The temperature of two outer surfaces of a composite slab, consisting of two materials having coefficients of thermal conductivity $$K$$ and $$2K$$ and thickness $$x$$ and $$4x$$, respectively are $$T_2$$ and $$T_1$$ $$(T_2 > T_1)$$. The rate of heat transfer through the slab, in a steady state is $$\left(\frac{A(T_2 - T_1)K}{x}\right)f$$, with $$f$$ equal to

In a steady state, the rate of heat transfer ($$H$$) through a composite slab consisting of two layers in series is constant throughout both layers. We can solve this using the thermal resistance analogy, where thermal resistance ($$R_{th}$$) is given by:

$$R_{th} = \frac{d}{KA}$$

Where:

 $$d$$ is the thickness of the material.

 $$K$$ is the coefficient of thermal conductivity.

 $$A$$ is the cross-sectional area of the slab (assumed to be the same for both layers).

Let's calculate the individual thermal resistances for the two slabs:

1. For the first slab layer:

$$\text{Thickness } (d_1) = x$$

$$\text{Thermal Conductivity } (K_1) = K$$

$$R_1 = \frac{x}{KA}$$

2. For the second slab layer:

$$\text{Thickness } (d_2) = 4x$$

$$\text{Thermal Conductivity } (K_2) = 2K$$

$$R_2 = \frac{4x}{(2K)A} = \frac{2x}{KA}$$

3. Equivalent Thermal Resistance ($$R_{eq}$$):

Since the two layers are joined in series, their individual thermal resistances simply add up:

$$R_{eq} = R_1 + R_2$$

$$R_{eq} = \frac{x}{KA} + \frac{2x}{KA}$$

$$R_{eq} = \frac{3x}{KA}$$

4. Rate of Heat Transfer ($$H$$):

The total rate of heat transfer through the entire composite slab driven by the outer temperature difference $$(T_2 - T_1)$$ is:

$$H = \frac{T_2 - T_1}{R_{eq}}$$

Substitute the value of $$R_{eq}$$ into the equation:

$$H = \frac{T_2 - T_1}{\left(\frac{3x}{KA}\right)}$$

$$H = \left( \frac{A(T_2 - T_1)K}{x} \right) \cdot \frac{1}{3}$$

5. Finding the value of $$f$$:

The problem states that the rate of heat transfer is given by:

$$H = \left( \frac{A(T_2 - T_1)K}{x} \right) f$$

Comparing our derived expression with the given equation, we find:

$$f = \frac{1}{3}$$

The value of $$f$$ is equal to $$\frac{1}{3}$$.