House Cool-Down and Warm-Up Time Calculator
Estimate how long a house stays warm without power, cold-house heating time, and heat-pump night-setback savings with a two-node air and building-mass model.
How long will my house stay warm without power? This calculator estimates how room air and the building's stored heat change during a heating interruption. It also estimates how long to heat a cold house and explores night setback with a heat pump. Engineers, students and homeowners can compare scenarios without building a detailed room-by-room simulation.
Choose power outage, cold-house warm-up or night setback. Enter the whole-house heat-loss coefficient, outdoor temperature and effective thermal storage. The heat-loss coefficient includes walls, roof, windows, floors and air exchange. If you already have a heat-load calculation, divide its heat loss by the corresponding indoor–outdoor temperature difference to estimate this coefficient. Check whether ventilation heat recovery would still operate during an outage.
For an outage, the default room threshold is 10 °C. Add continuously available backup heat to see how a small heater changes the crossing time. The comparison curve shows the same house without that heater. “Not reached” means the threshold is not crossed within the entered horizon; it does not mean the building is safe indefinitely.
The pipe-space result is deliberately a proxy. Its temperature lies between outdoor and room temperature, according to the indoor coupling fraction. A fraction of one represents a location at room temperature. A smaller fraction represents a colder, more exposed space. Reaching zero indicates a reason to investigate local freezing risk, not a prediction of when water freezes or a pipe bursts. Concealed pipes can be colder than a room sensor suggests.
For a cold house or dacha, both air and building mass begin at the entered cold temperature. The heater runs at its available thermal output until the room target is reached, then modulates to hold that target. A separate milestone shows when the mass is within 1 K of the target. Warm air alone can hide cold floors and walls.
For night setback, heating switches off until the lower setting is reached, holds that setting if necessary, then reheats at full output. The electricity comparison includes subsequent recovery of the building mass. Positive savings mean less electricity; negative savings mean more. A lower recovery COP can outweigh reduced overnight heat loss. This answers the night setback heat pump question for the entered assumptions, rather than prescribing one schedule for every house.
Play the elapsed-time control to follow the thermometer and house section. The curves remain available for the full selected horizon. Thermal parameters are effective values: measured cooling and heating records are better than construction labels alone. Constant weather, a single well-mixed zone and ideal heater modulation limit the estimate. Solar gains, wind changes, fuel limits, defrost and individual rooms are outside this model. Installation work belongs to a qualified person.
How it is calculated
The deterministic two-node energy balances are
Conductances are converted to kW/K, capacitances are in kWh/K, and time is in hours. This is the air–internal-mass RC topology in Bacher and Madsen (2011), Appendix A, model TiTm (attached p. 11), and the residential ETP model in PNNL-17694, printed pp. 19–20 (attached pp. 27–28). It is a transparent reduced model, not a reproduction of their fitted multi-node building.
Each constant-input phase uses the exact two-real-exponential solution. The fast and slow eigenvalues are calculated with a stable quadratic expression. Threshold times use bracketed bisection, not the chart's sampling interval. Charts include exact thermostat transitions, closely spaced samples on either side, and extra samples within short phases so brief full-output recovery is retained. A blank crossing is intentional: the threshold is not reached within the search horizon, or lies at/beyond the equilibrium. At zero mass coupling the air follows Newton's single-node law.
For warm-up, full output continues to the air target. Thereafter ideal modulation holds the air temperature and the mass relaxes exponentially. The mass milestone requires air recovery and a remaining mass deficit no greater than 1 K.
Setback starts from both nodes at the normal target. Heating is off, then holds the night setting if it is reached. Morning recovery uses the entered maximum output. Night heat is the envelope loss during the hold plus the mass energy change. Net heat savings subtract full reheat and the complete remaining mass deficit, . This is the exact integral of the later extra heating at held air temperature; both paths finish with equal stored energy. Normal and night heat, including that tail, use normal COP; full-output morning heat uses reheat COP. Negative electricity savings are retained.
The pipe-space estimate is . Its zero crossing is a local-air exposure indicator only. All diagram annotations use °C explicitly; physical input fields and numerical temperature cards support the viewer's metric/US conversion.
Assumptions
- One well-mixed air node and one slower internal-mass node; constant outdoor temperature, conductances and heat capacities.
- Air and mass start at the same temperature. Outage backup output is continuous; solar and occupant gains are omitted.
- UA includes transmission and operating air exchange. The envelope's storage is lumped into the internal mass; this approximation is weaker for thick exterior masonry.
