NONUNIQUENESS AND STABILITY FOR HEAT CONDUCTION THROUGH A DUPLEX HEAT EXCHANGER TUBE

Journal of Thermal Stresses ISSN: 0149-5739 (Print) 1521-074X (Online) Journal homepage: http://www.tandfonline.com/loi/uths20 NONUNIQUENESS AND STABILITY FOR HEAT CONDUCTION THROUGH A DUPLEX HEAT EXCHANGER TUBE J. R. Barber To cite this article: J. R. Barber (1986) NONUNIQUENESS AND STABILITY FOR HEAT CONDUCTION THROUGH A DUPLEX HEAT EXCHANGER TUBE, Journal of Thermal Stresses, 9:1, 69-78, DOI: 10.1080/01495738608961888 To link to this article: https://doi.org/10.1080/01495738608961888 Published online: 21 May 2007. Submit your article to this journal Article views: 14 View related articles Citing articles: 15 View citing articles Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalinformation?journalcode=uths20 Download by: [University of Michigan] Date: 03 January 2018, At: 08:22

NONUNIQUENESS AND STABILITY FOR HEAT CONDUCTION THROUGH A DUPLEX HEAT EXCHANGER TUBE J. R. Barber Department ofmechanical Engineering and Applied Mechanics University of Michigan Ann Arbor, Michigan 48109 An analysis is given for radial heat flow throu8,h the wall ofa duplex heat exchanger tube with a pressure or gap-dependent contact resistance at the interface. Thermal expansion ofthe tube changes the value ofthis resistance, and it is shown that for certain thermal conditions there will be more than one steady-state solution. The stability ofthese multiple solutions is then investigated using a perturbation method, and it is found that there is always an odd number of solutions, alternately stable and unstable. This kind of behavior has been observed in other thermoelastic contact problems, but this is the first example in which it has been exhibited in a case ofcontact between similar materials. INTRODUCTION When heat is conducted across an interface between two solids, the thermal distortion of the solids will generally influence the contact conditions and hence the heat conduction process. If the materials of the two solids are different, thermal rectification occurs-i.e., the thermal resistance of the interface depends upon the direction of heat flow [1,2]. For a contact between two half-spaces, interface resistance increases with heat flow into the material with the higher distortivity 8, defined by a(1 + v) &=-- k (I) where a, v, and k: are, respectively, coefficient of thermal expansion, Poisson's ratio, and thermal conductivity. Mathematical difficulties were encountered in early attempts to treat problems of this kind. In particular, the conventional boundary conditions of perfect thermal The author gratefully acknowledges support from the U.S. Army under contract DAAG 29-82-k 001. Journal of Thermal Stresses, 9:69-78, 1986 Copyright 1986 by Hemisphere Publishing Corporation 69

70 1. R. BARBER R contact seporotjon p- 0 _g Fig. 1 Typical variation of interface resistance, R with contact pressure p and gap g. contact (no resistance to heat flow, leading to continuity of temperature) in regions of mechanical contact and complete insulation (no heat flux) in regions of separation led to ill-posed boundary-value problems wherever the hotter solid had the higher distortivity [3-5]. An asymptotic analysis of the transitions between the various contact regions [5] suggests that the conventional boundary conditions can be safely applied when the heat flows in the opposite direction, but there is evidence that, in this case, the solution obtained is not necessarily unique [6]. In more recent papers [7,8], the author has probed this question by using more realistic boundary conditions that include a thermal contact resistance at the interface. It is anticipated that this resistance will increase monotonically with decreasing pressure in regions of contact [9] and be approximately linear with gap thickness in regions of separation (see Fig. 1). An idealization of this idea is the condition of "imperfect contact" introduced in [4] and subsequently used to solve a variety of thermoelastic contact problems [10-13]. Hitherto, it,has been assumed that these phenomena only arise in contact between two dissimilar materials, but the present paper will demonstrate their occurrence in an important heat transfer application involving similar materials. The author is grateful to Dr. M. G. Srinivasan for bringing this problem to his attention. STATEMENT OF THE PROBLEM We consider a duplex heat exchanger tube, as shown in Fig. 2, fabricated by shrinking one cylinder onto another, leaving the assembly in a state of prestress. Heat transfer occurs between the inner surface of the tube (radius r.) and a contained fluid Fig. 2 Geometry of the problem. The heat transfer coefficient is h, at radius r, and h, at radius r,. Both tubes are of the same material.

