Sources and References

• Incropera, DeWitt, Bergman and Lavine, Fundamentals of Heat and Mass Transfer, 7th and 8th editions - principal source for conduction, convection, radiation, heat-exchanger treatment, and dimensionless-group development.
• Cengel and Ghajar, Heat and Mass Transfer: Fundamentals and Applications - parallel coverage with abundant worked examples, especially for boiling and condensation.
• Kreith, Manglik and Bohn, Principles of Heat Transfer - strong on radiation exchange in grey and black enclosures, including Hottel’s zone method and mean-beam-length derivations.
• Holman, Heat Transfer - classical text with compact treatments of extended surfaces and heat-exchanger design.
• Kays, Crawford and Weigand, Convective Heat and Mass Transfer - graduate-style development of boundary-layer equations, momentum-heat transfer analogies, and turbulent-flow closures; source for ME 456 convection material.
• Siegel and Howell, Thermal Radiation Heat Transfer - definitive reference for participating-media radiation, view-factor algebra, and radiosity networks in ME 456.
• MIT OpenCourseWare 2.005 Thermal-Fluids Engineering I and 2.006 Thermal-Fluids Engineering II - lecture outlines and problem sets used to calibrate scope and emphasis.
• Stanford ME 131B Heat Transfer - syllabus and topic ordering cross-referenced for coverage of transient conduction and external forced convection.
• Mills and Coimbra, Basic Heat and Mass Transfer - supplementary reference for natural-convection correlations across geometries.
• Rohsenow, Hartnett and Cho (eds.), Handbook of Heat Transfer - consulted for boiling/condensation correlation constants and fouling-factor recommendations.

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ME 353 — Heat Transfer (1 & 2)

Heat transfer is the discipline that quantifies rates of thermal energy transport when a system is driven from equilibrium by temperature differences. Thermodynamics tells us how much energy is exchanged between two states; heat transfer tells us how fast and through what physical mechanism the transfer occurs. These notes consolidate the classical one-semester treatment of conduction, convection, and radiation with the second-semester extension into advanced convection, participating media, heat exchangers, and phase-change transport, so that a single document covers the material historically partitioned between ME 353 and ME 456.

1. Modes of Heat Transfer

Three physical mechanisms transport thermal energy. Conduction moves energy by molecular interaction — lattice vibrations in solids, collisions of molecules and free electrons in fluids — without bulk motion. Convection superimposes conduction on the advection of fluid parcels, so that energy rides the flow. Radiation transports energy via electromagnetic waves and requires no intervening medium; every body above absolute zero radiates.

The constitutive laws for each mode are written in rate form:

\[ q''_{\text{cond}} = -k \nabla T, \qquad q''_{\text{conv}} = h\left(T_s - T_\infty\right), \qquad q''_{\text{rad,net}} = \varepsilon \sigma \left(T_s^4 - T_{\text{sur}}^4\right). \]

Fourier’s law introduces the thermal conductivity \( k \) (W/m·K); Newton’s law of cooling introduces the convective coefficient \( h \) (W/m²·K); the Stefan–Boltzmann relation introduces the emissivity \( \varepsilon \) and the constant \( \sigma = 5.67 \times 10^{-8} \) W/m²·K⁴. All three rate laws are linear in a driving “potential” — a temperature gradient, a temperature difference, or a difference of fourth powers of absolute temperature — which motivates the thermal-resistance analogy used throughout the course.

2. The Heat Equation

Applying the first law to a differential control volume fixed in space, with Fourier’s law supplying the flux, yields the governing partial differential equation for the temperature field:

\[ \rho c_p \frac{\partial T}{\partial t} = \nabla \cdot \left(k \nabla T\right) + \dot{q}. \]

Here \( \dot{q} \) (W/m³) is a volumetric generation term. When conductivity is uniform and there is no generation, this becomes \( \partial T / \partial t = \alpha \nabla^2 T \), with thermal diffusivity \( \alpha = k / (\rho c_p) \). The diffusivity measures how rapidly a thermal disturbance propagates relative to how much energy the material stores, and it appears in every transient-conduction scale.

