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New analytical solution for heat transfer in insulated wires
International Journal of Thermal Sciences 49 (2010) 2391e2399
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New analytical solution for heat transfer in insulated wires Cristóbal Cortés, Luis I. Díez* Center of Research of Energy Resources and Consumptions (CIRCE), University of Zaragoza, María de Luna, 3, 50018 Zaragoza, Spain
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 October 2009 Received in revised form 26 July 2010 Accepted 27 July 2010 Available online 15 September 2010
In this paper we investigate the heat transfer from pin-fins of infinite length made of a high thermal conductivity core with a low thermal conductivity coating. Considering a two-dimensional thermal field in the coating and a one-dimensional field in the core, an exact, analytical solution is presented for this geometry, previously not available in the literature. This solution is obtained in the form of an infinite series, but its first term can serve as an excellent approximation of practical problems under a wide range of conditions. In particular, the new model is appropriate to assist design engineers in calculating heat transfer losses from insulated wires, which can be conceived as infinitely long composite fins. The first term approximation is at once different and improves upon the traditional one-dimensional formula. Both approximations are compared vs. the exact series solution in order to establish their limits of validity. We have computed heat transfer losses for different combinations of core/coating pairs and typical geometric ratios pertaining to commercial electric wires. The figures show that the new solution produces better results than the traditional approach, especially for Bi < 1 102 with relative errors below 1.35%. Thus, the new expedient proposed in this paper brings along a remarkable improvement, gaining accuracy and at the same time retaining a suitable simplicity. Ó 2010 Elsevier Masson SAS. All rights reserved.
Keywords: Composite fins Heat transfer Wires Infinite series Eigenvalues
1. Introduction Computation of heat transfer losses from electric wires, made of a high thermal conductivity core with a low thermal conductivity coating, is a relevant subject in wide engineering fields of electric power transport, electronics development and signals transmission, and of paramount importance when designing temperature measurement installations based on electric sensors. The conventional calculation relies on the assumption that heat conduction in the coating is locally 1D along the transverse direction, while heat conduction in the substrate is supposed to be 1D along the longitudinal direction. Nevertheless, a better solution can be obtained if the wire is conceived as an infinite coated fin. In this way, an exact analytical solution can be obtained. Then, the objective of this paper is twofold: 1) to survey an analytical solution for composite fins subjected to new boundary conditions, and 2) to apply the solution to a practical calculation and discuss its validity. The question of composite, or two-material, fins has received a scarce but continual attention in the heat transfer literature. As noted by Kraus et al. in their monograph [1], the first study is that of Barker [2], who solved the exact 2D problem for composite media in Cartesian and axial-cylindrical co-ordinates, i.e., for composite
* Corresponding author. Tel.: þ34 976 762 564; fax: þ34 976 732 078. E-mail address: [email protected] (L.I. Díez). 1290-0729/$ e see front matter Ó 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2010.07.012
straight fins and circular rods. Barker used the theory of orthogonal expansions [3] to obtain an infinite series solution and discovered a fundamental fact by analyzing its eigenvalues: under the conditions that warrant an approximate 1D treatment of ordinary finsdBi << 1, see [4] for a rigorous accountd, the infinite series converges to its first term and this approximation is equivalent to the usual 1D fin treatment simply by averaging the thermal conductivities in proportion to the cross-sectional areas. Lalot et al. [5] gave a rather contorted but interesting solution of the annular fin case, re-discovering essentially the same facts. Numerical treatment of the rectangular, constant thickness fin has been presented in [6], whereas the numerical studies in [7] have generalized the results of Barker for composite straight and annular fins of variable thickness. All these studies pattern their practical application from the case of a metal coating on a metal substrate. Two examples are galvanized or aluminized fins (zinc or aluminum on carbon steel) and copper fins covered with a protective coating of stainless steel. Even though only metals are involved, this translates into a relatively wide range of the conductivity ratio k1/k2, roughly from 0.1 to more than 10, and it is indeed surprising that a direct expedient, such as averaging k1 and k2, suffices for an accurate calculation under these circumstances. However, it forcibly must loose validity for the higher values of k1/k2 arising in other kind of applications. The most notable is a fouled or frosted finned heat exchanger, where the coating is
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Nomenclature An Ac Bi Cn e h Jn k L P Q Q q00 R r T X
x Yn
series of coefficients cross-sectional area of the fin, m2 Biot number series of coefficients convergence error convection coefficient, W m2 K1 Bessel function of first kind and order n ¼ 0, 1 thermal conductivity, W m1 K1 fin length, m fin perimeter, m heat transfer rate, W dimensionless heat transfer rate heat flux, W m2 dimensionless radial component of the temperature field fin radius, m; radial coordinate, m temperature, K dimensionless longitudinal component of the temperature field
a thermal insulator compared to the fin base medium. Also in this situation, the coating is of much larger thickness than in a bimetallic fin, which may contribute to complicate things. The pioneering study addressing this specific problem, by Epstein & Sandhu [8], examined two approximate, simple models that combine the “thermal resistances” of substrate and coating “in series” and “in parallel,” see Fig. 1. The latter is equivalent to use the area-average of conductivities, as in metal-coated fins. The former amounts to assume that heat conduction in the coating is locally 1D along the transverse direction, and to couple this idea with the usual approximate 1D model of an ordinary fin. Although accuracy was not assessed for either model, the “in series” calculation was recommended over the “in parallel” one, due to the fact that it gives the most conservative results for heat transfer. Devoted to the same class of situations, the work reported in [9] calculated the fin performance numerically and by an infinite series solution slightly simplified from the exact formulae. The poor performance of the “in parallel” model was also established, although the infinite series solution was not evaluated. More recently, the question has been revisited and put in order by Xia and Xacobi [10]. They show that, for the conditions usually
a
b
Fig. 1. “Thermal resistance” models for composite fins, from [8]: series model (a), parallel model (b).
longitudinal coordinate, m Bessel function of second kind and order n ¼ 0, 1
Greek letters relative error, % dimensionless radial coordinate K thermal conductivity ratio l eigenvalue q dimensionless temperature r radii ratio x dimensionless longitudinal coordinate
3 h
Subscripts 1 first value or term of the series; fin core 1t one-term approximation 2 fin coating N surrounding fluid b fin base m, n m-th, n-th value or term of the series p “in parallel” approximation s “in series” approximation
found in practice, it is appropriate to consider a simplified problem where the thermal field is fully two-dimensional in the coating, but approximately one-dimensional in the high-conductivity base fin. The analytical solution then demands a special application of the techniques of orthogonal series expansions; this is specifically developed in Cartesian co-ordinates, i.e., for rectangular straight fins. Furthermore it is shown that, for thermally slender fins, Bi << 1, the series converges to its first term, which leads as usual to a simple formula for the heat flux. This limit differs from the crosssectional average of conductivities, or “in parallel” model, so that previous work is confirmed, and the interesting fact is discovered that the exact, “2De2D” solution cannot deliver a proper approach. The reason is that its one-term limit implies a 1D field both in the core and the coating, something that can be appropriate for a metallic composite fin, but not so for a fin covered by an insulating layer. Although it might be anticipated that the new approximation coincides with the “in series” model of Fig. 1, in fact it doesn’t. Actually, the formula for the heat flux conveys an important gain in accuracy for Bi << 1, indicating that axial conduction in the frost layer is better accounted for. This theory has been extended to annular geometries in [11,12]. Deductions of “2De2D” solutions have been also reported by Gorobets [13] for finite, composite fins with non-uniform coating shapes, but with the different target of volume optimization. In this paper, we study a related class of situations, viz., the heat transfer from wires made of metallic cores coated by electrical insulation. In this case we are similarly at the higher end of the conductivity ratio, k1/k2 >> 1, with a large range of coating thicknesses, and dealing with thermally slender domains, Bi << 1. For this reason, we are going to adopt the same simplification of [10,11], i.e., a fully two-dimensional temperature field is prescribed in the isolator coating but approximately one-dimensional in the highconductivity core. As with the frosted fins, the calculation methods conventionally resorted to, see e.g. [14], essentially emanates from the “in series” concept. Giving the context, we develop here the case of infinitely long rods; although indicative, the results would need slight modification to be quantitatively applicable to a pin fin of finite length. In the Section 2 of the paper, we proceed with the analytical calculation and derive the exact 1De2D solution for composite
C. Cortés, L.I. Díez / International Journal of Thermal Sciences 49 (2010) 2391e2399
media in axial-cylindrical co-ordinates. It should be noted that this solution was not previously available in the literature. In Section 3, the one-term approximation of the new exact solution is obtained. The outcome is a simplified formula for the heat flux that is different from the “in parallel” and “in series” previous models, and improves upon both. This is established in Section 4 by considering a representative set of commercial designs possessing a variety of materials and geometric ratios. We finally compare in Section 5 the new formula against the conventional method, proving in this way that accuracy is greatly improved, while a considerable simplicity is retained.
