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Temperature transient after contact heating
SHORTER
926
TEMPORARY
COMMUNICATIONS
TRANSIENT AFTER CONTACT HEATING
RICHARD C. BAILIE
and LIANG-TSENG FAN
Kansas State University, Manhattan,
Kansas
(Received 3 1 August 1962 and in revisedform 15April 1963)
THIS
note gives equations representing transient temperature dis~ibutions in two heat-conducting, one-dimensional finite solid bars following contact heating. Both bars are assumed to have identical uniform physical properties which remain constant during the transient. The bars are subject to the foliowing conditions (Fig. 1). (1) Bar 1 with a length His initially at a constant ternperature 7;‘. (2) Bar 2 with a length L - H is initially at a constant temperature Ti(r,’ # T;). (3) Bars are brought into contact at time, t = 0. Heat is conducted only through the contact surface. (4) Contact is discontinued at time, I = t,. No heat is transferred to the surround~gs. One dimensional
heat conduction
is represented
by,
%T = k $7; ‘It
(1)
for 0 < f < to and subject to the conditions stated above is given by [I], ii 7-=X+2
1 exp (-&7’7) 71c n I,== 1
(2)
where D = g,
‘zii,
kt
T 1
T=giGFT,
I andX=!!. 2
The solutions of the conduction equation representing the temperature transient after discontinuing contact 0 > t0 or 7 > Q) are derived as follows: Bar 1: T _ 1 - -& X
co c n-1
7
where T’is the temperature, Z is the distance variable and k the thermal diffusivity. The solution of equation (1)
sin (nnX) cos (mm)
exp ( - ~WQ) - - .---n2
ru
u.
1
sin” nrX - _I P”X CG N n=t m.z, exp (-n%&) (sin nnX) [exp (-m”&l)] sin n(nX - nr) sin (nX - nr)a -+ nX- r?-- u* nX -- m
1 j
(3)
i
E
kos ?mrU~) ---
T
‘I n”X(1 - X) c
exp (-&+T~) (cos mu,)
exp (-- nW&J .-~~ .- 9
1
(sin2 naX) [exp (-m%+)]
I
~
1
---
1
n2(1 - X)” - nz2
where (7 z o,= x=B’ FIG. 1. Contact heat conduction bars. * This work was supported Foundation Grant, G-19857.
“=E2 kt
uetween two soua L = _... L-H’ by National
Science Fig. 2 illustrates some results of numerical computation.
SHORTER
COMMUNICATIONS
927
0.65 O-60-
FIG. 2. Temperature
distribution
represented
by equations (3) and (4); TV= 0.10.
The equations derived are equally valid for molecular diffusion. REFERENCES 1. H. S. CARSLAW and J. C. JAEGER, Conduction of Heat in Solids, 2nd Ed. Oxford University Press (1959).
THE TRANSITION FROM BLACK BODY TO ROSSELAND FORMULATIONS IN OPTICALLY THICK FLOWS7 R. GOULARD Purdue University,
Lafayette, Indiana
(Received 6 February 1963)
THE energy
radiated by optically thick flows through their boundaries is expressed sometimes as a black body fluxqR = oT4, sometimes as a Rosseland diffuse flux __. t The work described in this paper was supported by the National Science Foundation under grant No. G-23170.
@=
-
160T3 8T __ .3kR ay'
The purpose of this note is to establish, on the basis of two simple incompressible flows, the radiation parameters which govern the choice between these two formulations.
926
TEMPORARY
COMMUNICATIONS
TRANSIENT AFTER CONTACT HEATING
RICHARD C. BAILIE
and LIANG-TSENG FAN
Kansas State University, Manhattan,
Kansas
(Received 3 1 August 1962 and in revisedform 15April 1963)
THIS
note gives equations representing transient temperature dis~ibutions in two heat-conducting, one-dimensional finite solid bars following contact heating. Both bars are assumed to have identical uniform physical properties which remain constant during the transient. The bars are subject to the foliowing conditions (Fig. 1). (1) Bar 1 with a length His initially at a constant ternperature 7;‘. (2) Bar 2 with a length L - H is initially at a constant temperature Ti(r,’ # T;). (3) Bars are brought into contact at time, t = 0. Heat is conducted only through the contact surface. (4) Contact is discontinued at time, I = t,. No heat is transferred to the surround~gs. One dimensional
heat conduction
is represented
by,
%T = k $7; ‘It
(1)
for 0 < f < to and subject to the conditions stated above is given by [I], ii 7-=X+2
1 exp (-&7’7) 71c n I,== 1
(2)
where D = g,
‘zii,
kt
T 1
T=giGFT,
I andX=!!. 2
The solutions of the conduction equation representing the temperature transient after discontinuing contact 0 > t0 or 7 > Q) are derived as follows: Bar 1: T _ 1 - -& X
co c n-1
7
where T’is the temperature, Z is the distance variable and k the thermal diffusivity. The solution of equation (1)
sin (nnX) cos (mm)
exp ( - ~WQ) - - .---n2
ru
u.
1
sin” nrX - _I P”X CG N n=t m.z, exp (-n%&) (sin nnX) [exp (-m”&l)] sin n(nX - nr) sin (nX - nr)a -+ nX- r?-- u* nX -- m
1 j
(3)
i
E
kos ?mrU~) ---
T
‘I n”X(1 - X) c
exp (-&+T~) (cos mu,)
exp (-- nW&J .-~~ .- 9
1
(sin2 naX) [exp (-m%+)]
I
~
1
---
1
n2(1 - X)” - nz2
where (7 z o,= x=B’ FIG. 1. Contact heat conduction bars. * This work was supported Foundation Grant, G-19857.
“=E2 kt
uetween two soua L = _... L-H’ by National
Science Fig. 2 illustrates some results of numerical computation.
SHORTER
COMMUNICATIONS
927
0.65 O-60-
FIG. 2. Temperature
distribution
represented
by equations (3) and (4); TV= 0.10.
The equations derived are equally valid for molecular diffusion. REFERENCES 1. H. S. CARSLAW and J. C. JAEGER, Conduction of Heat in Solids, 2nd Ed. Oxford University Press (1959).
THE TRANSITION FROM BLACK BODY TO ROSSELAND FORMULATIONS IN OPTICALLY THICK FLOWS7 R. GOULARD Purdue University,
Lafayette, Indiana
(Received 6 February 1963)
THE energy
radiated by optically thick flows through their boundaries is expressed sometimes as a black body fluxqR = oT4, sometimes as a Rosseland diffuse flux __. t The work described in this paper was supported by the National Science Foundation under grant No. G-23170.
@=
-
160T3 8T __ .3kR ay'
The purpose of this note is to establish, on the basis of two simple incompressible flows, the radiation parameters which govern the choice between these two formulations.