Isoflux Nusselt Number and Slip Length Formulae for Superhydrophobic Microchannels

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
Ryan Enright ◽  
Marc Hodes ◽  
Todd Salamon ◽  
Yuri Muzychka
Keyword(s):  
Nusselt Number ◽  
Thermal Transport ◽  
Stokes Equations ◽  
Slip Length ◽  
Thermal Slip ◽  
Structured Surfaces ◽  
Transport Behavior ◽  
Hydrodynamic Slip ◽  
Flow Through ◽  
Plate Channel

We analytically and numerically consider the hydrodynamic and thermal transport behavior of fully developed laminar flow through a superhydrophobic (SH) parallel-plate channel. Hydrodynamic slip length, thermal slip length and heat flux are prescribed at each surface. We first develop a general expression for the Nusselt number valid for asymmetric velocity profiles. Next, we demonstrate that, in the limit of Stokes flow near the surface and an adiabatic and shear-free liquid–gas interface, both thermal and hydrodynamic slip lengths can be found by redefining existing solutions for conduction spreading resistances. Expressions for the thermal slip length for pillar and ridge surface topographies are determined. Comparison of fundamental half-space solutions for the Laplace and Stokes equations facilitate the development of expressions for hydrodynamic slip length over pillar-structured surfaces based on existing solutions for the conduction spreading resistance from an isothermal source. Numerical validation is performed and an analysis of the idealized thermal transport behavior suggests conditions under which superhydrophobic microchannels may enhance heat transfer.

Fully-Developed Thermal Transport in Microchannels With Streamwise Grooved Superhydrophobic Walls at Constant Heat Flux

2012 ◽  
Author(s):  
D. Maynes ◽  
J. Vanderhoff ◽  
G. Rosengarten
Keyword(s):  
Heat Flux ◽  
Nusselt Number ◽  
Thermal Transport ◽  
Average Nusselt Number ◽  
Temperature Jump ◽  
Slip Length ◽  
Relative Size ◽  
Constant Heat Flux ◽  
Constant Heat ◽  
Hydrodynamic Slip

This paper presents an analytical investigation of constant property, steady, fully-developed, laminar thermal transport in a parallel-plate channel comprised of metal superhydrophobic walls. The superhydrophobic walls considered here exhibit micro-ribs and cavities aligned in the streamwise direction. The cavities are assumed to be non-wetting and contain air, such that the Cassie-Baxter state is the interfacial state considered. The scenario considered is that of constant heat flux through the rib surfaces with negligible thermal transport through the air cavity interface. Closed form solutions for the local Nusselt number and local wall temperature are presented and are in the form of infinite series expansions. The analysis show the relative size of the cavity regions compared to the total rib and cavity width (cavity fraction) exercises significant influence on the aggregate thermal transport behavior. Further, the relative size of the rib and cavity module width compared to the channel hydraulic diameter (relative module width) also influences the Nusselt number. The spatially varying Nusselt number and wall temperature are presented as a function of the cavity fraction and the relative module width over the ranges 0–0.99 and 0.01–1.0, respectively. From these results the rib/cavity module averaged Nusselt number was determined as a function of the governing parameters. The results reveal that increases in either the cavity fraction or relative module width lead to decreases in the average Nusselt number and results are presented over a wide range of conditions from which the average Nusselt number can be determined for heat transfer analysis. Further, analogous to the hydrodynamic slip length, a temperature jump length describing the apparent temperature jump at the wall is determined in terms of the cavity fraction. Remarkably, it is nearly identical to the hydrodynamic slip length for the scenario considered here and allows straightforward determination of the average Nusselt number for any cavity fraction and relative rib/cavity module width.

Apparent Temperature Jump and Thermal Transport in Channels With Streamwise Rib and Cavity Featured Superhydrophobic Walls at Constant Heat Flux

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
D. Maynes ◽  
J. Crockett
Keyword(s):  
Nusselt Number ◽  
Thermal Transport ◽  
Average Nusselt Number ◽  
Temperature Jump ◽  
Slip Length ◽  
Relative Size ◽  
Constant Heat Flux ◽  
Apparent Temperature ◽  
Constant Heat ◽  
Hydrodynamic Slip

