Boelter, L M K

English

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NATIONAL COMMITTEE FOR AERONJkUTlt$
WAlr’mll luwom
ORIGINALLY ISSUED
December 1942 as
Advance
AN INVESTIGATION OF AIRCRAFT HEATERS
VI - HEAT TRANSFER EQUATIONS FOR THE
SINGLE PASS LONGITUDINAL EXCHANGER
By R. C. Martinelli, E. H. Morrin, and L. M. K. Boelter
University of California
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WASHINGTON LABORATORY
Eangley Fiel& Va.
NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution ,of
advance research results to an authorized group requiring them for the war effort. They were pre-
viously held under a security status but are now unclassified. Some of these reports were not tech-
nically edited. All have been reproduced without change in order to expedite general distribution.
https://ntrs.nasa.gov/search.jsp?R=19930092968 2020-06-17T02:04:43+00:00Z
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-- . .-———-.--— .— —-- -
MATIOMAL ADVI”SOW 00MMITTXI ~OR A=OEAUTIC S
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. A?VA3TOZ,Y)IU~IWMII -? OEt.T, ,,. . . .-
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. Am .l~$TI@AT~O~ OF AXIUIItWC =ATrn~oS , . ... .
. . . . . .,,
VI - HMT T~BSP3R 3QUATIONS FOE THU
.. . ... .. c .. . ..
sIlWL3 PASS LOIWITUD IBAL rnXCE4SGZE
,. ... . .. .
. . . ...”...
By E. C., Martinelli. S. H. lforr!n, and L. M. K. BndSter
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Presented herein iw an .annlysis of narallel flow and
eontrnflow sin~le pano Hemt dxabmgeru, together wit”h :
charts which allow the direct Bvsltinbton of 4he thermnl
perfnrnance of such units vithGut recourse to trial-and-
errnr teoknlquees The.Uo8 of-tkc..g~lation”a nnd chnrts ie
illustrated by eever~l exmiplou, ThA anmlysis indicates
that cne of the frequently st*+cfi Iqtitr.ictionn ~ri”the use
of the logarlthnic mean teuper-.~ure aitf>renoe” = that leg
that the exck.%n~;er must be rerfectly insulnted - ie not
always necess~mrym
SYNBCL S
A
8
area of heat trhnef~r surface; ft
,
‘Pa heat capaqity nf fsir at c~r.stant” mrfissure, Btu/lb ‘F
CP~ heat cabnait.?~~f exh~u$t:(’~s nt c“~n~t”nt nreswre,
Btu/13 “P . . .
1= ratio of the ener~”y’tr~nsferrer!. thrnuph the he~t
trqnsfer purface tn thet ~n.im=d by fluid n
‘R
rnt~o.?f the enerpy transferred thrnp~ho th~ heat
. . . s transfer .&urface-to(tb~%* loEt.bv fltiid g“ “ ‘ “
.. . .. .
q.”” rate if heat twmsfe$ thhou~h surf~ce separatin~ hbt
and cnld fluid, Btm/hr “
. . .
qi” ,; r~tk of””hent trnnsfer to” eurroundln~~ fron-flu:~ m,
n 3tu/hr .. ,, “. .. . .
.. .
.
. . ,.
. .
. . . :.:..
.. .
Ii, I 1~~——llm 1 -ml ,,-—m 1 -—,, ... ,— —.. —*
—.— .
2
~lg
u
(UA)
x
Ta
T al
‘aa
a?c
rate of heat transfer to surroundings from “fluid g,
Btu/hr
over-all conductance, Btu/hr fta ‘F
over-ail conductance. Btu/hr ‘F
coordinate measured along path of fluid flow, ft
mixed mean temperature of fluid a at any point x, ‘F
mixed mean temperature of fluid a at-point 1 of
exchanger (see fig. 2), ‘F
mixed mean temperature of fluid a
exchanger (see fig, 2), oF
at Point 2 &
mixed mean temperature of fluid g at any point x , ‘F
mixed mean temperature of fluid g at point 1 of
exchanger , ‘F
mixed &ean te,,,peratnre of fluid g at potnt 2 of .
exchanger , oF
Ka
function of
Wa Cp
----~ and ~ ~ c-.—- —— for 8 contra-
‘g ‘g cPg a a Pa
flow single pass exchanger, defined by equation (13).
function of
K :i CPA and #-— for a parallel
Fg Y~ Cpg a a Cpa
flow single pasa e;chnnger, defined by equation (9).
