Improved lumped analysis of Graetz problems with axial diffusion

  • L. S. de Barros
  • L. A. SphaierEmail author
Technical Paper


This paper proposes analytical approximations for solving extended Graetz problems with axial diffusion in infinite domains. A general formulation, valid for both parallel-plates channels and circular ducts, is employed, and a discontinuous Dirichlet condition is applied at the wall. The adopted methodology consists of transforming the 2D convection equation into simpler one-dimensional forms, using approximation rules provided by the Coupled Integral Equations Approach. This technique is employed for producing expressions for the bulk temperature, and different levels of approximations are analyzed. The results are compared with an exact analytical solution to the problem and an expression obtained with the classical lumped system analysis (CLSA). While the results obtained with the CLSA are demonstrated to be substantially discrepant compared to the exact solution, the proposed improved formulas are equivalently simple and are shown to provide very good estimates for calculating the bulk temperature distribution. In addition, estimates for the length through which heat is diffused backward are also calculated with the derived approximate formulations and a practically perfect agreement is obtained with the higher-order approximation and the exact solution data.


Lumped capacitance Heat convection Advection–diffusion Analytical solution Mathematical modeling 

List of symbols


Hermite approximation coefficient

\(c_{{\rm p}}\)

Specific heat


Diameter of channel spacing




General function

\(h_{{\rm i}}\)

Hermite domain of integration

\(H_{\alpha ,\beta }\)

Hermite approximation


Thermal conductivity


Characteristic length


Velocity profile


Dimensionless velocity profile


Cross-sectional averaged velocity

x, y

Spatial variables




Péclet number

Greek symbols

\(\alpha\), \(\beta\), \(\nu\)

Hermite approximation parameters


Geometry parameter


Heaviside function




Parameter in solution formula

\({\bar{\theta }}\)

Dimensionless averaged temperature


Dimensionless temperature

\(\theta _{{\rm m}}\)

Dimensionless bulk temperature

\(\xi\), \(\eta\)

Dimensionless spatial variables



The authors would like to acknowledge the financial support provided by the Brazilian Government funding agencies CAPES, CNPq, and FAPERJ.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Department of Mechanical Engineering – TEM/PGMECUniversidade Federal Fluminense – UFFNiteróiBrazil

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