ABSTRACTS

DETERMINING

THE

OXYGEN

HIGH-TEMPERATURE

COMBUSTION

FROM

TEST

COAL

PARTICLES

S.

DIFFUSIVITY

DATA

V. B u k h m a n

ON

THE

and

PRODUCTS

BURNING

E.

IN

OF

Nurekenov

UDC 533.15+662.62

It is shown in the a r t i c l e that, if data on both the combustion time and the t e m p e r a t u r e of p a r t i c l e s a r e given, a combined p r o c e s s i n g of these data can lead to an elimination of the s t o i c h i o m e t r i c r a t i o and yield a f o r m u l a for the oxygen diffusivity in h i g h - t e m p e r a t u r e combustion products: d

L3/4 [QI+Q _}_p~i

D~=0.375 p~ln (1 nu x~o)(q, - - q~) 1 -.{-L3/4

~

]

(q~--2q2) ,

(1)

with pp, T, d denoting r e s p e c t i v e l y the density, the combustion t i m e , and the d i a m e t e r of a p a r t i c l e ; p denoting the gas density; x denoting the r e l a t i v e m a s s concentration of oxygen; L = T ~ / T p denoting the ratio of s t r e a m t e m p e r a t u r e to p a r t i c l e t e m p e r a t u r e ; Q1, Q2 denoting r e s p e c t i v e l y the convective and the radiative t h e r m a l flux emanating f r o m a p a r t i c l e ; and ql, q2 denoting the heat of combustion which p r o d u c e s CO 2 and CO respectively. All quantities with the subscript ~ pertain to the boundary layer outside a particle and are referred to the temperature of the oncoming stream. Formula (i) has been derived from the heat balance and the material balance on the basis of diffusive combustion in a stationary medium, taking into account the Stefan flux as well as the nonisothermal conditions. Formula

(i) is used for evaluating the test data of various authors on the combustion

of coal dust.

When only data on the combustion rate of coal particles are given, the effect of kinetics and the indeterminacy of the stoichiometric ratio are eliminated from the analysis by processing the data obtained for temperatures above 1300~ It is assumed that combustion at the coal surface produces CO. Furthermore, the effect of secondary reactions at high temperatures is excluded from the analysis by taking data at low oxygen concentrations (02 < 3%) or at high values of the mass transfer coefficient (fine particles). Another advantage of data obtained at low concentrations is that they reflect combustion under almost nonisothermal conditions. The results indicate that at high temperatures (up to 1670~ the oxygen diffusivity in combustion duets can be described by a power equation with the exponent n = 1.75.

pro-

Kazakh S c i e n t i f i c - R e s e a r c h Institute of Power, A l m a - A t a . 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. 22, No. 5, pp. 930-940, May, 1972. Original a r t i c l e submitted June 24, 1971; a b s t r a c t s u b m i t ted N o v e m b e r 11, 1971. O 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 ~'est 17th Street, New York, N. Y. I00ll. 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, without written permission of the pt~blisher. A copy of this article is available from the publisher for $15.00.

652

SOLUTION PRODUCTS Z.

F.

