Sources and References

These notes synthesize material from the canonical transport-phenomena literature and publicly available open courseware. The Waterloo course CHE 312 was reorganized to absorb the mass-transfer content formerly taught in CHE 313, so the treatment below consolidates heat and mass transfer into a single, unified vector framework.

• Bird, R. B., Stewart, W. E., and Lightfoot, E. N. Transport Phenomena, 2nd edition.
• Incropera, F. P., DeWitt, D. P., Bergman, T. L., and Lavine, A. S. Fundamentals of Heat and Mass Transfer, 7th/8th editions.
• Welty, J. R., Wicks, C. E., Wilson, R. E., and Rorrer, G. L. Fundamentals of Momentum, Heat, and Mass Transfer.
• Cussler, E. L. Diffusion: Mass Transfer in Fluid Systems, 3rd edition.
• McCabe, W. L., Smith, J. C., and Harriott, P. Unit Operations of Chemical Engineering, 7th edition.
• Geankoplis, C. J. Transport Processes and Separation Process Principles.
• MIT OpenCourseWare 2.006 Thermal-Fluids Engineering II and 10.302 Transport Processes.
• Stanford University CHEMENG 120B Fundamentals of Heat and Mass Transport.
• Deen, W. M. Analysis of Transport Phenomena, 2nd edition, for the mathematical apparatus of separation of variables, Sturm–Liouville theory, and similarity solutions.

Chapter 1: Foundations of Heat and Mass Transport

1.1 Transport Phenomena as a Unified Subject

Chemical engineering problems routinely require the simultaneous accounting of momentum, energy, and chemical species as they move through solids, fluids, and interfaces. The underlying mathematics is almost identical in each case: a balance on a control volume produces a partial differential equation, a constitutive law closes that equation by relating a flux to a gradient, and boundary conditions describe how the system is coupled to its surroundings. The flux laws of Newton, Fourier, and Fick share a common linear form, and this shared structure is what makes the subject learnable as a single discipline rather than three disjoint ones.

Throughout these notes we treat heat and mass transfer in parallel, and we keep the vector notation that emphasizes the structural analogy. Only when empirical correlations or phase-change physics force us to specialize do we leave the abstract picture behind.

1.2 Species, Mixtures, and Velocities

Consider a mixture of \( N \) chemical species. For species \( A \) in a mixture let \( \rho_A \) be the mass concentration (mass of \( A \) per unit volume), \( \omega_A = \rho_A / \rho \) the mass fraction, \( c_A \) the molar concentration, and \( x_A = c_A / c \) the mole fraction. The species mass-average velocity is \( v_A \), the velocity one would measure by tagging molecules of \( A \) and following their average motion.

Two reference velocities are useful for describing the motion of the mixture as a whole. The mass-average velocity \( v \) and the molar-average velocity \( v^\ast \) are defined by

\[ v \;=\; \frac{1}{\rho}\sum_{A} \rho_A\, v_A, \qquad v^\ast \;=\; \frac{1}{c}\sum_{A} c_A\, v_A. \]

The diffusive flux of species \( A \) is its motion relative to one of these mixture velocities. The mass diffusive flux relative to \( v \) is

\[ j_A \;=\; \rho_A\,(v_A - v), \]

while the molar diffusive flux relative to \( v^\ast \) is

\[ J_A^\ast \;=\; c_A\,(v_A - v^\ast). \]

By construction \( \sum_A j_A = 0 \) and \( \sum_A J_A^\ast = 0 \): diffusion, as defined here, is motion with respect to a reference that moves with the bulk. The total fluxes \( n_A = \rho_A v_A \) and \( N_A = c_A v_A \) are the sum of a convective and a diffusive contribution,

\[ n_A \;=\; \rho_A v + j_A, \qquad N_A \;=\; c_A v^\ast + J_A^\ast. \]

1.3 Fick’s Law and the Diffusion Coefficient

For a binary mixture of species \( A \) and \( B \), Fick’s first law states that the diffusive flux is proportional to the concentration gradient,

\[ j_A \;=\; -\rho\, D_{AB}\, \nabla \omega_A, \qquad J_A^\ast \;=\; -c\, D_{AB}\, \nabla x_A. \]

The scalar \( D_{AB} \) is the binary diffusivity, with units \( \mathrm{m^2\,s^{-1}} \). For dilute gases Chapman–Enskog kinetic theory predicts \( D_{AB} \propto T^{3/2}/P \); for liquids the Stokes–Einstein estimate \( D_{AB} = k_B T /(6\pi\mu r_A) \) captures the essential temperature and viscosity dependence. Typical orders of magnitude are \( 10^{-5}\,\mathrm{m^2/s} \) in gases, \( 10^{-9}\,\mathrm{m^2/s} \) in liquids, and \( 10^{-12}\,\mathrm{m^2/s} \) in solids.

Multicomponent diffusion is governed by the Maxwell–Stefan equations, which are not needed for this course; we restrict attention to binary or pseudo-binary systems.

