1,720,982 research outputs found
Explicit formulas for forced convection in a shrouded longitudinal-fin heat sink with clearance
No full textWe consider laminar forced convection in a shrouded longitudinal-fin heat sink (LFHS) with tip clearance, as described by the pioneering study of (Sparrow, Baliga & Patankar 1978 J. Heat Trans. 100). The base of the LFHS is isothermal but the fins, while thin, are not isothermal, i.e. the conjugate heat transfer problem is of interest. Whereas Sparrow et al. numerically solved the fully developed flow and thermal problems for a range of geometries and fin conductivities, we consider the physically realistic asymptotic limit where the fins are closely spaced, i.e. the spacing is small relative to their height and the clearance above them. The flow problem in this limit was considered by (Miyoshi et al. 2024, J. Fluid Mech. 991, A2), and we consider the corresponding thermal problem. Usingmatched asymptotic expansions, we find explicit solutions for the temperature field (in both the fluid and fins) and conjugate Nusselt numbers (local and average). The structure of the asymptotic solutions provides further insight into the results of Sparrow et al.: the flow is highest in the gap above the fins, hence heat transfer predominantly occurs close to the fin tips. The new formulas are compared with numerical solutions and are found to be accurate for practical LFHSs. Significantly, existing analytical results for ducts are for boundaries that are either wholly isothermal, wholly isoflux or with one of these conditions on each wall. Consequently, this study provides the first analytical results for conjugate Nusselt numbers for flow through ducts
On spreading resistance for an isothermal source on a compound flux channel
No full textRecently, Jain [ASME J. Heat Mass Transfer, 220 (2024)] provided spreading-resistance formulas for an isothermal source on compound, orthotropic, semi-infinite, two-dimensional (axisymmetric) flux channels (tubes). The boundary condition (BC) in the source plane was a discontinuous convection (Robin) one. Along the source, a sufficiently large heat transfer coefficient was imposed to approximate an isothermal condition; elsewhere, it was set to zero, imposing an adiabatic BC. An eigenfunction expansion resolved the problem. Distinctly, we impose, precisely, a mixed isothermal-adiabatic BC in the source plane and use conformal maps to resolve the spreading resistance for the limiting case of a compound, isotropic flux channel. Our complimentary approach requires more time to compute the spreading resistance. However, it converges uniformly rather than pointwise, converges to the exact spreading resistance rather than one with an error, eliminates the Gibbs phenomenon at the edges of the source and fully resolves the square-root singularities in heat flux as the discontinuity in the BC is approached
One-dimensional analysis of gas diffusion-induced Cassie to Wenzel state transition
No full textWe develop a one-dimensional model for transient diffusion of gas between ridges into a quiescent liquid suspended in the Cassie state above them. In the first case study, we assume that the liquid and gas are initially at the same pressure and that the liquid column is sealed at the top. In the second one, we assume that the gas initially undergoes isothermal compression and that the liquid column is exposed to gas at the top. Our model provides a framework to compute the transient gas concentration field in the liquid, the time when the triple contact line begins to move down the ridges, and the time when menisci reach the bottom of the substrate compromising the Cassie state. At illustrative conditions, we show the effects of geometry, hydrostatic pressure, and initial gas concentration on the Cassie to Wenzel state transition.</p
Nusselt numbers for Poiseuille flow over isoflux parallel ridges accounting for meniscus curvature
No full textWe investigate forced convection in a parallel-plate-geometry microchannel with superhydrophobic walls consisting of a periodic array of ridges aligned parallel to the direction of a Poiseuille flow. In the dewetted (Cassie) state, the liquid contacts the channel walls only at the tips of the ridges, where we apply a constant-heat-flux boundary condition. The subsequent hydrodynamic and thermal problems within the liquid are then analysed accounting for curvature of the liquid-gas interface (meniscus) using boundary perturbation, assuming a small deflection from flat. The effects of this surface deformation on both the effective hydrodynamic slip length and the Nusselt number are computed analytically in the form of eigenfunction expansions, reducing the problem to a set of dual series equations for the expansion coefficients which must, in general, be solved numerically. The Nusselt number quantifies the convective heat transfer, the results for which are completely captured in a single figure, presented as a function of channel geometry at each order in the perturbation. Asymptotic solutions for channel heights large compared with the ridge period are compared with numerical solutions of the dual series equations. The asymptotic slip length expressions are shown to consist of only two terms, with all other terms exponentially small. As a result, these expressions are accurate even for heights as low as half the ridge period, and hence are useful for engineering applications.</p
