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We present a detailed treatment of dissipative quantum transport in carbon-nanotube field-effect transistors (CNT-FETs) using the nonequilibrium Green's function formalism. The effect of phonon scattering on the device characteristics of CNT-FETs is explored using extensive numerical simulation. Both intra- and intervalley scattering mediated by acoustic (AP), optical (OP), and radial-breathing-mode (RBM) phonons are treated. Realistic phonon dispersion calculations are performed using force- constant methods, and electron-phonon coupling is determined through microscopic theory. Specific simulation results are presented for (16,0), (19,0), and (22,0) zigzag CNTFETs, which are in the experimentally useful diameter range. We find that the effect of phonon scattering on device performance has a distinct bias dependence. Up to moderate gate biases, the influence of high-energy OP scattering is suppressed, and the device current is reduced due to elastic backscattering by AP and low-energy RBM phonons. At large gate biases, the current degradation is mainly due to high-energy OP scattering. The influence of both AP and high-energy OP scattering is reduced for larger diameter tubes. The effect of RBM mode, however, is nearly independent of the diameter for the tubes studied here.