Application of Superposition Principle for Solving a Nonlinear Energy Equation

Abstract

A new procedure to solve a nonlinear energy equation using the superposition principle is proposed. As an example of the utilization of this procedure, forced convection in a tube with temperature-dependent fluid properties was considered. The tube wall was maintained at uniform wall heat flux axially that varies with time, and the average fluid temperature at the outlet was calculated. This problem simulates convection heat transfer inside a solar collector tube. In the proposed procedure, the average fluid temperature at the outlet for a single heat pulse was determined for fluid properties evaluated at 15 different temperatures by solving the energy equation numerically assuming constant fluid properties and subsequently applying the superposition principle. The choice of the temperature at which fluid properties were evaluated as an important parameter in the simulation. This temperature was determined by using the inlet and outlet average fluid temperatures at the previous time-step multiplied by a weighting function. The average fluid temperature at the outlet obtained by this procedure was compared with the temperature obtained by solving the nonlinear energy equation using variable properties to determine the predictive accuracy of this procedure. The results for one-day operation of a sunny day with fluid velocity of 0.6 m/s, showed the highest root-mean-square (RMS) error of 0.25 K, and the highest mean absolute deviation (MAD) error of 0.16 K which agreed well with the result obtained by the numerical simulation of the nonlinear problem using variable properties.