Boundary conditions most commonly encountered are: prescribed surface temperature (first kind, or Dirichlet), prescribed surface flux (second kind, or Neumann, including the insulated case \( q'' = 0 \)), and convective exchange with a surrounding fluid (third kind, or Robin), \( -k \partial T / \partial n = h(T_s - T_\infty) \). A radiative boundary condition has the same form with \( h_r = \varepsilon \sigma (T_s + T_{\text{sur}})(T_s^2 + T_{\text{sur}}^2) \) replacing \( h \), linearising the fourth-power law around the working point.

3. Steady One-Dimensional Conduction

3.1 Plane wall

For a slab of thickness \( L \), conductivity \( k \), and no generation, integration of the heat equation gives a linear temperature profile and constant heat rate \( q = kA (T_1 - T_2) / L \). Defining a thermal resistance \( R_{\text{cond}} = L/(kA) \) allows direct analogy with Ohm’s law: the heat rate plays the role of current, the temperature difference the role of voltage. Composite walls with several layers in series produce an additive resistance \( \sum L_i /(k_i A) \), and parallel paths combine as parallel resistors. Contact resistance \( R''_{t,c} \) at mating interfaces accounts for the microscopic air gaps that impede conduction between nominally joined surfaces.

3.2 Radial systems

Integrating the heat equation in cylindrical coordinates, assuming purely radial conduction with constant \( k \) and no generation, yields a logarithmic temperature profile and resistance

\[ R_{\text{cyl}} = \frac{\ln(r_o / r_i)}{2\pi L k}. \]

For a sphere the equivalent result is \( R_{\text{sph}} = (1/r_i - 1/r_o)/(4\pi k) \). A classical subtlety for insulated pipes is the critical radius: adding insulation increases conductive resistance but decreases external convective resistance, and for small pipes the second effect can dominate. Setting \( dq/dr = 0 \) gives \( r_{\text{cr}} = k_{\text{ins}}/h \) for a cylinder and \( 2k_{\text{ins}}/h \) for a sphere; insulation thinner than \( r_{\text{cr}} \) counter-intuitively raises the heat loss.

3.3 Generation

With uniform volumetric generation \( \dot{q} \), a plane wall of half-thickness \( L \) cooled symmetrically to \( T_s \) has a parabolic profile \( T(x) = T_s + \dot{q}(L^2 - x^2)/(2k) \). The centreline temperature \( T_0 = T_s + \dot{q} L^2 /(2k) \) is often the design-limiting quantity, for example in nuclear fuel pins or current-carrying conductors. The cylindrical analogue has centreline temperature \( T_0 = T_s + \dot{q} r_o^2 / (4k) \).

4. Extended Surfaces (Fins)

When \( h \) is small, augmenting the convective surface with fins raises the heat rate for a given base temperature. Modelling a thin fin of constant cross-section with one-dimensional conduction along its length and convection to the ambient gives

\[ \frac{d^2 \theta}{dx^2} - m^2 \theta = 0, \qquad m^2 = \frac{hP}{k A_c}, \qquad \theta \equiv T - T_\infty, \]

where \( P \) is the perimeter and \( A_c \) the cross-sectional area. For an adiabatic tip the solution is hyperbolic-cosine, giving a base heat rate \( q_f = M \tanh(mL) \) with \( M = \sqrt{hPkA_c}\,\theta_b \). Fin efficiency \( \eta_f = q_f / (h A_f \theta_b) \) compares actual heat rate with the ideal case of a uniformly base-temperature fin, and fin effectiveness \( \varepsilon_f = q_f /(h A_{c,b} \theta_b) \) compares finned and unfinned bases. A rule of thumb is that fins are worthwhile when \( \varepsilon_f \gtrsim 2 \), which roughly requires \( k P /(h A_c) \gtrsim 4 \); fins do most work on the gas side of gas-to-liquid heat exchangers, where \( h \) is small.

Arrays of fins on a common base are analysed through the overall surface efficiency \( \eta_o = 1 - (NA_f / A_t)(1 - \eta_f) \), which then replaces the bare-wall convection term in the resistance network.

