If the house is electrically heated at a cost of $0.06/(kW hr), estimate the savings achieved with a double-pane compared to a single-pane window during December.
Determine the rate of heat loss through a double paned window, as shown in the sketch,
if the inside room temperature is 65°C and the average outside air is 0°C during
December. Neglect the effect of the window frame.
GIVEN
• A double-paned window as shown
• Inside room temperature (T?i) = 65°C
• Outside air temperature (T?o) = 0°C
• Cost of heating = $0.06/(kW hr)
FIND
(a) The rate of heat loss through a double paned window (b) The saving of double paning over single paning
ASSUMPTIONS
• Saving can be based on steady state analysis
• Inside and outside air is still
• Radiative heat transfer is negligible
PROPERTIES AND CONSTANTS
thermal conductivity of window glass (kg) = 0.81 W/(m K)
A single glazed window will be analyzed first
Since the temperature of the glass is unknown, an iterative procedure is required. For the first iteration, let Tg = 32.5°C.
for dry air
The Grashof number based on the window height is
Inside
Outside
Since GrH> 109, the Nusselt numbers are
Inside
Outside
The rate of convection inside and outside must be the same
Solving for the glass temperature
This is close enough to the initial guess that the heat transfer coefficients will not be re-calculated.
The thermal circuit for the single glazed window is shown below
where
The rate of heat transfer is given by
For the double glazed case, the temperature of the inside and outside panes must be estimated for the
first iteration. Let Tgo = 16°C and Tgi = 49°C (by symmetry).
for dry air
The Grashof numbers are
Inside
Outside
Enclosed space
Using the correlation in
Inside
Outside
Enclosed space: H/? = 0.8 m/0.05 m = 16. Although Pr < 1, the correlation in will be applied to estimate the Nusselt number for the enclosed space
The thermal circuit for the double glazed window is shown below
The thermal resistance of the glass (Rkg) is the same as the single glazed case. The remaining thermal resistances are
The rate of heat transfer is
The inner and outer glass temperatures can be checked by the convection equations for those surfaces
Inside
Outside
These are close enough to the initial guesses that another iteration is not warranted.
The savings of the double glazed over the single glazed window are
This one small double glazed window saves 6 cents per day.
COMMENTS
The two surfaces of each pane of glass will actually be at a slightly different temperature. This can be neglected because the convective resistance are an order of magnitude greater than the conductive resistance of the glass.
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