ABSTRACTS

THERMAL AT

CONDUCTIVITY

DIFFERENT N.

F.

OF

BENZENE

TEMPERATURES

Potienko

and

V. A.

AND

AND

TOLUOL

PRESSURES UDC 536.222

Tsymarnyi

We d e t e r m i n e d the e x p e r i m e n t a l coefficients of t h e r m a l conductivity of benzene and toluol at t e m p e r a t u r e s of up to 150 and 200~ r e s p e c t i v e l y , and p r e s s u r e s of up to 49.0 M N / m 2. The m e a s u r e m e n t s w e r e made to check the r e l i a b i l i t y of the a p p a r a t u s described p r e v i o u s l y [1], using the n o n s t e a d y - s t a t e method with a [[near heat s o u r c e of constant power to obtain new e x p e r i m e n t a l data a t elevated t e m p e r a t u r e s and p r e s s u r e s . Expansion of the w o r k i n g - p a r a m e t e r r a n g e r e q u i r e d a change in a n u m b e r of the units d e s c r i b e d e a r l i e r [1]. The cell was heated with a copper block bearing a Nichrome heating e l e m e n t and a photot h y r a t r o n r e g u l a t o r . The t e m p e r a t u r e in the block was m e a s u r e d with a platinum r e s i s t a n c e t h e r m o m e t e r (PTS-10) and an R-308 potentiometer. Control m e a s u r e m e n t s showed that the change in t e m p e r a t u r e in the cell did not exceed 10 -3 d e g / m i n . Since the duration of the e x p e r i m e n t was about 10 sec and the m a x i m u m r i s e in h e a t - s o u r c e t e m p e r a t u r e did not exceed one degree, this t e m p e r a t u r e - r e g i m e stability was quite adequate. The t e m p e r a t u r e gradient o v e r the height of the cell was no g r e a t e r than 10 -t d e g / m m . The m e a s u r i n g - c e l l design made it possible to conduct investigations at p r e s s u r e s of up to 49.0 MN

/ m 2. At elevated t e m p e r a t u r e s , the s o u r c e power could be reduced to 0.3 W / m and the sensitivity of the r e c o r d i n g s y s t e m i n c r e a s e d to 0.01 d e g / m m . In this case, the value of the product G r . P r [1], modified f o r n o n s t e a d y - s t a t e p r o c e s s e s , did not exceed 1000. The e x p e r i m e n t a l values of X for benzene and toluol w e r e c h a r a c t e r i z e d by an a v e r a g e d i v e r g e n c e f r o m smooth c u r v e s of ~0.5%; the m a x i m u m deviation was 1.8% and the calculated e r r o r of the method was 2.5%. The values of X obtained for toluol and benzene at their s a t u r a t i o n p r e s s u r e s w e r e c o m p a r e d with the r e s u l t s of other r e s e a r c h e r s , In m o s t c a s e s , the d i s c r e p a n c i e s w e r e +2.5-3.0%. In other studies [3, 4], the deviation at the limits of the t e m p e r a t u r e range investigated reached 4~c. The data of R a s t o r g u e v et al. [2] a r e 4-8% lower than ours at a t m o s p h e r i c p r e s s u r e and 10-12% lower at higher p r e s s u r e s . The d i s c r e p a n c y for benzene was 2-3% in m o s t c a s e s . Tables not included in this s u m m a r y give the smoothed values of ~ for benzene and toluol obtained by graphic p r o c e s s i n g of the e x p e r i m e n t a l data. The coefficients of t h e r m a l conductivity ~ for benzene and toluol at a t m o s p h e r i c p r e s s u r e and 30 ~ w e r e 0.1426 and 0.1318 W / m " deg, r e s p e c t i v e l y . LITERATURE 1. 2. 3. 4.

CITED

N . F . Potienko and V. A. T s y m a r n y i , M e a s u r e m e n t Techniques [in Russian] (1970). Yu. L. R a s t o r g u e v , B. A. G r i g o r ' e v , and G. F. Bogatov, I n z h . - F i z . Zh., 17, No. 5, 847 (1969). V . Z . Geller, Candidate's D i s s e r t a t i o n [in Russian], Moscow {1968). L . P . Filippov, Vest. MGU, No. 9 (1953).

Institute of Marine Engineers, Odessa. 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 M i Zhurnal, Vol. 20, No. 4, p. 733, 1971. Original a r t i c l e submitted F e b r u a r y 5, 1970; a b s t r a c t submitted July 27, 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 [or any purpose whatsoever without permission of the publisher, d copy of this article is available [rom the publisher [or $15.00.

530

EQUATION OF

CARBON

OF

STATE DIOXIDE

AND

THERMODYNAMIC

UP

A. A. Vasserman, and V. A. Tsymarnyi

TO E.

