Behavior of fast moving flow of compressible gas in cylindrical pipe in presence of cooling

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Title:
Behavior of fast moving flow of compressible gas in cylindrical pipe in presence of cooling = K voprosu o povedenii bystrodvizhushchegosya potoka szhimaemogo gaza v pryamoi tsilindricheskoi trube pri nalichii okhlzahdenia
Portion of title:
K voprosu o povedenii bystrodvizhushchegosya potoka szhimaemogo gaza v pryamoi tsilindricheskoi trube pri nalichii okhlazhdenia
Physical Description:
8 p. : ; 28 cm.
Language:
English
Creator:
Varshavsky, G. A
United States -- National Advisory Committee for Aeronautics
Publisher:
National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Mach number   ( lcsh )
Aeronautics   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Bibliography:
Includes bibliographic references (p. 8).
Statement of Responsibility:
by G.A. Varshavsky.
General Note:
"Technical memorandum 1274."
General Note:
"Report date September 1951."

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Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
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oclc - 76822893
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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1274


BEHAVIOR OF FAST MOVING FLOW OF COMPRESSIBLE GAS IN

CYLINDRICAL PIPE IN PRESENCE OF COOLING

By G. A. Varshavsky


INTRODUCTION

The investigation of the distribution of energy in the flow of
a compressible gas that moves without resistance in a straight
cylindrical pipe and gives off heat indicates that the existence of
a "thermal" Laval nozzle in the supersonic region is possible
(reference 1) (that is, Ba > 1, the cooling of the flow under
these conditions results in a rise in the Bairstow (Mach) number).
This interesting result is actually strongly distorted by the effect
of the resistance. For the simple case of gas cooling (heat con-
duction at the wall), the well-known relation between the resistance
and the heat transfer makes the existence of a "thermal" Laval
nozzle improbable. If only the heat transfer by contact is taken
into account, the existence of a "thermal" nozzle is impossible;
however, if radiation from the products of combustion is also
considered, the "thermal" nozzle is possible only in a narrow range
of high temperatures and for large dimensions of the nozzle (pipe
diameter).


1. SOLUTION OF EQUATIONS

Assumptions are based on the analysis of the simultaneous
solution of the following equations: the momentum equation (in
which the friction forces are taken into account by the usual
"hydraulic" resistance coefficient)

dw dp (1)
dx p dx 2D

the equation of continuity

wp = constant (2)


*'K Voprosu o Povedenii Bystrodvizhushchegosya Potoka Szhimaemogo
Gaza v Pryamoi Tsilindricheskol Trube pri Nalichii Okhlazhdenia."
Zhurnal teknicheskoi fiziki (U.S.S.R.). Vol. XVI, no. 4, 1946,
pp.413-416.






NACA TM 1274


and the heat-transfer equation

-Gc d T (1 + 2 Ba2) = a[Tl + "1Za2) -ti dS (3)

where w (m/sec) is the velocity of the flow at a certain section of
the pipe; p (kg/m2) is the static pressure at the same section;
T (OK) is the temperature of the gas; p ((kg)(sec2)/m4) is the
density; D (m) is the diameter of the pipe; C is the coefficient
of resistance; G (kg/sec) is the weight of the gas per second;
Cp (cal/(kg)(oC))is the specific heat of the gas; a (cal/(m2)(hr)(OC))
is the heat transfer coefficient; Ba = w/c is the Bairstow number
(Mach number); c (m/sec) is the local velocity of sound; t0 (OK)
is the temperature of the wall; and dS is an element of the pipe
area that corresponds to an element of the length dx.

The solution of this system for the general case was given by
the author and M. D. Weisman in 1934 (reference 2) and leads to
complicated expressions not capable of a clear qualitative analysis.
In investigating the problem of the formation of a "thermal" Laval
nozzle, the system was reduced by the author to nondimensional
variables and was solved under the assumption t0 = 0 (that is,
at a wall temperature negligibly small in comparison with the
stagnation temperature of the flowing gas).

The conclusions as to the possibility of formation of a
"thermal" Laval nozzle for tO = 0 will naturally be the more
favorable in this sense because for certain wall temperatures
comparable with the gas temperature, the intensity of cooling
will be less than in the case considered by the author.

The transformed system of equations is written as follows:

1a d(e) 1 dr2
--d P (1')
e d- k2 d-


S= constant (2')


ir1 + k-l\+
-e 2 + 4 Md (3' )



NACA comment: The symbol ti in equation (3) appears subse-
quently as tO. Brackets and braces in equations (3), (1'),
and (3') do not appear in the original version but are con-
sidered desirable in the interest of clarity.






NACA TM 1274


where 0 = T/TO is the nondimensional temperature (TO is the ini-
tial gas temperature); n = P/Po is the nondimensional pressure
(PO is the initial pressure); M = a/wgcpp is the Margulis
criterion ((g)(m)/sec2) is the gravitational acceleration);
S= Ba2; and f = x/D is the nondimensional length.

