ABSTRACTS

MAXIMUM TO

TEMPERATURE

ASYMMETRICAL

PRISM

OF N.

DROPS HEATING

RECTANGULAR Yu.

Taits

OF CROSS

and

A.

AND

G.

A

STRESSES PLATE

AND

DUE A

SECTION UDC 539.31:536,24

Sabel'nikov

When heating equipment is used in practice the metal is often heated asymmetrically; this is associated with imperfect design of the furnaces, or with the nature of the technological process. This factor can lead to the formation of considerable temperature drops (At) and dangerous 'magnitudes of thermal stresses (if). The present work solves the problem of determining the maximum temperature drops and the thermoelastic stresses in the case of asymmetrical heating of the plate and the prism of rectangular cross section in a medium with a constant gas temperature (tg = const). The value of the Fourier criterion has the form A.~ [cos ( ~ - - 82) - - 1]

l

(Fo)~m, 2n -- i~22 __ ~ In

AI.~ [1 -

~o~ (~1 -

~)]

(1)

when m a x i m u m values a r e r e a c h e d . The t h e r m o e l a s t i c s t r e s s e s in this c a s e a r e determined f r o m the equation

where a is the coefficient of t h e r m a l dfffusivity, m2/h; v is the Poisson r a t i o ; ~ is the coefficient of linear expansion, deg-1; E is the modulus of elasticity, kg/cm2; t a r is the a v e r a g e t e m p e r a t u r e of the metal, ~ 2B is the thickness of the plate; Vn and 6n a r e the roots of the c h a r a c t e r i s t i c equations; ~ is the duration of heating, h. By using (1) it is possible to calculate the m a x i m u m values of ~. Similar solutions a r e obtained f o r the c a s e of a s y m m e t r i c a l heating of a p r i s m of r e c t a n g u l a r c r o s s section.

The criterion relationships are represented out calculations.

in the form of graphs which are convenient for carrying

Metallurgical Institute, Dnepropetrovsk. T r a n s l a t e d f r o m Inzhenerno Fizicheskii Zhurnal, Vol. 19, No. 5, p. 944, N o v e m b e r , 1970. Original a r t i c l e submitted J a n u a r y 22, 1970; a b s t r a c t submitted March 23, 1970. 9 Consultants Bureau, a division of Plenum Publishing Corporation, 227 ~/est I7th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15,00.

1461

PIECEWISE-CONSTANT OF A MEDIUM Ya.

A.

PERIODIC

THERMAL

INFLUENCE

ON A S O L I D Levin

and

M.

S.

Shun

UDC 536.24

The solution of the third boundary problem of t h e r m a l conductivity is p r e s e n t e d f o r the c a s e of p i e c e wise-constant periodic laws of variation of the coefficient of heat t r a n s f e r and the t e m p e r a t u r e of the m e d i um with r e s p e c t to t i m e in a quasisteady state. The p r e s e n t work is an attempt to establish a region in which the principle of independence of the average t e m p e r a t u r e f o r the period tar(X, ~o) is c o r r e c t with a given a c c u r a c y in a quasisteady state f r o m the dimensions and thermophysical c h a r a c t e r i s t i c s of the body, We will introduce the following designations: ~ = T / T is dimensionless time; T = of variation of the coefficient of heat t r a n s f e r a s and the t e m p e r a t u r e of the medium t s, a r e dimensionless coordinates; R is half the thickness of the infinite plate; b = 2 v / P d , Predvoditel' c r i t e r i o n ; Bi s = ~sR/X is the Blot c r i t e r i o n , a is t h e coefficient of t h e r m a l is the n u m b e r of the part of the period.

~1 + ~2 is the period ~ = x / R and 0 = 0/R Pd = 2~R2/aT is the diffusion; s = 1, 2

If f , (~) = f~ ( - - x) --= t (x, T~),while y~(x)

4~+~(x)--t.~ and z ~ 0 ) = f,,+2(~)--ti 4 -- 4

t~ - - ta

'

then in the q u a s i s t e a d y - s t a t e s y s t e m y(x) = lira Yn (x), z(x) = lim zn (x-) and the functions y(x), z(x) a r e solutions of the s y s t e m 1

y ( x ) = l~-~-lb .I' qbx(x' O,-q)z(O) dO; 0 I

z (x) = 1-- ~ b S q5, (x, O, "rz)y (0) dO, 0

where Z2 .~- Bi2 q58(x, 0, z) = 2b Z k=l

and thiz h +

Bi, = 0 . izk

The solution is obtained in the f o r m

ta,,(~, 2) = ml~t_, +~@2

[1 + A (~i],

T r a n s l a t e d f r o m I n z h e n e r n o - F i z i c h e s k i i Zhurnal, Vol. 19, No. 5, pp. 945-946, November, 1970. Original a r t i c l e submitted J a n u a r y 28, 1969; a b s t r a c t submitted March 20, 1970.

