Heat Load Calculator

Fundamental Heat Transfer Principles

Heat transfer in buildings is governed by the fundamental laws of thermodynamics. Understanding these mechanisms is essential for accurate heat load calculations and effective HVAC system design.

**1. CONDUCTION - Heat Transfer Through Solids**

Conduction is the transfer of thermal energy through solid materials via molecular vibration and electron movement. In building envelopes, heat conducts from the warm interior through walls, roofs, floors, windows, and doors to the cold exterior.

**Fourier's Law of Heat Conduction:** Q = -k · A · (dT)/(dx)

For steady-state conduction through a uniform layer: Q = (k · A · Δ T)/(d)

Where: - **Q** = Heat transfer rate (W) - **k** = Thermal conductivity (W/m·K) - **A** = Area perpendicular to heat flow (m²) - **d** = Thickness of material (m) - **ΔT** = Temperature difference across material (K)

**Thermal Resistance (R-value):** R = (d)/(k) \text (m²·K/W)

For multi-layer assemblies, R-values add in series: R_total = R_inside,air + R_layer1 + R_layer2 + ... + R_outside,air

**Overall Heat Transfer Coefficient (U-value):** U = (1)/(R_total) \text (W/m²·K)

**Practical Example - Typical Wall Assembly:**

Consider a wall with: - Interior surface resistance: R = 0.13 m²·K/W - Gypsum board (12mm, k=0.17): R = 0.012/0.17 = 0.071 m²·K/W - Fiberglass insulation (100mm, k=0.04): R = 0.10/0.04 = 2.50 m²·K/W - Plywood sheathing (15mm, k=0.13): R = 0.015/0.13 = 0.115 m²·K/W - Brick veneer (100mm, k=0.72): R = 0.10/0.72 = 0.139 m²·K/W - Exterior surface resistance: R = 0.04 m²·K/W

Total: R = 0.13 + 0.071 + 2.50 + 0.115 + 0.139 + 0.04 = 2.995 m²·K/W

U-value: U = 1/2.995 = 0.334 W/m²·K

For this wall with 50 m² area and ΔT = 30 K: Q = U · A · Δ T = 0.334 × 50 × 30 = 501 \text W

**Material Thermal Conductivities:** - Air (still): 0.024 W/m·K - Expanded polystyrene (EPS): 0.033-0.040 W/m·K - Mineral wool: 0.035-0.042 W/m·K - Polyurethane foam: 0.022-0.028 W/m·K - Wood: 0.12-0.19 W/m·K - Concrete: 0.8-1.4 W/m·K - Brick: 0.6-1.0 W/m·K - Steel: 45-50 W/m·K - Glass: 0.8-1.0 W/m·K

**2. CONVECTION - Heat Transfer with Fluid Motion**

Convection involves heat transfer between a solid surface and a moving fluid (air, water). In buildings, convection occurs at interior and exterior surfaces.

**Newton's Law of Cooling:** Q = h · A · Δ T

Where: - **h** = Convective heat transfer coefficient (W/m²·K) - **A** = Surface area (m²) - **ΔT** = Temperature difference between surface and fluid (K)

**Convection Heat Transfer Coefficients:** - Still air (natural convection, interior): 2-5 W/m²·K - Moving air (forced convection, interior): 10-100 W/m²·K - Exterior surface (wind 5 m/s): 15-30 W/m²·K - Exterior surface (wind 10 m/s): 25-50 W/m²·K

**Surface Film Resistances:** Interior horizontal surface (heat flow up): R = 0.10 m²·K/W (h = 10 W/m²·K) Interior horizontal surface (heat flow down): R = 0.17 m²·K/W (h = 6 W/m²·K) Interior vertical surface: R = 0.13 m²·K/W (h = 7.7 W/m²·K) Exterior surface (15 mph wind): R = 0.03 m²·K/W (h = 34 W/m²·K)

**3. RADIATION - Electromagnetic Heat Transfer**

Radiation is the transfer of energy through electromagnetic waves. All objects emit thermal radiation proportional to the fourth power of their absolute temperature.