A A.

PRESSURE

PROPERTIES OF

2500

BARS

Golovskii,

UDC

536.71

The article presents new experimental data on the density of liquid carbon dioxide obtained with a device that has been described in [I]. On the basis of these quantities and the values of density taken from [2], we construct a reference grid, which is described analytically with high accuracy by an equation of state having the form P = A (T) p -~ B (T) 98 + C (T) pa @-D (T) pT.

The temperature functions in Eq. (i) are found by the method of squaring of isotherms similar to those described in [3], and are approximated by the equations

(1)

using operations

A (T) : 2387.3 + 3.1525T - - 16169. I02T "1 ~- 20186.10~T "2, B (T) = 2316 - - 10,02T @ 4032.102T" 1 __ 8885.10'~T" z,

(2)

C (T) = - - 4640 @ 15.6T, D ( T ) - - 1860-- 4.5T.

In the equation of state, the pressure is in bars, and the density is, kg/dm 3. A comparison of the calculated density values and the reference values showed that the standard deviations on the isotherms are 0.02-0.06%, and the maximum values are 0.05-0.13%. The deviation from the experimental values of density, as a rule, is within the limits ~:0.05%, and it reaches 0.10-0.13% only for five points. The equation quite satisfactorily represents the data on the density of liquid on the curves of saturation and solidification from the temperature of the triple point to I0 and -40~ respectively. Based on the equation of state, we calculate tables (given in the article) on the thermodynamic properties of carbon dioxide in the temperature range 220-320~ up to a pressure of 2500 bars. In order to calculate the calorific values, we integrate over the limits from the density of the boiling liquid to densities corresponding to the given pressures. Deviations of the calculated values of the isobaric specific heat from the experimental values [4] do not exceed 2%, decreasing in proportion to distance from the critical region. We also observe satisfactory agreement with the data of [5] based on the adiabatic differential Joule-Thomson effect. LITERATURE i. 2. 3. 4. 5.

CITED

A.A. Vasserman, E. A. Golovskii, and V. A. Tsymarnyi, Izmeritel'. Tekh., No. i0 (1970). A. Michels and C. Michels, Proc. Roy. Soc., A153, 201 (1935); A. Michels, C. Michels, and H. Wouters, Proc. Roy. Soe., A153, 214 (1935). A.A. Vasserman and V. A. Rabinovich, Thermophysiea[ Properties of Liquid Air and Its Components [in Russian], Standartov, Moscow (1968). S.L. Rivkin and V. M. Gurkov, Teplo6nergetika, No. I0 (1968). J.R. Roebuck, T. A. Murrell, and E. E. Miller, J. Amer. Chem. Soc., 64, 400 (1942).

Odessa Institute of Maritime Engineers. Translated from Inzhenerno-Fizicheskii Zhurnal, V o l . 20, No. 4, p. 734, April, 1971. Original article submitted May4, 1970; abstr~tct submitted July 20, 1970.

531

APPARATUS

OPERATING

THERMOPHYSICAL AT

HIGH

IN STEADY

STUDIES

PRESSURES

AND

STATE

IN G A S E O U S

FOR

MEDIUM

TEMPERATURES

M. A. P l o t n i k o v , A. A. A n t a n o v i c h , Y u . A. S a d k o v , V. V. P o p o v , a n d #.. A. K e m n i t s

UDC 536.62

In this a r t i c l e a r e p r e s e n t e d a d i a g r a m and the construction of a p p a r a t u s designed f o r conducting a wide r a n g e of p h y s i c o c h e m i c a l studies on gaseous media at p r e s s u r e s up to 12 kbar and t e m p e r a t u r e s to 3000~ A d i a g r a m of the a p p a r a t u s , which allows u n r e s t r i c t e d retention for a long t i m e of a fixed t e m p e r a t u r e in an i n e r t gas c o m p r e s s e d at the c o r r e s p o n d i n g p r e s s u r e s , is presented in Fig. 1. The a p p a r a t u s consists of a thick-walled f o r c e cylinder (1) calculated for the n e c e s s a r y inner p r e s sure. The inner wall of the f o r c e cylinder is s e p a r a t e d f r o m the t h e r m a l c h a m b e r (2) by a t h e r m a l insulation l a y e r (3). A w a t e r cooling s y s t e m of channels (4) is distributed in the thick wall of the f o r c e c y l inder. Heating of the inner volume of the t h e r m a l c h a m b e r is a c c o m p l i s h e d with an e l e c t r i c heating e l e m e n t (5). " T h e r m a l locks" (6) w e r e set in the ends of the c h a m b e r so that the heat conduction along the length of the a p p a r a t u s would not have a significant effect or~ the equilibrium nature of the t e m p e r a t u r e distribution along the axis of the t h e r m a l c h a m b e r . The r e s u l t s of t h e r m a l and endurance calculations of numerous v a r i a n t s of the a p p a r a t u s operating a c c o r d i n g to the d i a g r a m presented showed that the m o s t expedient of these conditions proved to be the use of pyrolytic graphite as the t h e r m a l insulating m a t e r i a l , which has both a c o m p a r a t i v e l y low t h e r m a l conductivity (0.7-1.1 W / m - deg), and an insignificantly s m a l l t h e r m a l expansion coefficient.