9 1973 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article c a n n o t be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $I5.00.

1462

where

b2(t2 -- t~)

[

J

Example. I f t i = 1 0 0 0 ~ t 2 = 1 0 0 ~ Bi l = 0 . 1 , Bi 2 = 1 , ~ - 1 = ~ - 2 = 0 . 5 , b = 0 . 0 1 ( P d = 6 2 8 ) , then, on the basis of (4) the relative e r r o r A(~) = -0.05%. Obviously the relationship (4) in this case expresses the principle of independence almost exactly.

1463

SOLUTION BY THE METHODS

OF NONLINEAR

COMBINED

HEAT-CONDUCTION

APPLICATION

AND FINITE

PROBLEMS

OF PERTURBATION

INTEGRAL

TRANSFORMS

A. M. Aizen

UDC 536.21

The t h r e e - d i m e n s i o n a l nonlinear heat-conduction equation is considered. It is assumed that the t h e r mal conductivity is l i n e a r l y dependent on t e m p e r a t u r e and that the Sturm condition is satisfied. It is shown that the solution can be reduced to a set of o r d i n a r y differential equations by expanding the r e q u i r e d solution in a s e r i e s and then making s u c c e s s i v e finite integral t r a n s f o r m s . It is shown that the kernel of the transformation for obtaining the unknown functions appearing in the expansion in powers of a small parameter does not depend on the order of the approximation. Thus explicit formulas for the temperatures can be obtained

t H e r e Pl, P2, and P3 a r e solutions of the corresponding S t u r m - L i o u v i l l e problems used to eliminate the spatial coordinates by the method of finite integral t r a n s f o r m s ; /~, v, and a are eigenvalues of these p r o b l e m s ; a 0 = k0/%o0 is thediffusivity, determined by the t e m p e r a t u r e - i n d e p e n d e n t parts of the t h e r m a l c o n ductivity and the volumetric heat capacity;

= j f0

I, +

+

+ r

0

K, = a~ .i Fo exp [% (bt~ + v~ + c =)-~] d~ ~

etc. In the l a t t e r relations f ~ i s the volumetric heat s o u r c e strength t r a n s f o r m e d by the s u c c e s s i v e applications of finite integral t r a n s f o r m s ; ~ * is the t r a n s f o r m e d initial condition, and F~ is the t r a n s f o r m e d r i g h t hand side of the f i r s t - a p p r o x i m a t i o n equation. The proposed method is illustrated by solving a nonlinear problem of the heating of a plate of finite thickness having boundary conditions of the f i r s t kind specified on its lateral f o r c e s .

All-Union S c i e n t i f i c - R e s e a r c h Planning and Design Institute of the P e t r o l e u m Refining and P e t r o l e u m C h e m i s t r y Industry, I
1464

THERMAL Yu.

STRESSES M.

Kolyano

IN A N U N B O U N D E D and

M.

V.

VISEOELASTIC

PLATE

Khomyak

UDC 539.377

The p a p e r deals with an unbounded v i s c o e l a s t i c plate of Kelvin or Maxwell material, which is heated by an immobile instantaneous line s o u r c e or by a s o u r c e whose output a l t e r s by a set amount at the initial instant. The solution is derived for the case ~2a < ~i, where u2 = oz/~u~, with ~ h e a t - t r a n s f e r factor for the side s u r f a c e s z = *5, X is the t h e r m a l conductivity, and a is the t h e r m a l diffusivity, ~l = 3 / ( 1 - 2 ~ ) ~ * , ~42 = EM/3GM~, d = 7?/GM is the relaxation time, ~?/Gk is the delay, ~? is viscosity, ~k is P o i s s o ~ ' s ratio for the Kelvin m a t e r i a l , Gk is the s h e a r modulus for that material, and E]VI and GM are Young's modulus and the s h e a r modulus for the Maxwell material. The solution is e x p r e s s e d via the functions Fo

M,(~, Fo, p ) =

~'~-~exp ~(Fo

~)--

d~, v = 0 ,

_+ 1, + 2 , + 3 . . . .

