ABSTRACTS

CALCULATING BOUNDARY HIGH

THE

LAYER

PRANDTL M. E.

HEAT

TRANSFER

OF VISCOUS

IN A TURBULENT

FLUID

WITH

A

NUMBER UDC 536.24:532.526

Podol'skii

As is well known, calculations of the heat t r a n s f e r in a turbulent boundary l a y e r yield too low values f o r the t h e r m a l flux when b a s e d on the t w o - l a y e r or the t h r e e - l a y e r model. It a p p e a r s that the t w o - l a y e r model will yield e n t i r e l y s a t i s f a c t o r y r e s u l t s at a high P r a n d t l n u m b e r , if the additional heat t r a n s f e r due to turbulent fluctuations in the l a m i n a r s u b l a y e r is taken into account. The f o r m u l a s h e r e a r e also quite s i m p l e and convenient f o r use on the c o m p u t e r . With turbulent heat t r a n s f e r taken into account, the f o r m u l a for the t h e r m a l conductivity k(r of the laminar sublayer becomes

~(, =~, + pcv~.

(1)

I

H e r e v t denotes the coefficient of turbulent v i s c o s i t y which, a c c o r d i n g to the L a n d a u - L e v i c h hypot h e s i s , will be c o n s i d e r e d p r o p o r t i o n a l to the fourth p o w e r of the distance f r o m the wall. The t h i c k n e s s of T the l a m i n a r s u b l a y e r will be denoted by 6 l and the dynamic velocity by v*. We then have for v t : v~ = n,14v.

(2)

In this f o r m u l a n

k 113

yv* --

6iv* --

and k is the proportionality factor between mixing length and distance from the wall. In combination with (1), formula (2) yields the following expression for the Stanton number (based on the velocity of the oncoming stream U and the temperature drop TO- Tf across the boundary layer):

c___f St =

~f~-I~- V ~ -

2T ; Cf=pU2 .

2

[* (Pr) Pr ]/4 --nl]

H e r e z denotes the f r i c t i o n a l s t r e s s at the wall and ~(Pr) is a function of the P r a n d t l number, q u a s i c o n stant when P r > 1 (~ ~8.4). F o r p i p e s , w h e r e the Stanton n u m b e r is usually b a s e d on the m e a n velocity v m and the m e a n - o v e r - t h e - f l o w t e m p e r a t u r e T O- T m, we have Stm=

(~mt%) (Us) l + ~ m V~-Tg [~ (pr) pr 3/4 - nt] Urn

~rn= V - ;

TO - - Trn

~rn = r o - r ~ '

*

81:

(3)

~ = p-~ "

Calculations by f o r m u l a (3) a r e in s a t i s f a c t o r y a g r e e m e n t with e x p e r i m e n t a l data and with values obtained by Snegova with the van D r e e s t f o r m u l a f o r the mixing length. Institute of Ship Construction, Leningrad. 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. 23, No. 1, pp. 154-166, July, 1972. Original a r t i c l e submitted May 13, 1971; a b s t r a c t submitted D e c e m b e r 20, 1971. 9 t974 Consultants Bureau, a division o f Plenum Publishing Corporation, 227 West 17th Street, New Yor]% N. u 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, withotLt written permission of the publisher. A copy of this article is available from the publisher for $15.00.

916

ENERGY

AND M A T E R I A L

INCLUDING

THE EFFECT

CHEMICAL

CONVERSION

HYDROTHERMAL

V. O.

N. A.

TRANSFER

PROCESSES

OF P R E S S U R E

AND

IN N A T U R A L

SYSTEMS

Kochergin, ]3alyshev,

UDC 532.72:536.242

V. N. V o i t a l y u k , a n d V. D. P a m p u r a

Natural p r o c e s s e s which occur during the interaction between hydrothermal solutions (or magnetic melts) and rocks deep inside the earth crust are governed by high rates of heat and mass t r a n s f e r [4] between the liquid and the solid phase [2]. The development of t em perat ure and humidity fields as a result of hydrothermal interactions is accompanied by a transport of chemical components within the geological system. The r a t e s of heat and mass t r a n s f e r determine the system kinetics and the character of the chemical reactions. Chemical conversions under natural conditions are accompanied by stronger and weaker th e rmal effects representing internal heat sources in the hydrothermal process. Because of the long development and formation period in the case of geological systems, such rat her weak thermal effects compound with time and may deform both the t e m p e r a t u r e and the humidity fields. F o r a quantitative study of the general laws govering the energy and the material t r a n s f e r in hydrothermal systems, it has become n e c e s s a r y to develop models simulating nearly actual conditions. The mathematical model proposed he re describes, to a closer approximation than e a r l i e r models [3], the natural hydrothermal pr oces s including both the dynamics and the thermal effects of chemical conversions, which, in the final analysis, r e p r e s e n t crystallization, reerystallization, dissolution of m in e rals, and metasomatic transformations of matrix rocks. The mathematical model of the pr oc e ss comprises a system of parabolic partial differential equations describing the interdependent heat and mass t r a n s f e r s , the diffusion of chemical components, and the p r e s sure changes in the interstitial solution of the (hydrothermal)solution-rocks system. A method is outlined for synthesizing the mathematical model on a digital computer, under constraints approaching the natural ones. The problem is solved by the two-dimensional finite-differences scheme [1] and the Runge-Kutta method [6], both needing to be evaluated for accuracy and stability of the solution to this system of equations. An e r r o r and convergence analysis of the solution shows that this procedure can be used successfully for solving not only individual equations [5] but also multidimensional systems of functionally coupled differential equations. F o r this reason, the results obtained by synthesis of the mathematical model describe accurately enough the general laws which govern the natural hydrothermal process and they are useful for studying the mechanism metamorphism

by which a mineralizing of rocks.

solution is transported,

LITERATURE 1. 2.

