Question

Consider a house that has a 10-m x 20-m base and a 4-m-high wall. All four walls of the house have an R-value of 2.31m2°C/W. The two 10-m x 4-m walls have no windows. The third wall has five windows made of 0.5-cm-thick glass (k = 0.78 W/m-K), 1.2m x 1.8min size. The fourth wall has the same size and number of windows, but they are double paned with a 1.5-cm-thick stagnant air space (k = 0.026 W/m-K) enclosed between two 0.5-cm-thick glass layers. The thermostat in the house is set at 24°C and the average temperature outside at that location is 8°C during the five-month-long heating season. Disregarding any direct radiation gain or loss through the windows and taking the heat transfer coefficients at the inner and outer surfaces of the house to be 7 and 18W/m2K, respectively, determine: (1) the average rate of heat transfer through the fourth wall with double paned window. (Hint: Real = R-value/Awal) (2) If the house is electrically heated and the price of electricity is ¥ l/kWh, determine the t of money this household will save per heating season by converting the single-pane windows in the third wall to double-pane windows (assume the heat loss in amoun the third wall with single-pane windows is 4534W).

Answer 1

NOTE: For the energy savings calculation, the solution uses the value 4534 W as the assumed heat loss from the third wall (in case of single-pane window) as mentioned in the question (part b). Although the calculated value (in part a) for the heat loss from the third wall is a little greater than that, which is 5224 W. If you want, you can also calculate the energy savings, using this value.

In case you have any doubt, feel free to ask.

Base area, A = 10m× 20m Height of wall, h 4m R-value, L = 2.31 m2 .℃ Window area. A-1.2mx1.8m Thermostat temperature 24°C Outside temperature -8°C Thermal conductivity, kir 0.026 W/m C Thermal conductivity for glass, k 0.78 W/m°C Imm Heat transfer coefficient for inside, h, 7 Wm2 Heat transfer coefficient for outside, h, 18 W/m2 Price of electricity, 1 kWh

For walls without windows; hA (7 W/m'°c)(10x4 m2) 0.003571 °C/w Lia RallkA R-value 2.31 m2 C/w (10x4 m) 0.05775 oC/w

h,A ( 18 W/m2°C)(10x4 m2 ) 0.00139 oc/w Ral -R+Rall+R 0.003571+0.05775+0.00139 0.0627 oC/w Heat transfer is given as Reotal (24-8)"C 0.0627°C/w 0 255.14w

Third wall with single pane window; = (7 Wm.oC)(20x4 m2) = 0.001786 °C/W Lia kA Rwal R- value 2.31 ㎡℃/W (20x4)-5(1.2x1.8)m2 = 0.033382 °C/W

Rglass glass kA 0.005 m (0.78 W/m2°c)(1.2x1.8)m2 0.002968 °C/w +5- glass +5 0.033382 0.002968 R 0.000583 °C/w egv (18 W/m2°C)(20x4 m2) 0.000694 °C/W

0.001786+0.000583+0.000694 0.0030626 °C/w Heat transfer is given as; total (24-8)℃ 0.0030626°C/W Q 5224.31 9 W Fourth wall with double- pane windows kA R-value 2.31 m2 C/w (20x4)-5(0.2x1.8)m2 0.033382 °C/w

R glass kA glass 0.005 m (0.78 W/m2C)1.2x1.8)m2 0.002968 °C/W Lai kA 0.015 m (0.026 W/m2 C)1.2x1.8)m2 0.267094 °C/W = 2 × 0.0029684 0.267094 0.27303 oC/w +5 R lRin 0.033382 0.27303 Rov = 0.02071 7 °C/W

0.001786+0.0207170.000694 0.023197 C/W Heat transfer will be total (24-8)°C 0.023197°C/w -689.74 W Power saved if the third wall had double-pane windows; double pane pane - 4534 W-689.74W - 3844.26 W 3.844 kW

Energy saved; -(3.844 kW) (5 x30x24 h) 1383934 kWh Money savings(Energy saved) (Unit cost of energy) (1383934 kWh)(#1 /kWh) Y13839.34