An Expression for Evaluating the Temperature Variation in a Solid Using Lumped

**LUMPED HEAT ANALYSIS:**

Lumped
Capacity System-Necessary Physical Assumptions

We know that a
temperature gradient must exist in a material if heat energy is to be conducted
into or out of the body. When Bi < 0.1, it is assumed that the internal
thermal resistance of the body is very small in comparison with the external
resistance and the transfer of heat energy is primarily controlled by the
convective heat transfer at the surface. That is, the temperature within the
body is approximately uniform. This idealised assumption is possible, if

(a) the
physical size of the body is very small,

(b) the
thermal conductivity of the material is very large, and

(c) the
convective heat transfer coefficient at the surface is very small and there is
a large

temperature difference across the fluid layer at the
interface.

**An Expression for Evaluating the
Temperature Variation in a Solid Using Lumped**

**Capacity Analysis**

Let us consider a small
metallic object which has been suddenly immersed in a fluid during a heat
treatment operation. By applying the first law of

Heat
flowing out of the body = Decrease in the internal thermal energy of

during
a time dt the body during that time dt

or,
hA_{s}( T -T_{¥} )dt = - pCVdT

where A_{s} is
the surface area of the body, V is the volume of the body and C is the specific
heat capacity.

or,
(hA/ rCV)dt = - dT /( T -T_{¥})

with
the initial condition being: at t = 0, T = T_{s}

The
solution is : ( T -T_{¥})/( T_{s} -T_{¥})
= exp(-hA / rCV)t

Fig. depicts the cooling of a body (temperature
distribution time) using lumped thermal capacity system. The temperature
history is seen to be an exponential decay.

We
can express

Bi
× Fo = (hL/k)×( at/L^{2})
= (hL/k)(k/ rC)(t/L^{2}) =
(hA/ rCV)t,

where
V / A is the characteristic length L.

And,
the solution describing the temperature variation of the object with respect to
time is given

(
T -T_{¥})/( T_{s} -T_{¥})
= exp(-Bi· Fo)

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Mechanical : Heat and Mass Transfer : Conduction : Lumped Heat Analysis |

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