3. 4. 5. 6.

the formation

of ore deposits,

and the

CITED

V. Vasov and G. Forsyte, Difference Methods of Solving Partial Differential Equations [Russian translation], Izd, IL, Moscow (1963). V. N. Koehergin, Inzh. Fiz. Zh., 13, No. 6 (1967). V. N. Koehergin and O. A. Balyshev, Inzh. Fiz. Zh., 15, No. 2 (1968). A. V. Lykov and Yu. A. Mikhaflov, Theory of Heat and Mass T r a n s f e r [in Russian], Gos~nergoizdat, Mo s co w- Leningrad (1963). V. V. Ryaben'kii and A. F. Filipov, On the Stability of Difference Equations [in Russian], Gostekhizdat, Moscow (1956). J. W. C a r t , J. Assoc. Comput. Mach., 5, 39 (1958). original article Submitted March 31, 1971; abstract submitted December 14, 1971. 917

MASS

TRANSFER

AROUND M.

DURING

A CIRCULAR Kh.

A TRANSVERSE

FLOW

CYLINDER

Kishinevskii

and

A.

A.

UDC 536.24:532.526+532.72

Mosyak

Most test data on heat and m a s s t r a n s f e r , at a high Prandtl number, between a c i r c u l a r cylinder and a s t r e a m flowing p e r p e n d i c u l a r l y to its axis have so f a r been obtained for a Reynolds number and a Prandtl n u m b e r not exceeding Re = 12,000 and P r = 2800 r e s p e c t i v e l y [1-5]. The authors have used the rotation method in an attempt to extend the r a n g e of both numbers. A stationa r y specimen was placed in a s t r e a m of liquid filling an annular channel of r e c t a n g u l a r c r o s s section and revolving around the v e r t i c a l axis. The lack of a free liquid surface in the revolving channel helped to r e duce the p e r t u r b a t i o n s due to i n t e r f e r e n c e between the specimen and the channel walls or bottom; this, in t u r n , made it possible to attain a high Reynolds n u m b e r during the flow around the specimen, Two r o t a r y devices of different s i z e s were used in the experiment. On one of them the t r a n s f e r coefficients were det e r m i n e d f r o m the loss of specimen m a s s , on the other they were determined f r o m the change in benzoic acid concentration in the solution. A s e r i e s of m a s s t r a n s f e r t e s t s was also p e r f o r m e d in a s t a t i o n a r y channel of r e c t a n g u l a r c r o s s section. Test s p e c i m e n s 0.0351 and 0.0085 m in d i a m e t e r with a 0.012-0.020 m height of the dissolving segment had been p r e p a r e d by molding benzoic acid powder under 1500 b a r s p r e s s u r e . Annular extensions were fastened to the s p e c i m e n s and only the center portion of benzoic acid with a g r a d e - 9 s u r f a c e p u r i t y was allowed to dissolve. The solvents were distilled water and aqueous 22.8-67.5% (weight) glycerine solutions with the concentration of benzoic acid v a r y i n g f r o m z e r o to 50-70% of the saturation level at 25 9 0.2~ The s t r e a m velocity at the critical point in front was calculated f r o m the difference between total and static p r e s s u r e at the stagnation point. The values of local m a s s t r a n s f e r coefficients were d e t e r m i n e d f r o m the change in the t r a n s v e r s e dimensions of the dissolving 0.0351 m cylindrical segments. It could be established, in this way, that the dissolution had o c c u r r e d u n i f o r m l y along the cylinder height and s y m m e t r i c a l l y with r e s p e c t to its axis. The local m a s s t r a n s f e r coefficients in the vicinity of the stagnation point determined thus were on the ave r a g e 5% lower than those calculated by the t h e o r e t i c a l f o r m u l a for P r -~oo [6]: Nu=1.332 t~e0'50Pr~

(1)

The m e a n - o v e r - t h e - d i a m e t e r m a s s t r a n s f e r coefficients within the r a n g e s Re = 4300-410,000 and P r = 930-200,000 c o r r e l a t e by the equation Nu=0 082 Re~ Pr~

(2)

F o r determining the Prandtl number, the velocity of the oncoming s t r e a m in f o r m u l a (2) was multiplied by a f a c t o r accounting for channel clogging and calculated by the method in [7]. NOTATION Pr Re Nu

is the P r a n d t l diffusion number; is the Reynolds number; is the Nusselt diffusion number. LITERATURE

1. 2. 3. 4. 5.

CITED

P. G r a s s m a n , N. Ibl, and I. Trueb, Chem. Ing. T e c h n . , No. 8, 529 (1961). R. Dobry and R. Finn, Industr. Eng. C h e m . , 48, 1541 (1956). V. P. Popov and N. A. P o k r y v a i l o , in: Heat and Mass T r a n s f e r during Chemical Conversions and P h a s e T r a n s f o r m a t i o n s [in Russian], Minsk (1968), p. 166. R. Sudhakara, G. I. V. Jagannadha Raju, and R. Venkata, Indian J. Technology, 6, 46 (1968). A. V. Lykov, Z. P. Shul'man, B. I. P u r i s , in: Heat and Mass T r a n s f e r [in Russian], Minsk (1968), Vol. 3, p. 54. Original a r t i c l e submitted September 28, 1971; a b s t r a c t submitted F e b r u a r y 2, 1972.

918

6. 7~

G. Schlichting, B o u n d a r y - L a y e r T h e o r y [Russian t r a n s l a t i o n ] , Izd. Nauka, Moscow (1969). Vliet and L e p p e r t , T r a n s . ASME Heat T r a n s f e r , 83C, No. 2, 76 (1961).

