Enter the indoor and outdoor temperatures, then the area and U-factor of each part (choosing a preset fills in a typical U). The ventilation rate, heating hours, COP, electricity rate and improvement comparison can be left blank (those items are then skipped).
The preset U-factors are typical values. If a window or door label or the building's energy calculations give a U-factor, change it to that value. Leave both the area and U-factor blank for parts you do not use. The "Room Volume and Ventilation Calculator" finds the ventilation rate from volume × air changes per hour.
Result and graph
Enter the indoor and outdoor temperatures and the area and U-factor of each part of the room on the left and press "Calculate". The result and a bar chart will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formulas and figures
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter the area and the U-factor \(U\) of each part of the room, such as walls, windows, ceiling (roof), floor and doors, and see right away how much heat escapes through each part (BTU/h) and in total, from the indoor-outdoor temperature difference
- Pick a U-factor from the presets (typical values for single-pane windows, double-pane windows, insulated walls and so on), and change it to the value from an NFRC window label or the plans
- A bar chart shows what percent of the heat escapes through each part, so priorities such as "the window is the weak spot" are clear at a glance
- Enter the ventilation rate (CFM) to add the ventilation heat loss (\(1.08 \times \mathrm{CFM} \times \Delta T\)) and get a guide to the heating capacity needed
- Enter the heating hours, COP and electricity rate for the daily heat (kWh) and electricity cost, and compare improvements such as "how much less heat escapes with low-E windows". A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are also on this page
This page estimates the heat loss of one room with the basic formulas for steady state, when the temperatures stay constant. It does not include heat from the sun, people and appliances, or the extra needed to warm up a cold room. Sizing heating equipment for a whole house in the US is done with an ACCA Manual J load calculation, and code compliance uses tools such as REScheck, so the room-by-room result here cannot replace them. The preset U-factors are typical values, so use the values from product labels or the plans when you have them.
What is this calculation used for?
Estimating the benefit of better windows (storm windows or replacement)
If you suspect the window makes the room cold, this formula can check it. In a 12 × 9 ft room (wall 130 ft² at U 0.079, double-pane window 33 ft² at U 0.83, ceiling and floor 108 ft² at U 0.044 and 0.088) with a 36 °F difference, the parts lose \(369.7 + 986.0 + 171.1 + 342.1 \approx 1869\) BTU/h, and the window, the smallest area, loses more than half of it.
Replacing it with a low-E double-pane vinyl window (about U 0.39) saves \((0.83 - 0.39) \times 33 \times 36 \approx 523\) BTU/h, about 24% of the total with ventilation (2,141 BTU/h). Entering the U-factors from the labels of storm windows or replacement windows lets you compare which job helps most. Real results also depend on installation and air leaks, so treat it as an estimate.
Finding where an old house loses heat
In an old house with no insulation, a room with 215 ft² of wall (about U 0.44), 43 ft² of single-pane window (about U 1.14), 130 ft² of ceiling (about U 0.62) and 130 ft² of floor (about U 0.53), at 68 °F inside and 20 °F outside (48 °F difference), loses \(4541 + 2353 + 3869 + 3307 \approx 14070\) BTU/h through the parts.
The shares are about 32% wall, 27% ceiling, 24% floor and 17% window, so the heat escapes everywhere, not only through the window. That points to priorities such as adding attic insulation or insulating under the floor. The preset values are rough, so use the values from an energy audit or the plans if you have them.
Getting a feel for heater size
A room with a total heat loss of 2,141 BTU/h needs about 2,100 BTU/h of heat (about 0.63 kW) to hold its temperature. That is the steady-state value. Warming a cold room quickly takes more, and real equipment is chosen with some margin.
Still, the formula tells you that a well-insulated room can get by with a small heater, while the uninsulated room above (about 14,000 BTU/h, over 4 kW) is far beyond a 1,500 W space heater (about 5,100 BTU/h). For a whole house, heating contractors size equipment with an ACCA Manual J load calculation. The "Air Conditioner Size Calculator" and the "Heating Cost Comparison Calculator" help with choosing equipment and comparing running costs.
Understanding ventilation and heating
Ventilating a 864 ft³ room at 0.5 air changes per hour is about 7 CFM, and with a 36 °F difference \(1.08 \times 7 \times 36 \approx 272\) BTU/h escapes through ventilation. In a well-insulated room, the loss through the parts is small, so ventilation becomes a bigger share, and in very tight, well-insulated homes it can approach 20% of the total.
Turning off ventilation would cut the loss, but it lets indoor air go stale and can cause condensation. Instead, a heat recovery ventilator (HRV or ERV) can recover the heat, and sealing air leaks cuts unplanned ventilation. The "Room Volume and Ventilation Calculator" finds the ventilation rate.
Estimating heating cost for a season
Heating a room with a total heat loss of 2,141 BTU/h for 8 hours a day takes \(2141.1 \times 8 \div 3412.14 \approx 5.02\) kWh of heat a day. With a heat pump of COP 3 that is about 1.67 kWh of electricity, or about $0.28 a day at $0.17/kWh, while a resistance heater (COP 1) would cost about $0.85.
Over a 150-day heating season that is about $43 with the heat pump and about $128 with resistance heat. Because the heat loss saved by an upgrade (BTU/h) turns directly into dollars, you can also get a rough idea of how many years an upgrade takes to pay for itself. The outdoor temperature changes day by day, so use an average winter temperature for cost estimates.
How this page relates to Manual J and energy codes
Whole-house tools use the same U × A idea as this formula, summed over the entire building envelope. Code compliance tools such as REScheck compare the envelope's total UA with a code-built house, and an ACCA Manual J load calculation sizes heating and cooling equipment with detailed rules for framing, buffer spaces, infiltration and local design temperatures.
This page is for estimating the heat loss of the room you are in, from areas you can measure and typical U-factors. Entering U-factors from labels or the plans makes it more accurate, but it is not meant for code compliance or official equipment sizing.
