Nusselt Numbers in Rectangular Ducts With Laminar Viscous Dissipation

1999 ◽  
Vol 121 (4) ◽  
pp. 1083-1087 ◽  
Author(s):  
G. L. Morini ◽  
M. Spiga
Keyword(s):  
Boundary Condition ◽  
Temperature Distribution ◽  
Viscous Dissipation ◽  
Integral Transforms ◽  
Navier Stokes ◽  
Brinkman Number ◽  
Balance Equations ◽  
Nusselt Numbers ◽  
Finite Integral ◽  
Axial Heat

In this paper, the steady temperature distribution and the Nusselt numbers are analytically determined for a Newtonian incompressible fluid in a rectangular duct, in fully developed laminar flow with viscous dissipation, for any combination of heated and adiabatic sides of the duct, in H1 boundary condition, and neglecting the axial heat conduction in the fluid. The Navier-Stokes and the energy balance equations are solved using the technique of the finite integral transforms. For a duct with four uniformly heated sides (4 version), the temperature distribution and the Nusselt numbers are obtained as a function of the aspect ratio and of the Brinkman number and presented in graphs and tables. Finally it is proved that the temperature field in a fully developed T boundary condition can be obtained as a particular case of the H1 problem and that the corresponding Nusselt numbers do not depend on the Brinkman number.

scholarly journals Viscous dissipation effect on laminar forced convection in a circular duct with adiabatic walls - preliminary numerical investigation

2018 ◽  
Vol 240 ◽  
pp. 03016
Author(s):  
Tomasz Janusz Teleszewski
Keyword(s):  
Heat Flux ◽  
Forced Convection ◽  
Viscous Dissipation ◽  
Heat Flux Density ◽  
Energy Equation ◽  
Brinkman Number ◽  
Circular Duct ◽  
Nusselt Numbers ◽  
Axial Heat ◽  
Laminar Forced Convection

In this work, an analysis of laminar forced convection in a pipe with heated and adiabatic walls for a Newtonian fluid with constant properties is performed by taking the viscous dissipation into account when the axial heat conduction in the fluid is neglected. The Nusselt number versus the Brinkmann, which is based on the total wall heat flux density, have been investigated. In order to determine the temperature field, an analytical solution describing the velocity field in the pipe was used, while the energy equation was determined by the boundary element method (BEM). The results of the calculations of Nusselt numbers as a function of the Brinkman number for different thermal insulation heights to the diameter of the circular duct were presented in the form of diagrams.

Fully Developed Laminar Forced Convection in a Circular Duct for Herschel–Bulkley Fluids With Viscous Dissipation and Axial Heat Conduction

2017 ◽  
Vol 139 (10) ◽  
Author(s):  
Rabha Khatyr ◽  
Jaafar Khalid-Naciri ◽  
Ali Il Idrissi
Keyword(s):  
Boundary Condition ◽  
Heat Conduction ◽  
Forced Convection ◽  
Viscous Dissipation ◽  
Circular Duct ◽  
Nusselt Numbers ◽  
Asymptotic Values ◽  
Axial Heat ◽  
Laminar Forced Convection ◽  
Axial Heat Conduction

The asymptotic behavior of laminar forced convection in a circular duct for a Herschel–Bulkley fluid of constant properties is analyzed. The viscous dissipation and the axial heat conduction effects in the fluid are both considered. The asymptotic bulk and mixing temperature field, and the asymptotic values of the bulk and mixing Nusselt numbers are determined for every boundary condition, enabling a fully developed region. In particular, it is proved that whenever the wall heat flux tends to zero, the asymptotic Nusselt number is zero. The obtained results are compared to other existing solutions in the literature for Newtonian and non-Newtonian cases.

scholarly journals Viscous Dissipation Effects in A Microchannel Caused by Oscillation of One Surface

