Logistic map is a simple non-linear demographic equation that exhibits chaotic behavior and is often used to illustrate how complex patterns arise from simple rules
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... both were applied in this wo rk only for co mparison purposes. The emp irical model is described by a very simp le equation (Table 1) wh ich has just one adjustable parameter and no physical mean ing. The diffusion model has some relation with the mass transfer phenomenon but it considers only the diffusion mechanism since the mass balance is applied to the solid phase while fluid phase is neglected. ...
... m the spline model the fo llo wing parameters for the CER period were estimated: the time span of the CER period (t CER ), the extraction rate for the CER period (M CER ), the mass ratio of ext ract in the supercritical phase at the bed outlet (Y CER ), the ext raction yield of the CER period (R CER ), and the solvent to feed mass ratio of the CER period (S/F CER ). The t CER and M CER (kg ext ract/s) are both adjustable parameters fro m the spline model (t CER and b 1 , respectively, of Supercritical Fluid Extraction from a Rice Bran Oil Byproduct according to Table 1). The Y CER (kg extract/kg CO 2 ) was obtained by dividing M CER by the mean solvent flow rate (Q CO2 , kg CO 2 /s). ...
... The spline model is a simple strategy to model extraction curves [43]. Despite corresponding to an empirical model [43], the experimental and modeled curves manage to define three regions: a constant rate of extraction (cer) associated with mass transfer by convection, a decrease in the rate of extraction (fer) described by control of mass transfer by diffusion and convection, and a period where the extraction rate is controlled by diffusion (dc) [31]. This stage is identified by the mass transfer rate, defined as Mcer, corresponding to parameter b1 of the spline model equations. ...
This characteristic pattern is often described using a bell-shaped or logistic curve and is closely associated with the concept of peak production, particularly in the oil and gas industry. A well-known application of this idea is the Hubbert Peak Theory, which predicts that for any given geographical area, the rate of petroleum extraction will rise, reach a maximum point, and then inevitably decline.
