(145d) Extension of Mean Age Theory to Transient Inlet Velocity Conditions

Mean age theory introduces a redefined perspective on time as a passive scalar variable within the advection-diffusion equation. This allows for the analysis of traditionally time-based variables, such as mean residence time, while utilizing a steady-state solution. Unlike conventional methods that demand time-consuming and computationally intensive transient solutions for the advection-diffusion equation, mean age theory offers a more efficient alternative. Even when a steady-state solution is desired, conventional methods necessitate transient calculations to model time-dependent tracer behavior. Computational fluid dynamic (CFD) solutions incorporating mean age theory offer additional benefits. They not only provide spatial and temporal resolution within the flow field but also offer insights beyond conventional velocity vector and contour plots. Moreover, they offer solutions in scenarios where traditional experimental measurements are challenging or impractical.

The conventional form of the mean age transport equation assumes a time-invariant velocity input boundary condition. In our work here, we derive a variation of the scalar transport equation that accommodates a time-dependent velocity input at the boundary. Beginning from the same advection-diffusion equation, we derive an expression analogous to the well-studied steady mean age transport equation. However, our derivation includes an additional term incorporating [d(velocity)/dt] to account for the time-dependent velocity. Consequently, our output provides mean age as a function of time over the period defined by the velocity input function. Notably, the average age over this period differs from that obtained when assuming a constant velocity equal to the average of the velocity function. Specifically, the time-averaged age at the exit combines the mean age derived from a constant velocity input with the contribution from the additional term containing the input velocity function, d(velocity)/dt.

To illustrate our approach, we apply it to the study of blood flow through a stenosed or constricted artery segment, where blood flow follows a pulse function rather than a constant velocity. The average mean age at the exit over the duration of one pulse differs from the mean age when the velocity input is defined as a constant value using the average velocity of the pulse. For instance, the average mean age with the pulse input was 0.080 secs, compared to 0.053 secs for the constant velocity input (refer to Figure 1 for pulse function and constant velocity). This disparity in mean age is attributed to the acceleration and deceleration of the blood, particularly affecting recirculation/holdup zones past the stenosis (see Figure 2), resulting in prolonged path lengths and greater residence times or mean age. The delay in a holdup region is more pronounced for pulsing velocity compared to constant velocity.

By computing mean age for each time step in the transient solution, our approach significantly reduces computational effort compared to computing mean residence time from time-dependent tracer analysis.