**Stefan-Boltzmann Law:** Q = \epsilon · \sigma · A · (T_1^4 - T_2^4)

Where: - **ε** = Emissivity (0-1, dimensionless) - **σ** = Stefan-Boltzmann constant = 5.67 × 10⁻⁸ W/m²·K⁴ - **T** = Absolute temperature (K)

**Linearized approximation for small temperature differences:** Q ≈ h_r · A · Δ T

Where h_r ≈ 4 \epsilon \sigma T_mean^3

For typical indoor temperatures (20°C = 293K): h_r ≈ 4 × 0.9 × 5.67 × 10^-8 × 293^3 ≈ 5.1 \text W/m²·K

**Emissivity Values:** - Polished aluminum: ε = 0.05 - Galvanized steel: ε = 0.25 - Concrete: ε = 0.90 - Brick: ε = 0.90-0.93 - Glass: ε = 0.84-0.90 - Wood: ε = 0.80-0.90 - Paint (non-metallic): ε = 0.90-0.95 - Low-E coating: ε = 0.05-0.20

**Significance in Windows:** Radiation is particularly important for windows. A standard double-glazed window loses heat through: - Conduction through glass: ~30% - Convection in air gap: ~20% - Radiation between panes: ~50%

Low-E (low emissivity) coatings reduce radiative heat transfer by 50-80%, dramatically improving window performance.

**4. COMBINED HEAT TRANSFER**

In real buildings, all three mechanisms occur simultaneously. The overall heat transfer combines them:

(1)/(U_total) = (1)/(h_inside) + (d_1)/(k_1) + (d_2)/(k_2) + ... + (1)/(h_outside)

Where the surface coefficients (h) include both convection and radiation: h_total = h_convection + h_radiation

**Complete Building Heat Balance Equation:** Q_heating = Q_transmission + Q_ventilation + Q_infiltration - Q_internal - Q_solar

Where: - **Q_transmission** = Heat loss through envelope (conduction + convection + radiation) - **Q_ventilation** = Heat loss from controlled air exchange - **Q_infiltration** = Heat loss from air leakage - **Q_internal** = Heat gains from occupants, lights, equipment (negative in winter) - **Q_solar** = Solar heat gains through windows (negative in winter)

**Steady-State vs. Dynamic Analysis:**

The equations above assume steady-state conditions (constant temperatures). In reality, buildings have thermal mass that stores and releases heat, creating dynamic behavior:

**Thermal Mass Effect:** - Lightweight construction (wood frame): Responds quickly to temperature changes - Heavyweight construction (concrete, masonry): Responds slowly, dampens temperature swings

**Time Lag:** Heat takes time to conduct through materials. For a 200mm concrete wall, the time lag can be 6-12 hours, meaning peak exterior temperatures affect interior loads hours later.

**Decrement Factor:** Heavy construction reduces peak loads. A concrete wall might reduce peak heat flux by 50-70% compared to calculated steady-state values.

For accurate heating load calculations, we use the steady-state approach with the coldest design conditions, ensuring adequate capacity for worst-case scenarios.

Conduction Heat Loss Through Building Envelope

The building envelope separates conditioned interior space from the exterior environment. Heat loss through the envelope represents 30-60% of total heating load in well-ventilated buildings, making it a critical component of heat load calculations.

**FUNDAMENTAL EQUATION:**

Q_conduction = A · U · Δ T

Where: - **A** = Surface area (m²) - **U** = Overall heat transfer coefficient (W/m²·K) - **ΔT** = Temperature difference between inside and outside (K)

**DETAILED U-VALUE CALCULATION:**

The U-value represents the rate of heat transfer through a composite assembly. It's the reciprocal of total thermal resistance:

U = (1)/(R_total) = (1)/(R_si) + Σ_i=1^n R_i + R_se

Where: - **R_si** = Interior surface resistance (m²·K/W) - **R_i** = Resistance of each layer (m²·K/W) - **R_se** = Exterior surface resistance (m²·K/W)

