If a roof is treated with a layer of thermal insulation

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If a roof is treated with a layer of thermal insulation material, the internal heat gain is reduced by 60%. The U-value of the roof slab (without thermal insulation) is 3 W m² /°C. Assuming a constant temperature difference between indoor and outdoor, the U-value of the thermal insulation layer in W m² /° C is ____________

1. 2
2. 5
3. 7
4. 11

Correct Option: A

We will start from the beginning with concept. U-value =	1	- value
	R

Following is the basic equation of heat flow,

Where,
q_x is rate of heat flow
k = Thermal conductivity
A = surface area of the wall
d_x = wall thickness
dT = temperature difference
R-value is inversely related to the thermal conductivity constant and is also related to thickness, thus :
So, the equation (1) becomes,
Let, R-value of thermal insulation material = R₁
R-value of the roof

= R₂ =	1	- value =	1	......(given)
	U		3

R-value of combined set = R_{1 + 2}
According to question,

	∆Q	=	A(T_f - T_i)	....(heat transfer rate when there is no insulation)
	∆t		R₂

0.4 ×		∆Q		=	A(T_f - T_i)	....(heat transfer rate with insulation)
		∆t			R_{1 + 2}

	1	=	R_{1 + 2}
	0.4		R₂

1	=	R_{1 + 2}
0.4			1
			3

	1	= 1.2
	R₂

Now as per addition of thermal resistance, R_{1 + 2} = R₁ + R₂

	1	= R₁ +	1	⇒ R₁ = 2
	1.2		3