The power and heat-stress conditions of geometrically similar engines are discussed. The advantages accruing from smaller cylinder dimensions are higher specific horsepower, lower weight per horsepower, lower piston temperature, and less frontal area, with reduced detonation tendency

Held, A

Riekert, P

English

NASA Technical Reports Server

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TECHNICAL MEMORAiVDUMS
WQ
H~AT !12RAlJS3’ERIN G-EOM3TRICALLy SI’IILAR CyLIi?D3RS
By P. Riekert and A. Held
Jahrbuch 1938 der Deutschen Luftfahrtforschung
I
‘Washington
May 3941
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NAT IOi?AL ADVISORY .COMMITTEI!lFOR AERONAUTICS!;.
TECHNI_C&L..MliMO~ANDUNi i?O, 977
... ...-.
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HEAT TRAHSFER IN GEOMETRICALLY SIMILAR CyLIfi~DEfiS*
By P, Riekert and A+ Held
SUMMARY
The power- and heat-stress conditions of geometri-
cally similar engines are discussed. The advantages ac-
cruing from smaller cylinder dimensions are higher spe-
cific horsepower,; lower weight per horsepower,, lower pis-
ton temperature, and less frontal area, with reduced deto-
nation tendency.
INTRODUCTION
Aside from the attempts toward units of high power,
the principal aims in aircraft-engine development are
directed. toward lower specific horsepower and lower head
resistance... In an engine of given design form, these
quantities, are affected by the size of the individual
cylinder, that is, by the type of subdivision of the to-
tal swept volume of the engine. In the following the
characteristics of geometrically similar cylinders and
engines built up of such cylinders are explored from the
standpoint of horsepower and heat’ transfer.
,, ,,
,, ,’.
POWER
,.
Not:ation.,
.
,.
Ti “force .,
*llLeistung und W5rmeabfuhr “bei geometrisc’h Uhnlichen
Zylindern. II Jahrbuch 1938 der deut$ch”en Luftfahrtforsch-
ung, pp. II 80-11 82.
‘n
2ni
‘i
Xi
Pmi
I’i
Vh i
‘i
The
i?ACA Technical Memorandum No. 97?
rpm
crank radius
power
mean effective pressure
piston area
swept volume
stroke
general power equation reads:
pmFsn pm~rhn
N=—— =
c c
(1)
The power increases with increasing mean effect pressure,
greater swept volume, and rising rpm, and is, in addition,
dependent upon quantity C, which compromises the engine
cycle - two or four stroke. T’he mean pressure pm is,
largely” determined by the fuel quantity combustible,in
the inducted air. volume conformably to chemical laws; it
can be increased ~oy supercharging the engine. A graphi-
cal view of th~se,tconditions is shown in figure 1:
~.he
straight lines with stroke s as paraueter, the abdc.issa
cm and the o.rdina’t~,form one. family, the. hyperbolas with
‘N/Vh as parameter;’ abscissa p= and n f“orm”another.
proceeding f“rom the mean piston speed ‘C’m = ~ = ~’
3s indication for the flow velocity of the gases through
the valves and hence for the charge of the cylinder, the
stroke s defines the rotational speed n. To illustrate:
For cm = 16 meters per second with s = 60 millimeters,
the speed is n = 8000 rpmc The rotational speed in con-
junction with the obtainable mean pressure, then, defines
N
,,
the specific horsepower pmn At ~ . 8000 rpm~=T. and
‘Pm= .10 kg/cm2, the specific horsepower is 90 horsepower
per liter f,or the four-stroke cycle and 180 horsepower per
lit’er for the two-stroke= ‘
.,. .
.. . . .. .
An arbitrary rise of Vh and n at oriceis not -
obtainable for mechanical reasons and because of the in-
—
NACA .Technical Memorandum No’. 977 3
-.. fer-i”b”r,cyl’ind-e”r’-ch-ar-ge“r-esulting from the flow resistance
in the gas passages. In order that these conditions may
be surveyed two geometrically similar cylinders (reference
1) with linear dimensions 10 and 11; must be considered.
l~hen equal admissible specific pressures are pOStU12tted
on piston, bearings, etc.$ the proportional value for the
effective forces Pi reads :
(’2)
By equal course of combustion process in both cylinders,
the bearing load due to the ignition pressures is the
same at all revolutions per minute ; whereas , the bearing
loads due to mass forces increase as the square with the
revolutions per minute and therefore become at aspiration,
that is, no-combustion pressures, decisive for the per-
missible bearing pressures. Thus (2) affords
PI mlrl nl 2_=
2 = A2P. morono
(3)
Hehce , because of the stipulated geometrical sirni.litud,e,
.
and
nl 1
—.=—
‘o A.