Mechanical power from thermocapillarity on superhydrophobic surfaces
No full textCrowdy et al. (2023 Phys. Rev. Fluids, vol. 8, 094201), recently showed that liquid suspended in the Cassie state over an asymmetrically spaced periodic array of alternating cold and hot ridges such that the menisci spanning the ridges are of unequal length will be pumped in the direction of the thermocapillary stress along the longer menisci. Their solution, applicable in the Stokes flow limit for a vanishingly small thermal Péclet number, provides the steady-state temperature and velocity fields in a semi-infinite domain above the superhydrophobic surface, including the uniform far-field velocity, i.e. pumping speed, the key engineering parameter. Here, a related problem in a finite domain is considered where, opposing the superhydrophobic surface, a flow of liquid through a microchannel is bounded by a horizontally mobile smooth wall of finite mass subjected to an external load. A key assumption underlying the analysis is that, on a unit area basis, the mass of the liquid is small compared with that of the wall. Thus, as shown, rather than the heat equation and the transient Stokes equations governing the temperature and flow fields, respectively, they are quasi-steady and, as a result, governed by the Laplace and Stokes equations, respectively. Under the further assumption that the ridge period is small compared with the height of the microchannel, these equations are resolved using matched asymptotic expansions which yield solutions with exponentially small asymptotic errors. Consequently, the transient problem of determining the velocity of the smooth wall is reduced to an ordinary differential equation. This approach is used to provide a theoretical demonstration of the conversion of thermal energy to mechanical work via the thermocapillary stresses along the menisci.</p
Solution of the Graetz-Nusselt problem for liquid flow over isothermal parallel ridges
No full textWe consider convective heat transfer for laminar flow of liquid between parallel plates that are textured with isothermal ridges oriented parallel to the flow. Three different flow configurations are analyzed: one plate textured and the other one smooth; both plates textured and the ridges aligned; and both plates textured, but the ridges staggered by half a pitch. The liquid is assumed to be in the Cassie state on the textured surface(s), to which a mixed boundary condition of no-slip on the ridges and no-shear along flat menisci applies. Heat is exchanged with the liquid either through the ridges of one plate with the other plate adiabatic, or through the ridges of both plates. The thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surface(s). Axial conduction is neglected and the inlet temperature profile is arbitrary. We solve for the three-dimensional developing temperature profile assuming a hydrodynamically developed flow, i.e., we consider the Graetz-Nusselt problem. Using the method of separation of variables, the thermal problem is essentially reduced to a two-dimensional eigenvalue problem in the transverse coordinates, which is solved numerically. Expressions for the local Nusselt number and those averaged over the period of the ridges in the developing and fully developed regions are provided. Nusselt numbers averaged over the period and length of the domain are also provided. Our approach enables the aforementioned quantities to be computed in a small fraction of the time required by a general computational fluid dynamics (CFD) solver.</p
Asymptotic Nusselt numbers for internal flow in the Cassie state
No full textWe consider laminar, fully developed, Poiseuille flows of liquid in the Cassie state through diabatic, parallel-plate microchannels symmetrically textured with isoflux ridges. Via matched asymptotic expansions, we develop expressions for (apparent hydrodynamic) slip lengths and Nusselt numbers. Our small parameter is the pitch of the ridges divided by the height of the microchannel. When the ridges are oriented parallel to the flow, we quantify the error in the Nusselt number expressions in the literature and provide a new closed-form result. It is accurate to and valid for any solid (ridge) fraction, whereas previous ones are accurate to and breakdown in the important limit when the solid fraction approaches zero. When the ridges are oriented transverse to the (periodically fully developed) flow, the error associated with neglecting inertial effects in the slip length is shown to be, where is the channel-scale Reynolds number based on its hydraulic diameter. The corresponding Nusselt number expressions' accuracies are shown to depend on the Reynolds number, Péclet number and Prandtl number in addition to. Manipulating the solution to the inner temperature problem encountered in the vicinity of the ridges shows that classic results for the thermal spreading resistance are better expressed in terms of polylogarithm functions.</p