DISCUSSION
Consider the diagram shown In figure 1 In which heat
is being transferred from fluid g to fluid a. Heat 18
kelng transferred also tb the surroundings from each fluid.
Let dq = rate of heat flow throu$h surface separating
the cool fluid (a) and the warm fluid (g)
dql% = rate of heat transfer from fluid (a) to
surroundin~s
d~ig = rate of heat transfer from fluid (g) to
surroundinfls
A heat balance ylelde:
,3
. . . . ..
. dTdq = (ML= - :g :Pgm &. ... .
sinae d T 1s a nefi~tivfivalue.
~. ,
The above equmtiens may be vrittan as:
dq = Ka Wa cpa d ‘a
(
dq~m
where . Ka. l+ --------- “-
~a ‘pa d ‘a )
(Ka is R number pe~ter than or equnl tm 1,)
nnd “ dqs - “K w c
.HUG!TG
(. dqz .wher~ ‘e=l- ---- ....5----~f! =+ d T~)
(1)
(2)
(3)
(4)
(5)
(6)
(Kg is a number flrep.tarthnn Pi eqllal to 1.)
The ratios KR nnd K LGy “DCdefined QE follows:[:
‘f4 = the reiti..rf thF en~rfy t“rn~sferred thr~u~h the
heat trargfor surfnce tr tkt trar.ef~rred
by the hct fluid (F),
If the ma.-nitudso of Km wid
‘F are cnnatnnt nlonfl
the length of the heat exchan~er; that is, the transfer of
heat to the surroundlnfta is a fixed fraction of tha net
rate of heat tranafer to the correspor.dinp fluid, the ef-
fect Is exactly equlv~.lent to ohanplnl: the heat capacity
of the flulds from
‘Pa and ‘PK
toKc mn~ xc,
he m~y be seen by means of equntions (3)aah (5). I; ?ge
ratio8 Ka nnd
‘u .
do not differ too m?t?at.lyfrom unity,
even thou~h they may vp.ry with length to snme extent, an
av9ra~e mal”nitude of Ka ,
‘e
may be utilized with small
error. For these conditions the lngnrithmlc m?an tempera-
ture difference may be used with confidence.
Hesselman (reference 1) has tre=ted this mroblem by con-
nldering the bent transfer to the surroundirips to bei in-
denendent of exchan~er lenpth the •m~~itude of which is
added t~ “r subtracted from the net rnte of heat trwnsfer
.- ,, m ,, ,,., -- ,,-,, ,-. n ,,-,, . m. ., ,1—- . . , - ,,
. . . .—— —... —.,——
—.
4“
of the corresponding fluid. “The resulting equaticns are
somewhat complex.
Expreeeing equations (3) and (5) in terms of the
over-all thermal conductance (U.~) thd following equation.
(see references 2, 3, an! 4) is ohta.ined:
( 1 1 ) d(l - ‘a)-u —— ————+ dA =‘a ‘a cPa Kg % cPg . (Tg- T~
When this expre$siog la integrated with r.eepect to h and
(Tg - 7a) , one obt~ins the exponential function of the tem-
~erature difference at the entrance. to the heat exchanger
(point 1) and at the exit end (point 2)
-( 1—---—- + )—J—-- (Uk )‘ga - ‘aa ‘a ‘a Cpa Kg Wg Cpg—.. ---— = e
‘gl - ‘al
From this equation tho rate of heat transfer through the
“surface separating flulde a and g csn be written
q=Ka ‘la cpa(Tgl-Tal)
1 (1+ ‘a ‘a cPaKWCg ~ Pg ) IJ
where the specific bents Cp Cp in equation “(B17) of
am g
reference 2 have been replaced by Ka Cpa nnd Kg Cpg.
The above equation may be written ae:
(‘a ‘a cpg) UA‘~ = ‘a ‘a c~a (Tgl - ‘al) @p -—--— (8)KWC )g ~ Pg Ka Wa cPa
The function (references 3 and 4)
5“[-~1()Ka Wa C P*l.+d UA——. ..t.‘g ‘~ %Jg Ka-wa-cpa .“ ,’”’:’ -op= L=@ —- -—( ) (9)‘a ‘a cpq.1+ ‘g % cPg “ .
18 plotted in figure 3.