**COMPREHENSIVE EXAMPLE - Residential Wall:**

**Wall Construction (exterior to interior):** 1. Exterior air film (15 mph wind): R_se = 0.030 m²·K/W 2. Vinyl siding (1mm, k=0.17): R = 0.001/0.17 = 0.006 m²·K/W 3. Building paper (1mm, k=0.06): R = 0.001/0.06 = 0.017 m²·K/W 4. Plywood sheathing (12mm, k=0.13): R = 0.012/0.13 = 0.092 m²·K/W 5. Mineral wool insulation (140mm, k=0.038): R = 0.140/0.038 = 3.684 m²·K/W 6. Polyethylene vapor barrier (0.2mm): Negligible thermal resistance 7. Gypsum drywall (12.5mm, k=0.17): R = 0.0125/0.17 = 0.074 m²·K/W 8. Interior air film: R_si = 0.130 m²·K/W

**Total Resistance:** R_total = 0.030 + 0.006 + 0.017 + 0.092 + 3.684 + 0.074 + 0.130 = 4.033 m²·K/W

**U-Value:** U = 1/4.033 = 0.248 W/m²·K

**Heat Loss Calculation:** For 120 m² of wall area with ΔT = 35 K (21°C inside, -14°C outside): Q = 0.248 × 120 × 35 = 1,041.6 W

**TYPICAL U-VALUES BY CONSTRUCTION TYPE:**

**Walls:** - Uninsulated brick wall (230mm solid brick): U = 2.2 W/m²·K - Uninsulated concrete wall (200mm): U = 3.0 W/m²·K - 2×4 wood frame, R-13 insulation: U = 0.45 W/m²·K - 2×6 wood frame, R-20 insulation: U = 0.30 W/m²·K - 2×6 wood frame, R-20 + R-5 exterior foam: U = 0.23 W/m²·K - Double-stud wall, R-40 insulation: U = 0.14 W/m²·K - Insulated concrete form (ICF), R-25: U = 0.22 W/m²·K - Structural insulated panel (SIP), 8" thick: U = 0.11 W/m²·K - Passive House standard wall: U = 0.10-0.15 W/m²·K

**Roofs/Ceilings:** - Uninsulated attic: U = 1.5-2.0 W/m²·K - R-19 fiberglass batts: U = 0.30 W/m²·K - R-30 blown cellulose: U = 0.19 W/m²·K - R-38 fiberglass: U = 0.15 W/m²·K - R-49 cellulose: U = 0.12 W/m²·K - R-60 blown insulation: U = 0.095 W/m²·K - Cathedral ceiling, R-30: U = 0.19 W/m²·K - Flat roof with R-30 rigid foam: U = 0.19 W/m²·K - Green roof with R-25: U = 0.23 W/m²·K

**Windows and Doors:** - Single-glazed, aluminum frame: U = 5.8-6.5 W/m²·K - Single-glazed, wood frame: U = 4.8-5.2 W/m²·K - Double-glazed, air gap, aluminum frame: U = 3.5-4.0 W/m²·K - Double-glazed, air gap, vinyl frame: U = 2.8-3.2 W/m²·K - Double-glazed, argon, low-E, vinyl frame: U = 1.8-2.2 W/m²·K - Triple-glazed, argon, low-E, vinyl frame: U = 1.2-1.6 W/m²·K - Triple-glazed, krypton, low-E2, fiberglass frame: U = 0.8-1.0 W/m²·K - Quadruple-glazed (passive house): U = 0.5-0.7 W/m²·K - Insulated steel door: U = 1.0-1.5 W/m²·K - Solid wood door (44mm): U = 2.8-3.2 W/m²·K - Insulated fiberglass door: U = 0.8-1.2 W/m²·K

**Floors:** - Uninsulated slab-on-grade: U = 1.0-1.5 W/m²·K (perimeter-based) - Insulated slab (R-10 perimeter): U = 0.5-0.7 W/m²·K - Floor over unheated basement, no insulation: U = 0.8-1.2 W/m²·K - Floor over basement, R-19: U = 0.30 W/m²·K - Floor over ventilated crawl space, R-19: U = 0.30 W/m²·K - Floor over outdoor space, R-30: U = 0.19 W/m²·K

