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.

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.

Confined, Milliscale Unsteady Laminar Impinging Slot Jets: Effects of Slot Width on Surface Stagnation Point Nusselt Numbers

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
Dae Hee Lee ◽  
Jong Ryeol Bae ◽  
Mira Ryu ◽  
Phil Ligrani
Keyword(s):  
Reynolds Number ◽  
Nusselt Number ◽  
Stagnation Point ◽  
Constant Heat Flux ◽  
Slot Width ◽  
Distance Ratio ◽  
Stagnation Region ◽  
Spatially Resolved ◽  
Nusselt Numbers ◽  
Plate Distance

The effects of slot width for confined, laminar impinging slot jets of millimeter-scale are considered, including experimental measurements of spatially resolved distributions of local Nusselt numbers measured on a constant heat flux surface. The effects of Reynolds number, nozzle-to-plate distance, and dimensional slot width on the local Nusselt number are investigated for slot nozzle width B values of 0.5 mm, 1.0 mm, and 1.5 mm. Reynolds numbers Re range from 120 to 200, nozzle-to-plate distances H/B vary from 0.75 to 12.5, and the nozzle aspect ratio y/B is 50. Observed are different stagnation point Nusselt number Nuo variations with Re, H/B, and B, where the onset of unsteadiness, and the intermittent flapping motion of the jet column are both associated with important variations to local, stagnation region Nusselt numbers Nuo, as experimental configuration and condition change. The variations of these stagnation-point Nusselt numbers associated with these two modes of unsteadiness are characterized by correlations which provide the dependence upon Reynolds number and normalized nozzle-to-plate distance ratio, H/B, for different dimensional values of B. Also presented are stagnation region Nusselt number variations, for steady, impingement jets at values of H/B less than 4.6–7.8. These are characterized by three separate regimes of behavior, each of which shows significantly different Nuo dependence upon Re, H/B, and B.

Laminar Slug Flow: Heat Transfer Characteristics With Constant Heat Flux Boundary

2009 ◽  
Author(s):  
P. A. Walsh ◽  
E. J. Walsh ◽  
Y. S. Muzychka
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Nusselt Number ◽  
Heat Transfer Performance ◽  
Constant Heat Flux ◽  
Transfer Performance ◽  
Segmented Flow ◽  
Constant Heat ◽  
Flux Boundary Condition ◽  
Flow Heat

The problem of elevated heat flux in modern electronics has led to the development of numerous liquid cooling devices which yield superior heat transfer coefficients over their air based counterparts. This study investigates the use of liquid/gas slug flows where a liquid coolant is segregated into discrete slugs, resulting in a segmented flow, and heat transfer rates are enhanced by an internal circulation within slugs. This circulation directs cooler fluid from the center of the slug towards the heated surface and elevates the temperature difference at the wall. An experimental facility is built to examine this problem in circular tube flow with a constant wall heat flux boundary condition. This was attained by Joule heating a thin walled stainless steel tube. Water was used as the coolant and air as the segregating phase. The flow rates of each were controlled using high precision syringe pumps and a slug producing mechanism was introduced for segmenting the flow into slugs of various lengths at any particular flow rate. Tube flows with Reynolds numbers in the range 10 to 1500 were examined ensuring a well ordered segmented flow throughout. Heat transfer performance was calculated by measuring the exterior temperature of the thin tube wall at various locations using an Infrared camera. Nusselt number results are presented for inverse Graetz numbers over four decades, which spans both the thermally developing and developed regions. The results show that Nu in the early thermally developing region are slightly inferior to single phase flows for heat transfer performance but become far superior at higher values of inverse Gr. Additionally, the slug length plays an important role in maximizing Nusselt number in the fully developed region as Nu plateaus at different levels for slugs of differing lengths. Overall, this paper provides a new body of experimental findings relating to segmented flow heat transfer in constant heat flux tubes without boiling. Put abstract text here.

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.

