8.1 Conduction, Convection, and Radiation

8.1 Conduction, Convection, and Radiation

Quick Answer: Conduction follows Fourier’s law q = −k dT/dx. Convection follows Newton’s law of cooling q = h (T_s − T_∞), with natural vs forced flow setting h. Radiation scales with T⁴ (Stefan–Boltzmann). Nu, Pr, and Re tell you how film coefficients depend on flow and fluid properties.

Domain C on the UPDA/MMUP Chemical exam covers transport phenomena—fluids (Chapter 7), heat (this chapter), and mass transfer (Chapter 9). Heat-transfer stems ask which mechanism dominates, how driving forces are written, what k or h means, and what Nu/Pr/Re represent. Qatar plant context (LNG, gas treating, refining, utilities) is full of exchangers, fired heaters, air coolers, and insulated lines—the same physics under exam MCQs.

The Three Modes of Heat Transfer

Important: Convection is not a fourth law of nature separate from conduction. At a solid–fluid interface, heat still conducts through a thin fluid film; h packages that complex film physics into one coefficient so engineers can write simple balances.

Conduction and Fourier’s Law

Fourier’s law (1-D, steady, constant k):

q_x = −k (dT/dx)

• q_x: heat flux (W/m²), energy per time per area in the x direction
• k: thermal conductivity (W/(m·K))
• The minus sign says heat flows from hot to cold (down the temperature gradient)

For a plane wall of thickness L, area A, faces at T₁ and T₂:

Q̇ = k A (T₁ − T₂) / L

Resistance form (useful when stacking layers later):

R_cond = L / (k A), Q̇ = ΔT / R_cond

k depends on material and temperature (and phase). On a short MCQ, treat k as given unless the stem highlights a metal vs insulator contrast.

Steady vs Unsteady Conduction (Awareness)

• Steady: temperature field fixed in time → Fourier flux is constant through a 1-D wall with no generation.
• Unsteady (lumped or transient): heating a thermowell, quenching a metal piece—time constants matter. UPDA items rarely demand full PDE solutions; they may ask whether a thick wall or low-k solid stores heat (thermal capacity ρ C_p L).

Convection and Newton’s Law of Cooling

Newton’s law of cooling:

q = h (T_s − T_∞)

or Q̇ = h A (T_s − T_∞)

• T_s: surface temperature
• T_∞: bulk fluid temperature far from the surface (or mixed mean in a duct)
• h: convective heat-transfer coefficient (W/(m²·K))

h is not a pure fluid property like k. It depends on fluid properties, velocity, geometry, and whether boiling/condensation occurs.

Natural vs Forced Convection

Exam cue: If the stem says “stagnant air,” “no fan,” or “thermosiphon without pump,” think natural convection and relatively low h. If a pump circulates cooling water, think forced and much higher h.

Resistance Form for a Film

R_conv = 1 / (h A), Q̇ = (T_s − T_∞) / R_conv

When wall conduction and two films appear in series, overall U (Section 8.2) multiplies these resistances.

Radiation Basics (Stefan–Boltzmann)

All surfaces emit thermal radiation. For a gray body in large surroundings:

q = ε σ (T_s⁴ − T_sur⁴)

• ε: emissivity (0 to 1); polished metals low, oxidized/painted surfaces higher
• σ: Stefan–Boltzmann constant 5.67×10⁻⁸ W/(m²·K⁴)
• T: absolute temperature (K, not °C)

Combined convection + radiation from a hot surface to air is often written with an effective h_rad so that q_total ≈ (h_conv + h_rad)(T_s − T_∞), where h_rad ≈ ε σ (T_s + T_∞)(T_s² + T_∞²) when surroundings ≈ fluid temperature. You need the idea—not a memorized expansion—for MCQs that ask which mode grows fastest as temperature rises (radiation, because of T⁴).

Combined Modes in Series and Parallel

Series (common wall): process fluid film → solid wall → utility fluid film. The same heat rate Q̇ passes each layer; temperature drops add. The largest resistance controls the duty for a fixed overall ΔT (Section 8.2).