EXPERIMENTAL MESHED E.

STUDY

OF

HEAT

TRANSFER

IN

MATRICES* I.

Mikhulin

and

Yu.

A.

Shevich

UDC 621.565.93.001.5

R e s u l t s a r e p r e s e n t e d of an e x p e r i m e n t a l study concerning the heat t r a n s f e r during a f o r c e d flow of a i r t h r o u g h f i n e - m e s h m a t r i c e s (dwire = 0.03-0.08 ram, dcell = 0.04-0.112 mm) of d o m e s t i c m a n u f a c t u r e . The t e s t s w e r e p e r f o r m e d under s t e a d y conditions with a continuous heat g e n e r a t i o n in the m e t a l by m e a n s of e l e c t r i c c u r r e n t . M a t r i c e s w e r e t e s t e d with v a r i o u s n u m b e r s of m e s h e s p e r packet, f r o m one to s e v e r a l hundred, i . e . , o v e r a wide r a n g e of the H/D e r a t i o (H denoting the packet length in the d i r e c t i o n of a i r flow and D e denoting the equivalent d i a m e t e r ) . It has been e s t a b l i s h e d that, as the r a t i o H/D e is m a d e l a r g e r , the r a t e of heat t r a n s f e r in m e s h e d m a t r i c e s d e c r e a s e s while the heat t r a n s f e r at a single m e s h is the s a m e as that at a cylinder in a t r a n s v e r s e s t r e a m , but the heat t r a n s f e r in a p a c k e t with m a n y m e s h e s does, to s o m e extend, a p p r o a c h that in a solid channel. The effect of the H / D e r a t i o and of o t h e r p a r a m e t e r s explains the wide d i s c r e p a n c y between known data on heat t r a n s f e r in m e s h e d m a t r i c e s . C r i t e r i a l equations a r e p r o p o s e d f o r d e s c r i b i n g the heat t r a n s f e r in f i n e - m e s h m a t r i c e s with v a r y i n g r e l a t i v e length and within the r a n g e of Reynolds n u m b e r Re = 10-500.

EVALUATING EVAPORATION I.

Ya.

THE

EFFECTS

AND

OF

UNSTEADY

CONDENSATION

Kolesnik

The solution to the equations d e s c r i b i n g u n s t e a d y condensate buildup and e v a p o r a t i o n p r o c e s s e s with a m o v a b l e i n t e r p h a s e b o u n d a r y is written in the f o r m f~r2=1--2~

"(-2- [

..... 1+ I / - f f ]

(1)

(2)

*Original a r t i c l e submitted April 20, 1971; a b s t r a c t submitted J a n u a r y 31, 1972. ~Original a r t i c l e submitted May 19, 1971; a b s t r a c t s u b m i t t e d J a n u a r y 10, 1972.

919

With the aid of expansions (1) and (2), the following f o r m u l a s a r e derived, to the f i r s t approximation with r e s p e c t to the small p a r a m e t e r v: for the r a t e of condensate buildup J**=]

{l-~-v~(x)},

(3)

for the evaporation rate J**=J ( I +ve~ (~)}, where (~) = ~

3

arctg }/'2~-" +

1 -{- 3~:

;

and J is the r a t e of q u a s i s t e a d y evaporation or condensate buildup. A c c o r d i n g to relation (3), even at T >- 1 the c o r r e c t i o n to the quasisteady flow r a t e during condensate buildup is a l m o s t twice as l a r g e as the c o r r e c t i o n suggested by an e a r l i e r f o r m u l a by other authors. As --* 0, the difference between both c o r r e c t i o n s i n c r e a s e s infinitely. The effect of unsteadiness on the p r o t e s s rate at some fixed ~ is d e t e r m i n e d e n t i r e l y by the value of the small dimensionless p a r a m e t e r v, i.e., by the t e m p e r a t u r e and the s u p e r s a t u r a t i o n . In the case of evaporation, the c o r r e c t i o n changes sign. It is positive at 9 within the interval 0 -- z < 71 (71 ~ 0.26) and is negative at 9 > ~l, while, according to the e a r l i e r formula, the c o r r e c t i o n is always negative, i . e . , the actual r a t e is always lower than the quasisteady rate. The difference between both c o r r e c t i o n s b e c o m e s appreciable, in this case, when ~---* 0 and at a time close to complete e.vaporation of a condensate. The f o r m u l a s derived h e r e are useful for studying and evaluating all basic effects related to unsteadiness r e s u l t i n g f r o m both a discontinuous concentration field at the initial instant of time and a movable int e r p h a s e boundary, they also indicate the limitations on the applicability of the quasisteady approximation in evaporation and condensate buildup theory.

THREE-DIMENSIONAL DURATION V.

G.

THERMAL

IN A T R A N S M I T T E N T Andreev

and

P.

I.

SHOCK

OF

FINITE

PLATE Ulyakov

UDC 539.3:536

The p r e s e n c e of high t e m p e r a t u r e gradients during a s h o r t - t i m e t h e r m a l shock r e q u i r e s that the analysis be based on the hyperbolic equation of heat conduction, which accounts for a finite velocity of heat propagation. Heat is t r a n s m i t t e d through d i e l e c t r i c lattices p r i n c i p a l l y by the conduction m e c h a n i s m , and the velocity of heat propagation is h e r e equal to the velocity of sound in the given medium. A propagation of t e m p e r a t u r e and s t r e s s p e r t u r b a t i o n s at the same velocity indicates that both t r a v e l on the same wave. When the initial conditions a r e of a d i s c r e t e kind ( m o m e n t a r y shock), t h e r e o c c u r s a jump in p r e s s u r e as well as in t e m p e r a t u r e and in density, with the r e s u l t that the equations of t h e r m o e l a s t i c i t y become inapplicable as f a r as determining the p a r a m e t e r s of the given medium under a discontinuity. In actual p r o c e s s e s a t h e r m a l shock is of a finite duration and the s t r e s s e s build up continuously behind the wave front. The dynamic p r o b l e m of t h e r m o e l a s t i c i t y is solved in this article by means of the Original article submitted August 13, 1971; a b s t r a c t submitted D e c e m b e r 30, 1971.

