Glossary

Biot number

The thermal Biot number, Bi =ℎ𝐿/𝑘𝑠$\mathrm{Bi}=hL/k_s$, compares internal conduction resistance with surface convection resistance. Here ℎ$h$ is the surface heat transfer coefficient, 𝑘𝑠$k_s$ is the solid thermal conductivity and 𝐿$L$ is a characteristic length of the solid. For the lumped-capacitance test, the usual definition is 𝐿 =𝐿𝑐 =𝑉/𝐴𝑠$L=L_c=V/A_s$, where 𝑉$V$ is the solid volume and 𝐴𝑠$A_s$ is the exposed heat-transfer area.

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Brinkman number

The dimensionless ratio

Br=𝜇𝑈2𝑘Δ𝑇∗,$$\mathrm{Br}=\frac{\mu U^2}{k\Delta T_*},$$

comparing viscous heat generation with conduction over the same transverse scale. Here 𝜇$\mu$ and 𝑘$k$ are fluid properties, 𝑈$U$ is the specified velocity scale, and Δ𝑇∗ >0$\Delta T_*>0$ is a characteristic temperature difference.

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Bulk temperature

The mixing-cup temperature that gives the same enthalpy flow as the actual fluid temperature profile. For constant properties it is weighted by the local mass flow, rather than being a simple area average.

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Colburn heat factor

The dimensionless factor 𝑗𝐻 =St Pr2/3$j_H=\mathrm{St}\,\mathrm{Pr}^{2/3}$. It combines heat transfer with a property-ratio correction. For suitable flows it is approximately 𝐶𝑓/2$C_f/2$; that analogy is an additional model, not its definition.

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Colburn mass factor

The dimensionless factor 𝑗𝐷 =St𝐷Sc2/3$j_D=\mathrm{St}_D\mathrm{Sc}^{2/3}$, using concentration-basis mass Stanton number. In matching flows it can be compared with the Colburn heat factor, 𝑗𝐻 =St𝐻Pr2/3$j_H=\mathrm{St}_H\mathrm{Pr}^{2/3}$. Equality of the factors is an empirical analogy with geometry and transport restrictions.

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Concentration boundary layer

The region near an interface where species concentration differs appreciably from its outer-fluid value. It has no sharp physical edge.

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Conduction

Conduction is the transfer of thermal energy within a medium due to a temperature gradient, through microscopic interactions without bulk motion of the medium. It occurs in solids, liquids and gases.

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Convection

Convection is the transfer of heat between a surface and a fluid in motion when they are at different temperatures. It involves the combined effects of conduction and bulk fluid motion.

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Convective mass transfer

Species transport involving molecular diffusion and bulk fluid motion. The flow changes the concentration field and the gradient driving interfacial transfer.

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Damköhler number

A dimensionless ratio of reaction rate to transport rate. For this first-order surface reaction and gas film,

Da=𝑘′′1𝑘𝑐=𝑘′′1𝛿𝐷𝐴𝐵.$$\mathrm{Da}=\frac{k_1''}{k_c}=\frac{k_1''\delta}{D_{AB}}.$$

Large Da$\mathrm{Da}$ means diffusion controls; small Da$\mathrm{Da}$ means surface reaction controls.

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Darcy friction factor

The dimensionless pressure-loss factor in

Δ𝑝=𝑓𝐷𝐿𝐷𝜌𝑢2𝑚2$$\Delta p=f_D\frac{L}{D}\frac{\rho u_m^2}{2}$$

for fully developed flow in a straight constant-diameter pipe. It is four times the Fanning factor. Entrance, fitting, elevation and acceleration contributions require separate treatment.

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Diffusive flux

Species transport rate per unit area relative to a specified mixture-average velocity. Its definition must identify both the mass or molar basis and the reference frame.

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Diffusivity

Diffusivity, 𝐷𝐴𝐵$D_{AB}$, is the material property relating the molar diffusive flux of A to its concentration gradient in the binary Fickian model at constant total concentration, 𝐽𝐴 = −𝐷𝐴𝐵𝑑𝑐𝐴/𝑑𝑧$J_A=-D_{AB}dc_A/dz$. Its SI unit is m2 s−1$\mathrm{m^2\,s^{-1}}$. It depends on the species pair, temperature, pressure and composition.