OF

THE

WITH

MIXING

MORE

PROBLEM

PRECISE

FOR

PETROLEUM

BOUNDARY

CONDITIONS UDC 622.692.43:532.72

Karimov

Recent industrial and l a b o r a t o r y e x p e r i m e n t s have shown that, during sequential pumping of various p e t r o l e u m products through the same pipeline, a c e r t a i n quantity of mixture (so-called p r i m a r y mixture) is f o r m e d a l r e a d y in the inner routes in the head pumping station. The quantity of this mixture, depending on the complexity and the length of suction routes as well as on the hydrodynamic conditions in the suction lines of pumps, m a y be of the same o r d e r as the quantity of mixture f o r m e d in the main segments of some pipelines. This is an important factor in the subsequent mixing of p e t r o l e u m products. The authors analyze a p r o b l e m the solution of which will allow one to e s t i m a t e the effect of p r i m a r y mixture on the mixing rate of s u c c e s s i v e l y pumped products flowing in a pipeline and to determine the size of the full m i x t u r e at the endpoint of a pipeline. It is a s s u m e d that at time t = 0 the p r i m a r y m i x t u r e occupies a pipeline s e g m e n t of length 2a f r o m the entrance point and that the e m p i r i c a l relation c 2(x, 0) = 1/2 (1 + sin (~-x/2a))represents the concentration distribution of liquid displaced in the zone of p r i m a r y m i x t u r e . In a s y s t e m of coordinates with the origin 0 at the volume c e n t e r of the p r i m a r y mixture and with the x - a x i s coinciding with the pipeline axis, then the mixing p r o c e s s at time t >> 0 can be d e s c r i b e d by the following s y s t e m s of equations based on the t h e o r y of turbulent diffusion:

Ocio_T~OxxO(KiOCi~ax], tE[0,~), xE[a, ~ ) at i : l, xc[--a~a], i = 2 , x C(-- o~ ,--a], i=3;

cl(x, 0)-~ 1; c~(x, 0)~

l-t-sin-~a ; c~(x,O) =0;

Oct (~, t) Ox -O;cl(a,t)=ce(a,t); Oq(a,t) ax

Oc~(a,t) --

-O~

ac2(--a,t) '

(1)

c2(--a,t)=c3(~a,t); ac~(--a,O.

ax

ax

(2)

Oc3 ( - c r

",

ax

~ o.

At t >> 0 it is legitimate to a s s u m e K i = K = eonst and the solution to the p r o b l e m will be c~(x, i)= 1 - - ~ eri a + e r f v ) T 8 e _~_[~-t*erf(u_iw)

~[e

e r f ( v - - i w ) + e '~ erf(v+iw)

q_ei~Perf(u_~iw)]};

(3)

l I-j----~[erI(--u)--erfvJ-j--~e-sin~----18e--w'{[e--i*er'(-u-iw) 1

1 [erf(_

c8 ( x , t) = ~" L

u)

w2

gx

+erf(--v) J--le-"{[e'r

Equations (3)-(5) describe the concentration distribution of liquid displaced in the product s e p a r a t i o n zone and, t h e r e f o r e , allow one to determine the size of other than standard m i x t u r e s as a function of time t and p a r a m e t e r s 2a, K. NOTATION

ci(x, t)

is the dimensionless concentration of liquid displaced f r o m a pipeline at a given s t r e a m z one;

u = (x-a)/2~fKt; v = (x + a)/2~fKt; w = w~fKt/2a; ~p = Ir(x + a)/2a; B a s h k i r " F o u r t i e t h A n n i v e r s a r y of the October Revolution" State University, Ufa. Original a r t i c l e submitted March 1, 1971; a b s t r a c t submitted November 4, 1971. 653

=

~(x-a)/2a; is the length of primary mixture zone; is the effective turbulent diffusivity, m2/sec; is the time counted from start of pumping, sec.

2a K t

RELATION

BETWEEN

CONDUCTIVITY

OF

THERMAL

AND

ELECTRICAL

GRAPHITE

A. I. Lutkov, B. and V. I. Volga*

K.

Dymov,

UDC 536.63.546.26-162

It is well known that the thermal conductivity and the electrical conductivity of graphite do not follow the Wiedemann-Frantz law. Heat transfer in graphite is effeeted by phonons, while electrical conduction is due to the flow of electrons and holes. Nevertheless, several authors have noted that at room temperature the product of thermal conductivity and electrical resistivity remains reasonably constant. No attempt was made so far to determine how these properties are related at high temperatures. The article presents the results of thermal conductivity k and electrical resistivity p measurements as well as calculated values of their product k. p within the 80-2500~ range of temperature for synthetic graphite with a density ranging from 1.0 to 2.26 g/cm 3. It has been established here that at low temperatures the values of k. p for various grades of graphite vary widely. At room temperature the values of k.p for the tested grades are closer together. Finally,at T > 1500~ the value of k 9 p is 0.34-0.38 V2/deg and independent of the temperature for all tested grades except those of the lowest density (i.0 g/era 3) and the highest density (2.26 g/era3).