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where the cross-sectional area and the perimeter of the core are Ac;1 ¼ pr12 and P1 ¼ 2pr1 . Boundary conditions at the fin base, x ¼ 0, and fin tip, x / N, are
T1 ð0Þ ¼ Tb
(3a)
T2 ðr; 0Þ ¼ Tb
(3b)
T1 ðNÞ ¼ TN
(4a)
T2 ðr; NÞ ¼ TN
(4b)
Convective transfer at the lateral boundary r ¼ r2 is imposed by
2. Problem description and analytical solution 2.1. Formulation of the problem
k2
Fig. 2 is a sketch of the wire geometry, represented as an infinitely long pin fin of constant, circular section. A metallic, conducting core of radius r1 and thermal conductivity k1 is covered by an insulating medium of radius r2 and conductivity k2. The conducting core is assumed passive, as in temperature measurement applications. The wire is heated through the base, x ¼ 0, and losses heat along its surface to a convective environment given by a coefficient h. The length of the wire is so large that we can consider the domain extending to x / N; exact temperature fields would show an axial-symmetric distribution, i.e., they would be given as a function of the longitudinal coordinate x and the radial coordinate r. The following simplifying assumptions, common in fin theory, are adopted: a) steady-state, b) no internal heat sources, c) two isotropic and homogeneous media with constant properties, d) uniform values of convection coefficient and boundary and fluid temperatures (which leads to linear boundary conditions), and e) nil contact resistance between the substrate and the coating. For ordinary values of the parameters, as shown in [10], we can assume that the thermal field is approximately 1D in the metallic core, T1(x) in 0 r r1, 0 x < N, whereas it remains fully 2D in the insulating layer, T2(r, x) in r1 r r2, 0 x < N. Since the core possesses a high thermal conductivity, conventional convective conditions lead to a thermally slender fin, for which the onedimensional approximation is warranted. However, this is not the case of the coating.
vT2 ¼ h½T2 ðr2 ; xÞ TN vr r¼r2
(5)
whereas temperature continuity at the interface is expressed by
T1 ðxÞ ¼ T2 ðr1 ; xÞ
(6)
The combination of the fin-like Eq. (1) with a fully elliptical problem seems strange, but it can be handled rather easily. By virtue of Eq. (6), Eq. (1) can be written as
k1 r1
v2 T2 ðr; xÞ vT2 ðr; xÞ þ 2k ¼ 0 2 vr r¼r1 vx2 r¼r1
(7)
so that it adopts the form of a boundary condition for the solution of Eq. (2) that gives the 2D temperature distribution T2 (x, r). The other three conditions necessary are already written as Eqs. (3b), (4b) and (5). Once solved for T2 (x, r), there is no need to integrate for the temperature T1(x). Since a one-dimensional field is assumed in the inner core, local evaluation of T1 is directly obtained by computing T2 at r ¼ r1, that is, it is simply given by Eq. (6). This also assures that, as a consequence of Eqs. (3b) and (4b), also Eqs. (3a) and (4a) are satisfied. Dimensionless variables are now introduced to streamline the nomenclature and represent a wider range of computed cases. We adopt the following definitions:
q ¼
T2 TN x r r k hr x ¼ h¼ r ¼ 1 K ¼ 1 Bi ¼ 2 Tb TN r2 k2 k2 r2 r2
(8)
By using them, Eq. (2) is transformed to 2.2. Analytical solution for the temperature distributions in the wire Governing equations for temperature in the core, T1, and the coating, T2, are:
k1 Ac;1
d2 T1 vT2 ¼ k P 2 1 vr r¼r1 dx2
1 v vT2 v2 T2 r þ ¼ 0 r vr vr vx2
(1)
1 v vq v2 q h þ 2 ¼ 0 h v h vh vx
(9)
which governs the dimensionless temperature q (h, x) in r h 1, 0 x < N. The set of boundary conditions given by Eqs. (3b), (4b), (5) and (7) becomes
qðh; 0Þ ¼ 1
(10a)
qðh; NÞ ¼ 0
(10b)
vq ¼ Bi qð1; xÞ vhh¼1
(10c)
(2)
rK
v2 q vq þ 2 ¼ 0 2 vhh¼r vx h¼r
(10d)
To solve this problem, we postulate a solution in the form
qðh; xÞ ¼ RðhÞXðxÞ: Substituting into Eq. (9), and after some algebra, Fig. 2. Problem of an infinitely long composite rod.
we get
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1 1 d dRðhÞ 1 d2 XðxÞ h ¼ ¼ const: RðhÞ h dh dh XðxÞ dx2
(11)
A solution with physical meaning requires that the axial part X(x) decrease along the coordinate, so that the constant of Eq. (11) has to be negative. In this way, the original partial differential equation separates in the following two ordinary differential equations
1 d2 XðxÞ 2 ¼ l >0 XðxÞ dx2 1 1 d dRðhÞ 2 h ¼ l >0 RðhÞ h dh dh
(12a)
qðh; xÞ ¼
An ½J0 ðln hÞ Cn Y0 ðln hÞexpðln xÞ
2 6 4
Z1
3
(12b)
1 2
r
using
Cn ¼
rK ln J0 ðrln Þ 2J1 ðrln Þ rK ln Y0 ðrln Þ 2Y1 ðrln Þ
An ¼ p
Bi
l2n
(15)
the
remaining
boundary
condition,
1þ
Bi2
1 BiY0 ðln Þ ln Y1 ðln Þ K 1 4 2 r2 K 2
(18)
l2n ln þ 2 2 ½BiY0 ðln Þ ln Y1 ðln Þ ½rK ln Y0 ðrln Þ2Y1 ðrln Þ
In the Appendix, the reader can consult the details about the resolution of the problem that yields the analytical solution given by Eqs. (13), (14), (15) and (18). Heat transfer from the wire can be calculated from the following equation:
and the eigenvalues ln satisfy the following eigenequation:
rK ln J0 ðrln Þ 2J1 ðrln Þ BiJ0 ðln Þ ln J1 ðln Þ ¼ rK ln Y0 ðrln Þ 2Y1 ðrln Þ BiY0 ðln Þ ln Y1 ðln Þ
upon
An:
(13)
(14)
it
(17)
qðh; 0Þ ¼ SAn Rn ðhÞ ¼ 1; Eq. (10a), we get the series of coefficients
n¼1
where the coefficients Cn are given by
2 2 hRm ðhÞRn ðhÞdh þ r2 KRm ðrÞRn ðrÞ7 5 ln lm ¼ 0
By
Solutions of Eqs. (12), combined into the temperature q ¼ XR, and with three of the four constants already determined by Eqs. (10b), (10c) and (10d), can be written as a function series in the form N X
a clear exploration of the values of the first ln, that can be otherwise feasibly computed by means of a numerical iteration. It can be shown in the appendix that the orthogonality property for the series of eigenfunctions Rn ðhÞ ¼ J0 ðhln Þ Cn Y0 ðhln Þ is derived from:
Q ¼ 2pr2 h
ZN ZN ½T2 ðr2 ;xÞTN dx ¼ 2pr22 hðTb TN Þ qð1; xÞdx 0
(19)
0
If the discrete values ln in this formula are substituted by a continuous variable l and we abbreviate its left and right sides as f(l) and g(l) respectively, the eigenvalues are the infinite zeros of the equation
To make this variable dimensionless, we cannot use the usual concept of efficiency, since it is by definition zero for an infinitely long fin. Instead, we define arbitrarily
f ðlÞ gðlÞ ¼ 0
Q ¼
(16)
Fig. 3 represents an example of the solution as one of the intersections on the l-axis of the functions f and g. This provides
Q ¼ 2pr22 hðTb TN Þ
ZN
qð1; xÞdx
(20)
0
Substitution of q (1, x) from Eq. (13) yields:
Q ¼
N X An
l n¼1 n
½J0 ðln Þ Cn Y0 ðln Þ
(21)
where An is given by Eq. (18), Cn by Eq. (14) and ln by Eq. (15). Equation (21) represents the exact analytical solution for the heat transfer losses of the wire, for an approximately one-dimensional thermal field in the core and two-dimensional in the insulation. 3. One-term approximation The use of the analytical solution is of course cumbersome due to the infinite series. However, as in ordinary fins, a useful approximation is valid for most practical problems, based on the fact that the small size and the reduced convective coefficient determine a thermally slender geometry, i.e., a situation with a Biot number Bi << 1. Therefore, paralleling the treatment of ordinary fins [4], the limit of the exact solution is sought under Bi / 0; it turns out that the series reduces then to its first term, thus leading to a simple, engineering formula for the heat flux. Given our definition of Bi, the condition Bi / 0 can be interpreted as a negligible radial gradient of temperature in the insulation, which adds up to the previous assumption in the same sense for the conducting core. By analyzing Eqs. (15) and (18), it can be shown that, for Bi / 0,
Fig. 3. Graphical determination of the roots of the eigenequation.
1 An / 0
n ¼ 1 n>1
(22a)
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On the one side, the “in parallel” approximation amounts to use the formula of a single-material rod with (k1Ac,1 þ k2Ac,2) replacing kAc. With our nomenclature, this gives
Qp ¼
Fig. 4. Heat fluxes in the wire according to the conventional “thermal resistances” “in series” modeling.
11
0
2
2Bi iC h A 1r2 2 2 r ðK 1Þ þ 1 þ Bi r lnrð1 KÞ þ 2
l1 /B @
(22b)
1 2 R1 ðhÞ ¼ J0 ðl1 hÞ C1 Y0 ðl1 hÞ/1 þ l1 lnðl1 hÞ/1 2
(22c)
Or, in other words, the series reduces to its first term and an analytical formula can be given for the first eigenvalue l1. It also happens that l1 / 0 when Bi / 0, which permits to show that the radial eigenfunction disappears; the one-term solution is also a 1D approximation. Equation (22b) can be further simplified, for usual values K >> 1 in electric wires, yielding the simpler limit:
1 2 2Bi l1 / 2 r ðK 1Þ þ 1 Bir2 Klnr
(22d)
1 q1t ðxÞ ¼ 1 þ l21;1t ln l1;1t h exp l1;1t x zexp l1;1t x 2
Q 1t ¼
1
l1;1t
¼
2Bi
(25)
It is worth reminding that this approach can be rigorously demonstrated to be the (one-term, one-dimensional) limit of the exact “2De2D” solution for Bi / 0. Therefore, the approach adopted here, that relies on a “1De2D” series solution and its limit under the same condition, produces a different but equally simple approximation that might retain the fact that the layer covering the fin is not a conducting medium but an insulator. This interpretation is reinforced by noting that the new formula given by Eq. (24) tends to Eq. (25) for a vanishingly small Bi, but only in the case that K is not very large, i.e., not when the inner material is much more conducting that the coating. On the other side, the traditional approach to an insulated wire (or frosted or fouled fin) assumes that the temperature distribution is longitudinally 1D in the core, but transversally 1D at each longitudinal coordinate in the coating. Fig. 4 sketches this approximate idea. The approximate temperature equation combines the fin balance with the “thermal resistances” of coating and external transfer disposed “in series:”
k1 pr12
d2 T1 T1 TN ¼ ¼ UPðT1 TN Þ lnðr2 =r1 Þ 1 dx2 þ 2pk2 h2pr2
(26)
Solving for T1dsubjected to the boundary conditions T1(0) ¼ Tb, T1(N) ¼ TNdleads logically to the same solution as a bare fin, with the quantity UP replacing hP. In particular, for the heat transfer loss,
2hr2 k1 hkr2 lnrr21 þ 1
!1 2
ðTb TN Þ
(27)
2
(23)
which in our dimensionless terms reads
12
r2 ðK 1Þ þ 1 Bir2 Klnr
12
2Bi
Qs ¼ pr1
Substituting these limits into Eqs. (13) and (21),
r2 ðK 1Þ þ 1
(24)
4. Discussion Although the one-term approximation is one-dimensional, cf. Eq. (23), it is not equivalent to previous models, and thus the new formula emerging in Eq. (24) differs from the traditional calculation of insulated wires.
