Most heat transfer problems are complex because temperature varies with both space and time

In lumped systems, temperature only varies with time (not space)

\(\quad T=f(x,y,z,t)\) \(\quad \rightarrow \quad\) \(\quad T=f(t)\)

Significantly simplifies the analysis!

Consider a body of arbitrary shape with

• At time $t=0$ the body is exposed to convective heat transfer from the outside:
• Surrounding temperature $T_\infty$
• Heat transfer coefficient $h$
• Surface area $A_s$

Energy transferred to body during $\text{d}t$:

\(\dot{Q}=h A_s (T_\infty-T) \text{d}t \) \(= m c_p \text{d}T\) \( =\rho V c_p \text{d}T\)

Because $T_\infty$ is constant, we may expand $\text{d}T$:

\(\dot{Q}=h A_s (T_\infty-T) \text{d}t = \rho V c_p \text{d}(T-T_\infty)\)

..Re-arranging:

$$ \frac{\text{d}(T-T_\infty)}{T-T_\infty} = -\frac{h A_s}{\rho V c_p} \text{d}t$$

Now, intergrate from $t=0$ where $T=T_i$ to $t$ at which $T=T(t)$:

\( \displaystyle\text{ln}\frac{T(t)-T_\infty}{T_i-T_\infty} = -\frac{h A_s}{\rho V c_p}t\)

Exponential on both sides gives lumped system equation:

\( \displaystyle \frac{T(t)-T_\infty}{T_i-T_\infty} = \text{e}^{-\frac{h A_s}{\rho V c_p}t}\)

Introducing $b=hA_s/(\rho V c_p)$, we obtain:

\( \displaystyle \frac{T(t)-T_\infty}{T_i-T_\infty} = \text{e}^{-\frac{h A_s}{\rho V c_p}t} = \text{e}^{-bt}\)

The time constant ($1/b$) describes the rate at which the system approaches the surrounding temperature $T_\infty$

• Large $b$: Temperature approaches $T_\infty$ quickly
• Small $b$: Temperature approaches $T_\infty$ slowly

We define a Biot number

• with the characteristic length $L_c=V/A_s$

If $\text{Bi}=0$: lumped system assumption is exact

If $\text{Bi}\leq 0.1$: lumped system assumption is resonable accurate

If $\text{Bi}> 0.1$: lumped system assumption is inaccurate