Parallel (same surface to surroundings): convection and radiation leave a hot outer surface simultaneously; fluxes add.

Inside solids: pure conduction. Inside flowing bulk fluid: energy is carried by advection; design still uses film coefficients at boundaries.

Worked Concept: Why Insulation Works

A bare steam line loses heat by outer convection (and radiation). Adding insulation:

1. Inserts a low-k solid layer → large R_cond
2. Outer surface temperature falls → both outer convection and radiation drop

Critical insulation thickness for small cylinders is a nuance sometimes in textbooks; for UPDA, know that insulation raises conduction resistance and usually cuts heat loss, improving energy efficiency and personnel protection.

Dimensionless Numbers: Nu, Pr, Re

Convective correlations rarely appear as raw tables on licensing exams. They appear as roles:

Reynolds Number — Flow Regime

Re = ρ v L / μ = v L / ν

• Compares inertial to viscous forces (Chapter 7)
• Laminar vs turbulent transitions (pipe Re ~ 2300 rule of thumb) change Nu dramatically
• Higher Re (forced flow) generally → higher h for the same fluid and size

Prandtl Number — Fluid Property Link

Pr = ν / α = μ C_p / k

Pr is a fluid property group (weakly T-dependent). It does not depend on equipment size or velocity.

Nusselt Number — Dimensionless Film Coefficient

Nu = h L / k_fluid

• Compares actual convective transfer to pure conduction across length L through the fluid’s k
• Correlations: Nu = f(Re, Pr, geometry) for forced convection; Nu = f(Gr, Pr) for natural convection (Grashof Gr encodes buoyancy)
• Solving for h: h = Nu · k_fluid / L

Exam trap: Nu uses fluid thermal conductivity in the definition, not the wall metal k. Metal k appears in conduction through the wall, not in Nu.

Worked Mini-Estimate: Order of h from Nu

Air flows over a tube with characteristic diameter D = 0.05 m. Suppose a correlation gives Nu ≈ 40, and k_air ≈ 0.028 W/(m·K).

h = Nu · k / D = 40 × 0.028 / 0.05 ≈ 22 W/(m²·K)

That is a plausible forced-air film coefficient. If the same Nu applied with liquid water (k ≈ 0.6 W/(m·K)):

h ≈ 40 × 0.6 / 0.05 ≈ 480 W/(m²·K)

Same Nu and size → much larger h because water conducts better in the film definition. Real Nu also changes with Re and Pr, but the illustration shows why liquid films often beat gas films.

Linking Modes to UPDA-Style Questions

Expect items that ask:

1. Which mechanism? Heat through a steel plate → conduction; heat from pipe OD to breezy air → convection (+ radiation if hot); furnace tube to flame/gas → radiation and convection.
2. What increases Q̇? Larger A, larger k or h, larger ΔT, thinner wall (for conduction-limited cases).
3. What does Nu mean? Dimensionless h; rises when turbulence improves mixing near the wall.
4. Natural vs forced? Forced flow raises Re and typically Nu and h.

Common Traps

• Using °C instead of K inside T⁴ radiation formulas
• Treating h as a constant material property like density
• Confusing k of the wall with k in Nu (fluid)
• Claiming radiation is always negligible outdoors on a 400 °C line—often false
• Mixing heat flux q (W/m²) with heat rate Q̇ (W)

Exam Workflow

1. Identify mode(s) and write the constitutive law.
2. Confirm driving force: dT/dx, (T_s − T_∞), or (T⁴ − T_sur⁴).
3. For convection, classify natural vs forced and whether phase change is implied.
4. Map Re → regime, Pr → fluid, Nu → h.
5. If multiple layers, switch to resistance / overall U thinking (Section 8.2).

Master these three modes and the three dimensionless roles before heat-exchanger LMTD arithmetic in Section 8.3—exchanger design assumes you already know what h and k mean.