2019 ◽  
Vol 1 (1) ◽  
pp. 13-17
Author(s):  
Chee Hao Hor ◽  
Chih Ping Tso ◽  
Gooi Mee Chen
Keyword(s):  
Boundary Condition ◽  
Temperature Field ◽  
Prandtl Number ◽  
Viscous Dissipation ◽  
Diffusion Rate ◽  
Lower Surface ◽  
Angular Frequency ◽  
Thermal Boundary Condition ◽  
Effect Of Temperature ◽  
Brinkman Number

The viscous dissipation effects in a microchannels caused by an oscillatory lower surface is investigated numerically. An asymmetric thermal boundary condition, particularly at upper plate insulated and lower plate with constant surface temperature is solved and analyzed in details graphically. Results reveal that effect of temperature field is strongly dependent on Brinkman number, while the thermal diffusion rate on the heat induced relies on the Prandtl number. The angular frequency has influence on the temperature field gradient.

Viscous dissipation effects on the limiting value of Nusselt numbers for a shear-driven flow through an asymmetrically heated annulus

2012 ◽  
Vol 226 (12) ◽  
pp. 2941-2949 ◽  
Author(s):  
Pranab Kumar Mondal ◽  
Sanchayan Mukherjee
Keyword(s):  
Nusselt Number ◽  
Viscous Dissipation ◽  
Newtonian Fluid ◽  
Axial Direction ◽  
Analytical Framework ◽  
Analytical Investigation ◽  
Interactive Effects ◽  
Brinkman Number ◽  
Nusselt Numbers ◽  
Flow Through

In this study, an analytical investigation for analyzing the effects of viscous dissipation on the limiting Nusselt number for a hydro-dynamically fully developed laminar shear-driven flow through an asymmetrically heated annulus of two infinitely long concentric cylinders has been made, where the inner cylindrical rod is moving in an axial direction at a constant speed. On the basis of some common assumptions, an analytical framework has been devised to explore the effects of viscous dissipation on the heat transfer characteristics for the flow of Newtonian fluid, and, consequently, closed-form expressions for the limiting Nusselt numbers are evaluated. In the analysis, focus has been given on the viscous dissipative effect due to the shear produced by the movable inner cylindrical rod apart from the viscous dissipation due to internal fluid friction for the flow of a Newtonian fluid. The interactive effects of the Brinkman number and degree of asymmetry on the limiting Nusselt number are analytically investigated. It is observed from this study that the limiting Nusselt number becomes independent of Brinkman number when both the walls of the annulus are kept at an equal temperature. Moreover, the temperature profile in the conduction limit obtained with the consideration of viscous dissipation effect provides a boundary condition required for solving energy equation including the axial conduction in the fluid.

Effect of viscous dissipation on laminar mixed convection in a horizontal channel

2002 ◽  
Vol 216 (3) ◽  
pp. 167-172 ◽  
Author(s):  
M M Salah El-Din
Keyword(s):  
Mixed Convection ◽  
Viscous Dissipation ◽  
Heat Fluxes ◽  
Horizontal Channel ◽  
Temperature Profiles ◽  
Brinkman Number ◽  
Thermal Boundary Conditions ◽  
Nusselt Numbers ◽  
Laminar Mixed Convection ◽  
Uniform Wall

Laminar mixed convection in a horizontal channel is studied analytically in the fully developed region when the effect of viscous dissipation cannot be neglected. Two kinds of thermal boundary conditions are considered: uniform wall temperatures and uniform wall heat fluxes. Velocity and temperature profiles and the Nusselt numbers have been determined in a closed form. The results show that when viscous dissipation is taken into consideration, temperature profiles and Nusselt numbers are significantly affected by the Brinkman number while velocity profiles are independent of the Brinkman number.

scholarly journals Thermal analysis of end pumped fiber lasers subjected to jacket fluid cooling

2019 ◽  
pp. 311-311
Author(s):  
Haj El ◽  
Han Bani ◽  
Israa Al-Sawafta ◽  
Ahmad Sedaghat ◽  
M. Alshabi ◽  
...  
Keyword(s):  
Temperature Distribution ◽  
Fiber Lasers ◽  
Brinkman Number ◽  
Dimensionless Form ◽  
Water Jacket ◽  
Cooling Method ◽  
Radial Heat ◽  
The Common ◽  
Fluid Cooling ◽  
Axial Heat

