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 T2 and T1 (T2 > T1). The rate of heat transfer through the slab, ?

To determine the rate of heat transfer through the composite slab, we can apply Fourier's law of heat conduction. This law states that the rate of heat transfer (Q) through a material is proportional to the temperature difference across it and inversely proportional to the thickness of the material. Let's analyze the setup you described step by step.

Understanding the Components

You have a composite slab made of two materials with differing thermal conductivities and thicknesses:

• Material 1: Coefficient of thermal conductivity = K, Thickness = x
• Material 2: Coefficient of thermal conductivity = 2K, Thickness = 4x

The outer temperatures are T2 (higher) and T1 (lower), where T2 > T1.

Applying Fourier's Law

The rate of heat transfer through each material can be expressed using the formula:

Q = (k * A * ΔT) / d

Where:

• Q = rate of heat transfer
• k = thermal conductivity of the material
• A = cross-sectional area
• ΔT = temperature difference across the material
• d = thickness of the material

Calculating Heat Transfer for Each Material

For Material 1, the temperature difference is ΔT1 = T2 - T1, and the thickness is x. Thus, the rate of heat transfer (Q1) can be written as:

Q1 = (K * A * (T2 - T1)) / x

For Material 2, since the temperature difference remains the same across the entire slab, we can say that the temperature drop across Material 2 is also ΔT = T2 - T1. The thickness here is 4x, so the rate of heat transfer (Q2) becomes:

Q2 = (2K * A * (T2 - T1)) / (4x)

This simplifies to:

Q2 = (K * A * (T2 - T1)) / (2x)

Using the Steady-State Condition

In a steady-state condition, the heat transfer rate must be equal through both materials; therefore, Q1 = Q2. Setting the two equations equal gives us:

(K * A * (T2 - T1)) / x = (K * A * (T2 - T1)) / (2x)

Solving for the Rate of Heat Transfer

By simplifying the equation, we can cancel out common terms:

1 = 1/2

This indicates that the heat transfer rate through the two materials is indeed equal when the steady-state condition is met. Thus, we can express the total heat transfer rate through the composite slab as:

Q = (1/Total Resistance) * (T2 - T1)

Calculating Total Resistance

The total thermal resistance (R_total) for the two materials can be calculated as:

• R1 = x / (K * A)
• R2 = (4x) / (2K * A) = (2x) / (K * A)

So the total resistance is:

R_total = R1 + R2 = (x / (K * A)) + ((2x) / (K * A)) = (3x) / (K * A)

Final Formula for Heat Transfer Rate

Now we substitute R_total back into the heat transfer equation:

Q = (T2 - T1) / R_total = (T2 - T1) * (K * A) / (3x)

In summary, the rate of heat transfer through the composite slab is given by:

Q = (K * A * (T2 - T1)) / (3x)

This formula allows you to calculate the rate of heat transfer based on the thermal properties of the materials and the temperature difference across the slab. Understanding these principles can help you analyze more complex thermal systems in the future!