HEAT CONDUCTION THROUGH DUPLEX TUBE 71 at temperature T 1 through a coefficient hi and between the outer surface (radius '2) and surrounding fluid at temperature T 2 through a coefficient h 2 At the interface (radius '0) there is a contact resistance R that varies with pressure or gap in a monotonic but otherwise unprescribed way, as shown in Fig. l. It is assumed that T 1 and T 2 are axisymmetric and that the axial variation is sufficiently slow for the problem to be treated as one of two-dimensional plane strain. The initial state of the interface can be described by a radial interference do or an initial gap go. STEADY-STATE SOLUTION In the steady state, the temperature at radius, is where = T + ~ + qo'o log ~ '0 < r < '2 2 A 2h2 k r r, A. i = -, i = 1, 2 '0 and qo is the heat flux per unit area at the interface (radius '0), taken as positive when directed outwards. These equations imply the existence of a temperature jump.1 at the interface given by where (2) (3) (4) (5) (6) and (7) We can solve this equation for qo, obtaining

72 J. R. BARBER (8) The temperature field described in Eqs. (2) and (3) can also be used to calculate the free thermal expansion of the two cylinders, i.e., the radial displacement u, that would occur in the absence of interface or fluid pressure. Substituting Eqs. (2) and (3) into equation (c, 151) of Timoshenko and Goodier [14], we find (9) for the inner cylinder and.for the outer. If the steady-state solution involves separation at the interface, the gap must be where and (10) g = U,2 - UrI - do (11) Substituting from Eqs. (9) and (10), and eiiminating qo using Eq. (8), we find A~ A~ c = log A - log A 2 (A~ - 1) 2 (A~ - I) I (12) (13) (14) If Eq. (12) defines a negative value of g, interference is implied and we must superpose an elastic field with an interface pressure p chosen to reduce this interference to zero. Elementary calculations [14, 28] show that in plane strain an interface pressure p produces a differential radial displacement (15)

HEAT CONDUCTION TIlROUGH DUPLEX TUBE 73 and hence, from Eq. (12) and the condition of zero interference, p is given by Following [7,8], Eq. (16) can be subsumed under Eq. (12) by defining (-C3P) as a continuation of g into negative values. (16) GRAPHIC SOLUTION The geometry of the problem demands that r2 > ro > '1 and with this condition it can be shown that c\ > C2 > O. It follows thatf is a monotonically increasing function of R that tends to C2 - (>0) when R - 0 and to 1 when R -+ 00. C\ The function / is therefore monotonic with g as shown in Fig. 3, and we can obtain a graphic solution of Eq. (12) by writing it in the form The right-hand side of this equation can be plotted on Fig. 3 as a straight line passing through the point (-do, 0) with a slope of [0:(1 + v)ro(t 2 - TI))-I. Intersections of this line with with /(g) represent possible steady-state solutions of the problem. We can conclude from Fig. 3 that there will be one and only one such intersection if (T 2 - T I ) < 0 (i.e. if the straight line has negative slope) or if do < d.. where d, is the intercept obtained by drawing the tangent to the f(g) curve at the point of maximum slope and extending it to intersect the g axis. In the case of an initial gap go the hitter condition becomes -go < d.. U do > d, and T 2 > T.. there will be some finite range of (T 2 - TI) in which three solutions are obtained, but on either side of which there will be a unique solution. (17) 9+<1" ao+vjrj't,-t) fig) / / <1" 9<O,P>O; contact 0 0>0; separation Fig. 3 Graphical representation of Eq. (17). For the case illustrated there are threesolutions, A, B, and C, of which B is unstable and A and C are stable [condition given in Eq. (39)].

74 J. R. BARBER If there are several points of maximum slope giving intercepts d.. d 2, etc., a more complex behavior is obtained, with a possibility of more then three solutions in some ranges of (T 2 - T.). However, for given conditions, the number of possible solutions is always odd. We note that the conventional boundary conditions of perfect thermal contact (R = 0, p > 0) and perfect separation (R - 00, g > 0) correspond to the two line segments f = cdc.. p > 0; and f = 1, g > O. If there is a positive gap go under isothermal conditions, there will always be a range of (negative) values of (T 2 - for which the straight line in Fig. 3 does not intersect either of these line segments. As in previous problems, a solution can only be obtained in these cases if the idealized f curve is completed by the inclusion of the line segment g = 0, C2/C. < f < I, which corresponds to imperfect contact as defined in [4, 5, 7]. STABILITY CONSIDERATIONS Following [7,8], we investigate the stability of the preceding steady-state solutions by determining the conditions under which a small perturbation in the temperature field can grow exponentially with time t. The perturbation must satisfy the transient heat conduction equation where K is the thermal diffusivity. The exponentially growing solution of this equation is T = e D![A 1 / o(l3r) + B,Ko(l3r)], T = e D1 [A 2 / o(13r) + B 2Ko( 13r)], r. < r < ro ro < r < r2 where 13 2 = al«and la, Ko are modified Bessel functions of first and second kind, respectively. The constants A.. A 2, B.. and 8 2 are to be determined from the boundary conditions at r = r.. rs, and roo At r = r.. r2 we have radiation into a medium at zero temperature (since the fluid temperature is not perturbed) and hence T.) (18) (19) (20) (21) (22) giving the two equations (23)

HEAT CONDUCTION THROUGH DUPLEX TUBE 75 where At r = ro. we require continuity in heat flux and hence in atfar between the cylinders. giving (25) (26) LINEAR PERTURBATION OF THE INTERFACE RESISTANCE EQUATION The final boundary condition is obtained by linearizing the relation in Eq. (5) between 1:1, R. and- qo for small perturbations about the steady state. Differentiating, we obtain The perturbation in heat flux, &qo. is given by and the perturbation in 1:1 is at &qo = - k - (r == ro) = - - zae Q1 ar r«81:1 == Ttr = r(;) - T(r == rt) from Eqs. (19) and (20). 'Solving Eq. (13) for R. we find k (27) (28) (29) (30) ro [C1- C2 ] R==- ---c k 1 - f I (31) and differentiating (32) We can find the change in gap 8g by substituting Eqs. (19 and 20) into (c 151) of [14]. using 11.3.25/7 of [15] obtaining

