A nonsteady heat diffusion problem with spherical symmetry

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Printed in U. S. A.

A Nonsteady Heat Diffusion Problem with Spherical Symmetry*

M. S. PLESSET AND S. A. ZWICK

California Institute of Technology, Pasadena, California (Received August 30, 1951)

A solution in successive approximations is presented for the heat diffusion across a spherical boundary with radial motion. The approximation procedure converges rapidly provided the temperature variations are appreciable only in a thin layer adjacent to the spherical boundary. An explicit solution for the tempera-ture field is given in the zero order when the temperatempera-ture at infinity and the temperatempera-ture gradient at the spherical boundary are specified. The first-order correction for the temperature field may also be found. It may be noted that the requirements for rapid convergence of the approximate solution are satisfied for the particular problem of the growth or collapse of a spherical vapor bubble in a liquid when the translational motion of the bubble is neglected.

I. INTRODUCTION

APROBLEM

countered if one considers the dynamics of a vaporof nonsteady heat diffusion is en-bubble in a liquid. As the size of the en-bubble changes, heat flows across the moving liquid-vapor interface. The liquid is assumed here to be nonviscous and in-compressible, and the thermal conductivity k, density p, and specific heat c of the liquid are assumed to have insignificant variation with temperature. The tempera-ture T in the liquid then satisfies the equation

1 dT

1

AT=

D dt

k

Here D= k/ pc is the thermal diffusivity of the liquid, n=n(t) is the heat source per unit volume in the liquid which is taken to be a function of time only, and dT/ dt

* This study was supported by the ONR.

denotes the particle derivative so that dT / dt= 8T/ at+v VT,

where v is the liquid velocity which in general varies with position and time.

II. FORMULATION OF THE PROBLEM

It is advantageous to transform Eq. (1) from Eulerian to Lagrangian coordinates. It will be assumed that the motion possesses spherical symmetry; i.e., the vapor bubble is spherical, and its radial motion is sufficiently rapid that any translational motion may be neglected. The Eulerian coordinates will be chosen as (r, t) with origin r= 0 at the center of the bubble. If R(t) is the bubble radius at time 1, Lagrange coordinates may be defined by

h= (1/3)[e

(2a)

1=1, _(2b)

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96

M. S

PLESSET AND S. A. ZWICK

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ZEROORDER SOLUTION

It is convenient to introduce in Eq. (7) a new time variable T defined by

T =

f

.R4 (t)dt.

o

Equation (7) -now becomes, to the first order in k/R3,

cry

1 au

it dn 4h a2U

_

D Or k dr R3 am

since

0 R4(1+ 4h/1!)

to this order. By the usual procedure of successive approximations, one 'sets

U U°±U'd-

T=T°-FT'+ -

,

mation in powers of the perturbation parameter lz/R°. where the superscript denotes 'the order of the approxi-The zero-order approximation, U°, is thus determined by

a2u0

auo h

=0,

(12)

---F-am D' Or

kdr

with the boundary conditions (6), (8), and (9). The appropriate solution of Eq. (12) is readily found by taking the Laplace transform on the variable T. If

u(h, s)-= U°(h, 7-)} f e--a*U°(h, 7-)dr,

0 cr(s) = n(7) , f(s)= { F(r) ; then

&u .s

its

dig D

it

---u= -

c:r(,$)7 (13)

where use has been made of Eq. (6) and the specification that n(0)=0. One has further, from Eqs. (8), and (9),

(du/ dh)h..-,,= (D/ k)cr(s), (14)

(d2U dh2) h = f(s) (15)

The required solution of Eq. (13) is

u= (D/ s)f(s)-exP{ - h( / D)i h(D/k)a-(i), so that

du

- (D/ s)i.f(s) exp - h(s/ D)i + (D/ k)o-(s). (16) dh

(a2u/0k2) &AI = (i/R2)(377 ar),-R(e)= F(t). (9) The diffusion problem thus defined can be solved by successive approximations, if the assumption is made that U and T vary appreciably only in the region

1K<R3(1), (10)

Physically, for the vapor bubble problem, this assump-tion is made plausible by the fact that not only is the heat capacity much greater in the liquid state than in the vapor statehut the thermal diffusivity is also much

smaller.

and Eq. (1) can now be written as

ak oh

D at

k

a (aT)

137.'.

If at t = 0, the temperature at r= co is To, then at a later time t the temperature at infinity is T.= To+ (D/ k)n(c), where n(0)=0. It is convenient to introduce a function

t) such that

au/ak= T- To,

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and Eq. (3) may then be written

a j

0

02y

1 aui

Ohl ak2

D at1

k

so that by integration

a2u

au

it

OW7 (5)

,Oh2 D at

k

where CO is an arbitrary function of time. From Eq. (4), U=

f

(T To)dit+ x(t),

and the function x(t) may be chosen so that 0(i)=0, and also so that

U(11, 0)=0, (6)

if it is assumed that the temperature at t = 0 is To every-where. Equation (5) now becomes

a2v

18(1

h

ea_

₍₇₎

DO!

k

-In addition to Eq. (6), one has the boundary condition at infinity

(8U/e3h),,_= (D/ k)n(t). (8)

The further boundary condition required to fix the

temperature field may be given by the temperature gradient at the bubble wall, r= R(1), as a function of time. One then has

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HEAT DIFFUSION PROBLEM

From Eq. (16), one finds

2-41du/dh}=

au./

an= T°(h, r) To

F(i.)

= (D1k)n(7) (D/

6f

0 (7 t)'

X exp 112 Id

4D(r)r

If one sets

0(h, 7)= T To (D/ k)n(r)= T T.,

then

F(r)

0°(h, 7)=

(D/r)if

(7.-0-1 h2

xexpi

14..