(4)
The limit,revolutions per minute are in the inverse ratio
of the liuear dimensions. The mean piston speeds decisive
for the cylinder charge have the proportional value
(5)
that is, they are equal and condition equal charge. From
the general power equation (1) there follows for equal
Pm
(6)
4 NAC4 Technioal Memorandum No. 977’
The power increases as the square of the length ratio.
Thus the power referred’ tounit ,of swept volume is as fel-
1.OWS:
..
. (7)
The specific horsepower units’ are in the inverse ratio of
the linear dimensions; hence they increase with decreasing
~ I,inear dimensions.
For t!le weight per unit of horsepower (kg/hp) the
equation rea,ds: ,..
. .
,. G1 NO
.— =+=?l (8)
. ..”
,N1 Go,,,:r,
With geometricall~~ s“imilar de~ign the weight per horse-”
power (kg/hp) increases with the linear dimensions.
Concerning the previous assumption that the con-
stancy of piston speed is decisive solely for the identi-
cal charge of similar cylinders, it is to be noted that,
s“trictly speaking; the flow in the gas passages is simi-
W’Llar only with cons,tant ,Reynolds .r+umber I?e = ~. However,
the resultant effect of t is’ relatively small. In prac-
tice, moreover, the reduction of weight per horsepower unit
is no longer possible in the ratio A when changing to
very small cylinder dimensions, since the wall thickness
of the cylinders or perhaps the spacing and thickness of
the cooling fins caanot be arbitrarily reduced. The exy
tent of the reduction in weight per horsepower unit is
therefore less than thatwhich corresponds to the length
ratio h. ,,
HEAT REMOVAL (fig. ,2)
,.,.. Here . ,
,$.
F w t) denotes the amount of inside area of the
cylinder at crank angle C#
Ti (9) gas te.,mperature at crank angle V
‘a outside temperature of coolant
NAC.A Technical Memorandum No. 977 5
n engine revolutions per minute .,.
.“
.–
9. ..,.,
““””-k (Q’;”“~~‘n) ‘co’efficient of heat transfer at crank angle 9 ,
A cylinder of the &eometrically similar bank trans-
fers, at crank setting Cp in time interval dt, to the
coolant the heat volume
dQ = r(~,l)k(~,~,n)[~i.(~) - Ta]dt (9)
With dq = ~ dt, the heat dissipation per cycle is, on
the assumption of independence of temperature
‘i and
‘a from 1 and n:
q=azw
<==
f
I’(~,l)k(~,,l,n)[Ti(?) - Ta]dV (lo)
‘n Cp=o
For the two-stroke cycle, a = 1, for the four-stroke,
a = 2. Referred to tine unit, the volume of heat removed
n
amounts to times; hence
z
Theoretically it is therefore necessary to differentiate
between the heat removal per cycle and per unit time.
If the same cylinder is considered at different revo-
lutions per minute and the coefficient of thermal conduc-
tivity k(q,lan) is assumed to be unaffected by n, the
heat volume
~ transferred. to the coolant per cycle de-
creases ‘oy increasing n and the heat transfer Q Ilperby
unit time remains tb.e same. But , since the power
equal mean pressure Pm 9 increases linearly with in~reas-
ing n together with the quantity of fuel inducted per
unit time, this means that a continuously decreasing
por$ion of the residuary heat not converted intopower
passes into the coolant and an increasing portibn into the
exhaust gases. In practice, however, k(y,l,n) increases
with increasing n, because of the improved coefficient
of heat transfer on the gas side, as is explained later on,
Inasmuch as the rise is slower than that of n, the heat
removal
~ Per Unit time is no lon:ger invariable, ?lthough
a displacement of the residuary”he’a”t in favor of the ex-
haust heat is still always present.
‘1
6 NACA Technical ~~emo&andum ITOO977
Proceeding with the conditions on different geometri-
cally similar cylinders, the functions F and. k can be
split in two “parts, one of which comprises the effect’ of
crank setting, the other the geometrical dimensions and
n; hence
(12)
Then ecuation (11). givt?S
$9= a,2Tr
r.