920

Laplace t r a n s f o r m a t i o n for a t h r e e - d i m e n s i o n a l t h e r m a l shock of finite duration. E x p r e s s i o n s a r e derived for the t e m p e r a t u r e s and the s t r e s s e s , while the p r o b l e m is solved simultaneously in the parabolic approximation. The quasistatic state of s t r e s s r e p r e s e n t s a special case (at c o -~oo). By introducing into the p r o b l e m a velocity of heat propagation equal to the velocity of sound, it has b e c o m e p o s s i b l e to eliminate the p h y s i c a l l y absurd a p p e a r a n c e of s t r e s s e s p r i o r to the a r r i v a l of a wave at a given point. An analysis of the solution shows that the amplitude of the t e m p e r a t u r e front decays exponentially with time. It subsequently does not affect the propagation of s t r e s s waves. A quasistatic s t r e s s field is e s t a b lished behind the front of an Mastic wave.

CONCERNING SPECIFIC Kh. and

THE

HEAT

AT

VALUE THE

I. Amirkhanov, G. V. S t e p a n o v

OF

THE

CRITICAL ]3.

G.

CONSTANT-VOLUME POINT

Alibekov,

On the b a s i s of available test data p e r t a i n i n g to the specific heat Cv of water, argon, xenon, nitrogen, and oxygen, this c o n s t a n t - v o l u m e specific heat is shown h e r e to have a finite value at the critical point. There is evidently no justification f o r letting it become infinite at that point, as some other authors have done by extrapolating e m p i r i c a l f o r m u l a s (the l o g a r i t h m i c or the p o w e r - l a w expression) to the c r i t i c a l point. It is shown, instead, that these f o r m u l a s d e s c r i b e with equal a c c u r a c y the t e s t r e s u l t s pertaining to the c r i t i c a l i s o e h o r and to adjacent i s o c h o r s , and that, t h e r e f o r e , the r e s u l t s of such an extrapolation would violate the laws of t h e r m o d y n a m i c s .

On the basis of a numerical evaluation of straight test data pertaining to the constant-volume specific heat of the substances examined most thoroughly, namely water and argon, another empirical formula is given to describe the trend of C v along the critical isochor, with a standard deviation smaller than that along the logarithmic curve. At the same time, the formula proposed here by the authors yields, by extrapolation, finite values for the constant-volume specific heat at the critical point. Graphs are shown also for the other substances listed here, indicating a closer agreement test data and the proposed formula of the

between

Cv a e x p [ - - b l T ~ - - T P'I ]. kind. The value of coefficients a and b for water and a r g o n a r e d e t e r m i n e d by the method of least s q u a r e s (for a r g o n a = 556.2 J / m o l e "degC and b = 2.487; for water a = 12.62 c a l / g - d e g C and b = 1.403 at T < T e r , a = 7.27 c a l / g - deg C and b = 1.668 at T > Tcr). It is aIso shown that the interpolation interval based on these f o r m u l a s c o v e r s a wider t e m p e r a t u r e range. F u r t h e r m o r e , test data on Cv along the s a t u r a t i o n line a r e also evaluated and a subsequent analysis shows that at the c r i t i c a l point the specific heat Cv r e m a i n s finite on the two-phase side as well as on the s i n g l e - p h a s e side, with a finite jump at that point.

Original a r t i c l e submitted May 18, 1970; a b s t r a c t submitted J a n u a r y 4, 1972.

921

EQUIVALENT

DIAMETER

NUMBER

AN E L L I P S O I D A L

FOR

REYNOLDS B.

NUMBER

M.

Re

AND MINIMUM

NUSSELT

PARTICLE

AT A

= 0*

Abramzon

UDC536.242

Calculating the convective heat t r a n s f e r at a p a r t i c l e of i r r e g u l a r shape r e d u c e s to the p r o b l e m of finding its equivalent d i a m e t e r de, i . e . , the d i a m e t e r of a sphere at which the heat t r a n s f e r rate under given conditions is the s a m e as at that p a r t i c l e . Three approximate definitions have been proposed in the technical l i t e r a t u r e for this equivalent d i a m e t e r : 1. the m e a n - v o l u m e d i a m e t e r d V = 3 ~

= de ,

2. the m e a n - s u r f a c e d i a m e t e r d S = ~

= de, and

3. the d i a m e t e r of a sphere whose volume to s u r f a c e r a t i o is the s a m e as that of the given p a r t i c l e : d e = 6V/S = d v / f (with V denoting the p a r t i c l e volume, S denoting the p a r t i c l e surface, and the a s p h e r i c i t y f a c t o r f = S/~d~). A solution is given h e r e to the p r o b l e m of heat and m a s s t r a n s f e r r a t e s at an ellipsoid i m m e r s e d in an infinitely l a r g e s t a t i o n a r y medium. The exactly calculated Nusselt n u m b e r is shown in Fig. 1 for a prolate ellipsoid (curve a) and an oblate ellipsoid (curve b) of r e v olution, as a function of the a s p h e r i c i t y factor. F o r comparison, c u r v e s I, 1I, and I~ r e p r e s e n t i n g r e s p e c t i v e l y the t h r e e approximate definitions of the equivalent d i a m e t e r have also been plotted h e r e (the Nusselt n u m b e r is based on the m e a n - v o l u m e d i a m e t e r of a particle).

l,lu

"\\ ~"~'~~ N

\

\

Thus, most a c c u r a t e is the method in which the m e a n - v o l ume d i a m e t e r of a particle is defined as its equivalent d i a m e t e r for approximate calculations.