(17') 4D(r

Thus, the difference between the temperature at the spherical boundary, 7'°(0, r), and the temperature at infinity, T., is given by

Jo' F(g-)di- .

00(0, r)= (D/ r)i

.

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o (7-0

One also has from Eq. (16) for h=0

0°(0,. 'T) = 2-1{ (D/s)V(s)1,

so that the inverse Laplace transform of Eq. (16) may be Written in an alternative form as

f 0°(0,

e8(k; [h/(47D)1]

0 (7 r)1

XexpiI

dr. (19)

4D (r

In terms of the original time variable, one has, for

example, from Eq. (18)

T°(0, t)T0=-2 (D/k)77(t)

ftR200(aT/ar),R(2)

(D/r)i

-

dx. (20)

jt

: R4(y)dy

If the variations in R(t) are sufficiently small, Eq. (20)

simplifies to T°(0, t) To= (D/k)n(i)

ft (avar),..R(4

.(D/r)i

- dx, (21) 0

0x) I

(17)

which represents the "plane approximation" obtained if the curvature of the boundary r= NO is completely neglected.

WITH SPHERICAL SYMMETRY

97-It may be noted, when

T°(0, r) To (D/k)n(r)=0°(0, r)

is a monotonic function of 7, that one obtains from Eq. (19) the inequality

0°(h, r)

<erfc{h/ (4Dr)il

60(0, r)

IV. FIRSTORDER CORRECTION TO THE SOLUTION

If one continues the procedure of successive approxi-mation, the right side of Eq. (11) is now considered as determined from the zero-order solution U°,- and the first-order correction U' is determined by

02U'

1 au'

4h a2u0

---= G(h, 7).

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Dôr

R3 8h2

The boundary conditions for U'(h, r) are

U' (h, 0)= (au /a1.0=(a2c /

O. (23)

From Eq. (4), the first-order correction, T', to the

temperature field is given by

T' = a ui / ak.

Denoting the Laplace transforms of U'(h, 7), G(h, 7) by v(h, s), g(h, s), respectively, one has

d2v s

----v= g(k, s),

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dh2 D

with the boundary conditions

(dv/ (d2v/dh2)1,.,0= O. (25)

By Eq. (22), G(0, r)= 0, and the solution to Eqs. (24) and (25) is readily found to be

v(hs)

so that dv 1

--dl 2

1D

leocemif

e.c./D)tex, s 2 s h h

±eh(810 f ei(811)?1 ex, s)dx

co h(810 f e..x,(81D)ig(X, S)C1X 0 co

j'

c(814")+,g(x,-fx(819)ig(x, s)dx

dx +e-h(")i f e-5(81116g(x, s)dx . (26)

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98

For h= 0, one has

I do\

k s)dx= { 2' (0, r)}.

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dhlr,0o

From Eq. (27), the first-order correction to the tem-perature at the spherical boundary is

T'(0,

r)=fc.'

dxf 7 G(x,

0 0

x exp[ xV4D(r

)]

X

[4713(7-- 811

By Eq. (22) and the definition of 00

4h ar(h, 7) G(h, 1-) =

10(r) Oh so that Eq. (28) becomes

a0°(x, T'(0, r) =

dxf

4X2 0 0 R3(t) ax

expr xV4Der

0]

X

[47D(7 9i

and, if it is permissible to interchange the order of the integrations, one gets

f0C0

a00(x,

T'(0, r):= 2fo

R3(E)DrDer 03Ji ax

X exp[ x2/4D (7 )]7C2dX. (29)

Differentiation of Eq. (19) gives

ae°(x, E) 1

if

t) a

Ox

(47D)i0

Ox

X Ix exp[x2/4D(Eradr.

Substitution of this relation in Eq. (29) leads to the result T)= 2

f

r

R3(E) 1E eV) i.)[

_3-14.

(T r)1

One may obtain the following inequalities from Eq. (30)

M. S. PLESSET AND S.

A. ZWICK

when R is a monotonic increasing function of

time-R3(T) (7"

14< T'

T)<

1

r7

r(0,

fr

r(0,

dr. (31)

0

R3(P)(r-When R is a monotonic decreasing function, the sense of the inequalities should be reversed. It may be noted in Eq. (31) that

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_{D\ I 7 0130,}

7

4= D

F (04,

I

Jo 0

where F(-) is defined by the boundary condition of Eq. (9). When R is a monotonic increasing function, Eq. (31) thus gives

F(r)4

(0, 7)<

f

F()4,

(32)

D 4D

R3 0 3R03 0

where R0= R(0). The upper bound in Eq. (32) repre-sents, of course, a poorer estimate than that given in Eq. (31).

V. CONCLUSION

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The approximations developed here have been applied by the authors to the problem of the growth of a vapor bubble in a superheated liquid. For this specific problem, one can examine in detail the validity of the assumption of the thin "thermal boundary layer" which has been justified previously only in general physical terms. Such an examination of the predicted temperature field shows that the zero-order approximation, as given by Eq. (17) or (17'), is sufficient. Therefore, an explicit expression for the first-order temperature correction at any point in the liquid has not been given here, although it may be found from Eq. (26). The first-order temperature correction at the boundary r= R(t) is given by a fairly simple expression, and the given bounds upon it provide a convenient estimate of the rapidity of the convergence of the approximation theory.

The approximation procedure presented here is not limited to heat diffusion across a spherical vapor bubble in a liquid. The theory applies without alteration to diffusion across any spherical boundary with radial motion in a fluid, provided the thin "thermal boundary layer" approximation is valid. It is to be emphasized that effects of any translational motion of the spherical boundary have not been considered. For the case of the

vapor bubble in a liquid, the solution is therefore

applicable only over time intervals so short that no significant translation can take place.