  Dense astrophysical environments like globular clusters and galactic nuclei
can host hyperbolic encounters of black holes which can lead to
gravitational-wave driven capture. There are several astrophysical models which
predict a fraction of binary black hole mergers to come from these
radiation-driven capture scenarios. In this paper we present the sensitivity of
a search towards gravitational-wave driven capture events for O3, the third
observing run of LIGO and Virgo. We use capture waveforms produced by numerical
relativity simulations covering four different mass ratios and at least two
different values of initial angular momentum per mass ratio. We employed the
most generic search for short-duration transients in O3 to evaluate the search
sensitivity in this parameter space for a wide range in total mass in terms of
visible spacetime volume. From the visible spacetime volume we determine for
the first time the merger rate upper limit of such systems. The most stringent
estimate of rate upper limits at 90\% confidence is
$0.2~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$ for an equal mass $200~M_\odot$
binary. Furthermore, in recent studies the event GW190521 has been suggested to
be a capture event. With this interpretation of GW190521, we find the merger
rate of similar events to be $0.47~\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$.