1.4 Fourier’s Law and Thermal Conductivity

The conductive heat flux in an isotropic medium is

\[ q \;=\; -k\, \nabla T, \]

where \( k \) is the thermal conductivity with units \( \mathrm{W\,m^{-1}\,K^{-1}} \). Typical values are \( 0.02 \) for gases, \( 0.6 \) for water, \( 0.1\text{–}1 \) for polymers, \( 10\text{–}400 \) for metals. In anisotropic media, \( k \) is a second-order tensor, but in these notes we always take \( k \) to be a scalar that may depend on temperature and composition.

The three mechanisms of heat transfer—conduction, convection, and radiation—are coupled through boundary conditions. Conduction is a diffusive molecular process described by Fourier’s law, convection is bulk motion described by an enthalpy flux \( \rho v \hat H \), and radiation is transfer by electromagnetic waves, whose flux between two gray surfaces is given by the Stefan–Boltzmann law \( q_{\mathrm{rad}} = \varepsilon\sigma(T_s^4 - T_{\mathrm{surr}}^4) \).

1.5 The Analogy Made Precise

The three molecular flux laws can be written in a common form, linking a flux to a gradient of something that is transported per unit mass.

Quantity	Flux law	Diffusivity
Momentum	\( \tau_{yx} = -\mu\, dv_x/dy \)	\( \nu = \mu/\rho \)
Energy	\( q_y = -\rho c_p \alpha\, dT/dy \)	\( \alpha = k/(\rho c_p) \)
Species	\( j_{A,y} = -\rho D_{AB}\, d\omega_A/dy \)	\( D_{AB} \)

The dimensionless groupings \( \nu/\alpha = \mathrm{Pr} \) (Prandtl), \( \nu/D_{AB} = \mathrm{Sc} \) (Schmidt), and \( \alpha/D_{AB} = \mathrm{Le} \) (Lewis) measure the relative magnitudes of these diffusivities and control the relative thicknesses of momentum, thermal, and concentration boundary layers.

Chapter 2: Microscopic Balances

2.1 Differential Mass Balance for a Species

Applying conservation of mass to an infinitesimal control volume for species \( A \),

\[ \frac{\partial \rho_A}{\partial t} + \nabla\!\cdot\! n_A \;=\; r_A, \]

where \( r_A \) is the mass rate of production of \( A \) by homogeneous reaction. Substituting \( n_A = \rho_A v + j_A \) and Fick’s law with constant \( \rho D_{AB} \),

\[ \frac{\partial \rho_A}{\partial t} + v\!\cdot\!\nabla \rho_A \;=\; D_{AB}\nabla^2 \rho_A + r_A. \]

In molar form with constant \( c D_{AB} \) and molar production rate \( R_A \),

\[ \frac{\partial c_A}{\partial t} + v^\ast\!\cdot\!\nabla c_A \;=\; D_{AB}\nabla^2 c_A + R_A. \]

When the medium is stagnant or effectively stationary (solid, quiescent liquid) and there is no reaction, this reduces to the diffusion equation

\[ \frac{\partial c_A}{\partial t} \;=\; D_{AB}\nabla^2 c_A, \]

structurally identical to the heat equation.

2.2 Differential Energy Balance

For a solid or an incompressible fluid with constant \( \rho \), \( c_p \), \( k \), the internal energy balance reduces to

\[ \rho c_p \!\left(\frac{\partial T}{\partial t} + v\!\cdot\!\nabla T\right) \;=\; k\nabla^2 T + \dot q_{\mathrm{gen}} + \Phi, \]

where \( \dot q_{\mathrm{gen}} \) is a volumetric heat source (reaction, Joule heating, absorbed radiation) and \( \Phi \) is the viscous dissipation, negligible for most engineering calculations. In a stationary medium the convective term vanishes and we recover Fourier’s conduction equation

\[ \frac{\partial T}{\partial t} \;=\; \alpha\nabla^2 T + \frac{\dot q_{\mathrm{gen}}}{\rho c_p}. \]

2.3 Shell Balance as a Derivation Tool

The vector form above is clean and general, but when the geometry is simple it is often faster to derive the governing ODE or PDE by a shell balance. For a thin shell of thickness \( \Delta r \) one writes rate in \( - \) rate out \( + \) generation \( = \) accumulation, divides by the shell volume, and takes the limit. This approach forces the student to be explicit about area factors in cylindrical (\( 2\pi r L \)) and spherical (\( 4\pi r^2 \)) coordinates, and is the standard path used in Bird–Stewart–Lightfoot for deriving the equations of change.

2.4 Boundary Conditions

Physical problems are closed by three classes of boundary condition.

• Dirichlet (first kind): the field is prescribed, e.g. \( T = T_s \) or \( c_A = c_{A,s} \) on a surface. This applies when a surface is held by a phase change (wet-bulb), a large reservoir, or a fast surface reaction.
• Neumann (second kind): the flux is prescribed, e.g. \( -k\,\partial T/\partial n = q_s'' \) or \( -D_{AB}\,\partial c_A/\partial n = N_{A,s} \). An insulated surface corresponds to \( \partial T/\partial n = 0 \).
• Robin (third kind): a flux is related linearly to the local field, as in convective cooling \( -k\,\partial T/\partial n = h(T_s - T_\infty) \) or interfacial mass transfer \( -D_{AB}\,\partial c_A/\partial n = k_c(c_{A,s} - c_{A,\infty}) \).

