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Carbon nanotubes (CNTs) are known for high thermal conductivity and have potential use as nano-radiators or heat exchangers. This paper focuses on the thermal performance of carpet-like arrays of vertically aligned CNTs on solid graphite substrates with the idea of investigating their behavior as a function of carpet dimensions and predicting their performance as thermal interface material (TIM) for electronic device cooling. Vertically aligned CNTs were grown on highly oriented pyrolytic graphite (HOPG) substrate, which creates a robust and durable all-carbon hierarchical structure. The multi-layer thermal analysis approach using Netzsch laser flash analysis system was used to evaluate their performance as a function of carpet height, from which their thermal properties can be determined. It was seen that the thermal resistance of the CNT array varies linearly with CNT carpet height, providing a unique way of decoupling the properties of the CNT carpet from its interface. This data was used to estimate the thermal conductivity of individual multi-walled nanotube strands in this carpet, which was about 35 W/m-K. The influence of CNT carpet parameters (aerial density, diameter, and length) on thermal resistance of the CNT carpet and its potential advantages and limitations as an integrated TIM are discussed.