Formulas and figures
Indoor-outdoor temperature difference
Standard notation
(the usual math form)
\(\Delta T\)
\(=\)
\(T_{in}\)
\(-\)
\(T_{out}\)
In words
(symbols replaced with words)
③
\(\Delta T\): temperature difference
\(=\)
①
\(T_{in}\): indoor temperature
\(-\)
②
\(T_{out}\): outdoor temperature
The formula in words
①
Subtract the
\(T_{out}\): outdoor temperature
from the
②
\(T_{in}\): indoor temperature
to get the
③
\(\Delta T\): temperature difference
Quick example
With the room kept at 68 °F and the outdoor temperature at 32 °F, the temperature difference is
temperature difference \(\Delta T\)
\(=\)
indoor (68 °F)
\(-\)
outdoor (32 °F)
\(68 - 32 = 36\,^{\circ}\mathrm{F}\)
Key idea
Heat flows from warm to cold, and how much flows is proportional to the temperature difference. That is why every calculation on this page starts from the difference between indoors and outdoors. In metric the difference is written in K (kelvin), which is the same size as a °C difference (20 °C minus 0 °C is 20 K). In US units it is simply a difference in °F. The outdoor temperature you choose changes the result a lot, so decide whether you are estimating for the coldest winter morning or for an average winter day. To size heating equipment, use the cold one (heating contractors use the local winter design temperature, often around 0 to 20 °F in much of the US). For energy costs, an average winter temperature makes more sense.
Heat loss through each part
Figure
Standard notation
(the usual math form)
\(q_i\)
\(=\)
\(U_i\)
\(\times\)
\(A_i\)
\(\times\)
\(\Delta T\)
In words
(symbols replaced with words)
④
\(q_i\): heat loss of part \(i\)
\(=\)
①
\(U_i\): U-factor
\(\times\)
②
\(A_i\): area
\(\times\)
③
\(\Delta T\): temperature difference
The formula in words
①
Multiply the
\(U_i\): U-factor (heat through 1 ft² per °F of difference)
by the
②
\(A_i\): area
and then by the
③
\(\Delta T\): temperature difference
to get the
④
\(q_i\): heat loss of part \(i\)
(heat escaping per hour, in BTU/h)
Quick example
Through 33 ft² of double-pane window in an aluminum frame (typical U-factor 0.83 BTU/(h·ft²·°F)), with a 36 °F difference, the heat escaping is
window heat loss \(q\)
\(=\)
U-factor (0.83)
\(\times\)
area (33 ft²)
\(\times\)
difference (36 °F)
\(0.83 \times 33 \times 36 \approx 986\,\mathrm{BTU/h}\)
Key idea
The U-factor \(U\) is how many BTU of heat a part lets through per hour, for each square foot and each °F of temperature difference (in metric, watts per m² per K). Multiply the \(U\) of each wall, window, ceiling and floor by its area and the temperature difference, and you have the heat escaping through each part. For example, an insulated wall (about 0.079) and a double-pane window (about 0.83) differ by about 10 times, so the same area of window loses about 10 times as much heat. Use the net wall area without windows and doors, and enter the windows as their own parts. Surfaces facing an attic or crawl space are usually a little less cold than the outdoors, so a detailed load calculation adjusts for that. This page uses the same temperature difference for all parts, so ceiling and floor losses come out a little high (on the safe side).
Ventilation heat loss
Standard notation
(the usual math form)
\(q_v\)
\(=\)
\(c\)
\(\times\)
\(V\)
\(\times\)
\(\Delta T\)
In words
(symbols replaced with words)
④
\(q_v\): ventilation heat loss
\(=\)
①
\(c\): ventilation factor
\(\times\)
②
\(V\): ventilation rate
\(\times\)
③
\(\Delta T\): temperature difference
The formula in words
①
Multiply the
\(c\): ventilation factor (usually 1.08 BTU/(h·CFM·°F))
by the
②
\(V\): ventilation rate (CFM, air exchanged per minute)
and then by the
③
\(\Delta T\): temperature difference
to get the
④
\(q_v\): ventilation heat loss
Quick example
For a 12 × 9 ft room with an 8 ft ceiling (864 ft³) ventilated 0.5 times per hour (864 × 0.5 ÷ 60 ≈ 7 CFM), with a 36 °F difference, the heat lost through ventilation is
ventilation heat loss \(q_v\)
\(=\)
factor (1.08)
\(\times\)
ventilation (7 CFM)
\(\times\)
difference (36 °F)
\(1.08 \times 7 \times 36 \approx 272.2\,\mathrm{BTU/h}\)
Key idea
Ventilation sends warm air outside and brings cold air in, so the heat needed to warm up the incoming air is lost. The factor 1.08 comes from the density of air (about 0.075 lb/ft³) × its specific heat (0.24 BTU per lb per °F) × 60 minutes per hour. (In metric, the same idea gives about 0.34 Wh per m³ per K.) The ventilation rate in CFM is room volume (ft³) × air changes per hour ÷ 60. Mechanical ventilation often uses about 0.35 to 0.5 air changes per hour, and the "Room Volume and Ventilation Calculator" can work it out. Air leaks (infiltration) follow the same formula. With a heat recovery ventilator (HRV or ERV), the loss is smaller by the amount of heat recovered.
