International Journal of Engineering Science 41 (2003) 1685–1698 www.elsevier.com/locate/ijengsci
Explicit solutions to the one-phase Stefan problem with temperature-dependent thermal conductivity and a convective term
Mar ııa F. Natale, Domingo A. Tarzia * Depto. Matema tica, FCE, Universidad Austral, CONICET, Paraguay 1950, S2000FZF Rosario, Argentina Received 28 October 2002; accepted 18 December 2002
Abstract
A one-phase Stefan problem for a semi-infinite material with temperature-dependent thermalpffiffi conduc- tivity and convective term with a constant temperature or a heat flux condition of the type q0= t ðq0 > 0Þ at the fixed face x ¼ 0 is studied. For both cases a parametric representation of the solution of the similarity type is also obtained. Ó 2003 Elsevier Science Ltd. All rights reserved.
AMS: 35R35; 80A22; 35C05 Keywords: Stefan problem; Free boundary problem; Movingboundary problem; Phase-changeprocess; Nonlinear thermal conductivity; Fusion; Solidification; Similarity solution
1. Introduction
We consider a Stefan problem for a semi-infinite region x > 0 with phase-change temperature hf [18]. It is required to determine the evolution of the movingphase separation x ¼ sðtÞ and the temperature distribution hðx; tÞ. The modelingof this kind of systems is a problem with a great mathematical and industrial significance. Phase-change problems appear frequently in industrial processes an other problems of technological interest [1,2,9–11,13–16,19,30]. A large bibliography on the subject was given in [29].
* Correspondingauthor. Tel.: +54-3414-814990; fax: +54-3414-810505. E-mail addresses: [email protected] (M.F. Natale), [email protected] (D.A. Tarzia).
0020-7225/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0020-7225(03)00067-3 1686 M.F. Natale, D.A. Tarzia / International Journal of Engineering Science 41 (2003) 1685–1698
Here, we consider a one phase-change process (one-phase Stefan problem) for a nonlinear heat conduction equation with a convective term. Owingto [26] we consider the followingfree boundary (fusion process) problem: oh o oh oh qc ¼ kðh; xÞ vðhÞ ; 0 < x < sðtÞ; t > 0 ð1Þ ot ox ox ox
oh q kðhð0; tÞ; 0Þ ð0; tÞ¼ p0ffiffi ; q > 0; t > 0 ð2Þ ox t 0
oh kðhðsðtÞ; tÞ; sðtÞÞ ðsðtÞ; tÞ¼ qlsðtÞ; t > 0 ð3Þ ox
hðsðtÞ; tÞ¼hf ; t > 0 ð4Þ
sð0Þ¼0 ð5Þ where the thermal conductivity kðh; xÞ and the velocity vðhÞ are given by
d vðhÞ¼qc ð6Þ 2ða þ bhÞ2
1 þ dx kðh; xÞ¼qc ð7Þ ða þ bhÞ2 and c, q and l are the specific heat, the density and the latent heat of fusion of the medium re- spectively, all of them are assumed to be constant with positive parameters a; b; d and a þ bhf > 0. The last condition guaranteed that v and k are well defined by maximal principle. This kind of nonlinear thermal conductivity or diffusion coefficients was considered in numerous papers, e.g. [4,7,8,17,21,23,27]. The nonlinear transport equation (1) arises in connection with unsaturated flow in heterogeneous porous media. If we set d ¼ 0 and b ¼ 0 in the free boundary problem (1)– (7) then we retrieve the classical one-phase Lam ee–Clapeyron–Stefan problem.pffiffi The first explicit solution for the one-phase Stefan problem was given in [18]. Here q0= t denotes the prescribed flux on the boundary x ¼ 0 which is of the type imposed in [28], where it was proven that the heat flux condition (2) on the fixed face x ¼ 0 is equivalent to the constant temperature boundary condition (43) for the two-phase Stefan problem for a semi-infinite material with constant thermal coefficients in both phases. This kind of heat flux at the fixed boundary x ¼ 0 was also considered in several applied problems, e.g. [3,12,22]. The goal of this paper is to determine which conditions on the parameters of the problem (in particular q0) must be satisfied in order to have an instantaneous phase-change process. In Section 2 we follow and improve [26] in the sense that the existence of an explicit solution of the problem (1)–(5) with nonlinear coefficients (6) and (7) is obtained for all data q0; q; c; l; hf ; a; b; d with the à restriction a þ bhf > 0. Moreover, this solution is given as a function of a parameter c which is M.F. Natale, D.A. Tarzia / International Journal of Engineering Science 41 (2003) 1685–1698 1687 given the unique solution of the transcendental equation (35). Although, if we replace the flux condition (2) for a new temperature boundary condition (43) on the fixed face x ¼ 0 we also obtain in Section 3 for all data the existence of solution which is given as a function of a parameter b which we prove it is the unique solution of the transcendental equation (71). This section is new with respect to [26]. In both cases, the explicit solution is given by a parametric representation of the similarity type. Other problems with nonlinear thermal coefficients in this subject are also given in [5,6,20,24,25].