Qs ¼
r2 K
12 (28)
2Bið1 BilnrÞ
Comparing Eq. (28) with Eq. (24) we see that, even for Bi / 0, they differ and would only provide similar results when r / 1, i.e., for a thin coating, or the trivial case when there is none. It is curious to note that the new approximation incorporates a slight radial variation given by a logarithm, Eq. (22d), exactly as in the elementary 1D solution of a cylindrical shell that is embedded in Eq. (26). However, in spite of this, the models are different. As a conclusion, the new one-dimensional approximation has produced a formula that can improve over the traditional “in series” concept, which can
Table 1 Case-studies considered. K
r
Application
Substrate
Coating
k1 (W/m K) [15e17]
k2 (W/m K) [15e17]
High-voltage wires
Aluminum, 99.5%
200
0.4
Electrolytic copper
Reticulated polyethylene Ethylene-propylene rubber
375
0.15
2500
Low-voltage wires
Tinned copper, 88% Cu Electrolytic copper
Reticulated polyethylene Ethylene-propylene rubber
60 375
0.4 0.15
150 2500
0.50e0.80
Heating cords
CoppereNickel Alloy, 80% Cu Copper brass, 75% Cu
Fluoride polymers Quartz fibers
12 25
0.16 0.03
75 750
0.20e0.40
Fouled or frosted rods
Stainless steel, 20% Cr-15% Ni Zinc
Ice Fouling, dirtiness
12.5 120
2.5 0.12
5 1000
0.10e0.90
500
0.25e0.60
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Table 2 Maximum fractional errors fn in the calculation of dimensionless heat transfer rates by a finite series (Bi ¼ 1 101).
r
K
f2
f3
0.10 0.20 0.25 0.50 0.80 0.90
5 75 500 150 2 500 1 000
0.00178 0.01196 0.00417 0.00142 0.00002 0.00001
0.00044 0.00047 0.00017 0.00007 0.00001 0.00000
Table 4 Relative errors of the different models. Low-voltage wires.
r
K
0.50
150
2 500
0.80
150
make sense based on the approximate nature of the latter. We elucidate this question in the next section.
5. Numerical results
Q n Q n1
(29)
n
where e is a given limit parameter representing the fractional correction that is deemed negligible. Table 2 summarizes the results for n ¼ 2, 3, for selected, extreme case-studies, and the highest Biot number, Bi ¼ 1 101, under which the slowest convergence can be expected. From those figures, it is clear that retaining three terms in the series suffices to assure an error always lower than 0.05%. Thus, the approximation of Eq. (21) is 3 X An
l n¼1 n
½J0 ðln Þ Cn Y0 ðln Þ
(30)
where the three eigenvalues and the constants are calculated as explained above.
r
K
0.20
75
750
0.40
75
750
0.25
K 500
2 500
0.60
500
2 500
3s ð%Þ
0.03 0.62 6.58 0.06 0.67 6.60
1.02 1.05 1.24 0.07 0.08 0.36
0.001 0.010 0.100 0.001 0.010 0.100
0.02 0.15 2.12 0.03 0.20 2.19
0.19 0.26 0.29 0.01 0.03 0.05
Bi
31t ð%Þ
3s ð%Þ
0.001 0.010 0.100 0.001 0.010 0.100
0.15 1.34 12.81 0.16 1.37 12.92
1.45 1.49 2.21 0.27 0.31 1.09
0.001 0.010 0.100 0.001 0.010 0.100
0.06 0.51 4.97 0.08 0.53 5.04
0.17 0.18 0.30 0.01 0.02 0.09
Bi
31t ð%Þ
3s ð%Þ
0.001 0.010 0.100 0.001 0.010 0.100
0.13 1.12 10.45 0.15 1.55 14.76
12.97 13.21 15.75 1.57 1.60 2.48
0.001 0.010 0.100 0.001 0.010 0.100
0.07 0.83 8.53 0.14 0.91 9.04
3.35 3.38 3.63 0.30 0.32 0.43
Tables 3e5 summarize the errors arising in the computation of the dimensionless heat transfer rates according to the exact solution, Eq. (30), the one-term approximation, Eq. (24), and the traditional “in series” model, Eq. (28). To quantify the results, we show the following percent relative errors:
31t ¼ 100 3s ¼ 100
Q 1t Q
(31a)
Q
Qs Q
(31b)
Q
As could be expected, comparison between the one-term approximation and the conventional formulation shows that the former is more appropriate for the lowest values of the Biot number. Ensuing errors 31t are lower than 0.15% for Bi ¼ 1 103 and lower than 1.55% for Bi ¼ 1 102; however they significantly Table 6 Relative errors of the different models. Fouled and frosted metallic rods.
Table 3 Relative errors of the different models. High-voltage wires.
r
31t ð%Þ
Table 5 . Relative errors of the different models. Heating cords.
In order to compare the new formula vs. the traditional model, we begin by selecting an adequate set of case-studies (see Table 1). With a view to the practical applications, different combinations of substrate/coating pairs and typical geometric ratios pertaining to electric wires are considered, taken from technical catalogues of the manufacturers [15e17]. We also include the different situation of a fouled or frosted metallic rod, which can also be treated by the formulation presented in this work. Taking into account typical dimensions and the low values of the heat transfer coefficient for losses to ambient air, we adopt three different values of the Biot number Bi ¼ 1 103, 1 102, 1 101, preserving a difference of two orders of magnitude between them. Next, in order to calculate the reference exact solution, we study the convergence of the series written in Eq. (21). If we call n the number of terms retained in a given approximation Q n ; a condition for it to be accurate enough can be
Qz
2 500
Bi 0.001 0.010 0.100 0.001 0.010 0.100
Fouled fins, K ¼ 1000
r
Bi
31t ð%Þ
0.10
0.001 0.010 0.100 0.001 0.010 0.100
0.13 1.98 18.59 0.02 0.11 1.05
4.70 4.82 7.26 0.00 0.00 0.01
0.001 0.010 0.100 0.001 0.010 0.100
0.02 0.40 3.98 0.02 0.09 0.99
78.10 78.30 79.99 2.25 2.28 2.31
0.90
Frosted fins, K ¼ 5a
0.10
0.90
3s ð%Þ
a Since K w 1, the limit used for the one-term approximation is the shown in Eq. (22b) instead of Eq. (22d).
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as an infinite series, has been demonstrated to converge with just retaining the three first terms. The conventional approach could be advisable however for Biot numbers approaching 0.1, and also because it always gives a conservative estimate, but it should be avoided in any case for conductivity ratios K below approx. 100. The general tendency, for all tested Biot numbers, is the lower the radii ratio r, the greater the conductivity ratio K for which the new formula improves the accuracy of the conventional calculation. Appendix Calculation of the problem relies on the determination of the temperature field
qðh; xÞ ¼ RðhÞXðxÞ
(A.1)
by the resolution of the differential system
Fig. 5. Comparison of relative errors as a function of dimensionless ratios (Bi ¼ 1 102). 1
1 d2 XðxÞ 2 ¼ l >0 XðxÞ dx2
(A.2)
1 1 d dRðhÞ 2 h ¼ l >0 RðhÞ h dh dh
(A.3)
increase for Bi ¼ 1 10 . Inversely, it can be said that the traditional approximation retains a remarkable validity up to high Biot numbers. The error of the one-term approximation diminishes when the radii ratio r increasesdthe size of the core increasesdor the conductivity ratio K decreasesdthe thermal conductivity of the coating increasesd. Table 6 displays the results when frosted or fouled metallic, infinite rods are simulated. The use of the one-term approximation is more accurate than in the case of wires, especially for the occurrence of a layer of frost on the rod; this is due to the small disparity in thermal conductivity between the coating and the substrate. For all cases computed, the one-term approximation tends to overestimate the heat transfer rates, opposite to the traditional approach. Comparing both methods, the new formula provides better results for the cases with low-to-moderate values of the radii ratio r. It can be also inferred that the lower the radii ratio, the greater the conductivity ratio K for which the one-term approximation improves upon the classical approach, and vice versa. This can be explained by the value of the product of dimensionless ratios rK: the classical approach is more appropriate when thin layers of very low thermal conductivity isolators are attached, in comparison to the conducting core. These tendencies can be observed in Fig. 5, where the relative errors 31t and 3s of some cases of Tables 3e5 are represented vs. each other.