5. Steady Two- and Three-Dimensional Conduction

When the one-dimensional idealisation is inadequate three tools are commonly used. Conduction shape factors \( S \) capture the geometry in tabulated form so that \( q = S k (T_1 - T_2) \) for canonical pairs of isothermal surfaces such as a buried pipe, an isothermal disc on a semi-infinite medium, or a spherical cavity in a solid. Separation of variables solves problems like the rectangular plate with three walls at \( T_1 \) and a fourth at \( T_2 \), producing a Fourier series whose eigenvalues encode the geometry. Finite-difference methods discretise the domain onto a grid; interior nodes in a Cartesian mesh with uniform spacing \( \Delta x \) satisfy \( T_{i+1,j} + T_{i-1,j} + T_{i,j+1} + T_{i,j-1} - 4T_{i,j} = 0 \), and boundary nodes require energy balances that absorb convective or radiative exchanges into modified coefficients.

6. Transient Conduction

6.1 Lumped capacitance

When internal conduction is fast compared with surface convection, the body responds as a single thermal mass and its temperature is spatially uniform. The energy balance gives

\[ \frac{T(t) - T_\infty}{T_i - T_\infty} = \exp\left[-\frac{hA_s}{\rho c_p V}\, t\right] = \exp(-t/\tau). \]

The criterion for validity is the Biot number \( \mathrm{Bi} = h L_c / k_s < 0.1 \), where \( L_c = V/A_s \) is a characteristic length. Physically, \( \mathrm{Bi} \) compares internal conductive resistance with external convective resistance; a small Biot means the inside of the body is well mixed thermally. The Fourier number \( \mathrm{Fo} = \alpha t / L_c^2 \) plays the role of a dimensionless time.

6.2 One-dimensional transient in simple shapes

When \( \mathrm{Bi} > 0.1 \) spatial gradients matter. For a plane wall, an infinite cylinder, and a sphere, separation of variables yields an infinite series of decaying exponentials governed by eigenvalues that depend on \( \mathrm{Bi} \). For \( \mathrm{Fo} > 0.2 \) the first term dominates and

\[ \theta^*(x^*, \mathrm{Fo}) = C_1 \exp(-\zeta_1^2 \mathrm{Fo})\, f(\zeta_1 x^*), \]

with tabulated eigenvalue \( \zeta_1 \) and constant \( C_1 \). Heisler and Gröber charts plot these results; modern practice simply tabulates \( \zeta_1, C_1 \) against \( \mathrm{Bi} \) and evaluates the one-term expression directly. The centreline temperature and the accumulated energy transfer \( Q/Q_o \) are read off as functions of \( \mathrm{Fo} \).

6.3 Semi-infinite solid

A body so thick that the thermal disturbance has not reached the far boundary during the time of interest is modelled as semi-infinite. For a sudden step in surface temperature the similarity solution is

\[ \frac{T(x,t) - T_s}{T_i - T_s} = \mathrm{erf}\!\left(\frac{x}{2\sqrt{\alpha t}}\right), \qquad q''_s(t) = \frac{k(T_s - T_i)}{\sqrt{\pi \alpha t}}. \]

Similar closed forms exist for a step in surface flux and for convective exchange with a fluid. A useful practical test: the thermal penetration depth scales as \( \delta_T \sim \sqrt{\alpha t} \), which bounds the applicability of the semi-infinite idealisation.

7. Fundamentals of Convection

7.1 Boundary layers

Viscous flow past a surface develops a velocity boundary layer of thickness \( \delta(x) \) within which the free-stream velocity is retarded from \( u_\infty \) to zero at the wall. Simultaneously, if wall and fluid are at different temperatures, a thermal boundary layer of thickness \( \delta_t(x) \) develops. At the wall

\[ q''_s = -k_f \left.\frac{\partial T}{\partial y}\right|_{y=0} = h(T_s - T_\infty), \]

so the convective coefficient is fixed by the wall-normal temperature gradient on the fluid side. Thin thermal boundary layers mean large gradients and large \( h \).

7.2 Governing equations and dimensionless groups

The boundary-layer equations for a two-dimensional incompressible flow with constant properties are the continuity, \( x \)-momentum, and energy equations. Non-dimensionalisation with a reference length \( L \) and velocity \( u_\infty \) yields three groups: the Reynolds number \( \mathrm{Re}_L = u_\infty L / \nu \), which compares inertia with viscosity and sets the transition to turbulence; the Prandtl number \( \mathrm{Pr} = \nu / \alpha \), which compares momentum and thermal diffusivities and therefore the relative thicknesses of velocity and thermal boundary layers; and the Nusselt number \( \mathrm{Nu}_L = h L / k_f \), a dimensionless convective coefficient. Empirical correlations almost universally take the form \( \mathrm{Nu} = f(\mathrm{Re}, \mathrm{Pr}) \).

