A cold-storage warehouse has an external wall of area made from concrete whose…

PHYSICS · Subjective · CBSE · Physics · Thermal Properties of Matter · Thermal Properties of Matter

Problem

A cold-storage warehouse has an external wall of area

100\,\text{m}^{2}

made from concrete whose intrinsic thermal resistance may be taken as

R_{0}=0.30\,\text{m}^{2}\,\text{K\,W}^{-1}

. To reduce operating costs the owner intends to add a layer of glass-wool insulation of thermal conductivity

0.040\,\text{W\,m}^{-1}\,\text{K}^{-1}

and thickness

x

. During the working season the temperature difference between indoor and outdoor air is maintained at

18\,\text{K}

for

180

days each year. The annual cost of electrical energy is Rs

6.0

per kWh. The installed cost of glass-wool is Rs

50

per square metre per centimetre of thickness. Assume the insulation has a service life of

10

years and ignore the time value of money. (a) Derive an expression for the combined annual cost per square metre,

C(x)

, consisting of (i) the annualised capital cost of the insulation layer and (ii) the annual energy cost due to heat conducted through the wall. (b) By differentiating

C(x)

with respect to

x

, obtain the thickness

x_{\text{opt}}

that minimises the total annual cost. (c) Calculate the numerical value of

x_{\text{opt}}

for the data given.

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