2. Solution of the free boundary problem with heat flux condition on the fixed face
We consider the problem (1)–(5). Takinginto account (6) and (7) we can put our problem as ! oh o 1 þ dx oh d ¼ þ ; 0 < x < sðtÞ; t > 0 ð8Þ ot ox ða þ bhÞ2 ox 2bða þ bhÞ
1 oh qà ð0; tÞ¼ p0ffiffi ; t > 0 ð9Þ ða þ bhð0; tÞÞ2 ox t
1 þ dsðtÞ oh ðsðtÞ; tÞ¼ asðtÞ; t > 0 ð10Þ ða þ bhðsðtÞ; tÞÞ2 ox
hðsðtÞ; tÞ¼hf ; t > 0 ð11Þ sð0Þ¼0 ð12Þ
à where a ¼ l=c and q0 ¼ q0=qc: We define the followingtransformations in the same way as in [26]: 8 hi 1 > y ¼ 2 ð1 þ dxÞ2 1 < d hi 1 > SðtÞ¼2 ð1 þ dsðtÞÞ2 1 ð13Þ :> d hðy; tÞ¼hðx; tÞ and
8 R > yà ¼ yÃðy; tÞ¼ y ða þ bhðr; tÞÞdr þðab þ aÞSðtÞ > SðtÞ < tà ¼ t hà ¼ 1 ð14Þ > aþb h :> à à S ¼ y y¼S ¼ðab þ aÞSðtÞ 1688 M.F. Natale, D.A. Tarzia / International Journal of Engineering Science 41 (2003) 1685–1698
In order to obtain an alternative expression for yà we compute Z oyà y oh o ¼ ða þ bhðSðtÞ; tÞÞSðtÞþ b o ðr; tÞdr þðab þ aÞSðtÞ t SðtÞ t Z ! y o 1 oh ¼ abSðtÞþ b dr o 2 o S t r a bh r; t r ð Þ ð þ ð ÞÞ ! 1 oh 1 oh ¼ abSðtÞþb ðy; tÞ ÀÁðSðtÞ; tÞ 2 o 2 o ða þ bhðy; tÞÞ y a þ bhðSðtÞ; tÞ y b oh ¼ ðy; tÞ ða þ bhðy; tÞÞ2 oy Z ! y o b oh b oh ¼ ðr; tÞ dr þ ð0; tÞð15Þ o 2 o 2 o 0 r ða þ bhðr; tÞÞ r ða þ bhð0; tÞÞ y Z y oh bqà ffiffi0 ¼ b o ðr; tÞdr þ p ð16Þ 0 t t then Z Z Z t y o bqà y yÃðy; tÞ¼ ða þ bhðr; tÞÞdr þ pffiffiffi0 ds þ ða þ bhðr; 0ÞÞdr or s Z0 0 0 y pffiffi à ¼ ða þ bhðr; tÞÞdr þ 2bq0 t ð17Þ 0 Now, applying(13) and (14) the problem (8)–(12) is transformed in a classical Stefan problem given by
ohà o2hà pffiffiffiffi ¼ ; 2bqà tà < yà < SÃðtÃÞ; tà > 0 ð18Þ otà oyÃ2 0
ohà pffiffiffiffi qÃb pffiffiffiffi ð2bqà tÃ; tÃÞ¼p0ffiffiffiffi hÃð2bqà tÃ; tÃÞ; tà > 0 ð19Þ oyà 0 tà 0
ohà ðSÃðtÃÞ; tÃÞ¼aÃSÃðtÃÞ; tà > 0 ð20Þ oyÃ
à à à à à à h ðS ðt Þ; t Þ¼hf ; t > 0 ð21Þ SÃð0Þ¼0 ð22Þ where
à ab à 1 a ¼ ; hf ¼ ð23Þ ðab þ aÞða þ bhf Þ a þ bhf M.F. Natale, D.A. Tarzia / International Journal of Engineering Science 41 (2003) 1685–1698 1689
Then, if we introduce the similarity variable: yà nà ¼ pffiffiffiffiffiffiffiffiffiffi ð24Þ 2cÃtà where cà is a dimensionless positive constant to be determined, and the solution is sought of the type hÃðyÃ; tÃÞ¼HÃðnÃÞ pffiffiffiffiffiffiffiffiffiffi ð25Þ SÃðtÃÞ¼ 2cÃtà then, we get that (18)–(22) yields sffiffiffiffi d2Hà dHà 2 þ cÃnà ¼ 0; bqà < nà < 1 ð26Þ dnÃ2 dnà 0 cà sffiffiffiffi! sffiffiffiffi! dHà 2 pffiffiffiffiffiffiffi 2 bqà ¼ 2cÃqÃbHà bqà ð27Þ dnà 0 cà 0 0 cÃ
