Mathematical model of the cooling of a plastic, thermoformed part

This paper presents research into the mechanism involved in the cooling of a plastic thermoformed part after it is formed onto a mold. The intent of this research is to develop a simple and practical mathematical model useful to small thermoforming companies without a large engineering staff that describes the transient heat conduction of the cooling process. The model should also be able to predict the temperature distribution within the thickness of the part during the cooling. This mathematical model, which began with simplified boundary conditions, was then compared to experimental cooling data and modified accordingly to properly fit that data and the actual boundary conditions of the cooling part. The research began by examining the cooling of a series of high molecular weight polyethylene thermoformed side panels for plastic, portable restrooms. These parts where chosen for this preliminary research because of their very simple, flat geometric shape that lends them to being modeled as simple plane walls in transient conduction. The shape of the parts also leads to near constant thickness over the vast majority of the part. Using the model of a plane wall in transient conduction, the governing partial differential equation was solved for two possible boundary conditions on the mold side of the part: constant imposed surface temperature and constant imposed surface heat flux. These two solutions were then compared to experimental data gathered on the temperature profile of the free surface of the part during a production environment. After the experimental data and simple mathematical models were compared the necessary changes to the assumed mold side boundary condition was made to adjust the mathematical model to the experimental data. The research found that the use of simple boundary conditions at the mold side of the part is incorrect. Neither the constant imposed surface temperature nor the imposed surface heat flux boundary conditions fit the data. Initial analysis of the experimental data showed that a time of 30 seconds into the cooling cycle an apparent change in that boundary condition occurs for the part and mold used to gather the data. Further analysis showed that the boundary condition begins as a constant surface heat flux and then changes to an imposed surface temperature that decays exponentially to the initial mold surface temperature. Using this boundary condition, a revised mathematical was developed that match the experimental data very well. The error of the new model compared to the experimental was less than 1.5% for all times during the cooling cycle.