It may be demonstrated readily that equation (7) Is
exactly equivalent “to the well-known form:
(lo)
A procedure similar to that outlined above yields the
equivalent expressions for contraflow, single pass exchang-
er. These are:
a Ka
where
@c =
Wa Cpa(.gl
I
‘al) ~c
(
Ka Wa cpa
——
‘4! ‘g cPg
UA
1
Ka.acpa
G~A
-—— ) (12)‘a ‘a cPa
-( )‘a ‘a cPa ~ UA——— - —--—Kg Wg cp Ka Wa cp1 -e g a
—--e
Kg Wg c
. ‘g
UA——
Ka Wa Cp
a
l
(13)
6Ecwation (13) Is plotted in figure 4. It may be
readily show-n again th~t sauation (11) is exactly equiva-
lent t~ the ezpresslon utilizlng the log mean temperature
difference (equation (10)). “ “
Equations (7) and (11) have the rreat advantage lf
being explicit solutions for q, while eauatlan (10) re-
quires a trial-and-error solution.
The net thermal energy gai~ed by fluid a and trans-
ferred from fluid g is readily obtainable from the
amount which Is tran~ferr”ed through the surface whioh
separates fluid a from fluid g:
. .
~a = :-
.“
a
(14)
(15)
In double tu~e, single-mass, pas-air -heat exchangers,
if the air Ie in the crntraz tub~, Ifa = 1, Kg Z.1. If
the air is tn the annuJar space, % = 1; Ka”5 1.
Inspection of fig~re~ 3 and ~ reveala that, far the
usual ranpe of rmriahl~~ f~l-md in exhaust gas-air heat
exchangers, there Is llttl~ ~ufiqriority o,f the contraflow
arraa~ement over the marallel flow arrnngament.
The following three examales illustrate how figures
3 and 4 may be eqloy~d to d“termine tke heat transfer in
-..
—- --. —.. . ——.-. -
——
I
7
~ingle pass, parallel flow or oontraflow heat exchangers:
The rate of ventilating air flow, Wa = 2000 lb!hr. .
.,.-. .
...
..-.
The rate of exhaust gas flew,. WF = 6000 ih/hr
Temperature of exhau~t gks entering exohanger,
% = 16000 r
Temperature of ventilating air entering exchanger,
‘a z .003”
The alr flows in the annular smace, and it Is esti-
mated that 10 percemt* of the heat transferred through the
heat transfer surfaoe Is transferred to the surrounilinget
that is, Ka ---- = 1.11 and Xg = 1.
Omgo
Determine:
1. The rate of heat flow through the transfer surface
2. The rate at which thermal enprpy is gained ?Iy the
alr
3. The answers to questicns 1 an? 2 for n rerfectly
Insulat-d heater
4. Yhe answers to au=stions 1, 2, and ~ Xor contra-
flow conditions
Soluttons:
(a) The ratio
‘a ‘a Cpa 1.11 X 2000 X 0.241------ --- = ------ ------ ------ - =
‘g ‘g *Pg 1 x 6000 x 0.267
0.335 (dimensionless)
(b) The ratio
UA
------- -- 25 Q.._-_q
= 0.466 (dimensionl~ss)
‘a ‘a cPa = i;;i-;-;OOO x 0..241
(c) The product .
(Tg$ - Tal) Ka Wa Cpa = 1600 X 1.11 X 2000 X 0.241
= 855,0!)0 Btu/hr
-.---- ----+- ------ y----- ------ ------ ------ ------ _----- --
*This amount usually can be calculated from a considera-
tion of the appropriate res~stances.
.8
..
11’rom figure 3 .
..”
@p = 0.346
Thus q = 00546 x ~55,000 = 296,000 3tu/hr trans-
ferred through the heat transfer surface.
The air ~ains “heat at a rate equal to:
If no heat transfer to the surroundlnge had occurred,
Ka = 1, instead of 1.11, and
“ ‘a ‘a Cpa s ~ ?02
-------- --
l .
‘% %% cPg,
UA
--------- = O.qlg
‘a ‘a cP~
Thus
Op = o.37g
~=qa’ 1600 x 2000 x 0.241 x o.~7g = 292,0co Btu/hr
The rate of heat transfer through the surface separat-
ing the two fluids is practically independent of small
rates of heat transfer to the surroundings. The slight
deCref36e in q (2 Percent) in the case of the adiabatic
exchanger follows from the reductlcn in log mean tempera-
ture difference resulting from the absence ef heqt trans-
fer to the surroundings.
