Home Heating

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Where are these houses losing the most heat to the environment? The picture on the left is taken with visible light, that on the right is taken with infrared light with a wavelength of about 10 μm.

Big Ideas: 
  • To maintain a building at a different temperature from the environment requires constant heating. In steady state, the power going into the building equals the power out.

Where are these houses losing the most heat to the environment? The picture on the left is taken with visible light, that on the right is taken with infrared light with a wavelength of about 10 μm.

To maintain a building at a temperature different from that of the immediate environment, the power input to the building has to be equal to the power output to its surroundings. Power input comes from intentional heating (furnace), unintentional heating (electrical appliances, lights, people), and solar radiation (particularly noticeable during sunny, cold days in winter). Power output arises from conduction and radiation to the outside world, and through ventilation losses as warm air leaks, or is ventilated, to the outside world.

The theory and practice of heating buildings is well written up in many places1, and I do not intend to reproduce all the information here. Instead we will approach the physics through a set of experiments. The first set of experiments explore how to model the cooling of thermal systems by plotting their energy loss against time. Under controlled experimental conditions the thermal time constant, a characteristic of thermal systems, can also be measured.

Thermal Time Constants

According to Newton's Law of cooling, the rate of heat loss from an object should be proportional to the difference in temperature between that object and its surroundings. Heat transfer occurs in three ways: conduction, convection, and radiation. Conduction is the transfer of heat from high temperature to low temperature by the direct contact of atoms or molecules, convection is the transfer of energy between a solid and a nearby gas or liquid in motion and radiation is the transfer of heat through space (for example heating of Earth by the Sun). It is possible to derive an expression for a thermal time constant if the process of heat transfer is limited to conduction, where convection and radiation are assumed to be negligible. Radiation, which follows Stefan-Boltzmann's Law (P=σAεT4) can be minimized if the emissivity (ε) of the container is low and convection can be minimized by eliminating environmental factors such as air currents.

Let's consider the case of a mass of water held in a thin-walled box of Styrofoam. The mass of water holds the heat and the Styrofoam transmits it in and out (rather slowly - it is important for this analysis that the thermal conductivity k of the Styrofoam be much less than that of the water). The box has a thickness d and a surface area A; the water has a mass m and a specific heat capacity c. The Styrofoam is thin enough that we don't have to consider its heat capacity, which is defined to be its specific heat capacity multiplied by its mass. The temperature of the water is T and that of the environment is T0.

Accourding to Fourier's Law of heat conduction, the rate of heat flow ΔQt from the water to the environment is given by

$ \dfrac{\Delta Q}{\Delta t}=\dfrac{kA(T-T_{o})}{d} $

and is measured in watts. If the water is cooler than the surroundings, it is negative.

However, there is a relationship between the change in heat ΔQ of an object and its change in temperature ΔT=T-T0 (if it doesn't undergo a phase change like boiling or freezing):

$ \Delta Q = mc \Delta T $

We can combine these two equations to discover how the temperature changes with time:

$ \dfrac{\Delta T}{\Delta t} = \dfrac{kA \Delta T}{mcd} $

So the rate of change of temperature difference is proportional to the temperature difference itself. This type of proportionality arises time and again in the sciences and the mathematical solution to this equation is an exponential function (See Figure 1).

$ \Delta T = \Delta T_{o}e^{-t/\tau} $

where ΔT0 is the initial temperature difference and τ is the thermal time constant.

In this case we find $ \tau $ to be

$ \tau = \dfrac{mdc}{kA} $

If you plot the natural logarithm of ΔT, you get a straight line whose slope is 1/$ \tau $.

$ \ln \Delta T = \ln( \Delta T_{o}e^{-t/ \tau} = -\dfrac{t}{\tau} + \ln( \Delta T_{o}) $

Figure 1. Example of an exponential decay with a time constant of 2 h. Note that the time constant can be found by drawing a tangent line at t =0 and seeing where it crosses the x-axis.

This method of finding the thermal time constant will only produce accurate results if the heat transfer mechanism is by conduction and if the outside temperature is constant. Experimentally this is very difficult to achieve. Thus in Experiment 1a, the actual value of the thermal time constant is not calculated. Instead amount of heat loss of an insulated container is measured and plotted using (ΔQ=mcΔT) and predicted using the law of conduction:

$ \dfrac{\Delta Q}{\Delta t}=\dfrac{kA(T-T_{o})}{d} $

Comparing the two plots will show whether the primary heat loss mechanism of the insulated constainer is indeed conduction. See Experiment 1a: Hot water cooling write-up for more details.

Resources
Lecture Notes: 
Home Heating Lecture Notes
Take Home Experiment: 
Home Heating Take-Home Experiments
Multiple Choice Problems: 
Home Heating Multiple Choice Questions

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