Equations that predict the dependence of regenerator effectiveness on heat conduction in the matrix both parallel to and perpendicular to the fluid flow are derived from numerical simulations. The equations developed use Biot numbers parallel to the direction of fluid flow (Bi(x)) and perpendicular to the fluid flow direction (Bi(y)), the ratio of heat capacity rates (C*), the heat capacity rate ratio (C(r)*) and the overall number of transfer units (NTU(o)) to characterize the regenerator performance. Comparison of numerical predictions with those obtained using the equations developed in this paper show excellent agreement. These equations enable designers to accurately account or two-dimensional conduction effects when regenerators are designed.