General computable functions can be described either functionally (in terms of closure of the coordinate functions, constant functions, composition, primitive recursion, and $\mu$-recursion), or in terms of a Turing machine.

I have only seen primitive recursion defined in the functional language, i.e. functions obtained by coordinates, constants, composition, primitive recursion.

Is there a similar type of machine model for primitive recursion?

I am aware of some (pedagogical) programming languages, such as Hofstadter's BLOOP, that are PR-complete, but this approach doesn't really look like a Turing machine to me.

machine model? Such formulations are beneficial mostly for philosophical reasons. For example, the advantage of the Turing-machine formulation of the Entscheidungsproblem is that it focuses all of the "infiniteness" in one place: the tape; everything else about the machine is finite. This laid to rest a lot of philosophical uncertainties in the 1930s about the role of the infinite ${\mathbb N}$ in $\mu$-recursion and insertion of characters into unboundedly-long $\lambda$-terms. I'm not sure there's any reason to prefer a machine model now that... – Adam Aug 17 '11 at 16:33