0

f o r which r e c u r r e n c e relations are established, while computed tables of values are given for the z e r o and f i r s t - o r d e r functions. The n u m e r i c a l r e s u l t s are p r e s e n t e d as graphs for an insulated plate of Maxwell material. The solution for ~2a > ~j has been given i n a published p a p e r (Inzh. Fiz. Zh., 17, No. 5, 1969).

P h y s i c a l Mechanics Institute A c a d e m y of Sciences of the Ukrainian SSR L'vov. T r a n s l a t e d f r o m I n z h e n e r n o - F i z i c h e s k i i Zhurrml, Vol. 19, No. 5, p. 948, November, 1970. Original a r t i c l e submitted D e c e m b e r 11, 1969; r e v i s i o n submitted May 5, 1970.

9 197J Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

1465

THE

NUMERICAL

SOLUTION

MULTIDIMENSIONAL V.

F.

Demchenko

OF MULTIPItASE

MODIFIED and

D.

STEFAN A.

AND

PROBLEMS

Kozlitin

UDC 536.248.2

The s i m p l e s t o n e - d i m e n s i o n a l model of m a s s t r a n s p o r t in a m o n o p h a s e b i n a r y s y s t e m f o r m i n g a sequence of continuous solid solutions h a s the f o r m

OC~ot

oxO (Di O-xCo~-), ~ , _ ~ ( t ) < x < ~ ( t ) ,

(I)

w h e r e ~i = ~i(t) is the position of the boundary between p h a s e s i and i + 1 (the equation of motion is a s s u m e d known), C i and D i a r e weighted concentrations and diffusion coefficients f o r one of the e I e m e n t s of the b i n a r y s y s t e m in phase i. At the boundary between the p h a s e s we have the following c o m p a t i b i l i t y conditions:

Ci (~,, t) = x,C,+l (~,, t), Di OC~ 9 -Ox--

OCi+l = d~i x=~i(l) -- Di+l Ox x=~i(t) dt [Ci+:-- Ci]x=}i(t)"

(2) (3)

The discontinuity in the unknown function at the i n t e r p h a s e boundary m a y be eliminated by introducing a new function - the m a s s - t r a n s p o r t potential, which is linked with the concentrations by m e a n s of the e q u a tion C~(x, t) = O~(x, t) u~ (x, t), w h e r e Pi - [I'I~j] is the solubility coefficient, u i = ui(x, t) the m a s s - t r a n s p o r t potential of the i - t h phase. ]=l

If we m a k e the change of v a r i a b l e (4) in (1)-(3) and i n t e r p r e t the discontinuities in the m a s s t r a n s p o r t potential fluxes at the i n t e r p h a s e boundaries as the p r e s e n c e of a s y s t e m of 5 - s h a p e d s o u r c e s , we can w r i t e the g e n e r a l i z e d equation d e s c r i b i n g the m a s s - t r a n s p o r t p r o c e s s throughout the whole of a m o n o p h a s e s y s t e m as

o

o (

o./,

at [pu] = -~x \ D ax / w h e r e p(x, t) = Pi(X, t), D(x, t) = DiP i, if xE[~i, ~i + l]-

S i m i l a r l y we can w r i t e the g e n e r a l i z e d equation in the multidimensional c a s e . Equation (5) m a k e s it p o s s i b l e to c o n s t r u c t d i f f e r e n c e s c h e m e s f o r d i r e c t calculation. The method of c o n s t r u c t i n g d i r e c t c a l c u l a tion s c h e m e s f o r Eq. (5) was d e s c r i b e d in [1]. In solving multidimensional p r o b l e m s the discontinuous functions p and D have to be subjected to smoothing in the neighborhood of the i n t e r p h a s e boundaries and a locally o n e - d i m e n s i o n a l method has to be applied to the smoothed equation. The above method was used to solve the p r o b l e m of the r e d i s t r i b u t i o n of an i m p u r i t y in a t h r e e - p h a s e s y s t e m consisting of a solid body, a liquid, and a solid body, s i m i l a r to the s y s t e m which o c c u r s in the zone r e f i n e m e n t c~ a m e t a l ; i t was a l s o used to c a l c u l a t e the c h e m i c a l inhomogeneity in the dendritic ( c e l lular) c h a r a c t e r ~ c r y s t a l l i z a t i o n [2].

