Question

The temperature of the rectangular solid (with length from -a to a in x-direction, width from -b to b in y-direction, height from -c to c in z-direction) is constant at T0 . The solid is suddenly immersed into a fluid. The temperature of the fluid is constant at T0 , and the convective coefficient of the fluid is h. Assume the properties of the solid are constant. Write down the partial differential equation of the temperature of the solid, initial and boundary conditions.

Answer 1

The partial differential equation for heat transfer in a rectangular solid with constant initial temperature and immersed in a fluid with convective heat transfer is Fourier's law of heat conduction. The initial condition is that the temperature is T0 throughout the solid, and the boundary conditions are described by convection at the solid's surface.

Step-by-step explanation:

Heat Transfer in a Rectangular Solid

For a rectangular solid with temperature initially constant at °To and then immersed in a fluid with the same temperature but with a convective heat transfer coefficient h, we need to write down the heat transfer partial differential equation as well as the initial and boundary conditions. The heat diffusion equation, assuming constant thermal properties, is given by Fourier's law of heat conduction as ρcp∂T/∂t = k(∂2T/∂x2 + ∂2T/∂y2 + ∂2T/∂z2). The initial condition is T(x,y,z,0) = T0. The boundary conditions considering convective heat transfer at the surface are −k∂T/∂n = h(T−Tfluid), where n is the outward normal to the surface, on all faces of the solid.

Initial condition: T(x,y,z,0) = T0

Boundary conditions: For the faces perpendicular to the x-axis (at x = ±a), y-axis (at y = ±b), and z-axis (at z = ±c), the conditions are −k∂T/∂x = h(T−T0), −k∂T/∂y = h(T−T0), and −k∂T/∂z = h(T−T0) respectively.