 ~ 1 0 - ~ E ~ Q ) 2 kc = 0 . 6 B t u / h r-
OF - f t I n o r d e r t o e l i m i n a t e , a s much a s p o s s i b l e , 2a = 0 . 0 2 f t / h r e r r o r s due t o d e v i a t i o n of t h e a n n u a l ambient tem- p e r a t u r e v a r i a t i o n from a s i n e wave, s u b s t i t u t i o n i s made f o r
?
from e q u a t i o n ( 1 6 ) . Thus0 -4 3 . 1 5 ~ 1 0 - ~ ~ Q s i n ( 7 . 1 8 x 10 ) 2 2
-
s i n ( 7 . 1 8 ~ 1 0 - ~ t + e ( 1 . 0 0 7 ) t ( l t 3 . 1 5 x ~ o - ~ E ~ Q ) 2 7L.
6 5 x 1 0 - E - Q s i n ( 0 . 2 6 2 t ) 1Comparison w i t h Measured Temperatures
F i g . 6 g i v e s measured t e m p e r a t u r e s f o r t h e
V
= 310,000 cu f t f o r a s t a t i o n 500 f t long f i r s t t h r e e weeks i n October i n a s t a t i o n of t h eD a t a f o r T o r o n t o Toronto Rapid T r a n s i t System. To be n o t e d i s t h e
A = 6 1 , 0 0 0 sq f t f o r a s t a t i o n 500 f t long
u n i f o r m i t y o f weekly b e h a v i o r compared w i t h t h e Rapid T r a n s i t
60
MON TUE W E D T H R F R I SAT SUN MON TUE WED THR F R I SAT SUN
L E G E N D
-
11 10162...
15110162--
8110162Fig. 7 Ambient air temperature (Bloor Street Meteorological Office, Toronto) October
1962
Fig.
6
Air temperatures i n the College Street Station,I
Toronto Rapid-Transit System, October 1962
I
T a b l e 1 Data for Calculation of Subway T e q p e r a t u r e s Station Section (L = 500ft)
TABLE 1
DATA FOR CALCULATION OF SUBWAY TEMPERATURES
STATION SECTION ( L = 500ft) 6 Example: Q = 1 . 5 x 10 cu ft/hr, qL = B O O ~ t u / h r - f t = 5 0 ° F , C A = 2 0 F d e g , C = B F d e g D T . (maximum) = 5 0 + ( 1 8 + 2 0 x 0 . 9 3 + B x 0 . 3 6 ) = 8 9 . 5 " F
5 2 12 2 4 6 8 10 12 14 16 18 2 0 2 2 2 4 M I D N I G H T NOON M I D N I G H T
I
1
COLLEGE STN SOUTH-
-
AVG 7 0 . 9 "F-
-
- - A M B I E N T AIR-
-
AVG 5 9 . 7 "F-
-I
I
Fig. 8 Average air temperatures for weekday operation, Toronto Rapid- Transit System
Fhis p a r t i c u l a r s e c t i o n of subway had a measured v e n t i l a t i o n r a t e of 2,400,000 c f h d u r i n g normal daytime o p e r a t i o n , Monday t o F r i d a y . An e s t i m a t e of t h e f a c t o r s E, and E2 c a n now be o b t a i n e d by comparing t h e a v e r a g e subway and ambient a i r tem- p e r a t u r e s . It w i l l be noted t h a t some of t h e r e c o r d s a r e m i s s i n g from F i g . 6 ; hence t e m p e r a t u r e s were averaged o n l y f o r Wednesday, Thursday and F r i d a y of t h e three-week p e r i o d . These average r e s u l t s a r e g i v e n i n F i g . 8 , from which t h e mean d a i l y subway a i r t e m p e r a t u r e , Wednesday t o F r i d a y ,
i s 70.9 F , w h i l e t h e average o u t s i d e a i r tempera- t u r e i s 59.7 F. The s t a t i o n t e m p e r a t u r e t h u s a v e r a g e s 1 1 . 2 deg F h i g h e r t h a n t h e ambient a i r t e m p e r a t u r e . The a m p l i t u d e of t h e d i u r n a l s t a t i o n t e m p e r a t u r e wave ( h a l f t h e d i f f e r e n c e between max- imum and minimum t e m p e r a t u r e ) i s 2.7 d e g r e e s F and t h e ambient a i r t e m p e r a t u r e a m p l i t u d e i s 5.5 d e g r e e s F. The c o e f f i c i e n t of s i n ( 0 . 2 6 2 t ) i n e q u a t i o n
( 2 4 )
i s t h e amplitude of t h e d i u r n a l subway a i r - t e m p e r a t u r e wave. E q u a t i n g t h i s t o t h e observed v a l u e of 2.7 deg F f o r Q = 2,400,000 g i v e s a v a l u e f o rEi
of 3 . 1 .h he
f a c t t h a t t h i s v a l u e i s s o h i g h s h o u l d n o t be cause f o r c o n c e r n a si t
r e p r e - s e n t s o n l y e m p i r i c a l i n c o r p o r a t i o n of t h e v a r i o u s t h e o r e t i c a l a p p r o x i m a t i o n s . ) The c o r r e s p o n d i n g v a l u e of eD i s 0.47 r a d i a n s , c o r r e s p o n d i n g t o a l a g of 1 . 8hr
behind t h e ambient a i r c y c l e . From F i g . 8 t h e observed l a g i s a b o u t 2.5 hr. With E2 e s t a b l i s h e di t
w i l l b e ' f o u n d t h a t e A i s o n l y 0.035 r a d i a n ; i . e.
