- Email: [email protected]
Approximate solution to Fredholm integral equations using linear regression and applications to heat and mass transfer
Engineering Analysis with Boundary Elements 36 (2012) 1278–1283
Contents lists available at SciVerse ScienceDirect
Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Approximate solution to Fredholm integral equations using linear regression and applications to heat and mass transfer Yiannos Ioannou a, Marios M. Fyrillas b,n, Charalabos Doumanidis a a b
Department of Mechanical Engineering, University of Cyprus, 1678 Nicosia, Cyprus Department of Mechanical Engineering, Frederick University, 1303 Nicosia, Cyprus
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 June 2011 Accepted 8 February 2012 Available online 29 March 2012
In this work we develop improved asymptotic solutions to one-dimensional Fredholm integral equations of the first kind using linear regression. For the cases under consideration the unknown function is the flux distribution along a strip, and the integral equation depends on a parameter or a number of parameters, i.e. the Pe´clet number, the Biot number, the dimensionless length scales etc. It is assumed that asymptotic solutions, with respect to the parameters, are available. We show that the asymptotic solutions can be improved and extended by relaxing the coefficients associated with them and applying regression analysis to yield best-fit coefficients. The asymptotic solutions may even be combined to obtain a matched asymptotic expansion. Explicit expressions for the coefficients, which can depend on a number of parameters, are obtained using regression analysis, i.e. by creating a variational principle for the Fredholm Integral Equation and employing the least squares method. The resulting expression, although it provides an approximate solution to the flux distribution, it is explicit and estimates accurately the overall transport rate. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Heat/mass transfer Integral equations Regression analysis Matched asymptotic expansions
1. Introduction In general, through a class of numerical methods known under the aliases boundary integral, boundary element, boundary integral-equation, panel and Green’s function methods, the classical scalar transport process in a medium [1] can be described by a kernel function (Green’s function G), which depends on a number of parameters (Pi) characterizing the material/process conditions, and the flux distribution (q) [2,3]. The resulting equation is a convolution integral equation known as a Fredholm integral equation [4]. This class of numerical methods offers the natural choice for inverse transport problems as it combines numerical simplicity and accuracy. Regarding the former, the reduction of the differential equation to an integral equation over the boundary reduces the dimensionality, hence the complexity, of the problem. Furthermore, the integral representation allows the consideration of an infinite-domain, direct calculation of the concentration gradient, and de-singularization of the singular points which translate into a high degree of accuracy [5–12,14–16,21]. Although the resulting integral equation is linear, it is notoriously difficult to solve and explicit solutions are only available for some special
n
Corresponding author. E-mail addresses: [email protected] (Y. Ioannou), [email protected] (M.M. Fyrillas), [email protected] (C. Doumanidis). 0955-7997/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2012.02.006
forms of the kernel [31]. Techniques for obtaining asymptotic solutions have appeared in the literature [8–12,42,43,14,17–20] however, it is often difficult to proceed to higher order or to obtain a matched asymptotic expansion. In natural and technological processes described by scalar (mass, energy, charge, etc.) transport phenomena, inverse transport problems have recently re-gained the keen interest of the scientific and engineering communities. Such inverse transport formulations seek the requisite input flux distribution (q), as well as the total flux (Q) required, in order to obtain a specified density distribution in the continuum (e.g. concentration, temperature, potential, etc.). The reason for this rekindled interest is due to both technical and mathematical progress in transport research: Technological means have become available over the past few decades for control of the flux distribution q, via high-bandwidth continuous scanning sources of lower dimensionality (e.g. robotic or servomechanism-guided plasma or laser heat torches [22–24], high-density infrared lamp strips [25], scanned electron or ion implantation beams [26], etc.). In addition, highly localized discrete stationary sources, at dimensions down to the nanoscale (e.g. nanoheaters [27,28]) and in custom-designed distributions or addressable multiplexed array configurations, are presently introduced for precise implementation of transport actuation in a variety of applications (self-heating, self-repaired materials, etc.). From the computational perspective, aside from off-line numerical simulation formulations (finite difference, finite element,
Y. Ioannou et al. / Engineering Analysis with Boundary Elements 36 (2012) 1278–1283
boundary value methods, etc.) and useful software tools that have become ubiquitous in engineering practice, there is also renewed attention to analytical techniques for inverse transport problems. Analytical solutions afford unique physical insights on the explicit effects of transport parameters, as well as computational efficiencies enabling their in-process technical implementation, in conjunction with the new actuation technologies mentioned above. Typical such analytical methods involve an optimization approach in approximating the desired density distribution in the continua by the action of combined/distributed elementary influxes (e.g. via a Green–Galerkin technique [29]). However, in an effort to avoid real-time numerical computation costs as much as possible during transport applications, there is a need for efficient optimization in the solution of the Fredholm integral equations (describing linear, stationary transport in a low-dimensionality continuum) by utilization of known analytical distribution functions in the asymptotic limits of the process parameters, i.e. corresponding to simpler, ideal physical problems. Motivated by such a necessity, in the next Section 2 we develop a method that can improve the existing asymptotic solutions of Fredholm integral equations of the first kind, through a least squares regression method. In the following Section 3, the method is implemented in a number of examples motivated from classical problems in heat and mass transfer.
2. Development of the least squares regression method
values of the parameter P, through which we can also obtain the total transport rate Z 1 Q ½P ¼ q½x; P dx: ð4Þ 0
The improved asymptotic solution that follows is motivated by the fact that numerical calculations demonstrate the persistence of the asymptotic functional relations 1=f 0 ½x and 1=f 1 ½x beyond the region of validity implied by the respective asymptotic analysis. Hence, this suggests a regression analysis where the solution is approximated as the sum of the two asymptotic solutions (2) and (3), a0m ½P a1m ½P þ , f 0 ½x f 1 ½x
qls ½x; P ¼
T s ½x ¼
Z
1
q½x0 ; P G½x,x0 ; P dx0 ,
ð1Þ
0
where G½x; P is the kernel i.e. the Green’s function associated with the problem, q½x; P is the unknown function i.e. the flux distribution, T s ½x is a known function i.e. the temperature distribution along a strip, and P is a dimensionless parameter (or parameters Pi) associated with the physical problem, i.e. the Pe´clet number, the Biot number, the length scales, etc. Here, we should point out that explicit solution for such equations are only available for some special forms of the kernel [31]. To proceed, we further assume that explicit forms of the unknown function q½x; P are available in the asymptotic limits of the parameter P, which we denote as q0 ½x; P 5 1 and q1 ½x; P b 1. In heat and mass transfer applications the functions q0 and q1 take the form [9–12,14] a0 ½P q0 ½x; P 51 ¼ f 0 ½x
ð2Þ
and q1 ½x; P b 1 ¼
0
where 1
0
G½x,x0 ; P 0 dx , f 0 ½x0
ð7Þ
and Z
g 1 ½x; P ¼
1 0
G½x,x0 ; P 0 dx : f 1 ½x0
ð8Þ
Minimizing the functional (6) leads to the following expressions for a0m ½P and a1m ½P G0 G12 G10 G1
a0m ½P ¼
G02 G12 G21
,
ð9Þ
,
ð10Þ
and a1m ½P ¼
G02 G1 G10 G0 G02 G12 G21
where G0 ½P ¼
Z
1
g 0 ½x; P T s ½x dx,
0
G1 ½P ¼
Z
1
g 1 ½x; P T s ½x dx,
0
G02 ½P ¼
ð3Þ
where f 0 ½x and f 1 ½x are explicit functions of x, and a0 ½P, a1 ½P are explicit functions of the parameter P. The expressions for q0 and q1 can be obtained through boundary layer analysis [9,32], the Wiener–Hopf technique [7,8,14], by asymptotic analysis [9–12], or combination of the above techniques. We are interested to obtain an approximate expression for the flux distribution q½x; P for arbitrary
Z
g 0 ½x; P ¼
G12 ½P ¼ a1 ½P , f 1 ½x
ð5Þ
where however, the coefficients a0m ½P and a1m ½P are now free expressions [33–36,20] that are going to be estimated using a variational principle. The integral equation may now be transformed into a regression problem [37,38]. We proceed using least squares, however one might attempt to use collocation, minimax or some other method. Here we should point out that the particular form of qls along with the choice of regression method may offer opportunities for addressing integral equations using numerical minimization techniques. Applying least squares regression leads to the following minimization problem associated with Fredholm integral equation (1): Z 1 minimize ða0m ½P g 0 ½x; P þ a1m ½P g 1 ½x,PT s ½xÞ2 dx, ð6Þ ða0m ,a1m Þ
In this section, we develop a method that can improve and extend existing asymptotic solutions of Fredholm integral equations of the first kind. The limitations and assumptions, associated with the applicability of the method, are outlined during the development of the method that follows. Consider a one-dimensional Fredholm integral equation of the first kind [4,7–12,14,30]. Without loss of generality we can assume the form
1279
G10 ½P ¼
Z
1
0
Z
ðg 0 ½x; PÞ2 dx,
1
0
Z 0
ðg 1 ½x; PÞ2 dx,
1
g 0 ½x; P g 1 ½x; P dx:
ð11Þ
The expressions for the coefficients a0m ½P and a1m ½P (Eqs. (9) and (10)), along with the integrals ((7), (8) and (11)), constitute an approximate explicit result for the unknown function q½x; P (Eq. (5)). In the next section, we compare results obtained through least squares regression, with both asymptotic
1280
Y. Ioannou et al. / Engineering Analysis with Boundary Elements 36 (2012) 1278–1283
and numerical results [2,6,9–12]. The latter were obtained through the collocation boundary element method.
