Physics quiz question: heat flow rate through window and shade

Physics 205A Quiz 7, fall semester 2011
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 14.56

"Choose the Best Honeycomb Shades"
Hunter Douglas
windowstotheworldinc.com/2014/05/best-honeycomb-shades/.

A Harvey Building Products vinyl casement window [*] has a thermal resistance of 0.33 K/watt. The temperature difference between inside and outside is 30° C. The SymphonyShades™ Virtuoso® window shade[**] is also claimed to have a thermal resistance of approximately 0.33 K/watt. If this window shade completely covers the window, the rate of heat conduction per time will be __________ of the original (uncovered) value.
(A) one-third.
(B) one-half.
(C) two-thirds.
(D) twice.
(E) (No change in rate of heat conduction.)

[*] R-factor = 0.33, ENERGY STAR rated, 19 square feet, harveybp.com/upload/products/literature/Harvey_AccessoryWindows_Brochure.pdf
[**] R-factor = 0.34, single-cellular light-filtering fabric, 19 square feet, symphonyshades.com/single_cell_shades.html.

Correct answer: (B)

The rate at which heat is conducted per time to the environment through the window given by:

Power_window = ((heat flow)/time)_window = ∆T/R_window,

where R_window is the thermal resistance of the window:

R_window = d_window/(κ_window·A_window),

which is already given as 0.33 K/watt, such that the heat flow per time conducted through the uncovered window is:

Power_window = (30° C)/(0.33 K/watt) = 90.909090 watts.

Thermal resistances are additive for materials that are stacked in series, so the heat flow per time through the window and shade will be:

Power_{window & shade} = ((heat flow)/time)_{window & shade} = ∆T/R_{window & shade},

Power_{window & shade} = ∆T/(R_window + R_shade),

Power_{window & shade} = (30° C)/(0.33 K/watt + 0.33 K/watt) = 45.45454545 watts.

Thus increasing the overall thermal resistance by a factor of two will reduce the heat flow per time to one-half of its original (uncovered) value.

Sections 70854, 70855
Exam code: quiz07h3A7
(A) : 9 students
(B) : 27 students
(C) : 8 students
(D) : 7 students
(E) : 1 student

Success level: 52%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.67