When atomic layer deposition (ALD) is performed on a porous material by using an organometallic precursor, minimum exposure time of the precursor for complete coverage becomes much longer since the ALD is limited by Knudsen diffusion in the pores. In the previous report by Min et al. (Ref. 23), shrinking core model (SCM) was proposed to predict the minimum exposure time of diethylzinc for ZnO ALD on a porous cylindrical alumina monolith. According to the SCM, the minimum exposure time of the precursor is influenced by volumetric density of adsorption sites, effective diffusion coefficient, precursor concentration in gas phase and size of the porous monolith. Here we modify the SCM in order to consider undesirable adsorption of byproduct molecules. $TiO_2$ ALD was performed on the cylindrical alumina monolith by using titanium tetrachloride ($TiCl_4$) and water. We observed that the byproduct (i.e., HCl) of $TiO_2$ ALD can chemically adsorb on adsorption sites, unlike the behavior of the byproduct (i.e., ethane) of ZnO ALD. Consequently, the minimum exposure time of $TiCl_4$ (~16 min) was significantly much shorter than that (~71 min) of DEZ. The predicted minimum exposure time by the modified SCM well agrees with the observed time. In addition, the modified SCM gives an effective diffusion coefficient of $TiCl_4$ of ${sim}1.78{ imes}10^{-2};cm^2/s$ in the porous alumina monolith.