A mathematical model based on the continuum mechanic concept has been developed to describe the profile of solid particles in a blast furnace with respect to the in-furnace conditions and characteristics, e.g., the shape and size of the deadman. The Navier-Stokes differential equation for multi-phase multi-dimensional space has been used to describe the behavior of existing phases. The equation has been modified to make it possible to describe the dual nature of the solid phase in the system by applying the concept of the solid surface stress to characterize the inter-granular surface interactions between particles. Since different phases co-exist in a blast furnace, the volume fraction plays an important role in a blast furnace. Therefore, the influence of three different packing densities (0.68, 0.71, and 0.74, respectively) on the profile of the flow in the upper part of a furnace down to the tuyeres level has been studied. It is shown that an increase in the volume fraction of the solid phase lead to a decrease in magnitude of the velocity. The decrease in the magnitude of the velocity due to an increase in the solid volume fraction will increase the resident time of the particles inside a blast furnace. In addition, it is shown that the solid phase velocity magnitude decreases from the throat to the belly of the furnace for the studied conditions. However, after belly the velocity magnitude increases. It is shown that the particle velocity increases in the upper part of the furnace up to the tuyeres level with a decrease in the packing density from 0.74 to 0.71 and 0.68 while all other parameters were keptconstant. The shrinkage in the size of the particles is not applied to the model. However it is plausible to say that an increase in the packing density of the bed decreases the velocity magnitude. This, in turn, increases the resident time of the particles.