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.

Effects of Curvature and Convective Heat Transfer in Curved Square Duct Flows

2006 ◽  
Vol 128 (5) ◽  
pp. 1013-1022 ◽  
Author(s):  
R. N. Mondal ◽  
Y. Kaga ◽  
T. Hyakutake ◽  
S. Yanase
Keyword(s):  
Heat Transfer ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Outer Wall ◽  
Heated Wall ◽  
Dean Number ◽  
Square Duct ◽  
Nusselt Numbers ◽  
Wide Range ◽  
Steady Solutions

Non-isothermal flows with convective heat transfer through a curved duct of square cross section are numerically studied by using a spectral method, and covering a wide range of curvature, δ, 0<δ≤0.5 and the Dean number, Dn, 0≤Dn≤6000. A temperature difference is applied across the vertical sidewalls for the Grashof number Gr=100, where the outer wall is heated and the inner one cooled. Steady solutions are obtained by the Newton-Raphson iteration method and their linear stability is investigated. It is found that the stability characteristics drastically change due to an increase of curvature from δ = 0.23 to 0.24. When there is no stable steady solution, time evolution calculations as well as their spectral analyses show that the steady flow turns into chaos through periodic or multi-periodic flows if Dn is increased no matter what δ is. The transition to a periodic or chaotic state is retarded with an increase of δ. Nusselt numbers are calculated as an index of horizontal heat transfer and it is found that the convection due to the secondary flow, enhanced by the centrifugal force, increases heat transfer significantly from the heated wall to the fluid. If the flow becomes periodic and then chaotic, as Dn increases, the rate of heat transfer increases remarkably.

Laminar Mixed Convection in a Shrouded Fin Array

1981 ◽  
Vol 103 (3) ◽  
pp. 559-565 ◽  
Author(s):  
S. Acharya ◽  
S. V. Patankar
Keyword(s):  
Secondary Flow ◽  
Forced Convection ◽  
Tip Clearance ◽  
Base Surface ◽  
Nusselt Numbers ◽  
Laminar Mixed Convection ◽  
Friction Factors ◽  
Wide Range ◽  
Laminar Forced Convection ◽  
The Heat Transfer Coefficient

An analytical study is made to investigate the effect of buoyancy on laminar forced convection in a shrouded fin array. Two heating conditions are considered; in one, the fins and the base surface are hotter than the fluid, and in the other, they are colder. The results are obtained numerically for a wide range of the governing buoyancy parameter. It is found that with a hot fin and base, the secondary flow pattern is mostly made up of a single eddy. The influence of buoyancy is significant and leads to Nusselt numbers and friction factors which are much higher than for pure forced convection. With a cold fin and base, the presence of a tip clearance between the fins and the shroud generates a multiple eddy pattern. The resulting stratification is responsible for the existence of high axial velocity and temperature in the clearance region relative to that in the inter-fin space. Compared to the hot fin case, the secondary flow is weaker, and therefore a relatively smaller increase in the friction factor is obtained. The Nusselt number is found to increase only in the absence of tip clearance. The distribution of the heat transfer coefficient along the fin and the base for both heating situations is found to be highly nonuniform.

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%.

Analytical Solution of the Thermal Entrance Heat Transfer Problem for Viscous Flow in Annulus

2016 ◽  
Vol 138 (7) ◽  
Author(s):  
E. K. Vachagina ◽  
A. I. Kadyirov
Keyword(s):  
Heat Transfer ◽  
Transfer Problem ◽  
Separation Of Variables ◽  
Flat Channel ◽  
Newtonian Flow ◽  
Full Agreement ◽  
Nusselt Numbers ◽  
Separation Of Variables Method ◽  
Thermal Entrance Region ◽  
Thermal Entrance

The Graetz–Nusselt problem with Brinkman extension is considered for steady-state laminar Newtonian flow in annuli. To solve the problem, a separation of variables method is used. In the limiting cases, the eigenvalues are in full agreement with the eigenvalues corresponding to flat channel and circular pipe. Useful formulas are represented to calculate the length of the thermal entrance region and Nusselt numbers in annuli.

Krypton 85 myocardial blood flow: precordial scintillation versus coronary sinus sampling

1965 ◽  
Vol 209 (4) ◽  
pp. 705-710 ◽  
Author(s):  
Michael D. Klein ◽  
Lawrence S. Cohen ◽  
Richard Gorlin
Keyword(s):  
Blood Flow ◽  
Myocardial Blood Flow ◽  
Coronary Sinus ◽  
Human Subjects ◽  
Radioactive Decay ◽  
Empirical Correction ◽  
Wide Range ◽  
Actual Flow ◽  
Flow Through ◽  
High Degree

Myocardial blood flow in human subjects was assessed by comparative simultaneous measurement of krypton 85 radioactive decay from coronary sinus and precordial scintillation. Empirical correction of postclearance background from precordial curves yielded a high degree of correlation between flows derived from the two sampling sites (r = .889, P < .001). Comparison of left and right coronary flows in nine subjects revealed similarity in flow through the two vessels over a wide range of actual flow values (r = .945, P < .001).

scholarly journals Development of Permeability Models for Saturated Fluid Flow across Arrays of Fibre Clusters

2002 ◽  
Vol 11 (3) ◽  
pp. 096369350201100
Author(s):  
E.M. Gravel ◽  
T.D. Papathanasiou
Keyword(s):  
Analytical Models ◽  
Dual Porosity ◽  
Hydraulic Permeability ◽  
Fibrous Media ◽  
Fibre Bundles ◽  
Hexagonal Packing ◽  
Composites Manufacturing ◽  
Starting Point ◽  
Wide Range ◽  
Flow Through

