Solutions to Transport Phenomena 2nd Revised Edition

2A.1: Thickness of a falling film 2A.2: Determination of capillary radius by flow measurement 2A.3: Volume flow rate through an annulus 2A.4: Loss of catalyst particles in stack gas 2B.1: Different choice of coordinates for the falling film problem 2B.2: Alternate procedure for solving flow problems 2B.3: Laminar flow in a narrow slit 2B.4: Laminar slit flow with a moving wall ("plane Couette flow") 2B.5: Interrelation of slit and annulus formulas 2B.6: Flow of a film on the outside of a circular tube 2B.7: Annular flow with inner cylinder moving axially 2B.8: Analysis of a capillary flowmeter 2B.9: Low-density phenomena in compressible tube flow 2B.10: Incompressible flow in a slightly tapered tube 2B.11: The cone-and-plate viscometer 2B.12: Flow of a fluid in a network of tubes 2C.1: Performance of an electric dust collector 2C.2: Residence time distribution in tube flow 2C.3: Velocity distribution in a tube 2C.4: Falling-cylinder viscometer 2C.5: Falling film on a conical surface 2C.6: Rotating cone pump 2C.7: A simple rate-of-climb indicator 2D.1: Rolling-ball viscometer 2D.2: Drainage of liquids

3A.1: Torque required to turn a friction bearing 3A.2: Friction loss in bearings 3A.3: Effect of altitude on air pressure 3A.4: Viscosity determination with a rotating-cylinder viscometer 3A.5: Fabrication of a parabolic mirror 3A.6: Scale-up of an agitated tank 3A.7: Air entrainment in a draining tank 3B.1: Flow between coaxial cylinders and concentric spheres 3B.2: Laminar flow in a triangular duct 3B.3: Laminar flow in a square duct 3B.4: Creeping flow between two concentric spheres 3B.5: Parallel-disk viscometer 3B.6: Circulating axial flow in an annulus 3B.7: Momentum fluxes for creeping flow into a slot 3B.8: Velocity distribution for creeping flow toward a slot 3B.9: Slow transverse flow around a cylinder 3B.10: Radial flow between parallel disks 3B.11: Radial flow between two coaxial cylinders 3B.12: Pressure distribution in incompressible fluids 3B.13: Flow of a fluid through a sudden contraction 3B.14: Torricelli's equation for efflux from a tank 3B.15: Shape of free surface in tangential annular flow 3B.16: Flow in a slit with uniform cross flow 3C.1: Parallel-disk compression viscometer 3C.2: Normal stresses at solid surfaces for compressible fluids 3C.3: Deformation of a fluid line 3C.4: Alternative methods of solving the Couette viscometer problem by use of angular momentum concepts 3C.5: Two-phase interfacial boundary conditions 3D.1: Derivation of the equations of change by integral theorems 3D.2: The equation of change for vorticity 3D.3: Alternate form of the equation of motion

4A.1: Time for attainment of steady state in tube flow 4A.2: Velocity near a moving sphere 4A.3: Construction of streamlines for the potential around a cylinder 4A.4: Comparison of exact and approximate profiles for flow along a flat plate 4A.5: Numerical demonstration of the von Kármán momentum balance 4A.6: Use of boundary-layer formulas 4A.7: Entrance flow in conduits 4B.1: Flow of a fluid with a suddenly applied constant wall stress 4B.2: Flow near a wall suddenly set in motion (approximate solution) 4B.3: Creeping flow around a spherical bubble 4B.4: Use of the vorticity equation 4B.5: Steady potential flow around a stationary sphere 4B.6: Potential flow near a stagnation point 4B.7: Vortex flow 4B.8: The flow field about a line source 4B.9: Checking solutions to unsteady flow problems 4C.1: Laminar entrance flow in a slit 4C.2: Torsional oscillatory viscometer 4C.3: Darcy's equation for flow through porous media 4C.4: Radial flow through a porous medium 4D.1: Flow near an oscillating wall 4D.2: Start-up of laminar flow in a circular tube 4D.3: Flows in the disk-and-tube system 4D.4: Unsteady annular flows 4D.5: Stream functions for three-dimensional flow