. ..
Q= 12k(l,n)w / f(P)k(V)[Ti(~?) - Tajd9 (13)an,,j
‘ C!)=()
..
..
For two similar cylinders the value of the integral is $
equal, giving the proportional value
,,
~ ,,2 k(l, ,ri,)
40 = “ k(lo,no)
or, for the amount of heat removal per unit surface,
. . .
q~ mi’, q)
~ = k(lo,no) (14)
The coe,ff,i.c’ientof thermal conductivity k(l,n.) is
(15)
1,, : .
k(l,n’)-= ,’ “,
.1
‘“.. -f-’J= ;a(”i:i—. ,
O@,n) a~ >>~~
., .;
where a~(l,n) denotes the average” value of the heat
transfer coefficient on the gas side per cycle, aK’ the
heat transfer coefficient on the’ coolant side, referred
to the smooth cylinder surface,’”hl<t jthe coefficient. o.f
heat conduction of the w~l~ and “8(1) the wall thickness.
(a~ and
~?U are” largely. unaffectedly 1 and, n). For
the” determination of’ UG’(~,~)”. the similitude relation,
.,
.’. .
with i?usselt!s eons,tant’~.i$& _:@- “;..,,
7.
and ~~clet?s constant
..-,~..
,,’’”.’:’;. ““.”’
,,, . . . .. ..— ... ..... .- . ------- .-
7
‘1
KACA Technical Memorandum No.. 9’77 7
W(’jt
P=— = wG~c GVG
.e-
.—...-ti.,- ‘-- ‘
. . ~G=” “d”~~””
,>
J
,, hG
are resorted to, where aG’ = — is the temperature con-
cGy G
ductility. factor with ~G as heat~conductivity factor of
,,
r the gas, CG as specific heat of gas, yG as specific
weight of gas,, and .wG as acharacteristic speed of the
gas cq,ntent. All quantities are “averages over a cycle.
.,
.“l?unction Q.”~ctin‘be expressed ,as,powe..rfunction
,,
.’.,
NU = w“’”C1(Pe),’? (16)
Nusselt !s*experimental value of M = 0.8 for the heat
transfer in a straight pipe of circular section serves as
a basis of the present study. !l?hen aG increases by the
assuneclly constant values CG, YG, and ?LG by equal cyl-
inder with increasing wG, hence increasing n. But the
“’increase by ‘W = 0.8 is slower, than that of n. Assuming
equal ~G, CG, and. ‘G for different cylinders’ also af-
Tords
; ,.
(17)
The piston speed can be, deal? with as characteristic speed
WG, ,
and gives with -=1
w~
o
,.,, .“,,
a G1
—-= AP-l. h-””z
CQ
(18)
o
l?rom this it is a~parent that for
tically M = 0..8,
p e 1’, that is, prac-
the heat transfer coefficient on the
gas side becomes smaller with increasing cylirider dimen-
sions. Also, 6(1) becomes greater by greater cylinder
dimensions. According to (14) and (15), the heat trans-
fer through the wall to the coolant per unit surface and
unit time therefore decreases on larger cylinders,
1
!n _..
*See reference 2.
-. —.—.
8 ITACA Technical Mem’or*dum ‘l?O.::..977
For equal piston spe,ed,:.t:he”,’,i.~rger cylinder gives off
a little less heat per unit surface than the smaller cyl-
inder because of the longer heat path. Since the area, as
well as the performance, and hence the inducted fuel vol-
ume in similar cylinders, increases in the ratio ?.2 the
heat volume inducted in the cylinder per unit cooling sur-
face is always identically great. The ptioportion of the
cooling heat on the total heat accordingly drops a little “
with increasing cylinder size, while the exhaust heat
rises. The same effect was observed above” for the iden-
tical cyliilder by increasing h. It generally occurs on
geometrically similar cyliizders, when the piston speed
rises (where, however, the nrinciple of similitude is
broken’ . As the charge hy ~igh ~~iston speeds can %e main-
tained only by supercb.arging, this possibility of raising
the piston speed is greater with the small cylinder be-
cause of the lower detonation tendency. In this respect
the small cylinder can also lecome su~erior in respect to
its cooling input per horsepower-hour as proved by experi-
ments on geometrically similar cylinders.
