Viscous heating effects on heat transfer characteristics of laminar compressible channel flow

Highlights

• Incompressible channel flow is analysed including viscous heating in energy equation.

• Temperature profiles and Nusselt numbers are derived as function of Brinkman number.

• CFD results support the analysis and enable its extension to compressible flow case.

• Local Nusselt number correlations are given for compressible flow case.

• Fluid heating and cooling, UHF and UWT BC, CR and PP cross-section are investigated.

Abstract

The present paper investigates viscous heating effects on heat transfer characteristics of laminar Newtonian compressible channel flow. At first, incompressible Poiseuille flow case is addressed analytically by including viscous dissipation term in energy conservation equation. Temperature distributions in the channel cross-section and local Nusselt numbers are derived as functions of the Brinkman number, highlighting the role of the viscous term. The analysis is numerically supported by a set of CFD simulations. Additional CFD results are employed to extend the analysis to compressible flow case resulting in dedicated local Nusselt number correlations, function of Brinkman and Mach numbers, and expansion-altered temperature profiles. The study assumes no-slip flow and no temperature-jump boundary conditions at the channel wall (Knudsen number $\text{Kn}<{10}^{-3}$), and addresses both fluid heating and cooling, both uniform heat flux (UHF) and uniform wall temperature (UWT) boundary condition (BC), and both circular (CR) and parallel-plate (PP) channel cross-section.

Introduction

Micro-fluidic devices are becoming relevant in many fields of application. This calls for a better understanding of both fluid flow and heat transfer mechanisms at such small scales. This is particularly true, for instance, in micro electro mechanical systems where due to the growing dissipated power densities at stake more efficient cooling is required.

While in conventionally-sized channels viscous heating is commonly negligible, heat transfer characteristics can be considerably altered by viscous dissipation effects at smaller scales. In particular, it is known that the Nusselt number is reduced as the Brinkman number grows in case of heated flow while, under certain conditions, even the opposite can be observed in cooled flow.

The relevance of viscous heating can be evinced from energy conservation equation where viscous dissipation term is proportional to fluid dynamic viscosity and velocity gradient. This means that in confined flows viscous heating may become important when fluid viscosity is particularly high and/or in case of (fast) micro-flows. It is also noted that, by this mechanism, heat is predominantly generated close to the channel walls, where velocity gradient is larger.

Heat transfer characteristics in channels and micro-channels including viscous heating effects have been addressed in the literature since the pioneering work of Brinkman [1] (cfr. Table 1), with the main aim of investigating temperature distributions and Nusselt numbers.

The works are mostly analytical and identify three main effects influencing heat transfer at small scales while being mostly negligible at larger ones. These are characterised by as many non-dimensional numbers and are

• rarefaction effects (for Knudsen numbers $\text{Kn}⪆{10}^{-3}$),

• viscous heating (for Brinkman numbers $\text{Br}⪆{10}^{-3}$ in modulus),

• axial conduction (for Péclet numbers $\text{Pe}=\text{Re}\text{Pr}⪅100$).

Other effects, seldomly addressed in the literature, are the conjugate heat transfer in presence of axially conductive walls [17], and the variation of fluid thermal properties [10]. To the author’s knowledge, effects of surface roughness and of compressibility have never been considered quantitatively. Commonly, the channel geometry investigated is either parallel-plate (PP) or circular (CR) cross-section, the wall thermal boundary condition is either uniform heat flux (UHF) or, less frequently, uniform wall temperature (UWT), the fluid is Newtonian, the flow laminar, incompressible, and either hydrodynamically and thermally fully-developed or thermally developing (Graetz problem).

Governing equations in the developing case consist in a Sturm–Liouville problem [2], [9] whose eigenvalues and eigenfunctions compose the terms of proper exponential series. The series sum fade to zero with axial distance, and summed to asymptotic values provides temperature profile and Nusselt number variation along the channel. The asymptotic behaviour alone, in presence of viscous heating, is investigated into more detail in [3] where asymptotically vanishing and non-vanishing heat fluxes are investigated for UWT boundary condition. In case of UWT, in fact, an isenthalpic condition is found in the fully-developed region where heat generated by viscous dissipation is balanced by wall heat flux which is therefore negative and does not tend to zero with the distance. In another interesting work [4] viscous heating in vertical channels in presence of gravity is addressed. The analytical solution is found by perturbation method. General criteria to draw the limit of significance for viscous dissipation effects in micro-channels are then proposed in [8].

In more recent works, rarefaction and viscous heating effects have been studied together giving birth to elaborated temperature profile and Nusselt number formulas function of both Brinkman and Knudsen numbers [15], [16], [20]. All these works address the slip flow regime and make use of first-order slip wall and temperature jump boundary condition. Second-order slip wall boundary condition, including thermal creep, is used in [18] instead. According to the widely accepted classification given in [22], different regimes of rarefaction can be identified, and slip flow regime falls in the range ${10}^{-3}<\text{Kn}<{10}^{-1}$. There, the law of continuum still holds given that slip flow and temperature jump boundary condition at the wall is adopted. Other recent analytical works also address axial conduction, together with rarefaction and viscous effects [11], [13], [19]. Numerical works are rarely found in this field of investigation: in [21] viscous heating effects on temperature profile and Nusselt number in the thermal entrance region of a rectangular channel with rounded corners are investigated numerically. In [12], [14] compressible flow numerical simulations including viscous heating are used to discuss the gas static temperature reduction due to expansion; results are given in terms of local or integral quantities such as wall temperature, static and total bulk temperature, heat flux.

In the present work, the focus is on heat transfer characteristics of laminar gas flow in micro-channels. Due to the high velocities that can be achieved under these circumstances compressibility cannot be neglected. At the same time, the problem scale is still large enough so that relatively high Reynolds numbers can be achieved. Axial conduction can thus be neglected and, similarly, rarefaction effects are unlikely to be meaningful.

At first, conservation equations are solved with the inclusion of viscous heating term for incompressible case, and a broad analysis on the impact of Brinkman number over Nusselt numbers and temperature profiles in the channel is made. Compared to the incompressible case, compressibility introduces extra terms in the governing equations so that a closed form analytical solution is no longer viable. A series of compressible CFD simulations with temperature-dependent fluid thermal properties is thus solved, spanning the whole laminar and subsonic region, and data on temperature profiles and Nusselt numbers collected. From this, compressible Nusselt number correlations, function of Mach and Brinkman numbers, are proposed by expanding the incompressible analytical solution previously found. Even though viscous heating effects across the channel might be globally negligible in terms of total temperature rise, Brinkman number is found to have an important role on local heat transfer characteristics both for compressible and incompressible flow.

In the current analysis, both CR and PP cross-section channel are addressed, both fluid heating and cooling, and both UHF and UWT boundary condition. UWT local Nusselt numbers and temperature profiles in a developed flow were never given in the literature unless for the asymptotic results discussed in [3] or derived from Graetz problem solution. To the author’s knowledge also compressibility effects, while investigated in some works for the evaluation of friction in adiabatic Fanno flows [23], [24], were never addressed in conjunction with viscous heating for the assessment of heat transfer characteristics in micro-channels.