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.

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.

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 Similarity Solution of Heat and Mass Transfer for Natural Convection over a Moving Vertical Plate with Internal Heat Generation and a Convective Boundary Condition in the Presence of Thermal Radiation, Viscous Dissipation, and Chemical Reaction

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
S. Mohammed Ibrahim ◽  
N. Bhashar Reddy
Keyword(s):  
Boundary Condition ◽  
Natural Convection ◽  
Differential Equations ◽  
Thermal Radiation ◽  
Viscous Dissipation ◽  
Chemical Reaction ◽  
Heat Generation ◽  
Vertical Plate ◽  
Internal Heat ◽  
Internal Heat Generation

Steady laminar natural convection flow over a semi-infinite moving vertical plate with internal heat generation and convective surface boundary condition in the presence of thermal radiation, viscous dissipation, and chemical reaction is examined in this paper. In the analysis, we assumed that the left surface of the plate is in contact with a hot fluid while the cold fluid on the right surface of the plate contains a heat source that decays exponentially with the classical similarity variable. We utilized similarity variable to transform the governing nonlinear partial differential equations into a system of ordinary differential equations, which are solved numerically by applying shooting iteration technique along fourth-order Runge-Kutta method. The effects of the local Biot number, Prandtl number, buoyancy forces, the internal heat generation, the thermal radiation, Eckert number, viscous dissipation, and chemical reaction on the velocity, temperature, and concentration profiles are illustrated and interpreted in physical terms. A comparison with previously published results on the similar special cases showed an excellent agreement. Finally, numerical values of physical quantities, such as the local skin-friction coefficient, the local Nusselt number, and the local Sherwood number, are presented in tabular form.

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