a0m ½H qls ½x; H ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ a1m ½H, xð1xÞ T s ½x ¼ 1,
3. Numerical examples In this section, we apply the method developed in the previous section ((5)–(11)) to six fundamental problems of heat and mass transfer. The results suggest that the approximate solution developed through least squares regression is quite successful in predicting the overall transport rate (Eq. (4)), and can be regarded as a Pade´ approximation [13]. In particular, a comparison with numerical results obtained through the collocation boundary element method [2,6,9–12], demonstrate that the deviation does not exceed 2%. In addition, the expression provides a best-fit approximation to the flux distribution.
where the parameter H is the dimensionless thickness of the slab. The relevant equations (Eq. (11)) take the form Z Z p 9xx0 9 1 1 1 ln½coth½4H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx0 dx, G0 ½P ¼ p 0 0 xð1xÞ G1 ½P ¼
G02 ½P ¼
3.1. Heat conduction in a slab due to an isothermal strip
0
G½x,x ; H ¼
1
p
ln½coth½ p 9xx0 9 4H
0
dx ,
Fig. 1. In the upper figure we show a schematic representation of the physical problem associated with heat conduction from a single isothermal strip (Section 3.1). The dots represent results obtained by least squares regression (Eqs. (5)–(11)), and the circles represent numerical results of the integral equation (Eq. (12)). In (a) we show the total transport rate (Shape Factor) as a function of the parameter H, while in (c) we show the flux distribution along the strip for H¼0.2. In (b) we have kept only one term, i.e. the a1m term, in the least squares regression. The choice of axes and the parameters is such that a direct comparison can be performed with Fig. 4 of Ref. [10].
Z
1
p
1
Z
0
G10 ½P ¼
p
1 0
2 Z 1
G12 ½P ¼
In the case of heat conduction in a solid rectangular slab embedded with an isothermal, strip [10,39–41], the formulation is as follows (Fig. 1):
ð12Þ
1 0
2 Z 1
p
0
p Z 0
1
Z
p 9xx0 9 ln½coth½4H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx0 x0 ð1x0 Þ
1 0
1 0
1
Z 0
1
2 Z 1
p 9xx0 9 dx0 dx, ln½coth½4H
Z
1 0
!2 dx, !2
p 9xx0 9 dx0 ln½coth½4H
dx,
p 9xx0 9 ln½coth½4H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx0 x0 ð1x0 Þ !
p 9xx00 9 dx00 dx: ln½coth½4H
In Fig. 1a we show results for the total transport rate Q (Eq. (4)) obtained through both the least squares regression (Eqs. (7)–(11)) and through direct numerical solution of the integral equation using the collocation method. There is excellent agreement between the results, and the error does not exceed 1%. In Fig. 1c, we also show results for the flux distribution q[x]. There is qualitative agreement between the results, however higher order terms must be included in the expansion in order to achieve quantitative accuracy. Finally, in Fig. 1b, we show results for the total transport rate Q obtained by keeping only one term (a1m ) in the linear regression approximation (12). Even in this oversimplified case, the overall transport rate is in good agreement with numerical calculations.
Fig. 2. In the upper figure we show a schematic representation of the physical problem associated with heat conduction due to periodic array of isothermal strips (Section 3.2). In (a) and (b) we show the total transport rate (Shape Factor) as a function of the parameters L/H and 1/H, respectively: (a) H¼10, (b) L/H¼ 5. The dots represent results obtained by least squares regression (Eqs. (5)–(11)), and the circles represent numerical results of integral equation (13). The choice of axes and the parameters is such that a direct comparison can be performed with Figs. 5 and 6 of Ref. [11].
Y. Ioannou et al. / Engineering Analysis with Boundary Elements 36 (2012) 1278–1283
3.2. Heat conduction in a slab due to a periodic array of isothermal strips The heat conduction in a slab embedded with an array of isothermal strips (Fig. 2) was treated in [11,15,16]. In the case of periodic array of strips the formulation is as follows [11]: ! 1 H 1 X 1 H ðxx0 Þ G½x,x0 ; H,L ¼ þ tanh 2pm cos 2pm , L L L pm¼1m a0m ½H,L qls ½x; H,L ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ a1m ½H,L, xð1xÞ T s ½x ¼ 1,
where H, L and Bi are the dimensionless numbers associated with the problem [12]. In Fig. 3 we show results for the resistance (R), defined as 1/Q (Eq. (4)), obtained through both the least squares regression method (Eqs. (7)–(11)) and through direct numerical solution of the integral equation using the collocation method. There is excellent agreement between the results and the error does not exceed 1% except for small values of the parameter H where it exceeds 2%. The reason is that an exponentially thin boundary layer develops close to the edges [11,14] and qls does not adequately describe this limit because an asymptotic expansion for H 51 is not available.
ð13Þ
where H and L are the dimensionless numbers associated with the problem defined in Fig. 2. In Fig. 2 we show results for the total transport rate Q (Eq. (4)) obtained through both least squares regression (Eqs. (7)–(11)) and direct numerical solution of the integral equation. There is excellent agreement between the results, and the error does not exceed 1%. 3.3. Heat convection from a slab embedded with a periodic array of isothermal strips The problem of two-dimensional heat conduction in a solid slab embedded with a periodic array of isothermal strips, where the surfaces of the slab are subjected to a convective heat transfer is outlined in [12]. The formulation is as follows (Fig. 3): 1þ HBi BiL 1 X 1 2pm þ BiL tanh 2pLmH 2pm cos ðxx0 Þ , 2 p mH mp BiL þ 2pm tanh L L m¼1
G½x,x0 ; H,L,Bi ¼
a0m ½H,L,Bi qls ½x; H,L,Bi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ a1m ½H,L,Bi, xð1xÞ T s ½x ¼ 1,
1281
ð14Þ
Fig. 3. In the upper figure we show a schematic representation of the physical problem associated with heat convection due to a periodic array of isothermal strips (Section 3.3). In the lower figures we show the resistance (1/Q) as a function of the parameter H: (a) L¼ 3, Bi ¼0.5, (b) L¼5, Bi¼ 20. The dots represent results obtained by least squares regression (Eqs. (5)–(11)), and the circles represent numerical results of integral equation (14). The choice of axes and the parameters is such that a direct comparison can be performed with Figs. 4b and 5 of Ref. [12].
