Finite differencing

TAK 2000 is a finite differencing analyzer. The user defines a thermal network in a manner analogous to an electrical system composed of resistors, voltages, capacitances, and currents. A thermal system is divided into nodes and connected by conductors. These conductors can represent conduction, convection, radiation, or mass flow. Constant and time/temperature-varying heat sources can be applied to the nodes. Examples of heat sources are: heaters, electrical dissipation, solar radiation, albedo, or earth IR. The network can be analyzed with a steady state and/or transient solution technique.

When running a steady state analysis, TAK 2000 computes the temperatures for all the nodes, the heat flows through all the conductors, and reports on any heaters and flow conductors that were modeled. When running a transient analysis, TAK 2000 computes the temperatures of each node as a function of time. The output can include temperature, incident heating, and values of time or temperature varying quantities.

In order to develop a thermal network and apply numerical techniques to its solution, it is necessary to subdivide the thermal system into a number of finite subvolumes called nodes. The thermal properties of each node are considered to be concentrated at the central nodal point of each subvolume. Each node represents two thermal network elements, a temperature (potential) and a capacitance (thermal mass) as shown.

The temperature, T, assigned to a node represents the average mass temperature of the subvolume. The capacitance, C, assigned to a node is computed from the thermophysical properties of the subvolume material evaluated at the temperature of the node. Capacitance is assumed to be concentrated at the nodal center of the subvolume. Because a node represents a lumping of parameters to a single point in space, the subvolume implied by the nodal temperature is linear as shown in Figure C and not a step function as illustrated in Figure B.

In a homogeneous material, the temperature at a point other than the nodal point may be approximated by interpolation between adjacent nodal points where the temperatures are known.

The error introduced by dividing a system into finite size nodes rather than volumes dx3 where dx approaches zero is dependent on numerous considerations: material thermal properties, boundary conditions, node size, node center placement, and time increment of transient calculations.

Conductors are the thermal math modeling network elements, which are used to represent the heat flow paths through which energy is transferred from one node to another node. The figure illustrates the element node temperatures (T), capacitances (C), and conductors (G) which comprise a thermal network.

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