t h e subway a i r t e m p e r a t u r e l a g s be- h i n d t h e a n n u a l o u t s i d e a i r t e m p e r a t u r e c y c l e b y o n l y about two d a y s . A si t
t u r n s o u t , t h e second term i n e q u a t i o n ( 2 5 ) i s n e g l i g i b l e f o r t h e Octo- b e r p e r i o d i n t h e Toronto Rapid T r a n s i t System, s o t h a t t h e f i r s t term r e p r e s e n t s t h e mean a n n u a l d i f - f e r e n c e i n t e m p e r a t u r e between t h e o u t s i d e a i r and t h e subway. S e t t i n gTi
-
To
= 1 1 . 2 deg F , t h e v a l u e f o r E i s found t o be 3.8.Maximum Subway Temperatures A t t a i n e d i n Midsummer With t h e h e l p of t h e f i g u r e s o b t a i n e d , t h e a p p r o x i m a t e t e n l p e r a t u r e s a s t h e y would o c c u r i n any c l i m a t e c a n now be e s t i m a t e d . I n Table 1 a r e g i v e n t h e mean a n n u a l d i f f e r e n c e s i n t e m p e r a t u r e between subway a i r and ambient a i r f o r d i f f e r e n t v a l u e s of qL, and t h e d i u r n a l and a n n u a l subway a i r - t e m p e r a t u r e a m p l i t u d e i n t e r m s of t h e ambient a m p l i t u d e f o r v a r i o u s v a l u e s of Q. V a l u e s f o r To, CA and CD c a n be o b t a i n e d from m e t e o r o l o g i c a l r e c o r d s . The v a l u e qL = 800 B t u / h r - f t i s t h e p r e s e n t l o a d on t h e T o r o n t o , Yonge S t r e e t , Rapid T r a n s i t System, qL = 1200 ~ t u / h r - f t i s a p o s s i b l e f u t u r e l o a d , and qL = 400 B t u / h r - f t h a s b e e n s u g - g e s t e d a s t h e p o s s i b l e l o a d f o r t r a i n s w i t h r e g e n - e r a t i v e r a t h e r t h a n r h e o s t a t i c b r a k e s .
DISCUSSION AND CONCLUSION
The a n a l y s i s of subway t e m p e r a t u r e s h a s shown t h a t i n s t e a d y o p e r a t i o n t h e v e n t i l a t i o n r a t e d e t e r m i n e s n o t o n l y maximal t e m p e r a t u r e s r e a c h e d i n summer, b u t a l s o t h e a m p l i t u d e of v a r i - a t i o n of t h e d a i l y t e m p e r a t u r e and t h e a n n u a l tem- p e r a t u r e wave. By a s i m p l e comparison w i t h meas- u r e d t e m p e r a t u r e s and v e n t i l a t i o n r a t e s
i t
h a s been found p o s s i b l e t o e s t i m a t e t e m p e r a t u r e s f o r o t h e r c l i m a t e s and f o r o t h e r v e n t i l a t i o n r a t e s . The e s t i m a t e s would a p p l y , of c o u r s e , o n l y t o t u n - n e l s s i m i l a r t o t h a t i n v e s t i g a t e d . The d a t a g i v e n i n T a b l e 1 s h o u l d be t r e a t e d w i t h c o n s i t i e r a b l e c a u t i o n , and a r e i n t e n d e d p r i n c i p a l l y f o r a p p r o x i - m a t i o n s i n t h e absence of more d e f i n i t i v e i n f o r m a - t i o n . I n a c t u a l f a c t , t h e t e m p e r a t u r e i n a t u n n e l v a r i e s from p o i n t t o p o i n t , b e i n g c l o s e t o ambient a i r t e m p e r a t u r e n e a r v e n t i l a t i o n o p e n i n g s and p a s - s e n g e r e x i t s . I n a d d i t i o n , t h e v a r i o u s s e c t i o n s of a t u n n e l have b o t h reciprocating a i r f l o w and some f l o w i n one d i r e c t i o n . Under t h e s e c o n d i t i o n s t h e r e i s a t e n d e n c y f o r t h e l o c a t i o n of t h e h i g h e s t t e m p e r a t u r e s t o be d i s p l a c e d i n t h e d i r e c t i o n of n e t f l o w .A n a l y s i s h a s shown t h a t v e r y l i t t l e h e a t i s