3.4. Mass transfer from a NAPL pool The two-dimensional problem of advection dispersion associated with a non-aqueous phase liquid (NAPL) pool was addressed using the boundary element method in [9]. A similar problem was addressed in [14] in the context of steady advection–diffusion around finite absorbers in two-dimensional potential flows. The formulation is as follows (Fig. 4): G½x,x0 ; Pe, L ¼
pffiffiffiffiffi Pe
p
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp½Peðxx0 Þ=2 K 0 ½ PeðPe=4 þ LÞ9xx0 9 dx0 ,
a0m ½Pe a1m ½Pe qls ½x; Pe, L ¼ 0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffi , x xð1xÞ C s ½x ¼ 1,
ð15Þ
where Pe is the Pe´clet number associated with the problem [9], i.e. Pe ¼ U ‘=D where U is the fluid velocity, ‘ is the length of the strip and D is the dispersion coefficient. In Fig. p 4 ffiffiffiffiffi we show results for the Sherwood number (Sh) defined as PeQ (Eq. (4)) obtained through both the least squares regression (Eqs. (7)–(11)), and through numerical solution of the integral equation. There is excellent agreement between the results and the error does not exceed 1%. In Fig. 4b, in view of the fact that L is not zero, we have used a different expression for the flux distribution in the linear
Fig. 4. In the upper figure we show a schematic representation of the physical problem associated with mass transfer from a NAPL pool into a uniform flow pffiffiffiffiffi (Section 3.4). In the lower figures we show plots of the Sherwood number ( PeQ ) as a function of the parameter Pe: (a) L ¼ 0, (b) L ¼ 10. The dots represent results obtained by least squares regression (Eqs. (5)–(11)), and the circles represent numerical results of integral equation (15). The choice of axes and the parameters is such that a direct comparison can be performed with Fig. 5 of Ref. [9].
1282
Y. Ioannou et al. / Engineering Analysis with Boundary Elements 36 (2012) 1278–1283
Shear flow (shear rate S) T =1 Insulated
1
0
Insulated x
10
10
10
10 10
10
10
10
10
10
Fig. 5. In the upper figure we show a schematic representation of the physical problem associated with mass transfer from a surface strip to shear flows (Section 3.5). In the lower figure we show a plot of the Nusselt number (Q) as a function of the parameter Pe. The dots represent results obtained by least squares regression (Eqs. (5)–(11)), and the circles represent numerical results of integral equation (16). The choice of axes and the parameters is such that a direct comparison can be performed with Fig. 6 of Ref. [5].
regression model pffiffiffiffi hpffiffiffiffiffiffiffii a0m ½Pe, L eLx qls ½x; Pe, L ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þa1m ½Pe, L Lerf Lx þ pffiffiffiffiffiffi px xð1xÞ motivated by the asymptotic analysis developed in Appendix B of Ref. [9]. 3.5. Heat/mass transfer from surface films to shear flows The problem of heat transfer (or mass transfer at low transfer rates) to a strip of finite length in a uniform shear flow is considered in [5,7,8]. The formulation is as follows [5,6,32,42]: Z 1 1 Ai½s 0 dw where s ¼ w2 =ðiwPeÞ2=3 , G½x,x0 ; Pe ¼ eiwðxx Þ 0 2p 1 ðiwPeÞ1=3 Ai ½s a0m ½Pe a1m ½Pe qls ½x; Pe ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ , x1=3 xð1xÞ T s ½x ¼ 1,
ð16Þ
where Pe is the Pe´clet number associated with the problem [5], i.e. Pe ¼ SL2 =k where k is the thermal conductivity, S is the constant shear rate and L is the length of the strip. In Fig. 5, we show results for the Nusselt number (Q, Eq. (4)) as a function of the Pe´clet number (Pe) obtained through both least squares regression (Eqs. (7)–(11)), and through numerical solution of the integral equation. There is excellent agreement between the results and the error does not exceed 1%. 3.6. Heat conduction in sliding solids
0
qls ½x; Pe1 ,Pe2 ,kr ¼ a1m ½Pe1 ,Pe2 T s ½x ¼ ðxePe2 x ðK 0 ½Pe2 x þ K 1 ½Pe2 xÞ þ ð1xÞePe2 ð1xÞ ðK 0 ½Pe2 ð1xÞK 1 ½Pe2 ð1xÞÞÞ,
ð17Þ
where the Pe´clet numbers Pe1 ,Pe2 (Pei ¼ V i ‘=2ai where Vi denotes the speed of the bodies, ‘ the contact length, and ai the thermal diffusivities) refer to body 1 and 2 respectively, and kr ¼ k2 =k1 is the ratio of the thermal conductivities [43]. In Fig. 6, we show results for the total transport rate (Q, Eq. (4)) as a function of the Pe´clet number (Pe1) obtained through both least squares regression (Eqs. (7)–(11)), and through numerical solution of the integral equation. There is excellent agreement between the results and the error does not exceed 1%. In Fig. 6 we also include the asymptotic result obtained by Yuen [43]; the comparison suggests that the least squares regression leads to a significant improvement of the asymptotic result.
4. Conclusions
The problem of heat conduction between sliding solids where heat energy is generated along the region of contact is considered (Fig. 6). The formulation is as follows [43]: G½x,x0 ; Pe1 ,Pe2 ,kr ¼ ePe1 ðxx Þ K 0 ½Pe1 9xx0 9 þ
Fig. 6. In the upper figure we show a schematic representation of the physical problem associated with heat conduction in sliding solids (Section 3.6). In the lower figure we show a plot of the Nusselt number (Q) as a function of the parameter Pe1 for Pe2 ¼0.01 and kr ¼ 1. The dots represent results obtained by least squares regression (Eqs. (5)–(11)), the circles represent numerical results of the integral equation, and the crosses are the asymptotic solution obtained by Yuen [43].
1 Pe2 ðxx0 Þ e K 0 ½Pe2 9xx0 9 kr
We have developed a method that improves and extends existing asymptotic solutions of Fredholm integral equations of the first kind. The method relies on describing the unknown function, i.e. the flux distribution, through a linear combination of known asymptotic approximations of the integral equation.