Total heat loss and the share of each part
Standard notation
(the usual math form)
\(Q\)
\(=\)
\(\sum q_i\)
\(+\)
\(q_v\)
\(p_i\)
\(=\)
\(q_i\)
\(\div\)
\(Q\)
\(\times\)
\(100\)
In words
(symbols replaced with words)
③
\(Q\): total heat loss
\(=\)
①
sum of the heat loss of each part \(q_i\)
\(+\)
②
\(q_v\): ventilation heat loss
⑤
\(p_i\): share of part \(i\)
\(=\)
④
\(q_i\): heat loss of part \(i\)
\(\div\)
\(Q\): total heat loss
\(\times\)
\(100\)
The formula in words
①
Add up the
\(q_i\): heat loss of each part
and add the
②
\(q_v\): ventilation heat loss
to get the
③
\(Q\): total heat loss
④
Divide each
\(q_i\): heat loss of a part
by the total \(Q\) and multiply by 100 to get the
⑤
\(p_i\): share of part \(i\)
(where the heat escapes)
Quick example
With 369.7 BTU/h through the wall, 986.0 through the window, 171.1 through the ceiling, 342.1 through the floor and 272.2 through ventilation, the total and the window's share are
total \(Q\)
\(=\)
parts (1,869.0 BTU/h)
\(+\)
ventilation (272.2 BTU/h)
\(369.7 + 986.0 + 171.1 + 342.1 = 1868.9 \approx 1869.0\,\mathrm{BTU/h}\)
\(1869.0 + 272.2 = 2141.2 \approx 2141.1\,\mathrm{BTU/h}\)
\(986.0 \div 2141.1 \times 100 \approx 46.1\,\%\)
Key idea
The total \(Q\) is how much heat leaves the room per hour if the heat is off, which is the same as the heat the heating system must keep putting out to hold the room temperature. Working out the share of each part shows where fixes pay off, such as "the window is small but loses almost half the heat". In the example 12 × 9 ft room, the window loses about 46%, so adding a storm window or replacing the window is the most effective fix. (The sums in the steps differ by 0.1 because each value is rounded. The calculator adds the unrounded values.) \(\sum\) (sigma) is the symbol for "add them all up".
Heating capacity needed and heat per day
Standard notation
(the usual math form)
\(P\)
\(=\)
\(Q\)
\(\div\)
\(12000\)
\(E\)
\(=\)
\(Q\)
\(\times\)
\(h\)
\(\div\)
\(3412.14\)
In words
(symbols replaced with words)
②
\(P\): heating capacity (tons)
\(=\)
①
\(Q\): total heat loss (BTU/h)
\(\div\)
\(12000\)
④
\(E\): heat per day (kWh)
\(=\)
\(Q\): total heat loss (BTU/h)
\(\times\)
③
\(h\): heating hours per day
\(\div\)
\(3412.14\)
The formula in words
①
The
\(Q\): total heat loss (BTU/h)
is the heating output needed. Dividing it by 12,000 gives the
②
\(P\): heating capacity (tons)
(1 ton = 12,000 BTU/h, the unit used for heat pumps).
③
Multiply the total \(Q\) by the
\(h\): heating hours per day
and divide by 3,412.14 (BTU in 1 kWh) to get the
④
\(E\): heat per day (kWh)
Quick example
For a room with a total heat loss of 2,141.1 BTU/h heated 8 hours a day, the heating capacity and the heat per day are
heat per day \(E\)
\(=\)
total (2,141.1 BTU/h)
\(\times\)
hours (8)
\(\div\)
\(3412.14\)
\(2141.1 \div 12000 \approx 0.18\ \text{tons}\)
\(2141.1 \times 8 \div 3412.14 \approx 5.02\,\mathrm{kWh}\)
Key idea
A heater keeps the room temperature steady by putting out as much heat as escapes. So the total heat loss (BTU/h) is a guide to the heating output needed to hold the temperature. Furnaces and space heaters are rated in BTU/h, and heat pumps are often sized in tons (1 ton = 12,000 BTU/h). A 1,500 W space heater puts out about 5,100 BTU/h. This is the steady-state value. Warming a cold room quickly takes more output, while heat from the sun, people and appliances helps. Real equipment is chosen with some margin, and a whole house is sized with an ACCA Manual J load calculation. BTU/h is heat per hour, and heat per hour × hours = heat for the day. Dividing BTU by 3,412.14 turns it into kWh, the unit on your electric bill.
Estimated electricity cost per day
Standard notation
(the usual math form)
\(C\)
\(=\)
\(E\)
\(\div\)
\(\mathrm{COP}\)
\(\times\)
\(u\)
In words
(symbols replaced with words)
④
\(C\): electricity cost per day
\(=\)
①
\(E\): heat per day
\(\div\)
②
heater \(\mathrm{COP}\)
\(\times\)
③
\(u\): electricity rate
The formula in words
①
Divide the
\(E\): heat per day (kWh)
by the
②
heater \(\mathrm{COP}\) (how many times more heat than electricity it puts out)
to get the electricity used per day (kWh), then multiply by the
③
\(u\): electricity rate ($/kWh)
to get the
④
\(C\): electricity cost per day
Quick example
To supply 5.02 kWh of heat a day (2,141.1 BTU/h × 8 hours) with a heat pump of COP 3 at $0.17/kWh, the electricity cost per day is
cost per day \(C\)
\(=\)
heat (5.02 kWh)
\(\div\)
COP (3)
\(\times\)
rate ($0.17/kWh)
\(5.02 \div 3 \approx 1.67\,\mathrm{kWh}\)
\(1.67 \times 0.17 \approx 0.28\)
Key idea
When heat is made up with electricity, a resistance heater such as a space heater or baseboard heater turns 1 kWh of electricity into 1 kWh of heat (COP = 1). A heat pump moves heat in from the outside air, so it delivers about 2 to 4 kWh of heat per kWh of electricity (COP is heating output ÷ power used, and it drops as the outdoor temperature falls). So for the same heat, the electricity cost is divided by the COP. The same room with a resistance heater would cost about \(5.02 \times 0.17 \approx 0.85\) dollars a day. This is for one day, so multiply by the number of days for a month or a season (the "Electricity Cost Calculator" can help). For a gas furnace, divide the heat by the energy in the fuel and the furnace efficiency (AFUE) to get the fuel used.
