The extended Graetz problem with piecewise constant wall heat flux for pipe and channel flows

2001 ◽  
Vol 44 (20) ◽  
pp. 3941-3952 ◽  
Author(s):  
B. Weigand ◽  
M. Kanzamar ◽  
H. Beer
Keyword(s):  
Heat Flux ◽  
Channel Flows ◽  
Wall Heat Flux ◽  
Piecewise Constant ◽  
Graetz Problem ◽  
Constant Wall Heat Flux ◽  
Extended Graetz Problem

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.

A Historical Misperception on Calculating the Average Convection Coefficient in Tubes With Constant Wall Heat Flux

2019 ◽  
Vol 141 (6) ◽  
Author(s):  
Ted D. Bennett
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Exact Solutions ◽  
Wall Heat Flux ◽  
First Principle ◽  
Convection Coefficient ◽  
Parallel Plates ◽  
Historical Approach ◽  
First Principle Calculations ◽  
Constant Wall Heat Flux

The historical approach to averaging the convection coefficient in tubes of constant wall heat flux leads to quantitative errors in short tubes as high as 12.5% for convection into fully developed flows and 33.3% for convection into hydrodynamically developing flows. This mistake can be found in teaching texts and monographs on heat transfer, as well as in major handbooks. Using the correctly defined relationship between local and average convection coefficients, eight new correlations are presented for fully developed and developing flows in round tubes and between parallel plates for the constant wall heat flux condition. These new correlations are within 2% of exact solutions for fully developed flows and within 6% of first principle calculations for hydrodynamically developing flows.

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