4

Z

3

" ....

"

4

"

....L

5

r

.-" ....

Fig. 1. D i a g r a m Of a p p a r a t u s

The i n e r t or neutral gas studied (argon, n i t r o gen, etc.) is pumped into the c h a m b e r under a p r e l i m i n a r y p r e s s u r e of 1-5 k b a r using a gas c o m p r e s s o r of A c a d e m i c i a n L. F. V e r e s h c h a g i n ' s construction. Subsequent i n c r e a s e in the gas p r e s s u r e is a c c o m plished by heating. Multiple t e s t s w e r e conducted on an e x p e r i m e n t a l model of the a p p a r a t u s on nitrogen at p r e s s u r e s up to 6 kbar and t e m p e r a t u r e s above 2000~ The r e s u l t s of the t e s t s indicate the reliability of the design and the full efficiency of all the construction e l e m e n t s of the apparatus.

Institute of H i g h - P r e s s u r e P h y s i c s , A c a d e m y of Sciences of the USS1R, 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. 20, No. 4, pp. 734-735, April, 1971. Original a r t i c l e submitted October 20, 1969; a b s t r a c t submitted August 1S, 1970.

532

CALCULATION THE

BASIS

OF OF

THE

THE

VISCOSITY

LAW

P . M. K e s s e l ' m a n , and A. P. Voloshin*

OF V.

OF

REAL

GASES

CORRESPONDING R.

ON

STATES

Kamenetskii,

UDC533.16

A method for calculating the viscosity coefficient of insufficiently investigated substances on the basis of the law of corresponding states is proposed. The characteristic volume and temperature values, which are generally variable, are used as the reduction parameters rather than the critical parameters ordinarily used in the well-known methods. These characteristic values are determined by using experimental data on the viscosity of the rarefied gas of the substance under investigation and of another, thoroughly investigated, substance, which is used as a standard. The method can be used for generalizing with sufficient accuracy the experimental data on the viscosity of rarefied gases with any molecular structure - from the simple "spherical ~ to complex polar gases. Moreover, it is shown that these data can be reliably extrapolated to the temperature range determined by the availability of data on the viscosity of the standard. The obtained reduction parameters are used to form reduced temperature and density as independent variables if it is necessary to calculate the viscosity coefficient of compressed gases. The calculation is then performed by means of the equation written for the standard substance. The above method was used for calculating the viscosity coefficient of heavy-water vapor in the temperature range 300-550~ at pressures up to 500 bar. The mean error of our results amounts to I-3%, while the maximum error does not exceed 6~c.

DISPERSION A

OF

DROPS

DIELECTRIC

ELECTRODES

MEDIUM IN

AN

OF

CONDUCTING

DURING ELECTRIC

V. P. Mardanenko, V. and V. G. Ben'kovskii~

RECHARGING

LIQUIDS

IN ON

FIELD G.

Emel'yanchenko,

UDC 532.529.6

The a r t i c l e , which is a continuation of [1], p r e s e n t s a n a l y t i c a l e v a l u a t i o n s of the v a l u e s of c h a r g e s and conditions of d i s p e r s i o n of d r o p s of conducting liquids in a d i e l e c t r i c m e d i u m in a u n i f o r m e l e c t r i c field d u r i n g r e c h a r g i n g on e l e c t r o d e s , and a l s o the r e s u l t s of e x p e r i m e n t s on d i s p e r s i o n on r e c h a r g i n g d r o p s of e l e c t r o l y t e s and s u r f a c e - a c t i v e a g e n t s (Table 1) in m i n e r a l oil. The f o r m u l a f o r the c h a r g e q of a drop, which is a p p r o x i m a t e d by an ellipsoid of r e v o l u t i o n s i m i l a r to that which was done in [2] on the a s s u m p t i o n that the s u r f a c e of a s e m i s p h e r o i d , which s i m u l a t e s the drop, and its s e m i m a j o r a x i s a r e equal to the s u r f a c e and s e m i m a j o r axis of the c o r r e s p o n d i n g ellipsoid, has the f o r m 2~e~or2Eo(V: 1 -}- (1 -- e'~) -[--(1 -- e~)~ -- 1) Iql = ' (1) ]~ (1 -- ee)~(n~ -- l)0learcth ~]e 1) -