MEASURING OF THE

DROPLETS

THE

DISPERSION IN

THE

THROAT

LIGHT-SCATTERING V. V. Ushakov, A. and B. A. Gusev~

AND OF

THE A

VELOCITIES VENTURI

BY

METHOD S.

Lagunov,

UDC

621.928.97

The mass distribution of droplets with respect to size was analyzed simultaneously at the center of the Venturi throat (section I) and behind the diffuser (section If), for various modes of water spraying after discharge from the center nozzle in the converging channel. The test conditions were such as to preclude either coagulation or fragmentation of droplets between the two illuminated sections. Functions gi (r) and g2(r) for one spray mode (v i = 120 m/sec, m = 0.02 kg/m 3, L = 35 ram) are shown in Fig. I. *Original article submitted December 9, 1970; abstract submitted November 15, 1971. ~Lenin Polytechnic Institute, Kharkov. Original article submitted July 16, 1971; abstract submitted November 15, 1971.

654

c~ (r) Fig. i. Mass distribution of droplets with respect to size: i) in the Venturi throat; 2) at the diffuser exit, for L = 35 mm and v I = 120 m/see.

2 4,0

0

60

r

A higher degree of d i s p e r s i o n in the s e g m e n t between sections I and II is explained by the gas c a r r y ing m o r e effectively s m a l l droplets than l a r g e droplets through the Venturi throat, while the velocity of all droplets is a l m o s t the s a m e and equal to the gas velocity in the diffuser. F r o m the continuity equation talel (r) dry l (r) 31 = m~g~ (r) dry 2 (r) $2,

(1)

and considering that vt(r ) ~ v 1 at r ~ 0 and v2(r ) ~ v2, we have vl (r) : vz gl (r)

g2 (r)

r

lira gz (r___) - ~ 0 gz (r) "

(2)

The calculation of vi(r ) by f o r m u l a (2) is e a s y b e c a u s e , within the range of r = 2-10 #, m e a s u r e m e n t s of function g(r) by the l i g h t - s c a t t e r i n g method a r e still v e r y a c c u r a t e and the ratio v l ( r ) / v 1 approaches unity v e r y slowly. NOTATION r g(r) v

v(r) L m mg (r) dr S

is is is is is is is is

the the the the the the the the

radius of droplet; m a s s distribution of droplets with r e s p e c t to size; velocity of gas at a given section; velocity of droplet with radius r; distance between discharge o r i f i c e s in nozzle and entrance to throat; s p r a y intensity; m a s s of droplets of radii within the r, r + dr range, p e r unit of illuminated volume; section a r e a of the Venturi.

Subscripts 1 2

r e f e r s to section I; r e f e r s to section I/.

EFFECT

OF

PROPAGATION

STRUCTURAL OF

POROSITY

ULTRASONIC

CAPILLARY-POROUS B. N. Stadnik, M. and L. N. Belyi

ON WAVES

MODEL

BODIES

F.

Kazanskii,

THE THROUGH

UDC

534.18

The p h y s i c o m e e h a n i c a l p r o p e r t i e s of c a p i l l a r y - p o r o u s bodies a r e quite a c c u r a t e l y described by a m o d e l - s y s t e m of s p h e r i c a l elastic p a r t i c l e s . The behavior of an e l e m e n t a r y c a p i l l a r y - p o r o u s cell is d e t e r mined by the elastic f o r c e s developing at the contact points between individual elements as well as by the friction f o r c e s appearing as a r e s u l t of the relative displacement of p a r t i c l e s .