- The illustrative defaults (UA 200 W/K, fast capacity 0.35 kWh/K, mass capacity 8 kWh/K and coupling 1500 W/K) are editable scenario assumptions, not a published house classification or a fit to RMI/SINTEF results.
- The exact phase solution, thermostat phase matching, pipe-space proxy and recovery energy accounting are derived here from the stated balances. The mass has no independent outdoor loss path.
- Heater output and COP are supplied at actual cold-weather conditions. Unlimited modulation is assumed; no compressor cycling, supply-temperature dynamics, resistance staging, defrost, or fuel depletion is simulated.
- Net setback savings concern one isolated setback event including complete subsequent mass recovery; they are not an annual saving or a tariff-cost estimate.
Where it applies
Use for sensitivity studies over hours to days, with parameters calibrated to the building where practical. Timing is highly sensitive to weather, leakage, occupied zones and effective mass. A cold start below freezing does not include latent heat or thawing.
A 10 °C room threshold is a reporting choice, not a safe-occupancy limit. RMI's February 2020 brief (not 2018) explains why vulnerability and envelope performance matter; SINTEF's 15 February 2023 example concerns a multi-room Oslo weather scenario. Those studies are context, not numerical benchmarks for these constant-weather defaults. Follow local emergency advice and measured temperatures.
The pipe-space fraction is user-assumed. This model cannot certify freeze protection, pipe-water temperature or burst time. Radiant-floor startup, humidity, condensation, ground coupling and room-to-room differences require a more detailed model. Installation work belongs to a qualified person.
The horizon is measured from the start of the displayed event. Morning recovery searches use only the time remaining after setback, and reported morning recovery times are measured from the end of setback. Energy savings are unavailable when recovery is not reached in that interval. “Mass within 1 K” is a storage indicator, not a full thermal-comfort metric.
References
- Bacher, P. and Madsen, H. (2011). Identifying suitable models for the heat dynamics of buildings. Energy and Buildings 43, 1511–1522. DOI: https://doi.org/10.1016/j.enbuild.2011.02.005. Open author copy: https://henrikmadsen.org/wp-content/uploads/2014/05/Journal_article_-_2011_-_Identifying_suitable_models_for_the_heat_dynamics_of_buildings.pdf
- PNNL (2008). GridLAB-D Technical Support Document: Residential End-Use Module Version 1.0, PNNL-17694, printed pp. 19–20. https://www.pnnl.gov/main/publications/external/technical_reports/PNNL-17694.pdf
- Ayyagari, Gartman and Corvidae (February 2020). Hours of Safety in Cold Weather. RMI. https://rmi.org/wp-content/uploads/2020/02/Hours-of-Safety-insight-brief.pdf
- Homaei, Hamdy and Andenæs (15 February 2023). Hvor lenge holder huset på varmen når strømmen går? SINTEF. https://www.sintef.no/siste-nytt/2023/hvor-lenge-holder-huset-pa-varmen-nar-strommen-gar/
- University of Utah, Linear Applications §2.5, printed p. 113 (attached p. 3): Newton cooling and the residence half-time example. https://www.math.utah.edu/~gustafso/s2015/2280/linearapplications2-5.pdf
Inputs
| Input | Symbol | Unit | Default |
|---|---|---|---|
| Scenario | scenario | Power outage / backup heater | |
| Outdoor temperature Held constant. Use a cold-night temperature, not the daily high. | T_out | °C | -10 |
| Whole-house heat-loss coefficient Transmission plus ventilation and infiltration, including heat recovery only if it remains active. From a heat-load result: UA = heat loss / indoor–outdoor temperature difference. | UA | W/K | 200 |
| Effective air / fast heat capacity Air plus rapidly responding contents. Illustrative default; this is a fitted effective capacitance, not all of the building mass. | C_air | kWh/K | 0.35 |
| Building-mass heat capacity Effective slower storage in walls, floors and furniture. Fit to measured cool-down data where possible; the default is an illustrative house. | C_mass | kWh/K | 8 |
| Air-to-mass thermal conductance Controls how quickly the mass exchanges heat with the room air. Zero is the isolated-mass limiting case, not a normal building. | H_mass | W/K | 1500 |
| Temperature when power fails Air and mass start at this same temperature, after a sustained heated period. | T_initial | °C | 21 |
| Room-temperature threshold Default 10 °C is a reporting threshold, not a safe-occupancy guarantee. Vulnerable occupants may need action much earlier. | T_limit | °C | 10 |