Institute of E l e c t r i c Welding, A c a d e m y of Sciences of the Ukrainian SSR. T r a n s l a t e d f r o m I n z h e n e r n o F i z i c h e s k i i Zhurnal, Vol. 19, No. 5, pp. 949-950, N o v e m b e r , 1970. Original a r t i c l e submitted D e c e m b e r 11, 1969; a b s t r a c t submitted April 7, 1970.

O 1973 Consultants Bureau, a division o[ Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

1466

LITERATURE 2.

CITED

V. F. Demchenko and S. S. Kozlitina, Inzh.-Fiz. Zh., 15, No. 4 (1968). Yu. A. Sterenbogen, V. F. Demchenko, and S. S. Kozlitina, Avtomaticheskaya

Svarka, No. 4 (1969).

1467

ANALOG-COMPUTER BETWEEN

SIMULATION

THE A.

E.

Pass

PARAMETERS and

OF OF

N.

P.

MOIST

Agafonov

THE

RELATIONS AIR UDC 533.1

It is f a i r l y t r o u b l e s o m e to c a l c u l a t e the t e m p e r a t u r e and humidity of a i r by t a b l e s , n o m o g r a m s , and analytical r e l a t i o n s h i p s , which a r e difficult to use, in o r d e r to choose the optimal c h a r a c t e r i s t i c s f o r a s y s t e m of air p r o c e s s i n g . The p a p e r h e r e a b s t r a c t e d p r e s e n t s a method we h a v e developed for calculating the p a r a m e t e r s of m o i s t a i r via electronic analog c o m p u t e r s . The initial p a r a m e t e r s a r e the b a r o m e t r i c p r e s s u r e , the a i r t e m p e r a t u r e , and s o m e one of the p a r a m e t e r s c h a r a c t e r i z i n g the humidity: the w a t e r v a p o r p r e s s u r e , the w a t e r content, the r e l a t i v e humidity, or the w e t - b u l b t h e r m o m e t e r t e m p e r a t u r e . The electronic apparatus enables one to d e t e r m i n e not only all the m i s s i n g p a r a m e t e r s c h a r a c t e r i z i n g the s t a t e of a i r , but also the p a r a m e t e r s at the s a t u r a t i o n point in this system. The model is b a s e d on general analytical r e l a t i o n s h i p s between the basic p a r a m e t e r s of m o i s t a i r , and the p a r a m e t e r c h a r a c t e r i z i n g the humidity is the w a t e r v a p o r p r e s s u r e {the p a r t i a l p r e s s u r e of w a t e r v a p o r in the air). The general electronic s y s t e m is built up f r o m p a r t i c u l a r models and is v e r y s i m p l e to o p e r a t e to d e t e r m i n e the changes in all the i n t e r e s t i n g p a r a m e t e r s on heating, cooling, drying, or humidification in air-conditioning equipment and in mixing various quantities of a i r ill different s t a t e s . /

One can t h e r e f o r e u s e existing s c h e m e s f o r a i r conditioning to choose the optimism conditions of o p e r a tion and to d e t e r m i n e the p a r a m e t e r via which the operation should be controlled; in designing an a i r c o n ditioning s y s t e m f r o m s c r a t c h , one can d e t e r m i n e the optimal design and e n e r g y c h a r a c t e r i s t i c s of the p a r t s in o r d e r to obtain the b e s t economic p e r f o r m a n c e .

Higher Marine Engineering College, Odessa. Translated from Inzhenerno-Fizichesldi Zhurnal, Vol. 19, No. 5, p. 951, November 1970. Original article submitted November 19, 1969; abstract submitted April 8, 1970. 9 1975 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West ]7th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

1468

DESIGN

OF K. M.