-

* T e c h n o l o g i c a l Institute of the R e f r i g e r a t i o n I n d u s t r y , Odessa. 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 Z h u r n a l , Vol. 20, No. 4, pp. 735-736, A p r i l , 1971. O r i g i n a l a r t i c l e submitted J a n u a r y 29, 1970; a b s t r a c t submitted J u l y 20, 1970. Institute of C h e m i s t r y of P e t r o l e u m and N a t u r a l Salts, A c a d e m y of S c i e n c e s of the Kazakh SSR, G u r ' e 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 Z h u r n a l , Vol. 20, No. 4, pp. 736-737, April, 1971. O r i g inal a r t i c l e s u b m i t t e d A p r i l 7, 1970; a b s t r a c t submitted J u l y 17, 1970. 533

TABLE 1. Values of ~ and k for Drops of Investigated Solutions

.103,Nim t 3 , 6

14.0

t

,2,0

t

32,2

L 54,0

f

-,5

where ~e -~ ~f2/~/3 - ~ / 1 + (1 - e 2) + (1 - e 2 ) 2, e =~/1 - b 2 / a 2 ; r is the radius of the drop; a, b, e a r e the semiaxes and e c c e n t r i c i t y of the ellipsoid; o~ is the coefficient of interphase surface tension; E 0 is the e x ternal field strength; eeo is the dielectric constant of the medium. The c o r r e s p o n d e n c e of this formula to the experimental data and to the calculation [3] for the case e - - 0 is indicated. Substitution of the value of q obtained into Eq. (6) f r o m [1] permits obtaining the condition of d i s persion of r e c h a r g i n g drops in the f o r m (2)

Eo > E c r = k (o~186o r) 1/2 .

where E~ r is the critical value of E 0 corresponding to d i s p e r s i o n of the drops, k = f(e). It is a s s u m e d that e = 0.952 for c r i t i c a l conditions [1]. Then k = f(e) is determined numerically: k = 0.3. A description of the experiments is given, f r o m which follows that the values of k (Table 1) found f r o m the a v e r a g e experimental data with a mean e r r o r of not m o r e than 12% for drops of all investigated solutions coincide, so that the a r i t h m e t i c mean value kav = 0.35 ~= 0.03. This r e s u l t is r e g a r d e d as s a t i s factory, and Eq. (2), where k = 0.3, is r e c o m m e n d e d for practical use. It follows f r o m Eq. (2) that the magnitude of the charge and intensity of d i s p e r s i o n of the r e c h a r g i n g drops, other conditions being equal, do not depend on the chemical composition. LITERATURE

I. 2. 3.

V.P. Mardanenko, B. F. Anisimov, V. G. Emel'yanchenko, and V. G. Ben'kovskii, Zh., 14, 5, 895 (1968). A.T.'-Nagizade, Izv. Akad. Nauk SSSR, I~nergetika i Transport, No. I, 156 (1966). N.N. Lebedev and I. P. Skal,skaya, Zh. Tekh. Fiz., 32, 3, 375 (1962).

A METHOD

OF

INELASTIC

PROPERTIES

OF

CITED

MATERIALS V.

M.

DETERMINING

AT HIGH

Baranov

THE

OF SMALL

ELASTIC

Inzh.-Fiz.

AND

SPECIMENS

TEMPERATURES

and Yu.

V.

Miloserdin

UDC 620.17

An ultrasonic r e s o n a n c e - p u l s e method for determining the elastic constants and the oscillation energy s c a t t e r i n g c h a r a c t e r i s t i c s of small specimens of isotropic m a t e r i a l s is described. M e a s u r e m e n t s a r e made on specimens in the f o r m of disks of diameter f r o m 6-8 m m to 50-60 m m and thickness f r o m 1 to 15 mm, for a thickness to diameter ratio f r o m 0.1 to 0.25. In this method s e v e r a l of the lower natural f r e quencies of oscillation of the specimens a r e measured~ f r o m which the n o r m a l modulus of elasticity the shear modulus, and P o i s s o n ' s ratio a r e calculated. F r o m the damping d e c r e m e n t of the oscillations the v i s c o s i t y of the m a t e r i a l s can be found. The apparatus for making these m e a s u r e m e n t s over a wide t e m p e r a t u r e range consists of two soundconducting rods of d i a m e t e r 6 m m and length 600 mm, between which the specimen is clamped. A piezoelectric r a d i a t o r and a piezoelectric r e c e i v e r a r e fixed to the free ends of the rods. The piezoelectric E n g i n e e r i n g - P h y s i c a l Institute, Moscow. Translated 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, 20, No. 4, p. 737, April, 1971. Original article submitted January 22, 1969; a b s t r a c t submitted June 2, 1970.

