§5 slice 5: (68) appliances_monthly_w — Appendix L13/L14/L16a

E_A = 207.8 × (TFA × N)^0.4714 (L13) chained through monthly factor
1 + 0.157 × cos(2π × (m - 1.78) / 12) (L14) then watts via × 1000 /
(24 × n_m) (L16a). Column A typical-gain form — 1.0× conversion. L16's
0.67× reduced form deferred (new-build DPER/TPER use).

Verified against Elmhurst U985-0001-000490 (68)m row to ≤5e-2 W
(display rounding from the (TFA × N)^0.4714 term).

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

packages/domain/src/domain/sap/worksheet/internal_gains.py

@@ -30,6 +30,12 @@ from dataclasses import dataclass
 from math import cos, exp, pi
 from typing import Final, Optional
 
+_DAYS_PER_YEAR: Final[float] = 365.0
+_APPLIANCES_E_A_COEFF: Final[float] = 207.8
+_APPLIANCES_E_A_EXPONENT: Final[float] = 0.4714
+_APPLIANCES_MONTHLY_AMPLITUDE: Final[float] = 0.157
+_APPLIANCES_MONTHLY_PHASE: Final[float] = 1.78
+
 
 _MONTHS_IN_YEAR: Final[int] = 12
 _DAYS_PER_MONTH: Final[tuple[int, ...]] = (31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31)
@@ -73,6 +79,38 @@ def cooking_monthly_w(*, n_occupants: float) -> tuple[float, ...]:
     return tuple(gain for _ in range(_MONTHS_IN_YEAR))
 
 
+def appliances_monthly_w(
+    *,
+    total_floor_area_m2: float,
+    n_occupants: float,
+) -> tuple[float, ...]:
+    """SAP 10.2 §5 line (68) — appliance gains in watts per month.
+
+    Appendix L equations L13, L14, L16a (Column A, typical gains):
+        E_A    = 207.8 × (TFA × N)^0.4714                                  [kWh/yr]
+        E_A,m  = E_A × [1 + 0.157 × cos(2π × (m - 1.78) / 12)] × n_m / 365 [kWh/mo]
+        G_A,m  = E_A,m × 1000 / (24 × n_m)                                 [W]
+
+    The cosine peaks ~end-of-January (m=1.78) and troughs ~end-of-July.
+    All electrical appliance energy stays as internal heat — full 1.0×
+    conversion. Column B's L16 (0.67×) reduced form is for new-build
+    DPER/TPER only.
+    """
+    e_a_annual = (
+        _APPLIANCES_E_A_COEFF
+        * (total_floor_area_m2 * n_occupants) ** _APPLIANCES_E_A_EXPONENT
+    )
+    monthly: list[float] = []
+    for m_idx, days in enumerate(_DAYS_PER_MONTH):
+        m = m_idx + 1
+        factor = 1.0 + _APPLIANCES_MONTHLY_AMPLITUDE * cos(
+            2.0 * pi * (m - _APPLIANCES_MONTHLY_PHASE) / _MONTHS_IN_YEAR
+        )
+        e_a_m_kwh = e_a_annual * factor * days / _DAYS_PER_YEAR
+        monthly.append(e_a_m_kwh * _KWH_TO_WH / (_HOURS_PER_DAY * days))
+    return tuple(monthly)
+
+
 def water_heating_gains_monthly_w(
     *,
     heat_gains_from_water_heating_monthly_kwh: tuple[float, ...],

packages/domain/src/domain/sap/worksheet/tests/test_internal_gains.py

@@ -13,6 +13,7 @@ Table 5a + Appendix L (lighting/appliances/cooking) + Appendix J Table 1b
 import pytest
 
 from domain.sap.worksheet.internal_gains import (
+    appliances_monthly_w,
     cooking_monthly_w,
     losses_monthly_w,
     metabolic_monthly_w,
@@ -121,3 +122,36 @@ def test_water_heating_gains_bridge_converts_kwh_per_month_to_watts() -> None:
     assert len(monthly) == 12
     for m, (actual, exp) in enumerate(zip(monthly, expected_w)):
         assert actual == pytest.approx(exp, abs=1e-3), f"month {m+1}"
+
+
+def test_appliances_gains_match_appendix_l13_l14_l16a_for_000490() -> None:
+    """SAP 10.2 Appendix L equations L13/L14/L16a:
+        E_A    = 207.8 × (TFA × N)^0.4714                                   [kWh/yr]
+        E_A,m  = E_A × [1 + 0.157 × cos(2π × (m - 1.78) / 12)] × n_m / 365  [kWh/mo]
+        G_A,m  = E_A,m × 1000 / (24 × n_m)                                  [W]
+
+    Verified against U985-0001-000490 worksheet (68)m row. For
+    TFA=66.06, N=2.1468:
+        E_A = 207.8 × 141.84^0.4714 ≈ 2147.8 kWh/yr
+        Jan factor = 1 + 0.157 × cos(2π × -0.78/12) ≈ 1.1441
+        E_A,Jan ≈ 208.66 kWh → G_A,Jan ≈ 280.46 W
+    matches worksheet 280.4965 to ≈0.04 W (display rounding from E_A^0.4714).
+
+    Table 5 Column A applies (rating + cooling); Column B's 0.67×
+    reduced-gain form (L16) is deferred to a future new-build slice.
+    """
+    # Arrange
+    tfa = 66.06
+    n = 2.1468
+    expected_w = (
+        280.4965, 283.4071, 276.0723, 260.4574, 240.7463, 222.2207,
+        209.8445, 206.9338, 214.2686, 229.8835, 249.5946, 268.1202,
+    )
+
+    # Act
+    monthly = appliances_monthly_w(total_floor_area_m2=tfa, n_occupants=n)
+
+    # Assert
+    assert len(monthly) == 12
+    for m, (actual, exp) in enumerate(zip(monthly, expected_w)):
+        assert actual == pytest.approx(exp, abs=5e-2), f"month {m+1}"