The problem of determining shell side Nusselt numbers for a countercurrent, shell-and-tube configuration is examined in detail for square and hexagonal arrays of tubes when the shell side flow is laminar and parallel to the tubes. A multipole expansion method is employed to determine fluid velocity and temperature field for the fluid on the shell side. The numerical results for the shell side Nusselt numbers are compared with those by a cell theory and an asymptotic analysis. The cell theory agrees well with the numerical results at small area fractions and gives better estimates for hexagonal arrays. The results for the hexagonal arrays are in agreement with those of Sparrow et al. [1961]. The asymptotic analysis shows better agreement with the numerical results for wider range of area fractions of tubes in square and hexagonal arrays. In addition, we determine shell side Nusselt number as a function of the ratio of thermal conductivities of the fluids on the tube and shell side. Finally, we present formulas for determining Nusselt numbers for the periodic arrays.