and then the temperature field is expressed as a function of four constants k1, k2, k3, k4 that have to be determined from the prescription of the boundary conditions:
6. Conclusions
qðh; xÞ ¼ ½k1 expðlxÞþk2 expðlxÞ½k3 J0 ðlhÞþk4 Y0 ðlhÞ
subjected to the four boundary conditions:
qðh; 0Þ ¼ 1
(A.4)
qðh; NÞ ¼ 0
(A.5)
vq ¼ Bi qð1; xÞ vhh¼1
(A.6)
rK
v2 q vq þ 2 ¼ 0 2 vhh¼r vx h¼r
(A.7)
The generic solutions of differential eqs. (A.2) and (A.3) are given by
XðxÞ ¼ k1 expðlxÞ þ k2 expðlxÞ
(A.8)
RðhÞ ¼ k3 J0 ðlhÞ þ k4 Y0 ðlhÞ
(A.9)
(A.10)
Substituting Eq. (A.10) in condition (A.5), it is mandatory that Computation of the heat transfer rate from an insulated wire has been performed, considered as a composite piece of infinite length, made of a high thermal conductivity core and a low thermal conductivity coating. To this purpose, an exact analytical solution has been obtained in the form of an infinite series, and assuming a thermal field that is 1D in the core and 2D in the coating. The one-term approximation for Bi << 1 has been also derived analytically, providing a new formula to compare against the conventional calculation. Practical cases of electric wires and frosted and fouled pin-fins have been calculated, in order to analyze the errors incurred with the two approximations. For the cases of Biot numbers below 0.01, and all along the whole range of parameters, the new approximation produces better results than the conventional approach, retaining at once a similar simplicity. Maximum relative errors are of 1.35% in comparison to the exact solution; this solution, obtained
k1 ¼ 0
(A.11)
to produce a finite value of the temperature at the end, limit tip of the wire. Combining Eq. (A.1) with conditions (A.6) and (A.7), we get the following expressions for the boundary conditions of the radial component of the temperature:
dR ¼ Bi Rð1Þ dhh¼1
(A.12)
dR 1 2 ¼ K rl RðrÞ dhh¼r 2
(A.13)
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Substituting Eq. (A.9) in condition (A.12), the relation between constants is given by
k4 ¼
BiJ0 ðlÞ lJ1 ðlÞ k lY1 ðlÞ BiY0 ðlÞ 3
(A.14)
(A.15)
where we have used the derivatives of the Bessel functions [18]:
dJ0 ðkzÞ ¼ kJ1 ðkzÞ dz
(A.16)
dY0 ðkzÞ ¼ kY1 ðkzÞ dz
(A.17)
rK lJ0 ðrlÞ 2J1 ðrlÞ BiJ0 ðlÞ lJ1 ðlÞ ¼ rK lY0 ðrlÞ 2Y1 ðrlÞ BiY0 ðlÞ lY1 ðlÞ
An ½J0 ðln hÞ Cn Y0 ðln hÞexpðln xÞ
(A.19)
n¼1
rK ln J0 ðrln Þ 2J1 ðrln Þ rK ln Y0 ðrln Þ 2Y1 ðrln Þ
(A.21)
dRn ðhÞ ¼ Bi Rn ð1Þ dh h¼1
(A.22)
1r2 KR ðrÞ n 2
Z1 þ
hRn ðhÞdh
r
An ¼ 1r2 KR2 ðrÞ n 2
Z1 þ
(A.29)
hR2n ðhÞdh
r
More additional algebra, that requires the introduction of the Wronskian of the Bessel functions [18],
J0 ðzÞY1 ðzÞ J1 ðzÞY0 ðzÞ ¼
2 zp
(A.30)
zY0 ðzÞdz ¼ zY1 ðzÞ
(A.32)
Z
Z
zJ02 ðzÞdz ¼
Z zY02 ðzÞdz ¼
i z2 h 2 J ðzÞ þ J12 ðzÞ 2 0
(A.33)
i z2 h 2 Y0 ðzÞ þ Y12 ðzÞ 2
zJ0 ðzÞY0 ðzÞdz ¼
(A.23)
1 d dR ðhÞ 2 h n þ ln Rn ðhÞ Rm ðhÞ h dh dh 1 d dR ðhÞ 2 h m Rn ðhÞ þ lm Rm ðhÞ ¼ 0 h dh dh
(A.34)
z2 ½J ðzÞY0 ðzÞ þ J1 ðzÞY1 ðzÞ 2 0
(A.35)
3
An ¼ p
1 Bi Y0 ðln Þ ln Y1 ðln Þ Bi2 K1 1þ 2 4 2 r2 K 2
Bi
l2n
ln
½Bi Y0 ðln Þ ln Y1 ðln Þ
2
þ
ln
½rK ln Y0 ðrln Þ 2Y1 ðrln Þ
2
(A.36) ðA:24Þ
After some algebra is done upon Eq. (A.24), and integrating in the domain of interest r < h < 1, we get the relation:
2 2 hRm ðhÞRn ðhÞdh þ r2 KRm ðrÞRn ðrÞ7 5 ln lm ¼ 0
Then, for n s m, it is required that:
(A.31)
brings along the sought value of the coefficients An:
The orthogonality of every pair of eigenfunctions Rn and Rm is expressed by:
1 2
zJ0 ðzÞdz ¼ zJ1 ðzÞ
Z
dRn ðhÞ 1 2 ¼ K rln Rn ðrÞ dh h¼r 2
r
(A.28)
(A.20)
1 d dR ðhÞ 2 h n þ ln Rn ðhÞ ¼ 0 h dh dh
6 4
An hRn ðhÞRm ðhÞ ¼ hRm ðhÞ
n¼1
Z
and the eigenvalues ln satisfy the eigenequation (A.18). We need now to obtain the coefficients An to close the solution of the problem. To this purpose, we check the orthogonality of the functions Rn ðhÞ ¼ J0 ðhln Þ Cn Y0 ðhln Þ. These functions comply with the differential equation of the radial component (A.3), and with the boundary conditions (A.12) and (A.13):
Z1
(A.27)
and the use of some additional properties of Bessel functions [18],
where the coefficients Cn are given by
2
N X
(A.18)
There are infinite solutions of the Eq. (A.18), the so-called eigenvalues ln (n ¼ 1 . N). Summing up, regrouping and recalling the constants, temperature field is given by
Cn ¼
An Rn ðhÞ ¼ 1
n¼1
Integration in the domain r < h < 1 and use of Eq. (A.26), along with some algebra, yield to:
From Eqs. (A.14) and (A.15), the eigenequation (15) is here obtained:
N X
(A.26)
Turning to the remaining boundary condition, Eq. (A.4) can be expressed as: N X
rK lJ0 ðrlÞ 2J1 ðrlÞ k 2Y1 ðrlÞ rK lY0 ðrlÞ 3
qðh; xÞ ¼
1 2
hRm ðhÞRn ðhÞdh ¼ r2 KRm ðrÞRn ðrÞ
r
and repeating the process in condition (A.13),
k4 ¼
Z1
(A.25)
The problem is now closed: the dimensionless temperature field in the coating is given by Eqs. (A.18), (A.19), (A.20) and (A.36). References [1] A.D. Kraus, A. Aziz, J.R. Welty, Extended Surface Heat Transfer. John Wiley, New York, 2000, Sect. 15.6. [2] J.J. Barker, The efficiency of composite fins, Nuclear Science and Technology 3 (1958) 300e312. [3] M.N. Ozisik, Heat Conduction. Wiley, New York, 1980.
C. Cortés, L.I. Díez / International Journal of Thermal Sciences 49 (2010) 2391e2399 [4] M. Levitsky, The criterion for validity of the fin approximation, International Journal of Heat and Mass Transfer 15 (1972) 1960e1963. [5] S. Lalot, C. Tournier, M. Jensen, Fin efficiency of annular fins made of two materials, International Journal of Heat and Mass Transfer 42 (1999) 3461e3467. [6] E.M.A. Mokheimer, M.A. Antar, J. Farooqi, S.M. Zubair, Analytical and numerical solution along with PC spreadsheets modeling for a composite fin, Heat and Mass Transfer 32 (1997) 229e238. [7] C. Cortés, L.I. Díez, A. Campo, Efficiency of composite fins of variable thickness, International Journal of Heat and Mass Transfer 51 (2008) 2153e2166. [8] N. Epstein, K. Sandhu, Effect of uniform fouling on total efficiency of extended heat transfer surfaces, in: Proceedings of the Sixth International Heat Transfer Conference, Toronto, Ontario, Canada, vol. 4, 1978, pp. 397e402. [9] H. Barrow, J. Mistry, D. Clayton, Numerical and exact mathematical analyses of two-dimensional rectangular composite fins, in: Proceedings of the Eight International Heat Transfer Conference, San Francisco, California, USA, vol. 2, 1986, pp. 367e372. [10] Y. Xia, A.M. Jacobi, An exact solution to steady heat conduction in a twodimensional slab on a one-dimensional fin: application to frosted heat
[11]
[12]
[13] [14] [15] [16] [17] [18]
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exchangers, International Journal of Heat and Mass Transfer 47 (2004) 3317e3326. A.D. Sommers, A.M. Jacobi, An exact solution to steady heat conduction in a two-dimensional annulus on a one-dimensional fin: application to frosted heat exchangers with round tubes, ASME Journal of Heat Transfer 128 (2006) 128e404. P. Tu, H. Inaba, A. Horibe, Z. Li, N. Haruki, Fin efficiency of an annular fin composed of a substrate metallic fin and a coating layer, ASME Journal of Heat Transfer 128 (2006) 851e854. V. Gorobets, Influence of coatings on thermal characteristics and optimum sizes of fins, Journal of Enhanced Heat Transfer 15 (2008) 65e80. H.D. Baker, E.A. Ryder, N.H. Baker, Temperature Measurement in Engineering, vol. 1, Omega Press, Stamford, Connecticut, 1975, (Chapter 7). General Cables Technology Corporation, www.generalcable.com. Groupe Omerin, www.omerincables.com. Times Microwave Systems Inc., www.timesmicrowave.com. M. Abramowitz, A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables Ninth Printing, Dover, NY, 1972.
