fins, Tvhich are welded and curved a t both ends. The fin thickness ii0.006 inch and the a l u m i y m employed has a conductivity of 110 B.t.u. pcr hour X foot X F. Thesteam side also has extended ~ excess steam it will be assurface, and by virtue of a large f l o of sumed that the thermal resistance on the steam side is negligible. The average air side coefficient mav be computed hyavariation of Equation 1,assuming thelogarithmic mean teniperaturedifference:
.y (%T;/km) 1 , 2
=
0.712 j [2
X 65.84'(110 djj.4
u.
1101
INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1948
x
0.0005)]1'* = 1.527
= 0.9
Method of Successive Approximations E (based on
k h
=
1
E) =
- 0.9
t a n h 1.527
l . j 7= 0.5958
X (1
- 0.5958)
h = 65.84/0.6362
p
=
0.6362
103.5
Repeating lvith 103.5 instead of 65.84, a closer approximation is ohtained, etc., viith the following results: Approximation
h
1st 2nd 3rd 4th 5th 6th
65.84 103.5 119.8
of the order of 15 to 30 minutes yet adequate acciiracy for most purposes. For higher accuracy, a check or additional approximation by the fmt method could now be employed. NOMENCLATURE
Use consistent units. A = total heat transfer area At = extended or fin heat transfer area c = specific heat of the fluid, assumed constant E = "fin" efficiency (Equation 4 ) h = local coefficient o! heat transfer, assumed constant h = effective mean coefficient of heat transfer, averaged over all the surface k = thermal conductivity of the metal in the fins m = thickness of the fins (both sides esposcd); for pins or strips use twice the ratio of cross section to perimeter - = X (211 ikm)' I?, dimensionless ratio with h R R = X (2h7km)l / z , dimensionless ratio with hq = total rate of heat transfer At1 = initial temperature difference from the fluid to the base of the fins Atz= final mean temperature diflerence from the fluid to the base of the fins Atm = over-all t'emperature difference from the fluid t o the base of the fins W = total mass flow rate of the fluid stream X = total width of fins from base to end, or to midline if there are no free ends
126.0 128.3 129.1
b. Direct Graphical Method. Reading from Figure 1 a t ( & / A ) = 0.9 and 9 ( 2 h / k n ) = 1.527, yields ( i / h ) = 0.509 approximately, or h = 65.84/0.509 = 129.4, with a time saving
LITERATURE C I T E D
(1) Harper, D. R.' 3rd, and Brown, W. B., Natl. Advisory Comm. .4eronaut., Rept. 158 (1922). RECEIVED January 14, 1948.
Heat Transfer with Extended Surface No Mixing Parallel to the Extended Surface W E N D E L L E. D U N N , JR.,A N D C H A R L E S F. BONILLA T H E J O H N S H O P K I N S UNIVER'SITY. B A L T I M O R E
c-
b
18. M D .
assuming constancy of c, r, and h, and neglecting heat conducted parallel to the direction of fluid flow.' If the fractional uncompleted temperature rise is called 2' and Q the dimensionless ratio (2Lh,lrc), Equation 1 may be simplified and abbreviated to: T = 6-Q 12)
I n heat transfer f r o m extended surface i t i s normally assumed t h a t complete instantaneous mixing occurs i n t h e fluid throughout each plane perpendicular t o t h e direction of fluid flow. Actually, mixing may be poor i n t h e direction parallel t o t h e fins and perpendicular t o t h e flow of t h e fluid, especially if t h e fins are wide i n t h a t direction. Accordingly, t h e equations for no mixing parallel t o rectangular fins of constant thickness have been derived and integrated. Additional assumptions include perfect mixing perpendicular t o t h e fins, no conduction i n t h e fins parallel t o t h e fluid flow or in t h e fluid, and constant fin base temperature, local coefficient of heat transfer, flow per u n i t fin w i d t h , specific heat, and m e t a l t h e r m a l conductivity. T h i s case places a lower l i m i t on t h e effectiveness of extended surface.
A more general case is that in which temperature drop is recognized t o exist along the fins in (only) the direction perpendicular to that of the fluid flow, and it is still assumed in spite of this fin t,emperature drop that the fluid temperature is constant in any plane perpendicular to its direction of flow. Heat transfer area a t the base of bhe fins is neglected. [If i t may not be neglected, see Equation 3, ( I ) ] . This case has been shown (2) to follow the law: - QtanhR (8)
THE
where R is the new dimensionless ratio X(2h/km)l'*. This is frequently considered the same as the first case, but with the surface area multiplied by a fin efficiency equal to (tan h R )
simplest case of heat t,ransfer involving extended surface is that " in ivhich the temperature is uniform throughout the extended surface, or fins, and complete mixing or heat conduction occurs in the fluid perpendicular t o the direction of fluid flow. This \youid mean, in Figure 1, that a t all places t , = T,, and that ta would vary with 1 but not with 2. The heat t,ransferred by the fin area shown could be expressed as:
~
=
e
R
~
R
'
X yet more general case would be that in which fluid temperature variation is recognized in the direction perpendicular t o the fluid f l o ~and parallel to the fins. The case of no mixing (or heat conduction) in that direction in the fluid has not been found in the literature, and is herein reported, for negligible area a t the base of the iins.
Vol. 40, No. 6
INDUSTRIAL AND ENGINEERING CHEMISTRY
L 102
DIRECTION OF FLUID FLOW
This nould
? T T T . T T T T
significant for 101-1 fin efficiencies, n ide fins, short fins, t f G -411 actual raze. mould, except ioi noncompliance with limiting assumptions that have been made in the derivations, fall betntmi these last two a b e o , depending on the amount of heat transfer hv t d t i i and molecular !’onduction perpendicuLar to the fluid flon and parallel to the fins.