U-factor and R-value
Figure
Standard notation
(the usual math form)
\(U\)
\(=\)
\(1\)
\(\div\)
\((\)
\(R_{si}\)
\(+\)
\(R\)
\(+\)
\(R_{se}\)
\()\)
In words
(symbols replaced with words)
④
\(U\): U-factor
\(=\)
\(1\)
\(\div\)
\((\)
①
\(R_{si}\): inside air film
\(+\)
②
\(R\): R-value
\(+\)
③
\(R_{se}\): outside air film
\()\)
The formula in words
①
Add the
\(R_{si}\): inside air film resistance
, the
②
\(R\): R-value of the materials (all layers)
and the
③
\(R_{se}\): outside air film resistance
, then divide 1 by the total to get the
④
\(U\): U-factor
Quick example
For a wall with R-13 fiberglass batts, taking the inside air film as 0.68 and the outside air film as 0.17 (a rough estimate with the insulation only)
U-factor \(U\)
\(=\)
\(1\)
\(\div\)
\((\)
inside (0.68)
\(+\)
insulation (13)
\(+\)
outside (0.17)
\()\)
\(0.68 + 13 + 0.17 = 13.85\ \mathrm{h{\cdot}ft^2{\cdot}{^\circ}F/BTU}\)
\(1 \div 13.85 \approx 0.072\ \mathrm{BTU/(h{\cdot}ft^2{\cdot}{^\circ}F)}\)
Key idea
The R-value printed on insulation (for example R-13 or R-38) is how well it resists heat, and the U-factor \(U\) is the opposite, how easily heat gets through. A wall is made of layers (drywall, insulation, sheathing, siding), and their R-values add up (like resistors in series). There is also some resistance between each surface and the air, called the air film. For walls, the usual values are about 0.68 inside and 0.17 outside (in h·ft²·°F/BTU, from the ASHRAE Handbook). Different values apply to ceilings, floors and other surfaces, so use the values from the energy calculations for the building if you have them. \(U\) is 1 divided by the total. In a real wall, the studs are a thermal bridge where the insulation is missing, so the whole-wall U is higher. A 2×4 wall with R-13 batts works out to about R-11 to R-12 overall, which is why this page's preset is about 0.079. For windows, the NFRC label already gives the whole-window U-factor, so enter it as is. (In metric, R-values are in m²·K/W, and 1 US R-value unit is about 0.176 m²·K/W.)
Heat loss saved by an improvement (comparison)
Standard notation
(the usual math form)
\(\Delta q\)
\(=\)
\((\)
\(U\)
\(-\)
\(U'\)
\()\)
\(\times\)
\(A\)
\(\times\)
\(\Delta T\)
In words
(symbols replaced with words)
⑤
\(\Delta q\): heat loss saved
\(=\)
\((\)
①
\(U\) before
\(-\)
②
\(U'\) after
\()\)
\(\times\)
③
\(A\): area
\(\times\)
④
\(\Delta T\): temperature difference
The formula in words
①
Subtract the
\(U'\): U-factor after the improvement
from the
②
\(U\): U-factor before
and multiply the difference by the
③
\(A\): area
and the
④
\(\Delta T\): temperature difference
to get the
⑤
\(\Delta q\): heat loss saved
Quick example
Replacing 33 ft² of double-pane aluminum window (about 0.83) with low-E double-pane vinyl windows (about 0.39), with a 36 °F difference, the heat loss saved is
heat loss saved \(\Delta q\)
\(=\)
\((\)
before (0.83)
\(-\)
after (0.39)
\()\)
\(\times\)
area (33 ft²)
\(\times\)
difference (36 °F)
\((0.83 - 0.39) \times 33 \times 36 \approx 522.7\,\mathrm{BTU/h}\)
Key idea
Heat loss is proportional to \(U\), so how far you can lower a part's \(U\) is exactly the benefit. The window in the example goes from 986.0 to 463.3 BTU/h, about 24% of the room's total (2,141.1 BTU/h). Enter the U-factor of each option, such as adding a storm window, replacing the glass, or replacing the whole window (ENERGY STAR windows have low U-factors, about 0.30 or less), and you can compare how much each one helps on the same basis. Real results also depend on installation quality and changes in air leaks, so treat this as an estimate.
A room's heat loss is the sum of U-factor \(U\) × area \(A\) × temperature difference \(\Delta T\) for each part, plus the ventilation loss, \(1.08 \times\) CFM \(\times \Delta T\). The total in BTU/h is a guide to the heating output needed to hold the room temperature (÷ 12,000 for tons), and the share of each part shows where the heat escapes. The U-factor is 1 divided by the total R-value, so thicker insulation or better windows lower it, and the difference × area × temperature difference is the heat loss an improvement saves.