conducted away i n t o t h e ground a b o u t t h e subway. The ground d o e s s e r v e a s a p a r t i a l h e a t r e s e r v o i r , however, d u r i n g t h e d i u r n a l t e m p e r a t u r e c y c l e , a s
i t
r e d u c e s t h e a m p l i t u d e of t h e t e m p e r a t u r e wave and c a u s e s t h e subway t e m p e r a t u r e t o be a b o u t 2.5 hr o u t of phase w i t h t h e ambient t e m p e r a t u r e . The ground s e r v e s o n l y a s a weak h e a t r e s e r v o i r w i t h r e g a r d t o t h e a n n u a l t e m p e r a t u r e wave, t h e phase s h i f t b e i n g a b o u t two d a y s .There h a s b e e n some s p e c u l a t i o n t h a t t h e tem- p e r a t u r e s i n a subway s y s t e m w i l l t e n d t o r i s e o v e r t h e y e a r s , even i n t h e a b s e n c e of a n i n c r e a s e i n h e a t l o a d . A s i m p l e c a l c u l a t i o n shows, however, t h a t l e s s t h a n one y e a r i s r e q u i r e d f o r t h e s t e a d y p e r i o d i c s t a t e t o be r e a c h e d . It s h o u l d be p o s s i b l e t o e v a l u a t e tempera- t u r e s more a c c u r a t e l y by employing t h e f u l l t h e o r y . C o n s i d e r a b l e t i m e would be r e q u i r e d t o make t h e c a l c u l a t i o n s , however, and a d e t a i l e d knowledge of t h e f l o w c h a r a c t e r i s t i c s i n t h e t u n n e l and e x i t s s h o u l d f i r s t be a v a i l a b l e f o r t h i s p u r p o s e . I n t h i s c o n n e c t i o n
i t
would p r o b a b l y prove s i m p l e r , f o r s p e c i a l s i t u a t i o n s , t o b u i l d a h e a t e d model a t a s c a l e of p e r h a p s 1 : 2 0 ; t e m p e r a t u r e s c o u l d t h e n be o b t a i n e d by measurement r a t h e r t h a n c a l c u l a t i o n . A s t h e p r i n c i p a l component of t h e t e m p e r a t u r e i st h a t which would o c c u r w i t h c o n s t a n t ambient a i r t e m p e r a t u r e , t h e model c o u l d be o p e r a t e d w i t h t h i s c o n d i t i o n . Su.ch a model c o u l d be b u i l t of s h e e t m e t a l and used s i m u l t a n e o u s l y t o d e t e r m i n e v e l o c i t y c o n d i t i o n s i n p a s s e n g e r a r e a s and a l s o i n d i c a t e n e c e s s a r y d e s i g n a l t e r a t i o n s p r i o r t o c o n s t r u c t i o n . An a n a l y s i s o f t h e v e n t i l a t i n g e f f e c t of t r a i n s i n t u n n e l s t h a t t a k e s a c c o u n t of t h e i n e r t i a of t h e a i r h a s shown t h a t t u n n e l a i r v e l o c i t i e s up t o 1/5 of t h e t r a i n v e l o c i t y o c c u r e v e n though t h e t r a i n c r o s s - s e c t i o n a l a r e a may be o n l y 1 / 4 t o 1/3 of t h e t u n n e l c r o s s - s e c t i o n . Both t h e t r a i n f r e - quency (headway) and t r a i n s c h e d u l i n g ( a s i t e s t a b - l i s h e s l o c a t i o n s where t r a i n s from o p p o s i t e d i r e c - t i o n s m e e t ) a f f e c t t h e v e n t i l a t i o n r a t e s of t h e t u n n e l s y s t e m . The v e n t i l a t i o n r a t e i n t u n n e l s w i t h o u t i n t e r m e d i a t e o p e n i n g s i s n o t s t r o n g l y a f - f e c t e d by t r a i n l e n g t h o r by moderate d i f f e r e n c e s i n f r i c t i o n a l r e s i s t a n c e and d r a g . A p a r t from t h e