C O M MUNI C A T lON
S I M I L A R I T Y SOLUTIONS OF T H E PLANE M E L T I N G PROBLEM W I T H TEMPERATURE-DEPEN D E N T T H E R M A L PROPERTIES
Time-independent similarity solutions of the plane freezing-melting problem exist, where the densities of the new and original phases are unequal, and the heat capacities and thermal conductivities of each phase a r e arbitrary functions of temperature. The solution reduces to the numerical integration of an initial-value problem proceeding in both directions from the immobilized phase boundary. For constant physical properties the classical exact solution i s obtained. A numerical example shows that the correction to this solution can be significant.
TIIF, essrntial dificuliy of the Stefan (81 problem. \vhich
is now a grneric term for a class of diffusion-controlled free boundary pt~nhletns.lice in the nonlinearity introduced by the presrncc of the moving boundary. It has been shoum ( I ) that masinium and minimum bounds for the plane meltingfrerzing prolilrm \vith timc-deprndent boundary conditions can be readily obtained Lvith the aid of a trancformation due to Zrncr (9). .A = t . ' 6 ( t ) , Ivhcre Y is the distance mrasured from the surface of thr srmi-inficitr elah. and 6 ( t ) is the thickness of t h e nc\vly formrd phase. I n this notr thc rxictrnce of self-similar solution? to the same problem whrn the thermal properties of hoth phasec are temprraiui-r-drprndrnl~ and where the phatrs have constant: although not nrcrccarily equal. deneitirs ic established by examining thr trancrorrned equations in A-spacr. It turns oiit that ciich sohitions rxict only \\-hen 6f-"* is a conctant. qn that the transrorniation is identical \vith that given by Roltzmann ( 7 ) . ti = t f - - ' / 2 . for the on?-dimrnsional heat equation \\-ith trmpri a t u r e - d r p r n d c n ~properties. T h e rstension to Stefantype pi-ol~lenisLvith relative motion of the phases is slight. but \\-orth\vhilr. This is drmonstrated by a numerical example. R c c r n t 1-rlatcd work ir>.cludrsthr contributions of Pattle (7) and Royri- ( 2 ) .\ v h o generalized the Bo1 t7mann similarity transformation to obtain exact colutions for thr single-phase heat conduc-tinn problrm ivhen thr thrrnial conductivity is proportionnl to a nonnrgative p o \ \ ~ of r the teinprt-ature. Statement of Problem
C:otiqidrr n srnii-infinitr slab of phase 1 . originally a t some uiiifoi-rii t m i p c r a ~ i i r r .7', . At tirnc. t = 0. corrcaponding to the i n c t a n r o l appearance of phase 2 . a conctant tcniperat~ire. 7#r. i , . npplicd a t the surface. \ = 0. T h e movinp boundary. locarrd a t h ( / j , i-ernains at thr transition teniperatiirr, 7-$. 'I'he dcncitieq of thc phasrq are aswined to be constant. although not ncrmwrily rqiial. '1 hr rriaci.owopic rncrqy equations describing thr trm]vi-atiii-rfirltl in r a r h pha;c a r r :
for phase 1 :
for phase 2 :
xvhere 1-1 is the velocity of phase 1 ari5ing from the difference in denqiry of the t\vo phases. The specific heats. c f . and thermal conductiviries. k j ( i = 1 . 2). are arbitrary. but specified, functions of temperature. T h e initial and boundary conditions are
6(0)
=
(3)
0
T , ( L0 )
=
T,
(4)
7'?(0. t )
=
T,
(5)
T,(& t)
=
T?(6.t'i
=
T,
(6)
Seglecting kinetic and potential energy effects. the conservation of energy at the free boundary reduces to a n enthalpy balance
\\here the plus and minus sign< iefex to melting. or free7ing. Noting that for conrtant densitieq
\ve
ohtain
In term? of X = rrarrangeirient.
i
6. these equations become. aftrr a clight
VOL. 3
NO. 2
MAY
1964
177
-¶VARIABLE PROPERTIES :CONSTANT PROPERTIES
--
I
Figure 1.
