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Propagation of thermal waves from the boundary of two media
PROPAGATIONOF THERMAL WAVES FROM THE BOUNDARY OF TWO MEDIA VOLNY
(RASPROSTRANIYE TEPLOVOI PMM VaZ.23,
OT QRANITSY DVUKH SRED)
No.5, 1959, pp. E.I.
991-992
ANDRIANKIN (loscowf
(Received
26
July
1958)
TQe problem of the propagation of heat from a plane boundary, and the decay of the discontinuity of temperature is considered, taking account of the change in phase of the material. An evaluation is carried out of the energy propagated into a medium with small thermal conductivity upon instantaneous evolution of heat at a point on the boundary of separation of two media. 1. Suppose that on the plane x = 0, (see Fig. l), the temperature is maintained. After being heated up to a temperature T*< To, the
Fig.
TO
1.
material is transformed into another phase state (region 2 in Fig. 1). The heat capacity c and the coefficient of thermal conductivity K for Ti T * are labeled by the index f (region i in Fig. 1). and for ?’ > 7’. by the index 2. The law of heat conduction
is described
x+, the condition
At the boundary of phase transition is written as: aT Xa ax X*_-O-‘I
I
by the equations
bT K
x*+0
d% = A dt
of heat balance
(2)
The problem is characterized by parameters, the units of which are expressed through the units of length L, tfme t. temperature T, and the
1420
Propagation
quantity
of
tkeraal
vavcs
fror
the
boundary
of
two media
1421
of heat Q: ci = QL+‘T- ,
eoi = QT-‘L-W,
Ts=T,
h = QL-8,
=Tl=T
Prom these parameters, the coordinate x and the time t. it is possible to form only one independent dimensionless variable and a series of dimensionless constants: Ct
rlt=z Therefore,
zx,il
1 %
[ equation
ClXlO
p
7=
[
-who
(1) is written dzi d ;i;j;?i(sJ
6
i
%
I
p-2
v=&.
’
(3)
as: dzi + ?i&i=O
(4)
i
If the initial temperature of the medium TO0 = 0, the boundary conditions fixing tl and 23 the front of the thermal wave XQ, and the boundary of the phase transition x., will be
Q G-l,@)= 0,
21 h,J
=
B,
2-4 (01 =
1,
2% @I%‘)
=
B
Integral curves of equation (4) were investigated in detail by Barenblatt in reference [ 3 1 , from the data of which it follows that there exists a unique integral curve satisfying at the end of the distance XQ, (t) the condition Ta = K (a T/h’ xl QD = 0. If TOO f 0, the solution is given by integral curves having at infinity a horizontal asymptote r.5 I. In analogous fashion, the problem of the decay of the temperature discontinuity is considered. If it be assumed that at the moment t = 0. two half-planes with temperatures Tl and T& are put in contact, the solution is self-similar. Due to self-similarity, the temperature on the boundary x = 0 remains constant. It should be determined from the condition of Continuity of the flow of heat at the section x = 0, for example, after solving the preceding problem for x > 0 and x < 0 for its dependence on the parameter TO. 2. We consider the problem of the instantaneous generation of heat on the boundary between two media with zero initial temperatures. We assume that K . = Koi +‘. If Kol = Ko2, cl = c2, then in the plane case f 1 1 the solution can be written:
1422
E. I.
Andriankin
a(k--I)
k+L (6)
{[
co=
2
The condition x = 0 for Kol f
Ko2,
of
buted
to the
according
equality Cl
of
f
temperature
requires
C2
Thereupon
the
to set
interesting at the
[l
of
two media
however,
that
only
ration
of
heat
pends
only
on
angle
by formulas
x > 0,
in the plane
should
be distri-
for
(71 with is
governed
kl
>
the
of
(6).
analogous of
boundary
heat
energy
by the is
of of
in which
it
is
x < 0.
It
is
one
the
to
=
into
case
of
(p1/p,)1’2,
generate
energy
instantaneous
two media
introduced
in the
on the discontinuity
law E2/E1
possible
The problem
on the
The quantity
it
k2,
in medium f.
(7)
and Q = 2 Q1 for
redistribution
from a point
8.
flow
energy
QI+ Qz= Q
given
formula
when the
instantaneously
and the
is for
2 Q,
I,
boundary
We notice,
served
solution Q =
to compare
gas dynamics
the
law
QI 1 Qz = 7, necessary
and heat
that
gene-
(kl = k2) de-
medium 2 is
con-
in time: el@w
x
AQY (k) = 2xQ In this,
sin 0 &I
there
F (4, 0) E2d<, T =
\
s n/a
[t(~~!“‘~-1]3”~k-i)
(5, 0) (8)
0
exists
a surface
through
which
no heat
flows.