**AREA CALCULATIONS:**

**Gross Wall Area Method:** Calculate total wall area, then subtract windows and doors: A_wall,net = (Perimeter × Height) - A_windows - A_doors

**Example - Rectangular House:** - Dimensions: 12m × 9m, Height: 2.7m - Perimeter: 2(12 + 9) = 42m - Gross wall area: 42 × 2.7 = 113.4 m² - Windows: 15 m² total - Doors: 4 m² total - Net wall area: 113.4 - 15 - 4 = 94.4 m²

**COMPLETE BUILDING ENVELOPE HEAT LOSS CALCULATION:**

**Example House Specifications:** - Location: Chicago, IL (Design temp: -20°C) - Indoor temp: 21°C - Temperature difference: ΔT = 41 K

**Surface Areas and U-values:** 1. Walls: 94.4 m², U = 0.30 W/m²·K 2. Windows: 15 m², U = 1.8 W/m²·K (double-glazed, low-E) 3. Doors: 4 m², U = 1.2 W/m²·K (insulated) 4. Ceiling: 108 m², U = 0.15 W/m²·K (R-38) 5. Floor (basement): 108 m², U = 0.35 W/m²·K (semi-heated basement)

**Heat Loss Calculations:** 1. Walls: Q = 94.4 × 0.30 × 41 = 1,161 W 2. Windows: Q = 15 × 1.8 × 41 = 1,107 W 3. Doors: Q = 4 × 1.2 × 41 = 197 W 4. Ceiling: Q = 108 × 0.15 × 41 = 665 W 5. Floor: Q = 108 × 0.35 × 41 = 1,551 W (using ΔT to outdoor temp; actual basement load is lower)

**Total Envelope Heat Loss: 4,681 W (4.68 kW)**

**KEY INSIGHTS:** - Windows (13% of area) account for 24% of heat loss - Floor losses significant when over unheated spaces - Improving window U-value from 2.8 to 1.8 saves 615 W (13% reduction) - Doubling ceiling insulation (R-38 to R-76, U=0.075) saves only 333 W

**FENESTRATION (WINDOWS & DOORS) CONSIDERATIONS:**

Windows are the weakest thermal link in the envelope: - Represent 10-20% of wall area but 25-40% of heat loss - U-values 5-20× higher than well-insulated walls - Also allow solar gains (benefit in winter, liability in summer) - Frame material and quality significantly impact performance

**Window Frame Performance:** - Aluminum (no thermal break): Frame U = 7-9 W/m²·K - Aluminum (thermal break): Frame U = 3-4 W/m²·K - Vinyl (hollow): Frame U = 2-3 W/m²·K - Vinyl (foam-filled): Frame U = 1.5-2 W/m²·K - Fiberglass: Frame U = 1.2-2 W/m²·K - Wood: Frame U = 1.5-2.5 W/m²·K

Frame represents 15-30% of window area, significantly impacting overall performance.

**ENERGY CODE REQUIREMENTS (Examples):**

**IECC Climate Zone 5 (Chicago, Minneapolis):** - Walls: U ≤ 0.29 W/m²·K (R-20) - Ceiling: U ≤ 0.14 W/m²·K (R-49) - Windows: U ≤ 2.0 W/m²·K - Doors: U ≤ 2.3 W/m²·K

**Passive House Standard (any climate):** - Walls: U ≤ 0.15 W/m²·K - Roof: U ≤ 0.12 W/m²·K - Windows: U ≤ 0.80 W/m²·K - Doors: U ≤ 0.80 W/m²·K

**OPTIMIZATION STRATEGIES:**

**Cost-Effectiveness Ranking:** 1. **Attic insulation:** Easiest and cheapest to upgrade (ROI < 5 years) 2. **Air sealing:** Low cost, high impact (complements insulation) 3. **Window upgrades:** Expensive but high heat loss reduction 4. **Wall insulation:** Expensive retrofit, best during renovation 5. **Foundation insulation:** Often overlooked, significant savings in cold climates

Ventilation and Infiltration Heat Loss

Air exchange between indoor and outdoor environments causes significant heat loss during heating season.