This paper presents an analytical investigation of constant property, steady, fully developed, laminar thermal transport in a parallel-plate channel comprised of metal superhydrophobic (SH) walls. The superhydrophobic walls considered here exhibit microribs and cavities aligned in the streamwise direction. The cavities are assumed to be nonwetting and contain air, such that the Cassie–Baxter state is the interfacial state considered. The scenario considered is that of constant heat flux through the rib surfaces with negligible thermal transport through the air cavity interface. Closed form solutions for the local Nusselt number and local wall temperature are presented and are in the form of infinite series expansions. The analysis show the relative size of the cavity regions compared to the total rib and cavity width (cavity fraction) exercises significant influence on the aggregate thermal transport behavior. Further, the relative size of the rib and cavity module width compared to the channel hydraulic diameter (relative module width) also influences the Nusselt number. The spatially varying Nusselt number and wall temperature are presented as a function of the cavity fraction and the relative module width over the ranges 0–0.99 and 0.01–1.0, respectively. From these results, the rib/cavity module averaged Nusselt number was determined as a function of the governing parameters. The results reveal that increases in either the cavity fraction or relative module width lead to decreases in the average Nusselt number and results are presented over a wide range of conditions from which the average Nusselt number can be determined for heat transfer analysis. Further, analogous to the hydrodynamic slip length, a temperature jump length describing the apparent temperature jump at the wall is determined in terms of the cavity fraction. Remarkably, it is nearly identical to the hydrodynamic slip length for the scenario considered here and allows straightforward determination of the average Nusselt number for any cavity fraction and relative rib/cavity module width.

scholarly journals Nusselt numbers for Poiseuille flow over isoflux parallel ridges accounting for meniscus curvature

2016 ◽  
Vol 811 ◽  
pp. 315-349 ◽  
Author(s):  
Toby L. Kirk ◽  
Marc Hodes ◽  
Demetrios T. Papageorgiou
Keyword(s):  
Nusselt Number ◽  
Poiseuille Flow ◽  
Numerical Solutions ◽  
Surface Deformation ◽  
Slip Length ◽  
Constant Heat Flux ◽  
Flux Boundary Condition ◽  
Nusselt Numbers ◽  
Dual Series Equations ◽  
Hydrodynamic Slip

We 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.

Effect of Meniscus Curvature on Apparent Thermal Slip

2016 ◽  
Vol 138 (12) ◽  
Author(s):  
Lisa Steigerwalt Lam ◽  
Marc Hodes ◽  
Georgios Karamanis ◽  
Toby Kirk ◽  
Scott MacLachlan
Keyword(s):  
Slip Length ◽  
Constant Heat Flux ◽  
Perturbation Approach ◽  
Thermal Slip ◽  
Flux Boundary Condition ◽  
Structured Surfaces ◽  
Cassie State ◽  
Heat Flux Boundary Condition ◽  
Approximate Expressions ◽  
Good Agreement

We analytically consider the effect of meniscus curvature on heat transfer to laminar flow across structured surfaces. The surfaces considered are composed of ridges. Curvature of the menisci, which separates liquid in the Cassie state and gas trapped in cavities between the ridges, results from the pressure difference between the liquid and the gas. A boundary perturbation approach is used to develop expressions that account for the change in the temperature field in the limit of small curvature of a meniscus. The meniscus is considered adiabatic and a constant heat flux boundary condition is prescribed at the tips of the ridges in a semi-infinite and periodic domain. A solution for a constant temperature ridge is also presented using existing results from a mathematically equivalent hydrodynamic problem. We provide approximate expressions for the apparent thermal slip length as function of solid fraction over a range of small meniscus protrusion angles. Numerical results show good agreement with the perturbation results for protrusion angles up to ± 20 deg.

Effective Temperature Jump Length and Influence of Axial Conduction for Thermal Transport Through Channels With Superhydrophobic Walls

2013 ◽  
Author(s):  
A. Cowley ◽  
D. Maynes ◽  
J. Crockett ◽  
B. W. Webb
Keyword(s):  
Thermal Transport ◽  
Peclet Number ◽  
Temperature Jump ◽  
Slip Length ◽  
Relative Size ◽  
Axial Conduction ◽  
Péclet Number ◽  
Jump Length ◽  
Wide Range ◽  
Hydrodynamic Slip

