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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