A mass-balanced, finite-difference solution to Münch's osmotically generated pressure-flow hypothesis is developed for the study of non-steady-state sucrose transport in the phloem tissue of plants. Major improvements over previous modeling efforts are the inclusion of wall elasticity, nonlinear functions of viscosity and solute potential, an enhanced calculation of sieve pore resistance, and the introduction of a slope-limiting total variation diminishing method for determining the concentration of sucrose at node boundaries. The numerical properties of the model are discussed, as is the history of the modeling of pressure-driven phloem transport. Idealized results are presented for a sharp, fast-moving concentration front, and the effect of changing sieve tube length on the transport of sucrose in both the steady-state and non-steady-state cases is examined. Most of the resistance to transport is found to be axial, rather than radial (via membrane transport), and most of the axial resistance is due to the sieve plates. Because of the sieve plates, sieve tube elasticity does not provide a significant enhancement to conductivity at high pressure, as previously suspected. The transit time of sucrose through a sieve tube is found to be inversely proportional to the square of the sieve tube's length; following that observation, it is suggested that 20 1-m-long sieve tubes could transport sucrose 20 times faster than a single 20 m sieve tube. Short sieve tubes would be highly sensitive to differentials between loading and unloading rate, and would require close cooperation with adjacent companion cells for proper function.