Heat Transfer

Lumped system assumption

Lecturer: Jakob Hærvig

What is the lumped system assumption and why?

  • Most heat transfer problems are complex because temperature varies in both with time and space
    • In lumped systems, temperature only varies with time and not space

    $$T=f(x,y,z,t)$$

    $$~\longrightarrow T=f(t)$$

  • Applying the lumped system assumption significantly simplifies the analysis
A roast beef (not lumped system)
A copper ball (lumped system)

Starting point of lumped system analysis

  • Consider a body of arbitrary shape with:
    • Mass $m$
    • Volume $V$
    • Density $\rho$
    • Specific heat capacity $c_p$
    • Initial internal temperature $T_i$
  • 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 $A_s$

Derivation of the lumped system equation

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

    • $$h A_s (T_\infty-T) \text{d}t = m c_p \text{d}T$$

    $$~=\rho V c_p \text{d}T$$

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

    • $$ h A_s (T_\infty-T) \text{d}t$$

    $$~= -h A_s (T-T_\infty) \text{d}t$$

    $$~= \rho V c_p \text{d}(T-T_\infty)$$

  • ..Re-arranging:

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

  • Now, intergrating from $t=0$ where $T=T_i$ to any time $t$ at which $T=T(t)$ we get:

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

  • Exponential on both sides gives lumped system equation: $$\frac{T(t)-T_\infty}{T_i-T_\infty} = \text{e}^{-\frac{h A_s}{\rho V c_p}t}$$

The time constant for lumped systems

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

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

  • The time constant ($1/b$) describes the rate at which the system approaches the surrounding temperature $T_\infty$
    • Large $b$ value, system approaches surrounding temperature quickly
    • Small $b$ value, system approaches surrounding temperature slowly

Validity of lumped system assumption

  • We define a Biot number:
    • $$ \text{Bi}=h L_c/k $$
    • 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

Example