At an interface between two phases, additional conditions enforce continuity of temperature (thermal equilibrium) and either continuity of flux or an interfacial resistance. For species, equilibrium at an interface is expressed through Henry’s law, partition coefficients, or vapor–liquid equilibrium relations.

Chapter 3: Steady Conduction and Diffusion in One Dimension

3.1 Cartesian Slab

For steady, source-free, one-dimensional conduction in a slab of thickness \( L \) with temperatures \( T_0 \) and \( T_L \) on its faces, \( d^2T/dx^2 = 0 \) yields a linear profile and a flux

\[ q'' \;=\; \frac{k(T_0 - T_L)}{L}. \]

This motivates the thermal-resistance concept, \( R_{\mathrm{cond}} = L/(kA) \), so the heat flow is \( \dot Q = \Delta T /R \). Convective boundary resistance \( 1/(hA) \), contact resistance \( R_c \), and radiative resistance combine in series and parallel like electrical resistors. For a plane wall with convection on both sides the total resistance is \( 1/(h_1 A) + L/(kA) + 1/(h_2 A) \).

The same mathematics applies to diffusion through a membrane at steady state: the molar flux of \( A \) is \( N_A = D_{AB}(c_{A,0} - c_{A,L})/L \), and permeability relations \( P = D_{AB} S \) (diffusivity times solubility) combine in series.

3.2 Cylindrical Geometry

Steady, radial conduction in a hollow cylinder satisfies \( (1/r)(d/dr)(r\,dT/dr) = 0 \). Integration gives

\[ T(r) \;=\; T_1 + \frac{T_2 - T_1}{\ln(r_2/r_1)}\,\ln\!\frac{r}{r_1}, \]

with flux \( \dot Q = 2\pi k L (T_1 - T_2)/\ln(r_2/r_1) \). The logarithmic profile means the temperature gradient is not uniform through the wall, and insulation effectiveness on a small pipe may even be counterproductive: the critical radius \( r_{\mathrm{cr}} = k_{\mathrm{ins}}/h \) marks the outer insulation radius beyond which further thickness reduces the total resistance.

3.3 Spherical Geometry

For a hollow sphere, \( (1/r^2)(d/dr)(r^2\,dT/dr) = 0 \), giving a \( 1/r \) profile and heat flow

\[ \dot Q \;=\; 4\pi k \frac{T_1 - T_2}{1/r_1 - 1/r_2}. \]

An identical equation governs steady diffusion in a stagnant spherical film around a dissolving particle or an evaporating droplet, a result central to Stefan’s problem and to the Frössling correlation for \( \mathrm{Sh} \) on a sphere.

3.4 Conduction with Heat Generation

For a slab with uniform volumetric generation \( \dot q \), \( k\,d^2T/dx^2 + \dot q = 0 \) gives a parabolic temperature profile. Symmetric cooling produces a maximum at the center, \( T_{\mathrm{max}} = T_s + \dot q L^2/(2k) \). Analogous problems arise in fixed-bed catalysis (heat of reaction) and in electrical wires (Joule heating). The corresponding mass-transfer problem is a first-order homogeneous reaction in a slab, governed by \( D_{AB}\,d^2c_A/dx^2 - k_1 c_A = 0 \), whose solution involves hyperbolic functions and leads to the Thiele modulus \( \phi = L\sqrt{k_1/D_{AB}} \) and the catalyst effectiveness factor.

3.5 Extended Surfaces: The Fin Equation

A fin of constant cross-section \( A_c \) and perimeter \( P \), with local temperature \( T(x) \) and ambient \( T_\infty \), satisfies

\[ \frac{d^2\theta}{dx^2} - m^2\theta \;=\; 0, \qquad m^2 = \frac{hP}{kA_c},\quad \theta = T - T_\infty. \]

Solutions are combinations of \( \cosh(mx) \) and \( \sinh(mx) \). For an infinite fin \( \theta = \theta_0 e^{-mx} \) and the heat dissipation is \( \dot Q = \sqrt{hPkA_c}\,\theta_0 \). For a fin of length \( L \) with adiabatic tip, \( \theta(x) = \theta_0 \cosh[m(L-x)]/\cosh(mL) \) and the fin efficiency \( \eta_f = \tanh(mL)/(mL) \) quantifies how close the fin comes to the ideal of being isothermal.

Chapter 4: Bessel Functions and Cylindrical Problems

4.1 Origin of Bessel’s Equation

In cylindrical coordinates with radial dependence only, the operator \( (1/r)(d/dr)(r\,d\cdot/dr) \) applied to a field with an added linear term produces Bessel’s equation. For example, a radial fin on a pipe, or cylindrical conduction with surface convection, leads to

\[ \frac{1}{r}\frac{d}{dr}\!\left(r\frac{d\theta}{dr}\right) - m^2\theta \;=\; 0, \]

which is the modified Bessel equation of order zero. The general solution is \( \theta(r) = C_1 I_0(mr) + C_2 K_0(mr) \), where \( I_0 \) grows and \( K_0 \) decays with increasing argument.