For gases \( \mathrm{Pr} \sim 0.7 \) so \( \delta \approx \delta_t \); for oils \( \mathrm{Pr} \gg 1 \) so \( \delta_t \ll \delta \); for liquid metals \( \mathrm{Pr} \ll 1 \), reversing the ordering.

7.3 Reynolds–Colburn analogy

Because momentum and thermal boundary layers obey equations of similar structure, the skin-friction and Stanton numbers are related. In the simplest form,

\[ \frac{C_f}{2} = \mathrm{St}\, \mathrm{Pr}^{2/3}, \qquad \mathrm{St} = \frac{\mathrm{Nu}}{\mathrm{Re}\,\mathrm{Pr}}. \]

This Chilton–Colburn analogy converts momentum-transfer measurements (pressure drop, friction factor) into heat-transfer predictions and underpins approximate turbulent-flow correlations where an exact analytical solution is out of reach.

8. Forced External Convection

8.1 Flat plate

For laminar flow over an isothermal flat plate the Blasius similarity solution yields \( \delta / x = 5 / \sqrt{\mathrm{Re}_x} \) and the local Nusselt number

\[ \mathrm{Nu}_x = 0.332\, \mathrm{Re}_x^{1/2}\, \mathrm{Pr}^{1/3}, \quad 0.6 \lesssim \mathrm{Pr} \lesssim 60. \]

Averaging over plate length \( L \) doubles the coefficient: \( \overline{\mathrm{Nu}}_L = 0.664\, \mathrm{Re}_L^{1/2}\, \mathrm{Pr}^{1/3} \). Transition to turbulence occurs near \( \mathrm{Re}_{x,c} \approx 5 \times 10^5 \); in the turbulent regime \( \mathrm{Nu}_x = 0.0296\, \mathrm{Re}_x^{4/5}\, \mathrm{Pr}^{1/3} \), and the mixed laminar/turbulent plate is handled by piecewise integration.

8.2 Cylinders and spheres in crossflow

For a cylinder the Churchill–Bernstein correlation,

\[ \overline{\mathrm{Nu}}_D = 0.3 + \frac{0.62\, \mathrm{Re}_D^{1/2}\, \mathrm{Pr}^{1/3}}{\left[1 + (0.4/\mathrm{Pr})^{2/3}\right]^{1/4}} \left[1 + \left(\frac{\mathrm{Re}_D}{282{,}000}\right)^{5/8}\right]^{4/5}, \]

covers six decades of \( \mathrm{Re}_D \cdot \mathrm{Pr} \). Spheres follow Whitaker: \( \overline{\mathrm{Nu}}_D = 2 + (0.4\, \mathrm{Re}_D^{1/2} + 0.06\, \mathrm{Re}_D^{2/3})\, \mathrm{Pr}^{0.4}\, (\mu_\infty/\mu_s)^{1/4} \). The additive “2” is the conduction limit for a sphere in a quiescent medium and is worth remembering as a sanity check. Tube banks, with either aligned or staggered arrays, are handled by Zukauskas-type correlations that introduce bank-geometry correction factors.

9. Forced Internal Convection

9.1 Hydrodynamic and thermal entry lengths

A fluid entering a pipe develops a velocity profile over a hydrodynamic entry length \( x_{fd,h}/D \approx 0.05 \mathrm{Re}_D \) in laminar flow, and similarly a thermal entry length \( x_{fd,t}/D \approx 0.05\, \mathrm{Re}_D\, \mathrm{Pr} \). Beyond these lengths the dimensionless profiles become invariant in \( x \).

9.2 Fully developed laminar flow

For a circular pipe with uniform surface heat flux the fully developed Nusselt number is \( \mathrm{Nu}_D = 4.36 \); with constant surface temperature it is \( \mathrm{Nu}_D = 3.66 \). These are exact results. The mean temperature of the fluid obeys

\[ \frac{d T_m}{dx} = \frac{q''_s\, P}{\dot{m} c_p}, \]

which integrates to a linear profile under constant flux and to an exponential approach to the wall temperature under constant \( T_s \): \( (T_s - T_m(L))/(T_s - T_{m,i}) = \exp(-P L \bar{h}/(\dot{m} c_p)) \).