Engineering Institute of the Light Industry, Kiev. Original article submitted May 2 0, 1971; abstract submitred November

ii, 197][. 655

By analyzing the propagation of an elastic wave along a linear chain of c a p i l l a r y - p o r o u s ceils, it is possible to show that, in the case of a negligible attenuation, the velocity of this wave does not depend on the size of p a r t i c l e s . A m e a s u r e m e n t of the propagation velocity of ultrasonic waves in fractionalized quartz sand has shown an insignificant (24%) i n c r e a s e in the velocity of a 70 kHz wave over a significant range (factor 7.6) of particle s i z e s . A linear chain of p a r t i c l e s behaves like a m e c h a n i c a l low-pass filter which does not t r a n s m i t elastic waves above the cutoff frequency. The attenuation of elastic waves at frequencies above cutoff is d e t e r mined not by internal friction but by s t r u c t u r a l p r o p e r t i e s of the c a p i l l a r y - p o r o u s body. It appears that the cutoff frequency of the model c a p i l l a r y - p o r o u s body here is proportional to the wave velocity and inversely p r o p o r t i o n a l to the particle size. A c c o r d i n g to calculations, the cutoff frequency of fractionalized quartz sand lies within the lower u l t r a s o n i c r a n g e .

CHOICE

OF THE

TIME

BOUNDARY-VALUE LIEBMANN

INTERVAL

PROBLEMS

IN SOLVING

BY THE

METHOD UDC 536.212

I . D. K o n o p l e v , R . V. M e r k t , and S. B. Tisheehkin

A wide range of s c i e n t i f i c - t e c h n i c a l p r o b l e m s can be reduced to solving equations of the following kind: 0

Ou

Ou

(i)

with boundary conditions 0u

(2)

u = f (xl, x~, x3, ~ini)

(3)

and initial conditions The e r r o r of the solution to Eq. (1) obtained by the Liebmann method depends, with all other f a c t o r s the s a m e , on the choice of space interval Ax and time interval AT. With an a l r e a d y selected space internal Ax, the a c c u r a c y of the solution will be affected principally by the choice of time interval AT. A s h o r t e r AT will r e s u l t in a b e t t e r a c c u r a c y , but the computations b e come m o r e laborious. Consequently, it is n e c e s s a r y to Aq a b derive to devise a p r o c e d u r e for selecting the time interval AT to ensure the required a c c u r a c y at a minimum c o m p u t a o,t8 tion effort.

i

o,,~

~

o/o qoe

In p r a c t i c a l simulations the time interval AT is usually taken equal to 0.01-0.1% of the transient period in a p r o c e s s . The interval is then lengthened during computations depending on the t i m e - v a r i a t i o n of the potential and on the basis of past experience. The a c c u r a c y is evaluated by a c o m p a r i son of s e v e r a l solutions obtained with different time i n t e r vals AT.

~56

'4-Z-

qo2 o

qo

t,a

z,,

o

q8

t,6

~o

Fig. I Institute of Maritime Engineers, mitted November 15, 1971. 656

Odessa.

An analysis of function u = f(T) at the endpoints, for various values of the f o r c e - t o - c a p a c i t y ratio, has yielded

Original article submitted April 29, 1971; abstract sub-

c r i t e r i a f o r s e l e c t i n g the t i m e i n t e r v a l f o r a g i v e n a c c u r a c y r e q u i r e m e n t . a quantity G O according to the definition

This ratio is characterized

by

6Ta O~

(4)

Axo

~+8Ax

"

