Fully Developed Transport Constants of Annular Tubes, With Application to the Entrance Region

2019 ◽  
Vol 12 (4) ◽  
Author(s):  
Ted D. Bennett
Keyword(s):  
Nusselt Number ◽  
Full Range ◽  
Analytic Solutions ◽  
Series Solutions ◽  
Nusselt Numbers ◽  
Exact Analytic Solutions ◽  
Constant Wall Heat Flux ◽  
Annular Tube ◽  
Tube Geometry ◽  
Generalized Correlation

Abstract Description of the laminar thermal entry problem in annular tubes has historically been limited to a few geometric cases that require piecing together classical Graetz series and Lévêque series solutions to span all values of the Graetz number. The current work uses a recently developed generalized correlation to describe the full range of Graetz numbers for any annular tube geometry. However, the correlation requires fully developed Nusselt number values that have only been accurately reported in tabular and graphical forms. Exact analytic solutions for the constant wall heat flux condition are developed in this work, and simplified correlations are proposed for all wall conditions that reproduce exact Nusselt number solutions to within ± 0.4%. Using these results, a modified version of the generalized Graetz problem correlation is developed to reproduce the most published Nusselt numbers for the thermal entry problem in an annular tube to be within ± 5%.

Exact Solutions to the Thermal Entry Problem for Laminar Flow Through Annular Ducts

2020 ◽  
Vol 142 (5) ◽  
Author(s):  
T. D. Bennett
Keyword(s):  
Separation Of Variables ◽  
Outer Wall ◽  
Series Solutions ◽  
Eigenvalues And Eigenfunctions ◽  
Nusselt Numbers ◽  
Wide Range ◽  
Flow Through ◽  
Laminar Forced Convection ◽  
Thermal Entrance ◽  
Annular Tube

Abstract The thermal entrance region for laminar-forced convection of a Newtonian fluid in an annular tube is solved by separation of variables using as many eigenvalues and eigenfunctions as needed to report exact results for a specified range of Graetz numbers. Results for the local and average Nusselt numbers are calculated for a wide range of inner to outer wall radius ratios and for convection to either the inner or outer wall, when the opposing wall is adiabatic. The present benchmark results are utilized to critically examine the accuracy of previous extended Lévêque series solutions that are convergent for short axial distances, and Graetz series solutions that are convergent for long axial distances, and to examine the performance of a new correlation for convection in annular tubes.

Experimental Investigation of Laminar Mixed Convection in Tubes With Longitudinal Internal Fins

1988 ◽  
Vol 110 (2) ◽  
pp. 366-372 ◽  
Author(s):  
I. M. Rustum ◽  
H. M. Soliman
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Friction Factor ◽  
Smooth Tube ◽  
Transfer Enhancement ◽  
Finned Tubes ◽  
Nusselt Numbers ◽  
Laminar Mixed Convection ◽  
Internal Fins ◽  
Tube Geometry

Experiments were performed to study the pressure drop and heat transfer characteristics for laminar flow in a smooth tube and four tubes with internal longitudinal fins, with emphasis on showing how the experimental results relate to previous analytical predictions. Measured quantities include the fully developed friction factor, local and fully developed Nusselt numbers. Good agreements were obtained between the friction factor results and previous analytical predictions, and between Nusselt number results for the smooth tube and previous experiments. Free convection is shown to have a strong influence on heat transfer in finned tubes and the results approach the forced-convection predictions as Rayleigh number decreases. Internal fins appear to retard the onset of significant free convective currents; however, once initiated, a faster rate of heat transfer enhancement occurs compared to the smooth tube. An empirical correlation of fully developed Nusselt number is presented taking into account the influences of tube geometry.