End pumped lasers are highly efficient lasers particularly in diode lasers using micro lenses. The common cooling method for end-pumped systems is using water jacket or copper tube surrounding the laser rod. In this paper, the temperature distribution within a water jacket and a fiber laser end pumped by a top hat beam is studied analytically. The temperature distribution is obtained by considering the radial heat convection with fully developed laminar flow neglecting the axial heat conduction. The effect of laser dimensions and the Brinkman number on the temperature distribution are presented. The results indicate that the temperature distribution is strongly dependent on the Brinkman number. The results are presented in dimensionless form so that they can be applied to any end-pumped laser rod and fluid types. The main output of this work is that it is better for cooling purposes to have low Br values.

Boundary Condition Design to Heat a Moving Object at Uniform Transient Temperature Using Inverse Formulation

2004 ◽  
Vol 126 (3) ◽  
pp. 619-626 ◽  
Author(s):  
Hakan Ertu¨rk ◽  
Ofodike A. Ezekoye ◽  
John R. Howell
Keyword(s):  
Boundary Condition ◽  
Temperature Distribution ◽  
Design Problem ◽  
Manufacturing Industry ◽  
Three Dimensional ◽  
Chemical Deposition ◽  
Iterative Solution ◽  
Conveyor Belt ◽  
Temperature History ◽  
Transient Temperature

The boundary condition design of a three-dimensional furnace that heats an object moving along a conveyor belt of an assembly line is considered. A furnace of this type can be used by the manufacturing industry for applications such as industrial baking, curing of paint, annealing or manufacturing through chemical deposition. The object that is to be heated moves along the furnace as it is heated following a specified temperature history. The spatial temperature distribution on the object is kept isothermal through the whole process. The temperature distribution of the heaters of the furnace should be changed as the object moves so that the specified temperature history can be satisfied. The design problem is transient where a series of inverse problems are solved. The process furnace considered is in the shape of a rectangular tunnel where the heaters are located on the top and the design object moves along the bottom. The inverse design approach is used for the solution, which is advantageous over a traditional trial-and-error solution where an iterative solution is required for every position as the object moves. The inverse formulation of the design problem is ill-posed and involves a set of Fredholm equations of the first kind. The use of advanced solvers that are able to regularize the resulting system is essential. These include the conjugate gradient method, the truncated singular value decomposition or Tikhonov regularization, rather than an ordinary solver, like Gauss-Seidel or Gauss elimination.

An analytical solution of a two-dimensional temperature field in a hollow cylinder under a time periodic boundary condition using Fourier series

2009 ◽  
Vol 223 (8) ◽  
pp. 1889-1901 ◽  
Author(s):  
G Atefi ◽  
M A Abdous ◽  
A Ganjehkaviri ◽  
N Moalemi
Keyword(s):  
Boundary Condition ◽  
Temperature Field ◽  
Fourier Series ◽  
Temperature Distribution ◽  
Analytical Solution ◽  
Hollow Cylinder ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Separation Of Variables ◽  
Two Dimensional

The objective of this article is to derive an analytical solution for a two-dimensional temperature field in a hollow cylinder, which is subjected to a periodic boundary condition at the outer surface, while the inner surface is insulated. The material is assumed to be homogeneous and isotropic with time-independent thermal properties. Because of the time-dependent term in the boundary condition, Duhamel's theorem is used to solve the problem for a periodic boundary condition. The periodic boundary condition is decomposed using the Fourier series. This condition is simulated with harmonic oscillation; however, there are some differences with the real situation. To solve this problem, first of all the boundary condition is assumed to be steady. By applying the method of separation of variables, the temperature distribution in a hollow cylinder can be obtained. Then, the boundary condition is assumed to be transient. In both these cases, the solutions are separately calculated. By using Duhamel's theorem, the temperature distribution field in a hollow cylinder is obtained. The final result is plotted with respect to the Biot and Fourier numbers. There is good agreement between the results of the proposed method and those reported by others for this geometry under a simple harmonic boundary condition.

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