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
New analytical solution for heat transfer in insulated wires Cristóbal Cortés, Luis I. Díez* Center of Research of Energy Resources and Consumptions (CIRCE), University of Zaragoza, María de Luna, 3, 50018 Zaragoza, Spain
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 October 2009 Received in revised form 26 July 2010 Accepted 27 July 2010 Available online 15 September 2010
In this paper we investigate the heat transfer from pin-fins of infinite length made of a high thermal conductivity core with a low thermal conductivity coating. Considering a two-dimensional thermal field in the coating and a one-dimensional field in the core, an exact, analytical solution is presented for this geometry, previously not available in the literature. This solution is obtained in the form of an infinite series, but its first term can serve as an excellent approximation of practical problems under a wide range of conditions. In particular, the new model is appropriate to assist design engineers in calculating heat transfer losses from insulated wires, which can be conceived as infinitely long composite fins. The first term approximation is at once different and improves upon the traditional one-dimensional formula. Both approximations are compared vs. the exact series solution in order to establish their limits of validity. We have computed heat transfer losses for different combinations of core/coating pairs and typical geometric ratios pertaining to commercial electric wires. The figures show that the new solution produces better results than the traditional approach, especially for Bi < 1 102 with relative errors below 1.35%. Thus, the new expedient proposed in this paper brings along a remarkable improvement, gaining accuracy and at the same time retaining a suitable simplicity. Ó 2010 Elsevier Masson SAS. All rights reserved.
Keywords: Composite fins Heat transfer Wires Infinite series Eigenvalues
1. Introduction Computation of heat transfer losses from electric wires, made of a high thermal conductivity core with a low thermal conductivity coating, is a relevant subject in wide engineering fields of electric power transport, electronics development and signals transmission, and of paramount importance when designing temperature measurement installations based on electric sensors. The conventional calculation relies on the assumption that heat conduction in the coating is locally 1D along the transverse direction, while heat conduction in the substrate is supposed to be 1D along the longitudinal direction. Nevertheless, a better solution can be obtained if the wire is conceived as an infinite coated fin. In this way, an exact analytical solution can be obtained. Then, the objective of this paper is twofold: 1) to survey an analytical solution for composite fins subjected to new boundary conditions, and 2) to apply the solution to a practical calculation and discuss its validity. The question of composite, or two-material, fins has received a scarce but continual attention in the heat transfer literature. As noted by Kraus et al. in their monograph [1], the first study is that of Barker [2], who solved the exact 2D problem for composite media in Cartesian and axial-cylindrical co-ordinates, i.e., for composite
* Corresponding author. Tel.: þ34 976 762 564; fax: þ34 976 732 078. E-mail address: [email protected] (L.I. Díez). 1290-0729/$ e see front matter Ó 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2010.07.012
straight fins and circular rods. Barker used the theory of orthogonal expansions [3] to obtain an infinite series solution and discovered a fundamental fact by analyzing its eigenvalues: under the conditions that warrant an approximate 1D treatment of ordinary finsdBi << 1, see [4] for a rigorous accountd, the infinite series converges to its first term and this approximation is equivalent to the usual 1D fin treatment simply by averaging the thermal conductivities in proportion to the cross-sectional areas. Lalot et al. [5] gave a rather contorted but interesting solution of the annular fin case, re-discovering essentially the same facts. Numerical treatment of the rectangular, constant thickness fin has been presented in [6], whereas the numerical studies in [7] have generalized the results of Barker for composite straight and annular fins of variable thickness. All these studies pattern their practical application from the case of a metal coating on a metal substrate. Two examples are galvanized or aluminized fins (zinc or aluminum on carbon steel) and copper fins covered with a protective coating of stainless steel. Even though only metals are involved, this translates into a relatively wide range of the conductivity ratio k1/k2, roughly from 0.1 to more than 10, and it is indeed surprising that a direct expedient, such as averaging k1 and k2, suffices for an accurate calculation under these circumstances. However, it forcibly must loose validity for the higher values of k1/k2 arising in other kind of applications. The most notable is a fouled or frosted finned heat exchanger, where the coating is
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Nomenclature An Ac Bi Cn e h Jn k L P Q Q q00 R r T X
x Yn
series of coefficients cross-sectional area of the fin, m2 Biot number series of coefficients convergence error convection coefficient, W m2 K1 Bessel function of first kind and order n ¼ 0, 1 thermal conductivity, W m1 K1 fin length, m fin perimeter, m heat transfer rate, W dimensionless heat transfer rate heat flux, W m2 dimensionless radial component of the temperature field fin radius, m; radial coordinate, m temperature, K dimensionless longitudinal component of the temperature field
a thermal insulator compared to the fin base medium. Also in this situation, the coating is of much larger thickness than in a bimetallic fin, which may contribute to complicate things. The pioneering study addressing this specific problem, by Epstein & Sandhu [8], examined two approximate, simple models that combine the “thermal resistances” of substrate and coating “in series” and “in parallel,” see Fig. 1. The latter is equivalent to use the area-average of conductivities, as in metal-coated fins. The former amounts to assume that heat conduction in the coating is locally 1D along the transverse direction, and to couple this idea with the usual approximate 1D model of an ordinary fin. Although accuracy was not assessed for either model, the “in series” calculation was recommended over the “in parallel” one, due to the fact that it gives the most conservative results for heat transfer. Devoted to the same class of situations, the work reported in [9] calculated the fin performance numerically and by an infinite series solution slightly simplified from the exact formulae. The poor performance of the “in parallel” model was also established, although the infinite series solution was not evaluated. More recently, the question has been revisited and put in order by Xia and Xacobi [10]. They show that, for the conditions usually
a
b
Fig. 1. “Thermal resistance” models for composite fins, from [8]: series model (a), parallel model (b).
longitudinal coordinate, m Bessel function of second kind and order n ¼ 0, 1
Greek letters relative error, % dimensionless radial coordinate K thermal conductivity ratio l eigenvalue q dimensionless temperature r radii ratio x dimensionless longitudinal coordinate
3 h
Subscripts 1 first value or term of the series; fin core 1t one-term approximation 2 fin coating N surrounding fluid b fin base m, n m-th, n-th value or term of the series p “in parallel” approximation s “in series” approximation
found in practice, it is appropriate to consider a simplified problem where the thermal field is fully two-dimensional in the coating, but approximately one-dimensional in the high-conductivity base fin. The analytical solution then demands a special application of the techniques of orthogonal series expansions; this is specifically developed in Cartesian co-ordinates, i.e., for rectangular straight fins. Furthermore it is shown that, for thermally slender fins, Bi << 1, the series converges to its first term, which leads as usual to a simple formula for the heat flux. This limit differs from the crosssectional average of conductivities, or “in parallel” model, so that previous work is confirmed, and the interesting fact is discovered that the exact, “2De2D” solution cannot deliver a proper approach. The reason is that its one-term limit implies a 1D field both in the core and the coating, something that can be appropriate for a metallic composite fin, but not so for a fin covered by an insulating layer. Although it might be anticipated that the new approximation coincides with the “in series” model of Fig. 1, in fact it doesn’t. Actually, the formula for the heat flux conveys an important gain in accuracy for Bi << 1, indicating that axial conduction in the frost layer is better accounted for. This theory has been extended to annular geometries in [11,12]. Deductions of “2De2D” solutions have been also reported by Gorobets [13] for finite, composite fins with non-uniform coating shapes, but with the different target of volume optimization. In this paper, we study a related class of situations, viz., the heat transfer from wires made of metallic cores coated by electrical insulation. In this case we are similarly at the higher end of the conductivity ratio, k1/k2 >> 1, with a large range of coating thicknesses, and dealing with thermally slender domains, Bi << 1. For this reason, we are going to adopt the same simplification of [10,11], i.e., a fully two-dimensional temperature field is prescribed in the isolator coating but approximately one-dimensional in the highconductivity core. As with the frosted fins, the calculation methods conventionally resorted to, see e.g. [14], essentially emanates from the “in series” concept. Giving the context, we develop here the case of infinitely long rods; although indicative, the results would need slight modification to be quantitatively applicable to a pin fin of finite length. In the Section 2 of the paper, we proceed with the analytical calculation and derive the exact 1De2D solution for composite
C. Cortés, L.I. Díez / International Journal of Thermal Sciences 49 (2010) 2391e2399
media in axial-cylindrical co-ordinates. It should be noted that this solution was not previously available in the literature. In Section 3, the one-term approximation of the new exact solution is obtained. The outcome is a simplified formula for the heat flux that is different from the “in parallel” and “in series” previous models, and improves upon both. This is established in Section 4 by considering a representative set of commercial designs possessing a variety of materials and geometric ratios. We finally compare in Section 5 the new formula against the conventional method, proving in this way that accuracy is greatly improved, while a considerable simplicity is retained.
