A finite difference numerical analysis is presented concerning the pressure drop and heat transfer in the steady flow of a eutectic ethylene glycol aqueous solution through a tube cooled by radiation from a rectangular fin into an environment at zero degrees Rankine. The temperature dependence of all fluid properties is taken into account. The solution predicts an unusual functional dependence of pressure drop on the mass flow rate: that for lower and lower values of the mass flow rate, the pressure drop through a tube reaches a minimum value and then begins to increase. The application to parallel network flows of the solution of the transport equations for the flow in a single tube is illustrated with two examples. The most accurate numerical solution is compared with a simplified numerical solution which neglects inlet effects and with an approximate analytical solution. The convergence of the numerical method and agreement with experimental data are demonstrated. Finally the FORTRAN programs developed are listed with flow charts and annotations.