Nusselt numbers for Poiseuille flow over isoflux parallel ridges for arbitrary meniscus curvature
No full textWe numerically compute Nusselt numbers for laminar, hydrodynamically, and thermally fully developed Poiseuille flow of liquid in the Cassie state through a parallel plate-geometry microchannel symmetrically textured by a periodic array of isoflux ridges oriented parallel to the flow. Our computations are performed using an efficient, multiple domain, Chebyshev collocation (spectral) method. The Nusselt numbers are a function of the solid fraction of the ridges, channel height to ridge pitch ratio, and protrusion angle of menisci. Significantly, our results span the entire range of these geometrical parameters. We quantify the accuracy of two asymptotic results for Nusselt numbers corresponding to small meniscus curvature, by direct comparison against the present results. The first comparison is with the exact solution of the dual series equations resulting from a small boundary perturbation (Kirk et al., 2017, "Nusselt Numbers for Poiseuille Flow Over Isoflux Parallel Ridges Accounting for Meniscus Curvature," J. Fluid Mech., 811, pp. 315-349). The second comparison is with the asymptotic limit of this solution for large channel height to ridge pitch ratio.</p
Solution of the extended Graetz-Nusselt problem for liquid flow over isothermal parallel ridges
No full textWe consider convective heat transfer for laminar flow of liquid between parallel plates. The configurations analyzed are both plates textured with symmetrically aligned isothermal ridges oriented parallel to the flow, and one plate textured as such and the other one smooth and adiabatic. The liquid is assumed to be in the Cassie state on the textured surface(s) to which a mixed boundary condition of no-slip on the ridges and no-shear along flat menisci applies. The thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surface(s). We solve for the developing three-dimensional temperature profile resulting from a step change of the ridge temperature in the streamwise direction assuming a hydrodynamically developed flow. Axial conduction is accounted for, i.e., we consider the extended Graetz-Nusselt problem; therefore, the domain is of infinite length. The effects of viscous dissipation and (uniform) volumetric heat generation are also captured. Using the method of separation of variables, the homogeneous part of the thermal problem is reduced to a nonlinear eigenvalue problem in the transverse coordinates which is solved numerically. Expressions derived for the local and the fully developed Nusselt number along the ridge and that averaged over the composite interface in terms of the eigenvalues, eigenfunctions, Brinkman number, and dimensionless volumetric heat generation rate. Estimates are provided for the streamwise location where viscous dissipation effects become important.</p
Fully developed flow through shrouded-fin arrays: exact and asymptotic solutions
No full textThe flow resistance, i.e. friction factor times Reynolds number , of longitudinal-fin heat sinks with or without clearance between a shroud and the tips of the fins is an important parameter in thermal design. This is because it dictates the caloric resistance of the heat sink, i.e. change in bulk temperature of the fluid flowing through it. When there is no clearance and the common and oft-valid assumption of negligible fin thickness is invoked, corresponds to simply that of a rectangular duct. However, with clearance, only numerical results are available as per the well-known study by Sparrow, Baliga and Patankar (ASME J. Heat Transfer, vol. 100, 1978). We develop analytical formulae for for fully developed flow with clearance. The exact solution is provided by an integral formula derived via conformal mappings. Additionally, simple formulae are derived via asymptotic expansions in three cases: (1) the fin spacing is small compared to the fin height and clearance; (2) the clearance is small compared to the fin spacing, which is small compared to the fin height; (3) the same as case (2) but valid for larger clearances. The different asymptotic formulae are compared to the exact formula, and together cover almost the entire relevant parameter range (for fin spacing and clearance) with errors of less than 15 %. A formula for the limiting case of no clearance is shown to be more accurate, for any fin spacing, than a widely used correlation from the literature.</p
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