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Drag coefficient

The dimensionless total drag

𝐶𝐷=𝐹𝐷12𝜌𝑈2𝐴ref,$$C_D=\frac{F_D}{\frac{1}{2}\rho U^2 A_{\mathrm{ref}}},$$

using the specified reference speed and area. For a bluff body the reference area is commonly its projected area normal to the flow. Total drag includes both wall shear and pressure forces; it is not a skin friction coefficient.

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Effectiveness factor

The actual overall reaction rate divided by the rate if the reacting region were everywhere at the reference surface concentration under the same kinetic and temperature assumptions. It measures the effect of internal concentration gradients.

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Equimolar counter-diffusion

Binary transport with equal and opposite total molar fluxes, 𝑁𝐴 = −𝑁𝐵$N_A=-N_B$. The net mixture molar flux is zero.

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Fanning friction factor

The dimensionless factor

𝑓𝐹=𝜏𝑤12𝜌𝑢2𝑚$$f_F=\frac{\tau_w}{\frac{1}{2}\rho u_m^2}$$

for internal flow, using mean speed 𝑢𝑚$u_m$. It has the same wall-shear normalization as the Skin friction coefficient. The Darcy factor is 𝑓𝐷 =4𝑓𝐹$f_D=4f_F$.

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Fin efficiency

The heat transferred by a fin divided by the heat it would transfer if its entire surface were at the base temperature, with the same surface coefficient and surroundings. For a uniform straight fin with insulated tip,

𝜂𝑓=tanh⁡(𝑚𝑓𝐿)𝑚𝑓𝐿,$$\eta_f=\frac{\tanh(m_fL)}{m_fL},$$

where

𝑚𝑓=√ℎ𝑃𝑘𝑠𝐴𝑐,$$m_f=\sqrt{\frac{hP}{k_sA_c}},$$

𝑃$P$ is exposed perimeter and 𝐴𝑐$A_c$ is the conduction cross-section. It is analogous to reaction effectiveness.

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Forced convection

Convection in which a pump, fan or imposed stream drives the fluid motion.

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Fourier number

The thermal Fourier number, Fo =𝛼𝑡/𝐿2$\mathrm{Fo}=\alpha t/L^2$, compares elapsed time 𝑡$t$ with the conduction diffusion time 𝐿2/𝛼$L^2/\alpha$. Thermal diffusivity is 𝛼 =𝑘𝑠/(𝜌𝐶𝑝)$\alpha=k_s/(\rho C_p)$. Use the same model length 𝐿$L$ as in the thermal Biot number. Mass diffusion has a separate Fourier number based on species diffusivity.

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Graetz number

For internal thermal development,

Gz𝑥=Re𝐷Pr𝐷𝑥=𝑈𝐷2𝛼𝑥,$$\mathrm{Gz}_x= \mathrm{Re}_D\mathrm{Pr}\frac{D}{x}=\frac{U D^2}{\alpha x},$$

using mean speed 𝑈$U$, tube diameter 𝐷$D$ and distance 𝑥 >0$x>0$ from the start of heating. Its inverse

𝑥𝐷Re𝐷Pr$$\frac{x}{D\mathrm{Re}_D\mathrm{Pr}}$$

is the dimensionless axial distance used in entrance solutions. Replace 𝛼$\alpha$ by 𝐷𝐴𝐵$D_{AB}$ for species development.

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Grashof number

The dimensionless thermal-buoyancy group

Gr𝐿=𝑔𝛽∣Δ𝑇∣𝐿3𝜈2.$$\mathrm{Gr}_L=\frac{g\beta|\Delta T|L^3}{\nu^2}.$$

It measures buoyancy relative to the viscous scale, with thermal expansion coefficient 𝛽$\beta$, kinematic viscosity 𝜈$\nu$ and the geometry’s specified length 𝐿$L$.

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Hatta number

A dimensionless measure of reaction relative to diffusion for a homogeneous reaction. These notes use the pool-depth convention

Ha=𝑚𝐿=𝐿√𝑘′′′1𝐷𝐴𝐵,$$\mathrm{Ha}=mL=L\sqrt{\frac{k_1'''}{D_{AB}}},$$

the pool depth divided by the reaction penetration length. In absorption film models the length is an equivalent film thickness instead of pool depth. Here 𝑘′′′1$k_1'''$ is a first-order volumetric coefficient in s−1$^{-1}$.

