Improvement of the exact analytical solutions for combined heat and mass transfer problems obtained with the Fourier method

• Mathematical instabilities of the established analytical solution using the Fourier method.
• Necessity of two analytical solutions with no information of the range of validity so far.
• The established tangent functions to find the eigenvalues are domain restricted.
• Rearranging this functions results in more eigenvalues and the mathematical instabilities vanish.
• Fourier method with complemented set of eigenvalues is physically valid for the whole range of flow length.

Grigor'eva and Nakoryakov presented an analytical solution for combined heat and mass transfer in laminar falling films by means of the Fourier method. The obtained solutions exhibited mathematical instabilities for small flow length, such as oscillations in the mass fraction profile and a mismatch of the inlet temperature. Grigor'eva and Nakoryakov explained these instabilities with the inconsistency of the inlet and boundary conditions and therefore an additional short term solution was introduced.In the present study the established tangent function, that is used to determine the eigenvalues within the Fourier method, is rearranged to a term without domain restrictions. Consequently, more eigenvalues are found, leading to a physical valid solution even for small flow lengths, matching the results of the short term solution perfectly.