By elementary transformations, the system (1'), (2'), and
(3') is reduced to the equation


de [( -4M) [] 1 + k (
di 1-0

that for constant values of and M may be integrated. By
simple transformations, corresponding expressions for 0 and n
can also be written. The relations (4'), (4''), and (4'') are
the result of the integration-


L0
[F


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7 f11 ^
1 V^O/


a 0 (l4XPO7
S\l+ F24M


7C
2 _11+ko 7-24M k(


I )4 7 -48M-
k4M-t +4M
k(4M- )00+4M


7C -48M
7C -24M
k(4M- ) 0+4M
k(4M- )0+4MJ


7Z -48M
:4M-)Po+4M 7-24M
:(4M-C)P4+4M


1The constant of the values C and M and the existence of the
normal relations between them for the case of large values of
Bairstow (Mach) number has been confirmed several times in both
Russian and other foreign literature (references 3 and 4).

2Hereinafter A = (k-l)/2.


S1
( = --1n
4M


(4')


(4")





(41 )





NACA TM 1274


2. ANALYSIS OF SOLUTION

(a) Radiation neglected. The hydrodynamic theory of heat
exchange gives the relation between s and M

C = 8M (5)

By substituting equation (5) in equation (4), the following expression
is obtained :

O- W1+4)(1-4) (6)
dt 2(B-1) ({)

A study of equation (6) shows the existence of three regions
of variation of 0:

1. 6 > 1. In this case, p drops along the pipe when
approaching 0 = 1.

2. 1/k < < 1. Here B increases when approaching 6 = 1.

3. p < 1/k. In this region 0 drops and approaches the normal
flow of an incompressible gas in the presence of cooling. The
behavior of p, 8, and A for particular cases is shown in
figures 1, 2, and 3.

(b) Radiation considered. In the case of the presence
of triatomic productsof combustion (water vapor and carbon dioxide)
in the flowing gas, a certain quantity of heat will be given to the
walls by radiation. This condition leads to an increase in a and
a breakdown of relation (5). Determination of the increase in a
that is required so that a "thermal" nozzle may exist in region 1
is made possible by equation (6). It is thus necessary that


M > (a)
4 1 + k


The application of relation (6) facilitates finding the heat stress
of the endothermal reaction and makes the existence of a thermal
nozzle fundamentally possible.






NACA TM 1274


that is

M> .L (7)
685

An increase in the value of a by 17 percent corresponds to equa-
tion (7) as compared with the value given by the hydrodynamic
theory of heat interchange.

The computations made for a nozzle with a 200-millimeter
diameter and by using the air products of gasoline combustion with
an excess coefficient equal to 1 indicate that for pO = 1 at
atmosphere, the coefficient of heat transfer increases by
10 percent because of the radiation. In working with a larger
air-fuel ratio or oxygen-fuel ratio, compositions for which a
increases by more than 17 percent are possible. This increase
in a will however occur over a small part of the pipe. After
a certain lowering in the temperature and a corresponding decrease
in the radiation, a drop in Ba along the pipe begins (as in the
usual case).


3. CONCLUSION

1. The hydrodynamic theory of heat exchange applied to the
investigation of the possibility of the formation of a "thermal"
Laval nozzle on cooling the gas by heat conduction at the wall
leads to a negative result; that is, the formation of such a
nozzle is impossible.

2. When radiation is considered for the case of the flow of
gasoline combustion products in an air-rich mixture, a certain part
of the pipe in the region of high temperatures may work as a thermal
nozzle.

After a certain lowering of the gas temperature, however, the
pipe will operate normally (with a drop in the Bairstow (Mach)
number along the pipe). The effect of an increase in Mach number
in this case holds only for: (a) relatively large diameters of the
pipe and (b) products of combustion obtained in an air-rich mixture.
The practical application of the thermal nozzle even under these
conditions is in the author's opinion impossible.

Translated by S. Reiss,
National Advisory Committee for Aeronautics.






6 NACA TM 1274


REFERENCES

1. Vulis, L. A.: On the Transformation of Energy in a Flow;'etc.
Otchet NII.

2. Varshavsky, G. A., and Weisman, M. D.: Bull. NIVK, 1, 1934.

3. Gukhman, A. A., Varshavsky, G. A., and Others: Jour. Tech.
Phys., vol. 4, 1934, p. 10.
4. June, Ingwar: Wirmeibergang und Reibungswiderstand bei
Gasstromung in Rohren bei hohen Geschwindigkeiten.
Forschungsheft 380, erganzung 3u Forschung auf dem Gebiete
des Ingenieurwesens, Bd. 7, Ausg. B, Sept./okt. 1936.







NACA TM 1274


Figure 1. Dependence of and. on t for 0.016 and. po 3.
Po





IIACA TM 1274


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