Symbols and terms
Symbols
| \(\Delta T\) | delta T | The indoor-outdoor temperature difference (°F, or K in metric). \(\Delta\) (delta) is the Greek letter for "difference" and \(T\) is from "temperature". \(\Delta T = T_{in} - T_{out}\). |
| \(T_{in}\) | T sub in | The indoor temperature (°F). The subscript "in" stands for inside. |
| \(T_{out}\) | T sub out | The outdoor temperature (°F). The subscript "out" stands for outside. For parts facing an attic or crawl space, the real temperature there is a little milder than outdoors. |
| \(i\) | i | The number of a part (1, 2, 3 and so on). From "index". "Part \(i\)" stands for "each part". |
| \(U_i\) | U sub i | The U-factor of part \(i\) (BTU/(h·ft²·°F)). The heat that passes through 1 ft² per hour for each °F of difference. The smaller it is, the better the part insulates. U is the customary symbol for it. |
| \(A_i\) | A sub i | The area of part \(i\) (ft²). From "area". For a wall, use the net area without windows and doors. |
| \(q_i\) | q sub i | The heat loss of part \(i\) (BTU/h). A quantity of heat is customarily written \(q\). \(q_i = U_i \times A_i \times \Delta T\). |
| \(c\) | c | The ventilation factor (BTU/(h·CFM·°F)), the heat lost per hour for each CFM and each °F, usually 1.08. It comes from the heat capacity of air (in metric, about 0.34 Wh/(m³·K)). From "capacity". |
| \(V\) | V | The ventilation rate (CFM, ft³ per minute). From "ventilation". Room volume × air changes per hour ÷ 60. |
| \(q_v\) | q sub v | The ventilation heat loss (BTU/h). The subscript v is for ventilation. \(q_v = c \times V \times \Delta T\). |
| \(\sum\) | sigma | The symbol for "add them all up" (capital Greek sigma). \(\sum q_i\) is the sum of \(q_i\) for all the parts. |
| \(Q\) | capital Q | The total heat loss (BTU/h). The heat loss of all the parts plus the ventilation heat loss, which is the heat the heating system must keep putting out. |
| \(p_i\) | p sub i | The share of part \(i\) (%). From "proportion". \(p_i = q_i \div Q \times 100\). |
| \(P\) | capital P | The heating capacity needed (tons). From "power". \(Q\) divided by 12,000 (1 ton = 12,000 BTU/h). |
| \(h\) | h | The heating hours per day. From "hour". |
| \(E\) | E | The heat needed per day (kWh). From "energy". \(E = Q \times h \div 3412.14\). |
| \(\mathrm{COP}\) | C O P | The coefficient of performance of a heater: how many times more heat than electricity it puts out. 1 for a resistance heater and about 2 to 4 for a heat pump. |
| \(u\) | lowercase u | The electricity rate ($/kWh). From "unit price". Not the same as the U-factor \(U\) (capital). |
| \(C\) | capital C | The estimated electricity cost per day ($). From "cost". \(C = E \div \mathrm{COP} \times u\). |
| \(R\) | R | The R-value of the materials (h·ft²·°F/BTU). From "resistance". The R-values of the layers add up. |
| \(R_{si}\) | R sub s i | The inside air film resistance. The subscript si stands for surface inside. For walls it is usually about 0.68 h·ft²·°F/BTU. |
| \(R_{se}\) | R sub s e | The outside air film resistance. The subscript se stands for surface exterior. For walls exposed to wind it is usually about 0.17 h·ft²·°F/BTU. |
| \(U'\) | U prime | The U-factor after the improvement (BTU/(h·ft²·°F)). The prime mark customarily marks "the changed value". |
| \(\Delta q\) | delta q | The heat loss saved by an improvement (BTU/h). The difference between the heat loss before and after, \(\Delta q = (U - U') \times A \times \Delta T\). |
| \(\mathrm{W}\) | watt | The metric unit of heat (energy) per second. 1 W = 1 joule per second ≈ 3.412 BTU/h. |
| \(\mathrm{K}\) | kelvin | A unit of temperature used for differences in metric. A difference of 1 K is the same as 1 °C, or 1.8 °F. |
| \(\mathrm{kWh}\) | kilowatt-hour | A unit of energy: 1 kW (1,000 W) for 1 hour, equal to 3,412.14 BTU. Electricity is billed in it. |
Terms
| heat loss | The heat that escapes from a room through the walls, windows, ceiling, floor and ventilation, measured per hour in BTU/h (or per second in watts). A heating system holds the room temperature by putting out as much heat as is lost. |
| U-factor | The heat a wall, window or other part lets through per hour for each square foot and each °F of difference (BTU/(h·ft²·°F)), also called U-value. The smaller it is, the better it insulates. It is 1 divided by the total R-value. Windows and doors show it on the NFRC label. |
| R-value | How well a layer resists heat (h·ft²·°F/BTU), printed on insulation as R-13, R-30 and so on. The larger it is, the better the layer insulates. The R-values of layers add up. |
| thermal conductivity | How easily heat travels through a material, often written as k (BTU·in/(h·ft²·°F)). The smaller it is, the better the material insulates. R-value = thickness ÷ conductivity. |
| air film | The resistance to heat flow between a surface and the air, given as an R-value. For walls, about 0.68 inside and 0.17 outside are usually used. Add them to the R-values of the materials before taking 1 ÷ total for the U-factor. |
| temperature difference | The difference between the indoor and outdoor temperatures. Heat flow is proportional to it, so every calculation on this page starts from it. |
| ventilation rate | The volume of air exchanged, in CFM (ft³ per minute) in the US. Room volume × air changes per hour ÷ 60. |
| air changes per hour | How many times the air in a room is replaced in an hour (ACH). Mechanical ventilation often uses about 0.35 to 0.5 ACH. |
| volumetric heat capacity | The heat needed to warm a unit volume of a material by one degree. For air it is density × specific heat, which gives the ventilation factor 1.08 BTU/(h·CFM·°F) (about 0.34 Wh/(m³·K) in metric). |
| steady state | When the indoor and outdoor temperatures stay constant and heat in and out are in balance. The formulas on this page assume it, so the extra heat to warm up a room and changing weather are not included. |
| heating load | The heat a heating system must put out to hold the room temperature. On this page it is the total heat loss. Detailed design calculations subtract heat from the sun and inside the house. |
| building envelope | The parts that enclose a building (walls, roof, ceiling, floor, windows and doors). A whole house's insulation performance is judged by the heat loss of the whole envelope. |