Tw
a:
fa
;
0 I
LLI
I
0
n
w
:
m
I
fa,2
which is satisfied b y :
be
=&
Figure 1. Flat Fin Showing Differential Element
-~
h
-4’(B
~ ! l = , e
+ b2)
i 14)
Substituting Equations 12 and 14 back into 10 yirlds a spwial solution:
-
(to
I’r)
c COS
(bz)
x
e
621 ____ -+ h’!
!16’
.4’@
HJ- the superpo*itinn thrnrem a more general solution is:
C
b:l
n=J: !fa
-
7’3,) =
(:,, c ,I.
ihrZxjx e
-4‘ ( B +
,164
’1:
n =0
T h e general expression for t , may be obtained by substituting Equation 16 and its clrrivativr with respect t o I into Equation 7 The result is:
,I8
’19
Referring t o Figure 1, a heat balance ilia? be written inr Thr differential fin area as follon s:
.ipplyirig kquation 18 t o 17 it is necersary a ~ i drutficient that
every value of ( b J )equal ( 2 n + 1 ) T,where n ma! he any integer. 2 including zero. .Applying Equation It1 til 16 WP rvrite: which gives; 3/fu
=
322
(2)
‘tw -
t a ) =, Bttu
-
to’
1 heat balance may also he written for the fluid:
and it is necessary for the couirie beries on the light to equal unity This can be accomplished by the Fourier cosine series hnvinq m its general coefficient ( 4 ) :
giving: where IL takes all the positive integral values including 0. Therefore we map write the final equations for (t,--T,) and (tu, - T , ) . Differentiating t ?%-ice : b_-i t , = A , a22
d”ty
+
a1622
a 2
Table I. Fractional Uncompleted Temperature Change for Unmixed Flow and Ratio for Mixed and Unmixed Flow versus Fin Length and W i d t h Parameters
X
a22
Substituting Equations 7 and 8 in 5:
K
i.0
3.0
1.41
2.0
0.539 0.989
0.629 0.982
0.295 0,959 0.164 0.916 0,0938 0.860 0,0538 0.813
0.408 0,935 0,276
0.870 0.181 eO.803
0.736 0.974 0.564 0,913 0.444 0.835 0.360 0.758
0.124 0.721
0,280 0.678
4.0
6.0
7.u
0.839 0.976 0.731 0.917 0.648 0.848 0.583 0.771 0.528 0.696
0.8YO 0.980 0.815 0.929 0.749 0.869 0.693 0.805 0.660 0.741
0.484 0.630 0.398 0.507 0.333 0.407 0.210 0.237 0.134 0.1367 0.0854 0.0789 0.0347 0.0262 0.0051
0.630 0.678 0.559
10.0
ia 9)
Equation 9 may be integrated b> making the assumption that 1, or (k- T,) is a product of functions of z and 1. Let (to
-
T u ) = F(z) X H ( 0
IO)
Differentiating with respect to L and to 1 as required, substituting in Equation 9, and separating functions of z and of I: 11)
Since z and I are independent variables, each side of Equation 11 must equal a constant, which may be called -b*. The left-hand member is satisfied by P(z)
=
C
COY
tbz)
(12)
The right-hand member equated to - b 2 mal- be rearranged to give: ”(1)
+ Ab ’ H(1) - 0 ‘@ + b2)
’
0,469 0,996
0.222 0.980 4 0.105 0.959 1 0.0510 0.932 i 0,0247 0.904 6 0,0117 0.868 8 0,0027 (1.827 2
0,031 0,747
0,083 0,227 0,658 0.601 0,036 0.146 0.549 0.483 0,0178 0 , 0 9 4 0,454 0.383 . . . . 0,032 0.278 0.216
9
0.681 0,370
0,612 0.278 0.487 io ,,. “ .... 0.213 0.752 0.571 0.386 1s 0,1075 0.589 0.387 0.219 0.0544 211 ., .. 0.449 0.262 o:iSio o:iziz 0.1237 0.0253 25 .... 0.357 o:i827 o:iOi3 o:o665 0.0672 0.00738 35 ... 0.210 0,’0829 0,’0383 0,’0208 0 , 0 2 2 1 50 0,1033 0 , 0 2 7 0 0.0083 0.0035 0.0037 100 0.0083 0,0005 0,0001 ..... 0,0088 0,670
.
....
!1
0.798 0.976 0.666 0.911 0.564 0.837 0.486 0.757 0.421
....
0.560
0.505 0.474 0.380 0.296 0.311 0.184 0.247 0.1145 0,1518 0.0443 0.0105 0,0001
0.917
0.986 0.866 0.946 0.826 0.898 0.783 0.856 0.763 0.794
l’,. Tm/T, Ty
Tm/3’, Tu Tm/T* Tu T,/T, Tu T,/T, Tu Tm/Tw
0.736 0.745 0.726 T, 0.619 Tm/7’, 0.677 Tu 0.543 T,,,/T,~ 0.570 Tu 0.391 Tm/T, 0.503 Tu 0.269 T~,.~T, 0.445 Tu 0.1845 T ~ ~ T , 0,348 TU 0.0672 T m , / T u 0.0280 T m / T , 0.0006 T m / T ,
-3
. I
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Tune 1948 '1
fa,.
- Tw)
(2n
I:
=n r , , - T,) = -__ tn:, - T w )
n=
-
= x.
-
I\
(I?
