The temperature of the two outer surfaces of a composite...

The temperature of the two outer surfaces of a composite slab, consisting of two materials having coefficients of thermal conductivity $K$ and $2 K$ and thickness $x$ and $4 x$ respectively are $T_{2}$ and $T_{1}\left(T_{2}>T_{1}\right)$. The rate of heat transfer through the slab, in a steady state is $\left(\frac{A\left(T_{2}-T_{1}\right) K}{x}\right) f$, with fequal to: (a) $]$ (b) $1 / 2$ (c) $2 / 3$ (d) $1 / 3$

The temperature of the two outer surfaces of a composite slab, consisting of two materials having coefficients of thermal conductivity $K$ and $2 K$ and thickness $x$ and $4 x$ respectively are $T_{2}$ and $T_{1}\left(T_{2}>T_{1}\right)$. The rate of heat transfer through the slab, in a steady state is $\left(\frac{A\left(T_{2}-T_{1}\right) K}{x}\right) f$, with
fequal to:
(a) $]$
(b) $1 / 2$
(c) $2 / 3$
(d) $1 / 3$

Key Concepts

Fourier's Law of Heat Conduction

Fourier's law quantifies the rate at which heat energy is transferred through a material. It relates the heat flux to the thermal conductivity of the material and the gradient in temperature. This law forms the basis for analyzing steady-state heat flow through materials, whether homogeneous or composed of multiple layers.

Thermal Conductivity

Thermal conductivity is a material property that indicates how well heat is conducted through a material. It directly affects the rate of heat transfer; materials with higher conductivity allow more heat to pass through for a given temperature difference and thickness. This concept is fundamental in designing systems where temperature regulation is crucial.

Thermal Resistance

Thermal resistance is the measure of a material's resistance to heat flow. It is defined as the ratio of the thickness of the material to the product of its thermal conductivity and the cross-sectional area through which heat is being transferred. In composite systems, the total thermal resistance is the sum of the individual resistances, analogous to electrical resistors in series.

Series Combination of Thermal Resistances

When multiple layers of materials are used in a system, each layer contributes its own thermal resistance to the overall heat transfer process. The total resistance is obtained by adding each individual layer's resistance. This concept is critical in analyzing composite slabs, as it allows for the determination of the effective rate of heat transfer through the entire system.

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