How does one pump that adds only about 200 mOsm at each level build a 1200 mOsm gradient, and how does ADH decide what the urine looks like?
Each button solves the steady state for a series of settings (your other settings stay as they are) and lists the result.
The science
Single effect and multiplication. In the thick ascending limb (TAL) the Na+-K+-2Cl- cotransporter and the Na+/K+ pump move NaCl out of a water-tight tube, so the lumen can be about ε = 200 mOsm more dilute than the interstitium at the same depth [typical textbook value]. The descending limb (DL) is water-permeable and solute-tight, so water leaves until it matches the interstitium. Because the two limbs are connected and fluid flows through them, each level adds the single effect on top of what the level above delivered: the small local ε is multiplied along the loop into a much larger difference between cortex and papilla.
Model equations (osmolality c in mOsm/kg, depth x in mm, nominal flow F): F dcd/dx = kd(ci − cd) (descending limb), −F dca/dx = −ka(ca − ci + ε) (ascending limb, flow upward), Fcd dcc/dx = kc(ci − cc) with kc ∝ ADH1.5, and the interstitium (with vasa recta) from ka(ca − ci + ε) = kd(ci − cd) + w kc(ci − cc) + kv(ci − c0); c0 = 300 is plasma/cortex. The tubes respond within seconds; the interstitium relaxes with a model time constant of 2 min, which in the coupled system means roughly one to two hours to build the gradient (Heun time stepping). Exact exponential cell steps are used along each tube, 100 depth nodes, interstitium relaxed in time.
Closed form (used as a check). With constant coefficients, no collecting duct and a uniform ε, deviations from the base state obey z′ = M z with a 2×2 matrix M that has one growing and one decaying eigenvalue λ±. The boundary conditions cd(0) = c0 and ca(L) = cd(L) give cd(L) − c0 = ε M12(eλ+L − eλ−L) / [eλ+L(M22 − λ− − M12) − eλ−L(M22 − λ+ − M12)]. The gain (ctip − c0)/ε is the multiplication factor; it depends on loop length, flow, pump strength and vasa recta washout, not on ε itself.
Flow, washout and blood flow. Faster tubular flow carries the gradient away (osmotic diuresis; the exchange lengths are F/k). More medullary blood flow (kv) removes solute from the interstitium and lowers the tip. A longer loop (desert rodents) reaches higher osmolality. Loop diuretics block the cotransporter, which cuts ε and the pump rate, so both gradient and concentrating ability collapse.
ADH and urea. ADH inserts AQP2 water channels in the collecting duct (kc rises) so water follows the gradient and urine osmolality approaches the papilla value; without ADH the duct stays water-tight and urine is dilute. In the inner medulla the thin ascending limb is passive, so its single effect is modelled as εIM = 420 · urea · (0.15 + 0.85 ADH) mOsm: urea recycled by UT-A transporters (which need ADH) provides the driving force. That coefficient is an illustrative closure, not a measured constant.
Urine volume. Osmole conservation in the solute-tight descending limb gives the flow leaving the loop Fa = F0 c0/cd(L) (F0 = 20 nL/min per nephron [typical]); the collecting duct then gives V = (1 − fd) Fa cstart/curine with 78% of the delivered osmoles reabsorbed in the distal tubule and cortical duct (illustrative). Whole-kidney flow assumes 2 million nephrons.
Honest simplifications. One solute (osmolality) and one long-looped nephron; a single interstitial compartment; constant mean flow in the exchange rates, while volume bookkeeping uses osmole conservation; no cortical nephrons, no axial diffusion, no real vasa recta countercurrent exchanger (it is a first-order sink with extra washout near the cortex); no K+/NH4+, no Na+ hormonal control. Rate constants were tuned so the default gives a tip of about 960 mOsm and urine of about 650 mOsm; they are model values, not measurements. Typical physiological urine osmolality is about 50 to 1200 mOsm/kg [typical; check against a physiology text]. Educational, not diagnostic, not medical advice.