w h e n the p o t e n t i a l of the m e d i u m u m = K7 f o r TE (0, Ta) and u m = K7a f o r T~ 0"a, ~'), w i t h the f i e l d d e s c r i b e d b y E q s . (1)- (3). T h e c u r v e s of A T v e r s u s Go in F i g . 1 a r e u s e f u l f o r s e l e c t i n g the t i m e i n t e r v a l a s a f u n c t i o n o f G Oand of t h e r e l a t i v e e r r o r A 0 / 0 . H e r e ~ = ~ - / r a i s the d i m e n s i o n l e s s t i m e and 0 = u / k ~ - a i s the d i m e n s i o n l e s s p o t e n t i a l . C u r v e s 1 - 6 h a v e b e e n p l o t t e d f o r a r e l a t i v e e r r o r A 0 / 0 = 0.2, 0.4, 0.5, 0.6, 0.8, and 1.0% r e s p e c t i v e l y . T h e g r a p h in F i g . l a i s to b e u s e d f o r V E (0, (0.8 + G0)/G0), the g r a p h in F i g . l b i s to b e u s e d f o r ~ E ((0.8 + G0)/G0, (5 + 1.6G0)/G0). NOTATION i s the p o t e n t i a l ; a r e the c o e f f i c i e n t s ; i s the s p a c e c o o r d i n a t e s in (1), (3); i s the t i m e ; i s the r a t e of c h a n g e of p o t e n t i a l .

U

6, # , w xi T K Subscripts s m

AN FOR

r e f e r s to s u r f a c e ; refers to medium.

EFFICIENT THE

NUMERICAL

INTEGRATION

TWO-DIMENSIONAL

TRANSIENT

HEAT-CONDUCTION VARIABLE-PROFILE V.

S.

Petrovskii

PROBLEM DISK and

SCHEME

CONCERNING

OF E.

A GAS E.

A

TURBINE

Denisov

UDC 6 2 1 . 4 3 8 - 2 5 4 : 5 3 6 . 2 . 0 0 1 . 2

T h e p r o b l e m i s f o r m u l a t e d f o r a d i s k w i t h a c e n t e r h o l e . T h e o r i g i n a l e q u a t i o n is 1 Ot ( 02t O~ . . . . . Or2 + 7

- - - - a

02t )

Ot

" o-7 - +

Oz~/' ~ > 0

(1)

The boundary conditions are at r-- r2

Ot a (r2) Or -~ [ t - - t B (r~)];

(2)

Ot

(3)

at r = r 1 ~ - r = qht ~-q~d'~(v)+%; at

Ot al (r) On -~, [ t - - tB~ (r)];

z = fl (r)

at z = f2 (r)

0t

On - -

a 2 (r) ~

[t - - tB,

(r)]

(4) (5)

(n d e n o t e s t h e n o r m a l to the l a t e r a l d i s k s u r f a c e ) , Institute of Aviation Engineering, Moscow. m i t r e d N o v e m b e r 25, 1971.

O r i g i n a l a r t i c l e s u b m i t t e d J u n e 14, 1971; a b s t r a c t s u b -

-

657

F o r a n u m e r i c a l integration the curved domain is replaced by a C a r t e s i a n domain with the aid of the fractional-linear transformation z-- f, (r) = ,. and n ,= a~& (~) _ :i (ri A c c o r d i n g l y , Eq. (1) and the boundary conditions b e c o m e O} [OB Ot O't O't O~t - g / . 74-+2B

(6)

1 (c)t

Ot )

+ D2 02tl

(7)

(B = a~l/ar; D = oq/ar); Ot

Ot

a t ~=r,. --~-+B at ~l=O

~ (rD [t -- t. (r~)];

(8)

n -=~lt+~'t~(~)+q~";

(9)

Ot *~1(r) /\ D O~l -- ;L [t--tB' (r)]C~

~t

a2 (r)

at:~q=fx D 0~l =

k

[t--tB'(r)lc~

(10)

/ \

(11)

The solution is sought at the nodes of a uniform grid resulting from the intersection of straight lines ~i = r 2 - i h ~ (i = 0, 1,2 . . . . . n) and ~?j = ]h~ (j = 0, 1,2 . . . . . m) for Tk = kh r (k = 0, 1,2 . . . . . s). The l e n g t h w i s e - c r o s s w i s e integration is p e r f o r m e d in the following m a n n e r . The interval of i n t e g r a tion with r e s p e c t to T is divided into two steps. Over the f i r s t half 0.5h~ Eq. (7) is integrated with r e s p e c t to c o h m m s (along the }-axis). The difference r e p r e s e n t a t i o n is used here for the derivatives with r e s p e c t in l a y e r k + 1/2 and for the derivatives with r e s p e c t to V in layer k. Over the remaining second half 0.5h~the integration is p e r f o r m e d with r e s p e c t to rows (along the 7?-axis). Here the difference r e p r e s e n t a t i o n is used for the derivatives with r e s p e c t to } in l a y e r k + 1/2 and for the derivatives with r e s p e c t to V in l a y e r k+l. The s y s t e m s of equations

