22-Mec-B2 Environmental Control in Buildings · December 2013
Question 8 of 8: Transmission heat loss through a frame dwelling wall
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. Professional Engineers of Ontario /
Engineers Canada annual examination 07-Mec-B2 Environmental Control in Buildings,
December 2013 — 3 hours, open book, non-communicating calculator permitted.
Eight problems of 20 points each; candidates answer any five.
ASHRAE psychrometric charts (SI and I-P) and an R-134a pressure–enthalpy diagram are
appended to the paper. All eight problems are solved below, because the set is
intended as a study resource.
Reference texts.
ASHRAE Handbook — Fundamentals (Ch. 1 Psychrometrics, Ch. 15 Fenestration,
Ch. 16 Ventilation and Infiltration, Ch. 18 Nonresidential Cooling and Heating Load Calculations,
Ch. 21 Duct Design);
McQuiston, Parker & Spitler, Heating, Ventilating, and Air Conditioning: Analysis and
Design, 6th ed.;
Çengel & Boles, Thermodynamics: An Engineering Approach, 9th ed. (R-134a
property tables);
ANSI/ASHRAE Standard 55 Thermal Environmental Conditions for Human Occupancy;
ANSI/ASHRAE Standard 62.1 Ventilation for Acceptable Indoor Air Quality.
Canadian practice: National Energy Code of Canada for Buildings (NECB) and CSA F280 where a
Canadian code reference is required.
Check — table-derived coefficients. This is an
open-book paper whose cover page instructs candidates to “submit a clear statement of the
assumption(s)” made. Where a solution needs a value that is read from a handbook table or
chart rather than given in the question — shading coefficients, air-space thermal
resistances, curtain-wall leakage rates, surface-averaged wind pressure coefficients, window and
door U-factors, CO2 generation rates — the value used is stated explicitly at the
point of use with its source. The method, not the table entry, is what the marks follow; a
different edition of the table shifts the arithmetic but not the answer structure.
Question 8: Transmission heat loss through a frame dwelling wall (20 points)
4 × (120 in. × 30 in.) = 100 ft², double glass, no thermal break
Door
—
80 in. × 32 in. = 17.8 ft², 1 3/8 in. solid core flush, no storm door
Net opaque wall area
$A_{\text{net}}$
1800 − 100 − 17.8 = 1682 ft²
Find. The total transmission heat loss through the wall assembly, including the windows
and the door, at the stated 62 °F design temperature difference.
Figure 8.1 — Wall section. Heat flows through two parallel paths between the sheathing and the plaster: the insulated cavity over 85% of the area, and the wood stud over the remaining 15%.
Approach. Build the wall U-factor by the parallel-path (isothermal-planes bypass)
method: sum the series resistances once through the insulated cavity and once through the wood stud, invert
each to a U-factor, and area-weight them 85/15. Then add the window and door areas at their own U-factors
and multiply the total UA by the temperature difference.
Assemble the series resistances of the insulated path. Using ASHRAE thermal
resistances (h·ft²·°F/Btu), with softwood at 1.25 per inch:
$$R = \underbrace{0.17}_{\text{outside film, 15 mph}}
+ \underbrace{0.86}_{\text{redwood } 22/32''}
+ \underbrace{0.98}_{\text{fir sheathing } 25/32''}
+ \underbrace{11.0}_{\text{3 in. rock wool}}
+ \underbrace{0.41}_{\text{gypsum lath + plaster}}
+ \underbrace{0.68}_{\text{inside film}} = 14.10$$
$$U_{\text{ins}} = \frac{1}{14.10} = 0.0709\ \text{Btu/(h}\cdot\text{ft}^2\!\cdot\!\text{°F)}$$
Repeat for the framing path. Through a stud the cavity resistance is that of
3.5 in. of softwood rather than the blanket:
$$R_{\text{stud path}} = 0.17 + 0.86 + 0.98 + \underbrace{4.38}_{3.5'' \text{ softwood}} + 0.41 + 0.68
= 7.47 \;\Rightarrow\; U_{\text{stud}} = 0.134$$
The stud path conducts nearly twice as readily as the insulated path — the thermal bridge the
15% figure is there to quantify.
Area-weight the two paths.
$$U_{\text{wall}} = (1 - f_{\text{stud}})U_{\text{ins}} + f_{\text{stud}}U_{\text{stud}}
= 0.85(0.0709) + 0.15(0.134) = \boxed{0.0804\ \text{Btu/(h}\cdot\text{ft}^2\!\cdot\!\text{°F)}}$$
Ignoring the framing entirely would give 0.0709, understating the wall loss by 12%.
Openings. The windows are $A_w = 4 \times (10 \times 2.5) = 100\ \text{ft}^2$; for
double glazing in a frame with no thermal break the ASHRAE fenestration table gives
$U_{\text{win}} = 0.62$. The door is $(80/12)(32/12) = 17.8\ \text{ft}^2$; a 1 3/8 in. solid-core flush
wood door without a storm door has $U_{\text{door}} = 0.39$. The net opaque wall is therefore
$$A_{\text{net}} = 1800 - 100 - 17.8 = 1682\ \text{ft}^2$$
Total conductance and heat loss. Summing the three UA products,
$$\Sigma UA = 1682(0.0804) + 100(0.62) + 17.8(0.39) = 135.2 + 62.0 + 6.9
= 204\ \text{Btu/(h}\cdot\text{°F)}$$
$$q = \Sigma UA\,(t_i - t_o) = 204 \times 62 = \boxed{12\,700\ \text{Btu/h}}\;(3.7\ \text{kW})$$
Interpret the split. The windows are 5.6% of the wall area but carry 30% of the
transmission loss, and the 15% of area occupied by studs carries a quarter of the opaque-wall loss. Both
are the standard lessons of a frame-wall calculation: glazing and thermal bridging dominate an otherwise
insulated envelope, which is why modern Canadian practice under the National Energy Code for Buildings
pushes towards thermally broken and low-e glazing, exterior continuous insulation over the studs, and
advanced framing at 24 in. centres. Note also that this is transmission loss only — the
infiltration load of Question 5(b) and the ceiling, floor and below-grade losses must be added before the
heating plant can be sized.
Check — table resistances. Layer resistances are ASHRAE values: outside air film 0.17 (winter, 15 mph), inside film 0.68 (still air, vertical surface, horizontal heat flow), softwood 1.25 per inch, 3 in. mineral (rock) wool blanket R-11, gypsum lath and plaster 0.41, and whole-assembly U-factors of 0.62 for double glass without a thermal break and 0.39 for a 1 3/8 in. solid-core flush wood door without a storm door. The 2×4 stud is taken at its actual 3.5 in. depth. Different table editions shift the total by a few per cent; the parallel-path structure of the answer does not change.
