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.

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.

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: \(q = -A(k_0 + k_1T) \frac{dT}{dx}\)

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.