Reaction-diffusion-advection equation in binary tree networks and optimal size ratio

A simple reaction-diffusion-advection equation is proposed in a dichotomous tree network to discuss an optimal network. An optimal size ratio $r$ is evaluated by the principle of maximization of total reaction rate. In the case of reaction-limited conditions, the optimal ratio can be larger than ${(1/2)}^{1/3}$ for a fixed value of branching number $n$, which is consistent with observations in mammalian lungs. We find furthermore that there is an optimal branching number $n_c$ when the Péclet number is large. Under the doubly optimal conditions with respect to the size ratio and branching number, the optimal value of $r$ is close to ${(1/2)}^{1/3}$.

Figure 1

(a) Dichotomous branching ducts. (b) Relation of the flow rate $Q$ and contraction rate $r$ for $n=23,a=b=\eta =1$, and $V_n=23$. (c) Relation of the flow rate $Q$ and total branching number $n$ for $a=b=\eta =1$ and $V_n=23$.

Figure 2

Numerical results for $k=100,K=1,D=D^{\prime }=0.1,\eta =1$, and $C_0=1$. (a) Profiles of $C_i$ for $n=26$ at $V_n=23,r={(1/2)}^{1/3}$, and $P=0$. The dashed curve is calculated using Eq. (7). (b) Relation of $n$ and $S$ for the parameter set of (a). The dashed line denotes $S\propto 1/n^{1/3}$. (c) Profiles of $C_i$ for $n=10$ and 26 at $V_n=23,r=0.81$, and $P=0.1$. (d) Relationship between $n$ and $S$ for the parameter set of (c). The dashed line denotes $S\propto 1/n^{0.55}$.

Figure 3

Numerical results for $k=0.1,K=1,D=100,D^{\prime }=0.1,\eta =1,P=10$, and $C_0=10$. (a) Profiles of $C_i$ for $n=15$ and 26 at $V_n=23$ and $r=0.8$. (b) Relationship between $n$ and $S$ for the same parameter set. The solid line denotes $S=\pi a^2{\left(2r^2\right)}^{n-1}(k/K)$ at $r=0.8$. The dashed line denotes $S\propto exp\left\{\right(log2/3\left)n\right\}$. (c) Relationship between $S$ and $r$ for $n=23$. (d) Ratio $r_m$ obtained by the maximization as a function of $n$.

Figure 4

Numerical results for $k=0.1,K=1,D=D^{\prime }=0.1,\eta =1,P=10$, and $C_0=10$. (a) Profiles of $C_i$ for $n=23$ at $V_n={10}^6$ and $r=0.818$. (b) Relationship between $i$ and $\text{Pe}$ for the same parameter set. The dashed line is $\text{Pe}=500e^{-0.47i}$. (c) Relationship between the optimal size ratio $r$ and $n$ at $V_n={10}^6$ (rhombi) and $V_n={10}^4$ (plus). (d) Relationship between the maximum reaction rate $S_m$ and $n$ at $V_n={10}^6$ (rhombi) and $V_n={10}^4$ (plus).