Transient Heat Transfer

The Lumped Capacitance Method

Used to consider the temperature of a body as uniform throughout the body, used for transient heat problems

Properties Include

• Uniform T at each time
• Transient (not steady state)
• Closed system
• ${\omega }^{\prime }=0$
• No energy generation

Constant properties $k,\rho ,C_p$

$\frac{dE}{dt}=Q^{\prime }-0$ (No work)

The solution to these problems is in the form $\theta ={\theta }_o{e}^{-bt}$

Derivation

Dr. Nishida does a derivation here that I won’t copy but he uses this boundary condition:
${\theta }_0=T_0-T_{\infty }@t=0$ then $C={\theta }_0$

To derive: ${\frac{T\left(t\right)-T\infty }{T_0-T_{\infty }}}_{}=e^{-bt}$ where $b={\frac{hAs}{\rho V{C}_p}}_{}=\frac{1}{\tau }$

Therefore, a lumped capacitance model exponentially decays in body temperature over time.

When can we use this?

For a lumped capacitance to be valid, thermal diffusion needs to happen faster than convection. We define the concept of a Biot number as follows…
$Bi=\frac{\text{Conduction resistance in body}}{\text{Conduction resistance at surface of body}}$
$Bi=\frac{\frac{L}{k{A}_s}}{\frac{1}{h{A}_s}}=h\frac{L}{k}\ll 1$
Another form of this is…
$Bi={\frac{hLc}{k}}_{}\le 0.1$
Where $L_c=\frac{V}{A_s}$

$L_C$ for common shapes have already been solved…

• Sphere $L_c=\frac{r}{3}$
• Cylinder $L_c=\frac{r}{2}$
• Wall $L_c=\frac{t}{2}$

Drawing connections

$\frac{\theta \left(t\right)}{{\theta }_i}=e^{\frac{-1}{R_{th}{C}_{th}}}$ is analogous to the time constant from AC Circuits.

Where $R_{th}{C}_{th}=\left(\frac{1}{h{A}_s}\right)\left(\rho V{c}_p\right)=\tau$

Calculating Cooling Times:
$\Delta T\left(t\right)=\left(T_i-T_{\infty }\right)e^{-\frac{t}{\tau }}$
Where remember that $\Delta T(t)=T\left(t\right)-T_{\infty }$ If you are solving for $T_{\infty }$!