Non-dimensional numbers are used in engineering and physics to analyze and compare different systems without the need to consider their specific dimensions. They provide valuable information about the behavior and characteristics of the system being studied.

In the context of solving a lumped parameter problem, which involves analyzing the behavior of a system as a whole rather than considering individual regions or components, the pair of non-dimensional numbers used is the Biot number and the Fourier number. Let's understand the significance of each of these numbers:

1. Biot number (Bi):
The Biot number is a dimensionless number that relates the heat transfer resistance inside a solid to the heat transfer resistance at the surface of the solid. It is defined as the ratio of the internal thermal resistance to the external thermal resistance. Mathematically, it can be expressed as:

Bi = hL / k

where h is the convective heat transfer coefficient, L is a characteristic length, and k is the thermal conductivity of the material.

The Biot number helps determine whether the heat transfer within the solid is primarily controlled by conduction or convection. A large Biot number indicates that convection dominates, while a small Biot number indicates that conduction dominates.

2. Fourier number (Fo):
The Fourier number is a dimensionless number that characterizes the transient heat conduction behavior in a solid. It represents the ratio of the time taken for heat conduction to occur over a characteristic time scale. Mathematically, it can be expressed as:

Fo = αt / L^2

where α is the thermal diffusivity of the material, t is time, and L is a characteristic length.

The Fourier number helps determine the rate at which heat is transferred within the solid. A large Fourier number indicates that heat conduction occurs rapidly, while a small Fourier number indicates that heat conduction occurs slowly.

The combination of the Biot number and the Fourier number is used in solving lumped parameter problems because they provide valuable information about the heat transfer behavior and the time required for heat conduction within the system. By analyzing these non-dimensional numbers, engineers and scientists can evaluate the transient behavior and design appropriate systems for efficient heat transfer.