However, in this case If k1 f k2. the problem is not self-similar. also it is possible to evaluate approximately the quantity of heat transferred into medium 2. We shall assume that in medium 1 the region of high temperature the law
and the
radius
of
the
heated
hemisphere
change
according
to
1 rQ, =
the
Assuming heat is
B (k,) t 3kl--1 ,
that K* << transmitted
K~,
when it
by means of
may be considered plane
waves,
that
we obtain
in medium 2
Propagation
tion
of
of
from
waves
the
boundary
of
which
media
1423
from the
soluthe
tmo
is determined derivative (J T/ Jx) of the propagation o?Ihiat from a wall
The value of the of the problem
temperature
thermal
1,
[ 2.5
is
T _ A (tl + 7)3/(i--9h‘1),
0
for AQ < Q.
BIBLIOGRAPHY I ..
Zel’ dovich,
1a.B.
teploprovodnosti, of
heat
Sb.
for
thermal
posviashch.
(Collection Ioffe. 2.
conductivity
of
to
the
the
P. Ia.,
3.
Dokl.
Barenblatt, G. I., gaza v poristoi in a porous
4.
Zel’ dovich, davleniia
porous
Sciences,
srede
anniversary U.S.S.R.,
v teorii encountered
SSSR,
Vol.
(On the PMM Vol.
NO. 6,
16,
No.
(On a nontheory of filtra1948.
motion 1,
1950.
A.F.
1950).
dvizheniiakh
non-steady
of
differential’nom
filtratsii in the
113,
0 neustanovivshikhsia
Akust.
G. I.,
tatsionarnoi of
birthday
pri
zhidkosti of
liquid
i and gas
1952.
Ia. B., Dvizhenie gaza pod deistviem kratkovremennogo (Motion of a gas under the action of short-duration
Barenblatt, tion
Nauk
medium).
pressure). 5.
Akad.
70th
Ob odnom nelineinom
uravnenii, vstrechaiushchemsia linear differential equation tion).
loffe.
Academy of
Polubarinova-Kochina,
tepla
propagation
according to temperature). Izd-vo Akad Nauk SSSR,
A.F.
70-letiiu
dedicated
Press
A. S., 0 rasprostranenii ot temperatury (On the
and Kompaneets, zavisiashchei
the
zh.
Vol.
No,
0 priblezhennom
fil’tratsii problem
medium).
2,
1,
reshenii
v poristoi of
PMM Vol.
No.
zadach
srede
one-dimensional 18,
1956.
3,
odnomernoi
nes-
(On an approximate
non-steady
filtration
soluin a
1954.
Translated
by D.T.W.
(RASPROSTRANIYE TEPLOVOI PMM VaZ.23,
OT QRANITSY DVUKH SRED)
No.5, 1959, pp. E.I.
991-992
ANDRIANKIN (loscowf
(Received
26
July
1958)
TQe problem of the propagation of heat from a plane boundary, and the decay of the discontinuity of temperature is considered, taking account of the change in phase of the material. An evaluation is carried out of the energy propagated into a medium with small thermal conductivity upon instantaneous evolution of heat at a point on the boundary of separation of two media. 1. Suppose that on the plane x = 0, (see Fig. l), the temperature is maintained. After being heated up to a temperature T*< To, the
Fig.
TO
1.
material is transformed into another phase state (region 2 in Fig. 1). The heat capacity c and the coefficient of thermal conductivity K for Ti T * are labeled by the index f (region i in Fig. 1). and for ?’ > 7’. by the index 2. The law of heat conduction
is described
x+, the condition
At the boundary of phase transition is written as: aT Xa ax X*_-O-‘I
I
by the equations
bT K
x*+0
d% = A dt
of heat balance
(2)
The problem is characterized by parameters, the units of which are expressed through the units of length L, tfme t. temperature T, and the
1420
Propagation
quantity
of
tkeraal
vavcs
fror
the
boundary
of
two media
1421
of heat Q: ci = QL+‘T- ,
eoi = QT-‘L-W,
Ts=T,
h = QL-8,
=Tl=T
Prom these parameters, the coordinate x and the time t. it is possible to form only one independent dimensionless variable and a series of dimensionless constants: Ct
rlt=z Therefore,
zx,il
1 %
[ equation
ClXlO
p
7=
[
-who
(1) is written dzi d ;i;j;?i(sJ
6
i
%
I
p-2
v=&.