dHà ð1Þ¼aÃcà ð28Þ dnÃ
à à H ð1Þ¼hf ð29Þ The solution of the differential equation (26) is given by rffiffiffiffi ! cà HÃðnÃÞ¼Aerf nà þ B ð30Þ 2 where A and B are two unknown coefficients. From (27) and (29) we get
hà A ¼ f qffiffiffi ð31Þ cà à p1ffiffi gbq0; p erf 2 à à p1ffiffi hf gbq0; p B ¼ qffiffiffi ð32Þ cà à p1ffiffi gbq0; p erf 2 where
gðx; pÞ¼erfðxÞþpRðxÞ; x > 0; p 2 R expð x2Þ ð33Þ RðxÞ¼ ; x > 0 x 1690 M.F. Natale, D.A. Tarzia / International Journal of Engineering Science 41 (2003) 1685–1698 and Z 2 x erfðxÞ¼pffiffiffi expð u2Þdu ð34Þ p 0
The unknown constant cà is determined by the remainingboundary condition (28) which yields the followingequation: rffiffiffi pffiffiffiffi 2 hà cà aà cà ¼ f qffiffiffi exp ; cà > 0 ð35Þ cà p à p1ffiffi 2 gbq0; p erf 2
Theorem 1. The free boundary problem (1)–(5) has a unique solution of the similarity type for all data q0; q; c; l; hf ; a; b; d. Moreover, the solution is given by 2 3 1 6 1 7 hðnÞ¼ 4 qffiffiffi a5 à b Aerf c nà þ B hi2 1 2 2 y d ð1 þ dxÞ 1 ð36Þ n ¼ pffiffiffiffiffiffi ¼ pffiffiffiffiffiffi 2ct 2ct " # 1 d pffiffiffiffiffiffi 2 sðtÞ¼ 1 þ 2ct 1 d 2 where " rffiffiffiffi !# Z à n cà n ¼ðab þ aÞ pffiffiffi Aerf r þ B dr ð37Þ 2 bqà 2 cà 0 and A; B and c are given by (31), (32) and (42) respectively. pffiffiffiffiffiffiffiffiffi Proof. First, we have to study the existence and uniqueness of the Eq. (35) If we define l ¼ cÃ=2, then l must be the solution of the followingequation: 1 1 hà gbqÃ; pffiffiffi ¼ gx; pffiffiffi f ; x > 0 ð38Þ 0 p p aÃ
It is easy to see that gð0; pÞ¼ 1, gðþ1; pÞ¼1 and ðog=oxÞðx; pÞ > 0, 8x > 0, 8p < 0: Moreover we have
hà 1=ða þ bh Þ a f f 1 39 à ¼ ¼ þ ð Þ a ab=ðab þ aÞða þ bhf Þ ab M.F. Natale, D.A. Tarzia / International Journal of Engineering Science 41 (2003) 1685–1698 1691 so the Eq. (38) can be written as 1 1 a gbqÃ; pffiffiffi ¼ gx; pffiffiffi 1 þ ; x > 0 ð40Þ 0 p p ab ÀÁffiffiffi à p Therefore, takinginto account that gbq0; 1= p < 1 then Eq. (40) has a unique solution for all data. From (25) we obtain the expression of SðtÞ given by pffiffiffiffiffiffi SðtÞ¼ 2ct ð41Þ where c is given by cà c ¼ ð42Þ ðab þ aÞ2
From (24), (25) and (30) we can obtain the parametric solution of the problem (1)–(5) given by (36) and (37). Note that sffiffiffiffi pffiffiffiffi 2bqà tà 2 nà j ¼ pffiffiffiffiffiffiffiffiffiffi0 ¼ bqÃ: n¼0 2cÃtà cà 0
