A house wall of area consists of three layers placed in series: • Interior drywall:…

PHYSICS · Multi-part · CBSE · Physics · Thermal Properties of Matter · Thermal Properties of Matter

Problem

A house wall of area 12 m212\,\text{m}^2 consists of three layers placed in series: • Interior drywall: thickness 1.3 cm1.3\,\text{cm}, thermal conductivity 0.17 W m−1 K−10.17\,\text{W}\,\text{m}^{-1}\,\text{K}^{-1} • Foam-board insulation: thickness xx (to be chosen), thermal conductivity 0.027 W m−1 K−10.027\,\text{W}\,\text{m}^{-1}\,\text{K}^{-1} • Exterior brick: thickness 9.0 cm9.0\,\text{cm}, thermal conductivity 0.72 W m−1 K−10.72\,\text{W}\,\text{m}^{-1}\,\text{K}^{-1} During winter, the indoor temperature is maintained at 19 ∘C19\,^{\circ}\text{C} while the outdoor temperature is −6 ∘C-6\,^{\circ}\text{C}. Building codes require that the steady–state heat loss through the wall not exceed 300 W300\,\text{W}. (a) Derive an inequality involving the unknown thickness xx that ensures compliance with the heat-loss requirement. (b) Determine the minimum insulation thickness xminx_{\text{min}} that satisfies the inequality. Express your answer in centimeters, to an appropriate number of significant figures. (c) Standard framing allows a maximum insulation space of 10 cm10\,\text{cm}. State whether the design with x=xminx = x_{\text{min}} is compatible with this construction constraint and justify your answer numerically. (d) Suppose inflation-resistant vacuum panels (thermal conductivity 0.0080 W m−1 K−10.0080\,\text{W}\,\text{m}^{-1}\,\text{K}^{-1}) become available. Keeping the wall within the same overall thickness limit of 10 cm10\,\text{cm} (including drywall and brick), propose one feasible redesign that meets the 300 W300\,\text{W} criterion and briefly explain its advantages and potential drawbacks.

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