3.2  Partial Lumped Parameter Modeling (PLPM)

In traditional lumped-parameter modeling (LPM), each element represents a whole — a complete subsystem with assumed uniform properties, like an entire thermal mass or electrical component. In this section, we explore a more granular approach: Partial Lumped Parameter Modeling (PLPM).

In PLPM, each lumped element represents only a part of a spatially distributed system. This allows us to model spatial variation by dividing a continuous domain into discrete segments, each governed by local energy balances. It's a form of spatial discretization—similar in spirit to finite element methods—but retains the simplicity and interpretability of LPM.

This method is particularly useful for modeling 1D heat conduction, where temperature varies along a single spatial dimension and the governing PDE is:

\[ \frac{\partial T(x,t)}{\partial t} = \alpha \frac{\partial^2 T(x,t)}{\partial x^2} \]

Rather than solving this equation directly, in this chapterwe approximate the domain with \(n\) thermal nodes connected by resistances and associated with thermal capacitances. This results in a system of ordinary differential equations (ODEs) that can be easily simulated.

Benefits of PLPM

Benefits of PLPM include the following:

• Converts PDE problems into ODE systems that are intuitive and solvable with basic numerical tools
• Resembles the structure of finite element models without requiring complex theoretical frameworks
• Easily extensible to systems with nonlinearities, control inputs, or time-varying parameters
• Highly useful in experimental, control, and practical settings

PLPM bridges the gap between simple system-level models and spatially distributed physical phenomena. In the next section, we will apply PLPM to a 1D conduction problem and simulate the resulting thermal dynamics. This leads to the 1D conduction lab exercise at the conclusion of this chapter, which can be simulated with PLPM.

Relation to Thermal Circuits

The method of thermal circuits is used extensively in Bergman, Lavine, and Incropera, 2019. In chapter 3 they use a steady-state model using thermal resistances and in chapter 5 they extend to transient analysis with thermal capacitances. These thermal circuits are lumped-parameter models, which we used in the previous chapter for modeling the thermal dynamics of a beaker of water. Thermal PLPMs are simply thermal circuits used to discretize thermal systems at a more detailed level, allowing for a more accurate representation of the thermal behavior of the system.