Unmodified Bessel functions \( J_\nu(x), Y_\nu(x) \) arise in oscillatory radial problems, most notably the separation-of-variables solutions of the transient diffusion and heat equations in cylindrical coordinates.

4.2 Properties of Bessel Functions

Key properties that are used repeatedly:

• \( J_0(0)=1 \), \( J_\nu(0)=0 \) for \( \nu>0 \), and \( Y_\nu \) is singular at the origin.
• \( J_0 \) has an infinite sequence of positive roots \( \zeta_n \), approximately \( \zeta_n \approx (n - 1/4)\pi \) for large \( n \).
• Orthogonality on \( [0,R] \) with weight \( r \): the eigenfunctions \( J_0(\zeta_n r/R) \) are mutually orthogonal for eigenvalues determined by the radial boundary condition.
• Recurrence and differentiation: \( dJ_0/dx = -J_1(x) \), \( x\,dJ_1/dx + J_1 = x J_0 \).

These properties allow transient cylindrical problems to be expanded in Fourier–Bessel series in exact analogy to Fourier series in Cartesian slab problems.

Chapter 5: Transient 1D Conduction and Diffusion

5.1 Lumped Capacitance

If the Biot number \( \mathrm{Bi} = hL_c/k \ll 0.1 \), internal conduction is fast compared to surface convection and the body is essentially isothermal. Its temperature evolves as

\[ \frac{T(t) - T_\infty}{T_0 - T_\infty} \;=\; \exp\!\left(-\frac{hA_s}{\rho c_p V}\,t\right). \]

The mass-transfer analog is a well-stirred vessel losing species \( A \) through a surface with convective coefficient \( k_c \): the same exponential decay applies to the concentration.

5.2 Separation of Variables for a Slab

Consider a slab \( -L \le x \le L \) initially at \( T_0 \) plunged into a fluid at \( T_\infty \) with convective coefficient \( h \). With \( \theta = (T-T_\infty)/(T_0-T_\infty) \) the governing problem is

\[ \frac{\partial \theta}{\partial t} \;=\; \alpha \frac{\partial^2\theta}{\partial x^2}, \qquad \theta(x,0)=1,\quad \frac{\partial\theta}{\partial x}\!\bigg|_{x=0}=0,\quad -k\frac{\partial\theta}{\partial x}\!\bigg|_{x=L}=h\,\theta(L,t). \]

Assuming \( \theta(x,t) = X(x)\,\tau(t) \) and dividing yields \( \tau'/\tau = \alpha X''/X = -\alpha\lambda^2 \), a common separation constant chosen so that time decays exponentially. The spatial ODE \( X'' + \lambda^2 X = 0 \) with the symmetry and Robin conditions gives a transcendental eigenvalue equation \( \zeta_n\tan\zeta_n = \mathrm{Bi} \) where \( \zeta_n = \lambda_n L \) and \( \mathrm{Bi} = hL/k \). Expansion in eigenfunctions produces

\[ \theta(x,t) \;=\; \sum_{n=1}^{\infty} C_n \cos\!\left(\zeta_n \tfrac{x}{L}\right) \exp\!\left(-\zeta_n^2\,\mathrm{Fo}\right), \]

where \( \mathrm{Fo} = \alpha t/L^2 \) is the Fourier number and \( C_n = 4\sin\zeta_n/(2\zeta_n + \sin 2\zeta_n) \). For \( \mathrm{Fo} > 0.2 \) a single term is typically sufficient, giving the Heisler-chart one-term approximation.

The cylinder and sphere follow the same logic; the cylinder’s radial ODE is a Bessel equation with eigenfunctions \( J_0 \), and the sphere’s radial ODE, after substitution \( u = r\theta \), reduces to a Cartesian-like problem with sine eigenfunctions.

5.3 Combination of Variables: The Semi-Infinite Solid

For a semi-infinite solid \( x \ge 0 \) initially at \( T_i \) with surface temperature suddenly raised to \( T_s \), the problem \( \theta_t = \alpha\theta_{xx} \) with \( \theta(x,0)=0 \), \( \theta(0,t)=1 \), \( \theta(\infty,t)=0 \) admits a similarity solution. Introducing \( \eta = x/(2\sqrt{\alpha t}) \), the PDE collapses to

\[ \theta'' + 2\eta\,\theta' \;=\; 0, \]

with solution

\[ \theta(\eta) \;=\; \mathrm{erfc}(\eta) \;=\; \frac{2}{\sqrt{\pi}}\int_\eta^\infty e^{-u^2}du. \]

The surface heat flux is \( q_s''(t) = k(T_s-T_i)/\sqrt{\pi\alpha t} \), decaying as \( t^{-1/2} \). The same similarity solution governs gas absorption into a quiescent liquid over short contact times, a result at the heart of the Higbie penetration theory of mass transfer.

5.4 Laplace Transforms for Linear PDEs

For problems with time-dependent boundary data, the Laplace transform \( \bar\theta(x,s) = \int_0^\infty e^{-st}\theta(x,t)\,dt \) converts the PDE into an ODE in \( x \) with \( s \) as a parameter. For the semi-infinite solid with surface condition \( \theta(0,t)=1 \) one obtains \( \bar\theta = e^{-x\sqrt{s/\alpha}}/s \), whose inverse is the complementary error function above. For prescribed surface flux \( q_s'' \) one finds instead that the surface temperature rises as \( \sqrt t \). Laplace transforms are particularly useful when the inhomogeneity is in the boundary condition, because separation of variables would require additional steps.