Y. Ioannou et al. / Engineering Analysis with Boundary Elements 36 (2012) 1278–1283
Subsequently, the integral equation is reformulated as a variational principle, i.e. least squares regression, and explicit expressions for the flux distribution can be obtained. The method has been applied to fundamental heat and mass transfer problems presented in the literature. The results suggest that the method is quite accurate, efficient, versatile and scalable. It has been successfully applied to problems with multiple parameters and to problems where limited asymptotic solutions are available. It is tempting to infer that the improved asymptotic solution is a matched asymptotic expansion, and that matching through least squares regression may not be limited to integral equations. More importantly, given that the integral equation formulation is the natural choice for formulating the inverse transport problem, and that the method provides tunable precision (via employing higher-order terms as needed), these make it invaluable for in-process actuation control at a trivial computational cost, especially for low-bandwidth (slow) transport processes. Towards such real-time implementation of the method in technical applications, two related problems will be attacked in future work. First, inevitable process disturbances and parameter alterations in real engineering settings are reflected in the current formulation as uncertainties in the kernel Green’s function, i.e. in the parameters of the asymptotic expressions eventually affecting the parameters in the least square solution. An approach dual to the control problem of actuation input addressed above, i.e. the observation of resulting output distributions for in-process identification of the requisite uncertain process parameters, will be tacked using similar least-squares optimization algorithms. Such a real-time observer/identification technology is enabled by technological progress in non-intrusive, non-destructive sensory measurement methods, e.g. infrared pyrometry, ultrasound, x-ray scanning microtomography, etc. Second, for relatively higherbandwidth (faster) transport conditions, a transient formulation of the method will be developed by addressing Fredholm integral equations of the second kind through appropriate least-squares optimization dynamic formulation and dynamic asymptotic expressions. These two future developments will enable application of the inverse/control methodology to manufacturing processes, such as selective laser sintering, etc.
Acknowledgment The work was funded by Porfyrios CHAP Glass Ltd. References [1] Incropera FP, DeWitt DP. Fundamentals of heat and mass transfer. John Wiley and Sons; 1990. [2] Brebbia CA. Boundary elements: an introductory course. 2nd ed. Computational Mechanics; 1992. [3] Carslaw HS, Jaeger JC. Conduction of heat in solids.Clarendon Press; 1959. [4] Porter D, Stirling DSG. Integral equations: a practical treatment, from spectral theory to applications.Cambridge University Press; 1990. [5] Ackerberg RC, Patel RD, Gupta SK. The heat/mass transfer to a finite strip at small Pe´clet numbers. J Fluid Mech 1978;86:49. [6] Stone HA. Heat/mass transfer from surface films to shear flows at arbitrary Peclet numbers. Phys Fluids A 1989;1:1112. [7] Springer SG, Pedley TJ. The solution of heat transfer problems by the Wiener– Hopf technique. I. Leading edge of a hot film. Proc R Soc London A 1973;333:347. [8] Springer SG. The solution of heat transfer problems by the Wiener–Hopf technique. II. Trailing edge of a hot film. Proc R Soc London A 1974;337:395. [9] Fyrillas MM. Advection–dispersion mass transport associated with a nonaqueous-phase liquid pool. J Fluid Mech 2000;413:49.
1283
[10] Fyrillas MM. Heat conduction in a solid slab embedded with a pipe of general cross-section: shape factor and shape optimization. Int J Eng Sci 2008;46:907. [11] Fyrillas MM. Shape factor and shape optimization for a periodic array of isothermal pipes. Int J Heat Mass Transfer 2010;53:982. [12] Fyrillas MM, Stone HA. Critical insulation thickness of a slab embedded with a periodic array of isothermal strips. Int J Heat Mass Transfer 2011;54:180. [13] Rajendran L. Two-point Pade´ approximation of mass transfer rate at microdisc in a channel flow for all Pe´clet numbers. Electochim Acta 2006;51:5407. [14] Choi J, Margetis D, Squires TM, Bazant MZ. Steady advection–diffusion around finite absorbers in two-dimensional flows. J Fluid Mech 2005;536: 155. [15] Homentcovschi D, Jude L. Potential due to some arbitrary charged strips between parallel earthed planes. Int J Eng Sci 1984;22:775. [16] De Mey G. Temperature distribution in floor heating systems. Int J Heat Mass Transfer 1980;23:1289. [17] Latta GE. The solution of a class of integral equations. Int J Ration Mech Anal 1956;5:821. [18] Lundgren T, Chiang D. Solution of a class of singular integral equations. Q Appl Math 1967;24:303. [19] Terry Hollands KG. The simplified-Fredholm integral equation solver and its use in thermal radiation. ASME J Heat Transfer 2010;132:1. [20] Tikhonov AN, Glasko VB. The approximate solution of Fredholm integral equations of the first kind. USSR Comput Math Math Phys 1964;55:236. [21] Fyrillas MM, Kontoghiorghes EJ. Numerical calculation of mass transfer from elliptical pools in uniform flow using the boundary element method. Transp Porous Media 2004;55:91. [22] Demetriou MA, Paskaleva A, Vayena O, Doumanidis H. Scanning actuator guidance scheme in a 1-D thermal manufacturing process. IEEE Trans Control Syst Technol 2003;11(5):757–64. [23] Hardjadinata G, Doumanidis C. Rapid prototyping by laser foil bonding and curing: thermomechanical modeling and process optimization. J Manuf Process 2001;3(2):108–19. [24] Ioannou Y, Fyrillas MM, Polychronopoulou K, Doumanidis C. Analytical model for geometrical characteristics control of laser sintered surfaces. Int J NanoManuf 2010;6:300. [25] Alaeddine M, Ranganathan R, Ando T, Doumanidis C. Modeling the melting and dissolution stages during thermal processing of intermetallic coatings from layered precursors. Trans ASME J Manuf Sci Eng 2005;127:148–56. [26] Zeng T, Doumanidis C, Hebb J, Brown D. Growth of ultrathin silicon dioxide films during rapid thermal oxidation. In: Proceedings of the 9th international conference on RTP 2001 advanced thermal processing of semiconductors; 2001. p. 287–95. [27] Jogdand H, Gulsoy G, Ando T, Chen J, Doumanidis CC, Gu Z, et al. Fabrication and characterization of nanoscale heating sources (‘‘nanoheaters’’) for nanomanufacturing. In: NSTI Nanotechnology Conference, Boston, MA, June 1–5, 2008, p. 280. [28] Cui Q, Jogland H, Chen J, Gus Z. Structures and ignition properties of nanoheaters formed by bimetallic Al–Ni reactive nanostructures. Mater Res Soc Symp Proc 2009;1148. [29] Alaeddine M, Doumanidis C. Distributed parameter thermal system control and observation by Green–Galerkin methods. Int J Numer Methods Eng 2004;61:1921–37. [30] Liu T, Campbell BT, Sullivan JP. Surface temperature of a hot film on a wall in shear flow. Int J Heat Mass Transfer 1994;37:2809. [31] Magnus W, Oberhettinger F. Formulas and theorems for the functions of mathematical physics. New York: Chelsea Publishing Company; 1949. [32] Le´vˆeque MA. Transmission de chaleur par convection. Ann Mines 1928;13: 283. [33] Chang CL, Gunzburger MD. A subdomain-Galerkin/least squares method for first-order elliptic systems in the plane. SIAM J Numer Anal 1990;27:1197. [34] Collatz L. The numerical treatment of differential equations. New York: Springer-Verlag; 1966. 184 and 220-221. [35] Hanke M. On a least squares collocation method for linear differentialalgebraic equations. Numer Math 1988;54:79. [36] Zwillinger D. Handbook of differential equations.Academic Press Inc; 1992. [37] Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical recipes. Cambridge University Press; 1992. [38] Kreyszig E. Advanced engineering mathematics.John Wiley & Sons; 1993. [39] Payvar P. Analytical solution of a new mixed boundary-value problem in heat conduction. Q J Mech Appl Math 1979;32:253. [40] Ranger KB. The solution of a new mixed boundary value problem by use of an integral transformation. Int J Eng Sci 1974;12:853. [41] Sadhal SS. Explicit solutions to a class of mixed boundary value problems. Int J Eng Sci 1981;19:1053. [42] Phillips GP. Heat and mass transfer from a film into steady shear flow. Q J Mech Appl Math 1990;43:135. [43] Yuen WYD. Heat conduction in sliding solids. Int J Heat Mass Transfer 1988;31:637.