o b v i o u s methods of a s s u r i n g maximum v e n t i l a t i o n i n t u n n e l s by minimizing t h e f l o w r e s i s t a n c e i n o p e n i n g s , t h e b e h a v i o r of t u n - n e l s y s t e m s h a v i n g v e n t i l a t i o n o p e n i n g s and s t a - t i o n s a t i n t e r v a l s a l o n g t h e l e n g t h p r o v e s t o be e x t r a o r d i n a r i l y complex. F o r t h i s c a s e v e r y l i t t l e p r e c i s e i n f o r m a t i o n a b o u t v e n t i l a t i o n c a n be ob- t a i n e d a n a l y t i c a l l y , and f o r s p e c i a l s i t u a t i o n s o n l y t h e method of models c a n be e x p e c t e d t o g i v e r e a s o n a b l y p r e c i s e d e s i g n i n f o r m a t i o n . I n t h i s c o n n e c t i o n
i t
might be p o i n t e d o u t t h a t model t e s t s have b e e n c a r r i e d o u t r e c e n t l y i n c o n n e c t i o n w i t h a u t o m o b i l e t u n n e l s(5,1,~),
b u t t o t h e a u t h o r ' s knowledge no work of t h i s k i n d h a s b e e n done f o r r a p i d - t r a n s i t s y s t e m s . Wheni t
i s r e - c a l l e d t h a t t h e c o s t of subway c o n s t r u c t i o n i scomparable t o t h a t of a i r c r a f t d e s i g n , where models a r e used e x t e n s i v e l y , t h e same method c o u l d b e a r c o n s i d e r a t i o n f o r subways.
ACKNOWLEDGMENT
The a u t h o r i s g r e a t l y i n d e b t e d t o Messrs. W. ti. P a t e r s o n , G e n e r a l Manager, J. T. Harvey, Chief E n g i n e e r , and F . C . P a t t i e , Mechanical E n g i n e e r ,
of t h e Subway C o n s t r u c t i o n Branch, Toronto T r a n s i t Commission, f o r t h e i r a s s i s t a n c e w i t h problems and d a t a p e r t a i n i n g t o subway v e n t i l a t i o n . T h i s paper i s a c o n t r i b u t i o n from t h e D i v i s i o n of B u i l d i n g R e s e a r c h of t h e N a t i o n a l R e s e a r c h C o u n c i l , Canada, and i s p u b l i s h e d w i t h t h e a p p r o v a l of t h e D i r e c t o r of t h e D i v i s i o n . REFERENCES
1 S. C . Mount, " V e n t i l a t i o n and Cooling i n London's Tube ail ways , ' I I n s t i t u t i o n of Heating and. V e n t i l a t i n g E n g i n e e r s J o u r n a l , January-February 1947, pp. 354-374, 4-93.
2 E. Brock, "Development of Formulae f o r C a l c u l a t i n g V e n t i l a t i o n f o r t h e Chicago Subways," J o u r n a l of t h e Western S o c i e t y of E n g i n e e r s , v o l .
48,
no. 2 , June 1943, pp. 7'6-91.3 W. E. Rasmus, " V e n t i l a t i n g t h e New
Chicago Subway," H e a t i n g , P i p i n g and A i r C o n d i t i o n - i n g , v o l . 1 5 , no.
8 ,
August 1943, pp. 393-399.4 W. E . Rasmus and E . Brock, " T r a i n P i s t o n A c t i o n V e n t i l a t i o n and Atmospheric C o n d i t i o n s i n Chicago Subways," T r a n s . ASME, v o l . 50, 1944, pp. 385-4-94.
5 A . H a e r t e r , " T h e o r e t i s c h e and experimen- t e l l e Untersuchunger Gber d i e ~ z f t u n g s a n l a g e n von S t r a s s e n t u n n e l n , " D i s s e r t a t i o n No. 3024, Swiss F e d e r a l I n s t i t u t e of Technology, Z u r i c h 1961.
6 H. S. Carslaw and J . C . J a e g e r , "Conduc- t i o n of Heat i n S o l i d s , " 2nd e d i t i o n , Clarenden P r e s s , 1959.
7 A . L. O l i v e r and H. C . Gurner, "Water Model T e s t s on Road Tunnel V e n t i l a t i o n , " DSIR, Glasgow, F l u i d s Report no. 49, 1956.
