When to use the Lumped Capacitance method in Heat Transfer problems?

[Sabra_a]

Homework Statement

What is the main assumption to apply the lumped capacitance method? What particular non dimensional number and what mathematical condition represent this assumption? What is the meaning of the nominator and the denominator in this non-dimensional number?

Relevant Equations

Energy released
Biot number

I attached below my attempt to solve the question.

 

[Chestermiller]

Homework Statement:: What is the main assumption to apply the lumped capacitance method? What particular non dimensional number and what mathematical condition represent this assumption? What is the meaning of the nominator and the denominator in this non-dimensional number?
Relevant Equations:: Energy released
Biot number

I attached below my attempt to solve the question.
View attachment 254413View attachment 254414

In a system like this, the asymptotic internal resistance to heat at long times is proportional to L/k and the external resistance is proportional to 1/h. The lumped parameter approach is valid if the external resistance dominates, or when $$\frac{1}{h}>>\frac{L}{k}$$or when $$\frac{k}{Lh}>>1$$ or, equivalently, when Bi << 1.