o,oL+/ -

+

vmi

+

o,oL+/- = 1-i+1

(12) 9

a r e solved by the elimination method. Here ai, bi, ci, aj, ~., and c: are independent of T J J and Fk+l/2 contain derivatives with r e s p e c t to ~ and ~ in the difference r e p r e s e n t a t i o n . J LITERATURE .

(13)

for tows

Functions

Fik

CITED

A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical P h y s i c s [in Russian], Nauka, Moscow

(1966).

SOLVING

CERTAIN

ON A N A N A L O G

PROBLEMS

IN THERMOPHYSICS

COMPUTER

K. I. Bogatyrenko, V. E . P r o k o f ' e v ,

O. T . I I ' c b e n k o , a n d O . N. S u e t i n

UDC

537.212.001

The solution of great many application problems in thermophysics by simulation is fraught with serious difficulties. The reason is that the use of existing models is essentially limited to solving direct Lenin Polytechnic Institute, Kharkov. Original article submitted June 15, 1971; a b s t r a c t submitted November 17, 1971.

658

p r o b l e m s in field t h e o r y . At the s a m e time, m o s t p r a c t i c a l p r o b l e m s are f o r m u l a t e d m o r e in t e r m s of r e v e r s e , i n v e r s e , or inductive p r o b l e m s . Consequently, often only an insignificant p a r t of the solution t o an application p r o b l e m can be obtained by simulation and the main p a r t of the p r o b l e m m u s t be solved by other m e a n s and methods of computer engineering. This a r t i c l e deals with the design and p r i n c i p l e s of analog c o m p u t e r s y s t e m s {ACS) for a d i r e c t solution of c e r t a i n application p r o b l e m s in t h e r m o p h y s i c s . The solution p r o c e s s is in this case a conti~uous one and does not r e q u i r e a u x i l i a r y computations. A typical functional s c h e m a t i c d i a g r a m of an analog c o m p u t e r s y s t e m is shown here containing an R C - n e t w o r k for simulating the test object and a device for setting the boundary conditions (DSBC). The latter component r e p r e s e n t s a known model for solving d i r e c t p r o b l e m s in field t h e o r y . An essential f e a ture of this analog c o m p u t e r s y s t e m is a control unit (CU) which, together with the DSBC sets the boundary conditions a c c o r d i n g to the values of the derivatives of field c h a r a c t e r i s t i c s obtained on the computing unit. In other words, the r e s u l t s of solving a d i r e c t p r o b l e m in field t h e o r y operationally on the R C - n e t w o r k are used for the e l e c t r o n i c simulation of such boundary conditions which will ensure that the analyzed p r o c e s s continues in the d e s i r e d direction. F o r illustration, the authors d e s c r i b e an analog c o m p u t e r s y s t e m designed on the basis of USM-1 c o m p u t e r elements and show the r e s u l t s of solving on it one p r a c t i c a l t h e r m o p h y s i c s p r o b l e m . The p a r t i t u l a r p r o b l e m c o n c e r n s a heat c a r r i e r flowing f r o m inlet to outlet between two coaxial c y l i n d e r s , and the t e m p e r a t u r e - t i m e c h a r a c t e r i s t i c is sought which will e n s u r e a transition of this s y s t e m f r o m initial to operating state in m i n i m u m time. The operating state of this s y s t e m is defined by the m a x i m u m s t e a d y state t e m p e r a t u r e of the heat c a r r i e r . The control p r o c e s s m u s t s a t i s f y the r e q u i r e m e n t that the relative displacement between the free ends of the cylinders do not exceed the allowable limit, this displacement being equal to the difference between their t h e r m a l elongations. Such a p r o b l e m is often encountered in applications as, for example, determining the s t a r t u p conditions in a s t e a m turbine. The optimum t e m p e r a t u r e - t i m e curve for the heat c a r r i e r thus found consists of two distinct r a n g e s . Within one range the t e m p e r a t u r e r e a c h e s its m a x i m u m , the other range c o r r e s p o n d s to a displacer, aent of the controlled s y s t e m to the boundary value of the m e a s u r e d p a r a m e t e r . The p r o p o s e d s t r u c t u r e of such an analog c o m p u t e r s y s t e m yields a basis for solving m o r e complicated application p r o b l e m s . The model s t r u c t u r e is here determined by the p r o b l e m formulation and should r e f l e c t the nature of the c o n s t r a i n t s as well as the r e q u i r e m e n t imposed on the simulated p r o c e s s .