CRYSTALLINE GRAVITY

2009 ◽  
Vol 24 (18n19) ◽  
pp. 3243-3255 ◽  
Author(s):  
GERARD 't HOOFT
Keyword(s):  
Finite Number ◽  
Lorentz Group ◽  
Degrees Of Freedom ◽  
Holonomy Group ◽  
Discrete Subgroup ◽  
Analytic Solutions ◽  
Space Time ◽  
Exact Analytic Solutions ◽  
Finite Domains ◽  
Dimensional Crystal

Matter interacting classically with gravity in 3+1 dimensions usually gives rise to a continuum of degrees of freedom, so that, in any attempt to quantize the theory, ultraviolet divergences are nearly inevitable. Here, we investigate a theory that only displays a finite number of degrees of freedom in compact sections of space-time. In finite domains, one has only exact, analytic solutions. This is achieved by limiting ourselves to straight pieces of string, surrounded by locally flat sections of space-time. Next, we suggest replacing in the string holonomy group, the Lorentz group by a discrete subgroup, which turns space-time into a 4-dimensional crystal with defects.

scholarly journals Exact analytical solutions of continuously graded models of flat lenses based on transformation optics

2017 ◽  
Vol 30 (4) ◽  
pp. 639-646 ◽  
Author(s):  
Mariana Dalarsson ◽  
Raj Mittra
Keyword(s):  
Magnetic Fields ◽  
Longitudinal Direction ◽  
Middle Layer ◽  
Analytic Solutions ◽  
Transformation Optics ◽  
Confluent Hypergeometric Functions ◽  
Graded Index ◽  
Exact Analytic Solutions ◽  
Electric And Magnetic Fields ◽  
Physical Insight

We present a study of exact analytic solutions for electric and magnetic fields in continuously graded flat lenses designed utilizing transformation optics. The lenses typically consist of a number of layers of graded index dielectrics in both the radial and longitudinal directions, where the central layer in the longitudinal direction primarily contributes to a bulk of the phase transformation, while other layers act as matching layers and reduce the reflections at the interfaces of the middle layer. Such lenses can be modeled as compact composites with continuous permittivity (and if needed) permeability functions which asymptotically approach unity at the boundaries of the composite cylinder. We illustrate the proposed procedures by obtaining the exact analytic solutions for the electric and magnetic fields for one simple special class of composite designs with radially graded parameters. To this purpose we utilize the equivalence between the Helmholtz equation of our graded flat lens and the quantum-mechanical radial Schr?dinger equation with Coulomb potential, furnishing the results in the form of Kummer confluent hypergeometric functions. Our approach allows for a better physical insight into the operation of our transformation optics-based graded lenses and opens a path toward novel designs and approaches.

scholarly journals New Soliton Solutions for the Higher-Dimensional Non-Local Ito Equation

2021 ◽  
Vol 10 (1) ◽  
pp. 374-384
Author(s):  
Mustafa Inc ◽  
E. A. Az-Zo’bi ◽  
Adil Jhangeer ◽  
Hadi Rezazadeh ◽  
Muhammad Nasir Ali ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Solution Process ◽  
Soliton Solutions ◽  
Analytic Solutions ◽  
Bright Solitons ◽  
Exact Analytic Solutions ◽  
Higher Dimensional ◽  
Non Local ◽  
Free Parameters ◽  
Ito Equation

Abstract In this article, (2+1)-dimensional Ito equation that models waves motion on shallow water surfaces is analyzed for exact analytic solutions. Two reliable techniques involving the simplest equation and modified simplest equation algorithms are utilized to find exact solutions of the considered equation involving bright solitons, singular periodic solitons, and singular bright solitons. These solutions are also described graphically while taking suitable values of free parameters. The applied algorithms are effective and convenient in handling the solution process for Ito equation that appears in many phenomena.

Spatially Resolved Heat Transfer and Friction Factors in a Rectangular Channel With 45-Deg Angled Crossed-Rib Turbulators

2003 ◽  
Vol 125 (3) ◽  
pp. 575-584 ◽  
Author(s):  
P. M. Ligrani ◽  
G. I. Mahmood
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Nusselt Number ◽  
Rectangular Channel ◽  
Test Surface ◽  
Spatially Resolved ◽  
Nusselt Numbers ◽  
Friction Factors ◽  
Smooth Channel ◽  
Rib Turbulators