| Continuous backup heat delivered Thermal output that remains available during the outage. Try 1.5 kW for a small resistance heater with an adequate independent supply. No thermostat or fuel exhaustion is modelled. | P_backup | kW | 0 |
| Pipe-space indoor coupling fraction Proxy temperature = outdoor + fraction × (room − outdoor). 1 means a pipe space at room temperature; smaller values mean greater outdoor exposure. This is not a pipe-freezing calculation. | pipe_factor | 0.8 | |
| Initial cold-house temperature Air and mass both start cold. This captures a house or dacha that has been unheated for a long period. | T_cold | °C | 5 |
| Normal indoor target | T_target | °C | 21 |
| Available heating output Delivered thermal capacity at the entered outdoor temperature and operating water/air temperature. Use the heat pump's cold-weather output, not its electrical input or mild-weather nameplate. | P_heat | kW | 8 |
| Night thermostat setting | T_setback | °C | 17 |
| Setback duration | night_h | h | 8 |
| COP during normal / night holding Delivered heat divided by electrical input. Include auxiliaries consistently. The same COP is used during the mass-recovery tail. | COP_normal | 3 | |
| COP during full-output reheat May be lower at high flow temperature or with resistance backup. Set equal to normal COP to isolate the effect of reduced heat loss. | COP_reheat | 2.5 | |
| Calculation / chart horizon | horizon_h | h | 96 |
| Elapsed time ▶ Animates the house and thermometer. Curves and milestone searches are cached independently of this input. | clock_h | h | 8 |
Results
With the default inputs above, the calc gives:
| Result | Value |
|---|---|
| Room-air milestone | Room threshold: 15.1 h |
| Mass / pipe-space milestone | Pipe proxy to 0 °C: 37.1 h |
| Energy / backup comparison | Without backup: 15.1 h |
| Room air at elapsed time | 13.3 °C |
| Building mass at elapsed time | 16.2 °C |
| Interpretation | Pipe freezing may occur earlier locally; this is a space-temperature proxy. |
Checked against sources
Each case below was calculated by this calc and agrees with the value from its source (within the tolerance where one is shown).
Bacher–Madsen Table 3: two-node decay modes
Source: Bacher and Madsen (2011), Table 3, TiTh column: Ci=1.36, Ch=0.309 kWh/K, Ria=5.31 and Rih=0.639 K/kW; published time constants 0.16 h and 8.9 h. Mapping the heater node to the unforced mass gives the identical homogeneous system; forcing location does not affect its eigenvalues. This tests the two-node dynamics, not their measured building trajectory. Their p. 1518 cautions that this smaller model is an incomplete building fit. Tolerance covers rounded source data.
Inputs: Effective air / fast heat capacity = 1.36 kWh/K; Building-mass heat capacity = 0.309 kWh/K; Whole-house heat-loss coefficient = 188.3 W/K; Air-to-mass thermal conductance = 1,565 W/K; the others as above.
| Result | Source | This calc |
|---|---|---|
| Fast response time constant | 0.16 ± 0.03 h | 0.1602 h |
| Slow response time constant | 8.9 ± 0.03 h | 8.9 h |
Utah residence: k = 0.5 per hour, published 1.4 h half-time
Source: University of Utah Linear Applications §2.5, printed p. 113 / attached p. 3: 50°F outdoors, 70°F initially, 60°F midpoint; k=0.5 h^-1 has published rounded half-time 1.4 h. H_mass=0 isolates the single-node limiting case; UA/C_air=0.5 h^-1. Temperatures are affine-converted from Fahrenheit. The exact half-time is ln(2)/0.5; expected 1.4 remains the source's rounded number.
Inputs: Air-to-mass thermal conductance = 0 W/K; Effective air / fast heat capacity = 0.4 kWh/K; Outdoor temperature = 10 °C; Temperature when power fails = 21.111 °C; Room-temperature threshold = 15.556 °C; the others as above.
| Result | Source | This calc |
|---|---|---|
| Outage: time until room reaches its threshold | 1.4 ± 0.05 h | 1.386 h |
| Fast response time constant | 2 h | 2 h |
Utah residence: k = 0.05 per hour, published 14 h half-time
Source: Same independent University of Utah published example, slow end of its k range: k=0.05 h^-1 and rounded 14 h half-time. The exact value is ln(2)/0.05; the tolerance reflects rounding, not a modification of the physics.
Inputs: Air-to-mass thermal conductance = 0 W/K; Effective air / fast heat capacity = 4 kWh/K; Outdoor temperature = 10 °C; Temperature when power fails = 21.111 °C; Room-temperature threshold = 15.556 °C; the others as above.
| Result | Source | This calc |
|---|---|---|
| Outage: time until room reaches its threshold | 14 ± 0.5 h | 13.863 h |
| Fast response time constant | 20 h | 20 h |
Responses