§4 slice 6: lines (45)m energy content + (46)m distribution loss

(45)m = 4.18 × V_d,m × n_m × (52 − Tcold[m]) / 3600    [kWh/month]
                                          Appendix J equation J14
  (46)m = 0.15 × (45)m                    spec §4 step 7 (normal systems)
        = 0                                (instantaneous at point of use,
                                            hot water codes 907 / 909)

4.18 J/(g·K) is the specific heat of water; / 3600 converts to kWh. The
J14 transform converts daily L of hot water at delivery temperature into
the monthly sensible-heat requirement.

Both Elmhurst non-RR fixtures use a combi boiler from a central system
(neither 907 nor 909), so distribution loss is the full 15 % of (45)m.
Lodged LINE_45_M and LINE_46_M arrays on both fixtures for forward use.

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

packages/domain/src/domain/sap/worksheet/tests/_elmhurst_worksheet_000474.py

@@ -189,3 +189,11 @@ LINE_44_M_DAILY_HW_USAGE_L: tuple[float, ...] = (
     110.1180, 107.7045, 104.8007, 100.4697, 96.9221, 93.1204,
     91.7218, 94.6138, 97.6550, 101.6380, 106.0332, 109.8446,
 )
+LINE_45_M_HW_ENERGY_CONTENT_KWH: tuple[float, ...] = (
+    174.4000, 153.3698, 161.0747, 137.5380, 130.4758, 114.5024,
+    110.9296, 117.1517, 120.4183, 137.9218, 151.0638, 171.9901,
+)
+LINE_46_M_DISTRIBUTION_LOSS_KWH: tuple[float, ...] = (
+    26.1600, 23.0055, 24.1612, 20.6307, 19.5714, 17.1754,
+    16.6394, 17.5728, 18.0628, 20.6883, 22.6596, 25.7985,
+)

packages/domain/src/domain/sap/worksheet/tests/_elmhurst_worksheet_000490.py

@@ -177,3 +177,11 @@ LINE_44_M_DAILY_HW_USAGE_L: tuple[float, ...] = (
     118.6172, 116.0171, 112.8890, 108.2238, 104.4023, 100.3071,
     98.8008, 101.9162, 105.1922, 109.4827, 114.2171, 118.3228,
 )
+LINE_45_M_HW_ENERGY_CONTENT_KWH: tuple[float, ...] = (
+    187.8607, 165.2069, 173.5062, 148.1530, 140.5456, 123.3393,
+    119.4910, 126.1935, 129.7125, 148.5670, 162.7232, 185.2648,
+)
+LINE_46_M_DISTRIBUTION_LOSS_KWH: tuple[float, ...] = (
+    28.1791, 24.7810, 26.0259, 22.2229, 21.0818, 18.5009,
+    17.9237, 18.9290, 19.4569, 22.2851, 24.4085, 27.7897,
+)

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

@@ -20,6 +20,8 @@ from domain.sap.worksheet.water_heating import (
     annual_average_hot_water_l_per_day,
     annual_average_hot_water_other_uses_l_per_day,
     assumed_occupancy,
+    distribution_loss_monthly_kwh,
+    energy_content_of_hot_water_monthly_kwh,
     hot_water_baths_monthly_l_per_day,
     hot_water_mixer_showers_monthly_l_per_day,
     hot_water_other_uses_monthly_l_per_day,
@@ -400,6 +402,61 @@ def test_annual_average_hot_water_matches_elmhurst_line_43(fixture) -> None:  #
     )
 