**Ventilation Heat Loss Formula:** Q_ventilation = \dotV · \rho · c_p · Δ T = (V · ACH · \rho · c_p · Δ T)/(3600)

Where: - **V** = Room volume (m³) - **ACH** = Air changes per hour (h⁻¹) - **ρ** = Air density = 1.2 kg/m³ (at sea level, 20°C) - **c_p** = Specific heat capacity of air = 1005 J/(kg·K) - **ΔT** = Temperature difference (K) - **3600** = Conversion factor from seconds to hours

**Simplified Formula:** Q_ventilation = 0.34 · V · ACH · Δ T \text (Watts)

Where 0.34 = (1.2 × 1005) / 3600

**Air Change Rates (ACH):** - **Modern airtight homes:** 0.1-0.3 ACH (with mechanical ventilation) - **Standard residential:** 0.3-0.5 ACH - **Older homes:** 0.5-1.5 ACH - **Leaky construction:** 1.5-3.0 ACH - **Commercial spaces:** 1.0-2.0 ACH (minimum for air quality) - **Restaurants/kitchens:** 15-20 ACH - **Laboratories:** 6-12 ACH

**Infiltration vs. Ventilation:** - **Infiltration:** Uncontrolled air leakage through cracks and gaps - **Ventilation:** Controlled air exchange for indoor air quality

With heat recovery ventilation (HRV), the effective ventilation load is reduced: Q_ventilation,HRV = Q_ventilation · (1 - \eta_HRV)

Where η_HRV = heat recovery efficiency (typically 70-95%)

Internal Heat Gains and Occupant Load

Occupants, lighting, and equipment generate heat that reduces heating requirements.

**Occupant Heat Gains:** Q_occupants = n · q_s

Where: - **n** = Number of occupants - **q_s** = Sensible heat per person (W)

**Heat Generation by Activity Level:** - **Seated, quiet:** 75 W/person (65 W sensible + 10 W latent) - **Seated, light work:** 95 W/person (75 W sensible + 20 W latent) - **Standing, light activity:** 115 W/person (85 W sensible + 30 W latent) - **Walking, moderate activity:** 140 W/person (95 W sensible + 45 W latent) - **Heavy work/exercise:** 200-400 W/person

**Other Internal Gains:** - **Lighting:** 5-15 W/m² (depending on lighting type) - **Computers/electronics:** 50-100 W per workstation - **Kitchen appliances:** 500-3000 W - **Manufacturing equipment:** Varies widely

**Note:** During heating season, internal gains reduce the net heating load, but they are often conservatively omitted from calculations to ensure adequate capacity during unoccupied periods or equipment shutdown.

Thermal Bridges and Edge Effects

Thermal bridges are localized areas of higher heat flow through the building envelope, occurring at structural elements, corners, and junctions.

**Common Thermal Bridge Locations:** - Wall-floor junctions - Wall-roof connections - Window and door frames - Balcony connections - Steel columns penetrating insulation - Structural elements (beams, studs)

**Linear Thermal Transmittance:** Q_thermal\_bridge = Σ (\psi_i · L_i · Δ T)

Where: - **ψ** = Linear thermal transmittance (W/m·K) - **L** = Length of thermal bridge (m) - **ΔT** = Temperature difference (K)

**Typical ψ-values:** - Insulated wall-floor junction: 0.05-0.15 W/m·K - Poorly insulated junction: 0.5-1.0 W/m·K - Window frame perimeter: 0.03-0.10 W/m·K

**Impact on Heat Load:** Thermal bridges can increase overall heat loss by 15-30% compared to calculations based solely on U-values. Modern design aims to minimize thermal bridging through: - Continuous insulation layers - Thermal breaks in metal frames - Careful detailing at junctions - Advanced framing techniques

Design Temperature and Climate Factors

Selecting appropriate design temperatures is critical for proper system sizing.

**Design Temperature Selection:**

Outdoor design temperature is typically chosen as the temperature that is exceeded during 99% or 99.6% of the coldest month hours.