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Heat flux

Heat flux is the rate of heat transfer per unit area normal to the direction of transfer. Its SI unit is W m−2$\mathrm{W\,m^{-2}}$. A signed heat flux specifies the direction of transfer relative to the chosen coordinate or surface normal.

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Heat transfer coefficient

The heat transfer coefficient, ℎ$h$, is the proportionality coefficient between the convective heat flux at a surface and the temperature difference between the surface and the fluid away from it, 𝑞 =ℎ(𝑇𝑠 −𝑇∞)$q=h(T_s-T_\infty)$. Its SI unit is W m−2 K−1$\mathrm{W\,m^{-2}\,K^{-1}}$. It depends on the fluid properties, surface geometry and flow conditions.

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Heterogeneous reaction

A reaction at an interface between phases, such as a gas reacting at a catalyst surface. It supplies a species-flux boundary condition.

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Homogeneous reaction

A reaction occurring within a phase volume. Its production or consumption rate appears as a volumetric source in the species balance.

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Hydraulic diameter

The length 𝐷ℎ =4𝐴𝑐/𝑃$D_h=4A_c/P$ used to characterise duct flow, where 𝐴𝑐$A_c$ is the flow area and 𝑃$P$ is the wetted perimeter. It equals the diameter for a circular pipe.

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Lewis number

The dimensionless ratio

Le=𝛼𝐷𝐴𝐵=ScPr,$$\mathrm{Le}=\frac{\alpha}{D_{AB}}=\frac{\mathrm{Sc}}{\mathrm{Pr}},$$

where 𝛼$\alpha$ is the fluid’s thermal diffusivity and 𝐷𝐴𝐵$D_{AB}$ is the species diffusivity, both in m2 s−1$\mathrm{m^2\,s^{-1}}$.

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Lumped capacitance

Lumped capacitance is a transient heat-transfer model in which the temperature of a body is uniform in space at each instant. The body stores thermal energy as its temperature changes. The model applies when internal conduction resistance is small compared with surface heat-transfer resistance.

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Mass Biot number

The dimensionless resistance ratio Bi𝑚 =𝑘𝑐𝐿/𝐷𝐴𝐵$\mathrm{Bi}_m=k_cL/D_{AB}$, comparing internal species-diffusion resistance with external film resistance on the same concentration basis. Here 𝑘𝑐$k_c$ is expressed on the interior basis. It is the species counterpart of the thermal Biot number.

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Mass Fourier number

The dimensionless time Fo𝑚 =𝐷𝐴𝐵𝑡/𝐿2$\mathrm{Fo}_m=D_{AB}t/L^2$, elapsed time divided by the species-diffusion time 𝐿2/𝐷𝐴𝐵$L^2/D_{AB}$. Use wall half-thickness or cylinder or sphere radius for the finite-body solutions here. It corresponds to the thermal Fourier number with 𝐷𝐴𝐵$D_{AB}$ replacing 𝛼$\alpha$.

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Mass fraction

The mass of a species divided by the total mixture mass. It is dimensionless, and all species mass fractions sum to one.

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Mass Grashof number

The composition-buoyancy group

Gr𝐷=𝑔∣Δ𝜌𝐷∣𝐿3𝜌𝜈2,$$\mathrm{Gr}_D=\frac{g|\Delta\rho_D|L^3}{\rho\nu^2},$$

where the density difference is due only to composition at a common temperature and pressure. Use the correlation’s reference density and characteristic length.

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Mass Rayleigh number

The composition natural-convection group

Ra𝐷=Gr𝐷Sc=𝑔∣Δ𝜌𝐷∣𝐿3𝜌𝜈𝐷𝐴𝐵.$$\mathrm{Ra}_D=\mathrm{Gr}_D\mathrm{Sc} =\frac{g|\Delta\rho_D|L^3}{\rho\nu D_{AB}}.$$

It combines composition buoyancy, momentum diffusion and species diffusion, using the same reference state and length in both factors.

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Mass Stanton number

The dimensionless species-transfer coefficient

St𝐷=𝑘𝑐𝑈=ShRe Sc,$$\mathrm{St}_D=\frac{k_c}{U}=\frac{\mathrm{Sh}}{\mathrm{Re}\,\mathrm{Sc}},$$

using a concentration-basis 𝑘𝑐$k_c$ in m s⁻¹ and the same speed and length throughout. It compares surface transfer velocity with stream velocity and corresponds to the thermal Stanton number.