534

radiator is excited 10-30 times/sec by high-frequency pulses of duration 200-300 #sec, with a variable filling frequency. The signals from the piezoelectric receiver are amplified and observed on the screen of an oscilloscope. The effect of clamping the specimen on the intensity of excitation of the natural frequencies corresponding to different shapes of the oscillations and on the measurement accuracy is considered. The method of calculating the elastic constants from the results of the measurements is described in detail, and appropriate theoretical tables are presented. The accuracy of determining the moduli of elasticity is estimated to be within 2-2.5~c. As an illustration of the use of this method results are presented for the elastic constants and viscosity of specimens of heat-resistant constructional steels KhI8N9T, EI-787, and VZh-98, in the temperature range 293-1200~ made on specimens of diameter 18 mm and thickness 3 mm in the frequency range 130-210 kHz. From the values of the elastic moduli obtained the velocities of propagation of longitudinal and transverse elastic waves are calculated. The activation energy of the processes connected with the absorption of oscillation energy, calculated from the viscosity-temperature curve, is i.i, 1.15, and 1.4 eV, respectively, for the above materials.

FLOW

OUTSIDE

AXISYMMETRIC SEMIBOUNDED G.

THE

TURBULENT

JET

DISCHARGING

REGION INTO

OF

AN

A

SPACE Ya.

Borodyanskii

UDC

532.522

Flow outside the region of an axisymmetric jet was considered earlier by L. D. Landau on the assumption of potentiality of motion. It can be shown, however, that the character of a secondary flow induced by an axisymmetric turbulent jet can be investigated without the assumption of potentiality. The smallness of the velocities of thesecondary flow permit neglecting the inertia terms in the Navier-Stokes equations and reduces the problem to integration of approximate Stokes equations: 1

Op

v

(?Dq)

1

Op

v

p

OP

R" sin 0

aO

p

O0

sin 0

OD~ OR

The solution is sought in the f o r m : r = Rf (cos 0). The c o n s t a n t s of i n t e g r a t i o n a r e d e t e r m i n e d f r o m the conditions on the b o u n d a r i e s of the region: the conditions when 0 = 01 r e f l e c t a d h e s i o n of the liquid on the n o z z l e wail, i . e . , v a n i s h i n g of both v e l o c i t y components; the conditions when 0 = 00 r e f l e c t closing of the s o l u t i o n s in the t u r b u l e n t jet and in the i n v e s t i g a t e d r e g i o n . The solution of the p r o b l e m of an a x i s y m m e t r i c t u r b u l e n t s o u r c e with a "new" P r a n d t l dependence f o r t u r b u l e n t s h e a r s t r e s s is used f o r flow in a t u r b u l e n t jet. On the b a s i s of the a n a l y t i c a l s o l u t i o n obtained for the s t r e a m function the c a l c u l a t e d s t r e a m l i n e s and d i s t r i b u t i o n of the p e r i p h e r a l v e l o c i t y of the s e c o n d a r y flow in the r e g i o n (01 = 45 ~ 0o = 12~ a r e constructed. R a r e f a c t i o n of the p r e s s u r e , a s the solution shows, d e c r e a s e s with d i s t a n c e f r o m the s o u r c e (as 1 /R3), w h e r e a s with a p o t e n t i a l c h a r a c t e r of the s e c o n d a r y flow the r a t e of d e c r e a s e is l e s s (as l / R 2 ) .

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. 20, No. 4, p. 738, A p r i l , 1971. a r t i c l e s u b m i t t e d D e c e m b e r 3, 1969; a b s t r a c t s u b m i t t e d S e p t e m b e r 9, 1970.

Original

535

CONVECTIVE PIPE

HEAT

OF

TRIANGULAR

Sh.

Nuriddinov

TRANSFER

IN

A

PRISMATIC

SECTION

UDC 532.517.2:536.244

Heat t r a n s f e r during l a m i n a r flow of a viscous i n c o m p r e s s i b l e liquid inside a s t r a i g h t semiinfinite p r i s m a t i c pipe (channel) with a c r o s s section in the f o r m of an equilateral triangle with consideration of f r i c t i o n and other internal s o u r c e s of heat was investigated. The p r o b l e m is reduced to the solution of the equation

(1)

Oy

c7

with boundary conditions

'T(k, y,z~lz_~ = T o : c o n s t ,

(2)

T(x, y,z) lF=f(z),

w h e r e F is the boundary of the t r i a n g u l a r region. In d i m e n s i o n l e s s coordinates the equation and boundary conditions have the f o r m

or

Io'-r

o'-7"x b~q(~,n, I ) .