| UA | The sum of U × A over the whole building envelope (BTU/(h·°F)). Code compliance tools such as REScheck compare it with a code-built house. It covers the whole building, not the single room this page estimates. |
| Manual J | The ACCA (Air Conditioning Contractors of America) standard method for calculating a home's heating and cooling loads, used to size furnaces, heat pumps and air conditioners. It goes into much more detail than this page. |
| thermal bridge | A spot where heat gets through more easily because the insulation is interrupted, such as studs, headers and metal fasteners. Whole-wall U-factors account for the framing. |
| buffer space | An unheated space such as an attic, crawl space or garage that is milder than the outdoors. Detailed calculations use a smaller temperature difference for parts facing it. This page does not, so ceilings and floors come out a little high. |
| attic | The space between the ceiling and the roof. Ceiling insulation is laid here, and heat escaping through the ceiling goes into the attic first, which is a little milder than outdoors. |
| continuous insulation | A layer of rigid insulation over the outside (or inside) of the studs, in addition to the insulation between them. It is not broken by the studs, so it lowers the wall's U-factor a lot. |
| openings | Windows, doors and other openings in a wall. Use the wall area minus the openings, and enter windows and doors as their own parts. |
| single pane | A window with one layer of glass. It insulates poorly, and in an aluminum frame the U-factor is often over 1.1 BTU/(h·ft²·°F). |
| double pane | Two layers of glass with air or gas between them, also called insulated glass. It lets through much less heat than single pane. |
| low-E | A thin metallic coating on window glass that reflects heat. Low-E stands for low emissivity. Low-E double-pane glass in a vinyl frame brings the window U-factor down to about 0.3 to 0.4. |
| vinyl frame | A window frame made of vinyl (PVC). Aluminum conducts heat well, so vinyl or wood frames give a lower whole-window U-factor than aluminum frames. |
| storm window | An extra window added inside or outside an existing window. It is a common way to lower a window's U-factor, and you can use the comparison inputs to see its effect. |
| COP | The coefficient of performance: how many times more heat than electricity a heater puts out. 1 for a resistance heater and about 2 to 4 for a heat pump. It is heating output ÷ power used. |
| heat pump | A system that moves heat from the outside air (or the ground) into the house. It delivers the same heat with less electricity than a resistance heater. |
| solar heat gain | Heat that comes in as sunlight through windows. In winter it offsets some of the heat loss, but this page's estimate does not include it. |
| internal heat gain | Heat from people, appliances and lights. It helps the heating, so design calculations subtract it, but this page's estimate does not include it. |
| infiltration | Air leaking in and out through gaps around windows, doors and other cracks. It causes heat loss by the same formula as ventilation, so in a leaky house, enter a higher ventilation rate to get closer to reality. |
| heat recovery ventilator | A ventilation unit (HRV, or ERV that also moves moisture) that passes heat from the outgoing air to the incoming air. The ventilation heat loss is smaller by the heat it recovers. |
Good to know before you start
Here is what helps to understand before you start, so that you can use the calculations on this page with a clear understanding.
| Multiplying and dividing decimals (Grade 5 to 6) |
|
| Percents (Grade 6) |
|
| Converting units (Grade 6 and middle school science) |
|
| Proportional relationships (Grade 6 to 7) |
|
| Reciprocals (Grade 6) |
|
| How heat moves (elementary and middle school science) |
|
| Power and energy (middle school science) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table for the temperature difference
| Indoor temperature (°F) | 68 |
| Outdoor temperature (°F) | 32 |
| Temperature difference ΔT (°F) | =B1-B2 |
Table for the heat loss of one part
| U-factor U (BTU/(h·ft²·°F)) | 0.83 |
| Area A (ft²) | 33 |
| Temperature difference ΔT (°F) | 36 |
| Heat loss q (BTU/h) | =B1*B2*B3 |
Table for the ventilation heat loss
| Ventilation factor c (BTU/(h·CFM·°F)) | 1.08 |
| Ventilation rate V (CFM) | 7 |
| Temperature difference ΔT (°F) | 36 |
| Ventilation heat loss qv (BTU/h) | =B1*B2*B3 |
Table for the total heat loss and the share of each part
| Wall heat loss (BTU/h) | 369.72 |
| Window heat loss (BTU/h) | 986.04 |
| Ceiling heat loss (BTU/h) | 171.072 |
| Floor heat loss (BTU/h) | 342.144 |
| Ventilation heat loss (BTU/h) | 272.16 |
| Total heat loss Q (BTU/h) | =SUM(B1:B5) |
| Window share (%) | =B2/B6*100 |
Table for the heating capacity and heat per day
| Total heat loss Q (BTU/h) | 2141.136 |
| Heating hours per day h | 8 |
| Heating capacity P (tons) | =B1/12000 |
| Heat per day E (kWh) | =B1*B2/3412.14 |
Table for the estimated electricity cost per day
| Heat per day E (kWh) | 5.02 |
| Heater COP | 3 |
| Electricity rate u ($/kWh) | 0.17 |
| Electricity cost per day C ($) | =B1/B2*B3 |
Table for the U-factor from R-values
| Inside air film Rsi (h·ft²·°F/BTU) | 0.68 |
| R-value of the materials R (h·ft²·°F/BTU) | 13 |
| Outside air film Rse (h·ft²·°F/BTU) | 0.17 |
| U-factor U (BTU/(h·ft²·°F)) | =1/(B1+B2+B3) |
Table for the heat loss saved by an improvement
| U-factor before U | 0.83 |
| U-factor after U' | 0.39 |
| Area A (ft²) | 33 |
| Temperature difference ΔT (°F) | 36 |
| Heat loss saved Δq (BTU/h) | =(B1-B2)*B3*B4 |
After you paste, the upper rows of column B are the inputs and the last rows are the calculated results.
SUM(B1:B5) adds up B1 through B5 (the Σ in the formulas).
B3 in the 1st table is 36, B4 in the 2nd table is about 986.0, B4 in the 3rd table is about 272.2, the 4th table gives about 2,141.1 in B6 and about 46.1 in B7, the 5th table gives about 0.18 in B3 and about 5.02 in B4, B4 in the 6th table is about 0.28, B4 in the 7th table is about 0.072, and B5 in the 8th table is about 522.7. Just replace the numbers in column B with your own. To add parts, add rows to the 4th table and widen the SUM range.
SUM(B1:B5) adds up B1 through B5 (the Σ in the formulas).
