This paper deals with approximate analytical solutions to the two-region one-dimensional model describing the charging process of stratified thermal storage tanks with variable inlet temperature as well as momentum-induced mixing. The inlet temperature, which is allowed to increase arbitrarily except that it always remains higher than (or equal to) the temperature at the tank top, is broken down into step changes and intervals of continuous change. Assuming that each continuous interval can be approximated as a finite number of piecewise linear functions, there is admitted an analytical solution for the perfectly mixed region. Based on the principle of superposition, the temperature profiles in the plug flow region (assuming both semi-infinite and adiabatic ends) are successfully derived, in terms of well-defined functions. The effect of the end condition proves to be negligible under the practical operating conditions. For a quadratic variation of inlet temperature, the approximate solution modeled on a moderate number of pieces agrees well with the corresponding closed-form solution.