5.5 Sturm–Liouville Theory and Orthogonality

A regular Sturm–Liouville problem on \( [a,b] \) has the form

\[ \frac{d}{dx}\!\left[p(x)\frac{d\phi}{dx}\right] + \left[q(x) + \lambda w(x)\right]\phi \;=\; 0, \]

with homogeneous boundary conditions of the type \( \alpha \phi + \beta\phi' = 0 \). For \( p>0, w>0 \), classical theorems guarantee that the eigenvalues \( \lambda_n \) form a countable increasing sequence with \( \lambda_n \to \infty \), that the eigenfunctions \( \phi_n \) are orthogonal with weight \( w \), and that any reasonably smooth function on \( [a,b] \) can be expanded as \( f(x) = \sum_n c_n \phi_n(x) \) with

\[ c_n \;=\; \frac{\int_a^b w(x) f(x)\phi_n(x)\,dx}{\int_a^b w(x)\phi_n^2(x)\,dx}. \]

This framework unifies Fourier sine/cosine series (Cartesian slab), Fourier–Bessel series (cylinder), and Legendre series (sphere with angular dependence). Every separation-of-variables problem in this course fits the Sturm–Liouville mold, and the coefficients of expansions such as the Heisler-series solution of Section 5.2 are computed from this inner-product formula.

Chapter 6: Steady Two-Dimensional Conduction

6.1 Laplace’s Equation in a Rectangle

Consider steady conduction in a rectangle \( 0\le x\le a \), \( 0\le y\le b \) with three sides at \( T_1 \) and the top at \( T_2 \). With \( \theta = (T-T_1)/(T_2-T_1) \), \( \nabla^2\theta = 0 \), \( \theta = 0 \) on three sides, \( \theta = 1 \) on \( y=b \). Separation \( \theta(x,y) = X(x)Y(y) \) gives \( X'' + \lambda^2 X = 0 \), \( Y'' - \lambda^2 Y = 0 \). The \( x \)-eigenfunctions are \( \sin(n\pi x/a) \), and \( Y_n(y) = \sinh(n\pi y/a) \) after applying the \( y=0 \) condition. Summing and applying the \( y=b \) condition via Fourier coefficients,

\[ \theta(x,y) \;=\; \frac{2}{\pi}\sum_{n=1}^{\infty} \frac{(-1)^{n+1}+1}{n}\,\frac{\sinh(n\pi y/a)}{\sinh(n\pi b/a)}\,\sin\!\left(\frac{n\pi x}{a}\right). \]

The same series solves a 2D steady diffusion problem with analogous boundary data.

6.2 Superposition and Conduction Shape Factors

Any steady 2D problem with four inhomogeneous Dirichlet boundaries can be decomposed into four subproblems each with one nonzero boundary. Summing their solutions by linearity gives the full solution. For common geometries (buried pipe, square duct, disk on semi-infinite medium), tabulated shape factors \( S \) give \( \dot Q = Sk\Delta T \) without needing to solve the PDE.

6.3 Laplace-Transform Approach

When a steady 2D problem has one infinite direction (e.g. a semi-infinite strip), a Laplace transform in the finite direction or a Fourier transform in the infinite direction produces an ODE in the remaining variable. These techniques are especially useful when the solution decays at infinity, so that boundary conditions reduce the transform to a single-arbitrary-constant form.

Chapter 7: Convective Heat Transfer

7.1 Boundary Layers and the Reynolds, Prandtl, Nusselt Groups

When a fluid flows over a heated surface, the thermal boundary layer has thickness \( \delta_T \) related to the momentum boundary layer thickness \( \delta \) through the Prandtl number. Dimensional analysis of the boundary-layer equations shows that the local Nusselt number \( \mathrm{Nu}_x = h_x x/k \) is a function of Reynolds number \( \mathrm{Re}_x = \rho U_\infty x/\mu \) and Prandtl number \( \mathrm{Pr} = \nu/\alpha \) only, in fully developed regimes. For mass transfer the analog is \( \mathrm{Sh} = k_c L/D_{AB} = f(\mathrm{Re},\mathrm{Sc}) \).

Reynolds’s analogy, in its simplest form, states that when \( \mathrm{Pr} = 1 \) (or \( \mathrm{Sc} = 1 \)) and pressure gradients are absent, \( \mathrm{St} = C_f/2 \). The Chilton–Colburn \( j \)-factor correction extends the analogy to \( 0.6 \le \mathrm{Pr} \le 60 \) via

\[ j_H \;=\; \mathrm{St}\,\mathrm{Pr}^{2/3} \;=\; j_D \;=\; \mathrm{St}_m \,\mathrm{Sc}^{2/3} \;=\; \frac{C_f}{2}. \]

7.2 External Flow Correlations

For laminar flow over a flat plate (\( \mathrm{Re}_x < 5\times 10^5 \), \( \mathrm{Pr} \gtrsim 0.6 \)),

\[ \mathrm{Nu}_x \;=\; 0.332\,\mathrm{Re}_x^{1/2}\mathrm{Pr}^{1/3},\qquad \overline{\mathrm{Nu}}_L \;=\; 0.664\,\mathrm{Re}_L^{1/2}\mathrm{Pr}^{1/3}. \]

For turbulent flow over a flat plate,

\[ \mathrm{Nu}_x \;=\; 0.0296\,\mathrm{Re}_x^{4/5}\mathrm{Pr}^{1/3}, \]

with appropriate mixed correlations for the full plate if the transition occurs within it. Flow across a single cylinder uses the Hilpert or Churchill–Bernstein correlation; flow across tube banks uses Zukauskas; spheres use Whitaker or Frössling.

