Transient heat convection within a spherical film

TRANSIENT HEAT CONVECTION WITHIN A SPHERICAL FILM (0 NESTATSIONARNOI TEPLOVOI SHAROVOM SLOE) PMM Vo1.22,

No.3,

(Received

heat

solution

is

within

fluid

convection

1958,

V

pp.419-423

G. SEVRUK (Perm)

I.

An approximate

KONVEKTSII

25

April

obtained

for

trapped

1955)

the

problem

between

of

weak transient

two concentric

spherical

walls. which are maintained at constant temperature. The assumption is that at the instant of starting, the fluid is at rest, and is at a uniform temperature temperature

everywhere

and velocity

approximations, Rayleigh vection

of

expressed

number. on the

of

differing

from that

the

fluid

found

in these cooling

of

1. Statement heat convection

of Problem. We will

centric

of

spheres

temperature at constant We will

in fluid

confined

radius

of

well

of

terms the

the

fluid

deal within

is

the walls.

the

these

zero

The

and first

in powers

of

effect

con-

of

the

reviewed.

with the problem of transient the space bounded by two con-

R1 and R2 (RI < R2),

the

to

approximations,

T1, if, at the initial temperature T2 f T1. make use of

are

as expansions

Finally,

rate

but

instant

maintained

the

known equations

fluid

of

L?T’ at + (v VT’)

at constant

was at rest

free

and was

convection

= xA’7”,

&VV

=

[l

I. (i.ij

0

temperature and pressure where v is the velocity of the fluid, T’ and p’, reckoned from the values which they would have had under conditions of mechanical and thermal equilibrium at some mean temperature, p is the mean density

of

of

viscosity,

the

kinematic fluid,

the

and g is

We eliminate

p’

liquid,

v,

thermal the

6,

acceleration

by applying

x,

respectively,

expansion

due to

operation

587

are

and thermal

rot

the

gravity. to

(1.1);

coefficients

conductivity

thus

of

588

1. G. Sevruk

We introduce

the dimensionless 1

u=-v, where

1 = R2 -

coordinates

T’ Tz-Tl

5=

x -

Using

RI.

this

in what follows,

equations

of

the

a

variables,

notation

t+t

1

the

at the

(1.3)

nondimensional following

time

N,,E

$>

_

-!_ rot [u rot u] = IV&V,, NFr (‘t’p, -f)

div u = 0,

g@(Tz---l)P

(Grashof

-

V2

-

tw

[VTkl

(Prandt’l

No)

k=-!%

No)

(1.5)

g

If we reckon the value T of the fluid temperature from T1, then T1 and r = (TT1)/(T2 - T1). It follows that the temperature boundaries of t!je field is Zero at any O-.".. Divan _-.I"I,*I inatnnt nf timo _^ '_..._(

at the

initial

instant

it

is

r(r,

for

the

problem

T’ = T at whilst ....-....,Y

unity: t)=

The velocity of the fluid times,whilst at the initial stated

and

dimensionless

problem:

~7 = W?$ + N,,

Tthe

for

we arrive

rot u + rot rot rot u -

z

r

rl=-,

’

0,

T (E, 0) = 1

u at the boundary instant it is zero

f1.Q

is taken over the

to be zero at al. whole fluid as

at the outset: u(F, t) = u (r, 0) = 0

(1.7)

Assuming the ated from (1.4)

convection to be weak the required values can be evaluas a power series in Grashof numbers 12 1. As the Gras-

hof

enters

number only

called

the Rayleigh

expanded

in a series

u = 11,(XrriVcr)+

the

equations

number, of

the Rayleigh

the

problem we are

If we limit ourselves to the mat ions, we obtain the following

zero

and first

equations

)

Rayleigh

of

the

ATi = N,, 2

2. Zero

It less

Approximation. (1.9), heat

follows distance

Using

It

is

evident

describes the process of conduction(to be explicit from symmetry from the

spherical

centre

coordinates

combination for

can be

that

the

the

that

+ (UiDT”)

(1.9)

the

zero

(1.10)

approximation,

cooling of the heated T2 is taken > T1).

