Heat Transfer

Key points

1. 01

   Conduction moves heat via vibrational energy or free electrons, with no movement of matter; the sign on Fourier's law only tracks direction — $\hat{Q}$ itself always comes out positive.

2. 02

   Heat crossing a jacketed wall moves in three steps: convection (bulk fluid to wall), conduction (through the wall), convection (wall to bulk fluid) — driven by two separate $\Delta T$s, one per convective step.

3. 03

   Nusselt number ($Nu = hD/k_{fb}$) is the convective-to-conductive heat transfer ratio; Prandtl number ($Pr = C_p\mu_b/k_{fb}$) is the momentum-diffusivity-to-thermal-diffusivity ratio — the calculation chain runs $Pr \to Nu \to h \to \hat{Q}$, backwards from what you actually want.

4. 04

   Overall heat transfer coefficient $U$ avoids measuring $\Delta T$ at each wall interface: $1/U = 1/h_1 + x/k + 1/h_2$ combines both film coefficients with the wall's own conduction resistance.

5. 05

   $\Delta T_{LMTD} = (\Delta T_2 - \Delta T_1)/\ln(\Delta T_2/\Delta T_1)$. For fixed terminal temperatures, $\Delta T_1 + \Delta T_2$ is identical for parallel and counter-current flow, but counter-current keeps $\Delta T_1$ and $\Delta T_2$ closer together — so its LMTD is always $\geq$ parallel flow's, never smaller.

6. 06

   The two arrangements only tie when one fluid doesn't change temperature at all (a condensing vapor or boiling liquid) — otherwise counter-current always wins.

7. 07

   Metabolic heat tracks oxygen consumption — Cooney's correlation: $\approx 460\,\text{kJ/mol}\ O_2$ ($\approx 14\,\text{kJ/g}\ O_2$).

8. 08

   Scale-up hurts cooling because $V \propto D^3$ but cooling area $A \propto D^2$, so $A/V \propto 1/D$; external jackets stay the default (and are preferred for shear-sensitive cultures), while internal coils add area at large scale at the cost of extra shear and working volume.