Contents lists available at SciVerse ScienceDirect
Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Approximate solution to Fredholm integral equations using linear regression and applications to heat and mass transfer Yiannos Ioannou a, Marios M. Fyrillas b,n, Charalabos Doumanidis a a b
Department of Mechanical Engineering, University of Cyprus, 1678 Nicosia, Cyprus Department of Mechanical Engineering, Frederick University, 1303 Nicosia, Cyprus
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 June 2011 Accepted 8 February 2012 Available online 29 March 2012
In this work we develop improved asymptotic solutions to one-dimensional Fredholm integral equations of the first kind using linear regression. For the cases under consideration the unknown function is the flux distribution along a strip, and the integral equation depends on a parameter or a number of parameters, i.e. the Pe´clet number, the Biot number, the dimensionless length scales etc. It is assumed that asymptotic solutions, with respect to the parameters, are available. We show that the asymptotic solutions can be improved and extended by relaxing the coefficients associated with them and applying regression analysis to yield best-fit coefficients. The asymptotic solutions may even be combined to obtain a matched asymptotic expansion. Explicit expressions for the coefficients, which can depend on a number of parameters, are obtained using regression analysis, i.e. by creating a variational principle for the Fredholm Integral Equation and employing the least squares method. The resulting expression, although it provides an approximate solution to the flux distribution, it is explicit and estimates accurately the overall transport rate. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Heat/mass transfer Integral equations Regression analysis Matched asymptotic expansions
1. Introduction In general, through a class of numerical methods known under the aliases boundary integral, boundary element, boundary integral-equation, panel and Green’s function methods, the classical scalar transport process in a medium [1] can be described by a kernel function (Green’s function G), which depends on a number of parameters (Pi) characterizing the material/process conditions, and the flux distribution (q) [2,3]. The resulting equation is a convolution integral equation known as a Fredholm integral equation [4]. This class of numerical methods offers the natural choice for inverse transport problems as it combines numerical simplicity and accuracy. Regarding the former, the reduction of the differential equation to an integral equation over the boundary reduces the dimensionality, hence the complexity, of the problem. Furthermore, the integral representation allows the consideration of an infinite-domain, direct calculation of the concentration gradient, and de-singularization of the singular points which translate into a high degree of accuracy [5–12,14–16,21]. Although the resulting integral equation is linear, it is notoriously difficult to solve and explicit solutions are only available for some special
n
Corresponding author. E-mail addresses: [email protected] (Y. Ioannou), [email protected] (M.M. Fyrillas), [email protected] (C. Doumanidis). 0955-7997/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2012.02.006
forms of the kernel [31]. Techniques for obtaining asymptotic solutions have appeared in the literature [8–12,42,43,14,17–20] however, it is often difficult to proceed to higher order or to obtain a matched asymptotic expansion. In natural and technological processes described by scalar (mass, energy, charge, etc.) transport phenomena, inverse transport problems have recently re-gained the keen interest of the scientific and engineering communities. Such inverse transport formulations seek the requisite input flux distribution (q), as well as the total flux (Q) required, in order to obtain a specified density distribution in the continuum (e.g. concentration, temperature, potential, etc.). The reason for this rekindled interest is due to both technical and mathematical progress in transport research: Technological means have become available over the past few decades for control of the flux distribution q, via high-bandwidth continuous scanning sources of lower dimensionality (e.g. robotic or servomechanism-guided plasma or laser heat torches [22–24], high-density infrared lamp strips [25], scanned electron or ion implantation beams [26], etc.). In addition, highly localized discrete stationary sources, at dimensions down to the nanoscale (e.g. nanoheaters [27,28]) and in custom-designed distributions or addressable multiplexed array configurations, are presently introduced for precise implementation of transport actuation in a variety of applications (self-heating, self-repaired materials, etc.). From the computational perspective, aside from off-line numerical simulation formulations (finite difference, finite element,
Y. Ioannou et al. / Engineering Analysis with Boundary Elements 36 (2012) 1278–1283
boundary value methods, etc.) and useful software tools that have become ubiquitous in engineering practice, there is also renewed attention to analytical techniques for inverse transport problems. Analytical solutions afford unique physical insights on the explicit effects of transport parameters, as well as computational efficiencies enabling their in-process technical implementation, in conjunction with the new actuation technologies mentioned above. Typical such analytical methods involve an optimization approach in approximating the desired density distribution in the continua by the action of combined/distributed elementary influxes (e.g. via a Green–Galerkin technique [29]). However, in an effort to avoid real-time numerical computation costs as much as possible during transport applications, there is a need for efficient optimization in the solution of the Fredholm integral equations (describing linear, stationary transport in a low-dimensionality continuum) by utilization of known analytical distribution functions in the asymptotic limits of the process parameters, i.e. corresponding to simpler, ideal physical problems. Motivated by such a necessity, in the next Section 2 we develop a method that can improve the existing asymptotic solutions of Fredholm integral equations of the first kind, through a least squares regression method. In the following Section 3, the method is implemented in a number of examples motivated from classical problems in heat and mass transfer.
2. Development of the least squares regression method
values of the parameter P, through which we can also obtain the total transport rate Z 1 Q ½P ¼ q½x; P dx: ð4Þ 0
The improved asymptotic solution that follows is motivated by the fact that numerical calculations demonstrate the persistence of the asymptotic functional relations 1=f 0 ½x and 1=f 1 ½x beyond the region of validity implied by the respective asymptotic analysis. Hence, this suggests a regression analysis where the solution is approximated as the sum of the two asymptotic solutions (2) and (3), a0m ½P a1m ½P þ , f 0 ½x f 1 ½x
qls ½x; P ¼
T s ½x ¼
Z
1
q½x0 ; P G½x,x0 ; P dx0 ,
ð1Þ
0
where G½x; P is the kernel i.e. the Green’s function associated with the problem, q½x; P is the unknown function i.e. the flux distribution, T s ½x is a known function i.e. the temperature distribution along a strip, and P is a dimensionless parameter (or parameters Pi) associated with the physical problem, i.e. the Pe´clet number, the Biot number, the length scales, etc. Here, we should point out that explicit solution for such equations are only available for some special forms of the kernel [31]. To proceed, we further assume that explicit forms of the unknown function q½x; P are available in the asymptotic limits of the parameter P, which we denote as q0 ½x; P 5 1 and q1 ½x; P b 1. In heat and mass transfer applications the functions q0 and q1 take the form [9–12,14] a0 ½P q0 ½x; P 51 ¼ f 0 ½x
ð2Þ
and q1 ½x; P b 1 ¼
0
where 1
0
G½x,x0 ; P 0 dx , f 0 ½x0
ð7Þ
and Z
g 1 ½x; P ¼
1 0
G½x,x0 ; P 0 dx : f 1 ½x0
ð8Þ
Minimizing the functional (6) leads to the following expressions for a0m ½P and a1m ½P G0 G12 G10 G1
a0m ½P ¼
G02 G12 G21
,
ð9Þ
,
ð10Þ
and a1m ½P ¼
G02 G1 G10 G0 G02 G12 G21
where G0 ½P ¼
Z
1
g 0 ½x; P T s ½x dx,
0
G1 ½P ¼
Z
1
g 1 ½x; P T s ½x dx,
0
G02 ½P ¼
ð3Þ
where f 0 ½x and f 1 ½x are explicit functions of x, and a0 ½P, a1 ½P are explicit functions of the parameter P. The expressions for q0 and q1 can be obtained through boundary layer analysis [9,32], the Wiener–Hopf technique [7,8,14], by asymptotic analysis [9–12], or combination of the above techniques. We are interested to obtain an approximate expression for the flux distribution q½x; P for arbitrary
Z
g 0 ½x; P ¼
G12 ½P ¼ a1 ½P , f 1 ½x
ð5Þ
where however, the coefficients a0m ½P and a1m ½P are now free expressions [33–36,20] that are going to be estimated using a variational principle. The integral equation may now be transformed into a regression problem [37,38]. We proceed using least squares, however one might attempt to use collocation, minimax or some other method. Here we should point out that the particular form of qls along with the choice of regression method may offer opportunities for addressing integral equations using numerical minimization techniques. Applying least squares regression leads to the following minimization problem associated with Fredholm integral equation (1): Z 1 minimize ða0m ½P g 0 ½x; P þ a1m ½P g 1 ½x,PT s ½xÞ2 dx, ð6Þ ða0m ,a1m Þ
In this section, we develop a method that can improve and extend existing asymptotic solutions of Fredholm integral equations of the first kind. The limitations and assumptions, associated with the applicability of the method, are outlined during the development of the method that follows. Consider a one-dimensional Fredholm integral equation of the first kind [4,7–12,14,30]. Without loss of generality we can assume the form
1279
G10 ½P ¼
Z
1
0
Z
ðg 0 ½x; PÞ2 dx,
1
0
Z 0
ðg 1 ½x; PÞ2 dx,
1
g 0 ½x; P g 1 ½x; P dx:
ð11Þ
The expressions for the coefficients a0m ½P and a1m ½P (Eqs. (9) and (10)), along with the integrals ((7), (8) and (11)), constitute an approximate explicit result for the unknown function q½x; P (Eq. (5)). In the next section, we compare results obtained through least squares regression, with both asymptotic
1280
Y. Ioannou et al. / Engineering Analysis with Boundary Elements 36 (2012) 1278–1283
and numerical results [2,6,9–12]. The latter were obtained through the collocation boundary element method.