= rD (r, t),

5

of

Number approxi-

problem;

div u1 = 0

~rotul+rotrotrotul=[Orok],

equation molecular

in the looking

: =*,Cl)+ Tit.(.V,,,aVv,,) + fz (XFr‘YGr)~ + . . . ($.S)

u2 (NprN&Z+...,

azlj = M,, 2;

of

quantities number:

where

r is

the

fluid

dimension-

sphere.

first

equation

(1.9)

is

by

as follows

Transient

heat

convection

in

spherical

film

589

In order to be consistent with the postulated initial and boundary conditions, we require that the solution of (2.1) satisfies the following conditions:

Solving

by the method of Fourier.

k%” )sinnn(r

-t

A,exp(-

no =

we get -rl)

ra co9 nx)’ (2.2)

A,== &-

i > ( Pr ?%==l 3. Firat Approximation. 1. We will first find the velocity of the fluid. To do this we have to solve equations (1.10) taking (2.2) into account with the corresponding initial and boundary conditions.

Note that the fluid motion is symmetrical with respect to the vertical diameter of the sphere. We will introduce the dimensionless stream function $ = $!f fr, 6. t). For the components of the velocity ml in the spherical

system,

we will

have:

(3.1) We will

look for

a stream function

$ in the form

J, = f (r, t) sin28 Now putting obtain

(3.2) ii’ 2i

p

( yr--

From conditions f (rs,

1) =

f P2,

into

4

=

(3.2)

and the result

of this

into

(1. lo),

^“.‘. ,, ,_< .-%Z” 01 0 (LJ _z_+._$~~_i!Xj-r~ gx ) - Zt\f

(1.7) 0,

(3.11,

we get f’

h

0

the =

f’

following (r2,

2) =

we (3.3)

conditions: f(r,

0,

0) = f’(r,

0) = 0

(3.4)

(both here, and in what follows, the prime denotes differentiation with respect to r). We will look for a solution to the nonhomogeneous linear differential equation (3.5) in the following form: f(r, t) = 5

R, (r)Sk 01

wnere

eigen functions of the homogeneous problem corresponding homogeneous problem (3.3). (3.4). whilst Sk(t) are, so far, functions.

are

In

(3.5)

k-1

..L-_-

equation

(3.6)

Xl(k = 1.2.. ) are roots

of the equation

to the donundetermined

I.G. Sevruk

590

3 (r12 + rs2) cos h + (ra3 - r13) ‘xsin A -

and the constants

sin h

3 -

x

= 6rlr,

(3.7)

ck have the values

(3.8) c,(g)

=

kg k

f&o

(J is the Bessel Rk(r) (k = 1,2,..

= -

f-h,, (a& J-l,* O&b) - J-a,* ch,) iIf,

Ck@fQ -

bJ

.d- ~cpJ,,, (akd + C,(W_l,r (wl ak2

function). We may note that the system of functions ) is complete (see Ref. [ 3 1 1.

It can be seen on checking that the functions Rk(r) and (2R,/r* - R,‘? for k f II are mutually orthogonal and therefore if we multiply both sides Of (3.5) by (2Rk/r2 - Rk) and integrate with respect to r within the limits r1 and r2 we obtain

To find the integral on the left-hand side of (3.9) we multiply both sides Of (3.3) by Rk and integrate with respect to r from r1 to r2 [ 4 1 . Integrating by parts we shift the differentiating operation from the rearrangement we obtain the function f t0 Rk, and, after straightforward equation (&?n)

Now we solve the equation (3.19) for Gk(‘). taking the initial ditions (3.4) into account. The result of the calculation is t

COn-

(3.11) Hk (1) exp (akBt)dt s 0 expression for the integral is put into (3.9) and th’e second Introducing the Value Of Sk(t) SO there is solved for Sk(t). ._ -. into (3.5), we then arrive at a solution of equation (3.3) which satisfies all the conditions (3.4) in the form:

G, tt) = - exP (- h,‘t)

This equation obtained formally

f (r, t) = C A’, (4 R, W = -

5 h-=1

R, (r)

ak2exp (-- akat) * . Hk (t) exp (Irk%)dt f qg (0 0

(3.12)

Transient

Putting (3.12) as a series.

into

heat convection

(3.2),

in spherical

the stream function

We find the velocity components to the first numbers from the formulas

fila

591

can also

be expressed

approximation

in Rayleigh

(3.13) 2. We will now turn our attention to finding the fluid temperature. The second equation in (1.9), which in the first approximation yields the convective portion of the temperature distribution, can be written in spherical coordinates [(bearing in mind (3.1) and (3.2) 1 in the form

The initial and the boundary conditions of the problem (see section l), also taking into account the conditions under which the zero approximation temperature equation (see section 2) was solved. give We will

71 (r1, 8, t>= *I (r2, 6, q = 0, look for a solution of equation 71 =(p(r,

Now putting (3.16) into function gS must satisfy

(3.14)

T1 (f,

6,

(3.14)

0)

=

(3.15)

0

in the form (3.16)

t)cos6

we arrive

at the equation

which the

(3.17) From (3.15),

we have the following cp(rr, t) = ‘P(r2,

t)

Using the method of separation the problem (3. Y?) and (3.18).