where 7'1
=
TIIX. t ) for h
Freezing of copper melt initially a t fusion temperature
2 1 . and
l ' h e boundary conditions become
Equation 11 was obtained by Landau (6) and Horvay (.5j. The Similarity Solution
.At this point we consider the possibility of finding particular solutions to the system Lvhich are independent of time. Thus, if I', = ?',(A) only, it follows from Equation 14 that d6*,'dt is a conctant. Equations 10 and 11 now yield two secondorder ordinary differential equations, on setting the left-hand sides equal to zero. The boundary conditions 1 2 to 1 4 and the initial conditions, 4, are consistent with the similarity requirements. although these cannot be met in any finite space domain. Letting d62,'dt = 4a2, the equations and boundary conditions become
For any preselected values of a and d?2( I ) , ' d h . Equation 19 determines d l ' l ( 1 ) 'dX. With thesr initial values a t X = 1 . togetherwith Equation 17. integration of Equations 16 and 17 proceeds outward in both dirrctions. From Equation 18 one then obtains the values of 7 and T , corresponding to the assumed initial slopes. An iterative process is required to determine the solution For. a given 7 ,and 7-,. If the physical properties of the phases are constant. the solution of Equations 15 to 19 is readily obtained, and is identical with that given by Carslaw and Jaegrr ( 3 ) ,obtained in a diffrrent manner. Numerical Example
'4s an example. the solutions for the freezing of a srmiinfinite melt of copper, originally at the fusion temperature, upon being subjected to a surface temperature of 3' C., are illustiated in Figure 1 O n e solution is obtained using the published thermal conductivity and specific heat values for solid copper as functions of temperaturr, and the other by evaluating the thermal properties a t the average solid temperature. For this case, the difference in the freezing rates a t any instant is of the order of 10% Nomenclature
heat capacity a t constant pressure of phase z
G,
=
k
= thermal conductivity
t
=
time
7' = tempeiature T , = transition temperature 7 , = wall temperature 'I',,,= temperature of original phase a t
LTl 178
l&EC FUNDAMENTALS
=
velocity of original phase
t =
0
x
= distance from plate
a
=
6 -1
x
(5) Horvay. G.: J . Heat Transfer 82, 37 (1960). (6) Landau, H. G . , Quart. Appl. M a t h . 8, 81 (1950). ( 7 ) Pattle. R. E.; Quart. .Mech. Appi. M a t h . 12, 408 (1959). (8) Stefan. J.. .4nn. Phys. Chem. (Wiedemann) ( N . F . ) , 42, 269 (1891). (9) Zener, C., J . .4,b,bi.Phys. 20, 950 (1949).
growth parameter, Equation 15 = distance from plate to interface = latent heat of fusion =
X I 6
p
=
density
w
= P?
T. D. HAMILL
.Veu Yurk CniLersitj AYeu Y o r k . AV Y
PI
SCBSCRIPTS = original phase = nexvly generated phase
S. G. BANKOFF
.\\'brthuestern C n i w s i t l ELanston, Ill
1 2
literature Cited
(1) Boltzmann, L.. Ann. Physrk (.Y.F.) 53, 959 (1894). ( 2 ) Boyer. R. H.. J . .\lath. Phjs. 40, 41 (1961). (3) Carslaw, H. S.. Jaeger. J . C.. "Conduction of Heat in Solids." 2nd ed.. p. 285. Oxford Univ. Prrss, London. 1959. (4) Hamill. T. D.. Bankoff. S. G.. A.I.17h.E. J . 9, '741 (1963).
RECEIVED for review December 28, 1962 ~ X X P T E D January 6, 1964 Technical report G-14773. Work supported by a U. S. Atomic Energy Commission fellowship and a National Science Foundation grant.
COMM UNICATION
T R A N S I E N T MASS TRANSFER IN A FIXED BED Transient mass transfer from a flowing fluid to a fixed b e d of particles has been studied mathematically for the conditions where either external diffusion, intraparticle diffusion, or surface adsorption processes control the rate.
In adsorption, ion-exchange, or extraction, the last two processes particularly can offer significant
resistances. A mathematical solution i s presented which takes all three resistances into account. The result, based upon a first order reversible surface adsorption process, i s in the form of an infinite integral expressed in terms of four dimensionless parameters: +
R(Xp,/DJ'
*, (D,/kJ), lO/Q,), (Ox