Although, according to the foregoing, the heat removal
of the large cylinder is” lower by equal piston speed, the
temperature conditions act in its disfavor, as may be seen
from .the ensuing study of heat removal on the piston; For
simplicity, assume that. each uait, area of the piston l)ot-
tom receives the same amount of heat for unit time and :
that the total heat flow in the piston bottom is in radial
direction, the axial flow being discounted. Then the con- ‘
ditions in this substitute system are exactly the same as
in an infinitely long homogeneous :cylinder of radius R,
in which the heat volume y{ per unit space and time is
produced and at whose jacket with heat transfer coeffi-
cient aq< the coolant of temperature Tx borders. The
temperature distribution in this cylinder satisfies the
linear differential equation of the second order
d2 T +ldT w
.—
d r2 rdr+~=”o
(19)
the solution of which, with the given boundary conditions,
is (reference 2)
.
,. .: .! ::.. ... , .:. .’
..’.
. . ..
.:. . . .. ... -
., .”.....’-
.’
.,,
.. . ..... .. . ..----- . .
.. . ...- -----
..
-A
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. J
EACA Technical Memorandum No. 977 9
The temperature in the cylinder axis is accor~%qglz...
,.,-—...-—..,.
...----
.-.’ W“R2 ~ +..”2AW
‘A (
= TK +— —
)
(21)
4A~’ @
2+,1>1
Now — ; hence within the scope’,of the study, it
aKR ,.
approximates to
if R
TTT + —
‘A= -. 2a
K ,,
(22)
The heat volume in the piston per unit surface with which
M is identical, decreases S1OW17 with increasing cylii~der
size according to previous arguments. Since R rises in
~r~R
proportion to increasing cylinder size, — and hence
%
‘A increase also with the size of the cylinder (by equal
coolant temperature
~~). The piston in lar+e cylinders
becomes accordingly hotter than in small cylinders, as
proved by tests. (piston cooling on large cylinders).
,. ,.
. .
POKER AND COOLIiVG CONDITIONS OF A COMPLETE ENGINE
An engine with specified power Jf can be designed
either for high speed with small stroke volume or slow
speed with large stroke volume. If the revolutions per
minute are pushed to the limits imposed by the mechanical
stresses and the flow resistances, the performances of the
individual cylinders are as A=+,
(6).
according to equation
Hence it requires a proportional number of cylinders
L-2 by equal total N. The cooling surfaces of the sep-
arate cylinders are in the ratio h2; hence for 1“-2
cylinders the same cooling surface is available. The heat
given off on the coolant by the engine with large cylinder
is therefore somewhat less than from the high-speed engine
with small cylinders because of the smaller heat removal
by larger cylinders. But of substantially negative influ-
ence on the engine operation are the higher piston temper-
atures and the greater detonation tendency. Since the
total stroke volume and the total weight of the cylinders
on different engines have the ratio ~ , the same holds
true for the weight per horsepower and the horsepower per
cubic inch, as on the separate cylinder; that is, both
—.
- ... . ....— .
10 NAC.A .Technical Memorandum .Noi.‘97’7
quantities are..more favorable with s:maller”dimensions and
more cylinders than with large individual cylinders. The
frontal areas, decisive for the air. resistance, are also
more favorable in” the ratio h? on the smaller engines.
Translation by J. Vanie”r,
National Advisory Committee
for Aeronautics.
..
1. LIJ~~, 0.: Dynamic Similitude in Internal-Combustion
fingines. T“.M. do. 978, NACA, 1941..
2. itusselt, Wilhelm: Der ‘Wlirrneiibergangim Rohr. ‘Z.VD.I
vol. 61, no. 33, Aug. 18, 1917, pp. 685-89.
3. Gr”6ber and Erk: Die Grund,gesetze der Wirmefi.bertragung,
2nd cd., 1933, p. 108, (Berlin).
,’
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r
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~!
NACA Technical Memorandum No. 977
,-
=.-.
(
14000
12000
I
I 6000
d
.qj(x)
2300
0
-- ,-
.,,.. . ..
Pm —+ kg/cm2
Figs. 1,2
,.
stroke
stroke
Figre l.- Characteristic curves of reciprocal
[
I
Figu.re’2.-
F(y,l)
hp/1
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Heat transfer in geometrically similar cylinders

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