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Mass transfer coefficient

A coefficient relating species flux to a specified interfacial-to-bulk driving difference. On a molar-concentration basis,

𝑁𝐴=𝑘𝑐(𝑐𝐴,𝑠−𝑐𝐴,𝑏)$$N_A=k_c(c_{A,s}-c_{A,b})$$

and 𝑘𝑐$k_c$ has units m s−1$\mathrm{m\,s^{-1}}$; other driving-force bases use different units.

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Mole fraction

The molar amount of a species divided by the total molar amount of the mixture. It is dimensionless, and all species mole fractions sum to one.

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Molecular diffusion

Species transport relative to the bulk mixture caused by molecular motion and a thermodynamic driving force. In the simple Fickian models used here, the driving force is a concentration gradient.

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Natural convection

Convection driven by buoyancy arising from fluid density differences in a gravitational field. Temperature or composition differences can create the density differences.

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Number of transfer units

For heat exchange with a fixed wall temperature, the number of transfer units is

NTU𝐻=¯¯¯¯ℎ𝐴𝑠˙𝑚𝐶𝑝.$$\mathrm{NTU}_H=\frac{\bar h A_s}{\dot m C_p}.$$

It compares surface transfer conductance with stream heat-capacity rate. For a species with fixed wall concentration on a consistent concentration basis,

NTU𝑚=¯¯¯¯𝑘𝑐𝐴𝑠𝑄,$$\mathrm{NTU}_m=\frac{\bar k_c A_s}{Q},$$

where 𝑄$Q$ is volumetric flow rate. Both are dimensionless and describe the exponential approach to the wall state in these constant-property models.

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Nusselt number

The dimensionless heat transfer coefficient Nu =ℎ𝐿/𝑘𝑓$\mathrm{Nu}=hL/k_f$. It compares surface heat transfer with conduction across a fluid length 𝐿$L$ and uses the fluid conductivity 𝑘𝑓$k_f$.

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Partition coefficient

An equilibrium ratio of concentrations in two phases, for example

𝐾=𝑐∗𝐴,inside𝑐∗𝐴,outside.$$K=\frac{c_{A,\mathrm{inside}}^*}{c_{A,\mathrm{outside}}^*}.$$

It is dimensionless only when numerator and denominator use the same concentration units and basis. Its direction must be specified before converting a film coefficient.

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Péclet number

The ratio of advection to diffusion over the same length 𝐿$L$ and speed 𝑈$U$. For heat,

Pe𝐿=𝑈𝐿𝛼=Re𝐿Pr.$$\mathrm{Pe}_L=\frac{UL}{\alpha}=\mathrm{Re}_L\mathrm{Pr}.$$

For a species,

Pe𝑚,𝐿=𝑈𝐿𝐷𝐴𝐵=Re𝐿Sc.$$\mathrm{Pe}_{m,L}=\frac{UL}{D_{AB}}=\mathrm{Re}_L\mathrm{Sc}.$$

The diffusivity must match the transported quantity. See the thermal scale estimate and the species scale estimate.

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Prandtl number

The dimensionless ratio Pr =𝜈/𝛼$\mathrm{Pr}=\nu/\alpha$, comparing momentum diffusivity with thermal diffusivity.

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Pseudo steady state

An approximation that neglects accumulation in a rapidly adjusting transport region while a slower boundary or system state changes with time. Its validity requires separated transport and system time scales.

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Radiation

Thermal radiation is the emission of energy as electromagnetic waves by matter at a nonzero absolute temperature. Energy transfer by radiation does not require a material medium.

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Rayleigh number

The thermal natural-convection group

Ra𝐿=Gr𝐿Pr=𝑔𝛽∣Δ𝑇∣𝐿3𝜈𝛼,$$\mathrm{Ra}_L=\mathrm{Gr}_L\mathrm{Pr} =\frac{g\beta|\Delta T|L^3}{\nu\alpha},$$

using the same length and properties in both factors. It measures buoyancy relative to momentum and thermal diffusion.

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Reynolds number

The dimensionless ratio

Re=𝜌𝑢𝐿𝜇=𝑢𝐿𝜈,$$\mathrm{Re}=\frac{\rho uL}{\mu}=\frac{uL}{\nu},$$

comparing inertial and viscous effects. The velocity and characteristic length must suit the flow geometry.

