Equation (4.8) is valid when the heat conductivity is independent on the direction. How would a generalization look like for the case of heat conductivity being different in the three coordinate directions? Which physical properties, e.g., of biomass material, could cause such difference?

First, let's understand the heat conduction equation for an isotropic material. It can be given by Fourier's law of heat conduction in the following form: \[ \nabla \cdot (\kappa \nabla T) = \frac{\partial T}{\partial t}, \] where \(\kappa\) is the heat conductivity, \(T\) is the temperature, and \(\nabla\) represents the gradient operator.

If the material is anisotropic and the heat conductivity varies in each direction, we can modify the equation using anisotropic heat conductivity tensor \(\boldsymbol{\kappa}\): \[ \nabla \cdot (\boldsymbol{\kappa} \nabla T) = \frac{\partial T}{\partial t}, \] where \(\boldsymbol{\kappa}\) is now a 3x3 matrix, including the heat conductivity in each coordinate direction \(\kappa_x\), \(\kappa_y\), and \(\kappa_z\): \[ \boldsymbol{\kappa} = \begin{bmatrix} \kappa_x & 0 & 0 \\ 0 & \kappa_y & 0 \\ 0 & 0 & \kappa_z \end{bmatrix}. \]

Biomass materials such as wood and fibrous materials have anisotropic properties due to their intrinsic structure. These materials consist of fibers or grain patterns that have higher heat conductivity along their direction compared to the perpendicular directions. This difference causes the heat conductivity to be different in each coordinate direction. A few reasons for varying heat conductivities in different directions for biomass materials are: 1. The alignment of fibers or grains in the material, which creates a preferential path for heat transfer along one axis. 2. The presence of air pockets or voids in the material, which can slow heat transfer in certain directions. 3. The variations in the composition of the material, which can affect heat conductivity in different directions. In conclusion, for cases when the heat conductivity is different in each coordinate direction, we must modify the heat conduction equation using the anisotropic heat conductivity tensor \(\boldsymbol{\kappa}\). Biomass materials such as wood and fibrous materials have anisotropic properties that cause the heat conductivity to vary in different directions.