9.3 Turbulent flow

Turbulent pipe flow (\( \mathrm{Re}_D \gtrsim 2300 \), practically \( > 10^4 \)) is modelled by the Dittus–Boelter correlation \( \mathrm{Nu}_D = 0.023\, \mathrm{Re}_D^{4/5}\, \mathrm{Pr}^{n} \) with \( n = 0.4 \) for heating and \( n = 0.3 \) for cooling, valid for \( 0.7 \le \mathrm{Pr} \le 160 \) and sufficiently long tubes. The Gnielinski correlation,

\[ \mathrm{Nu}_D = \frac{(f/8)(\mathrm{Re}_D - 1000)\,\mathrm{Pr}}{1 + 12.7\,(f/8)^{1/2}\,(\mathrm{Pr}^{2/3} - 1)}, \]

with \( f \) from the Moody chart, offers better accuracy over a wider range. Non-circular ducts are handled by using the hydraulic diameter \( D_h = 4 A_c / P \) in place of \( D \), with tabulated Nusselt numbers for common geometries in the laminar regime.

10. Free (Natural) Convection

When density differences from temperature gradients drive the flow, the governing group is the Rayleigh number \( \mathrm{Ra}_L = g \beta (T_s - T_\infty) L^3 /(\nu \alpha) \), the product of the Grashof and Prandtl numbers, with \( \beta \) the coefficient of thermal expansion. For a vertical isothermal plate the Churchill–Chu correlation spans laminar and turbulent regimes:

\[ \overline{\mathrm{Nu}}_L = \left\{0.825 + \frac{0.387\, \mathrm{Ra}_L^{1/6}}{\left[1 + (0.492/\mathrm{Pr})^{9/16}\right]^{8/27}}\right\}^2. \]

Horizontal plates separate into hot-up / cold-down and hot-down / cold-up configurations, each with its own correlation. Horizontal cylinders, spheres, inclined plates, and enclosures (annular and rectangular) each have specific correlations — the common theme is \( \overline{\mathrm{Nu}} = C\, \mathrm{Ra}^n \) with \( n \approx 1/4 \) in the laminar regime and \( n \approx 1/3 \) in the turbulent regime.

11. Radiation: Fundamentals

11.1 Blackbody radiation

A blackbody is an idealised surface that absorbs all incident radiation and emits the theoretical maximum at every wavelength and direction. Planck’s distribution gives its spectral emissive power,

\[ E_{b,\lambda}(T) = \frac{C_1}{\lambda^5\left[\exp(C_2/(\lambda T)) - 1\right]}, \]

with \( C_1 = 3.742 \times 10^8 \) W·μm⁴/m² and \( C_2 = 1.439 \times 10^4 \) μm·K. Integrating over all wavelengths yields Stefan–Boltzmann: \( E_b = \sigma T^4 \). Wien’s displacement law \( \lambda_{\max} T = 2898 \) μm·K locates the spectral peak; at \( T = 5800 \) K (sun) the peak falls in the visible, at \( T = 300 \) K it falls near 10 μm in the far infrared. The fractional emission within a band \( [0, \lambda] \) is tabulated as \( F_{0 \to \lambda T} \), a dimensionless function of the product \( \lambda T \).

11.2 Real surfaces

Real surfaces are characterised by spectral and directional emissivity \( \varepsilon_{\lambda,\theta} \), absorptivity \( \alpha_\lambda \), reflectivity \( \rho_\lambda \), and transmissivity \( \tau_\lambda \), with \( \alpha_\lambda + \rho_\lambda + \tau_\lambda = 1 \). A diffuse surface emits and reflects independent of direction; a grey surface has \( \alpha_\lambda = \varepsilon_\lambda = \) constant over the spectrum of interest. Kirchhoff’s law, \( \varepsilon_\lambda = \alpha_\lambda \) at every wavelength and direction, links emission and absorption when the surface is in thermal equilibrium with its environment; the integrated total values are equal only for grey surfaces or when incident radiation has the same spectral shape as blackbody emission at \( T_s \).