This paper presents a numerical investigation of thermal transport in a parallel-plate channel comprised of superhydrophobic walls. The scenario analyzed in this paper is laminar, fully developed, steady flow with constant properties. The superhydrophobic walls considered here have alternating micro-ribs and cavities aligned perpendicular to the flow direction. The cavities are assumed to be non-wetting and contain air. The thermal transport through the ribs is considered to have a constant heat flux while the thermal transport through the air/fluid interface over the cavity is considered to be negligible. Numerical results have been obtained over a range a Peclet numbers, cavity fractions, and relative rib/cavity widths. Results were also obtained where axial conduction was neglected and these results are compared to previous analytical work with excellent agreement. When the influence of axial conduction is not neglected, however, the results for local wall temperatures and Nusselt numbers show departure from the previous analytical results. The departure is more pronounced at low Peclet numbers and at large relative channel diameters. This paper provides a comparison over a wide range of parameters that characterize the overall influence of axial conduction. In general, the results show that the relative size of the cavity compared to the total rib/cavity module width (cavity fraction) and the flow Peclet number have a significant impact on the total thermal transport properties. Also, the rib/cavity module width compared to the hydraulic diameter affects the overall thermal transport behavior. Lastly, this paper explores the concept of a temperature jump length which is analogous to the hydrodynamic slip length. The ratio of temperature jump length to hydrodynamic slip length is presented in terms of cavity fraction, Peclet number, and relative size of the rib cavity module.

Effect of Evaporation and Condensation at Menisci on Apparent Thermal Slip

2015 ◽  
Vol 137 (7) ◽  
Author(s):  
Marc Hodes ◽  
Lisa Steigerwalt Lam ◽  
Adam Cowley ◽  
Ryan Enright ◽  
Scott MacLachlan
Keyword(s):  
Heat Transfer ◽  
Heat Transfer Coefficient ◽  
Transfer Coefficient ◽  
Slip Length ◽  
Interfacial Heat Transfer Coefficient ◽  
Conformal Map ◽  
Thermal Slip ◽  
Structured Surfaces ◽  
Evaporation And Condensation ◽  
Solid Liquid

We semi-analytically capture the effects of evaporation and condensation at menisci on apparent thermal slip lengths for liquids suspended in the Cassie state on ridge-type structured surfaces using a conformal map and convolution. An isoflux boundary condition is prescribed at solid–liquid interfaces and a constant heat transfer coefficient or isothermal one at menisci. We assume that the gaps between ridges, where the vapor phase resides, are closed systems; therefore, the net rates of heat and mass transfer across menisci are zero. The reduction in apparent thermal slip length due to evaporation and condensation relative to the limiting case of an adiabatic meniscus as a function of solid fraction and interfacial heat transfer coefficient is quantified in a single plot. The semi-analytical solution method is verified by numerical simulation. Results suggest that interfacial evaporation and condensation need to be considered in the design of microchannels lined with structured surfaces for direct liquid cooling of electronics applications and a quantitative means to do so is elucidated. The result is a decrease in thermal resistance relative to the predictions of existing analyses which neglect them.

Effect of core hydrophobicity on the electrophoresis of pH-regulated soft particles

Soft Matter ◽  
2021 ◽  
Vol 17 (11) ◽  
pp. 3074-3084
Author(s):  
Bharti ◽  
Partha P. Gopmandal ◽  
R. K. Sinha ◽  
H. Ohshima
Keyword(s):  
Inner Core ◽  
Theoretical Study ◽  
Slip Length ◽  
Hydrodynamic Slip ◽  
Soft Particles

We propose a theoretical study on the electrophoresis of pH-regualted soft particles considering the effect of hydrodynamic slip length of the hydrophobic inner core.

scholarly journals Transient Pressure-Driven Electroosmotic Flow through Elliptic Cross-Sectional Microchannels with Various Eccentricities

Computation ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 27
Author(s):  
Nattakarn Numpanviwat ◽  
Pearanat Chuchard
Keyword(s):  
Laplace Transform ◽  
Electroosmotic Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Mathieu Functions ◽  
Navier Stokes Equations ◽  
Flow Rates ◽  
Elliptic Cross ◽  
Flow Through ◽  
Relative Errors

The semi-analytical solution for transient electroosmotic flow through elliptic cylindrical microchannels is derived from the Navier-Stokes equations using the Laplace transform. The electroosmotic force expressed by the linearized Poisson-Boltzmann equation is considered the external force in the Navier-Stokes equations. The velocity field solution is obtained in the form of the Mathieu and modified Mathieu functions and it is capable of describing the flow behavior in the system when the boundary condition is either constant or varied. The fluid velocity is calculated numerically using the inverse Laplace transform in order to describe the transient behavior. Moreover, the flow rates and the relative errors on the flow rates are presented to investigate the effect of eccentricity of the elliptic cross-section. The investigation shows that, when the area of the channel cross-sections is fixed, the relative errors are less than 1% if the eccentricity is not greater than 0.5. As a result, an elliptic channel with the eccentricity not greater than 0.5 can be assumed to be circular when the solution is written in the form of trigonometric functions in order to avoid the difficulty in computing the Mathieu and modified Mathieu functions.

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