The definition of Reynolds number (Re) in a Taylor-Couette flow for a shear-thinning fluid is discussed in this paper. Since the shear-thinning property causes spatial distribution of fluid viscosity in a Taylor-Couette flow reactor (TCFR), a method to determine Re by using a numerical simulation is suggested. The effective viscosity (eta (eff)) in Re was the average viscosity using a weight of dissipation function eta(eff) = Sigma(N)(i=1) (gamma)over dot(i)eta(i)Delta V-i/Sigma(N)(i=1) (gamma)over dot(i)(2)Delta V-i, where N is the total mesh number, eta (i) (Pa center dot s) is the local viscosity, (1/s) is the local shear-rate, and Delta V (i) (m(3)) is the local volume for each cell. The critical Reynolds number, Re (cr), at which Taylor vortices start to appear, was almost the same value with the Re (cr) obtained by a linear stability analysis for Newtonian fluids. Consequently, Re based on eta (eff) could be applicable to predict the occurrence of Taylor vortices for a shear-thinning fluid. In order to understand the relation between the rotational speed of the inner cylinder and the effective shear rate that resulted in eta (eff), a correlation equation was constructed. Furthermore, the critical condition at which Taylor vortices appear was successfully predicted without further numerical simulation.