HEAT

F. I.

EXCHANGERS

Nechitailo, Slobodskoi,

WITH

COMPOUND

FLOW

V. A. Safonov, and A. I. Yakovlev

UDC 536.27

Using the equations of heat b a l a n c e and heat t r a n s f e r in an e l e m e n t a r y s e g m e n t of a t w o - p a s s heat exchanger and the solution of a s y s t e m of differential equations we d e t e r m i n e the c u r r e n t value of the t e m p e r a t u r e in the direct and counter flow p a s s e s of a heat exchanger with compound flow: t2i -- C~S1em'x + C2S2em=x+ t~', t2ii = Clem*X+ C~em'x'F ti', where

(~II A_ kI

~I

~/~( ~tI

/~tI /

~I

kI

~II) 2

S~ ~ -- 1 - - --WA-ml.2; C1 2 =

9

~72

~I/~II W2

--

t; -

'

t; -

s~,, (t"i -

t;)

(Sx ~.- - $2,1) e~I'2l

A new nondtmensional f o r m of the equation between the p a r a m e t e r s has been derived f o r a heat e x changer with compound flow Z-_'lk~t_[A~+ W1 X

+0.5A+0.5

4 R2

k~] -~ {[ ki~ In

1--RP .....

A~-~- R kn

•

1--R + - -R

1 -- P --RP

1--P--RP

+ 0.5A--0.5

@

4

+ 0"5A -- O'5 ( A + R~

- - +R 0 , 5 A +

(

4

A2+ R~

ki )o.5

kH

]

k, )o-~]-~ kH

A~+-~-kII) j I'

w h e r e l is the length of the heat exchanger, W1 is the w a t e r equivalent of the coolant in the interpipe space, k I, kii a r e the products of the heat t r a n s f e r coefficients with the p e r i m e t e r s of the f i r s t and second p a s s e s ,

A=,+

"

ki+ kH

I

kr

R k~

1 . R-

R '

tl-t~

t~--t~ '

P

t;--t~

ti - l ~

; ti' t;, t~, t~ a r e the t e m p e r a t u r e s at the inlet and

outlet in the interpipe s p a c e and the main p a s s a g e . The dependences l = /(R, P, ki/kii) w e r e derived on a c o m p u t e r and reduced to the f o r m of n o m o g r a m s l = I(R, P) f o r ki/kii = 0.5, 1, 2, 3. U s i n g t h e s e n o m o g r a m s we can calculate, Without involving s u c c e s s i v e a p p r o x i m a t i o n s , both design and c h e c k (operational) calculations. Analysis of the above equations shows that f o r p a r t i c u l a r combinations of R and P an i n c r e a s e in the intensity of heat t r a n s f e r in the d i r e c t flow leads not to a reduction, but to an i n c r e a s e , in the total length of the heat e x c h a n g e r . It is p a r t i c u l a r l y i m p o r t a n t to take into account the above equations when the h e a t - t r a n s f e r c o e f f i c i e n t s or the h e a t - e x c h a n g e r s u r f a c e s in the counter flows a r e significantly d i s s i m i l a r (for e x a m p l e , when s o m e of the tubes in a tubular heat exchanger a r e clogged or when the heat exchange conditions a r e different in the d i r e c t and counter flows of a heat e x c h a n g e r with coaxial coolant flow).

Aviation Institute, K h a r ' k o v . T r a n s l a t e d f r o m I n z h e n e r n o - F i z i c h e s k i i Zhurnal, Vol. 19, No. 5, pp. 952-953, N o v e m b e r , 1970. Original a r t i c l e submitted October, 1969; a b s t r a c t submitted April 1, 1970.

9 1975 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

1469

INVESTIGATION OF

A

GAS

iV[. E.

OF

A VERTICAL

STABILIZED

FLOW

SUSPENSION

Dogin

UDC 532.529.5:628.567.8

A d i m e n s i o n l e s s equation f o r calculating the r e s i s t a n c e of a m a t e r i a l - c o n d u c t i n g pipeline f o r a vertical stabilized flow of a gas suspension was e s t a b l i s h e d on the b a s i s of e x p e r i m e n t a l and t h e o r e t i c a l i n v e s t i g a tions [1]. The e x p e r i m e n t a l investigations w e r e c a r r i e d out on a pneumatic t r a n s p o r t i n g device the principal c h a r a c t e r i s t i c f e a t u r e s of which w e r e : g r e a t length of the vertical m a t e r i a l - c o n d u c t i n g pipeline, wide r a n g e of variation of the p a r a m e t e r s of the t w o - p h a s e flow, and t h e i r automatic r e c o r d i n g . The device c o n s i s t e d of two v e r t i c a l s t e e l m a t e r i a l - c o n d u c t i n g pipelines, 27 m high, with an inside d i a m e t e r of the pipes of 125 and 70 r a m . The initial section of the pipelines, of length 150-260 d i a m e t e r s , p r o v i d e d in all m o d e s an a c c e l e r a t i o n of the solid component being t r a n s p o r t e d to the m a x i m u m s t e a d y speed. The a i r velocity at the e n t r a n c e of the device was v a r i e d f r o m 5 to 50 m / s e c . The m a x i m u m capacity of the device r e a c h e d 30 tons/h. The analysis of the e x p e r i m e n t a l r e s u l t s was b a s e d on the p r i n c i p l e of additivity of the r e s i s t a n c e s of the conveying m e d i u m and solid component being t r a n s p o r t e d . o n the b a s i s of the investigations the d i m e n s i o n l e s s equation f o r calculating the r e s i s t a n c e coefficient of the v e r t i c a l pipeline with a stabilized t w o - p h a s e flow has the f o r m i D \0.97