Dual porosity fibrous media are important in a number of applications, ranging from bioreactor design and transport in living systems to composites manufacturing. In the present study we are concerned with the development of predictive models for the hydraulic permeability ( Kp) of various arrays of fibre bundles. For this we carry out extensive computations for viscous flow through arrays of fibre bundles using the Boundary Element Method (BEM) implemented on a multi-processor computer. Up to 350 individual filaments, arranged in square or hexagonal packing within bundles, which are also arranged in square of hexagonal packing, are included in each simulation. These are simple but not trivial models for fibrous preforms used in composites manufacturing – dual porosity systems characterised by different inter- and intra-tow porosities. The way these porosities affect the hydraulic permeability of such media is currently unknown and is elucidated through our simulations. Following numerical solution of the governing equations, ( Kp) is calculated from the computed flowrate through Darcy's law and is expressed as function of the inter- and intra-tow porosities (φ, φt) and of the filament radius ( Rf). Numerical results are also compared to analytical models. The latter form the starting point in the development of a dimensionless correlation for the permeability of such dual porosity media. It is found that the numerically computed permeabilities follow that correlation for a wide range of φ i, φt and Rf.

On LES Methods Applied to Seal Geometries

2012 ◽  
Author(s):  
James Tyacke ◽  
Richard Jefferson-Loveday ◽  
Paul Tucker
Keyword(s):  
Maximum Error ◽  
Labyrinth Seal ◽  
Navier Stokes ◽  
Final Solution ◽  
Complex Flow ◽  
Eddy Simulation ◽  
Wide Range ◽  
Large Eddy ◽  
Rans Models ◽  
Flow Through

Nine Large Eddy Simulation (LES) methods are used to simulate flow through two labyrinth seal geometries and are compared with a wide range of Reynolds-Averaged Navier-Stokes (RANS) solutions. These involve one-equation, two-equation and Reynolds Stress RANS models. Also applied are linear and nonlinear pure LES models, hybrid RANS-Numerical-LES (RANS-NLES) and Numerical-LES (NLES). RANS is found to have a maximum error and a scatter of 20%. A similar level of scatter is also found among the same turbulence model implemented in different codes. In a design context, this makes RANS unusable as a final solution. Results show that LES and RANS-NLES is capable of accurately predicting flow behaviour of two seals with a scatter of less than 5%. The complex flow physics gives rise to both laminar and turbulent zones making most LES models inappropriate. Nonetheless, this is found to have minimal tangible results impact. In accord with experimental observations, the ability of LES to find multiple solutions due to solution non-uniqueness is also observed.

CFD Simulation of Fluid Flow Through Porous Media: Application to Decomposed Gases Flow Through Foam Pattern in Lost Foam Casting

Materials ◽  
2003 ◽  
Author(s):  
Sayavur I. Bakhtiyarov ◽  
Ruel A. Overfelt
Keyword(s):  
Cfd Simulation ◽  
Reynolds Numbers ◽  
Phase Flow ◽  
Flow Through Porous Media ◽  
Foam Material ◽  
Element Analysis ◽  
Wide Range ◽  
Flow Through ◽  
Foam Pattern ◽  
Lost Foam

Numerical simulation of decomposed gases through foam pattern was conducted using finite element analysis. A new kinetic model is proposed for gaseos phase flow between molten metal and foam material. The computations were performed for a wide range of Reynolds numbers. The results of the simulations are compared with the experiemental data obtained in this study.

The Graetz Problem for the Ellis Fluid Model

2018 ◽  
Vol 74 (1) ◽  
pp. 15-24 ◽  
Author(s):  
N. Ali ◽  
M.W.S. Khan
Keyword(s):  
Energy Equation ◽  
Fluid Model ◽  
Separation Of Variables ◽  
Surface Heat ◽  
Thermal Boundary Conditions ◽  
Nusselt Numbers ◽  
Graetz Problem ◽  
Developing Flow ◽  
Sturm Liouville

AbstractThe determination of temperature and auxiliary quantities such as local and average Nusselt numbers for thermally developing flow is referred as the Graetz problem. In the classical Graetz problem, the fluid entering the tube or channel is Newtonian in nature. Here, an extension of the classical Graetz problem is presented by assuming that the fluid entering the tube or channel obeys the Ellis constitutive equation. The energy equation for the considered problem is solved using the separation of variables technique supplemented with the MATLAB routine bvp4c for computation of the eigenvalues and numerical solution of the associated Sturm-Liouville boundary value problem. The problem is solved for two types of thermal boundary conditions, namely, uniform surface temperature and uniform surface heat flux for both flat and circular geometries. Expressions for bulk mean temperature and local and average Nusselt numbers are presented and discussed through tables and graphs.

scholarly journals Two conjugate convection boundary-layers of counter forced flow

2018 ◽  
Vol 22 (2) ◽  
pp. 835-846
Author(s):  
Mohamed Mosaad
Keyword(s):  
Heat Transfer ◽  
Boundary Layers ◽  
Conjugate Heat Transfer ◽  
Transfer Problem ◽  
Forced Flow ◽  
Relative Importance ◽  
Counter Flow ◽  
Wide Range ◽  
Convection Boundary ◽  
Laminar Forced Convection

In this study, the conjugate heat transfer problem of two laminar forced convection boundary-layers of counter flow on the opposite sides of a conductive wall is analyzed by employing the integral method. The analysis is conducted in a dimensionless framework to generalize the solution. The dimensionless parameters affecting the thermal interaction between the two convection layers are deduced from the analysis. These parameters give a measure of the relative importance of interactive heat transfer modes. Mean Nusselt number data are obtained for a wide range of the main affecting parameters.

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