Question

Consider steady one-dimensional heat conduction in a plane wall in which the thermal conductivity varies linearly. The error involved in heat transfer calculations by assuming constant thermal conductivity at the average temperature is \((a)\) none, \((b)\) small, or \((c)\) significant.

Short answer

The error involved in assuming constant thermal conductivity when it varies linearly depends on the range of variation and the values of the thermal conductivity constants. However, for most engineering materials, the thermal conductivity does not vary significantly within the range of temperatures of interest, and the error involved is generally small. So, the likely answer is (b) small.

Step-by-step solution

Heat transfer equation with constant thermal conductivity

When assuming constant thermal conductivity, the one-dimensional heat conduction equation can be expressed by Fourier's law as follows: \(q = -kA\frac{dT}{dx}\) where q is the heat transfer rate, k is the thermal conductivity, A is the cross-sectional area, and dT/dx is the temperature gradient along the x-direction.

Heat transfer equation with linearly varying thermal conductivity

When thermal conductivity varies linearly, it can be represented as a function of temperature: \(k(x) = k_0 + k_1T(x)\) where k_0 is the thermal conductivity at a reference temperature T_0, k_1 is the linear coefficient, and T(x) is the temperature along the x-direction. Now, the heat conduction equation becomes: \(q = -A(k_0 + k_1T) \frac{dT}{dx}\)

Comparing the results

When comparing these two equations, we can see that: (1) If \(k_1\) is zero or very small, i.e., the thermal conductivity is almost constant, the error involved in assuming a constant thermal conductivity is negligible. In this case, the answer is (a) none. (2) If \(k_1\) is not too large, but thermal conductivity varies somewhat linearly, the error involved in assuming a constant thermal conductivity is small and still acceptable. In this case, the answer is (b) small. (3) If \(k_1\) is large and thermal conductivity varies significantly, the error involved in assuming a constant thermal conductivity is significant and can lead to errors in calculations. In this case, the answer is (c) significant. Thus, without knowing the exact range of \(k_1\) or the values of \(k_0\) and \(k_1\), we can't provide a definite answer to the question. However, for most engineering materials, the thermal conductivity does not vary significantly within the range of temperatures of interest, and the error involved in assuming constant thermal conductivity is generally small. Therefore, the answer is likely (b) small.

Key concepts

Thermal Conductivity

Thermal conductivity is a critical property in the context of heat conduction. It is a measure of how well a material can conduct heat. This property is represented by the symbol \( k \) and is typically given in units of watts per meter-kelvin (W/m·K). Materials with high thermal conductivity, such as metals, are good conductors of heat because they allow heat to move through them quickly. On the other hand, materials with low thermal conductivity, such as rubber, are good insulators because they slow down the transfer of heat.
Understanding thermal conductivity is crucial for designing systems involving heat transfer, such as electronics cooling or thermal insulation in buildings. Engineers need to consider the thermal conductivity of materials to manage heat flow efficiently in these applications. It is important to note that thermal conductivity can vary with temperature and may not always remain constant across a temperature range.

Fourier's Law

Fourier’s law of heat conduction is a fundamental principle that describes how heat energy is transferred through a material. The law is mathematically expressed as \( q = -kA\frac{dT}{dx} \), where \( q \) represents the rate of heat transfer, \( k \) is the thermal conductivity, \( A \) is the cross-sectional area perpendicular to the heat transfer direction, and \( \frac{dT}{dx} \) is the temperature gradient.
According to Fourier's law, the heat transfer rate is proportional to the negative of the temperature gradient; as the temperature difference increases, so does the heat flow. The negative sign indicates that heat moves from high to low temperatures, consistent with the second law of thermodynamics. Fourier’s law forms the basis for describing heat transfer in many engineering applications, helping predict how heat will flow through different materials and structures.

Temperature Gradient

The temperature gradient is an important concept when discussing heat conduction. It is the rate of change of temperature with respect to position in a material, denoted by \( \frac{dT}{dx} \) in one-dimensional analysis. In simpler terms, a temperature gradient indicates how much the temperature changes over a certain distance.
A high temperature gradient means there's a significant change in temperature over a short distance, which often results in a high rate of heat transfer. Conversely, a low temperature gradient indicates a slow or gradual change in temperature over a distance, leading to a lower rate of heat transfer.
Understanding temperature gradients is essential for managing heat flow, as it allows engineers to predict how quickly or slowly heat will move through materials. In practical applications, this knowledge is used to ensure efficient thermal management in systems ranging from household appliances to industrial machinery.

Linear Thermal Conductivity

Linear thermal conductivity refers to a situation where the thermal conductivity of a material changes linearly with temperature, as described by the equation \( k(x) = k_0 + k_1T(x) \). Here, \( k_0 \) is the conductivity at a reference temperature, and \( k_1 \) is the coefficient representing how the conductivity changes with temperature.
When thermal conductivity varies linearly, it means that as the temperature changes, the conductivity will either increase or decrease steadily. This can significantly impact heat transfer calculations, especially if there's a large difference between the reference and actual temperatures.
In many real-world applications, such variations in thermal conductivity are considered when conducting precise engineering analyses. While assuming a constant thermal conductivity is sufficient for many standard calculations, accounting for linear variations is crucial when high accuracy in heat transfer predictions is required. This ensures designs are safe and efficient in applications such as process industries where temperatures vary widely.