For a contraflow exchanger from figure 4
*C with heat transfer to surroundings = 0.~57
Thus, q = 0.357 x g55,009 = 306,000 I?tu/hr
qa = 275,(3GO Btu/hr
@c with no heat transfer to surroundings = 0.z~9
q = qa = 300,000 Btu/hr
#9
ThuE the increase In thermal output due to contra-
$10.W, over parallel flow, Is about 3 percent, for the con-
dition’s atak’ed. “ “’- —-.
The mixed mean .temperatur’e of the fluids’ leaving the
exchanger gor any of the conditions ~tated, of..caurse, can
be readily calculated, “.
CONCLUSIONS
. .
1. The use of the logarithmic mean temperature dif-
ference as the heat flow potential in exchangers which
have heat flow to the surroundings is Justified when this
flow of heat is constant along the length of the exchanger
or when it is variable with length hut is small,
.
2. Heat transfer in eingle pass, parallel flow or
contraflow heat exchangers may be computed directly with
the aid of the eo.uations and curves given.
3. For the case of a“heat exchanger utilizing the hot
exhaust gases from an airplane en~lne, very little advan-
tage is gained by using the contraflow arrangement.
University of California,
Berkeley, C~lif.
10
, RE1’EREHCES
Iiesselmann, K.: Der Einflu9s der W&rmeverluste auf
“ Doppelroh v~rmeaustauscher.
i
~eitschr, far die
Gesamte K lte-Induetrle, April 1928. .
Martinelli, R. C., Weinberg, E. B., Mnrrin, E. H., and
Roelter, L. M. l?.: An Invegtlgation of Aircraft
Heaters. IV - Measured and Predicted Performance
of Longitudinally Finned ?uhes. NACA ARR, Oct.
1942.
(?rober, l?.: F~rmeflbertragung. Julius Spriri~er,
(Yerlin), 1926, p. 112.
Boelter, L. M. K., Cherry, V. H., and John90n, H. A.:
Sumplement~.ry dnat Transfnr Notes. Univ.
Pr;ss,
of Cal If.
Berkeley, Calif., >d cd., lQb2, p. XTV-3.
Martinelli, R. C., Weinberg,” E. B., Morrln, E. H., and
Boglter, L. M. K.: Au Investigation of Alrcrnft
Eesters. III - Mea~l~red and Pradlcted Performance
of Double Tube Heat Exchangers. I?ACA ARR, Oct.
~94~.
BIBIJIO(JRAPIZY
Makar, ~. L., and Thornbur~, M. W. : Economic Features of
Heat ExcLang6r D@sign. Mech. Eng., VOI. 46, no. 12,
Dec. 1924, p. 891.
NACA FIGS. 1,2
6
.
FIGURE 1. HEAT BALANCE
791’
‘%2
‘raz
%1 ‘
Parallel Flaw Contra Flow I
FIGURE 2. FLOW DIAGRAMS
NACA
4P
I
FtG.
FIGURE 3. FUNCTION @p F- SINGLE PASS 6
—
0.64-- —PARALLEL FLOW -HEAT EXCHANGERS_
\
0.60.
\
0.56. ~
\
0.52. \
\
o.4e. \
\
\ \o.44-
\ \
o.40- \ \ <
. \ \.
0.36” \ ‘ \ \
. \
1 \
0.32. 1 ‘ 1.i .
\ ‘ \ 1 i
L
-
\
o.2e
1
1 I ~ Y
1.
1 1 .
I ~ 1
0.24. ~
- ~ ~
\
\
~ _
— _
~ _
— _
—
— _
0.20. =0.25
— _
_ =o. 20
0.16. — ~ — —
=0.1 5
0.12.
,=0.10
o.oe, .=o. oe
I
=0.06
0.04
0.0
0 0.2 0.4 0.6 o.e 1.0 1.2 1.4 1.6
3
Ka WaGPO
KQWgGpg
1111
NA CA
0.64
0.6C
0.56
0.52
0.48
0.44
0.40
0.36
Q
0.32
0.28
0.24
0.20
0.16
0.12
0.08
0.04
0.0
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
KaWa CDa
Kg WgCM



An investigation of aircraft heaters VI : heat transfer equations for the single pass longitudinal exchanger

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