7.3 Internal Flow

For fully developed laminar flow in a circular tube with constant wall heat flux, \( \mathrm{Nu}_D = 4.36 \); with constant wall temperature, \( \mathrm{Nu}_D = 3.66 \). In the thermal entry region the Hausen correlation gives

\[ \mathrm{Nu}_D \;=\; 3.66 + \frac{0.0668\,(D/L)\mathrm{Re}_D\mathrm{Pr}}{1 + 0.04\,[(D/L)\mathrm{Re}_D\mathrm{Pr}]^{2/3}}. \]

For fully developed turbulent flow the Dittus–Boelter equation,

\[ \mathrm{Nu}_D \;=\; 0.023\,\mathrm{Re}_D^{4/5}\mathrm{Pr}^{n}, \]

with \( n=0.4 \) for heating and \( n=0.3 \) for cooling, is the standard working equation in the range \( 0.7 < \mathrm{Pr} < 160 \), \( \mathrm{Re}_D > 10^4 \). The Sieder–Tate correction \( (\mu/\mu_s)^{0.14} \) accounts for property variation when wall and bulk viscosities differ strongly. Gnielinski’s correlation improves accuracy at lower Reynolds numbers.

Energy balance on a tube gives \( T_m(x) \) along the tube. For constant \( T_s \),

\[ \frac{T_s - T_m(L)}{T_s - T_{m,i}} \;=\; \exp\!\left(-\frac{\pi D L\,\bar h}{\dot m c_p}\right), \]

the cornerstone relation for heat-exchanger design by the LMTD method.

Chapter 8: Heat Transfer with Phase Change

8.1 Pool Boiling

The boiling curve for water at \( 1\,\mathrm{atm} \) exhibits four regimes as the wall excess temperature \( \Delta T_e = T_s - T_{\mathrm{sat}} \) increases: natural convection, nucleate boiling, transition boiling, and film boiling. The Rohsenow correlation for the nucleate regime,

\[ q_s'' \;=\; \mu_\ell h_{fg}\!\left[\frac{g(\rho_\ell - \rho_v)}{\sigma}\right]^{1/2}\!\left[\frac{c_{p,\ell}\Delta T_e}{C_{sf}\,h_{fg}\,\mathrm{Pr}_\ell^{n}}\right]^{3}, \]

depends on an empirical surface–fluid constant \( C_{sf} \). The critical heat flux (CHF), which marks the transition away from nucleate boiling and is the design-limiting quantity for avoiding burnout, is predicted by Zuber’s correlation

\[ q_{\mathrm{CHF}}'' \;=\; 0.149\,h_{fg}\rho_v\!\left[\frac{\sigma g(\rho_\ell-\rho_v)}{\rho_v^2}\right]^{1/4}. \]

8.2 Film Condensation

For quiescent laminar film condensation on a vertical plate, Nusselt’s analysis gives

\[ \bar h_L \;=\; 0.943\!\left[\frac{g\rho_\ell(\rho_\ell-\rho_v)k_\ell^3 h_{fg}'}{\mu_\ell L(T_{\mathrm{sat}}-T_s)}\right]^{1/4}, \]

where \( h_{fg}' = h_{fg} + 0.68 c_{p,\ell}(T_{\mathrm{sat}}-T_s) \) corrects for subcooling. Horizontal tube banks use a modified correlation, and dropwise condensation (when achievable) gives coefficients an order of magnitude higher. Horizontal cylinders carry the factor \( 0.729 \) in place of \( 0.943 \) and use \( D \) in place of \( L \).

Chapter 9: Heat Exchangers

9.1 Configurations and Overall Coefficient

Common configurations are double-pipe (parallel or counter flow), shell-and-tube (with baffles to induce crossflow), and crossflow with one or both streams mixed or unmixed. The overall heat transfer coefficient between the two fluids is

\[ \frac{1}{UA} \;=\; \frac{1}{h_i A_i} + \frac{R_{f,i}}{A_i} + \frac{\ln(D_o/D_i)}{2\pi k L} + \frac{R_{f,o}}{A_o} + \frac{1}{h_o A_o}, \]

including tube-side and shell-side convective resistances, fouling factors, and conduction through the tube wall.

