A novel model for the blood system is postulated focusing on the flow rate and pressure distribution inside the arterioles and venules of the pulmonary acinus. Based upon physiological data it is devoid of any ad hoc constants. The model comprises nine generations of arterioles, venules, and capillaries in the acinus, the gas exchange unit of the lung. Blood is assumed incompressible and Newtonian and the blood vessels are assumed inextensible. Unlike previous models of the blood system, the venules and arterioles open up to the capillary network in numerous locations along each generation. The large number of interconnected capillaries is perceived as a porous medium in which the flow is macroscopically unidirectional from arterioles to venules openings. In addition, the large number of capillaries extending from each arteriole and venule allows introduction of a continuum theory and formulation of a novel system of ordinary, nonlinear differential equations which governs the blood flow and pressure fields along the arterioles, venules, and capillaries. The solution of the differential equations is semianalytical and requires the inversion of three diagonal, 9 × 9 matrices only. The results for the total flow rate of blood through the acinus are within the ballpark of physiological observations despite the simplifying assumptions used in our model. The results also manifest that the contribution of the nonlinear convection term of the Navier-Stokes equations has little effect (less than 2%) on the total blood flow entering/leaving the acinus despite the fact that the Reynolds number is not much smaller than unity at the proximal generations. The model makes it possible to examine some pathological cases. Here, centri-acinar and distal emphysema were investigated yielding a reduction in inlet blood flow rate.