\ %

Fig. 1

DIMENSIONLESS RATE FLUIDS

PRESSURE

CHARACTERISTICS IN

THE

CHANNEL

COMPOUND-DISPLACEMENT V. I. Yankov, L . M. S. A . B o s t a n d z h i y a n , and V V. K i s e l e v

HEAD OF OF

VERSUS

FLOW

NON-NEWTONIAN A

SCREW

PUMP~

Beder, V. I. B o y a r c h e n k o ,

UDC 678.02:532.135

I s o t h e r m a l steady flow of an anomalously viscous fluid with a p o w e r - l a w rheological c h a r a c t e r i s t i c is analyzed, under conditions in the channel of a compound-displacement s c r e w pump. F o r m u l a s are derived for the m a x i m u m axial and t r a n s v e r s e p r e s s u r e gradient. Knowing these values, one can introduce d i m e n s i o n l e s s p r e s s u r e g r a d i e n t s and flow r a t e s for plotting the p r e s s u r e head v e r s u s flow rate c h a r a c t e r i s t i c s of s c r e w pumps with various pitch angles and fluids with v a r i o u s v i s c o s i t y anomalies - c h a r a c t e r i s t i c s which are invariant with r e s p e c t to fluid v i s c o s i t y , tangential velocity, and s c r e w dimensions.

*Original article submitted May 12, 1971; abstract submitted January 5, 1972. i'Original article submitted March 16, 1971; abstract submitted January 7, 1972.

922

With the aid of t h e s e c h a r a c t e r i s t i c s and a u x i l i a r y g r a p h s , one can calculate the p r e s s u r e and the p o w e r r e q u i r e m e n t s of a s c r e w p u m p without the u s e of c o m p u t e r s .

ADIABATIC THE

FLOW

CHANNEL

SCREW

OF

PUMP V. S.

OF

I. A.

A

A

NON-NEWTONIAN

FLUID

IN

COMPOUND-DISPLACEMENT

* Yankov, L. Bostandzhiyan,

M.

Beder, and V.

UDC I.

678.02:532.135

Boyarchenko

Adiabatic flow of a fluid with a power-law rheological characteristic in the channel of a screw pump or any other screw machine is analyzed, taking into account the circulatory flow in such a channel. It is assumed that no heat transfer occurs with the ambient medium and that the fluid temperature varies only along the channel, as a result of dissipative heating, but remains constant in two other mea surements and, on this basis, convenient engineering formulas are derived for the temperature, the pressure drop, and the power requirement. An equation is obtained for the pump efficiency, it is also shown how the pump efficiency depends on the viscosity anomaly and on the screw pitch angle.

ACCOUNTING

FOR

THROTTLE Yu.

TYPE V.

TWO-DIMENSIONAL DUST

Rzheznikov

FLOW and

FLOW

IN

A

METERS E.

V.

Lifshits

UDC 681.121:621.6.04

The a u t h o r s analyze the p e r f o r m a n c e of a t h r o t t l e device (Fig. 1), a flat channel (width H0) with an axially mounted e x p e l l e r (width H) consisting of a s m a l l head and a sufficiently long plate (length l) behind it, in a d u s t y s t r e a m . The head and the plate a r e subjected to f o r c e s Qt and Q2 r e s p e c t i v e l y , r e s u l t i n g f r o m p r e s s u r e , friction, and impact of p a r t i c l e s . The p h y s i c a l p r o p e r t i e s of both p h a s e s a r e a s s u m e d constant o v e r the specific channel s e g m e n t , the impact of dust p a r t i c l e s on the plate is a s s u m e d inelastic, and the s t r e a m is c o n s i d e r e d steady and i n c o m p r e s s i b l e . In o r d e r to account for the t w o - d i m e n s i o n a l i t y of the flow h e r e , t h e s t r e a m is divided into two regions in e a c h of which the flow is a s s u m e d o n e - d i m e n s i o n a l : in one region (width H~ f a r in front of the expeller) the dust p a r t i c l e s collide with the plate, in the other r e gion they do not collide~ The a i r - v e l o c i t y field is plotted on the b a s i s of potential-flow calculations, w h e r e upon the t r a j e c t o r i e s of dust p a r t i c l e s a r e found by integrating the equation of motion for single s p h e r i c a l such p a r t i c l e s . F r o m the equations of m a s s , m o m e n t u m , and e n e r g y c o n s e r v a t i o n (with l o s s e s during p a r t i c l e a c c e l e r a t i o n and d e c e l e r a t i o n accounted for) at sections 0-0, 1-1, and 2-2, one obtains analytical e x p r e s s i o n s f o r Q1, Q2, and Q J Q I =II(~, 6), with u denoting the weight concentration, 5 denoting the dim e n s i o n of a single p a r t i c l e , *All-Union S c i e n t i f i c - R e s e a r c h Institute of Synthetic F i b e r s , Kalinin. Moscow Branch, Institute of C h e m i c a l P h y s i c s , A c a d e m y of Sciences of the USSR, Moscow. Original a r t i c l e submitted April 12, 1971; a b s t r a c t s u b m i t t e d J a n u a r y 7, 1972. ~F. E. D z e r z h i n s k i All-Union Institute of Heat Engineering, Moscow. Original a r t i c l e submitted S e p t e m b e r 8, 1971; a b s t r a c t submitted J a n u a r y 10, 1972. 923

l

I

/7

7

J

"If

Fig. 1. Ratio of f o r c e s Q2/QI = II as a function of the dust concentration #: 5 -~ 0 (1), 5 = 5 D m ( 2 ) , 5 = 1 0 ~ m (3), 5 = 15 ~m (45, ~ = 20 # m (55,

6=30

m(6), ~-*oo(75.

