The Solution of Temperature Development in the Entrance Region of an MHD Channel by the B. G. Galerkin Method

1969 ◽  
Vol 91 (2) ◽  
pp. 212-220 ◽  
Author(s):  
R. C. LeCroy ◽  
A. H. Eraslan
Keyword(s):  
Eigenvalue Problem ◽  
Galerkin Method ◽  
Wall Temperature ◽  
Mathematical Problem ◽  
Considerable Effect ◽  
Constant Wall Temperature ◽  
Axial Conduction ◽  
Temperature Development ◽  
Constant Wall Heat Flux ◽  
Plate Channel

The general mathematical problem of MHD thermal entrance regions is formulated for a parallel plate channel by including Joule heating, viscous dissipation, and the effect of axial conduction. The associated eigenvalue problem is solved by the B. G. Galerkin method and the results are presented for constant wall temperature and constant wall heat flux conditions. It is shown that the particular method has distinct computational advantages over the classical form of solutions. The constant wall temperature case is investigated by employing the solutions of the eigenvalue problem and it is concluded that the axial conduction has considerable effect on the temperature development for low values of Peclet number.

Laminar Forced Convection in Transversely Corrugated Microtubes

2019 ◽  
Vol 141 (3) ◽  
Author(s):  
F. Talay Akyildiz ◽  
Dennis A. Siginer
Keyword(s):  
Heat Flux ◽  
Analytical Solution ◽  
Forced Convection ◽  
Wall Temperature ◽  
Wall Heat Flux ◽  
Bulk Temperature ◽  
Straight Tube ◽  
Computational Domain ◽  
Constant Wall Temperature ◽  
Constant Wall Heat Flux

Forced convection heat transfer in fully developed laminar flow in transversely corrugated tubes is investigated for nonuniform but constant wall heat flux as well as for constant wall temperature. Epitrochoid conformal mapping is used to map the flow domain onto the unit circle in the computational domain. The governing equations are solved in the computational domain analytically. An exact analytical solution for the temperature field is derived together with closed form expressions for bulk temperature and Nusselt number for the case of the constant heat flux at the wall. A variable coefficient Helmholtz eigenvalue problem governs the case of the constant wall temperature. A novel semi-analytical solution based on the spectral Galerkin method is introduced to solve the Helmholtz equation. The solution in both constant wall heat flux and constant wall temperature case is shown to collapse onto the well-known results for the circular straight tube for zero waviness.

Constant Wall Temperature Nusselt and Poiseuille Numbers in Rectangular Microchannels

2007 ◽  
Author(s):  
Jennifer van Rij ◽  
Tim Ameel ◽  
Todd Harman
Keyword(s):  
Aspect Ratio ◽  
Boundary Conditions ◽  
Viscous Dissipation ◽  
Wall Temperature ◽  
Rectangular Channel ◽  
Slip Flow ◽  
Second Order ◽  
Constant Wall Temperature ◽  
Axial Conduction ◽  
Accommodation Coefficients

Slip flow convective heat transfer and friction loss characteristics are numerically evaluated for constant wall temperature rectangular microchannels. The effects of rarefaction, accommodation coefficients, aspect ratio, second-order slip boundary conditions, axial conduction, and viscous dissipation with flow work are each considered. Second-order slip boundary conditions, axial conduction, and viscous dissipation with flow work effects have not been studied previously for rectangular channel slip flows. The effects of each of these parameters on the numerically computed convective heat transfer rate and friction loss are evaluated through the Nusselt number and Poiseuille number respectively. The numerical results are obtained using a continuum-based computational fluid dynamics algorithm that includes second-order slip flow and temperature jump boundary conditions. Numerical results for the three-dimensional, fully developed Nusselt and Poiseuille numbers are presented as functions of Knudsen number, first- and second-order velocity slip and temperature jump coefficients, aspect ratio, Brinkman number, and Peclet number. Effects of rarefaction, accommodation coefficients, and aspect ratio are consistent with previously reported analytical results for rectangular channel constant wall temperature flows. The effects of second-order slip terms, axial conduction and viscous dissipation are also shown to significantly affect the Nusselt and Poiseuille numbers.

