Approximate analytical solution of the Graetz problem

Abstract

The study on velocity and temperature distribution in fluid flows is important both for the theory and practical applications. The design of efficient heat-exchange equipment, the development of heat and thermomechanical modes of product treatment, the determination of heat losses in the pipeline systems include the need to determine the velocity and temperature fields in fluid flows. The key aspects of the method have been considered with help of using the example of solving the Graetz problem for parallel and cylindrical channels.It is shown that finding the solution to a partial differential equation with respect to the temperature function can be reduced to integrating an ordinary differential equation with respect to the new unknown function q(η) which is the law of temperature change in the center of the channel. The combined use of the heat balance integral method and additional boundary characteristics made it possible to obtain a simple in form analytical solution to the problem under consideration. It is noted that the accuracy of the solutions obtained depends on the number of approximations used, i.e. the number of terms of the approximating series.When using only one term, i.e. already in the first approximation, the relative error of the method is not more than 8 percent in the range of the longitudinal coordinate change 0.1 ≤ η < ∞ and decreases to 4 percent in the second approximation. The analytical form of the solutions obtained provides analyzing the isotherms fields, calculating the average temperature, the Nusselt number, etc.