Effects of clumping on temperature I: externally heated clouds
  We present a study of radiative transfer in dusty, clumpy star-forming regions. A series of self-consistent, 3-D, continuum radiative transfer models are constructed for a grid of models parameterized by central luminosity, filling factor, clump radius, and face-averaged optical depth. The temperature distribution within the clouds is studied as a function of this parameterization. Among our results, we find that: (a) the effective optical depth is smaller in clumpy regions than in equivalent homogeneous regions; (b) penetration of radiation is drive by the fraction of open sky (FOS) -- which measures the fraction of solid angle which is devoid of clumps; (c) FOS increases as clump radius increases and filling factor decreases; (d) for FOS > 0.6-0.8 the sky is sufficiently open that the temperature is relatively insensitive to FOS; (e) the physical process by which radiation penetrates is streaming between clumps; (f) filling factor dominates the temperature distribution for large optical depths, and at small clump radii for small optical depths; (g) at lower optical depths, the temperature distribution is most sensitive to filling factors of 1-10 per cent, in accordance with many observations; (h) direct shadowing can be important approximately one clump radius behind a clump. 
S. D. Doty, R. A. Metzler, and M. L. Palotti