12. View Factors and Their Algebra

The view factor \( F_{ij} \) is the fraction of radiation leaving surface \( i \) that is intercepted by surface \( j \). For diffuse surfaces it is purely geometric. Three algebraic rules are sufficient to solve most problems:

Rule	Statement
Reciprocity	\( A_i F_{ij} = A_j F_{ji} \)
Summation	\( \sum_{j=1}^{N} F_{ij} = 1 \) for an enclosure
Superposition	\( F_{i,(j+k)} = F_{ij} + F_{ik} \) when surfaces \( j \) and \( k \) are disjoint

For two-dimensional geometries the crossed-strings method gives closed-form view factors directly from surface widths and the lengths of connecting strings. Tabulated charts cover coaxial discs, perpendicular rectangles with a common edge, and parallel rectangles. A surface that cannot see itself (a convex body, for example) has \( F_{ii} = 0 \); a concave surface has \( F_{ii} > 0 \).

13. Radiation Exchange in Enclosures

For a diffuse, grey enclosure the radiosity \( J_i = \varepsilon_i E_{b,i} + \rho_i G_i \) represents all radiation leaving surface \( i \). The net rate at which radiation leaves surface \( i \) is

\[ q_i = \frac{E_{b,i} - J_i}{(1 - \varepsilon_i)/(\varepsilon_i A_i)}, \]

which is a “surface resistance”. Between any two surfaces the space resistance is \( 1/(A_i F_{ij}) \), so that \( q_{i \to j} = (J_i - J_j) /[1/(A_i F_{ij})] \). Assembling the network and enforcing energy balance at every surface gives a linear system for the \( J_i \). Special cases worth memorising:

• Two-surface enclosure: \( q_{12} = \sigma(T_1^4 - T_2^4) / \left[(1-\varepsilon_1)/(\varepsilon_1 A_1) + 1/(A_1 F_{12}) + (1-\varepsilon_2)/(\varepsilon_2 A_2)\right]. \)
• Small object in a large enclosure: \( q = \varepsilon_1 A_1 \sigma(T_1^4 - T_2^4) \).
• Parallel infinite planes with radiation shield of emissivity \( \varepsilon_3 \) inserted between them reduces exchange by a factor that Hottel’s zone network makes transparent.

For re-radiating surfaces (adiabatic, such that \( q_i = 0 \)) the surface floats at a temperature such that \( J_i = E_{b,i} \), and they can be treated as pure space-resistance nodes.

14. Participating Media

Most industrial gases (N₂, O₂) are essentially transparent in the infrared, but CO₂, H₂O vapour, and combustion products emit and absorb selectively. The monochromatic intensity along a path through an absorbing–emitting (non-scattering) gas obeys the Beer–Lambert law: \( I_\lambda(s) = I_\lambda(0) \exp(-\kappa_\lambda s) \), where \( \kappa_\lambda \) is the spectral absorption coefficient. Integrating over wavelength and path and combining with the source term gives the total gas emissivity \( \varepsilon_g \) and absorptivity \( \alpha_g \), which Hottel tabulated against gas temperature, path length \( L \), and partial pressure. For a black enclosure containing a grey gas of emissivity \( \varepsilon_g \),

\[ q = A_s \sigma \left(\varepsilon_g T_g^4 - \alpha_g T_s^4\right). \]

Mean beam lengths \( L_e \) reduce three-dimensional geometries to a single characteristic path so that Hottel’s charts remain applicable.

15. Heat Exchangers

15.1 Classification

Heat exchangers transfer energy between two streams. The dominant classes are double-pipe, shell-and-tube (with various shell/tube pass combinations), crossflow (with one or both streams mixed or unmixed), compact (e.g., finned plate-fin cores), and plate-and-frame. A global overall heat transfer coefficient \( U \) lumps convection on each side, wall conduction, and fouling:

\[ \frac{1}{UA} = \frac{1}{(\eta_o h A)_h} + R''_{f,h}/A_h + R_{wall} + R''_{f,c}/A_c + \frac{1}{(\eta_o h A)_c}, \]

with fouling resistances accounting for scale, corrosion, and biofilm accumulation over service life.

15.2 LMTD method

For a parallel-flow or counterflow exchanger with constant \( U \) and specific heats, the heat rate is \( q = U A\, \Delta T_{\mathrm{lm}} \), where the log-mean temperature difference is

\[ \Delta T_{\mathrm{lm}} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}. \]

Counterflow yields the largest \( \Delta T_{\mathrm{lm}} \) for given terminal temperatures. Other configurations carry a correction factor \( F \le 1 \) read from charts parameterised by the dimensionless capacity and temperature ratios \( R \) and \( P \); \( q = U A F \Delta T_{\mathrm{lm,cf}} \).

