# Lumped analysis

Let us consider an arbitrarily shaped object with volume V, surface area As, and a uniform initial temperature of Ti as shown in the figure on the right. At time t= 0, the arbitrarily shaped object is exposed to a fluid with temperature of ${T_\infty }$( ${T_\infty } < {T_i}$) and the convective heat transfer coefficient between the fluid and the arbitrarily shaped object is h. Since the object is cooled from the surface, it is expected that the temperature at the center will be higher than that at the surface. We will consider a special case in this subsection that the temperature in the arbitrarily shaped object is uniform at any time – referred to as the lumped capacitance method. Qualitatively speaking, this assumption is valid when the thermal conductivity of the arbitrarily shaped object is very large or when its size is very small. The quantitative criterion about the validity of lumped capacitance method will be discussed later in this subsection. The convective heat transfer from the surface is $q = h{A_s}(T - {T_\infty })$

Similar to what we did for heat transfer from an extended surface, the heat loss due to surface convection can be treated as an equivalent heat source $q''' = - \frac{q}{V} = - \frac{{h{A_s}(T - {T_\infty })}}{V}$

Since the temperature is assumed to be uniform, the energy equation for heat conduction becomes $\rho {c_p}\frac{{\partial T}}{{\partial t}} = - \frac{{h{A_s}(T - {T_\infty })}}{V} \qquad \qquad(1)$

which is subject to the following initial condition $T = {T_i},{\rm{ }}t = 0 \qquad \qquad(2)$

Introducing excess temperature, $\vartheta = T - {T_\infty }$, eqs. (1) and (2) become $\frac{{d\vartheta }}{{dt}} = - \frac{{h{A_s}}}{{\rho V{c_p}}}\vartheta \qquad \qquad( 3)$ $\vartheta = {\vartheta _i},{\rm{ }}t = 0 \qquad \qquad(4)$

Integrating eq. (3) and determining the integral constant using eq. (4), the solution becomes $\frac{\vartheta }{{{\vartheta _i}}} = {e^{ - t/{\tau _t}}} \qquad \qquad(5)$

where ${\tau _t} = \frac{{\rho V{c_p}}}{{h{A_s}}} \qquad \qquad(6)$

is referred to as the thermal time constant. At time t = τt, the ratio of excess temperature and initial excess temperature is θ / θi = e − 1 = 0.368. The value of the thermal time constant is a measure of how fast the temperature of the object reacts to its thermal environment.

The cooling process requires transferring heat from the center of the object to the surface and the further transfering heat away from the surface by convection. When the lumped capacitance method is employed, it is assumed that the conduction resistance within the object is negligible compared with the convective thermal resistance at the surface, therefore, the validity of the lumped analysis depends on the relative thermal resistances of conduction and convection. The conduction thermal resistance can be expressed as ${R_{cond}} = \frac{{{L_c}}}{{k{A_c}}}$

where Lc is the characteristic length and Ac is the area of heat conduction. The convection thermal resistance at the surface is ${R_{conv}} = \frac{1}{{h{A_s}}}$

Assuming As = Ac, one can define the Biot number, Bi, as the ratio of the conduction and convection thermal resistances ${\rm{Bi = }}\frac{{{R_{cond}}}}{{{R_{conv}}}} \simeq \frac{{h{L_c}}}{k} \qquad \qquad( 7)$

If the characteristic length is chosen as Lc = V / As, the lumped capacitance method is valid when the Biot number is less than 0.1, or the conduction thermal resistance is one order of magnitude smaller than the convection thermal resistance at the surface.

## References

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.