Study of an Impingement Cooling Jet Array for Turbine Blade Cooling With Single and Double Exit Cases

2013 ◽  
Author(s):  
Michael Keenan ◽  
Ryo S. Amano ◽  
Shichuan Ou
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Reynolds Numbers ◽  
Constant Heat Flux ◽  
Flux Boundary Condition ◽  
Turbine Blade Cooling ◽  
Impingement Jet ◽  
Flow Phenomena ◽  
Cfd Analyses ◽  
Jet Array

A study was conducted on convective heat transfer of a 55 impingement jet array (5×11) with a constant heat flux boundary condition. A spatial variation in a time-averaged Nusselt number, as well as a spanwise time-averaged Nusselt number, are presented for jet Reynolds numbers of 4,000, 8,000, 12,000, and 15,000 for jet to target standoff distances of z/D = 3, 4 and 5. For each of these configurations the exit flow was varied to include both a single exit and a double exit configuration. In all cases, the computed Nusselt number correlates well with the experimentally measured results. The local and spanwise averaged Nusselt number distributions are presented as a function of the jet Reynolds number. Several complex heat transfer and flow phenomena were clarified through extensive computational investigation by using CFD analyses.

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.

Natural Convection in a Rectangular Porous Cavity With Constant Heat Flux on One Vertical Wall

1984 ◽  
Vol 106 (1) ◽  
pp. 152-157 ◽  
Author(s):  
V. Prasad ◽  
F. A. Kulacki
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Numerical Solutions ◽  
Vertical Wall ◽  
Local Heat ◽  
Constant Heat Flux ◽  
Heated Wall ◽  
Width Ratio ◽  
Constant Heat ◽  
Nusselt Numbers

Numerical solutions for two-dimensional, steady, free convection are presented for a rectangular cavity with constant heat flux on one vertical wall, the other vertical wall being isothermally cooled. The horizontal walls are insulated. Results are presented in terms of streamlines and isotherms, local and average Nusselt numbers at the heated wall, and the local heat flux at the cooled wall. Flow patterns are observed to be quite different from those in the case of a cavity with both vertical walls at constant temperatures. Specifically, symmetry in the flow field is absent and any increase in applied heat flux is not accompanied by linearly proportional increase in the temperature on the heated wall. Also, for low Prandtl number, the heat transfer rate based upon the mean temperature difference is higher as compared to experimental results for the isothermal case. Heat transfer results, further, indicate that the average Nusselt number is correlated by a relation of the form Nu = constant Ra*mAn, where Ra* is the Rayleigh number and A the height-to-width ratio of the cavity.

scholarly journals Nusselt Numbers for Poiseuille Flow Over Isoflux Parallel Ridges for Arbitrary Meniscus Curvature

2018 ◽  
Vol 140 (8) ◽  
Author(s):  
Simon Game ◽  
Marc Hodes ◽  
Toby Kirk ◽  
Demetrios T. Papageorgiou
Keyword(s):  
Poiseuille Flow ◽  
Channel Height ◽  
Geometrical Parameters ◽  
Asymptotic Results ◽  
Nusselt Numbers ◽  
Large Channel ◽  
Dual Series Equations ◽  
Collocation Spectral Method ◽  
Pitch Ratio ◽  
Oriented Parallel

We 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.

Thermal Measurements in Rectangular Microchannels

2002 ◽  
Author(s):  
Aristotel Popescu ◽  
James R. Welty ◽  
David Pfund ◽  
David Rector
Keyword(s):  
Heat Transfer ◽  
Aspect Ratio ◽  
Single Channel ◽  
Constant Heat Flux ◽  
Working Fluid ◽  
Flux Boundary Condition ◽  
Scale Theory ◽  
Nusselt Numbers ◽  
Micro Channels ◽  
Macro Scale

This paper reports on an experimental study of heat transfer in high aspect ratio (width/depth), rectangular micro-channels. A single channel with width of 10 μm was cut into polycarbonate spacers of various thicknesses, resulting in channel depths of 128 μm, 263 μm and 521 μm. Heat transfer experiments were performed with a constant heat flux boundary condition applied at the surface of the channel. Experiments conducted for refrigerant R-124 working fluid in the range Re = 300 − 900 and Pr = 3.6 − 3.8 showed small or no departure from macro-scale predictions for channels with hydraulic diameters larger than 500 μm. Results for the 80:1 aspect ratio channel showed a significant departure from theoretical predictions. Experimental values of local Nusselt numbers were approximately 25 percent lower than predicted using macro-scale theory.

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