  Coagulation in the dense interstellar medium (ISM) is an important process
that determines the size of the largest grains. We use submillimetre galaxies
(SMGs) as laboratories of grain coagulation, since some of them host the
densest ISM on a galactic scale among various populations of galaxies known. We
examine how large the grains can be in such dense environments based on the
mean ISM density estimated from the observed typical dust mass density in SMGs.
We also consider local density enhancement based on a model of supersonic
turbulence, which is expected from strong stellar feedback. In the unlimited
coagulation model, in which we do not impose any coagulation threshold
velocity, grains as large as $\sim 30~\mu$m can form under the observationally
estimated mean gas density if the Mach number of turbulence is
$\mathcal{M}\gtrsim 3$. We exclude this possibility since the observed
emissivity index $\beta\simeq 2$ in the far infrared (FIR) indicates that such
large grains cannot actively form in SMGs. This means that coagulation does not
proceed in an unlimited way: 30-$\mu$m grains should have velocities larger
than the coagulation threshold. If we use a coagulation threshold (upper limit)
grain velocity ($\sim 0.08$ km s$^{-1}$) taken from a theoretical study, grains
likely grow only up to $\mu$m size, which is small enough not to affect the FIR
emissivity index. The above results indicate that SMGs can be used to constrain
the physical processes relevant to coagulation.
