Heat transfer asymptote in laminar flow of non-linear viscoelastic fluids in straight non-circular tubes

Abstract

The fully developed thermal field in constant pressure gradient driven laminar flow of a class of non-linear viscoelastic fluids with instantaneous elasticity in straight pipes of arbitrary contour ∂D with constant wall flux is investigated. The non-linear fluids considered are constitutively represented by a class of single mode, non-affine constitutive equations. The driving forces can be large. Asymptotic series in terms of the Weissenberg number Wi are employed to expand the field variables. A continuous one-to-one mapping is used to obtain arbitrary tube contours from a base tube contour ∂D₀. The analytical method presented is capable of predicting the velocity and temperature fields in tubes with arbitrary cross-section. Heat transfer enhancement due to shear-thinning is identified together with the enhancement due to the inherent elasticity of the fluid. The latter is to a very large extent the result of secondary flows in the cross-section but there is a component due to first normal stress differences as well. Increasingly large enhancements are computed with increasing elasticity of the fluid as compared to its Newtonian counterpart. Order of magnitude larger enhancements are possible even with slightly viscoelastic fluids. The coupling between inertial and viscoelastic non-linearities is crucial to enhancement. Isotherms for the temperature field are discussed for non-circular contours such as the ellipse and the equilateral triangle together with the behavior of the average Nusselt number Nu, a function of the Reynolds Re, the Prandtl Pr and the Weissenberg Wi numbers. Analytical evidence for the existence of a heat transfer asymptote in laminar flow of viscoelastic fluids in non-circular contours is given for the first time. Nu becomes asymptotically independent from elasticity with increasing Wi, Nu = f(Pe, Wi) → Nu = f(Pe). This asymptote is the counterpart in laminar flows in non-circular tubes of the heat transfer asymptote in turbulent flows of viscoelastic fluids in round pipes. A different asymptote corresponds to different cross-sectional shapes in straight tubes. The change of type of the vorticity equation governs the trends in the behavior of Nu with increasing Wi and Pe. The implications on the heat transfer enhancement is discussed in particular for slight deviations from Newtonian behavior where a rapid rise in enhancement seems to occur as opposed to the behavior for larger values of the Weissenberg number where the rate of increase is much slower. The asymptotic independence of Nu from elasticity with increasing Wi is related to the extent of the supercritical region controlled by the interaction of the viscoelastic Mach number M and the Elasticity number E, which mitigates and ultimately cancels the effect of the increasingly strong secondary flows with increasing Wi to level off the enhancement. The physics of the interaction of the effects of the Elasticity E, viscoelastic Mach M, Reynolds Re and Weissenberg Wi numbers on generating the heat transfer enhancement is discussed.

Introduction

Experimental findings concerning heat transfer characteristics of aqueous polymer solutions flowing in straight tubes point at considerable enhancement as compared to its Newtonian counterpart driven by the same conditions and in the same geometry. Specifically, it is reported that heat transfer results for viscoelastic aqueous polymer solutions are considerably higher in flows fully developed both hydrodynamically and thermally, as much as by an order of magnitude depending primarily on the constitutive elasticity of the fluid and to some extent on the boundary conditions, than those found for water in laminar flow in rectangular ducts, Hartnett and Kostic [1], [2]. Heat transfer phenomena in laminar flow of non-linear fluids has not been the subject of many investigations with the exception of round pipes and the case of inelastic shear-thinning fluids in tubes of rectangular cross-section in spite of the widespread use of some specific contours in industry such as flattened elliptical tubes. This statement is true for all cross-sectional shapes for both steady and unsteady phenomena including quasi-periodic flows. Heat transfer with viscoelastic fluids has been declared to be a new challenge in heat transfer research in the early nineties, Hartnett [3], but progress has been limited since that time. The physics of the phenomenon has not been entirely clarified.

Highly enhanced heat transfer to aqueous solutions of polyacrylamide and polyethylene of the order of 40–45% as compared to the case of pure water in flattened copper tubes was observed by Oliver [4] and later by Oliver and co-workers as early as 1969. Recent numerical investigations in rectangular cross-sections of Gao and Hartnett [5], [6], Naccache and Souza Mendes [7], Payvar [8] and Syrjala [9] establish the connection between the enhanced heat transfer observed and the secondary flows induced by viscoelastic effects. The former researchers as well as Naccache and Souza Mendes predict for instance viscoelastic Nusselt numbers as high as three times their Newtonian counterparts. Gao and Hartnett [5] report numerical results in rectangular contours which provide evidence that the stronger the secondary flow (as represented by the dimensionless second normal stress coefficient Ψ₂) the higher the value of the heat transfer (as represented by the Nusselt number Nu) regardless the combination of thermal boundary conditions on the four walls. Constant heat flux is imposed everywhere on the heated walls in their numerical experiments with the remaining walls being adiabatic. The combination of boundary conditions plays some role in the enhancement reported with the largest enhancement occurring when two opposing walls are heated. Despite these efforts heat transfer characteristics of viscoelastic fluids in steady laminar flow in rectangular tubes remains very much an open question.