’
(3)
as: dzi + ?i&i=O
(4)
i
If the initial temperature of the medium TO0 = 0, the boundary conditions fixing tl and 23 the front of the thermal wave XQ, and the boundary of the phase transition x., will be
Q G-l,@)= 0,
21 h,J
=
B,
2-4 (01 =
1,
2% @I%‘)
=
B
Integral curves of equation (4) were investigated in detail by Barenblatt in reference [ 3 1 , from the data of which it follows that there exists a unique integral curve satisfying at the end of the distance XQ, (t) the condition Ta = K (a T/h’ xl QD = 0. If TOO f 0, the solution is given by integral curves having at infinity a horizontal asymptote r.5 I. In analogous fashion, the problem of the decay of the temperature discontinuity is considered. If it be assumed that at the moment t = 0. two half-planes with temperatures Tl and T& are put in contact, the solution is self-similar. Due to self-similarity, the temperature on the boundary x = 0 remains constant. It should be determined from the condition of Continuity of the flow of heat at the section x = 0, for example, after solving the preceding problem for x > 0 and x < 0 for its dependence on the parameter TO. 2. We consider the problem of the instantaneous generation of heat on the boundary between two media with zero initial temperatures. We assume that K . = Koi +‘. If Kol = Ko2, cl = c2, then in the plane case f 1 1 the solution can be written:
1422
E. I.
Andriankin
a(k--I)
k+L (6)
{[
co=
2
The condition x = 0 for Kol f
Ko2,
of
buted
to the
according
equality Cl
of
f
temperature
requires
C2
Thereupon
the
to set
interesting at the
[l
of
two media
however,
that
only
ration
of
heat
pends
only
on
angle
by formulas
x > 0,
in the plane
should
be distri-
for
(71 with is
governed
kl
>
the
of
(6).
analogous of
boundary
heat
energy
by the is
of of
in which
it
is
x < 0.
It
is
one
the
to
=
into
case
of
(p1/p,)1’2,
generate
energy
instantaneous
two media
introduced
in the
on the discontinuity
law E2/E1
possible
The problem
on the
The quantity
it
k2,
in medium f.
(7)
and Q = 2 Q1 for
redistribution
from a point
8.
flow
energy
QI+ Qz= Q
given
formula
when the
instantaneously
and the
is for
2 Q,
I,
boundary
We notice,
served
solution Q =
to compare
gas dynamics
the
law
QI 1 Qz = 7, necessary
and heat
that
gene-
(kl = k2) de-
medium 2 is
con-
in time: el@w
x
AQY (k) = 2xQ In this,
sin 0 &I
there
F (4, 0) E2d<, T =
\
s n/a
[t(~~!“‘~-1]3”~k-i)
(5, 0) (8)
0
exists
a surface
through
which
no heat
flows.
However, in this case If k1 f k2. the problem is not self-similar. also it is possible to evaluate approximately the quantity of heat transferred into medium 2. We shall assume that in medium 1 the region of high temperature the law
and the
radius
of
the
heated
hemisphere
change
according
to
1 rQ, =
the
Assuming heat is
B (k,) t 3kl--1 ,
that K* << transmitted
K~,
when it
by means of
may be considered plane
waves,
that
we obtain
in medium 2
Propagation
tion
of
of
from
waves
the
boundary
of
which
media
1423
from the
soluthe
tmo
is determined derivative (J T/ Jx) of the propagation o?Ihiat from a wall
The value of the of the problem
temperature
thermal
1,
[ 2.5
is
T _ A (tl + 7)3/(i--9h‘1),
0
for AQ < Q.
BIBLIOGRAPHY I ..
Zel’ dovich,
1a.B.
teploprovodnosti, of
heat
Sb.
for
thermal
posviashch.
(Collection Ioffe. 2.
conductivity
of
to
the
the
P. Ia.,
3.
Dokl.
Barenblatt, G. I., gaza v poristoi in a porous
4.
Zel’ dovich, davleniia
porous
Sciences,
srede
anniversary U.S.S.R.,
v teorii encountered
SSSR,
Vol.
(On the PMM Vol.
NO. 6,
16,
No.
(On a nontheory of filtra1948.
motion 1,
1950.
A.F.
1950).
dvizheniiakh
non-steady
of
differential’nom
filtratsii in the
113,
0 neustanovivshikhsia
Akust.
G. I.,
tatsionarnoi of
birthday
pri
zhidkosti of
liquid
i and gas
1952.
Ia. B., Dvizhenie gaza pod deistviem kratkovremennogo (Motion of a gas under the action of short-duration
Barenblatt, tion
Nauk
medium).
pressure). 5.
Akad.
70th
Ob odnom nelineinom
uravnenii, vstrechaiushchemsia linear differential equation tion).
loffe.
Academy of
Polubarinova-Kochina,
tepla
propagation
according to temperature). Izd-vo Akad Nauk SSSR,
A.F.
70-letiiu
dedicated
Press
A. S., 0 rasprostranenii ot temperatury (On the
and Kompaneets, zavisiashchei
the
zh.
Vol.
No,
0 priblezhennom
fil’tratsii problem
medium).
2,
1,
reshenii
v poristoi of
PMM Vol.
No.
zadach
srede
one-dimensional 18,
1956.
3,
odnomernoi
nes-
(On an approximate
non-steady
filtration
soluin a
1954.
Translated
by D.T.W.