, r/o~\3

[ or ~]

(3) (4)

w h e r e b is the altitude of the equilateral triangle. The Laplace i n t e g r a l t r a n s f o r m with r e s p e c t to the longitudinal coordinate ~ is used. obtained in the region of the t r a n s f o r m s is solved by the B u b n o v - G a l e r k i n method.

The p r o b l e m

The final solution is presented in the f o r m of the s u m of products of polynomials and exponential functions, w h e r e b y the polynomials depend on the v a r i a b l e s ~ and ~, and the exponential functions only on ~. The following c a s e s a r e investigated in detail. 1. A constant t e m p e r a t u r e r e g i m e is maintained on the pipe wall and the liberation t e r n a l s o u r c e s is uniform o v e r the e n t i r e volume of the flow. The solutions a r e second and third approximations. Simple and sufficiently a c c u r a t e f o r m u l a s a r e t e r m i n i n g local Nusselt n u m b e r s (in the a b s e n c e of internal s o u r c e s and e n e r g y change of the Nusselt n u m b e r along the flow is presented graphically.

of heat by infound in the obtained for d e dissipation). The

2. The t e m p e r a t u r e at the pipe entrance coincides with the wall t e m p e r a t u r e and the power of the internal heat s o u r c e s is negligibly small, i.e., heat t r a n s f e r is due only to the heat of friction. Stabilization of the t e m p e r a t u r e profile along the liquid flow {on the plane of the m e r i d i a n section) is presented graphically. F o r m u l a s a r e obtained for the r e l a t i v e heat flux on the pipe walls. I n v e s t i g a t i o n s of the f o r m u l a s obtained showed that the heat flux in the middle of the face is m a x i m u m and in the c o r n e r regions is m i n i m u m , which is consistent with the data of other authors. 3. The wall t e m p e r a t u r e is a linear function of the longitudinal coordinate and the power of the int e r n a l heat s o u r c e s is constant over the e n t i r e volume of the flow. A g r a p h of t e m p e r a t u r e s t a b i l ization along the liquid flow in the a b s e n c e of internal s o u r c e s and e n e r g y dissipation is presented. 4.

The wall t e m p e r a t u r e i n c r e a s e s exponentially, and the power of the internal heat s o u r c e s is c o n stant.

The c o n v e r g e n c e of the a p p r o x i m a t e solutions to the exact solution is discussed.

Tadzhik Polytechnic Institute, Dushanbe. 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. 20, No. 4, pp. 738-739, April, 1971. Original article submitted D e c e m b e r 16, 1969; a b s t r a c t submitted May 5, 1970.

536

A

METHOD

OF

INVOLVING ON

THE

HEAT

NONLINEAR

A.

Brobkin

CONDUCTION

PROBLEMS

HEAT-TRANSFER

BOUNDARIES L.

We

SOLVING

OF and

A L.

RELATIONS

BODY S.

UDC

Krylova

536.2.01

consider the solution of the heat conduction differential equation OT

027

R2 OFo =--Ox~]-" -k

OT 0.< x ._.<1R, Ox . . . .

le - - 1 x

(1)

for the b o u n d a r y c o n d i t i o n s

(2)

OT

r(x. 0 ) = T a, O T - ( 0 ' F ~

( 3}

OT

~-aT-. (~' Fo) = q [T(.~, Fo)].

F o r the c a l c u l a t i o n of t e m p e r a t u r e f i e l d s of b o d i e s , for e x a m p l e , the t e m p e r a t u r e f i e l d s i n i n d u s t r i a l f u r n a c e s , the b o u n d a r y c o n d i t i o n (3) i s n o n l i n e a r : ~,

An exact solution of the system

OF 4 (R, Fo) = cz[Tb -- r (R, Fo)] ~- ~ Ire --T4 (R, Fo)]. Ox

{4)

(1), (2), and (4) is not known.

In this connection, it may be possible to have an approximate solution, which satisfies exactly only the heat conduction equation and the initial conditions. In such a solution the boundary conditions are satisfied discretely at g time instants, taken on the interval Fo, the evolution time of the process. The quantity g determines the degree of approximation tO the exact solution, and as g ~ % the approximate solution becomes coincident with the exact solution. An approximate solution can be obtained, which is based on a generalized notation for the temperature field in bodies for boundary conditions of the first kind. We approximate the temperature variation law on the surface of the body by a polynomial n~g

n

T (R, Fo) ~ T Oq- 7 ~ AnF~

(S}

F o r a plate the p r o b l e m r e d u c e s to s o l v i n g a s y s t e m of g e q u a t i o n s i n g u n k n o w n s An: n~g

2 ~ ~ AnO~,n (Foi) = q (Foi),

i = 1,2,3, ...,g.

