On a cold winter day, the outside wall of a home is exposed to air temperature of –2°C when the inside temperature of room is at 22°C. As a result of this temperature gradient, there is heat loss through the wall to the outside. Consider the convective heat transfer coefficients for the air inside the room and at the outside wall surface to be respectively 12.0 and 28.0 W/(m2 K). If the composite room wall is modelled as a plane wall with thermal resistance per unit area of 0.5 m2 K/W, determine the temperature at the outer surface of the wall as well as the rate of heat flow through the wall per unit area. If the homeowner were to consider using a fiberglass insulation on the inside wall surface for reducing this heat loss by 50 %, what is the required thickness of this layer and the

Question

outside wall temperature for this case?



GIVEN

• Heat transfer through a plane wall

• Air temperature

? Inside wall (Ti) = 22°C

? outside wall (To) = –2°C

• Heat transfer coefficient

? Inside wall ( ci h ) = 12 W/(m2 K)

? Outside wall ( co h ) = 28 W/(m2 K)

• Thermal resistance of a unit area (A Rw) = 0.5 (m2 K)/W FIND (a) Temperature of the outer surface of the wall (Two) (b) Rate of heat flow through the wall per unit area (q/A) (c) Thickness of fiberglass insulation inside wall surface (d) Outside wall temperature with fiberglass insulation

ASSUMPTIONS

• One dimensional heat flow

• Steady state has been reached

PROPERTIES AND CONSTANTS

Thermal conductivity

Fiber glass (k) =0.035 W/m K

SKETCH

Answer 1

The thermal circuit for the wall is shown below



The rate of heat transfer can be used to calculate the temperature of the outer surface of the wall,

therefore part (b) will be solved first.

(b) The heat transfer situation can be visualized using the thermal circuit shown above. The total heat

transfer through the wall,



The three thermal resistances are in series, therefore



The heat flow through the wall is



(a) The temperature of the outer surface of the wall can be calculated by examining the

convective heat transfer from the outside of the wall





With fiber glass included as insulation, total resistance becomes



For reducing heat loss by 50 % the overall resistance should be doubled, thus





The temperature of the outer surface of the wall can be calculated by examining the convective heat

transfer from the outside of the wall