Turbulent Forced Convection Inside a Parallel-Plate Channel With Periodic Variation of Inlet Temperature

1989 ◽  
Vol 111 (4) ◽  
pp. 882-888 ◽  
Author(s):  
W. S. Kim ◽  
M. N. O¨zis¸ik
Keyword(s):  
Eigenvalue Problem ◽  
Forced Convection ◽  
Parallel Plate ◽  
Periodic Variation ◽  
Bulk Temperature ◽  
Inlet Temperature ◽  
Complex Eigenvalue ◽  
Phase Lag ◽  
Constant Wall Temperature ◽  
Plate Channel

The analysis of heat transfer in turbulent forced convection subject to a periodically varying inlet temperature leads to a nonclassical Sturm–Liouville type eigenvalue problem for which no known solution is available. In this work a new methodology is developed to alleviate the need for the solution of a complex eigenvalue problem in the analysis of turbulent forced convection inside a parallel-plate channel with a periodicially varying inlet temperature and a uniform constant wall temperature. In this approach, the problem is transformed to the solution of a system of coupled ordinary differential equations in the complex domain, which could readily be solved. For the cases considered it is demonstrated that the solutions obtained from the decoupled system, referred to as the lowest-order solution, produce sufficiently accurate results. The variation of the amplitudes and phase lag of both fluid bulk temperature and the wall heat flux along the channel is investigated and a simple approximate analytic formula is developed for determining the variation of the phase lag for the bulk temperature along the channel.

Heat-Transfer Effects on the Developing Laminar Flow Inside Vertical Tubes

1966 ◽  
Vol 88 (2) ◽  
pp. 214-222 ◽  
Author(s):  
W. T. Lawrence ◽  
J. C. Chato
Keyword(s):  
Heat Flux ◽  
Transition Process ◽  
Constant Heat Flux ◽  
Wall Heat Flux ◽  
Axial Position ◽  
Fluid Viscosity ◽  
Constant Heat ◽  
Constant Wall Temperature ◽  
Constant Wall Heat Flux ◽  
Vertical Tubes

A numerical method was developed for the calculation of entrance flows in vertical tubes for the cases of upflow or downflow and constant wall heat flux or constant wall temperature. The solutions were in excellent agreement with experimental data obtained with water flowing upward in a vertical heated tube. The results show that both the density and the viscosity have to be treated as nonlinear functions of temperature. Consequently, for the constant heat flux condition, the velocity and temperature profiles constantly change and never reach “fully developed” states. The transition to turbulent flow was also studied. The experimental measurements demonstrated that the transition process depends on the developing velocity profiles. For the constant heat flux case, transition will always occur at some axial position. For a given entrance condition, the distance to transition is fixed by the fluid flow rate and the wall heat flux. For the experimental results, a tentative transition criterion was obtained, which depends only on the velocity profile shape, fluid viscosity, and the entrance Reynolds number.

Second law analysis of compressible flow through a diffuser subjected to constant wall temperature

2010 ◽  
Vol 7 (1) ◽  
pp. 110
Author(s):  
Mohammad Hasan Arshad ◽  
Ramazan Kahraman ◽  
Ahmet Z. Sahin ◽  
Rached Ben Mansour
Keyword(s):  
Compressible Flow ◽  
Wall Temperature ◽  
Second Law ◽  
Constant Wall Temperature ◽  
Second Law Analysis ◽  
Flow Through

EFFECTS OF THERMAL CREEP ON COOLING OF MICROFLOWS IN SHORT MICROCHANNELS WITH CONSTANT WALL TEMPERATURE

2012 ◽  
Vol 23 (11) ◽  
pp. 1250072 ◽  
Author(s):  
ALI AMIRI-JAGHARGH ◽  
HAMID NIAZMAND ◽  
METIN RENKSIZBULUT
Keyword(s):  
Wall Temperature ◽  
Temperature Jump ◽  
Control Volume ◽  
Velocity Slip ◽  
Temperature Gradients ◽  
Thermal Creep ◽  
Inlet Temperature ◽  
Constant Wall Temperature ◽  
Aspect Ratios ◽  
Wide Range

Fluid flow and heat transfer in the entrance region of rectangular microchannels of various aspect ratios are numerically investigated in the slip-flow regime with particular attention to thermal creep effects. Uniform inlet velocity and temperature profiles are prescribed in microchannels with constant wall temperature. An adiabatic section is also employed at the inlet of the channel in order to prevent unrealistically large axial temperature gradients due to the prescribed uniform inlet temperature as well as upstream diffusion associated with low Reynolds number flows. A control-volume technique is used to solve the Navier–Stokes and energy equations which are accompanied with appropriate velocity slip and temperature jump boundary conditions at the walls. Despite the constant wall temperature, axial and peripheral temperature gradients form in the gas layer adjacent to the wall due to temperature jump. The simultaneous effects of velocity slip, temperature jump and thermal creep on the flow and thermal patterns along with the key flow parameters are examined in detail for a wide range of cross-sectional aspect ratios, and Knudsen and Reynolds numbers. Present results indicate that thermal creep effects influence the flow field and the temperature distribution significantly in the early section of the channel.

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