Exponential cloud collector growth ($R=R_0{e}^{{\alpha }_{R}t}$)

If a spherical collector falls with $u_R=k_{R}R$ through constant liquid-water content $L$ and efficiency $E$, neglecting condensation and small-droplet fall speed, then $\dot{R}={\alpha }_{R}R$ with ${\alpha }_R=k_{R}LE/(4{\rho }_l)$. A uniform updraft gives $h(t)=Ut-(k_R{R}_0/{\alpha }_R)(e^{{\alpha }_{R}t}-1)$. The formula is valid only while the assumed cloud conditions hold.