Advection Dominated Accretion Flows. A Toy Disk Model
  A toy model of a disk undergoing steady state accretion onto a black hole is presented. The disk is in a hydrostatic equilibrium for all radii r > r_{in}, with the inner disk radius located between the marginally stable and marginally bound orbits: r_{ms} > r_{in} > r_{mb}. Matter flows from the disk through a narrow cusp at r_{ms} and falls freely into the black hole, carrying with it no thermal energy. At radii larger than r_{out} the disk is assumed to radiate away all locally generated heat, and therefore the disk is geometrically thin for r > r_{out}. We assume that no heat generated in the inner disk, with r_{out} > r > r_{in} can be radiated away, i.e. the disk is 100% advective, and it becomes geometrically thick in this range of radii. All enthalpy of the thick disk is used up to press the inner disk radius towards the marginally bound orbit, and to lower the efficiency of conversion of accreted mass into radiation generated only for r > r_{out}, by assumption.   Conservation laws of mass, angular momentum and energy make it possible to calculate the inner thick disk radius r_{in} for any specified value of its outer radius r_{out}. As the nature of disk viscosity is not known there is some freedom in choosing the shape of the thick disk, subject to several general conditions, which include the hydrostatic equilibrium everywhere for r > r_{in}. The main purpose of this toy model is to emphasize the effect the disk thickness has on lowering the energetic efficiency of a black hole accretion. 
Bohdan Paczynski