We model the infrared to submillimeter spectral energy distribution of 11 nearby galaxies of the KINGFISH sample using Spitzer and Herschel data and compare model extrapolations at 870um (using different fitting techniques) with LABOCA 870um observations. We investigate how the differences between predictions and observations vary with model assumptions or environment. At global scales, we find that modified blackbody models using realistic cold emissivity indices (beta_c=2 or 1.5) are able to reproduce the 870um observed emission within the uncertainties for most of the sample. Low values (beta_c<1.3) would be required in NGC0337, NGC1512 and NGC7793. At local scales, we observe a systematic 870um excess when using beta_=2.0. The beta_c=1.5 or the Draine and Li (2007) models can reconcile predictions with observations in part of the disks. Some of the remaining excesses occur towards the centres and can be partly or fully accounted for by non-dust contributions such as CO(3-2) or, to a lesser extent, free-free or synchrotron emission. In three non-barred galaxies, the remaining excesses rather occur in the disk outskirts. This could be a sign of a flattening of the submm slope (and decrease of the effective emissivity index) with radius in these objects.
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