Spatially resolved Nusselt numbers, spatially averaged Nusselt numbers, and friction factors are presented for a stationary channel with an aspect ratio of 4 and angled rib turbulators inclined at 45 deg with perpendicular orientations on two opposite surfaces. Results are given at different Reynolds numbers based on channel height from 10,000 to 83,700. The ratio of rib height to hydraulic diameter is .078, the rib pitch-to-height ratio is 10, and the blockage provided by the ribs is 25% of the channel cross-sectional area. Nusselt numbers are given both with and without three-dimensional conduction considered within the acrylic test surface. In both cases, spatially resolved local Nusselt numbers are highest on tops of the rib turbulators, with lower magnitudes on flat surfaces between the ribs, where regions of flow separation and shear layer reattachment have pronounced influences on local surface heat transfer behavior. The augmented local and spatially averaged Nusselt number ratios (rib turbulator Nusselt numbers normalized by values measured in a smooth channel) vary locally on the rib tops as Reynolds number increases. Nusselt number ratios decrease on the flat regions away from the ribs, especially at locations just downstream of the ribs, as Reynolds number increases. When adjusted to account for conduction along and within the test surface, Nusselt number ratios show different quantitative variations (with location along the test surface), compared to variations when no conduction is included. Changes include: (i) decreased local Nusselt number ratios along the central part of each rib top surface as heat transfer from the sides of each rib becomes larger, and (ii) Nusselt number ratio decreases near corners, where each rib joins the flat part of the test surface, especially on the downstream side of each rib. With no conduction along and within the test surface (and variable heat flux assumed into the air stream), globally-averaged Nusselt number ratios vary from 2.92 to 1.64 as Reynolds number increases from 10,000 to 83,700. Corresponding thermal performance parameters also decrease as Reynolds number increases over this range, with values in approximate agreement with data measured by other investigators in a square channel also with 45 deg oriented ribs.

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.

Nusselt Numbers and Flow Structure on and Above a Shallow Dimpled Surface Within a Channel Including Effects of Inlet Turbulence Intensity Level

2004 ◽  
Vol 127 (2) ◽  
pp. 321-330 ◽  
Author(s):  
P. M. Ligrani ◽  
N. K. Burgess ◽  
S. Y. Won
Keyword(s):  
Nusselt Number ◽  
Flow Structure ◽  
Turbulence Intensity ◽  
Local Nusselt Number ◽  
Channel Height ◽  
Intensity Level ◽  
Stagnation Temperature ◽  
Channel Inlet ◽  
Local Flow ◽  
Nusselt Numbers

Experimental results from a channel with shallow dimples placed on one wall are given for Reynolds numbers based on channel height from 3,700 to 20,000, levels of longitudinal turbulence intensity from 3% to 11% (at the entrance of the channel test section), and a ratio of air inlet stagnation temperature to surface temperature of approximately 0.94. The ratio of dimple depth to dimple print diameter δ∕D is 0.1, and the ratio of channel height to dimple print diameter H∕D is 1.00. The data presented include friction factors, local Nusselt numbers, spatially averaged Nusselt numbers, a number of time-averaged flow structural characteristics, flow visualization results, and spectra of longitudinal velocity fluctuations which, at a Reynolds number of 20,000, show a primary vortex shedding frequency of 8.0Hz and a dimple edge vortex pair oscillation frequency of approximately 6.5Hz. The local flow structure shows some qualitative similarity to characteristics measured with deeper dimples (δ∕D of 0.2 and 0.3), with smaller quantitative changes from the dimples as δ∕D decreases. A similar conclusion is reached regarding qualitative and quantitative variations of local Nusselt number ratio data, which show that the highest local values are present within the downstream portions of dimples, as well as near dimple spanwise and downstream edges. Local and spatially averaged Nusselt number ratios sometimes change by small amounts as the channel inlet turbulence intensity level is altered, whereas friction factor ratios increase somewhat at the channel inlet turbulence intensity level increases. These changes to local Nusselt number data (with changing turbulence intensity level) are present at the same locations where the vortex pairs appear to originate, where they have the greatest influences on local flow and heat transfer behavior.

Sign in / Sign up

Export Citation Format

Share Document