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where the cross-sectional area and the perimeter of the core are Ac;1 ¼ pr12 and P1 ¼ 2pr1 . Boundary conditions at the fin base, x ¼ 0, and fin tip, x / N, are
T1 ð0Þ ¼ Tb
(3a)
T2 ðr; 0Þ ¼ Tb
(3b)
T1 ðNÞ ¼ TN
(4a)
T2 ðr; NÞ ¼ TN
(4b)
Convective transfer at the lateral boundary r ¼ r2 is imposed by
2. Problem description and analytical solution 2.1. Formulation of the problem
k2
Fig. 2 is a sketch of the wire geometry, represented as an infinitely long pin fin of constant, circular section. A metallic, conducting core of radius r1 and thermal conductivity k1 is covered by an insulating medium of radius r2 and conductivity k2. The conducting core is assumed passive, as in temperature measurement applications. The wire is heated through the base, x ¼ 0, and losses heat along its surface to a convective environment given by a coefficient h. The length of the wire is so large that we can consider the domain extending to x / N; exact temperature fields would show an axial-symmetric distribution, i.e., they would be given as a function of the longitudinal coordinate x and the radial coordinate r. The following simplifying assumptions, common in fin theory, are adopted: a) steady-state, b) no internal heat sources, c) two isotropic and homogeneous media with constant properties, d) uniform values of convection coefficient and boundary and fluid temperatures (which leads to linear boundary conditions), and e) nil contact resistance between the substrate and the coating. For ordinary values of the parameters, as shown in [10], we can assume that the thermal field is approximately 1D in the metallic core, T1(x) in 0 r r1, 0 x < N, whereas it remains fully 2D in the insulating layer, T2(r, x) in r1 r r2, 0 x < N. Since the core possesses a high thermal conductivity, conventional convective conditions lead to a thermally slender fin, for which the onedimensional approximation is warranted. However, this is not the case of the coating.
vT2 ¼ h½T2 ðr2 ; xÞ TN vr r¼r2
(5)
whereas temperature continuity at the interface is expressed by
T1 ðxÞ ¼ T2 ðr1 ; xÞ
(6)
The combination of the fin-like Eq. (1) with a fully elliptical problem seems strange, but it can be handled rather easily. By virtue of Eq. (6), Eq. (1) can be written as
k1 r1
v2 T2 ðr; xÞ vT2 ðr; xÞ þ 2k ¼ 0 2 vr r¼r1 vx2 r¼r1
(7)
so that it adopts the form of a boundary condition for the solution of Eq. (2) that gives the 2D temperature distribution T2 (x, r). The other three conditions necessary are already written as Eqs. (3b), (4b) and (5). Once solved for T2 (x, r), there is no need to integrate for the temperature T1(x). Since a one-dimensional field is assumed in the inner core, local evaluation of T1 is directly obtained by computing T2 at r ¼ r1, that is, it is simply given by Eq. (6). This also assures that, as a consequence of Eqs. (3b) and (4b), also Eqs. (3a) and (4a) are satisfied. Dimensionless variables are now introduced to streamline the nomenclature and represent a wider range of computed cases. We adopt the following definitions:
q ¼
T2 TN x r r k hr x ¼ h¼ r ¼ 1 K ¼ 1 Bi ¼ 2 Tb TN r2 k2 k2 r2 r2
(8)
By using them, Eq. (2) is transformed to 2.2. Analytical solution for the temperature distributions in the wire Governing equations for temperature in the core, T1, and the coating, T2, are:
k1 Ac;1
d2 T1 vT2 ¼ k P 2 1 vr r¼r1 dx2
1 v vT2 v2 T2 r þ ¼ 0 r vr vr vx2
(1)
1 v vq v2 q h þ 2 ¼ 0 h v h vh vx
(9)
which governs the dimensionless temperature q (h, x) in r h 1, 0 x < N. The set of boundary conditions given by Eqs. (3b), (4b), (5) and (7) becomes
qðh; 0Þ ¼ 1
(10a)
qðh; NÞ ¼ 0
(10b)
vq ¼ Bi qð1; xÞ vhh¼1
(10c)
(2)
rK
v2 q vq þ 2 ¼ 0 2 vhh¼r vx h¼r
(10d)
To solve this problem, we postulate a solution in the form
qðh; xÞ ¼ RðhÞXðxÞ: Substituting into Eq. (9), and after some algebra, Fig. 2. Problem of an infinitely long composite rod.
we get
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C. Cortés, L.I. Díez / International Journal of Thermal Sciences 49 (2010) 2391e2399
1 1 d dRðhÞ 1 d2 XðxÞ h ¼ ¼ const: RðhÞ h dh dh XðxÞ dx2
(11)
A solution with physical meaning requires that the axial part X(x) decrease along the coordinate, so that the constant of Eq. (11) has to be negative. In this way, the original partial differential equation separates in the following two ordinary differential equations
1 d2 XðxÞ 2 ¼ l >0 XðxÞ dx2 1 1 d dRðhÞ 2 h ¼ l >0 RðhÞ h dh dh
(12a)
qðh; xÞ ¼
An ½J0 ðln hÞ Cn Y0 ðln hÞexpðln xÞ
2 6 4
Z1
3
(12b)
1 2
r
using
Cn ¼
rK ln J0 ðrln Þ 2J1 ðrln Þ rK ln Y0 ðrln Þ 2Y1 ðrln Þ
An ¼ p
Bi
l2n
(15)
the
remaining
boundary
condition,
1þ
Bi2
1 BiY0 ðln Þ ln Y1 ðln Þ K 1 4 2 r2 K 2
(18)
l2n ln þ 2 2 ½BiY0 ðln Þ ln Y1 ðln Þ ½rK ln Y0 ðrln Þ2Y1 ðrln Þ
In the Appendix, the reader can consult the details about the resolution of the problem that yields the analytical solution given by Eqs. (13), (14), (15) and (18). Heat transfer from the wire can be calculated from the following equation:
and the eigenvalues ln satisfy the following eigenequation:
rK ln J0 ðrln Þ 2J1 ðrln Þ BiJ0 ðln Þ ln J1 ðln Þ ¼ rK ln Y0 ðrln Þ 2Y1 ðrln Þ BiY0 ðln Þ ln Y1 ðln Þ
upon
An:
(13)
(14)
it
(17)
qðh; 0Þ ¼ SAn Rn ðhÞ ¼ 1; Eq. (10a), we get the series of coefficients
n¼1
where the coefficients Cn are given by
2 2 hRm ðhÞRn ðhÞdh þ r2 KRm ðrÞRn ðrÞ7 5 ln lm ¼ 0
By
Solutions of Eqs. (12), combined into the temperature q ¼ XR, and with three of the four constants already determined by Eqs. (10b), (10c) and (10d), can be written as a function series in the form N X
a clear exploration of the values of the first ln, that can be otherwise feasibly computed by means of a numerical iteration. It can be shown in the appendix that the orthogonality property for the series of eigenfunctions Rn ðhÞ ¼ J0 ðhln Þ Cn Y0 ðhln Þ is derived from:
Q ¼ 2pr2 h
ZN ZN ½T2 ðr2 ;xÞTN dx ¼ 2pr22 hðTb TN Þ qð1; xÞdx 0
(19)
0
If the discrete values ln in this formula are substituted by a continuous variable l and we abbreviate its left and right sides as f(l) and g(l) respectively, the eigenvalues are the infinite zeros of the equation
To make this variable dimensionless, we cannot use the usual concept of efficiency, since it is by definition zero for an infinitely long fin. Instead, we define arbitrarily
f ðlÞ gðlÞ ¼ 0
Q ¼
(16)
Fig. 3 represents an example of the solution as one of the intersections on the l-axis of the functions f and g. This provides
Q ¼ 2pr22 hðTb TN Þ
ZN
qð1; xÞdx
(20)
0
Substitution of q (1, x) from Eq. (13) yields:
Q ¼
N X An
l n¼1 n
½J0 ðln Þ Cn Y0 ðln Þ
(21)
where An is given by Eq. (18), Cn by Eq. (14) and ln by Eq. (15). Equation (21) represents the exact analytical solution for the heat transfer losses of the wire, for an approximately one-dimensional thermal field in the core and two-dimensional in the insulation. 3. One-term approximation The use of the analytical solution is of course cumbersome due to the infinite series. However, as in ordinary fins, a useful approximation is valid for most practical problems, based on the fact that the small size and the reduced convective coefficient determine a thermally slender geometry, i.e., a situation with a Biot number Bi << 1. Therefore, paralleling the treatment of ordinary fins [4], the limit of the exact solution is sought under Bi / 0; it turns out that the series reduces then to its first term, thus leading to a simple, engineering formula for the heat flux. Given our definition of Bi, the condition Bi / 0 can be interpreted as a negligible radial gradient of temperature in the insulation, which adds up to the previous assumption in the same sense for the conducting core. By analyzing Eqs. (15) and (18), it can be shown that, for Bi / 0,
Fig. 3. Graphical determination of the roots of the eigenequation.
1 An / 0
n ¼ 1 n>1
(22a)
C. Cortés, L.I. Díez / International Journal of Thermal Sciences 49 (2010) 2391e2399
2395
On the one side, the “in parallel” approximation amounts to use the formula of a single-material rod with (k1Ac,1 þ k2Ac,2) replacing kAc. With our nomenclature, this gives
Qp ¼
Fig. 4. Heat fluxes in the wire according to the conventional “thermal resistances” “in series” modeling.
11
0
2
2Bi iC h A 1r2 2 2 r ðK 1Þ þ 1 þ Bi r lnrð1 KÞ þ 2
l1 /B @
(22b)
1 2 R1 ðhÞ ¼ J0 ðl1 hÞ C1 Y0 ðl1 hÞ/1 þ l1 lnðl1 hÞ/1 2
(22c)
Or, in other words, the series reduces to its first term and an analytical formula can be given for the first eigenvalue l1. It also happens that l1 / 0 when Bi / 0, which permits to show that the radial eigenfunction disappears; the one-term solution is also a 1D approximation. Equation (22b) can be further simplified, for usual values K >> 1 in electric wires, yielding the simpler limit:
1 2 2Bi l1 / 2 r ðK 1Þ þ 1 Bir2 Klnr
(22d)
1 q1t ðxÞ ¼ 1 þ l21;1t ln l1;1t h exp l1;1t x zexp l1;1t x 2
Q 1t ¼
1
l1;1t
¼
2Bi
(25)
It is worth reminding that this approach can be rigorously demonstrated to be the (one-term, one-dimensional) limit of the exact “2De2D” solution for Bi / 0. Therefore, the approach adopted here, that relies on a “1De2D” series solution and its limit under the same condition, produces a different but equally simple approximation that might retain the fact that the layer covering the fin is not a conducting medium but an insulator. This interpretation is reinforced by noting that the new formula given by Eq. (24) tends to Eq. (25) for a vanishingly small Bi, but only in the case that K is not very large, i.e., not when the inner material is much more conducting that the coating. On the other side, the traditional approach to an insulated wire (or frosted or fouled fin) assumes that the temperature distribution is longitudinally 1D in the core, but transversally 1D at each longitudinal coordinate in the coating. Fig. 4 sketches this approximate idea. The approximate temperature equation combines the fin balance with the “thermal resistances” of coating and external transfer disposed “in series:”
k1 pr12
d2 T1 T1 TN ¼ ¼ UPðT1 TN Þ lnðr2 =r1 Þ 1 dx2 þ 2pk2 h2pr2
(26)
Solving for T1dsubjected to the boundary conditions T1(0) ¼ Tb, T1(N) ¼ TNdleads logically to the same solution as a bare fin, with the quantity UP replacing hP. In particular, for the heat transfer loss,
2hr2 k1 hkr2 lnrr21 þ 1
!1 2
ðTb TN Þ
(27)
2
(23)
which in our dimensionless terms reads
12
r2 ðK 1Þ þ 1 Bir2 Klnr
12
2Bi
Qs ¼ pr1
Substituting these limits into Eqs. (13) and (21),
r2 ðK 1Þ þ 1
(24)
4. Discussion Although the one-term approximation is one-dimensional, cf. Eq. (23), it is not equivalent to previous models, and thus the new formula emerging in Eq. (24) differs from the traditional calculation of insulated wires.
