It has been proposed that highly concentrating solar power such as the one below can be used to process materials economically when it is desirable to heat the material surface rapidly without significantly heating the bulk. In one such process for case hardening low-cost carbon steel, the surface of a thin disk is to be exposed to concentrated solar flux.

Question

The distribution of absorbed solar flux on the disk is given by



where r is the distance from the disk axis and q''max and Ro are parameters that describe

the flux distribution. The disk diameter is 2Rs, its thickness is Zs, its thermal conductivity

is k and its thermal diffusivity is a. The disk is initially at temperature Tinit and at time

t = 0 it is suddenly exposed to the concentrated flux. Derive the set of explicit difference

equations needed to predict how the disk temperature distribution evolves with time. The

edge and bottom surface of the disk are insulated and reradiation from the disk is

neglected.



GIVEN

- Steel disk exposed to concentrated solar flux

FIND

(a) Explicit difference equations that describe evolution of disk temperature

Answer 1

The problem is a two-dimensional cylindrical geometry in the coordinates r and z. There are no

gradients in the circumferential direction. Let there be N radial nodes and M axial nodes as shown in

the sketch below. Then the size of the control volumes and the node locations are given by



There are a total of N x M control volumes and each has the shape of a ring with rectangular cross-

section. We need to develop an energy balance equation for each control volume. First, let us

determine the volume and surface area of each control volume since these will be needed in the energy

balance equations.



The top or bottom face surface area of each control volume is Afi



Now, the volume of each control volume is just



Except for the control volume at node i = 1, each control volume has two curved surfaces, an outer

surface and an inner surface, see sketch below.



The surface area of the outer curved surface is



The surface area of the inner surface is



By definition



(In the above notation for Acii, the first i in the subscript refers to the inner curved surface and the

second i is the node index.)

The control volumes along the exposed surface absorb solar flux given by the equation in the problem

statement. We need to integrate this flux equation over each control volume to determine the solar

energy absorbed for each control volume. The following equation expresses this





Carrying out the integration and simplifying we find



We are now in a position to evaluate the energy balance for each control volume. We actually only

need to develop 9 unique difference equations. These are for the interior nodes, the nodes at the four

corners, and the nodes on the axis, and on the three outer surfaces.

The explicit energy balance equation for all interior nodes is



Solving for the node temperatures



For the interior nodes along the axis we have



Solving for the node temperatures



For the node on the top of the axis



Solving for the node temperature



For the node on the bottom of the axis



For the interior nodes along the outer curved surface



Solving for the node temperatures



For the node on the top of the outer curved surface



Solving for the node temperature



For the node on the bottom of the outer curved surface



Solving for the node temperature



For the interior nodes on the bottom surface