CYLINDRICAL WAVES

AND SPHERICAL

IN THE

VELOCITY I. K.

OF

SHORT-TIME HEAT

Naval

and

THERMOELASTIC DOMAIN

WITH

A FINITE

PROPAGATION P.

F.

Sabodash

UDC 539.377:536.49

The authors analyze the excitation of o n e - d i m e n s i o n a l cylindrical and s p h e r i c a l t h e r m o e l a s t i c waves in elastic and isotropie cylindrical (spherical) l a y e r s of finite thickness as a r e s u l t of t e m p e r a t u r e jumps at the inside s u r f a c e of such l a y e r s . The initial t e m p e r a t u r e of a l a y e r is equal to the t e m p e r a t u r e of the ambient m e d i u m . The t h e r m o p h y s i c a l p r o p e r t i e s are a s s u m e d constant. On the a s s u m p t i o n that the t e m p e r a t u r e field follows the hyperbolic equation of heat conduction, the p r o b l e m r e d u c e s to integrating a s y s t e m of uncoupled differential equations (in dimensionless variables) c3~ v 8~ vv 3~ ~e

Original article submitted June 14, 1971; abstract submitted November

25, 1971.

659

0~0

~

O0

0~0

O0

o~-z+ T 9 0-T=M~r~ + ~

(2)

,

inside the l a y e r with the boundary conditions Ov

v

Ov

v

u~

at 0(~, x ) = H ( x )

a t : t ; = t o ; 0(~, x ) = O a t

~=Pol

and z e r o initial conditions av

00 = o -at x~.~o,

~ = - ~ y = o = 0--7 where . O=

T ~ To ~ , Tr TQ

cr --, a

[=

(3,t + 2~) czTr ~='

iZH-2~)~o

T~

c2t --, a

v:--,

u ro

M--

c

,

cq

(3~,+ 2~) aTo '

~-

(~,+2~)~o

, 7-~,+2~,r176176

v = 1 for a c y l i n d r i c a l wave and v = 2 for a s p h e r i c a l wave. System (1)-(2) is solved with the aid of the unilateral Laplace t r a n s f o r m a t i o n with r e s p e c t to time. An exact t e m p e r a t u r e distrs has been obtained for the inside of a sphere. In the case of a c y l inder, an exact t r a n s f o r m a t i o n is difficult b e c a u s e of the unwieldy e x p r e s s i o n s for the t e m p e r a t u r e and the d i s p l a c e m e n t in t e r m s of the Laplace o p e r a t o r . An a s y m p t o t i c t r a n s f o r m a t i o n is shown here for large values of the Laplace o p e r a t o r , c o r r e s p o n d i n g to s h o r t time p e r i o d s , and the solution b e c o m e s valid in the region of incident and multiply reflected (at both boundary surfaces) t h e r m o e l a s t i c wave fronts. NOTATION

T(r, t)

is the t e m p e r a t u r e ; is the radial component of the elastic disPlacement vector; is the t h e r m a l diffusivity; is the velocity Of heat propagation; is the radial coordinate; is the time; are the Lain6 constants; is the velocity of elastic waves; is the t h e r m a l expansivity; is the Heaviside function.

u(r, t) a eq r

t e oz

H(r)

A

THEOREM

ON

TRANSFORMATION O.