B3 in the 1st table is 36, B4 in the 2nd table is about 986.0, B4 in the 3rd table is about 272.2, the 4th table gives about 2,141.1 in B6 and about 46.1 in B7, the 5th table gives about 0.18 in B3 and about 5.02 in B4, B4 in the 6th table is about 0.28, B4 in the 7th table is about 0.072, and B5 in the 8th table is about 522.7. Just replace the numbers in column B with your own. To add parts, add rows to the 4th table and widen the SUM range.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table for the temperature difference
| Indoor temperature (°F) | 68 |
| Outdoor temperature (°F) | 32 |
| Temperature difference ΔT (°F) | =B1-B2 |
Table for the heat loss of one part
| U-factor U (BTU/(h·ft²·°F)) | 0.83 |
| Area A (ft²) | 33 |
| Temperature difference ΔT (°F) | 36 |
| Heat loss q (BTU/h) | =B1*B2*B3 |
Table for the ventilation heat loss
| Ventilation factor c (BTU/(h·CFM·°F)) | 1.08 |
| Ventilation rate V (CFM) | 7 |
| Temperature difference ΔT (°F) | 36 |
| Ventilation heat loss qv (BTU/h) | =B1*B2*B3 |
Table for the total heat loss and the share of each part
| Wall heat loss (BTU/h) | 369.72 |
| Window heat loss (BTU/h) | 986.04 |
| Ceiling heat loss (BTU/h) | 171.072 |
| Floor heat loss (BTU/h) | 342.144 |
| Ventilation heat loss (BTU/h) | 272.16 |
| Total heat loss Q (BTU/h) | =SUM(B1:B5) |
| Window share (%) | =B2/B6*100 |
Table for the heating capacity and heat per day
| Total heat loss Q (BTU/h) | 2141.136 |
| Heating hours per day h | 8 |
| Heating capacity P (tons) | =B1/12000 |
| Heat per day E (kWh) | =B1*B2/3412.14 |
Table for the estimated electricity cost per day
| Heat per day E (kWh) | 5.02 |
| Heater COP | 3 |
| Electricity rate u ($/kWh) | 0.17 |
| Electricity cost per day C ($) | =B1/B2*B3 |
Table for the U-factor from R-values
| Inside air film Rsi (h·ft²·°F/BTU) | 0.68 |
| R-value of the materials R (h·ft²·°F/BTU) | 13 |
| Outside air film Rse (h·ft²·°F/BTU) | 0.17 |
| U-factor U (BTU/(h·ft²·°F)) | =1/(B1+B2+B3) |
Table for the heat loss saved by an improvement
| U-factor before U | 0.83 |
| U-factor after U' | 0.39 |
| Area A (ft²) | 33 |
| Temperature difference ΔT (°F) | 36 |
| Heat loss saved Δq (BTU/h) | =(B1-B2)*B3*B4 |
The same formulas as in Excel work as they are (SUM has the same name). Copy the whole table, paste it into cell A1, and replace the numbers in column B with your own.
How to calculate it in Python
t_in = 68 # indoor temperature (F)
t_out = 32 # outdoor temperature (F)
# each part: (name, U-factor [BTU/(h*ft2*F)], area [ft2]); change the U-factors to values from labels or the plans
parts = [
("Wall", 0.079, 130),
("Window", 0.83, 33),
("Ceiling", 0.044, 108),
("Floor", 0.088, 108),
]
ventilation_cfm = 7 # ventilation rate (CFM); 0 if not used
vent_factor = 1.08 # ventilation factor (BTU/(h*CFM*F))
heating_hours = 8 # heating hours per day
cop = 3 # heater COP (1 for a resistance heater)
price_per_kwh = 0.17 # electricity rate ($/kWh)
improved = ("Window", 0.39) # improvement: (name of the part, U-factor after U')
delta_t = t_in - t_out # temperature difference (F)
losses = [(name, u * area * delta_t) for name, u, area in parts] # heat loss of each part q_i (BTU/h)
envelope = sum(q for _, q in losses) # heat loss through the parts
vent = vent_factor * ventilation_cfm * delta_t # ventilation heat loss q_v (BTU/h)
total = envelope + vent # total heat loss Q (BTU/h)
tons = total / 12000 # heating capacity (tons)
energy_kwh = total * heating_hours / 3412.14 # heat per day E (kWh)
cost = energy_kwh / cop * price_per_kwh # electricity cost per day C ($)
print(f"Temperature difference: {delta_t} F")
for name, q in losses:
print(f"{name}: {q:.1f} BTU/h ({q / total * 100:.1f} %)")
print(f"Ventilation: {vent:.1f} BTU/h ({vent / total * 100:.1f} %)")
print(f"Total: {total:.1f} BTU/h -> heating capacity {tons:.2f} tons")
print(f"Heat per day: {energy_kwh:.2f} kWh, electricity cost per day: ${cost:.2f}")
# improvement: dq = (U - U') x A x dT
for name, u, area in parts:
if name == improved[0]:
saved = (u - improved[1]) * area * delta_t
print(f"Improving the {name} to U={improved[1]} saves {saved:.1f} BTU/h ({saved / total * 100:.1f} % of the total)")
It runs with the standard library only. Add or remove parts in the parts list, and the same formulas work for any number of parts. Replace the temperatures, parts, ventilation, hours and rate at the top with your own numbers and run it (the example gives a total of 2,141.1 BTU/h and 0.18 tons, 5.02 kWh and about $0.28 a day, and 522.7 BTU/h saved by the window upgrade).