)

w h e r e X and ~0 a r e the r e s i s t a n c e coefficients of the t w o - p h a s e m i x t u r e and conveying m e d i u m ; g is the m a s s flow concentration; D and d a r e the d i a m e t e r s of the pipeline and p a r t i c l e s ; F r t is the F r o u d e n u m b e r f o r the t r a n s p o r t e d p a r t i c l e s . The second addend includes, in addition to l o s s e s due to f r i c t i o n and i m p a c t of p a r t i c l e s , the l o s s e s due to lifting the solid component. The generalizing c h a r a c t e r of the equation obtained and the n u m e r i c a l value of its coefficient C w e r e e s t a b l i s h e d in e x p e r i m e n t a l investigations of the pneumatic t r a n s p o r t of dustlike, powdery, and g r a n u l a r m a t e r i a l s of nine i t e m s differing s u b s t a n t i a l l y in f r a c t i o n a l composition, hydraulic and g e o m e t r i c s i z e of the p a r t i c l e s , and t h e i r density. The t r a n s p o r t r e g i m e s c o v e r e d a wide r a n g e of v a r i a t i o n s of the 'mass concentration (2 < g < 90), speed of t r a n s p o r t , and density of the conveying m e d i u m . F o r m a t e r i a l s the pneumatic t r a n s p o r t of which is not accompanied by the f o r m a t i o n of a f i l m f r o m the t r a n s p o r t e d p a r t i c l e s on the wall of the pipeline, the coefficient C is equal to 22.5 910 -2 . The standard deviation in t r e a t i n g the e x p e r i m e n t a l data was 9 8.7%. A special s e r i e s of e x p e r i m e n t s on pneumatic t r a n s p o r t in rough pipes m a d e p o s s i b l e an evaluation of the effect of r e l a t i v e r o u g h n e s s of a pipeline on the value of the r e s i s t a n c e coefficient of a t w o - p h a s e mixture.

Institus of R a i l r o a d - T r a n s p o r t a t i o n Engineers, G o m e l ' T r a n s l a t e d f r o m I a z h e n e r n o - F i z i c h e s k i i Zhurnal, V o l . 19, No. 5, pp. 954-955, N o v e m b e r , 1970. Original a r t i c l e submitted N o v e m b e r 11, 1969; a b s t r a c t s u b m i t t e d M a r c h 23, 1970. 9 1973 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

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LITERATURE 1.

CITED

Z. R. Gorbis, Heat Transfer of Dispersed Through Flows [in Russian], Energiya, Moscow-Leningrad (1964).

1471

INVESTIGATION CONTOUR A.

OF

THE

OF NATURAL P.

REGION

OF STABILITY

CIRCULATION

Proshutinskii

and

R.

A.

