修改体积导数计算逻辑，补充完善index.md

tests/cryo_tank/test_properties.py

@@ -6,6 +6,7 @@ sys.path.insert(0, "src")
 
 from cryo_tank.properties import (
     ln2_rho, ln2_h, ln2_u, ln2_T_from_u,
+    ln2_drho_dT_const_p, ln2_du_dT_const_p,
     he_u, he_h, he_cp, he_cv,
 )
 from cryo_tank.config import P_WORKING
@@ -32,6 +33,21 @@ class TestLN2Properties:
         T_recovered = ln2_T_from_u(u, P_WORKING)
         assert abs(T_recovered - T_orig) < 0.01
 
+    def test_ln2_density_derivative_matches_finite_difference(self):
+        T = 78.0
+        step = 1e-3
+        actual = ln2_drho_dT_const_p(T, P_WORKING)
+        expected = (
+            ln2_rho(T + step, P_WORKING)
+            - ln2_rho(T - step, P_WORKING)
+        ) / (2.0 * step)
+        assert actual < 0.0
+        assert abs(actual - expected) / abs(expected) < 1e-5
+
+    def test_ln2_internal_energy_derivative_is_positive(self):
+        du_dT = ln2_du_dT_const_p(78.0, P_WORKING)
+        assert du_dT > 0.0
+
 
 class TestHeliumProperties:
     """Helium properties."""

tests/cryo_tank/test_tank_model.py

@@ -12,10 +12,9 @@ from cryo_tank.config import (
 )
 
 
-def _make_tank():
+def _make_tank(**overrides):
     """Create a CryoTank with default config and MLI heat leak."""
-    heat_leak = MLIHeatLeak(A_total=A_TOTAL, q_mli=1.0)
-    return CryoTank(
+    kw = dict(
         V_total=V_TOTAL, H_tank=H_TANK,
         P_work=P_WORKING,
         T_init=T_INIT, ullage_fraction=ULLAGE_FRACTION,
@@ -23,8 +22,10 @@ def _make_tank():
         mdot_out_ln2=MDOT_OUT_LN2,
         T_in_he=T_IN_HE,
         h_conv=H_CONV_SURFACE, T_env=T_ENV,
-        heat_leak_model=heat_leak,
+        heat_leak_model=MLIHeatLeak(A_total=A_TOTAL, q_mli=1.0),
     )
+    kw.update(overrides)
+    return CryoTank(**kw)
 
 
 class TestGeometry:
@@ -75,3 +76,37 @@ class TestInitialState:
         info = tank.derive(y0)
         P_He = info['P_He']
         assert abs(P_He - P_WORKING) / P_WORKING < 1e-12
+
+
+class TestVolumeRate:
+
+    def test_ullage_volume_rate_includes_liquid_thermal_expansion(self):
+        tank = _make_tank(
+            mdot_in_ln2=0.0,
+            mdot_out_ln2=0.0,
+            h_conv=0.0,
+        )
+        y0 = tank.initial_state()
+        info = tank.derive(y0)
+
+        Q_liq_to_ull = 0.0
+        Q_leak = tank.heat_leak_model.compute(info['T_liq'], tank.T_env)
+        A_wet, A_dry = tank.wetted_areas(info['liquid_level'])
+        Q_leak_liq = Q_leak * A_wet / (A_wet + A_dry)
+        dU_liq_dt = tank._liquid_energy_rate(
+            info, Q_liq_to_ull, Q_leak_liq
+        )
+
+        dV_ull_dt = tank._ullage_volume_rate(
+            info, tank.dm_liq_dt, dU_liq_dt
+        )
+
+        y_next = y0.copy()
+        dt = 1.0
+        y_next[1] += dU_liq_dt * dt
+        finite_difference = (
+            tank.derive(y_next)['V_ull'] - info['V_ull']
+        ) / dt
+
+        assert dV_ull_dt < 0.0
+        assert abs(dV_ull_dt - finite_difference) / abs(finite_difference) < 1e-5