**Examples by Climate Zone:** - **Minneapolis, MN (Cold):** -28°C (99.6%), -24°C (99%) - **Chicago, IL (Cold):** -22°C (99.6%), -19°C (99%) - **New York, NY (Mixed):** -15°C (99.6%), -12°C (99%) - **Atlanta, GA (Warm):** -8°C (99.6%), -6°C (99%) - **Phoenix, AZ (Hot-Dry):** 2°C (99.6%), 4°C (99%)

**Indoor Design Temperature:** - **Residential:** 20-22°C (68-72°F) - **Offices:** 20-22°C (68-72°F) - **Retail:** 18-20°C (64-68°F) - **Warehouses:** 15-18°C (59-64°F) - **Hospitals:** 21-24°C (70-75°F)

**Temperature Difference:** Δ T = T_indoor - T_outdoor,design

Example: Indoor 21°C, Outdoor -20°C → ΔT = 41 K

**Climate Factors:** - **Wind speed:** Increases infiltration and convective heat loss from exterior surfaces - **Humidity:** Affects latent heat requirements (usually minor for heating) - **Ground temperature:** Important for basement and slab-on-grade heat loss - **Solar radiation:** Reduces heating requirements during sunny periods

Ground Heat Loss

Heat loss through floors and basements requires special consideration due to ground thermal properties.

**Slab-on-Grade Heat Loss:** Q_slab = P · \psi_ground · Δ T

Where: - **P** = Perimeter length (m) - **ψ_ground** = Ground heat transfer coefficient (W/m·K) - **ΔT** = Temperature difference (K)

Typical ψ_ground values: - Uninsulated slab: 1.0-1.5 W/m·K - Insulated slab edge: 0.3-0.5 W/m·K

**Basement Heat Loss:** Basement walls and floors lose heat to ground, but ground temperature is higher than outdoor air: - At 2m depth: Ground temp typically 8-15°C year-round (depends on climate) - Use modified ΔT based on ground temperature, not outdoor air temperature

**Simplified Basement Calculation:** Q_basement = (A_walls · U_walls + A_floor · U_floor) · (T_indoor - T_ground)

Safety Factors and System Sizing

Heat load calculations require safety factors to account for uncertainties and ensure adequate capacity.

**Safety Factor Components:**

**1. Design Temperature Margin:** Use 99% or 99.6% design temperature plus additional 2-3°C margin for extreme events.

**2. System Capacity Factor:** Multiply calculated load by 1.10-1.15 (10-15% margin) to account for: - Calculation uncertainties - Building envelope degradation over time - Actual vs. theoretical U-values - Unmodeled thermal bridges

**3. Pickup Load:** For buildings with night setback or intermittent heating, add 15-25% to recover from reduced temperature: Q_pickup = 1.15 \text to 1.25 × Q_design

**4. Distribution Losses:** For ducted or hydronic systems, account for heat loss from distribution: - Well-insulated ducts in conditioned space: +5% - Uninsulated ducts in unconditioned space: +15-25% - Piping losses: +5-10%

**Total Sizing:** Q_equipment = Q_calculated × (1 + f_safety) × (1 + f_pickup) × (1 + f_distribution)

**Example:** - Calculated load: 10,000 W - Safety factor: 10% (1.10) - Pickup factor: 20% (1.20) - Distribution: 10% (1.10) - Equipment size: 10,000 × 1.10 × 1.20 × 1.10 = 14,520 W

**Important:** Excessive oversizing (>25%) leads to: - Short cycling and reduced efficiency - Poor humidity control - Increased capital costs - Reduced equipment lifespan

Step-by-Step Calculation Procedure

**Complete Heat Load Calculation Process:**

**Step 1: Building Information** - Collect floor plans with dimensions - Identify wall, roof, floor constructions - Note window and door locations/sizes - Determine building orientation - Assess air tightness

**Step 2: Design Conditions** - Select indoor design temperature (typically 20-22°C) - Determine outdoor design temperature for location - Calculate design temperature difference (ΔT)

**Step 3: Conduction Heat Loss** For each building element: - Calculate area (A) - Determine U-value - Calculate: Q = A × U × ΔT - Sum all elements

**Step 4: Ventilation/Infiltration Loss** - Calculate building volume (V) - Determine air change rate (ACH) - Calculate: Q_vent = 0.34 × V × ACH × ΔT