We remark that the temperature at the fixed face x ¼ 0 is constant in time which origins the followingsection. Ã
3. Solution of the free boundary problem with temperature boundary condition on the fixed face
Now, we consider the problem (1)–(5) but the condition (2) will be replaced by the following temperature boundary condition ðh0 > hf Þ:
hð0; tÞ¼h0; t > 0 ð43Þ
We can define the same transformations (13) and (14) as were done for the previous problem but now we get Z Z oyà y oh b oh y oh b oh ¼ b ðr; tÞdr þ ÀÁð0; tÞ¼b ðr; tÞdr þ ð0; tÞ: o o 2 o o 2 o t 0 t a þ bhð0; tÞ y 0 t ða þ bh0Þ y
Then Z Z ! Z t y o b oh y À Á yÃðy; tÞ¼ ða þ bhðr; sÞÞdr þ ð0; sÞ ds þ a þ bhðr; 0Þ dr o 2 o 0 0 s ða þ bh Þ y 0 Z Z 0 y b t oh ¼ ða þ bhðr; tÞÞdr þ ð0; sÞds ð44Þ 2 o 0 ða þ bh0Þ 0 y 1692 M.F. Natale, D.A. Tarzia / International Journal of Engineering Science 41 (2003) 1685–1698
Our problem becomes (20)–(22) and Z ohà o2hà b tà oh ¼ ; ð0; sÞds < yà < SÃðtÃÞ; tà > 0 ð45Þ o o à 2 o t y ða þ bh0Þ 0 y Z ! b tà oh hà ð0; sÞds; tà ¼ hà ð46Þ 2 o 0 ða þ bh0Þ 0 y
à à where hf and a are given by (23) and
à 1 1 1 h0 ¼ ¼ ¼ ð47Þ a þ bhð0; tÞ a þ bhð0; tÞ a þ bh0
It easy to see that we have a classical Stefan problem so that the free boundary must be of the type pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi SÃðtÃÞ¼ 2cÃtà SðtÞ ¼ 2ct ð48Þ where cà (i.e. c) is a dimensionless constant to be determined. If we introduce the similarity variable (24) and we propose the solution of the type (25) then the problem (20)–(22) and (45) and (46) yields (28), (29) and Z d2Hà dHà b tà oh þ cÃnà ¼ 0; pffiffiffiffiffiffiffiffiffiffi ð0; sÞds < nà < 1; tà > 0 ð49Þ Ã2 à 2 à à o dn dn ða þ bh0Þ 2c t 0 y Z ! b tà oh Hà pffiffiffiffiffiffiffiffiffiffi ð0; sÞds ¼ hÃ; tà > 0 ð50Þ 2 à à o 0 ða þ bh0Þ 2c t 0 y
à From (50) we must necessarily have that there exist a constant n0 such that
Z à t oh ða þ bh Þ2 pffiffiffiffiffiffiffiffiffiffi 0; s ds nà 0 2cÃtà 51 o ð Þ ¼ 0 ð Þ 0 y b
Therefore (49) and (50) can be written as
d2Hà dHà þ cÃnà ¼ 0; nà < nà < 1 ð52Þ dnÃ2 dnà 0
à à à à H ðn0Þ¼h0; t > 0 ð53Þ M.F. Natale, D.A. Tarzia / International Journal of Engineering Science 41 (2003) 1685–1698 1693
The solution of the problem (28), (29), (52), (53) is given by rffiffiffiffi ! cà HÃðnÃÞ¼A0 erf nà þ B0; nà < nà < 1 ð54Þ 2 0