N.

THE

FINITE AND

Kharin

and

INTEGRAL

ITS V.

HANKEL

APPLICATION

S. Blinov,

UDC 519.47

The authors c o n s i d e r the finite integral Hankel t r a n s f o r m a t i o n of the f i r s t o r d e r , for which the following t h e o r e m is proved: Gubldn Institute of the P e t r o c h e m i c a l and Natural Gas Industry, Moscow. Original article submitted July 2, 1970; a b s t r a c t submitted November 9, 1971.

660

THEOREM.

Let functions

f(r) and g(r) satisfy the Dirichlet conditions !

on a close interval

[0, i]

r

(1) J r

r

, 0

*

Then 1 ~ if f ( r ) - - F(an), (2)

g (r)~ F(an)

2--; an

~F(b0), n = 0 2 ~ if f ( r ) ~ ( F ( b n ) ' n = 1,2 . . . . .

n

is known, then

t

i 4

g (0--

40

iI gF(bn)

(3)

b: ~

~176

F(bo), n = 1, 2 . . . . ;

3 ~ if f ( r ) ~ F ( c n ) , 1

Y (gtz) g,(r) ~-- Cn2

J1 (cn) s r[ (r) dr, hc=

(4)

0

with an, bn, and e n denoting the roots of characteristic the t h i r d kind r e s p e c t i v e l y .

equations

corresponding

to the first, the second,

and

C O R O L L A R Y 1. If the s u m f j ( r ) of the s e r i e s ~n~ 1 JoJ~(a.) (anr)

h(O=~

(a. ) " F ,~2j

(]=o,

I,a ....

)

(5)

is known, then the s u m fj+i(r) of s e r i e s (5) c a n be c a l c u l a t e d b y the f o r m u l a ]

9 dr

f'"

: .I-;- J < (') r

I6)

0

C O R O L L A R Y 2. If the s u m f j ( r ) of the s e r i e s

h (4 = 2

Jg (b~)

F(bn)

(j = o, ~, 2 , . . . )

(7)

is known, then fl+l(r) --d

i

i

r .t' ff[(rjdr-~-

r

COROLLARY

3. If the sum

+f

(8)

#.

0

0

fj (r) o f the series

Zo (Cnr) /j(,)

2

F(c.)

(/= o, 1, 2 ....

s~(~.l+S~(c.)

)

(9)

is known, then 1

r

I

h§ ( r ) = j'-~-,S'rf'(")dr + V1 r

COROLLARY different intervals

0

4. If the sum fj(r) of one of the series of r, then also fi+~ (r) =

,fie)

~/+1 (r) ~(0 ,/+~ (r)

S

fly (r) dr.

( 1 O)

0

(5), (7), (9) is defined by different functions (R~r.~l), ( o ~ . r". ~ " ~),

on

(ii)

661

where

1 r R ,j-pl~(e)"(r, = ~ d'~'- S rl~.e). (r) dr -- In r rr i) (r) dr -}- r (r), r

R R r ~drf,)

,(i)

(') =

-7-. q} (') e +r}g r

Here the value of function r

(12)

o

(e).

(13)

0

depends on the kind of the s u m m e d s e r i e s 0, 1

1_~2~ rn~i (r) dr, ~(r) =

o

(14)

I

0 The f i r s t value of r c o r r e s p o n d s to a summation of a s e r i e s like (5), the second value to a summation of a s e r i e s like (7), and the third value to a s u m m a t i o n of a s e r i e s like (9). The application of these r e s u l t s is illustrated in the p r o b l e m of heating a t h e r m a l l y insulated cylinder f r o m v a r i a b l e - p o w e r s o u r c e s u n i f o r m l y distributed within a c e r t a i n region of the cylinder.

662