= Solution
The <mass> and surface <area> of the spherical droplet are $m=(4\pi/3)\rho r^3$ and $A=4\pi r^2$. Constant surface <mass accretion rate> $\beta$ means
$$
\dot m=4\pi r^2\beta=4\pi\rho r^2\dot r.
$$
Therefore $\dot r=\beta/\rho=\mu$, and the initial radius gives
$$
\boxed{r(t)=r_0+\mu t,\qquad\mu=\frac\beta\rho.}
$$
Here $[\beta]=\mathrm{kg\,m^{-2}\,s^{-1}}$, $[\rho]=\mathrm{kg\,m^{-3}}$, and $[\mu]=\mathrm{m\,s^{-1}}$. This <surface-fed growth of an accreting sphere> law assumes constant droplet <mass density>.
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