à 0 0 à where the unknown coefficients n0; A ; B and c must satisfy the followingequations: rffiffiffiffi! cà A0 erf nà þ B0 ¼ hÃ; t > 0 ð55Þ 0 2 0 sffiffiffiffi 2 cà aà pffiffiffi exp ¼ p ð56Þ cà 2 A0 rffiffiffiffi! cà A0 erf þ B0 ¼ hà ð57Þ 2 f
Firstly, we shall obtain the followingpreliminary result.
à Lemma 1. There exists a constant q0 > 0 such that
oh qà ð0; tÞ¼ p0ffiffi ; 8t > 0 ð58Þ oy t where A0 cÃnÃ2 à ffiffiffi 0 q0 ¼ p Ã3 exp ð59Þ b ph0 2
Proof. From (14), (44), (25) and (54) we have oh A0ða þ bh Þ cÃnÃ2 ð0; tÞ¼ qffiffiffi0 exp 0 ; t > 0 ð60Þ o pffiffiffiffiffi 2 y 0 à cà 0 2 b pt A erf n0 2 þ B then we find that
0 à Ã2 à A ða þ bh0Þ c n0 q0 ¼ qffiffiffi 2 exp ð61Þ pffiffiffi à 0 à c 0 2 b p A erf n0 2 þ B 1694 M.F. Natale, D.A. Tarzia / International Journal of Engineering Science 41 (2003) 1685–1698 or equivalently A0 cÃnÃ2 à ffiffiffi 0 q0 ¼ p Ã3 exp ð62Þ b ph0 2
0 Ã Ã Moreover, from (56) we have that A > 0, then q0 > 0:
Remark 1. We have that Z bhÃ2 t oh nà ¼ pffiffiffiffiffiffiffiffi0 ð0; sÞds 0 à o 2c t 0 y and then for Lemma 1, we obtain sffiffiffiffi 2 nà ¼ bqÃhÃ2 ð63Þ 0 0 0 cÃ
0 0 Ã Ã Now we have to solve the system (55)–(57), (62) and (63) where A ; B ; c and n0: From (55) and (57) we have
hà hà A0 ¼ qffiffiffi0 f qffiffiffi ð64Þ Ã cà cà erf n0 2 erf 2 qffiffiffi qffiffiffi à à cà à cà hf erf n0 2 h0erf 2 B0 ¼ qffiffiffi qffiffiffi ð65Þ Ã cà cà erf n0 2 erf 2
Now, we have to solve the followingsystem:
expð b2Þ ¼ a ðerfðzÞ erfðbÞÞ ð66Þ b 1
expð z2Þ ¼ a ðerfðzÞ erfðbÞÞ ð67Þ z 2 where rffiffiffiffi rffiffiffiffi cà cà b ¼ ; z ¼ nà ; 2 0 2 ffiffiffi ffiffiffi Ãp p a p a pða þ bh0Þ a1 ¼ à à ¼ < 0 ð68Þ h0 hf ðab þ aÞðh0 hf Þ M.F. Natale, D.A. Tarzia / International Journal of Engineering Science 41 (2003) 1685–1698 1695 ffiffiffi ffiffiffi Ãp p h0 p pða þ bhf Þ a2 ¼ à à ¼ > 0 ð69Þ h0 hf bðh0 hf Þ
Then, we have now to solve (66) and
a b expðb2Þ¼ 2 z expðz2Þð70Þ a1
From (66) we have z ¼ erf 1ðW ðbÞÞ where function W is defined by
1 W ðxÞ¼ expð x2ÞþerfðxÞ a1x
with W ðx1Þ¼ 1 and b > x1: Then we can write (67) in the followingway
2 b expðb Þ¼BðbÞ; b > x1 ð71Þ where function B is defined by
a2 1 1 2 BðxÞ¼ erf ðW ðxÞÞ expððerf ðW ðxÞÞÞ Þ for x > x1: a1
Takinginto account that B is a decreasingfunction, Bðx1Þ¼þ1, Bðþ1Þ ¼ 1, then there exists 1 a unique solution b > x1 of Eq. (71) and then we have z ¼ erf ðW ðbÞÞ:
Finally, resumingthe previous results we have the followingtheorem.