 
+@pytest.mark.parametrize("fixture", (_w000474, _w000490), ids=("000474", "000490"))
+def test_energy_content_of_hot_water_matches_elmhurst_line_45(fixture) -> None:  # type: ignore[no-untyped-def]
+    """SAP10.2 §4 line (45)m via Appendix J equation J14:
+        (45)m = 4.18 × V_d,m × n_m × (52 − Tcold[m]) / 3600   [kWh/month]
+
+    4.18 J/g/K is the specific heat of water, dividing by 3600 converts
+    to kWh. Both fixtures heat hot water to 52°C from mains-temperature
+    cold water (Table J1 mains column).
+    """
+    # Arrange / Act
+    monthly_kwh = energy_content_of_hot_water_monthly_kwh(
+        monthly_hot_water_l_per_day=fixture.LINE_44_M_DAILY_HW_USAGE_L,
+        cold_water_temps_c=TABLE_J1_TCOLD_FROM_MAINS_C,
+    )
+
+    # Assert
+    for m, (actual, exp) in enumerate(zip(monthly_kwh, fixture.LINE_45_M_HW_ENERGY_CONTENT_KWH)):
+        assert actual == pytest.approx(exp, abs=1e-2), f"month {m+1}"
+
+
+@pytest.mark.parametrize("fixture", (_w000474, _w000490), ids=("000474", "000490"))
+def test_distribution_loss_matches_elmhurst_line_46_for_non_instantaneous(
+    fixture,  # type: ignore[no-untyped-def]
+) -> None:
+    """SAP10.2 §4 line (46)m: for non-instantaneous systems
+        (46)m = 0.15 × (45)m
+
+    Both Elmhurst fixtures run a combi boiler with hot water from a
+    main central system (not codes 907/909 instantaneous), so distribution
+    loss is 15% of the energy content."""
+    # Arrange / Act
+    monthly_loss = distribution_loss_monthly_kwh(
+        monthly_energy_content_kwh=fixture.LINE_45_M_HW_ENERGY_CONTENT_KWH,
+        is_instantaneous_at_point_of_use=False,
+    )
+
+    # Assert
+    for m, (actual, exp) in enumerate(zip(monthly_loss, fixture.LINE_46_M_DISTRIBUTION_LOSS_KWH)):
+        assert actual == pytest.approx(exp, abs=1e-3), f"month {m+1}"
+
+
+def test_distribution_loss_zero_for_instantaneous_point_of_use_water_heating() -> None:
+    """SAP10.2 §4: hot water codes 907 and 909 are instantaneous at the
+    point of use — no distribution pipework, so (46)m is zero for every
+    month regardless of (45)m."""
+    # Arrange / Act
+    loss = distribution_loss_monthly_kwh(
+        monthly_energy_content_kwh=(100.0,) * 12,
+        is_instantaneous_at_point_of_use=True,
+    )
+
+    # Assert
+    assert all(v == pytest.approx(0.0, abs=1e-9) for v in loss)
+
+
 def test_assumed_occupancy_floor_at_n_eq_1_for_small_dwellings() -> None:
     """Appendix J piecewise definition: TFA ≤ 13.9 m² → N=1 exactly. A
     tiny studio flat at the boundary is the most common trigger."""

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

@@ -190,6 +190,45 @@ def total_hot_water_monthly_l_per_day(
     return tuple(s + b + o for s, b, o in zip(showers, baths, other_uses))
 
 
+def energy_content_of_hot_water_monthly_kwh(
+    *,
+    monthly_hot_water_l_per_day: tuple[float, ...],
+    cold_water_temps_c: tuple[float, ...],
+) -> tuple[float, ...]:
+    """SAP 10.2 §4 line (45)m via Appendix J equation J14:
+        (45)m = 4.18 × V_d,m × n_m × (52 − Tcold[m]) / 3600    [kWh/month]
+
+    Sensible heat to raise the monthly hot water volume from Tcold[m] to
+    the 52 °C delivery temperature. 4.18 J/(g·K) is the specific heat of
+    water; dividing by 3600 converts J/g to Wh/g (= kWh/kg, since 1 L of
+    water ≈ 1 kg).
+    """
+    return tuple(
+        4.18 * vd * n * (_HOT_DELIVERY_TEMPERATURE_C - tcold) / 3600.0
+        for vd, n, tcold in zip(
+            monthly_hot_water_l_per_day, _DAYS_IN_MONTH, cold_water_temps_c
+        )
+    )
+
+
+def distribution_loss_monthly_kwh(
+    *,
+    monthly_energy_content_kwh: tuple[float, ...],
+    is_instantaneous_at_point_of_use: bool,
+) -> tuple[float, ...]:
+    """SAP 10.2 §4 line (46)m.
+
+    Distribution loss is a flat 15% of (45)m unless the water heating is
+    instantaneous at the point of use (Table 4a hot water codes 907 and
+    909) — those have no distribution pipework so the loss is zero. Heat
+    networks still incur the 15% loss whether or not a hot water cylinder
+    is present (spec §4 step 7).
+    """
+    if is_instantaneous_at_point_of_use:
+        return tuple(0.0 for _ in range(12))
+    return tuple(0.15 * e for e in monthly_energy_content_kwh)
+
+
 def _days_weighted_average(monthly: tuple[float, ...]) -> float:
     """Σ value[m] × n_m / 365 — used by Appendix J equations J4 and J9."""
     return sum(v * d for v, d in zip(monthly, _DAYS_IN_MONTH)) / _DAYS_IN_YEAR