What is meant by a lumped capacity? What are the physical assumptions necessary for a lumped capacity unsteady state analysis to apply?
The Correct Answer and Explanation is:
Correct Answer:
A lumped capacity (or lumped system) refers to an assumption in unsteady state (transient) heat transfer analysis where the temperature within an object is assumed to be spatially uniform at any given instant in time. This simplifies the analysis because the object can be treated as having a single, time-dependent temperature, rather than requiring the use of partial differential equations to account for temperature gradients inside the object.
Explanation:
In thermal analysis, particularly in transient (time-dependent) heat transfer problems, the lumped capacity model is a simplifying assumption that considers an object to have uniform temperature throughout its volume at any moment. This is often valid when the object conducts heat internally much faster than it exchanges heat with its surroundings.
The key parameter used to determine if this assumption is valid is the Biot number (Bi), defined as: Bi=hLckBi = \frac{hL_c}{k}
where:
- hh is the convective heat transfer coefficient (W/m²·K),
- LcL_c is the characteristic length of the object (typically volume/surface area),
- kk is the thermal conductivity of the object (W/m·K).
For the lumped system approximation to be valid, Bi must be less than 0.1. This condition implies that the internal thermal resistance is negligible compared to the surface resistance to heat transfer.
Physical Assumptions Required:
- Uniform internal temperature: The object has high thermal conductivity so that any internal temperature gradients are negligible.
- Small Biot number: Ensures surface resistance dominates and internal resistance is minimal.
- Homogeneous material properties: The material’s thermal conductivity and heat capacity do not vary significantly.
- Negligible internal heat generation: Heat sources within the object are either absent or uniformly distributed.
- Constant convective environment: The surrounding temperature and heat transfer coefficient are steady or known functions of time.
This model is commonly used for small metal objects, electronic components, and other bodies with high thermal conductivity exposed to convection or radiation. It drastically simplifies the mathematical treatment of transient heat transfer.