15.3 Effectiveness–NTU method

When outlet temperatures are unknown the effectiveness–NTU formulation is more convenient. Define the heat-capacity rates \( C_h = \dot{m}_h c_{p,h}, C_c = \dot{m}_c c_{p,c} \), \( C_{\min} \) and \( C_{\max} \) their smaller and larger values, and \( C_r = C_{\min}/C_{\max} \). The maximum conceivable heat rate is \( q_{\max} = C_{\min} (T_{h,i} - T_{c,i}) \), attained by an infinitely long counterflow device. The effectiveness is \( \varepsilon = q / q_{\max} \), and the number of transfer units is \( \mathrm{NTU} = UA/C_{\min} \). For counterflow,

\[ \varepsilon = \frac{1 - \exp[-\mathrm{NTU}(1 - C_r)]}{1 - C_r \exp[-\mathrm{NTU}(1 - C_r)]}, \qquad C_r < 1. \]

When \( C_r = 0 \) (phase change on one side) this collapses to \( \varepsilon = 1 - e^{-\mathrm{NTU}} \). Analogous closed forms exist for parallel-flow and shell-and-tube configurations; crossflow uses tabulated or chart-based expressions. Practical design proceeds by selecting a configuration, estimating \( U \), computing \( \mathrm{NTU} \) needed for target \( \varepsilon \), and sizing \( A \) accordingly — then iterating as properties refine.

16. Boiling Heat Transfer

When a liquid is exposed to a surface at \( T_s > T_{\text{sat}} \), vapour is generated at the wall. Pool boiling of saturated water at 1 atm is the canonical case. Plotting heat flux against excess temperature \( \Delta T_e = T_s - T_{\text{sat}} \) on log axes traces Nukiyama’s characteristic curve with four regimes:

1. Free convection boiling (\( \Delta T_e \lesssim 5 \) K) — single-phase buoyant flow dominates.
2. Nucleate boiling (\( 5 \lesssim \Delta T_e \lesssim 30 \) K) — isolated bubbles, then columns and slugs; the industrially useful regime because fluxes are high and the surface stays wetted. Rohsenow’s correlation,

\[ q''_s = \mu_l h_{fg} \left[\frac{g(\rho_l - \rho_v)}{\sigma}\right]^{1/2} \left(\frac{c_{p,l} \Delta T_e}{C_{sf} h_{fg} \mathrm{Pr}_l^n}\right)^3, \]

captures the cubic dependence on superheat, with \( C_{sf} \) and \( n \) surface-fluid empirical constants. 3. Transition boiling (\( \Delta T_e \) between \( \sim 30 \) and \( \sim 120 \) K for water) — unstable, with the surface alternating between wet and vapour-blanketed. 4. Film boiling (\( \Delta T_e \gtrsim 120 \) K) — continuous vapour film insulates the surface; the heat flux rises again, now with significant radiative contribution through the film.

The critical heat flux (CHF) at the peak of the nucleate curve is given by Zuber: \( q''_{\max} = C\, h_{fg}\, \rho_v\, [\sigma g(\rho_l - \rho_v)/\rho_v^2]^{1/4} \), with \( C \approx \pi/24 \). Exceeding CHF under constant-flux conditions causes the surface to jump abruptly to the film-boiling branch, often destroying the equipment (“burnout”). Flow boiling in tubes adds advection; the local heat-transfer coefficient varies dramatically with quality and mass flux, and the Chen correlation superposes a nucleate-boiling and a forced-convection contribution.

17. Condensation Heat Transfer

When vapour contacts a cooler surface it condenses. In film condensation a continuous liquid film drains under gravity; Nusselt’s 1916 analysis of laminar film condensation on a vertical plate gives the local coefficient

\[ h_x = \left[\frac{g \rho_l (\rho_l - \rho_v) h'_{fg} k_l^3}{4 \mu_l (T_{\text{sat}} - T_s) x}\right]^{1/4}, \]

with the modified latent heat \( h'_{fg} = h_{fg} + 0.68\, c_{p,l}(T_{\text{sat}} - T_s) \) accounting for subcooling of the film. The average coefficient is \( 4/3 \) times the local value at \( x = L \). Beyond a film Reynolds number \( \mathrm{Re}_\delta \approx 30 \) the film becomes wavy, and turbulent above \( \approx 1800 \); correlations of Kutateladze and Labuntsov extend Nusselt’s result. Horizontal tubes and tube banks require their own geometry-specific forms.