a0m ½H qls ½x; H ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ a1m ½H, xð1xÞ T s ½x ¼ 1,
3. Numerical examples In this section, we apply the method developed in the previous section ((5)–(11)) to six fundamental problems of heat and mass transfer. The results suggest that the approximate solution developed through least squares regression is quite successful in predicting the overall transport rate (Eq. (4)), and can be regarded as a Pade´ approximation [13]. In particular, a comparison with numerical results obtained through the collocation boundary element method [2,6,9–12], demonstrate that the deviation does not exceed 2%. In addition, the expression provides a best-fit approximation to the flux distribution.
where the parameter H is the dimensionless thickness of the slab. The relevant equations (Eq. (11)) take the form Z Z p 9xx0 9 1 1 1 ln½coth½4H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx0 dx, G0 ½P ¼ p 0 0 xð1xÞ G1 ½P ¼
G02 ½P ¼
3.1. Heat conduction in a slab due to an isothermal strip
0
G½x,x ; H ¼
1
p
ln½coth½ p 9xx0 9 4H
0
dx ,
Fig. 1. In the upper figure we show a schematic representation of the physical problem associated with heat conduction from a single isothermal strip (Section 3.1). The dots represent results obtained by least squares regression (Eqs. (5)–(11)), and the circles represent numerical results of the integral equation (Eq. (12)). In (a) we show the total transport rate (Shape Factor) as a function of the parameter H, while in (c) we show the flux distribution along the strip for H¼0.2. In (b) we have kept only one term, i.e. the a1m term, in the least squares regression. The choice of axes and the parameters is such that a direct comparison can be performed with Fig. 4 of Ref. [10].
Z
1
p
1
Z
0
G10 ½P ¼
p
1 0
2 Z 1
G12 ½P ¼
In the case of heat conduction in a solid rectangular slab embedded with an isothermal, strip [10,39–41], the formulation is as follows (Fig. 1):
ð12Þ
1 0
2 Z 1
p
0
p Z 0
1
Z
p 9xx0 9 ln½coth½4H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx0 x0 ð1x0 Þ
1 0
1 0
1
Z 0
1
2 Z 1
p 9xx0 9 dx0 dx, ln½coth½4H
Z
1 0
!2 dx, !2
p 9xx0 9 dx0 ln½coth½4H
dx,
p 9xx0 9 ln½coth½4H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx0 x0 ð1x0 Þ !
p 9xx00 9 dx00 dx: ln½coth½4H
In Fig. 1a we show results for the total transport rate Q (Eq. (4)) obtained through both the least squares regression (Eqs. (7)–(11)) and through direct numerical solution of the integral equation using the collocation method. There is excellent agreement between the results, and the error does not exceed 1%. In Fig. 1c, we also show results for the flux distribution q[x]. There is qualitative agreement between the results, however higher order terms must be included in the expansion in order to achieve quantitative accuracy. Finally, in Fig. 1b, we show results for the total transport rate Q obtained by keeping only one term (a1m ) in the linear regression approximation (12). Even in this oversimplified case, the overall transport rate is in good agreement with numerical calculations.
Fig. 2. In the upper figure we show a schematic representation of the physical problem associated with heat conduction due to periodic array of isothermal strips (Section 3.2). In (a) and (b) we show the total transport rate (Shape Factor) as a function of the parameters L/H and 1/H, respectively: (a) H¼10, (b) L/H¼ 5. The dots represent results obtained by least squares regression (Eqs. (5)–(11)), and the circles represent numerical results of integral equation (13). The choice of axes and the parameters is such that a direct comparison can be performed with Figs. 5 and 6 of Ref. [11].
Y. Ioannou et al. / Engineering Analysis with Boundary Elements 36 (2012) 1278–1283
3.2. Heat conduction in a slab due to a periodic array of isothermal strips The heat conduction in a slab embedded with an array of isothermal strips (Fig. 2) was treated in [11,15,16]. In the case of periodic array of strips the formulation is as follows [11]: ! 1 H 1 X 1 H ðxx0 Þ G½x,x0 ; H,L ¼ þ tanh 2pm cos 2pm , L L L pm¼1m a0m ½H,L qls ½x; H,L ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ a1m ½H,L, xð1xÞ T s ½x ¼ 1,
where H, L and Bi are the dimensionless numbers associated with the problem [12]. In Fig. 3 we show results for the resistance (R), defined as 1/Q (Eq. (4)), obtained through both the least squares regression method (Eqs. (7)–(11)) and through direct numerical solution of the integral equation using the collocation method. There is excellent agreement between the results and the error does not exceed 1% except for small values of the parameter H where it exceeds 2%. The reason is that an exponentially thin boundary layer develops close to the edges [11,14] and qls does not adequately describe this limit because an asymptotic expansion for H 51 is not available.
ð13Þ
where H and L are the dimensionless numbers associated with the problem defined in Fig. 2. In Fig. 2 we show results for the total transport rate Q (Eq. (4)) obtained through both least squares regression (Eqs. (7)–(11)) and direct numerical solution of the integral equation. There is excellent agreement between the results, and the error does not exceed 1%. 3.3. Heat convection from a slab embedded with a periodic array of isothermal strips The problem of two-dimensional heat conduction in a solid slab embedded with a periodic array of isothermal strips, where the surfaces of the slab are subjected to a convective heat transfer is outlined in [12]. The formulation is as follows (Fig. 3): 1þ HBi BiL 1 X 1 2pm þ BiL tanh 2pLmH 2pm cos ðxx0 Þ , 2 p mH mp BiL þ 2pm tanh L L m¼1
G½x,x0 ; H,L,Bi ¼
a0m ½H,L,Bi qls ½x; H,L,Bi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ a1m ½H,L,Bi, xð1xÞ T s ½x ¼ 1,
1281
ð14Þ
Fig. 3. In the upper figure we show a schematic representation of the physical problem associated with heat convection due to a periodic array of isothermal strips (Section 3.3). In the lower figures we show the resistance (1/Q) as a function of the parameter H: (a) L¼ 3, Bi ¼0.5, (b) L¼5, Bi¼ 20. The dots represent results obtained by least squares regression (Eqs. (5)–(11)), and the circles represent numerical results of integral equation (14). The choice of axes and the parameters is such that a direct comparison can be performed with Figs. 4b and 5 of Ref. [12].
