Conduction
Conduction
Concepts
R-value
U-value
Fourier's Law
Thought experiment
Given a block of material sitting between two temperatures, what heat flow do you expect?
Fourier's Law
dimensions of energy per time or power
dimensions of area
dimensions of power per distance per degree
is the temperature difference
is the thickness of the material
Fourier's Law Differential Form
dimensions of power per unit area
dimensions of power per distance per degree
Fourier's Law, Buildings Form
heat transfer dimensions of power
dimensions of power per area per degree temperature
dimension of area
dimension of temperature
Conductivity

R value and U value
Building materials publish an R-value
Sometimes published as a U-value
Conduction

Conduction

Conduction

Units
in US units R value is
in SI units R value is
Parallel
If you have two conducting surfaces in parallel, the U-values add
In parallel, the heat can take either path
Series
If you have two conducting surfaces in series, the U-values add
according to
In series, the heat must take both paths
Bathtub model of heat flow
What is the input?
Now the drain is faster with greater temperature
Typical R and U values
Window range (US R-values)
R-1 for single pane
R-12.5 for more advanced windows
Wall range
R-3.4 (2x4 no inssulation)
R-12.7 (2x4 R-13 insulation)
R-34.6 (2x6 R-21 insulation)
Cylindrical Insulator
Note that this R does not have the same dimensions as R-value. It has dimensions of temperature difference per unit power.

Spherical Insulator
Note that this R does not have the same dimensions as R-value. It has dimensions of temperature difference per unit power.

Typical UA values
Residential Home ?
Commercial Building ?
Cooler ?
Down Jacket ?
Activities
R-Value of SIP
Are the components of the SIP in parallel or series?
How do we find the properties of each?
Wood 0.15 watts per kelvin per meter
Polyurethane foam 0.02 watts per kelvin per meter
4.5 inch panel 13.8 R value
Convert R-values
Convert between a US R-value and a metric R-value.

Once you learn how to do this, you can use the value you calculate as a conversion factor. This will help you convert more quickly.
Estimate Wall Loss in ETC
What is the R-value?
What is the total area of walls?
How much power do we need to maintain the ETC one degree above the
outside temperature?
Calculus Approach to Heat Loss through conduction
We calculate the temperature as a function of time of a heated object that loses heat to its surroundings through an insulation. We start with a lumped mass approximation with conductive heat loss through an insulator. Using the heat capacity of the object we have the relation
Where , the heat capacity is the product of the density, , the volume of the object , and the specific heat capacity of the material .
Over a small time interval , the heat lost by the object as heat conducts away is the product of the temperature difference , the thermal conductivity of the insulation, and the time interval .
Substituting, we get
which we rearrange and integrate
with initial conditions at and at . Integrating, we get
We exponentiate both sides and get
The equation shows an exponential decay in temperature starting at , the initial temperature of the object, decaying to .
Discrete approach
We can also do this numerically with a discrete time period.
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