Nusselt Numbers for Fully-Developed Flow Between Parallel Plates With One Plate Textured With Isothermal Parallel Ridges

2016 ◽  
Author(s):  
Georgios Karamanis ◽  
Marc Hodes ◽  
Toby Kirk ◽  
Demetrios T. Papageorgiou
Keyword(s):  
Boundary Condition ◽  
Nusselt Number ◽  
Mixed Boundary ◽  
Three Dimensional ◽  
Liouville Problem ◽  
Textured Surface ◽  
First Eigenvalue ◽  
Parallel Plates ◽  
Fully Developed Flow ◽  
Streamwise Location

We develop a semi-analytical solution for the Nusselt number for fully-developed flow of liquid between parallel plates, one of which is textured with isothermal parallel ridges. The opposite plate is smooth and adiabatic. The liquid is assumed to be in the Cassie state on the textured surface, on which a mixed boundary condition of no slip on the ridges and no shear along menisci applies. An existing solution for the velocity field is valid. The thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surface. Given the nature of the isothermal boundary condition, the analysis concerns a three-dimensional developing temperature profile, and the results are obtained for a streamwise location that tends to infinity. We assume that the temperature field is governed by an infinite sum of the product of a function of the streamwise coordinate and a second function of the spanwise co-ordinates. The latter functions are eigenfunctions satisfying a two-dimensional Sturm-Liouville problem from which the eigenvalues follow. The fully-developed Nusselt number follows from the first eigenvalue.

scholarly journals Solution of the Graetz–Nusselt Problem for Liquid Flow Over Isothermal Parallel Ridges

2017 ◽  
Vol 139 (9) ◽  
Author(s):  
Georgios Karamanis ◽  
Marc Hodes ◽  
Toby Kirk ◽  
Demetrios T. Papageorgiou
Keyword(s):  
Boundary Condition ◽  
Temperature Profile ◽  
Mixed Boundary ◽  
Separation Of Variables ◽  
Three Dimensional ◽  
The Other ◽  
Inlet Temperature ◽  
Textured Surface ◽  
Parallel Plates ◽  
Time Required

We 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.

scholarly journals Solution of the Extended Graetz–Nusselt Problem for Liquid Flow Over Isothermal Parallel Ridges

2018 ◽  
Vol 140 (6) ◽  
Author(s):  
Georgios Karamanis ◽  
Marc Hodes ◽  
Toby Kirk ◽  
Demetrios T. Papageorgiou
Keyword(s):  
Boundary Condition ◽  
Viscous Dissipation ◽  
Heat Generation ◽  
Mixed Boundary ◽  
Nonlinear Eigenvalue Problem ◽  
Separation Of Variables ◽  
Infinite Length ◽  
Textured Surface ◽  
Parallel Plates ◽  
Volumetric Heat Generation

We 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.

Free Convection Between Series of Vertical Parallel Plates With Embedded Line Heat Sources

1991 ◽  
Vol 113 (1) ◽  
pp. 108-115 ◽  
Author(s):  
S. H. Kim ◽  
N. K. Anand ◽  
L. S. Fletcher
Keyword(s):  
Boundary Condition ◽  
Nusselt Number ◽  
Surface Temperature ◽  
Mass Flow Rate ◽  
Mass Flow ◽  
Flow Rate ◽  
Parallel Plates ◽  
Cold Surface ◽  
Maximum Increase ◽  
Hot Surface

Laminar free convective heat transfer in channels formed between series of vertical parallel plates with an embedded line heat source was studied numerically. These channels resemble cooling passages in electronic equipment. The effect of a repeated boundary condition and wall conduction on mass flow rate (M), maximum surface temperature (θh,max and θc,max), and average surface Nusselt number (Nuh and Nuc) is discussed. Calculations were made for Gr*=10 to 106, K=0.1, 1, 10, and 100, and t/B=0.1 and 0.3. The effect of a repeated boundary condition decreases the maximum hot surface temperature and increases the maximum cold surface temperature. The effect of a repeated boundary condition with wall conduction increases the mass flow rate. The maximum increase in mass flow rate due to wall conduction is found to be 155 percent. The maximum decrease in average hot surface Nusselt number due to wall conduction (t/B and K) occurs at Gr*=106 and is 18 percent. Channels subjected to a repeated boundary condition approach that of a symmetrically heated channel subjected to uniform wall temperature conditions at K≥100.