3.4. Mass transfer from a NAPL pool The two-dimensional problem of advection dispersion associated with a non-aqueous phase liquid (NAPL) pool was addressed using the boundary element method in [9]. A similar problem was addressed in [14] in the context of steady advection–diffusion around finite absorbers in two-dimensional potential flows. The formulation is as follows (Fig. 4): G½x,x0 ; Pe, L ¼
pffiffiffiffiffi Pe
p
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp½Peðxx0 Þ=2 K 0 ½ PeðPe=4 þ LÞ9xx0 9 dx0 ,
a0m ½Pe a1m ½Pe qls ½x; Pe, L ¼ 0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffi , x xð1xÞ C s ½x ¼ 1,
ð15Þ
where Pe is the Pe´clet number associated with the problem [9], i.e. Pe ¼ U ‘=D where U is the fluid velocity, ‘ is the length of the strip and D is the dispersion coefficient. In Fig. p 4 ffiffiffiffiffi we show results for the Sherwood number (Sh) defined as PeQ (Eq. (4)) obtained through both the least squares regression (Eqs. (7)–(11)), and through numerical solution of the integral equation. There is excellent agreement between the results and the error does not exceed 1%. In Fig. 4b, in view of the fact that L is not zero, we have used a different expression for the flux distribution in the linear
Fig. 4. In the upper figure we show a schematic representation of the physical problem associated with mass transfer from a NAPL pool into a uniform flow pffiffiffiffiffi (Section 3.4). In the lower figures we show plots of the Sherwood number ( PeQ ) as a function of the parameter Pe: (a) L ¼ 0, (b) L ¼ 10. The dots represent results obtained by least squares regression (Eqs. (5)–(11)), and the circles represent numerical results of integral equation (15). The choice of axes and the parameters is such that a direct comparison can be performed with Fig. 5 of Ref. [9].
1282
Y. Ioannou et al. / Engineering Analysis with Boundary Elements 36 (2012) 1278–1283
Shear flow (shear rate S) T =1 Insulated
1
0
Insulated x
10
10
10
10 10
10
10
10
10
10
Fig. 5. In the upper figure we show a schematic representation of the physical problem associated with mass transfer from a surface strip to shear flows (Section 3.5). In the lower figure we show a plot of the Nusselt number (Q) as a function of the parameter Pe. The dots represent results obtained by least squares regression (Eqs. (5)–(11)), and the circles represent numerical results of integral equation (16). The choice of axes and the parameters is such that a direct comparison can be performed with Fig. 6 of Ref. [5].
regression model pffiffiffiffi hpffiffiffiffiffiffiffii a0m ½Pe, L eLx qls ½x; Pe, L ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þa1m ½Pe, L Lerf Lx þ pffiffiffiffiffiffi px xð1xÞ motivated by the asymptotic analysis developed in Appendix B of Ref. [9]. 3.5. Heat/mass transfer from surface films to shear flows The problem of heat transfer (or mass transfer at low transfer rates) to a strip of finite length in a uniform shear flow is considered in [5,7,8]. The formulation is as follows [5,6,32,42]: Z 1 1 Ai½s 0 dw where s ¼ w2 =ðiwPeÞ2=3 , G½x,x0 ; Pe ¼ eiwðxx Þ 0 2p 1 ðiwPeÞ1=3 Ai ½s a0m ½Pe a1m ½Pe qls ½x; Pe ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ , x1=3 xð1xÞ T s ½x ¼ 1,
ð16Þ
where Pe is the Pe´clet number associated with the problem [5], i.e. Pe ¼ SL2 =k where k is the thermal conductivity, S is the constant shear rate and L is the length of the strip. In Fig. 5, we show results for the Nusselt number (Q, Eq. (4)) as a function of the Pe´clet number (Pe) obtained through both least squares regression (Eqs. (7)–(11)), and through numerical solution of the integral equation. There is excellent agreement between the results and the error does not exceed 1%. 3.6. Heat conduction in sliding solids
0
qls ½x; Pe1 ,Pe2 ,kr ¼ a1m ½Pe1 ,Pe2 T s ½x ¼ ðxePe2 x ðK 0 ½Pe2 x þ K 1 ½Pe2 xÞ þ ð1xÞePe2 ð1xÞ ðK 0 ½Pe2 ð1xÞK 1 ½Pe2 ð1xÞÞÞ,
ð17Þ
where the Pe´clet numbers Pe1 ,Pe2 (Pei ¼ V i ‘=2ai where Vi denotes the speed of the bodies, ‘ the contact length, and ai the thermal diffusivities) refer to body 1 and 2 respectively, and kr ¼ k2 =k1 is the ratio of the thermal conductivities [43]. In Fig. 6, we show results for the total transport rate (Q, Eq. (4)) as a function of the Pe´clet number (Pe1) obtained through both least squares regression (Eqs. (7)–(11)), and through numerical solution of the integral equation. There is excellent agreement between the results and the error does not exceed 1%. In Fig. 6 we also include the asymptotic result obtained by Yuen [43]; the comparison suggests that the least squares regression leads to a significant improvement of the asymptotic result.
4. Conclusions
The problem of heat conduction between sliding solids where heat energy is generated along the region of contact is considered (Fig. 6). The formulation is as follows [43]: G½x,x0 ; Pe1 ,Pe2 ,kr ¼ ePe1 ðxx Þ K 0 ½Pe1 9xx0 9 þ
Fig. 6. In the upper figure we show a schematic representation of the physical problem associated with heat conduction in sliding solids (Section 3.6). In the lower figure we show a plot of the Nusselt number (Q) as a function of the parameter Pe1 for Pe2 ¼0.01 and kr ¼ 1. The dots represent results obtained by least squares regression (Eqs. (5)–(11)), the circles represent numerical results of the integral equation, and the crosses are the asymptotic solution obtained by Yuen [43].
1 Pe2 ðxx0 Þ e K 0 ½Pe2 9xx0 9 kr
We have developed a method that improves and extends existing asymptotic solutions of Fredholm integral equations of the first kind. The method relies on describing the unknown function, i.e. the flux distribution, through a linear combination of known asymptotic approximations of the integral equation.
