Lumped capacitance model
A lumped-parameter model reduces a thermal system to a number of discrete “lumps” and assumes that the temperature difference inside each lump is negligible. This approximation is useful to simplify otherwise complex differential heat equations. It was developed as a mathematical analog of electrical capacitance.
To determine the number of lumps the Biot number (Nb), a dimensionless parameter of the system, is used. If the Biot number is less than 0.1 for a solid object, then the entire material will be nearly the same temperature with the dominant temperature difference will be at the surface.
The single capacitance approach can be expanded to involve many resistive and capacitive elements, with Nb < 0.1 for each lump. As the Biot number is calculated based upon a characteristic length of the system, the system can often be broken into a sufficient number of sections, or lumps, so that the Biot number is acceptably small.
Some characteristic lengths of thermal systems are:
- Plate: thickness
- Fin: thickness/2
- Long cylinder: diameter/4
- Sphere: diameter/6
For arbitrary shapes, it may be useful to consider the characteristic length to be volume / surface area.
Example
Consider a flat wall divided into n lumps; each lump is connected to its neighbours by conductive resistance elements and the edge lumps are connected to the temperatures at infinity by convective elements. At each node, the current into the node - current out gives the rate of heat storage, and the current flowing through each resistor is given by the temperature difference / Rn.