Table of Contents
The lumped capacitance method is a simplified approach used in transient heat transfer analysis. It assumes that the temperature within a solid body remains uniform during the process, which is valid when the Biot number is small. This method allows for easier calculation of temperature changes over time without solving complex partial differential equations.
Principle of the Lumped Capacitance Method
The core assumption of this method is that the temperature within the object is spatially uniform at any given time. This simplifies the heat transfer problem to a first-order ordinary differential equation. The method is applicable when the internal resistance to heat conduction is much less than the external resistance to heat transfer.
Mathematical Formulation
The temperature change of the object over time is described by:
m cp (dT/dt) = – h A (T – T∞)
where m is mass, cp is specific heat capacity, h is the convective heat transfer coefficient, A is the surface area, T is the temperature of the object, and T∞ is the ambient temperature.
Solution and Applications
The solution to the differential equation provides the temperature as a function of time:
T(t) = T∞ + (Ti – T∞) e– (h A / m cp) t
This approach is useful for quick estimations in processes such as cooling or heating of small objects, where internal temperature gradients are negligible. It is commonly applied in engineering problems involving thermal analysis of solids with high thermal conductivity.