To estimate the apex field-enhancement factor associated with an array of carbon nanotubes (CNTs) on a planar cathode surface, the model of the floating spheres between parallel anode and cathode plates was proposed. An approximate formula for the enhancement factor was derived showing that the intertube distance of a CNT array critically affects the field emission. When the intertube distance is less than the height of the tube, the enhancement factor decreases rapidly with decreasing distance in qualitative agreement with the numerical simulation. Considering the field-emission current density, the field emission can be optimized when the intertube distance is comparable with the tube height, in accordance with the results from the experiments. Finally, the influence of the anode-cathode distance on the enhancement factor was also discussed, showing that the anode-cathode distance has little effect on the field emission from a CNT array. Thus, we can reduce the theshold voltage to some extent by decreasing the anode-cathode distance.