Heat & Mass Transfer

Steady conduction is heat transfer through a solid (or stagnant fluid) with temperature field independent of time. Fourier’s law $q = -kA\frac{dT}{dx}$ relates heat rate to the temperature gradient. Plane walls, cylinders, and spheres are solved with thermal resistances $R_{\mathrm{th}}$ in series/parallel networks (RC Sachdeva / GATE HT).

Fins extend surface area to enhance convection. For a straight fin of uniform cross-section, $\dfrac{d^2\theta}{dx^2}-m^2\theta=0$ with $m=\sqrt{hP/(kA_c)}$. Tip conditions give standard $q_{\mathrm{fin}}$ formulas; efficiency $\eta_f=q_{\mathrm{fin}}/q_{\max}$ (RC Sachdeva / Incropera).

Convection is quantified by Newton's law q = hA(T_s − T_∞); h follows from Nusselt correlations Nu = hL/k. Forced convection uses Nu = C·Reᵐ·Prⁿ, natural convection uses Nu = f(Ra), per RC Sachdeva.

A black body emits E = σT⁴ (σ = 5.67×10⁻⁸ W/m²K⁴); a grey body emits εσT⁴, and exchange between surfaces uses q = σA₁F₁₂(T₁⁴ − T₂⁴). View factors obey reciprocity A₁F₁₂ = A₂F₂₁, per RC Sachdeva.

Under steady flow with constant $U$ and specific heats, heat exchanger duty is $q=UA\,\Delta T_{\mathrm{lm}}\,F$ where $\Delta T_{\mathrm{lm}}$ is the log-mean temperature difference. For counterflow $\Delta T_{\mathrm{lm}}=(\Delta T_a-\Delta T_b)/\ln(\Delta T_a/\Delta T_b)$; cross/shell-and-tube use correction $F$ (RC Sachdeva / GATE HT).