Y. Ioannou et al. / Engineering Analysis with Boundary Elements 36 (2012) 1278–1283
Subsequently, the integral equation is reformulated as a variational principle, i.e. least squares regression, and explicit expressions for the flux distribution can be obtained. The method has been applied to fundamental heat and mass transfer problems presented in the literature. The results suggest that the method is quite accurate, efficient, versatile and scalable. It has been successfully applied to problems with multiple parameters and to problems where limited asymptotic solutions are available. It is tempting to infer that the improved asymptotic solution is a matched asymptotic expansion, and that matching through least squares regression may not be limited to integral equations. More importantly, given that the integral equation formulation is the natural choice for formulating the inverse transport problem, and that the method provides tunable precision (via employing higher-order terms as needed), these make it invaluable for in-process actuation control at a trivial computational cost, especially for low-bandwidth (slow) transport processes. Towards such real-time implementation of the method in technical applications, two related problems will be attacked in future work. First, inevitable process disturbances and parameter alterations in real engineering settings are reflected in the current formulation as uncertainties in the kernel Green’s function, i.e. in the parameters of the asymptotic expressions eventually affecting the parameters in the least square solution. An approach dual to the control problem of actuation input addressed above, i.e. the observation of resulting output distributions for in-process identification of the requisite uncertain process parameters, will be tacked using similar least-squares optimization algorithms. Such a real-time observer/identification technology is enabled by technological progress in non-intrusive, non-destructive sensory measurement methods, e.g. infrared pyrometry, ultrasound, x-ray scanning microtomography, etc. Second, for relatively higherbandwidth (faster) transport conditions, a transient formulation of the method will be developed by addressing Fredholm integral equations of the second kind through appropriate least-squares optimization dynamic formulation and dynamic asymptotic expressions. These two future developments will enable application of the inverse/control methodology to manufacturing processes, such as selective laser sintering, etc.
Acknowledgment The work was funded by Porfyrios CHAP Glass Ltd. References [1] Incropera FP, DeWitt DP. Fundamentals of heat and mass transfer. John Wiley and Sons; 1990. [2] Brebbia CA. Boundary elements: an introductory course. 2nd ed. Computational Mechanics; 1992. [3] Carslaw HS, Jaeger JC. Conduction of heat in solids.Clarendon Press; 1959. [4] Porter D, Stirling DSG. Integral equations: a practical treatment, from spectral theory to applications.Cambridge University Press; 1990. [5] Ackerberg RC, Patel RD, Gupta SK. The heat/mass transfer to a finite strip at small Pe´clet numbers. J Fluid Mech 1978;86:49. [6] Stone HA. Heat/mass transfer from surface films to shear flows at arbitrary Peclet numbers. Phys Fluids A 1989;1:1112. [7] Springer SG, Pedley TJ. The solution of heat transfer problems by the Wiener– Hopf technique. I. Leading edge of a hot film. Proc R Soc London A 1973;333:347. [8] Springer SG. The solution of heat transfer problems by the Wiener–Hopf technique. II. Trailing edge of a hot film. Proc R Soc London A 1974;337:395. [9] Fyrillas MM. Advection–dispersion mass transport associated with a nonaqueous-phase liquid pool. J Fluid Mech 2000;413:49.
1283
[10] Fyrillas MM. Heat conduction in a solid slab embedded with a pipe of general cross-section: shape factor and shape optimization. Int J Eng Sci 2008;46:907. [11] Fyrillas MM. Shape factor and shape optimization for a periodic array of isothermal pipes. Int J Heat Mass Transfer 2010;53:982. [12] Fyrillas MM, Stone HA. Critical insulation thickness of a slab embedded with a periodic array of isothermal strips. Int J Heat Mass Transfer 2011;54:180. [13] Rajendran L. Two-point Pade´ approximation of mass transfer rate at microdisc in a channel flow for all Pe´clet numbers. Electochim Acta 2006;51:5407. [14] Choi J, Margetis D, Squires TM, Bazant MZ. Steady advection–diffusion around finite absorbers in two-dimensional flows. J Fluid Mech 2005;536: 155. [15] Homentcovschi D, Jude L. Potential due to some arbitrary charged strips between parallel earthed planes. Int J Eng Sci 1984;22:775. [16] De Mey G. Temperature distribution in floor heating systems. Int J Heat Mass Transfer 1980;23:1289. [17] Latta GE. The solution of a class of integral equations. Int J Ration Mech Anal 1956;5:821. [18] Lundgren T, Chiang D. Solution of a class of singular integral equations. Q Appl Math 1967;24:303. [19] Terry Hollands KG. The simplified-Fredholm integral equation solver and its use in thermal radiation. ASME J Heat Transfer 2010;132:1. [20] Tikhonov AN, Glasko VB. The approximate solution of Fredholm integral equations of the first kind. USSR Comput Math Math Phys 1964;55:236. [21] Fyrillas MM, Kontoghiorghes EJ. Numerical calculation of mass transfer from elliptical pools in uniform flow using the boundary element method. Transp Porous Media 2004;55:91. [22] Demetriou MA, Paskaleva A, Vayena O, Doumanidis H. Scanning actuator guidance scheme in a 1-D thermal manufacturing process. IEEE Trans Control Syst Technol 2003;11(5):757–64. [23] Hardjadinata G, Doumanidis C. Rapid prototyping by laser foil bonding and curing: thermomechanical modeling and process optimization. J Manuf Process 2001;3(2):108–19. [24] Ioannou Y, Fyrillas MM, Polychronopoulou K, Doumanidis C. Analytical model for geometrical characteristics control of laser sintered surfaces. Int J NanoManuf 2010;6:300. [25] Alaeddine M, Ranganathan R, Ando T, Doumanidis C. Modeling the melting and dissolution stages during thermal processing of intermetallic coatings from layered precursors. Trans ASME J Manuf Sci Eng 2005;127:148–56. [26] Zeng T, Doumanidis C, Hebb J, Brown D. Growth of ultrathin silicon dioxide films during rapid thermal oxidation. In: Proceedings of the 9th international conference on RTP 2001 advanced thermal processing of semiconductors; 2001. p. 287–95. [27] Jogdand H, Gulsoy G, Ando T, Chen J, Doumanidis CC, Gu Z, et al. Fabrication and characterization of nanoscale heating sources (‘‘nanoheaters’’) for nanomanufacturing. In: NSTI Nanotechnology Conference, Boston, MA, June 1–5, 2008, p. 280. [28] Cui Q, Jogland H, Chen J, Gus Z. Structures and ignition properties of nanoheaters formed by bimetallic Al–Ni reactive nanostructures. Mater Res Soc Symp Proc 2009;1148. [29] Alaeddine M, Doumanidis C. Distributed parameter thermal system control and observation by Green–Galerkin methods. Int J Numer Methods Eng 2004;61:1921–37. [30] Liu T, Campbell BT, Sullivan JP. Surface temperature of a hot film on a wall in shear flow. Int J Heat Mass Transfer 1994;37:2809. [31] Magnus W, Oberhettinger F. Formulas and theorems for the functions of mathematical physics. New York: Chelsea Publishing Company; 1949. [32] Le´vˆeque MA. Transmission de chaleur par convection. Ann Mines 1928;13: 283. [33] Chang CL, Gunzburger MD. A subdomain-Galerkin/least squares method for first-order elliptic systems in the plane. SIAM J Numer Anal 1990;27:1197. [34] Collatz L. The numerical treatment of differential equations. New York: Springer-Verlag; 1966. 184 and 220-221. [35] Hanke M. On a least squares collocation method for linear differentialalgebraic equations. Numer Math 1988;54:79. [36] Zwillinger D. Handbook of differential equations.Academic Press Inc; 1992. [37] Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical recipes. Cambridge University Press; 1992. [38] Kreyszig E. Advanced engineering mathematics.John Wiley & Sons; 1993. [39] Payvar P. Analytical solution of a new mixed boundary-value problem in heat conduction. Q J Mech Appl Math 1979;32:253. [40] Ranger KB. The solution of a new mixed boundary value problem by use of an integral transformation. Int J Eng Sci 1974;12:853. [41] Sadhal SS. Explicit solutions to a class of mixed boundary value problems. Int J Eng Sci 1981;19:1053. [42] Phillips GP. Heat and mass transfer from a film into steady shear flow. Q J Mech Appl Math 1990;43:135. [43] Yuen WYD. Heat conduction in sliding solids. Int J Heat Mass Transfer 1988;31:637.