=

0,

conditions ‘p

of variables

(PI

for 0 =

the function

d: (3.18)

0)

we find

the solution

of

Here, Ds (r) = (eir)-“’ Ir_*,, (‘irl) J*/s(Eir)- Js,, (E*rl)J_*/, (EirfI

(3.20)

are efgen functions of the homogeneous problem corresponding to the nonhomogeneous problem (3.17)_(3.18). and c i are roots of the equation,

Therefore temperature

in the first

approximation

we get for

the dimensionless

I.G.

592

where r o is

obtained

from

Sevruk

formula

(2.21,

and r 1 from

formulas

(3.16)

and

(3.1). Flow.

Cooling

Q flowing

4.

Heat

through

the

instance,

over

For this,

the

Time.

time

t from the

we make use of

where x is the coefficient bounding spherical surface surface.

In the

distribution

the

of of

approximation

is

1. We will now find the quantity of over a finite time interval t (for

boundary

given

start

the

fluid

cooling

process).

formula

heat conductivity of the fluid. ($1 is the the film, n is the outward normal to this we are

by the

of

heat

dealing

with,

this

fluid

temperature

formula

T = TX + Pa -

T,f (70 + ~~~~~~~1)

(4.2)

where T 0 and I 1 are determined from equations (2.2), If we put (4.2) into (4.11 and integrate we find;

Q=-+a--~)

;

(3.16)

and (3.1).

“n2[exp (- $+-i]

(4.3)

TX==%

Equation

(4.3)

asymptotically fluid. 2.

shows

to

In the

the

study

of

or “equalization” If fluid. the

i.e. heat

is

lost -

heat

by the

T,).

we find

have much effect

heat

that

problems,

conduction

is

in the

case

on the

the

increases

the

volume

the

cooling

our

first

of

time

of heat

approxi

transfer

main controlling

of

factor

the in

process.

in the

from the fluid

is

solution

of

determined

functions of the coordinates the rapidity of temperature

and exponential change is mainly

the

absolute

sum which

in time

V is

quantity.

solution not

fluid

where

transfer

an important

as may be seen

distribution

heat

does

molecular

transfer

Indeed, ture

the

convection

the

Cx/x)Y(TZ

transient

time

we study

mation,

that

value

has the

least

value

the

problem,

by the

sum of

functions determined of

the

the

of time. by that

coefficient

tempera-

products

of

of

Obviously term in t. The

reciprocal time. It

of this coefficient can be used to express the “equalization” appears that the temperature “equalization” time is given by for purely molecular thermal conduction, 12/xrr2 for both cases, i.e.,

t = and for

the

case

when thermal

convection

affects

the

temperature

distri-

Transient

heat

in

convection

spherical

film

593

bution. We should be able to obtain a graphic picture of the heat the axis of symmetry of process if, in any plane containing drew out streamlines and isothermal8 for various instants of ever this was not done because the formulas obtained are so

transfer the flow, we time. Howunwieldy.

The author thanks I. G. Shapshnikov for making available the subject of the work, and S.I. Melnik and V.S. Sorokin for some valuable criticisms.

BIBLIOGRAPHY 1.

Landau, L. and Lifshits, E. I Mekhanika Continuous Media). GITTL. 1954.

2.

Shaposhnikov, I. G., K teorii theory of weak convection). 1952.

3.

Kamke. E., (Guide

4.

Sprauochnik to

the

po

slaboi Zh.

sred

(Mechanics

differentsinl’nyr Equations).

IL,

uravneiniaa

1951.

Grinberg, G. A., Izbrannye voprosy matematicheskoi teorii electricheskikh i magnitnikh iavlenii (Some selected problems in the mathematical theory of electrical and magnetic problems). Itv. Akad.

Nauk

SSSR

of

konvektsii (Contribution to the Fiz. Vol. 22, No. 5, p. 826,

Tekh.

obyknovennyn

Common Differential

sp~oshnykh

1948.

Translated

by V.H.E.