Qs ¼
r2 K
12 (28)
2Bið1 BilnrÞ
Comparing Eq. (28) with Eq. (24) we see that, even for Bi / 0, they differ and would only provide similar results when r / 1, i.e., for a thin coating, or the trivial case when there is none. It is curious to note that the new approximation incorporates a slight radial variation given by a logarithm, Eq. (22d), exactly as in the elementary 1D solution of a cylindrical shell that is embedded in Eq. (26). However, in spite of this, the models are different. As a conclusion, the new one-dimensional approximation has produced a formula that can improve over the traditional “in series” concept, which can
Table 1 Case-studies considered. K
r
Application
Substrate
Coating
k1 (W/m K) [15e17]
k2 (W/m K) [15e17]
High-voltage wires
Aluminum, 99.5%
200
0.4
Electrolytic copper
Reticulated polyethylene Ethylene-propylene rubber
375
0.15
2500
Low-voltage wires
Tinned copper, 88% Cu Electrolytic copper
Reticulated polyethylene Ethylene-propylene rubber
60 375
0.4 0.15
150 2500
0.50e0.80
Heating cords
CoppereNickel Alloy, 80% Cu Copper brass, 75% Cu
Fluoride polymers Quartz fibers
12 25
0.16 0.03
75 750
0.20e0.40
Fouled or frosted rods
Stainless steel, 20% Cr-15% Ni Zinc
Ice Fouling, dirtiness
12.5 120
2.5 0.12
5 1000
0.10e0.90
500
0.25e0.60
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Table 2 Maximum fractional errors fn in the calculation of dimensionless heat transfer rates by a finite series (Bi ¼ 1 101).
r
K
f2
f3
0.10 0.20 0.25 0.50 0.80 0.90
5 75 500 150 2 500 1 000
0.00178 0.01196 0.00417 0.00142 0.00002 0.00001
0.00044 0.00047 0.00017 0.00007 0.00001 0.00000
Table 4 Relative errors of the different models. Low-voltage wires.
r
K
0.50
150
2 500
0.80
150
make sense based on the approximate nature of the latter. We elucidate this question in the next section.
5. Numerical results
Q n Q n1
(29)
n
where e is a given limit parameter representing the fractional correction that is deemed negligible. Table 2 summarizes the results for n ¼ 2, 3, for selected, extreme case-studies, and the highest Biot number, Bi ¼ 1 101, under which the slowest convergence can be expected. From those figures, it is clear that retaining three terms in the series suffices to assure an error always lower than 0.05%. Thus, the approximation of Eq. (21) is 3 X An
l n¼1 n
½J0 ðln Þ Cn Y0 ðln Þ
(30)
where the three eigenvalues and the constants are calculated as explained above.
r
K
0.20
75
750
0.40
75
750
0.25
K 500
2 500
0.60
500
2 500
3s ð%Þ
0.03 0.62 6.58 0.06 0.67 6.60
1.02 1.05 1.24 0.07 0.08 0.36
0.001 0.010 0.100 0.001 0.010 0.100
0.02 0.15 2.12 0.03 0.20 2.19
0.19 0.26 0.29 0.01 0.03 0.05
Bi
31t ð%Þ
3s ð%Þ
0.001 0.010 0.100 0.001 0.010 0.100
0.15 1.34 12.81 0.16 1.37 12.92
1.45 1.49 2.21 0.27 0.31 1.09
0.001 0.010 0.100 0.001 0.010 0.100
0.06 0.51 4.97 0.08 0.53 5.04
0.17 0.18 0.30 0.01 0.02 0.09
Bi
31t ð%Þ
3s ð%Þ
0.001 0.010 0.100 0.001 0.010 0.100
0.13 1.12 10.45 0.15 1.55 14.76
12.97 13.21 15.75 1.57 1.60 2.48
0.001 0.010 0.100 0.001 0.010 0.100
0.07 0.83 8.53 0.14 0.91 9.04
3.35 3.38 3.63 0.30 0.32 0.43
Tables 3e5 summarize the errors arising in the computation of the dimensionless heat transfer rates according to the exact solution, Eq. (30), the one-term approximation, Eq. (24), and the traditional “in series” model, Eq. (28). To quantify the results, we show the following percent relative errors:
31t ¼ 100 3s ¼ 100
Q 1t Q
(31a)
Q
Qs Q
(31b)
Q
As could be expected, comparison between the one-term approximation and the conventional formulation shows that the former is more appropriate for the lowest values of the Biot number. Ensuing errors 31t are lower than 0.15% for Bi ¼ 1 103 and lower than 1.55% for Bi ¼ 1 102; however they significantly Table 6 Relative errors of the different models. Fouled and frosted metallic rods.
Table 3 Relative errors of the different models. High-voltage wires.
r
31t ð%Þ
Table 5 . Relative errors of the different models. Heating cords.
In order to compare the new formula vs. the traditional model, we begin by selecting an adequate set of case-studies (see Table 1). With a view to the practical applications, different combinations of substrate/coating pairs and typical geometric ratios pertaining to electric wires are considered, taken from technical catalogues of the manufacturers [15e17]. We also include the different situation of a fouled or frosted metallic rod, which can also be treated by the formulation presented in this work. Taking into account typical dimensions and the low values of the heat transfer coefficient for losses to ambient air, we adopt three different values of the Biot number Bi ¼ 1 103, 1 102, 1 101, preserving a difference of two orders of magnitude between them. Next, in order to calculate the reference exact solution, we study the convergence of the series written in Eq. (21). If we call n the number of terms retained in a given approximation Q n ; a condition for it to be accurate enough can be
Qz
2 500
Bi 0.001 0.010 0.100 0.001 0.010 0.100
Fouled fins, K ¼ 1000
r
Bi
31t ð%Þ
0.10
0.001 0.010 0.100 0.001 0.010 0.100
0.13 1.98 18.59 0.02 0.11 1.05
4.70 4.82 7.26 0.00 0.00 0.01
0.001 0.010 0.100 0.001 0.010 0.100
0.02 0.40 3.98 0.02 0.09 0.99
78.10 78.30 79.99 2.25 2.28 2.31
0.90
Frosted fins, K ¼ 5a
0.10
0.90
3s ð%Þ
a Since K w 1, the limit used for the one-term approximation is the shown in Eq. (22b) instead of Eq. (22d).
C. Cortés, L.I. Díez / International Journal of Thermal Sciences 49 (2010) 2391e2399
2397
as an infinite series, has been demonstrated to converge with just retaining the three first terms. The conventional approach could be advisable however for Biot numbers approaching 0.1, and also because it always gives a conservative estimate, but it should be avoided in any case for conductivity ratios K below approx. 100. The general tendency, for all tested Biot numbers, is the lower the radii ratio r, the greater the conductivity ratio K for which the new formula improves the accuracy of the conventional calculation. Appendix Calculation of the problem relies on the determination of the temperature field
qðh; xÞ ¼ RðhÞXðxÞ
(A.1)
by the resolution of the differential system
Fig. 5. Comparison of relative errors as a function of dimensionless ratios (Bi ¼ 1 102). 1
1 d2 XðxÞ 2 ¼ l >0 XðxÞ dx2
(A.2)
1 1 d dRðhÞ 2 h ¼ l >0 RðhÞ h dh dh
(A.3)
increase for Bi ¼ 1 10 . Inversely, it can be said that the traditional approximation retains a remarkable validity up to high Biot numbers. The error of the one-term approximation diminishes when the radii ratio r increasesdthe size of the core increasesdor the conductivity ratio K decreasesdthe thermal conductivity of the coating increasesd. Table 6 displays the results when frosted or fouled metallic, infinite rods are simulated. The use of the one-term approximation is more accurate than in the case of wires, especially for the occurrence of a layer of frost on the rod; this is due to the small disparity in thermal conductivity between the coating and the substrate. For all cases computed, the one-term approximation tends to overestimate the heat transfer rates, opposite to the traditional approach. Comparing both methods, the new formula provides better results for the cases with low-to-moderate values of the radii ratio r. It can be also inferred that the lower the radii ratio, the greater the conductivity ratio K for which the one-term approximation improves upon the classical approach, and vice versa. This can be explained by the value of the product of dimensionless ratios rK: the classical approach is more appropriate when thin layers of very low thermal conductivity isolators are attached, in comparison to the conducting core. These tendencies can be observed in Fig. 5, where the relative errors 31t and 3s of some cases of Tables 3e5 are represented vs. each other.