**Step 5: Ground Heat Loss** - Calculate slab perimeter heat loss - Calculate basement heat loss (if applicable) - Use appropriate ground temperatures

**Step 6: Thermal Bridges** - Identify major thermal bridges - Calculate linear heat loss - Add 10-20% to envelope losses as approximation

**Step 7: Internal Gains (Optional)** - Calculate occupant heat gains - Estimate lighting and equipment gains - Subtract from heating load (conservative: omit these)

**Step 8: Apply Safety Factors** - Add capacity margin (10-15%) - Add pickup load if using setback (15-25%) - Add distribution losses (5-25%)

**Step 9: Equipment Selection** - Select heating equipment matching calculated capacity - Verify equipment efficiency ratings - Consider modulating or staged equipment for better control

Worked Example

Inputs

• area: 150
• uValue: 0.35
• tempDiff: 35
• volume: 375
• airChanges: 0.5
• occupants: 4
• heatPerPerson: 75

Result: Total Load: 3,431.25 W (3.43 kW)

Explanation

**Example: Residential Living Space Heat Load Calculation**

**Given Parameters:** - Surface Area (A): 150 m² (walls, windows, roof combined) - U-value: 0.35 W/m²·K (well-insulated construction) - Temperature Difference (ΔT): 35 K (Indoor: 21°C, Outdoor: -14°C) - Room Volume (V): 375 m³ - Air Changes per Hour (ACH): 0.5 h⁻¹ - Number of Occupants: 4 people - Heat per Person: 75 W (seated, quiet activity)

**Step-by-Step Calculation:**

**1. Conduction Heat Loss:** Q_conduction = A · U · Δ T Q_conduction = 150 \text m² × 0.35 \text W/m²·K × 35 \text K Q_conduction = 1,837.5 \text W

**2. Ventilation Heat Loss:** Q_ventilation = (V · ACH · \rho · c_p · Δ T)/(3600) Q_ventilation = (375 × 0.5 × 1.2 × 1005 × 35)/(3600) Q_ventilation = (7,909,687.5)/(3600) = 2,197.13 \text W

Or using simplified formula: Q_ventilation = 0.34 · V · ACH · Δ T Q_ventilation = 0.34 × 375 × 0.5 × 35 = 2,231.25 \text W

**3. Occupant Heat Generation (Internal Gain):** Q_occupants = n · q_s Q_occupants = 4 × 75 = 300 \text W

**4. Net Heating Load:** Q_total = Q_conduction + Q_ventilation + Q_occupants Q_total = 1,837.5 + 2,231.25 + 300 Q_total = 4,368.75 \text W = 4.37 \text kW

**Component Breakdown:** - **Conduction Loss:** 1,837.5 W (42.1% of total load) - **Ventilation Loss:** 2,231.25 W (51.1% of total load) - **Occupant Load:** 300 W (6.8% of total load)

**Key Insights:** 1. Ventilation heat loss (51%) is the dominant component, highlighting the importance of air tightness and heat recovery systems 2. With good insulation (U=0.35), conduction losses are well-controlled at 42% 3. Occupant gains offset heating requirements by ~7% 4. For equipment selection with safety factors (15%), specify a heater with capacity: 4.37 kW × 1.15 = 5.03 kW minimum

**Recommendations:** - Consider installing heat recovery ventilation (HRV) with 80% efficiency to reduce ventilation load from 2,231 W to 446 W - This would reduce total load to 2,584 W (41% reduction) - Payback period for HRV typically 5-10 years depending on energy costs

Small Apartment

Inputs

• area: 80
• uValue: 0.4
• tempDiff: 30
• volume: 200
• airChanges: 0.4
• occupants: 2
• heatPerPerson: 75

Result: Total Load: 2,142 W (2.14 kW)

Explanation

Small, moderately insulated apartment: - Conduction: 80 × 0.4 × 30 = 960 W - Ventilation: 0.34 × 200 × 0.4 × 30 = 816 W - Occupants: 2 × 75 = 150 W - Total: 1,926 W + 15% safety = 2,142 W