3

0,75f ~ ,

z

\, o,z5

o,5

/,o

~,5

2p.(C2--~'cD+ (1 §

-I=

f 7~:,. ~ no--r1

+ Vt / car-- 1 --1--~2 co(l--7-(~ '~! ' c~ i - - o 1--O ]

c denoting the r a t i o of dust velocity to a i r velocity, s u b s c r i p t s 0, 1, 2 r e f e r r i n g to the r e s p e c t i v e sections shown on the d i a g r a m , p r i m e sign ('5 r e f e r r i n g to the dust jet whose p a r t i c l e s collide with the expeller, 7 = i - H~/H0, ~ = 1 - H/H0, and ~ denoting the friction coefficient. Quantities ct, c2, c~, and 7 a r e functions of the p a r t i c l e dimension. Calculations have been made for 5, 10, 15, 20, and 30 ~m p a r t i c l e s and I I - e h a r a c t e r i s t i c s of this dust flow m e t e r have been plotted (Fig. 1), for application to carbon dust m e a s u r e m e n t s in dusts feeding the b u r n e r s of boilers a g g r e g a t e s , a c c o r d i n g to which concentration m e a s u r e m e n t s based on determining the p a r a m e t e r 1I a r e sufficiently sensitive up to ~ = 1. One r e s u l t of p r a c t i c a l importance is that TI does not depend much on the p a r t i c l e dimensions, as long as 5 < 15 ~m, which has been revealed in t e s t s with t h r o t tle-type dust flow m e t e r s [1, 2] but has not yet been p r o p e r l y explained. The values of f o r e e s Ql and Q2 as well as of the p r e s s u r e d r o p s obtained f o r this type of dust flow m e t e r on the basis of a two-dimensional flow differ appreciably f r o m the values obtained by the conventional one-dimensional method. LITERATURE

1~

A, A. Shatil', Teplo6aergetika, No. 8 (1957).

2.

C. G r a c z y k , Chemik [Polish], 14, Nos. 7-8 (19615.

924

CITED

EVALUATING

THE

MEASUREMENTS

ACCURACY IN

Ya.

P.

OF

GAMMA-RAY

SEDIMENTARY

Boikova

and

N.

DEPOSITS* P.

UDC 627.157

Glazkov

N e w ways to improve the accuraey of g a m m a - r a y measurements with a transmittance geometry in sedimentary deposits are explored. The effective cross sections of sandstone molecules (sandstone is the principal component of sedimentary deposits) are estimated, for this purpose, in terms of g a m m a - r a y attenuation due to Compton scattering and due to photoelectric absorption within the 20-1300 keV energy band. On the basis of calculations, the relative attenuation is plotted as a function of sediment concentration, along with the linear attenuation factor and the measurement error as functions of the g a m m a - r a y energy. F r o m these graphs one m a y conclude that the sensitivity and the accuracy of sediment measurements by the g a m m a - r a y method can be improved 50 times to 0.i g/liter, ff a soft 22 keV cadmium-109 source" is used instead of a 1300 keV cobalt-60 source.

TEMPERATURE A MOVING R.

I.

HEAT

DISTRIBUTION

IN A

CYLINDER

WITH

SOURCE~

Blyakhman

UDC 536.21

The transient temperature field in a semiinfinite circular cylinder is determined analytically for the ease where a point source moves at constant velocity along a chord of the end section. The source output, which varies arbitrarily with time, is incorporated into the specific power of heat sources in terms of improper Dirac functions. The thermophysical properties are considered constant. It is assumed, furthermore, that the heat transfer between the cylindrical surface and the ambient m e d i u m follows Newton's law and that heat dissipation from the end surface is negligible. The problem is solved with the aid of Fourier-Hankel integral transformations. The solution, obtained in universal coordinates and extended to a cylinder of finite length, is suitable for computer-aided calculations of temperature field during material processing operations.

* Original article submitted June 24, 1970; abstract submitted F e b r u a r y 2, 1972. $Original article submitted August 31, 1970; abstract submitted F e b r u a r y 2, 1972.

925

APPLICATION

OF

TO AN ANALYSIS COMPOSITE

THE OF

FOURPOLE

NETWORK

TEMPERATURE

FIELDS

THEORY IN

BODIES

Lo A . S a v i n t s e v a , E. a n d N. N. T a r n o v s k i i

D.

Strelova,

UDC 536.2

A p r o c e d u r e is outlined for n u m e r i c a l l y analyzing the heat t r a n s m i s s i o n through a composite body in whose component p a r t s the t e m p e r a t u r e fields a r e either uniform or one-dimensional, with heat being t r a n s f e r r e d f r o m one generating s u r f a c e to another and to the ambient medium. The theoretical model, where the p r o p e r t i e s of the body and the p r o c e s s e s in the body are idealized, is t r e a t e d as a t h e r m a l network. The l a t t e r is r e p r e s e n t e d by an equivalent circuit d i a g r a m with lumped t h e r m a l impedances, t h e r mal c u r r e n t s o u r c e s , and t e m p e r a t u r e - d i f f e r e n c e s o u r c e s . The t h e r m a l circuit is calculated on the basis of Ohm's law and Kirchhoff's laws, allowing a determination of mean t e m p e r a t u r e s at the s u r f a c e s of the body elements and at the interfaces between the elements. The interface t e m p e r a t u r e s can then be used for d e t e r m i n i n g the t e m p e r a t u r e distribution in the body elements. The circuit r e p r e s e n t a t i o n of elements (bodies) with one-dimensional t e m p e r a t u r e fields and with : heat dissipation at the generating s u r f a c e s is based on the e l e c t r i c fourpole network theory. A c o m m o n f o r m of elements in a composite body, n a m e l y a disc of uniform thickness with a center hole, is now analyzed. The disc is surrounded by a medium and by bodies at given t e m p e r a t u r e s . These a r e all r e p l a c e d by a fictitious medium whose t e m p e r a t u r e is equal to the weighted average of all given t e m p e r a t u r e s . The heat t r a n s f e r p r o c e s s e s in such a s y s t e m a r e r e p r e s e n t e d by t h r e e t h e r m a l conduct a n c e s f o r m i n g a T - n e t w o r k and by a s o u r c e of t e m p e r a t u r e difference. The latter c o r r e s p o n d s to the given t e m p e r a t u r e of the fictitious medium. The analytical e x p r e s s i o n s for the conductances in the circuit a r e derived f r o m the exact solution to the equation of heat conduction* ro = 2 ~ (~z~+cz.)[K~(m0 5 (m~)-- I~ (m~)K~(m~)1,