How to write it in LaTeX and other math languages (copy and paste)
Indoor-outdoor temperature difference
ΔT = T_in − T_out
\Delta T = T_{in} - T_{out}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>Δ</mi><mi>T</mi><mo>=</mo>
<msub><mi>T</mi><mi>in</mi></msub><mo>−</mo><msub><mi>T</mi><mi>out</mi></msub>
</mrow>
</math>
Delta T = T_(in) - T_(out)
tIn - tOut
DeltaT := T_in - T_out;
DeltaT = T_in - T_out;
ΔT = T_in − T_out
Heat loss through each part
qᵢ = Uᵢ × Aᵢ × ΔT
q_i = U_i \times A_i \times \Delta T
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>q</mi><mi>i</mi></msub><mo>=</mo>
<msub><mi>U</mi><mi>i</mi></msub><mo>×</mo>
<msub><mi>A</mi><mi>i</mi></msub><mo>×</mo>
<mi>Δ</mi><mi>T</mi>
</mrow>
</math>
q_i = U_i xx A_i xx Delta T
u*a*deltaT
q := U*A*DeltaT;
q = U*A*DeltaT;
q_i = U_i × A_i × ΔT
Ventilation heat loss
q_v = c × V × ΔT
q_v = c \times V \times \Delta T
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>q</mi><mi>v</mi></msub><mo>=</mo>
<mi>c</mi><mo>×</mo><mi>V</mi><mo>×</mo>
<mi>Δ</mi><mi>T</mi>
</mrow>
</math>
q_v = c xx V xx Delta T
c*v*deltaT
q_v := c*V*DeltaT;
q_v = c*V*DeltaT;
q_v = c × V × ΔT
Total heat loss and the share of each part
Q = Σ qᵢ + q_v, pᵢ = qᵢ ÷ Q × 100
Q = \sum_{i} q_i + q_v,\quad p_i = \frac{q_i}{Q} \times 100
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>Q</mi><mo>=</mo>
<munder><mo>∑</mo><mi>i</mi></munder><msub><mi>q</mi><mi>i</mi></msub>
<mo>+</mo><msub><mi>q</mi><mi>v</mi></msub>
<mo>,</mo>
<msub><mi>p</mi><mi>i</mi></msub><mo>=</mo>
<mfrac><msub><mi>q</mi><mi>i</mi></msub><mi>Q</mi></mfrac>
<mo>×</mo><mn>100</mn>
</mrow>
</math>
Q = sum q_i + q_v, p_i = q_i / Q xx 100
{Total[q] + qv, q/(Total[q] + qv)*100}
Q := add(q[i], i = 1 .. n) + q_v; p[i] := q[i]/Q*100;
Q = sum(q) + q_v; p = q / Q * 100;
Q = ∑ q_i + q_v, p_i = q_i/Q × 100
Heating capacity needed and heat per day
P = Q ÷ 12000, E = Q × h ÷ 3412.14
P = \frac{Q}{12000},\quad E = \frac{Q \times h}{3412.14}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>P</mi><mo>=</mo><mfrac><mi>Q</mi><mn>12000</mn></mfrac>
<mo>,</mo>
<mi>E</mi><mo>=</mo><mfrac><mrow><mi>Q</mi><mo>×</mo><mi>h</mi></mrow><mn>3412.14</mn></mfrac>
</mrow>
</math>
P = Q / 12000, E = Q xx h / 3412.14
{q/12000, q*h/3412.14}
P := Q/12000; E := Q*h/3412.14;
P = Q/12000; E = Q*h/3412.14;
P = Q/12000, E = Q × h/3412.14
Estimated electricity cost per day
C = E ÷ COP × u
C = \frac{E}{\mathrm{COP}} \times u
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>C</mi><mo>=</mo>
<mfrac><mi>E</mi><mi>COP</mi></mfrac>
<mo>×</mo><mi>u</mi>
</mrow>
</math>
C = E / COP xx u
e/cop*u
C := E/COP*u;
C = E/COP*u;
C = E/COP × u
U-factor and R-value
U = 1 ÷ (R_si + R + R_se)
U = \frac{1}{R_{si} + R + R_{se}}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>U</mi><mo>=</mo>
<mfrac><mn>1</mn><mrow>
<msub><mi>R</mi><mi>si</mi></msub><mo>+</mo><mi>R</mi><mo>+</mo><msub><mi>R</mi><mi>se</mi></msub>
</mrow></mfrac>
</mrow>
</math>
U = 1 / (R_(si) + R + R_(se))
1/(rSi + r + rSe)
U := 1/(R_si + R + R_se);
U = 1/(R_si + R + R_se);
U = 1/(R_si + R + R_se)
Heat loss saved by an improvement (comparison)
Δq = (U − U') × A × ΔT
\Delta q = (U - U') \times A \times \Delta T
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>Δ</mi><mi>q</mi><mo>=</mo>
<mrow><mo>(</mo><mi>U</mi><mo>−</mo><msup><mi>U</mi><mo>′</mo></msup><mo>)</mo></mrow>
<mo>×</mo><mi>A</mi><mo>×</mo><mi>Δ</mi><mi>T</mi>
</mrow>
</math>
Delta q = (U - U') xx A xx Delta T
(u - uNew)*a*deltaT
Delta_q := (U - U_new)*A*DeltaT;
Delta_q = (U - U_new)*A*DeltaT;
Δq = (U − U') × A × ΔT
How to have ChatGPT do the calculation
You are an assistant for home heat loss calculations. Do the following calculation by actually running Python code, and base your answer only on the numbers from the run (do not answer from mental math or guesses). A room is kept at 68 °F with the outdoor temperature at 32 °F. The U-factor U and area A of each part are: Wall: 'U = 0.079 BTU/(h·ft²·°F), A = 130 ft² / Window: U = 0.83, A = 33 ft² / Ceiling: U = 0.044, A = 108 ft² / Floor: U = 0.088, A = 108 ft²' The ventilation rate is 7 CFM (ventilation factor 1.08 BTU/(h·CFM·°F)), the heating time is 8 hours a day, the heater COP is 3, and the electricity rate is $0.17/kWh. Find each of the following. 1. The temperature difference ΔT and the heat loss of each part q = U × A × ΔT (BTU/h) 2. The ventilation heat loss 1.08 × CFM × ΔT (BTU/h) 3. The total heat loss (BTU/h) and the share of each part and of ventilation (%) 4. The heating capacity in tons (total ÷ 12,000), the heat per day in kWh (total × hours ÷ 3,412.14), and the electricity cost per day from the COP and rate ($) 5. The heat loss saved (BTU/h) by improving the window to U = 0.39, and its share of the total (%) Show the formulas you used and the numbers from the run.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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