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Natural convection with heat transfer in porous media has been subject of extensive study in engineering due to its numerous applications. A case of particular interest is the Bénard-type flow.The paper aims to discuss this issue.

Based on the network simulation method in order to solve this problem, a numerical model is proposed.

Nusselt-Rayleigh correlation is determined for a broad range of Rayleigh, the dimensionless number that influences the solution, above and below the threshold which separates the conduction and convection pure mechanisms.

Based on the network simulation method.

This study aims to define and measure the capabilities applied by firms to waste-related patents and their relations with the businesses economic performance to support decision-making towards a circular economy (CE).

A model of cause-and-effect relationships between firms' waste-related patents and the firm' capabilities was defined within the dynamic capabilities' theoretical framework. Empirical results were obtained by applying partial least squares structural equation modelling to a sample of 2,216 Spanish firms that hold 120,406 patents.

Results revealed the importance of the innovation capabilities of firms related to patenting, such as collaborative innovation, persistence in patenting or the capabilities to collaborate with research institutes, as drivers of level of waste patents to improve the businesses economic performance.

The systemic nature of the CE at the firm level suggests future research focused on the environmental divergence that appears when the innovation on waste fall outside the regular domain of its industry. Another topic to be investigated is related to the full text of patents that could improve the results of this study.

The definition of indicators to measure investments in the CE is complex, but it is necessary to assess progress in the closing of material loops at a micro level and to report the investments in waste-related patents in a circular model to the stakeholders involved in the economic management of the company.

Measuring CE-related patents and the specific capabilities needed for patenting in a circular framework is an understudied topic, and this study opens a specific line of inquiry enhancing the knowledge of CE within the dynamic capabilities' theoretical framework.