and then the temperature field is expressed as a function of four constants k1, k2, k3, k4 that have to be determined from the prescription of the boundary conditions:
6. Conclusions
qðh; xÞ ¼ ½k1 expðlxÞþk2 expðlxÞ½k3 J0 ðlhÞþk4 Y0 ðlhÞ
subjected to the four boundary conditions:
qðh; 0Þ ¼ 1
(A.4)
qðh; NÞ ¼ 0
(A.5)
vq ¼ Bi qð1; xÞ vhh¼1
(A.6)
rK
v2 q vq þ 2 ¼ 0 2 vhh¼r vx h¼r
(A.7)
The generic solutions of differential eqs. (A.2) and (A.3) are given by
XðxÞ ¼ k1 expðlxÞ þ k2 expðlxÞ
(A.8)
RðhÞ ¼ k3 J0 ðlhÞ þ k4 Y0 ðlhÞ
(A.9)
(A.10)
Substituting Eq. (A.10) in condition (A.5), it is mandatory that Computation of the heat transfer rate from an insulated wire has been performed, considered as a composite piece of infinite length, made of a high thermal conductivity core and a low thermal conductivity coating. To this purpose, an exact analytical solution has been obtained in the form of an infinite series, and assuming a thermal field that is 1D in the core and 2D in the coating. The one-term approximation for Bi << 1 has been also derived analytically, providing a new formula to compare against the conventional calculation. Practical cases of electric wires and frosted and fouled pin-fins have been calculated, in order to analyze the errors incurred with the two approximations. For the cases of Biot numbers below 0.01, and all along the whole range of parameters, the new approximation produces better results than the conventional approach, retaining at once a similar simplicity. Maximum relative errors are of 1.35% in comparison to the exact solution; this solution, obtained
k1 ¼ 0
(A.11)
to produce a finite value of the temperature at the end, limit tip of the wire. Combining Eq. (A.1) with conditions (A.6) and (A.7), we get the following expressions for the boundary conditions of the radial component of the temperature:
dR ¼ Bi Rð1Þ dhh¼1
(A.12)
dR 1 2 ¼ K rl RðrÞ dhh¼r 2
(A.13)
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C. Cortés, L.I. Díez / International Journal of Thermal Sciences 49 (2010) 2391e2399
Substituting Eq. (A.9) in condition (A.12), the relation between constants is given by
k4 ¼
BiJ0 ðlÞ lJ1 ðlÞ k lY1 ðlÞ BiY0 ðlÞ 3
(A.14)
(A.15)
where we have used the derivatives of the Bessel functions [18]:
dJ0 ðkzÞ ¼ kJ1 ðkzÞ dz
(A.16)
dY0 ðkzÞ ¼ kY1 ðkzÞ dz
(A.17)
rK lJ0 ðrlÞ 2J1 ðrlÞ BiJ0 ðlÞ lJ1 ðlÞ ¼ rK lY0 ðrlÞ 2Y1 ðrlÞ BiY0 ðlÞ lY1 ðlÞ
An ½J0 ðln hÞ Cn Y0 ðln hÞexpðln xÞ
(A.19)
n¼1
rK ln J0 ðrln Þ 2J1 ðrln Þ rK ln Y0 ðrln Þ 2Y1 ðrln Þ
(A.21)
dRn ðhÞ ¼ Bi Rn ð1Þ dh h¼1
(A.22)
1r2 KR ðrÞ n 2
Z1 þ
hRn ðhÞdh
r
An ¼ 1r2 KR2 ðrÞ n 2
Z1 þ
(A.29)
hR2n ðhÞdh
r
More additional algebra, that requires the introduction of the Wronskian of the Bessel functions [18],
J0 ðzÞY1 ðzÞ J1 ðzÞY0 ðzÞ ¼
2 zp
(A.30)
zY0 ðzÞdz ¼ zY1 ðzÞ
(A.32)
Z
Z
zJ02 ðzÞdz ¼
Z zY02 ðzÞdz ¼
i z2 h 2 J ðzÞ þ J12 ðzÞ 2 0
(A.33)
i z2 h 2 Y0 ðzÞ þ Y12 ðzÞ 2
zJ0 ðzÞY0 ðzÞdz ¼
(A.23)
1 d dR ðhÞ 2 h n þ ln Rn ðhÞ Rm ðhÞ h dh dh 1 d dR ðhÞ 2 h m Rn ðhÞ þ lm Rm ðhÞ ¼ 0 h dh dh
(A.34)
z2 ½J ðzÞY0 ðzÞ þ J1 ðzÞY1 ðzÞ 2 0
(A.35)
3
An ¼ p
1 Bi Y0 ðln Þ ln Y1 ðln Þ Bi2 K1 1þ 2 4 2 r2 K 2
Bi
l2n
ln
½Bi Y0 ðln Þ ln Y1 ðln Þ
2
þ
ln
½rK ln Y0 ðrln Þ 2Y1 ðrln Þ
2
(A.36) ðA:24Þ
After some algebra is done upon Eq. (A.24), and integrating in the domain of interest r < h < 1, we get the relation:
2 2 hRm ðhÞRn ðhÞdh þ r2 KRm ðrÞRn ðrÞ7 5 ln lm ¼ 0
Then, for n s m, it is required that:
(A.31)
brings along the sought value of the coefficients An:
The orthogonality of every pair of eigenfunctions Rn and Rm is expressed by:
1 2
zJ0 ðzÞdz ¼ zJ1 ðzÞ
Z
dRn ðhÞ 1 2 ¼ K rln Rn ðrÞ dh h¼r 2
r
(A.28)
(A.20)
1 d dR ðhÞ 2 h n þ ln Rn ðhÞ ¼ 0 h dh dh
6 4
An hRn ðhÞRm ðhÞ ¼ hRm ðhÞ
n¼1
Z
and the eigenvalues ln satisfy the eigenequation (A.18). We need now to obtain the coefficients An to close the solution of the problem. To this purpose, we check the orthogonality of the functions Rn ðhÞ ¼ J0 ðhln Þ Cn Y0 ðhln Þ. These functions comply with the differential equation of the radial component (A.3), and with the boundary conditions (A.12) and (A.13):
Z1
(A.27)
and the use of some additional properties of Bessel functions [18],
where the coefficients Cn are given by
2
N X
(A.18)
There are infinite solutions of the Eq. (A.18), the so-called eigenvalues ln (n ¼ 1 . N). Summing up, regrouping and recalling the constants, temperature field is given by
Cn ¼
An Rn ðhÞ ¼ 1
n¼1
Integration in the domain r < h < 1 and use of Eq. (A.26), along with some algebra, yield to:
From Eqs. (A.14) and (A.15), the eigenequation (15) is here obtained:
N X
(A.26)
Turning to the remaining boundary condition, Eq. (A.4) can be expressed as: N X
rK lJ0 ðrlÞ 2J1 ðrlÞ k 2Y1 ðrlÞ rK lY0 ðrlÞ 3
qðh; xÞ ¼
1 2
hRm ðhÞRn ðhÞdh ¼ r2 KRm ðrÞRn ðrÞ
r
and repeating the process in condition (A.13),
k4 ¼
Z1
(A.25)
The problem is now closed: the dimensionless temperature field in the coating is given by Eqs. (A.18), (A.19), (A.20) and (A.36). References [1] A.D. Kraus, A. Aziz, J.R. Welty, Extended Surface Heat Transfer. John Wiley, New York, 2000, Sect. 15.6. [2] J.J. Barker, The efficiency of composite fins, Nuclear Science and Technology 3 (1958) 300e312. [3] M.N. Ozisik, Heat Conduction. Wiley, New York, 1980.
C. Cortés, L.I. Díez / International Journal of Thermal Sciences 49 (2010) 2391e2399 [4] M. Levitsky, The criterion for validity of the fin approximation, International Journal of Heat and Mass Transfer 15 (1972) 1960e1963. [5] S. Lalot, C. Tournier, M. Jensen, Fin efficiency of annular fins made of two materials, International Journal of Heat and Mass Transfer 42 (1999) 3461e3467. [6] E.M.A. Mokheimer, M.A. Antar, J. Farooqi, S.M. Zubair, Analytical and numerical solution along with PC spreadsheets modeling for a composite fin, Heat and Mass Transfer 32 (1997) 229e238. [7] C. Cortés, L.I. Díez, A. Campo, Efficiency of composite fins of variable thickness, International Journal of Heat and Mass Transfer 51 (2008) 2153e2166. [8] N. Epstein, K. Sandhu, Effect of uniform fouling on total efficiency of extended heat transfer surfaces, in: Proceedings of the Sixth International Heat Transfer Conference, Toronto, Ontario, Canada, vol. 4, 1978, pp. 397e402. [9] H. Barrow, J. Mistry, D. Clayton, Numerical and exact mathematical analyses of two-dimensional rectangular composite fins, in: Proceedings of the Eight International Heat Transfer Conference, San Francisco, California, USA, vol. 2, 1986, pp. 367e372. [10] Y. Xia, A.M. Jacobi, An exact solution to steady heat conduction in a twodimensional slab on a one-dimensional fin: application to frosted heat
[11]
[12]
[13] [14] [15] [16] [17] [18]
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exchangers, International Journal of Heat and Mass Transfer 47 (2004) 3317e3326. A.D. Sommers, A.M. Jacobi, An exact solution to steady heat conduction in a two-dimensional annulus on a one-dimensional fin: application to frosted heat exchangers with round tubes, ASME Journal of Heat Transfer 128 (2006) 128e404. P. Tu, H. Inaba, A. Horibe, Z. Li, N. Haruki, Fin efficiency of an annular fin composed of a substrate metallic fin and a coating layer, ASME Journal of Heat Transfer 128 (2006) 851e854. V. Gorobets, Influence of coatings on thermal characteristics and optimum sizes of fins, Journal of Enhanced Heat Transfer 15 (2008) 65e80. H.D. Baker, E.A. Ryder, N.H. Baker, Temperature Measurement in Engineering, vol. 1, Omega Press, Stamford, Connecticut, 1975, (Chapter 7). General Cables Technology Corporation, www.generalcable.com. Groupe Omerin, www.omerincables.com. Times Microwave Systems Inc., www.timesmicrowave.com. M. Abramowitz, A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables Ninth Printing, Dover, NY, 1972.