Heat transfer Study Guide

📖 Core Concepts Heat transfer – movement of thermal energy due to a temperature difference (conduction, convection, radiation, phase‑change). Thermal equilibrium – state where contacting bodies share the same temperature; no net heat flow. Heat flux, \(q\) – rate of heat flow per unit area (W·m⁻²); vector quantity. Heat‑transfer coefficient, \(h\) – relates \(q\) to the surface‑to‑fluid temperature difference: \(q = h\,(T{\text{surface}}-T{\infty})\). Thermal conductivity, \(k\) – material property governing conductive resistance (W·m⁻¹·K⁻¹). Biot number, \(Bi = \dfrac{hLc}{k}\) – ratio of internal conductive resistance to external convective resistance; \(Bi<0.1\) ⇒ lumped behavior. Rayleigh number, \(Ra = \dfrac{g\beta\Delta T L^{3}}{\nu\alpha}\) – compares buoyancy‑driven convection to thermal diffusion; \(Ra \gtrsim 10^{3}\!-\!10^{4}\) → convection dominates. Stefan‑Boltzmann law – radiant heat flux \(\Phi = \varepsilon\sigma\left(T{\text{surf}}^{4}-T{\text{surr}}^{4}\right)\). Latent heat – heat absorbed/released during a phase change at constant temperature (boiling, condensation, melting). 📌 Must Remember Fourier’s law (1‑D steady conduction): \(q = -k\,\dfrac{dT}{dx}\). Newton’s law of cooling (convective): \(q = h\,(T{\text{surface}}-T{\infty})\) – linear only for modest \(\Delta T\). Biot‑number criterion: \(Bi<0.1\) → treat whole object as a single temperature (lumped capacitance). Critical heat flux: maximum \(q\) before transition from nucleate to film boiling (drastic heat‑transfer drop). Heat‑exchanger effectiveness: Counter‑flow > Cross‑flow > Parallel‑flow because of larger mean temperature difference. Thermal resistance: \(R{\text{th}} = \dfrac{\Delta T}{q\,A}\); higher \(R{\text{th}}\) = better insulation. Emissivity vs reflectivity: Reflectivity \(= 1 - \varepsilon\) for radiant barriers. 🔄 Key Processes Steady‑state conduction Set up geometry, apply Fourier’s law, integrate \(\int{T1}^{T2} dT = -\dfrac{q}{k}\int{x1}^{x2} dx\). Transient conduction (1‑D slab) Use heat equation \(\displaystyle \frac{\partial T}{\partial t}= \alpha \nabla^{2}T\). Apply appropriate boundary/initial conditions → analytical (Fourier series) or numerical solution. Natural convection onset Compute \(Ra\). If \(Ra > 10^{3}\!-\!10^{4}\), buoyancy drives flow; use correlations for \(Nu\) vs \(Ra\). Forced convection heat‑transfer coefficient Determine Reynolds number \(Re = \dfrac{\rho v L}{\mu}\). Use empirical \(Nu = C\,Re^{m}Pr^{n}\) → \(h = \dfrac{Nu\,k}{L}\). Radiation heat loss Evaluate \(\Phi\) with surface emissivity \(\varepsilon\) and temperatures (K). Lumped capacitance analysis Write energy balance: \( \rho V cp \dfrac{dT}{dt} = -hA (T - T{\infty})\). Solve → exponential decay \(T(t)-T{\infty}= (T0-T{\infty})e^{-t/\tau}\) with \(\tau = \dfrac{\rho V cp}{hA}\). Boiling heat‑transfer regimes Increase heat flux → nucleate boiling (high \(q\)). Reach critical heat flux → film boiling (low \(q\)). 🔍 Key Comparisons Conduction vs Convection Mechanism: Molecular collisions (solid) vs bulk fluid motion + diffusion. Dominance: Low \(Ra\) → conduction; high \(Ra\) → convection. Natural vs Forced Convection Drive: Buoyancy (density change) vs external fan/pump. Typical \(Nu\) correlation: \(Nu \sim Ra^{1/4}\) (natural) vs \(Nu \sim Re^{0.8}Pr^{0.33}\) (forced). Parallel‑flow vs Counter‑flow heat exchangers Temperature profile: Same direction → smaller ΔT; opposite direction → larger ΔT, higher effectiveness. Nucleate vs Film boiling Surface condition: Liquid contacts surface (high \(q\)) vs vapor layer insulating surface (low \(q\)). ⚠️ Common Misunderstandings “Newton’s law of cooling is always linear.” – It fails for large \(\Delta T\) where radiative terms become significant. “Low Biot number means no temperature gradients inside the solid.” – It only justifies the lumped approximation; small but non‑zero gradients may still exist. “Higher thermal conductivity always means better heat‑transfer performance.” – In convection‑dominated situations, surface resistance (h) can dominate. “Radiation only matters at high temperatures.” – Even moderate temperatures can yield noticeable radiative loss if emissivity is high and surface area large. 🧠 Mental Models / Intuition “Thermal resistance ladder” – Treat a multilayer wall like electrical resistors in series: \(R{\text{total}} = \sum Ri\). Larger \(R\) → slower heat flow. “Convection as “stirring” the fluid.” – Forced convection = adding a “spoon” to speed up heat removal; natural convection = fluid “rises” on its own when heated. “Phase change as a heat “buffer.” – Latent heat absorbs or releases large energy without temperature change, analogous to a capacitor storing charge. 🚩 Exceptions & Edge Cases Radiation non‑linearity – For \(\Delta T\) > 50 K, the \(T^{4}\) term dominates; Newton’s law no longer accurate. Material‑dependent \(k(T)\) – Most solids have weak temperature dependence, but polymers and gases can vary enough to require variable‑\(k\) analysis. Biot number borderline (0.1–0.3) – Lumped model gives only approximate results; consider a simple 1‑D transient conduction model instead. Critical heat flux – Specific to geometry and pressure; exceeding it abruptly switches to film boiling. 📍 When to Use Which Lumped‑capacitance → \(Bi < 0.1\) and geometry simple (sphere, plate). Fourier’s law (steady) → Temperature field not changing with time, known \(k\). Transient heat equation → Rapid heating/cooling, \(Bi\) moderate or geometry complex. Newton’s law of cooling → Small to moderate \(\Delta T\), convection‑dominant, surface properties known. Radiation term → High surface temperature, high emissivity, or vacuum environment. Counter‑flow HX design → Maximize effectiveness when inlet‑outlet temperature constraints are tight. 👀 Patterns to Recognize \(Ra > 10^{4}\) → convection dominates – look for buoyancy‑driven flow in vertical walls or heated plates. Sudden drop in \(q\) after a peak – sign of reaching critical heat flux (transition to film boiling). Linear temperature profile in a wall – indicates steady‑state conduction with constant \(k\). Exponential temperature decay in time – lumped system response. 🗂️ Exam Traps Choosing \(h\) from Newton’s law for large \(\Delta T\) – the answer will be wrong because radiative heat loss is ignored. Confusing \(Ra\) with \(Re\) – \(Ra\) involves buoyancy and thermal diffusivity; \(Re\) involves inertia and viscosity. Assuming parallel‑flow heat exchangers are as efficient as counter‑flow – they have lower mean temperature difference; the effectiveness will be over‑estimated. Neglecting emissivity when using Stefan‑Boltzmann law – assuming \(\varepsilon=1\) yields a larger \(\Phi\) than a real surface. Applying lumped analysis to a thick plate with high \(k\) but large surface area – \(Bi\) may be >0.1; temperature gradients cannot be ignored. --- All statements are drawn directly from the provided outline.