A model of the rat nephron (Weinstein. Am J Physiol Renal Physiol 308: F1098–F1118, 2015) has been extended with addition of medullary vasculature. Blood vessels contain solutes from the nephron model, plus additional species from the model of Atherton et al. (Am J Physiol Renal Fluid Electrolyte Physiol 247: F61–F72, 1984), representing hemoglobin buffering. In contrast to prior models of the urine-concentrating mechanism, reflection coefficients for DVR are near zero. Model unknowns are initial proximal tubule pressures and flows, connecting tubule pressure, and medullary interstitial pressures and concentrations. The model predicts outer medullary (OM) interstitial gradients for Na⁺, K⁺, CO₂, and , such that at OM-IM junction, the respective concentrations relative to plasma are 1.2, 3.0, 2.7, and 8.0; within IM, there is high urea and low , with concentration ratios of 11 and 0.5 near the papillary tip. Quantitative similarities are noted between K⁺ and urea handling (medullary delivery and permeabilities). The model K⁺ gradient is physiologic, and the urea gradient is steeper due to restriction of urea permeability to distal collecting duct. Nevertheless, the predicted urea gradient is less than expected, suggesting reconsideration of proposals of an unrecognized reabsorptive urea flux. When plasma K⁺ is increased from 5.0 to 5.5 mM, Na⁺ and K⁺ excretion increase 2.3- and 1.3-fold, respectively. The natriuresis derives from a 3.3% decrease in proximal Na⁺ reabsorption and occurs despite delivery-driven increases in Na⁺ reabsorption in distal segments; kaliuresis derives from a 30% increase in connecting tubule Na⁺ delivery. Thus this model favors the importance of proximal over distal events in K⁺-induced diuresis.