9.2 Log-Mean Temperature Difference

For a double-pipe exchanger with either parallel or counter flow, the integrated energy balance gives

\[ \dot Q \;=\; UA\,\Delta T_{\mathrm{lm}}, \qquad \Delta T_{\mathrm{lm}} \;=\; \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1/\Delta T_2)}, \]

where \( \Delta T_1 \) and \( \Delta T_2 \) are the temperature differences at the two ends. Counter-flow is always more efficient than parallel-flow and can in principle achieve \( T_{h,\mathrm{out}} < T_{c,\mathrm{out}} \). For multi-pass and cross-flow arrangements, a correction factor \( F \) is applied, \( \dot Q = F\,UA\,\Delta T_{\mathrm{lm,cf}} \), where the reference \( \Delta T_{\mathrm{lm,cf}} \) is computed as if the exchanger were counter-flow.

9.3 Effectiveness–NTU Method

When outlet temperatures are unknown the LMTD approach requires iteration. The \( \varepsilon\text{–}\mathrm{NTU} \) method avoids this. With capacity rates \( C_h = \dot m_h c_{p,h} \), \( C_c = \dot m_c c_{p,c} \), let \( C_{\min},C_{\max} \) and \( C_r = C_{\min}/C_{\max} \). Define

\[ \varepsilon \;=\; \frac{\dot Q}{\dot Q_{\max}} \;=\; \frac{\dot Q}{C_{\min}(T_{h,i} - T_{c,i})},\qquad \mathrm{NTU} \;=\; \frac{UA}{C_{\min}}. \]

Then \( \varepsilon = f(\mathrm{NTU}, C_r) \) for each flow arrangement. For a counter-flow exchanger,

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

with appropriate limits when \( C_r = 0 \) (phase change on one side) or \( C_r = 1 \). Corresponding expressions tabulated in Incropera cover parallel, shell-and-tube, and cross-flow cases.

Chapter 10: Interfacial Mass Transfer and Continuous Contactors

10.1 Two-Film Theory

At a gas–liquid interface the two-film model postulates thin stagnant films on each side through which transfer is by molecular diffusion. Overall mass transfer coefficients \( K_G \) and \( K_L \) (gas- and liquid-phase based) relate the flux to the overall driving force in terms of partial pressures or concentrations and combine the individual film coefficients \( k_G \), \( k_L \) with the Henry’s-law slope \( m = y^\ast/x \):

\[ \frac{1}{K_G} \;=\; \frac{1}{k_G} + \frac{m}{k_L},\qquad \frac{1}{K_L} \;=\; \frac{1}{k_L} + \frac{1}{m k_G}. \]

When \( m \) is small (very soluble gas, e.g. ammonia in water), the gas-side resistance dominates; when \( m \) is large (sparingly soluble, e.g. oxygen), the liquid-side resistance dominates.

10.2 Higbie Penetration and Surface-Renewal Theories

Higbie’s penetration theory models the liquid as parcels that expose themselves at the interface for a characteristic contact time \( t_c \), during which transient diffusion gives \( k_L = 2\sqrt{D_{AB}/(\pi t_c)} \), i.e. \( k_L \propto D_{AB}^{1/2} \). Danckwerts’s surface-renewal theory replaces the fixed \( t_c \) with a distribution of ages, yielding the same \( D^{1/2} \) scaling with \( k_L = \sqrt{s D_{AB}} \) where \( s \) is the surface-renewal rate. Both theories contrast with the two-film theory’s prediction \( k_L \propto D_{AB}^{1} \), and experimental evidence typically falls between the two.

10.3 Continuous Differential Contactors

A packed absorption column with dilute solute obeys, at each axial position \( z \),

\[ N_A a\,dz \;=\; \text{moles of } A \text{ transferred per unit tower volume per unit time}, \]

with \( a \) the wetted specific interfacial area (m\( ^2 \)/m\( ^3 \)). Writing a differential balance on the gas and using \( K_G a \) as the volumetric overall coefficient,

\[ G\,dy \;=\; -K_G a\,(y - y^\ast)\,dz. \]

Integration gives the standard design equation

\[ z \;=\; H_{OG}\,N_{OG}, \qquad H_{OG} = \frac{G}{K_G a},\qquad N_{OG} = \int_{y_1}^{y_2}\frac{dy}{y - y^\ast}, \]

where \( H_{OG} \) is the height of a transfer unit (HTU) and \( N_{OG} \) is the number of transfer units (NTU). For dilute systems with a linear equilibrium line \( y^\ast = m x \), and with the operating line set by the liquid–gas balance, \( N_{OG} \) has the Colburn closed form

\[ N_{OG} \;=\; \frac{1}{1 - A^{-1}}\ln\!\left[\left(1 - A^{-1}\right)\frac{y_1 - m x_2}{y_2 - m x_2} + A^{-1}\right], \]

with absorption factor \( A = L/(mG) \). Stripping columns use the analog with \( S = mG/L \). Minimum solvent flow rate \( L_{\min} \) is set by pinching at the rich end of the operating line against the equilibrium curve.

10.4 Distillation as Continuous Contacting

Packed distillation columns are analyzed with the same HTU/NTU framework but with two operating lines (rectifying and stripping) and a \( q \)-line defining the feed condition. Tray columns are analyzed by McCabe–Thiele stepping between operating lines and the \( y = K x \) equilibrium curve; a detailed treatment is the subject of CHE 330, but the analogy with absorption is explicit.