Although these studies establish a basis for the mechanism of viscoelastic heat transfer the results are at best indicative of the prevailing trends for slight deviations from Newtonian behavior. Gao and Hartnett [5] and Naccache and Souza Mendes [7], as well as Payvar [8] and Syrjala [9] use the Reiner-Rivlin and Criminale–Ericksen–Filbey (CEF) constitutive equations, respectively, to characterize the fluid behavior. Both of these constitutive structures are open to an array of criticism and are hardly representative of any viscoelastic fluid in complex flows except in the case of slight deviations from Newtonian behavior for both equations, and when the flow is viscometric in which case the CEF equation is exact.

The shear rate dependent viscosity of purely viscous fluids (negligible relaxation time) is also responsible for enhanced and reduced heat transfer in the case of shear-thinning and shear–thickening fluids, respectively, as shown by Gingrich et al. [10] and others. But there is evidence in the literature that in the case of viscoelastic fluids the effect of the shear rate dependent viscosity on heat transfer enhancement is at least two orders of magnitude smaller when compared to the influence of the secondary flow, Naccache and Souza Mendes [7]. Thus the latter remains the dominant mechanism for enhanced heat transfer.

In this paper we investigate the heat transfer behavior of a class of non-affine and non-linear viscoelastic fluids in straight tubes of non-circular contour in pressure gradient driven laminar flow and under constant wall flux conditions. The velocity field is obtained via hierarchical regular perturbation problems derived through the expansion of the field variables into asymptotic series in terms of the Weissenberg number Wi, Siginer and Letelier [11]. The solution for the thermal field is presented in this paper. The thermal field at the zeroth order represents the temperature distribution in a Newtonian fluid in a tube of arbitrary contour with constant wall flux. The thermal field at the first order is null, a consequence of a null first order velocity field. At the second order the thermal field is altered separately by shear-thinning and elasticity with additive superposed contributions. The longitudinal velocity field is further changed at the third order with a corresponding change in the thermal field due to elasticity, but more importantly at this order a secondary flow triggered by unbalanced second normal stresses brings large changes to the temperature distribution and heat dissipation.

The physics governing the asymptotic behavior of Nu and its independence from elasticity with increasing Wi is clarified and discussed. The physics is based on the change of type of the vorticity equation. The analysis leading to the criteria governing the change of type for the class of non-affine quasilinear fluids with interpolated Maxwell type convected derivatives and instantaneous elasticity of interest in this paper is similar to the analysis for upper convected Maxwell (UCM) fluids. When the viscoelastic Mach number, the ratio of a characteristic velocity of the fluid to that of the speed of shear waves into rest, exceeds one M² = ReWi > 1 a supercritical region develops near the centerline of the tube with a compatible elliptic region near the wall. Inertial terms can only be neglected for small values of the Mach number as the analysis in this paper will also make it clear. When the Mach number M ≫ 1 the spread of the hyperbolic region is small if the dimensionless Elasticity number E = Wi/Re is large and vice versa. A complete discussion of the cause and effect relationship of the interaction of the viscoelastic Mach number, the Elasticity number and the Peclet number on the heat transfer enhancement in relation to the extent of the hyperbolic region is given.

For each given non-circular cross-section the enhancement curves Nu = f(Pe, Wi) level off at a higher critical Wi_c with increasing Pe or inertia Re. Beyond the boundary Wi_c = f_c(Pe), that is for Wi > Wi_c at fixed Pe enhancement asymptotically approaches a constant value and is no longer a function of the elasticity, Nu = f(Pe). This asymptotic behavior is quite similar to Virk’s asymptote for drag reduction in turbulent flow in round tubes under isothermal conditions, Virk et al. [12]. The counterpart of Virk’s asymptote for heat transfer with viscoelastic fluids in turbulent flow in round straight tubes was shown experimentally by Hartnett [3]. We show for the first time in this paper the existence of a similar asymptote Nu = f(Re) or more generally Nu = f(Pe) in laminar flow of non-linear viscoelastic fluids in non-circular straight tubes. A different asymptote exists for each cross-sectional shape separating the region where the enhancement is a function of both Wi and Re (at constant Pr) at low Wi from the region where it is a function of Re alone at higher Wi.