Conduction

Fourier’s Law

The fundamental formula for heat conduction in a material is Fourier’s Law.

q=kT q = -k \nabla T

  • qq has dimensions of power per unit area
  • kk has dimensions of power per distance per degree
  • TT is the temperature over space
  • \nabla computes the direction and magnitude of the greatest temperature change

If you don’t speak vector calculus (most don’t) the formula tells you that

  • the direction of heat energy flow is in the direction of decreasing temperature
  • the amount of heat energy flow is proportional to a material property, kk, and how abrupt the temperature change is.
  • the minus sign tells us heat flows from hot to cold

Consequences

If a material is very hot on one side and cool on the other, there will be a large amount of heat flow.

If a material has a uniform temperature, there is no heat flowing.

Conduction

Fourier’s law covers many complex situations, but we can simplify this for our purposes.

Our simplification is called one-dimensional, steady-state conduction and it works for many real-life applications including buildings.

If we have

  • a material (the bar in the middle)
  • negligible heat loss on the edges of material
  • a steady warm temperature on one side
  • a steady cool temperature on one side
  • and have waited until the temperature is no longer changing

We can use the following formula:

q=kAΔTL q = \frac{k A \Delta T}{L}

Where qq is the heat power flowing across the material, kk is the conductivity, AA is the cross-section area of the material, and ΔT\Delta T is the temperature difference on the sides of the material, and LL is the length of the material in the same direction as the heat flow.

What this means is that if I know the properties of a building wall and the inside and outside temperature, I can predict how much power is flowing across it.

Fourier’s Law, Buildings Form

Walls and buildings have several elements (walls, sheetrock, insulation, lumber, etc.). We can combine these into a single value called the UA value to perform power estimations for an entire building.

q=UAΔT q = U A \Delta T

  • qq heat transfer dimensions of power
    • watts or BTU/hour
  • UU dimensions of power per area per degree temperature
    • watts/square meter/degree K
    • BTU/hour/square foot/degree F
  • AA dimension of area
    • square meters
    • square feet
  • TT dimension of temperature
    • Kelvin/Celsius
    • Fahrenheit
Quantity Dimensions Metric Units Imperial Units
qq heat transfer power watts BTU / hour
UU power per area per temperature difference watts / square meter / K BTU / hour / square foot / F
RR area times temperature difference per power square meter \cdot K / watt square foot \cdot F \cdot hour / BTU
AA area square meters square feet
ΔT\Delta T temperature difference Kelvin or Celsius Fahrenheit

R-value

U=1R U = \frac{1}{R}

RR has

  • dimensions of area times temperature difference divided by power
  • square meters times celsius per watt
  • square feet times fahrenheit times hour per BTU

Formula and Unit Rodeo

The equation above is presented in several forms with different units.

q=UAΔT q = UA \Delta T

q=1RAΔT q = \frac{1}{R} A \Delta T

Conductivity

This graph shows the range of conductivities for different materials.

We can calculate the U-value of a material from

U=kL U = \frac{k}{L}

Examples

You have a wall with a UA value of 5 watts per Kelvin that is 25C on the surface of one side and 5C on the surface of the other side.

If the wall is at steady-state, how much heat is flowing in watts.

q=UAΔT q = UA\Delta T q=5w/K(25C5C) q = 5 w/K \cdot (25C-5C) q=100W q = 100 W