ri=ro/{m2 [Io (mi) K1(m2)-~ Ko(mi)/1 (m2)]-- 1}, Y2=Yo](mi [K1 (m~)Io (m~)--}-I i (mi) Ko(m~)]-- 1}, mi=r~ V ~

' m2=r~V E ~

'"

where r 2 and r I denote the outside and the inside (hole) radius of the disc r e s p e c t i v e l y ; I0, I1, K0, K1 are modified z e r o t h - o r d e r and f i r s t - o r d e r B e s s e l and MacDonald functions r e s p e c t i v e l y . The p r o p o s e d method is f u r t h e r illustrated in the calculation of the t e m p e r a t u r e field in a s y s t e m c o m p o s e d of a rod and a c i r c u l a r disc.

*The t h e r m a l conductivity of the disc m a t e r i a l ~ and the coefficients of heat t r a n s f e r f r o m the upper and f r o m the lower fiat s u r f a c e to the fictitious medium d u and d l a r e functions of neither the coordinates nor the t e m p e r a t u r e . Original a r t i c l e submitted June 10, 1971; a b s t r a c t submitted D e c e m b e r 20, 1971.

926

DETERMINING OF

THE

ICE

CONTENT

IN ROCKS

A MASSIF

A. V. R a s h k i n , N. G. a n d Y u . M. V e d y a e v

Shuvalov,

UDC 551.491.7

A deficiency of both l a b o r a t o r y and field methods of determining the ice content in r o c k s is that the r o c k s t r u c t u r e b e c o m e s distorted during t e s t s and that the l a t t e r a r e c u m b e r s o m e in view of the need for c o s t l y drilling operations. The method p r o p o s e d h e r e , by which the ice content is d e t e r m i n e d f r o m the amount of e n e r g y drawn f r o m a line s o u r c e for phase t r a n s f o r m a t i o n within a cylindrical m a s s i f between two or s e v e r a l p a r a l l e l c r e v i c e s , will eliminate this deficiency. A test was p e r f o r m e d in one of the d i s t r i c t s p e r m a n e n t l y f r o z e n with a seasonal 2 m deep thaw layer. T h e r m o c o u p l e s were installed in two c r e v i c e s (10 m deep and 0.3 m apart) and their emfs were r e c o r d e d by m e a n s of a model G Z P - 4 2 m i r r o r g a l v a n o m e t e r with a model P P - 6 3 potentiometer. In one c r e v i c e was also installed a linear e l e c t r i c h e a t e r with a power PN = 93 W p e r running m e t e r . The ice content i was calculated f r o m the equation of heat balance i=

O'S6PN'~--Qnkg/ m3 79.6V

(i)

with the heat Qn, which is expended on heating the m a s s i f within the radius rli m = r ~ ' / ~ of the t e m p e r a t u r e field induced by the heat s o u r c e , calculated f r o m the t e m p e r a t u r e change At at a distance r f r o m the h e a t e r [1]:

At=

0-86PN'~ i 2a~,

--1 --x, x e dx,

(2)

r

2VZ,.

Here V denotes the volume of thawed m a s s i f p e r unit c r e v i c e length (m3/l. m.), m denotes the r a d i us of the thaw zone, T denotes the thawing time, i . e . , the period of h e a t e r action (h), and T1 denotes the lag time of the t e m p e r a t u r e wave c o r r e s p o n d i n g to r 1 : r = 0.3. The values of t h e r m a l conductivity k and t h e r m a l diffusivity a were selected in a c c o r d a n c e with the m a s s i f lithology. A c o m p a r i s o n between r e s u l t s of ice content determinations and data obtained by p r o s p e c t i n g has d e m o n s t r a t e d the applicability of the p r o p o s e d method to geocryological field studies. An engineering v e r sion of this method will facilitate field d e t e r m i n a t i o n s of t h e r m a l c h a r a c t e r i s t i c s by means of a constantpower cylindrical probe or by means of pulse heating under the exclusion of phase t r a n s f o r m a t i o n s . LITERATURE 1.

2.

CITED

A. D. M e i s e n e r , in: F r e e z i n g P h e n o m e n a in Soils [Russin translation], Izd. IL, Moscow (1955). Z. A. N e r s e s o v a , in: P a p e r s on L a b o r a t o r y Studies of Fozen Soils [in Russian], Izd. Akad. Nauk SSSR (1963), Ed. 1.

Original a r t i c l e submitted J a n u a r y 30, 1970; a b s t r a c t submitted J a n u a r y 4, 1972.

927

SYMMETRY QUANTITIES D.

P.