The Numerical Prediction of Turbulent Flow and Heat Transfer in The Entrance Region of a Parallel Plate Duct

1976 ◽  
Vol 98 (4) ◽  
pp. 594-600 ◽  
Author(s):  
A. F. Emery ◽  
F. B. Gessner
Keyword(s):  
Heat Transfer ◽  
Turbulent Flow ◽  
Flow Behavior ◽  
Three Dimensional ◽  
Axial Flow ◽  
Entrance Region ◽  
Parallel Plates ◽  
Fully Developed Flow ◽  
Flow And Heat Transfer ◽  
Corner Flows

Velocity and temperature profiles were computed for turbulent flow, both in the entrance region and the fully developed state, in a duct with heated parallel plates. By starting the calculations at the duct inlet and using a finite difference technique and a three-dimensional mixing length originally defined for corner flows, it was possible to predict axial flow behavior and the nonasymptotic approach to fully developed flow with and without associated heat transfer.

Experimental Study on the Effect of Low Acute Side Angles on Heat Transfer in Rhombic Shaped Microchannel

2018 ◽  
Vol 141 (2) ◽  
Author(s):  
Sunil K. Dwivedi ◽  
Sandip K. Saha
Keyword(s):  
Heat Transfer ◽  
Boundary Condition ◽  
Experimental Study ◽  
Heat Flux ◽  
Nusselt Number ◽  
Three Dimensional ◽  
Constant Heat Flux ◽  
Numerical Results ◽  
Transfer Model ◽  
Design And Optimization

This experimental study on rhombic shaped microchannels was conducted to understand the effect of a low acute side angle on the Nusselt number and compare the results with the published numerical results for H1 (axially constant heat flux and circumferentially constant temperature) and H2 (constant axial and circumferential wall heat flux) boundary conditions. The hydraulic and heat transfer characteristics of the rhombic geometry with a side angle of 30 deg for different mass flow rates and heat flux inputs are obtained using a three-dimensional (3D) conjugate heat transfer model, which is validated with the experimental results. It is found that the average Nusselt number obtained from the experimental and numerical results can be approximated closely with that computed using the H1 boundary condition. The local Nusselt number of hydrodynamically and thermally developed regions obtained from the numerical analysis is compared with a correlation for the H1 boundary condition. These results will be useful in design and optimization of a rhombic shaped microchannel for electronic cooling applications.

Evaluation of Nusselt Number for a Flow in a Parallel Plates Using MHD Second Order Slip Model

2021 ◽  
Author(s):  
Hatice Simsek
Keyword(s):  
Magnetic Field ◽  
Boundary Condition ◽  
Nusselt Number ◽  
Slip Flow ◽  
Mhd Flow ◽  
Second Order ◽  
Parallel Plates ◽  
First Order ◽  
Condition Model ◽  
Boundary Condition Model

Abstract In this study, two separate boundary condition models, as proposed by Beskok and Karniadakis [1] and Deissler [2], widely preferred for the second order boundary condition, were used. These two proposed boundary condition models were solved in the presence of a magnetic field moving normal to the plate surface in magneto-hydrodynamic (MHD) flow between micro-parallel plates with constant wall heat flux. The energy equation for the second-order temperature jump boundary condition, taking into account the momentum and viscous dissipation, as well as the corresponding Nusselt value were solved analytically in slip flow regime.The flow of an incompressible viscous flow between fixed micro-parallel plates with electrical conductivity is assumed to be constant, laminar, hydrodynamically and thermally developed. The closed form solutions for the temperature field and the fully developed Nusselt number are derived as a function of the Magnetic parameter (MHD), Knudsen number and Brinkman number and shown graphically and in a tabular form. The second order boundary condition model proposed by Deissler [2] predicts the Nusselt number to be at lower values when compared to the first order boundary condition model, and the second order boundary condition model proposed by Beskok and Karniadakis [1] predicts the Nusselt number to be at higher values than that of the first order boundary condition model. Moreover, increasing the magnetic field parameter M, led to higher Nusselt values in the slip flow model proposed by both Deissler [2] and Beskok and Karniadakis [1] compared to that when M = 0.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

Sign in / Sign up

Export Citation Format

Share Document