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Schmidt number

The dimensionless ratio Sc =𝜈/𝐷𝐴𝐵$\mathrm{Sc}=\nu/D_{AB}$, where 𝜈 =𝜇/𝜌$\nu=\mu/\rho$ is the fluid’s kinematic viscosity, or momentum diffusivity, and 𝐷𝐴𝐵$D_{AB}$ is the species diffusivity. Here 𝜇$\mu$ is dynamic viscosity and 𝜌$\rho$ is fluid density.

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Semi-infinite medium

A semi-infinite medium is an idealised body that extends indefinitely in one direction away from its exposed surface. Its temperature approaches the initial temperature far from that surface. The model also approximates a finite body during the period when the surface temperature disturbance has not significantly reached another boundary.

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Sherwood number

The dimensionless mass transfer coefficient Sh =𝑘𝑐𝐿/𝐷𝐴𝐵$\mathrm{Sh}=k_cL/D_{AB}$, comparing surface transfer with diffusion across the fluid length 𝐿$L$. The coefficient 𝑘𝑐$k_c$ must use a concentration difference on the same basis as the species diffusivity. It is the mass counterpart of Nusselt number.

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Skin friction coefficient

The dimensionless wall shear

𝐶𝑓=𝜏𝑤12𝜌𝑈2,$$C_f=\frac{\tau_w}{\frac{1}{2}\rho U^2},$$

using the specified reference speed 𝑈$U$. A local coefficient uses local wall shear; an average coefficient uses its surface average. It excludes pressure drag.

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Species continuity equation

A local species balance relating accumulation, the divergence of species flux and net production by reaction.

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Stagnant carrier

A species with zero total flux past the chosen fixed observer, such as 𝑁𝐵 =0$N_B=0$. Its diffusive and bulk contributions can still be nonzero and cancel each other.

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Stanton number

The thermal Stanton number is

St=ℎ𝜌𝑈𝐶𝑝=NuRe Pr,$$\mathrm{St}=\frac{h}{\rho U C_p} =\frac{\mathrm{Nu}}{\mathrm{Re}\,\mathrm{Pr}},$$

using the same speed and length in all groups. It compares the surface heat-transfer coefficient with the fluid’s heat-capacity flux per unit area. Use consistent local or average quantities.

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Stefan-flow enhancement factor

For A diffusing through stagnant B in the stated binary gas film,

𝐹𝑆=1𝑦𝐵𝑀=𝑃𝑝𝐵𝑀$$F_S=\frac{1}{y_{BM}}=\frac{P}{p_{BM}}$$

multiplies the equimolar-diffusion flux based on the same endpoint concentration difference. It accounts for bulk molar motion. The local multiplier 1/(1 −𝑦𝐴)$1/(1-y_A)$ is distinct from this integrated factor.

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Thermal boundary layer

The region next to a surface where the fluid temperature differs appreciably from the outer fluid temperature. Its thickness depends on heat diffusion and fluid motion.

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Thermal conductivity

Thermal conductivity, 𝑘$k$, is a material property that relates conductive heat flux to the temperature gradient through Fourier’s law. For one-dimensional conduction, 𝑞 = −𝑘 𝑑𝑇/𝑑𝑥$q=-k\,dT/dx$. Its SI unit is W m−1 K−1$\mathrm{W\,m^{-1}\,K^{-1}}$.

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Thermal diffusivity

Thermal diffusivity, 𝛼$\alpha$, is the ratio of thermal conductivity to volumetric heat capacity, 𝛼 =𝑘/(𝜌̂𝐶𝑝)$\alpha=k/(\rho\widehat C_p)$. It characterises the rate at which temperature disturbances spread through a material by conduction. Its SI unit is m2 s−1$\mathrm{m^2\,s^{-1}}$.

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Thermal resistance

Thermal resistance, 𝑅$R$, is the ratio of the temperature difference across a heat-transfer process to the resulting heat-transfer rate, 𝑅 =Δ𝑇/˙𝑄$R=\Delta T/\dot Q$. Its SI unit is K W−1$\mathrm{K\,W^{-1}}$.

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Total molar flux

Molar transport rate of a species per unit area past a fixed observer. It includes transport by bulk motion and diffusion, and has units such as mol m−2 s−1$\mathrm{mol\,m^{-2}\,s^{-1}}$.

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Velocity boundary layer

The region next to a surface where viscous effects make the fluid velocity differ appreciably from the outer flow. Its conventional edge is not a sharp physical boundary.

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