(6)

9/~ r z = l

To solve the system we obtain the constants A n . The temperature on the plate surface is determined from Eq. (5); the temperature at the center, and the temperature averaged with respect to the mass, are given, respectively, by the equations n~g

r (~, Fo) -- r (0. Fo) -- ~ A~r

(Fo),

(7)

n=l n~g

ray (Fo) - -

ro = 2 ~

A~r

(Fo).

(8)

n=I

N o m o g r a m s a r e t h e n c o n s t r u c t e d for the f u n c t i o n s ~ l , n ( F o ) , ~2,n(Fo), ~4,n(Fo). Studies show that it i s s u f f i c i e n t to r e s t r i c t g to the v a l u e g = 2 f o r t h e r m a l l y t h i n b o d i e s and g = 3 for m a s s i v e b o d i e s . M o r e o v e r the e r r o r of the m e t h o d ml.5%. The method c a n be u s e d f o r c o m p o u n d and n o n l i n e a r b o u n d a r y c o n d i t i o n s , and a l s o for b o u n d a r y c o n d i t i o n s i n i m p l i c i t f o r m .

V. I. Lenin Ivanovskii Energetics Institute. 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 Z h u r n a l , Vol. 20, No. 4, pp. 739-740, April, 1971. Original article s u b m i t t e d S e p t e m b e r 9, 1969; a b s t r a c t s u b m i t t e d J u l y 1, 1970. 537

ACCURACY

OF

THE

PROBLEMS

ON

COMBINED

S. V. D a v y d o v

SOLUTION

OF

BOUNDARY-VALUE

MODELS

UDC 681.142.334

The solution of two- and t h r e e - d i m e n s i o n a l b o u n d a r y - v a l u e p r o b l e m s can be obtained by the method of e l e c t r i c a l modeling with the use of combined models in which the physical field is r e p r e s e n t e d by a continuous e l e c t r i c a l l y conducting m e d i u m in combination with d i s c r e t e e l e m e n t s . This c l a s s includes in p a r ticular models of the type " e l e c t r i c a l l y conducting p a p e r - r e s i s t o r s " used for solving p r o b l e m s of unsteady heat conduction by the method of s u c c e s s i v e i n t e r v a l s (Liebmann method). The connection of r e s i s t o r s with conducting p a p e r at nodes is a c c o m p l i s h e d via contact e l e c t r o d e s of finite size which a r e g e n e r a l l y round with a d i a m e t e r D = 0.1 of the spacing of the nodes. An i n c r e a s e of the c u r r e n t density in the neighborhood of the e l e c t r o d e s is a c c o m p a n i e d by distortion of the potential lines in the continuous medium, which introduces a c e r t a i n e r r o r into the solution. To reduce the e r r o r of the d i s c r e t e c u r r e n t supply, it is suggested to use models with r e l a t i v e l y enlarged dimensions of the contact e l e c t r o d e s D = 0.3 with a v e r a g e values of the s p a c e and t i m e intervals. The change of the effective r e s i s t a n c e of the conducting paper in the p r e s e n c e of the enlarged cont a c t e l e c t r o d e s can be taken into account by introducing a c o r r e c t i o n into the value of the specific r e s i s tance of the conducting paper, which e n t e r s into the f o r m u l a s f o r calculating the d i s c r e t e e l e m e n t s of the model, p a r t i c u l a r l y the t i m e r e s i s t o r s R T. The magnitude of the c o r r e c t i o n can be d e t e r m i n e d f r o m the e x p e r i m e n t a l r e l a t i o n s between the d i m e n s i o n l e s s effective r e s i s t a n c e of a s q u a r e of conducting p a p e r and the s i z e and shape of the contact electrodes. Another way of reducing the effect of d i s c r e t e n e s s is to change the magnitude of the r e s i s t o r s s u p plied to the inside nodes. The magnitude of the c o r r e c t i o n f o r the effect of d i s c r e t e n e s s can be d e t e r m i n e d e x p e r i m e n t a l l y . However, in this c a s e it is r e c o m m e n d e d to use enlarged e l e c t r o d e s in the boundary regions. It is shown with r e f e r e n c e to two-dimensional p r o b l e m s for which the t h e o r e t i c a l solution is known that the methods proposed in the a r t i c l e for reducing the effect of d i s c r e t e n e s s of e l e c t r o d e s p e r m i t r e ducing the modeling e r r o r f r o m 8-7 to about 2% of the m a x i m u m potential difference for a v e r a g e values of the s p a c e and t i m e i n t e r v a l s . Modeling was done on the E G D A - 9 / 6 0 e l e c t r o i n t e g r a t o r with a specialized a t t a c h m e n t for solving t w o - d i m e n s i o n a l p r o b l e m s of unsteady heat conduction.