Dropwise condensation, in which the condensate forms discrete droplets that roll off the surface rather than a continuous film, yields coefficients an order of magnitude higher but is difficult to sustain because surface promoters degrade. Most industrial design assumes film condensation for conservatism.

18. Practical Design Considerations

Every heat-transfer design must negotiate four families of trade-offs. Thermal performance vs. pumping power: higher \( h \) usually requires higher velocity, which increases \( \Delta P \) as roughly \( V^2 \) or \( V^3 \); the exchanger’s goodness factor \( j/f \) (Colburn to friction) quantifies the compromise. Capital vs. operating cost: a larger area reduces NTU demand per unit surface and lowers pumping losses, but raises material and footprint costs. Fouling: design clean-side velocities above the threshold at which particulates deposit, specify cleaning access, and budget a fouling factor; expect \( U \) to degrade by 20–40 % over service life. Material compatibility and fabrication: working-fluid chemistry constrains material choice (copper for water, titanium for seawater, stainless or nickel alloys for acids), and phase-change devices require careful treatment of thermal stresses, vibration, and two-phase instabilities.

19. Worked Synthesis: A Cooled Electronic Component

Consider a silicon die dissipating \( P = 30 \) W, mounted on a finned aluminium heat sink with air forced across it at \( u_\infty = 5 \) m/s. A full analysis illustrates how the material in these notes combines into a design calculation. First, the air-side flow is characterised by \( \mathrm{Re}_L \) at the sink length, placing it in the laminar or turbulent external-flow regime and giving \( \bar{h} \) through Section 8. Second, the finned array is reduced to an overall efficiency \( \eta_o \) using the single-fin efficiency from Section 4 and the base/fin area ratio. Third, the overall resistance from die junction to ambient is the series combination of die-to-case conduction, thermal-interface-material contact, spreading resistance in the sink base, and \( 1/(\eta_o \bar{h} A_t) \) at the finned surface. Fourth, the junction temperature is \( T_j = T_\infty + P R_{\text{tot}} \). A typical result with \( T_\infty = 30 \)°C, \( R_{\text{tot}} \approx 1.8 \) K/W gives \( T_j \approx 84 \)°C, safely below the \( 100 \)°C silicon limit; sensitivity analysis identifies the finned convective resistance as dominant and directs design effort there (thinner denser fins, higher velocity, or a forced-liquid cold plate if air proves insufficient).

20. Summary Table of Dimensionless Groups

Group	Definition	Physical meaning
Reynolds \( \mathrm{Re} \)	\( \rho V L / \mu \)	Inertia / viscous forces
Prandtl \( \mathrm{Pr} \)	\( \nu/\alpha \)	Momentum / thermal diffusivity
Nusselt \( \mathrm{Nu} \)	\( h L / k_f \)	Dimensionless convective coefficient
Stanton \( \mathrm{St} \)	\( h/(\rho V c_p) \)	Heat transferred / convected
Biot \( \mathrm{Bi} \)	\( h L_c / k_s \)	Internal / external conduction resistance
Fourier \( \mathrm{Fo} \)	\( \alpha t / L_c^2 \)	Dimensionless time for conduction
Grashof \( \mathrm{Gr} \)	\( g\beta \Delta T L^3 / \nu^2 \)	Buoyancy / viscous forces
Rayleigh \( \mathrm{Ra} \)	\( \mathrm{Gr}\,\mathrm{Pr} \)	Driver of natural convection
Jakob \( \mathrm{Ja} \)	\( c_{p,l}\Delta T / h_{fg} \)	Sensible / latent energy in phase change
Bond \( \mathrm{Bo} \)	\( g(\rho_l - \rho_v) L^2 / \sigma \)	Gravity / surface-tension forces
NTU	\( UA / C_{\min} \)	Size of heat exchanger
Effectiveness \( \varepsilon \)	\( q / q_{\max} \)	Heat exchanger performance