Heat exchanges are used widely and often consist of a set of tubes in parallel. These tubes are fitted with fins along their length spaced regularly, which couples them elastically, to improve their thermal performance. There is a need to ensure that the lowest natural frequencies of the tube-fin systems are kept above the frequency band of the base excitation, to increase fatigue life. Because the tube-fin assembly consists of 100s-1000s of components, FEM based modal analysis is computationally expensive at initial stages of design search, which inevitably requires running codes for a very large number of parametric combinations. Here, a Rayleigh quotient-based approximation to estimate the lowest few natural modes of the system is developed. The approach makes use of an ansatz consistent with the geometry of the problem, with only a few undetermined degrees-of-freedom that are resolved variationally. Further, a continuum model that aims to smear the elastic and inertia properties of the structure are developed in the spirit of effective medium theory. The method affords computational efficiency that does not scale with the number of components, and hence is suitable for repetitive design evaluations and optimisation. A Python based GUI is also developed for deployment in industry.