DURING Shugam

OF

THE

BOILING UDC 532.501.34

This w o r k is devoted to e x p e r i m e n t a l and analytical r e s e a r c h on t h e boundaries of the stability region of a contour of natural c i r c u l a t i o n during boiling. The phenomenon of dynamic instability, e x p r e s s e d in the f o r m of nondamped vibrations of the working p a r a m e t e r s , is c h a r a c t e r i s t i c f o r a whole s e r i e s of s y s t e m s such such as uniflow b o i l e r s , n u c l e a r w a t e r - c o o l e d w a t e r - m o d e r a t e d boiling r e a c t o r s etc. E x p e r i m e n t a l r e s e a r c h on the boundaries of the s t a b i l i t y region was c a r r i e d out on a special t e s t r i g designed f o r studying the h y d r o d y n a m i c s of t w o - p h a s e flows. The circulation contour of this t e s t - r i g was f o r m e d by a lifting section, heated by an e l e c t r i c c u r r e n t , a c o m p a r a t i v e l y long nonheated lifting tube, a s e p a r a t i o n column, and, finally, a lowering b r a n c h . The outlet of the s y s t e m on the stability boundary was achieved either by i n c r e a s i n g the heating of the w a t e r at the inlet to the heated section to the s a t u r a t i o n t e m p e r a t u r e in the c a s e of constant p r e s s u r e and heat e m i s s i o n , or by reducing the p r e s s u r e in the c a s e of constant r e m a i n i n g p a r a m e t e r s . As shown by the e x p e r i m e n t s c a r r i e d out (Fig. 1), the region of unstable flow of the c i r c u l a t i o n contour in the coordinates p r e s s u r e ( P ) - h e a t i n g (A~u) is situated within a t r i a n g l e f o r m e d on the one hand by the axis of heating of the heat c a r r i e r to the s a t u r a t i o n t e m p e r a t u r e at the inlet to the heated p a r t (ordinate), and on the other hand, b y the s t r a i g h t lines which r e s p e c t the boundary of the region obtained f r o m the e x p e r i m e n t s . P r e s s u r e i n c r e a s e l e a d s to approach to the upper and l o w e r houndaries of the instability region; when it r e a c h e s a c e r t a i n value t h e s e interlock. I n c r e a s e of the density of the t h e r m a l flux leads to d i s p l a c e m e n t of this point into the high p r e s s u r e region.

% %,

%. \

.%

%

Fig. 1. Boundaries of the stability region in the plane of the p a r a m e t e r s A~u--P w h e r e n = 1 (A~au, ~ P, b a r ) : 1 - 8 ) t h e o r e t i c a l and e x p e r i m e n t a l c u r v e s r e s p e c t i v e l y , f o r q = 1.27 m V / m 2, 1.72, 2.20, 2.60; 9) e x p e r i m e n t a l c u r v e f o r q = 2.90 m V / m 2.

-, .....:~.~,..-

I0

0

N

40

O0

P

All-Union Central S c i e n t i f i c - R e s e a r c h Institute of Complex Automation, Moscow. T r a n s l a t e d f r o m I n z h e n e r n o - F i z i c h e s k i i Zhurnal, Vol. 19, No. 5, pp. 956-957. Original a r t i c l e submitted D e c e m b e r 16, 1969; a b s t r a c t submitted May 16, 1970. 1973 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

1472

Researchwas carried out on the influence of the resistance of the contour on its stability. periments have shown, the increase of this parameter is a stabilizing factor.

As the ex-

In addition to e x p e r i m e n t a l r e s e a r c h oa the stability of the contour, analytical r e s e a r c h was also c a r r i e d out on the s t a b i l i t y b o u n d a r i e s using a f r e q u e n c y c r i t e r i o n . The a m p l i t u d e - p h a s e c h a r a c t e r i s t i c of an open s y s t e m was obtained f r o m equations of c o n s e r v a t i o n of m a s s , energy, and m o m e n t u m , written in integral f o r m and c o m p l e m e n t e d by the e x p r e s s i o n f o r l e a k a g e of v a p o r , which is a s s u m e d to be constant with r e s p e c t to t i m e . The m a i n allowances m a d e on converting the initial equations a r e a s s u m p t i o n s about the a b s e n c e of s u r f a c e boiling, constancy with r e s p e c t to t i m e and the level of the r e g i o n of the physical p a r a m e t e r s of w a t e r and v a p o r , and the c o n s t a n c y of the t e m p e r a t u r e of the wall of the tube with r e s p e c t to t i m e . M o r e o v e r , the r e l a t i o n s h i p between the actual s p e e d s of the v a p o r and w a t e r and the coordinate w e r e a p p r o x i m a t e d by linear functions. Research on the stability of the investigated model was carried out by linear approximation. Comparison of theoretical and experimental data was included in the comparison of the boundaries of the stability region obtained by means of calculation according to the amplitude-phase characteristic of the open system, by using the frequency criterion of stability, and experimentally (Fig. I). In addition, the values of the frequency on the stability boundary mentally were compared; these also agreed quite satisfactorily.

obtained from calculation and experi-

1473