76 J. R. BARBER _ [A)hlll(hIZ) - B)hIK1(h1Z) - A/t(z) + BtK1(Z)]} hi - 1 (33) CHARACTERISTIC EQUATION Finally, we substitute for R, ar, qo, 8qo, and lid from Eqs. (31), (32), (8), (28), and (30), respectively, into Eq. (27) and eliminate AI> BI> A 2, and B 2 using Eqs. (23), (24), and (26), to obtain the characteristic equation where A given steady-state solution of Eq. (12) will be unstable if arsmall perturbation grows exponentially, and this will be the case if Eq, (34) has one or more real roots other than zero. It can be shown that G(z) is an even function that varies monoton-. ically with z in the range z > 0 and that G(z) < 0 if z is sufficiently large, for all physically meaningful values of the parameters. It follows that a real positive root will occur if (34) Lt G(z) > 0...0 Expanding the Bessel functions in Eq. (34) at small values of the argument, we find that the terms containing the factor (1 - f) cancel and (37) Hence, the solution will be unstable if

HEAT CONDUcnON THROUGH DUPLEX TUBE f' > ------- a(i + v)(t 2 - T,)ro I 77 (39) since (CI - C2) > 0, (see above). Referring to Fig. 3, it is clear that this criterion describes those intersections at which the f(g) curve crosses (g + do)/a(l + v)ro(t 2 - T 1 ) from below with increasing g. Thus, solution B is unstable, while A and C are stable. More generally, there will always be an odd number of steady-state solutions that are alternately stable and unstable. If there is only one solution, it is always stable. We also note that as in [7,8] it is possible to define an "energy" function u(i + v)ro(t 2 - T 1 ) Jfdg - ~ (g + t4i V(g) = ------------ such that the condition for a steady-state solution is and the condition for stability is av -=0 ag Thus, in all respects, the problem treated in this paper exhibits the same behavior as the examples combined in [7,8], despite the fact that there is no difference in distortivity between the two contacting bodies. It is interesting to speculate whether the bodies being multiply-connected is an essential factor in causing this behavior with similar materials. REFERENCES (40) (41) (42) I. A. M. Clausing, Heat transfer at the interface of dissimilar metals-the influence of thermal strain, Int. J. Heat Mass Transfer vol. 9, pp. 791-801, 1966. 2. J. R. Barber, The effect of thermal distortion on constriction resistance, Int. J. Heat Mass Transfer. vol. 14, pp. 751-766,1971. 3. J. R. Barber, Indentation of the semi-infinite elastic solid by a hot sphere, Int. J. Mech. Sci., vol. 15, pp. 813-819, 1973. 4. J. R. Barber, Contact problems involving a cooled punch, J. Elasticity. vol. 8, pp. 409-423, 1978. 5. M. Comninou and J. Dundurs, On the Barber boundary conditions for thermoelastic contact, J. App/. Mech., vol. 46, pp. 849-853, 1979. 6. M. Comninou and J. Dundurs, On lack of uniqueness in heat conduction through a solid to solid interface, ASME J. Heal Transfer. vol. 102, pp. 319-323, 1980.

78 J. R. BARBER 7. J. R. Barber, J. Dundurs, and M. Comninou, Stability considerations in thermoelastic contact, J. AppJ. Mech., vol. 47, pp. 871-874,1980. 8. J. R. Barber, Stability of thermoelastic contact for the Aldo model, J. Appl. Mech., vol. 48, pp. 555-558, 1981. 9. M. G. Cooper, B. B. Mikic, and M. M. Yovanovich, Thermal contact conductance, Int. J. Heal Mass Transfer, vol. 12, pp. 279-300, 1969. 10. J. R. Barber, Indentation of an elastic half-space by a cooled flat punch, Q. J. Mech. AppJ. Math., vol. 35, pp. 141-154, 1982. 11. M. Comninou, J. Dundurs, and J. R. Barber, Planar Hertz contact with heat conduction, J. Appl, Mech.. vol. 48, pp. 549-554, 1981. 12. M. Comninou, J. R. Barber, and J. Dundurs, Heat conduction through a flat punch, J. Appl. Mech.. vol. 48, pp. 871-875, 1981. 13. J. R. Barber and M. Comninou, The penny-shaped interface crack with heat flow: II Imperfect contact, J. Appl, Mech., vol. SO, pp. 770-776, 1983. 14. S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3d ed., McGraw-Hill, New York, 1970. IS. M. Abramowitz and I. A. Stegun, eds., Handbook of mathematical functions, National Bureau of Standards, Washington, D.C. 1970. Request reprints from J. R. Barber. Received February 20, 1985 Communicated by Donald E. Carlson