10.5 Liquid–Liquid Extraction

In liquid–liquid extraction the operating diagram uses mass-ratio coordinates \( X = x_A/(1-x_A) \) and \( Y = y_A/(1-y_A) \) for immiscible solvents, and the design equations have the same HTU/NTU form. The extraction factor \( E = mS/F \) plays the role of the absorption factor. Ternary systems with partial solvent miscibility are handled by the Hunter–Nash graphical method on triangular diagrams.

Chapter 11: Analogies, Dimensional Analysis, Simultaneous HMT

11.1 Chilton–Colburn Analogy Revisited

For turbulent flow, the Chilton–Colburn analogy links momentum, heat, and mass transfer coefficients through the \( j \)-factors defined in Section 7.1. Practically, a correlation for \( f/2 \) implies a correlation for \( \mathrm{Nu} \) and for \( \mathrm{Sh} \), and measured values of one transfer coefficient can be converted to estimates of the others. The analogy is weakened by strong property variation, wall injection, or recirculating flows, and is not used when radiation is significant.

11.2 Dimensional Analysis and Buckingham π

The Buckingham \( \pi \) theorem states that a physical relation among \( n \) dimensional variables involving \( k \) independent dimensions can be reduced to a relation among \( n-k \) dimensionless groups. For forced convection over a sphere the variables \( h, D, \rho, \mu, c_p, k, U \) with four dimensions (M, L, T, \( \Theta \)) reduce to three dimensionless groups, conventionally \( \mathrm{Nu} = f(\mathrm{Re},\mathrm{Pr}) \). The same procedure applied to mass transfer gives \( \mathrm{Sh} = f(\mathrm{Re},\mathrm{Sc}) \), and applied to natural convection introduces the Grashof number \( \mathrm{Gr} = g\beta\Delta T L^3/\nu^2 \) with \( \mathrm{Nu} = f(\mathrm{Gr},\mathrm{Pr}) \).

A nonexhaustive list of dimensionless groups used in this course:

Group	Definition	Meaning
\( \mathrm{Re} \)	\( \rho U L/\mu \)	inertia/viscous
\( \mathrm{Pr} \)	\( \nu/\alpha \)	momentum/thermal diffusion
\( \mathrm{Sc} \)	\( \nu/D \)	momentum/species diffusion
\( \mathrm{Le} \)	\( \alpha/D \)	thermal/species diffusion
\( \mathrm{Nu} \)	\( hL/k \)	convection/conduction at surface
\( \mathrm{Sh} \)	\( k_c L/D \)	convection/diffusion for mass
\( \mathrm{Bi} \)	\( hL/k_s \)	external convection/internal conduction
\( \mathrm{Fo} \)	\( \alpha t/L^2 \)	dimensionless time
\( \mathrm{Gr} \)	\( g\beta\Delta T L^3/\nu^2 \)	buoyancy/viscous
\( \mathrm{Ra} \)	\( \mathrm{Gr\,Pr} \)	natural-convection driving
\( \mathrm{St} \)	\( h/(\rho c_p U) \)	heat transferred/convected

11.3 Simultaneous Heat and Mass Transfer

When evaporation, condensation, absorption with heat effects, drying, or cooling-tower operation are analyzed, heat and mass transfer are coupled at the interface. A wet-bulb thermometer reaches a steady state at which the sensible heat flux from the gas equals the latent heat carried by evaporating water:

\[ h(T_\infty - T_{wb}) \;=\; k_c \rho_v'\,[\,\omega_{v,s}(T_{wb}) - \omega_{v,\infty}\,]\,h_{fg}. \]

Using the Chilton–Colburn analogy \( h/k_c \approx \rho c_p\,\mathrm{Le}^{2/3} \), this reduces to a single equation for \( T_{wb} \) in terms of ambient humidity and temperature, the psychrometric relation underlying humidification and drying operations. Analogous couplings arise in cooling-tower design (Merkel equation) and in nonisothermal absorption where heat of solution cannot be ignored.

Drying operations are divided into a constant-rate period, where liquid water migrates to the surface faster than the ambient can evaporate it, so the surface remains wet at \( T_{wb} \), and a falling-rate period, where internal moisture transport limits the rate and the surface warms and dries. The transition moisture content depends on the porous structure and initial saturation.

11.4 Concluding Perspective

The material of CHE 312 (merged with CHE 313) knits together three threads. First, a set of linear constitutive laws that let us write a closed PDE for \( T \) or \( c_A \). Second, the mathematical apparatus—ODE techniques, Bessel and Legendre functions, separation of variables, Sturm–Liouville theory, Laplace transforms, similarity solutions—that lets us solve the resulting boundary-value problems analytically in the canonical geometries. Third, empirical correlations and film-theoretic models that cover the turbulent and multiphase regimes where the mathematics alone cannot reach. The unifying idea is that a flux is driven by a gradient, and that the same partial differential equation, with the same boundary-condition structure, describes phenomena as different as a cooling ingot, a dissolving drug tablet, and a packed absorption column. A student who internalizes this structure finds that problems that seemed to belong to separate courses (heat exchanger design, absorber design, catalyst effectiveness) are all variants of the same mathematical theme.