evaluation of the residence time distribution (rtd) for flow in ducts with velocity profile of two independent variables

The correct information on rtd can help in system design and evaluation. the rtd corresponding to the velocity profile is known only for certain cases, where the velocity profile depends on one coordinate only. in this research, a general procedure for the derivation of rtd corresponding to a known velocity profile is introduced. the rtd of laminar flows in different ducts as elliptic, equilateral triangular, moon-shaped, and rectangular ducts are derived. also, it is shown that the final rtd for laminar flow in any duct, can be estimated using relation e(θ) = k θmin/θn which is similar to laminar flow in the pipe, with their own dimensionless minimum time, θmin, where is defined as the required time for traveling the duct with the maximum velocity in a unit of space-time. the values of k and n are calculated to meet the condition of ∫ꝏθmin e(θ)dθ = ∫ꝏθmin θe(θ)dθ =1. besides, the values of θmin for different cross-sections are studied. the results show that the rtd of elliptic ducts is precisely similar to the pipe flow. in the case of other shape ducts, the proposed model shows a suitable estimate of the numerical values. the previously published experimental data and precise analytical solutions agree with the proposed model with an acceptable consistency, except for very little time say θmin < θ < 0.7.