  (Abridged) Protoplanetary disks are vital objects in star and planet
formation, possessing all the material which may form a planetary system
orbiting the new star. We investigate the synthesis of complex organic
molecules (COMs) in disks to constrain the achievable chemical complexity and
predict species and transitions which may be observable with ALMA. We have
coupled a 2D model of a protoplanetary disk around a T Tauri star with a
gas-grain chemical network including COMs. We compare compare synthesised line
intensities and calculated column densities with observations and determine
those COMs which may be observable in future. COMs are efficiently formed in
the disk midplane via grain-surface chemical reactions, reaching peak
grain-surface fractional abundances 1e-6 - 1e-4 that of the H nuclei number
density. COMs formed on grain surfaces are returned to the gas phase via
non-thermal desorption; however, gas-phase species reach lower fractional
abundances than their grain-surface equivalents, 1e-12 - 1e-7. Including the
irradiation of grain mantle material helps build further complexity in the ice
through the replenishment of grain-surface radicals which take part in further
grain-surface reactions. There is reasonable agreement with several line
transitions of H2CO observed towards several T Tauri star-disk systems. The
synthesised line intensities for CH3OH are consistent with upper limits
determined towards all sources. Our models suggest CH3OH should be readily
observable in nearby protoplanetary disks with ALMA; however, detection of more
complex species may prove challenging. Our grain-surface abundances are
consistent with those derived from cometary comae observations providing
additional evidence for the hypothesis that comets (and other planetesimals)
formed via the coagulation of icy grains in the Sun's natal disk.