OF

FUNDAMENTAL

AND

THERMODYNAMIC

RELATIONS

Kolodnyi

UDC 536.70

The s y m m e t r y of fundamental t h e r m o d y n a m i c quantities and r e l a t i o n s between t h e m is r e v e a l e d and analyzed as a s p e c i a l c a s e of s y m m e t r y a p p a r e n t in v a r i o u s b r a n c h e s of the natural s c i e n c e s [1]. Such a s y m m e t r y can s e r v e as a tool f o r d i s c o v e r i n g new quantities and r e l a t i o n s as well as f o r simplifying the d e r i v a t i o n and the p r e s e n t a t i o n of n u m e r o u s t h e r m o d y u a m i c equations, e s p e c i a l l y differential ones [2 ]. A total of 16 quantities and 24 r e l a t i o n s (definitions, equations of the F i r s t law, and equations of the Second law of t h e r m o d y n a m i c s } a r e c l a s s i f i e d into 10 kinds (of two t y p e s : A and B) of 4 e l e m e n t s each. Quantities A' include volume V, e n t r o p y S, p r e s s u r e P, and t e m p e r a t u r e T; quantities A" include latent e n e r g y (PV) and bonded e n e r g y (TS), e a c h with a "+" and a " - " sign; quantities A " include work, heat, available work [3], and f r e e heat [2]; quantities B" include internal energy, enthalpy, f r e e enthalpy, and f r e e energy. T y p e - A equations r e l a t e quantities A"A ', A~A ~, A~B ~, and A " A ' ; t y p e - B equations r e late quantities B"A n' and B"A'. F o u r e l e m e n t s in each kind a r e a r r a n g e d and consecutive n u m b e r s a r e a s s i g n e d to t h e m : 1, 3, 5, 7 to t y p e - A quantities and 2, 4, 6, 8 to t y p e - B quantities. The e l e m e n t s with the s a m e index n u m b e r f o r m a figure; the r e s u l t a r e configurations of 8 f i g u r e s (type A) with 7 e l e m e n t s each and 4 f i g u r e s (type B) with 3 e l e m e n t s e a c h (shown in Table 1). The s y m m e t r y c o n s i s t s in an i n v a r i a n c e of configurations with r e s p e c t to a group of 8 t r a n s f o r m a tions [1, p. 33] mapping one figure into another of the s a m e type with the kind of e l e m e n t retained. Since a figure c o n s i s t s of quantities as well as r e l a t i o n s , hence a change of quantities ~ccording to s o m e rule (code) by a c e r t a i n t r a n s f o r m a t i o n yields an a u t o m a t i c mapping of the r e l a t i o n s and equations d e r i v e d f r o m them, which s u b s t a n t i a t e s the method p r o p o s e d by the author f o r t r a n s f o r m i n g the equations of t h e r m o d y n a m i c s [2]. In addition to the null t r a n s f o r m a t i o n , by which each figure is m a p p e d into itself (with a p o s s i b l e change of sign at the quantities of the p a i r s P, V and T, S), t h e r e a r e t h r e e b a s i c kinds of t r a n s f o r m a t i o n s (I, H, and III) c o r r e s p o n d i n g to the analogies: between single and r e p e t i t i v e p r o c e s s e s (I), between adiabatic and i s o t h e r m a l p r o c e s s e s (II), between m e c h a n i c a l and t h e r m a l quantities (II1). The l a s t of t h e s e analogies is of a p u r e l y f o r m a l c h a r a c t e r , and extending it to the r o l e of s y m m e t r i c quantities would lead to s u b s t a n tial e r r o r s . The r e m a i n i n g t r a n s f o r m a t i o n s r e p r e s e n t combinations ( s u c c e s s i v e applications) of the f o r e going o n e s : I I x I = I x l l , IIIxI=IIxIII, IIIxII=IxHI, IIIxIIxI=IxIIxIIL Distinctive is the s y m m e t r y of I, which e n c o m p a s s e s not only the fundamental r e l a t i o n s but a l s o the equations of state for an ideal gas, the p o l y t r o p i c equations and, s p e c i f i c a l l y , the adiabatic equation. Table 2 l i s t s the n u m b e r s of f i g u r e s into which f i g u r e s 1-8 a r e mapped by each of the 8 t r a n s f o r m a tions. Table 3 shows the combinations and also that any of t h e m r e s u l t s only in t r a n s f o r m a t i o n s among those 8, which, t h e r e f o r e , f o r m a group. The s y m m e t r y in t h e r m o d y n a m i c s is of the s a m e f o r m as the g e o m e t r i c a l s y m m e t r y in a fourtho r d e r plane with 8 t r a n s f o r m a t i o n s : 4 r e f l e c t i o n s and 4 r o t a t i o n s . Table 4 shows the t h e r m o d y n a m i c configurations s y m b o l i c a l l y , as a s t a r of 8 r a y s 45 ~ a p a r t with odd (type A) and even (type ]3) ones alternating, a l s o the c o r r e s p o n d e n c e between the 8 t h e r m o d y n a m i c and the 8 g e o m e t r i c a l t r a n s f o r m a t i o n s . LITERATURE 1.

2. 3.

G. D. D. the

CITED

Weil, S y m m e t r y [Russian t r a n s l a t i o n ] , Izd. Nauka, Moscow (1968). P. Kolodnyi, Inzh. Fiz. Z h . , 17___,No. 6 (1969). P. Kolodnyi, "Available work in a t h e r m o d y n a m i c p r o c e s s , " in: S c i e n t i f i c - R e s e a r c h P a p e r s of Textile Institute in Tashkent [in Russian], T a s h k e n t (1956), Ed. 3.

Institute of the Textile and the Light I n d u s t r i e s , Tashkent. a b s t r a c t s u b m i t t e d F e b r u a r y 2, 1972.

928

Original a r t i c l e submitted June 19, 1970;