V. I. Lenin Ivanovo Power Institute. 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. 20, No. 4, pp. 740-741, April, 1971. Original a r t i c l e submitted J a n u a r y 29, 1970; a b s t r a c t submitted August 3, 1970.

538

PRINCIPAL

BOUNDARY-VALUE

THERMOELASTICITY D.

V.

FOR Grilitskii

and

PROBLEMS A

OF

CIRCULAR

I. N.

DISK

UDC 539.377

Osiv

The a u t h o r s i n v e s t i g a t e d a c i r c u l a r is o t r o p i c d i s k r e f e r r e d to a r e c t a n g u l a r C a r t e s i a n c o o r d i n a t e s y s t e m with the o r i g i n at the c e n t e r of the disk, at s o m e point of which is located a s t a t i o n a r y c o n c e n t r a t e d heat s o u r c e with i n t e n s i t y W. Mixed conditions for the t e m p e r a t u r e T(z, z) a r e a s s i g n e d at the edge of the d i s k OY (t, t)

-

-

-

[~ ( t ) ,

(1)

t ~ L~,

On

(2)

Tti, ~ = h (t), ~C L~, and h o m o g e n e o u s conditions f o r s t r e s s e s ~r + iT,0 = [(t),

tC L,

(3)

t c L.

(4)

or for displacements § iv = g (t), The b a s e s of the d i s k a r e a s s u m e d h e a t - i n s u l a t e d . The s t a t i o n a r y t e m p e r a t u r e field and s t r e s s e s of the disk a r e d e t e r m i n e d by m e a n s of the r e l a t i o n ships of the t w o - d i m e n s i o n a l t h e o r y of t h e r m o e l a s t i c i t y [1, 2]. The f u n c t i o n d e t e r m i n i n g the t e m p e r a t u r e in the d i s k is p r e s e n t e d in the f o r m T (z, z) - Re [A 1~(z--~0) + Fo (z)].

(5)

D e t e r m i n a t i o n of the function of the c o m p l e x v a r i a b l e F0(z ) h o l o m o r p h i c in the r e g i o n of the d i s k is r e d u c e d to a s i n g u l a r i n t e g r a l equation of the f i r s t kind whose solution in the g e n e r a l c a s e is r e p r e s e n t e d in q u a d r a t u r e s . The t h e r m o e l a s t i c s t a t e f o r a h a f t - p l a n e u n d e r the effect of a c o n c e n t r a t e d s t a t i o n a r y heat s o u r c e , both with b o u n d a r y conditions of the mixed type f o r t e m p e r a t u r e and h o m o g e n e o u s for the m e c h a n i c a l c h a r a c t e r i s t i c was obtained by p a s s a g e to the limit. F o r c e r t a i n p a r t i c u l a r v a l u e s of the initial data the r e s u l t s of the a r t i c l e coincide with those obtained e a r l i e r in [3, 4]. The thermoelastic state of the disk and half-plane in real variables is determined, the distribution of temperature and stresses at the edge of the disk are also presented.

and graphs of

NOTATION

z=x+iy f(t), f~(t), f2(t), g'(t)

~/an t=x+iy

is the complex variable; are the known functions satisfying condition H; is the derivative with respect to the normal; is a point located at the edge of the disk;

n

L1 = f__Zaajbj

is the set of arcs on L arranged so that, by-passing L counterclockwise, , an, b n are found in the indicated sequence;

points at, bl,

a2, b 2 .... L2 = L - L I ; ~ r , ~-r 0 U,

A

V

a r e the n o r m a l and tangential s t r e s s c o m p o n e n t s on L; a r e the r a d i a l and tangential d i s p l a c e m e n t s of points of the edge L; is a coefficient d e t e r m i n e d by the i n t e n s i t y of the heat s o u r c e , t h i c k n e s s , and c o e f ficient of i n t e r n a l heat c o n d u c t i v i t y of the disk.

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 Z h u r n a l , Vol. 20, No. 4, pp. 741-742, A p r i l , 1971. Original a r t i c l e s u b m i t t e d N o v e m b e r 13, 1969; a b s t r a c t submitted May 19, 1970.

539

LITERATURE 1.

2. 3. 4.

540

CITED

N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity [in Russian], Nauka, Moscow (1966). G. N. Savin, Distribution of Stresses Near Holes [in Russian], Naukova Dumka, Kiev (1968). A. D. Khanov, Inzh.-Fiz. Zh., 11, No. 5 (1966). E. Melan and G. Parkus, Thermal Stresses Caused by Temperature Fields [in Russian], Fizmatgiz, Moscow (1958).