Theorem 2. The free boundary problem (1), (43), (3)–(5) has a unique solution of a similarity type for all data q0; q; c; l; hf ; a; b; d. Moreover, the solution is given by
2 3 1 6 1 7 hðnÞ¼ 4 qffiffiffi a5 b 0 cà à 0 A erf 2 n þ B hi 1 2 1 dx 2 1 y d ð þ Þ ð72Þ n ¼ pffiffiffiffiffiffi ¼ pffiffiffiffiffiffi 2ct 2ct 2 ! 3 rffiffiffiffi 2 1 ct sðtÞ¼ 4 1 þ d 15 d 2 1696 M.F. Natale, D.A. Tarzia / International Journal of Engineering Science 41 (2003) 1685–1698 where
" rffiffiffiffi !# Z à n cà n ¼ðab þ aÞ A0 erf r þ B0 dr nà 2 jn¼0 ð73Þ cà c ¼ ðab þ aÞ2 and the coefficients A0 and B0 are given by
bðh h Þ A0 ¼ f 0qffiffiffi qffiffiffi ð74Þ Ã cà cà ða þ bhf Þða þ bh0Þ erf n0 2 erf 2 qffiffiffi qffiffiffi à cà cà ða þ bh0Þerf n0 2 ða þ bhf Þerf 2 B0 ¼ qffiffiffi qffiffiffi ð75Þ Ã cà cà ða þ bhf Þða þ bh0Þ erf n0 2 erf 2
pffiffiffiffiffiffiffiffiffi à 2 à à where c ¼ 2b , n0 ¼ 2=c erfðW ðbÞÞ and b is the unique solution of the Eq. (71).
Remark 2. The case without the convective term in (1), that is d ¼ 0, it cannot be obtained from what we did previously for the case d 6¼ 0 and taking d ! 0; because the transformation x ! y through (13) is the identity since
h i 2 1 lim ð1 þ dxÞ2 1 ¼ x; 8x > 0: d!0 d
Then, the case d ¼ 0 must be solved by usingother techniques which will be developed in a forthcomingpaper.
4. Conclusion
For a one-phase Stefan problem for a semi-infinite material with temperature-dependent thermal conductivity (7) and a convective term (6) with a heat flux (2) or a constant temperature boundary condition (43) a parametric representation of the similarity type is obtained. For the heat flux boundary condition we have followed and improved [26] obtainingthe existence theorem in Section 2. Moreover a new temperature boundary condition is also studied obtainingthe correspondingexistence theorem in Section 3. For all cases their explicit solutions are givenas a functions of a parameter which is defined as the unique solution of a transcendental equation. M.F. Natale, D.A. Tarzia / International Journal of Engineering Science 41 (2003) 1685–1698 1697
Acknowledgement
This paper has been partially sponsored by the project ‘‘Free Boundary Problems for the Heat- Diffusion Equation’’ from CONICET-UA, Rosario (Argentina).
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