Modelling a fin-and-tube heat exchanger as porous media based on Volume Averaging Theory (VAT), specific geometry can be accounted for in such a way that the details of the original structure can be replaced by their averaged counterparts and the VAT based governing equations can be solved for a wide range of heat exchanger designs. To complete the VAT based model, proper closure is needed, which is related to a local friction factor and a heat transfer coefficient of a Representative Elementary Volume (REV). The present paper describes an effort to model a fin-and-tube heat exchanger based on VAT and obtain closure for the model. Experiment data and correlations for the air side characteristics of fin-and-tube heat exchangers from the published literature were collected and rescaled using the ‘porous media’ length scale suggested by VAT. The results were surprisingly good, collapsing all the data onto a single curve for friction factor and Nusselt number, respectively. It was shown that using the ‘porous media’ length scale is very beneficial in collapsing complex data yielding simple heat transfer and friction factor correlations and that by proper scaling, closure is a function of the porous media, which further generalizes macro scale porous media equations. The current work is a step closer to our final goal, which is to develop a universal fast running computational tool for multiple-parameter optimization of heat exchangers.

The present paper describes an effort to obtain closure for a Volume Averaging Theory (VAT) based model of a plane fin heat sink (PFHS) with scale-roughened surfaces by evaluating the closure terms for the model using Computer Fluid Dynamics (CFD). Modeling a PFHS as porous media based on VAT, specific geometry can be accounted for in such a way that the details of the original structure can be replaced by their averaged counterparts and the VAT based governing equations can be efficiently solved for a wide range of parameters. To complete the VAT based model, proper closure is needed, which is related to a local friction factor and a heat transfer coefficient of a Representative Elementary Volume (REV). The terms in the closure expressions are complex and sometimes relating experimental data to the closure terms is difficult. In this work we use CFD to obtain detailed solutions of flow and heat transfer through an element of the scale-roughened heat sink and use these results to evaluate the closure terms needed for a fast running VAT based code, which can then be used to solve the heat transfer characteristics of a higher level heat sink. The objective is to show how heat sinks can be modeled as a porous media based on Volume Averaging Theory and how CFD can be used in place of a detailed, often formidable, experimental effort to obtain closure for a VAT based model. 

Modelling of fin-and-tube heat exchangers based on Volume Averaging Theory (VAT) requires proper closure of the VAT based governing equations. Closure can be obtained from reasonable lower scale solutions of a CFD code, which means the tube row number chosen should be large enough so that the closure can be evaluated for a representative elementary volume (REV) that is not affected by the entrance or re-circulation at the outlet of the fin gap. To determine the number of tube rows, three-dimensional numerical simulations for plate fin-and-tube heat exchangers were performed, with the Reynolds number varying from 500 to 6000 and the number of tube rows varying from 1 to 9. A clear perspective of the variations of both overall and local fiction factor and Nusselt number as the tube row number increases are presented. These variation trends are explained from the view point of the Field Synergy Principle (FSP). Our investigation shows that 4+1+1 tube rows is the minimum number to get reasonable lower scale solutions. A computational domain including 5+2+2 tube rows is recommended, so that the closure formulas for drag resistance coefficient and heat transfer coefficient could be evaluated for the sixth and seventh elementary volumes to close the VAT based model.