5A.1: Pressure drop needed for laminar-turbulent transition 5A.2: Velocity distribution in turbulent pipe flow 5B.1: Average flow velocity in turbulent tube flow 5B.2: Mass flow rate in a turbulent circular jet 5B.3: The eddy viscosity expression in the viscous sublayer 5C.1: Two-dimensional turbulent jet 5C.2: Axial turbulent flow in an annulus 5C.3: Instability in a simple mechanical system 5D.1: Derivation of the equation of change for the Reynolds stresses 5D.2: Kinetic energy of turbulence

6A.1: Pressure drop required for a pipe with fittings 6A.2: Pressure difference required for flow in pipe with elevation change 6A.3: Flow rate for a given pressure drop 6A.4: Motion of a sphere in a liquid 6A.5: Sphere diameter for a given terminal velocity 6A.6: Estimation of void fraction of a packed column 6A.7: Estimation of pressure drops in annular flow 6A.8: Force on a water tower in a gale 6A.9: Flow of gas through a packed column 6A.10: Determination of pipe diameter 6B.1: Effect of error in friction factor calculations 6B.2: Friction factor for flow along a flat plate 6B.3: Friction factor for laminar flow in a slit 6B.4: Friction factor for a rotating disk 6B.5: Turbulent flow in horizontal pipes 6B.6: Inadequacy of mean hydraulic radius for laminar flow 6B.7: Falling sphere in Newton's drag-law region 6B.8: Design of an experiment to verify the f vs. Re chart for spheres 6B.9: Friction factor for flow past an infinite cylinder 6C.1: Two-dimensional particle trajectories 6C.2: Wall effects for a sphere falling in a cylinder 6C.3: Power input to an agitated tank 6D.1: Friction factor for a bubble in a clean liquid

7A.1: Pressure rise in a sudden enlargement 7A.2: Pumping a hydrochloric acid solution 7A.3: Compressible gas flow in a cylindrical pipe 7A.4: Incompressible flow in an annulus 7A.5: Force on a U-bend 7A.6: Flow-rate calculation 7A.7: Evaluation of various velocity averages from Pitot tube data 7B.1: Velocity averages from the 1/7 power law 7B.2: Relation between force and viscous loss for flow in conduits of variable cross section 7B.3: Flow through a sudden enlargement 7B.4: Flow between two tanks 7B.5: Revised design of an air duct 7B.6: Multiple discharge into a common conduit 7B.7: Inventory variations in a gas reservoir 7B.8: Change in liquid height with time 7B.9: Draining of a cylindrical tank with exit pipe 7B.10: Efflux time for draining a conical tank 7B.11: Disintegration of wood chips 7B.12: Criterion for vapor-free flow in a pipeline 7C.1: End corrections in tube viscometers 7D.1: Derivation of the macroscopic balances from the equations of change

8A.1: Flow of a polyisoprene solution in a pipe 8A.2: Pumping of a polyethylene oxide solution 8B.1: Flow of a polymeric film 8B.2: Power law flow in a narrow slit 8B.3: Non-Newtonian flow in an annulus 8B.4: Flow of a polymeric liquid in a tapered tube 8B.5: Slit flow of a Bingham fluid 8B.6: Derivation of the Buckingham-Reiner equation 8B.7: The complex-viscosity components for the Jeffreys fluid 8B.8: Stress relaxation after cessation of shear flow 8B.9: Draining of a tank with an exit pipe 8B.10: The Giesekus model 8C.1: The cone-and-plate viscometer 8C.2: Squeezing flow between parallel disks 8C.3: Verification of Giesekus viscosity function 8C.4: Tube Flow for the Oldroyd 6-Constant Model 8C.5: Chain Models with Rigid-Rod Connectors

9A.1: Prediction of thermal conductivities of gases at low density 9A.2: Computation of the Prandtl numbers for gases at low density 9A.3: Estimation of the thermal conductivity of a dense gas 9A.4: Prediction of the thermal conductivity of a gas mixture 9A.5: Estimation of the thermal conductivity of a pure liquid 9A.6: Calculation of the Lorenz number 9A.7: Corroboration of the Wiedemann-Franz-Lorenz law 9A.8: Thermal conductivity and Prandtl number of a polyatomic gas 9A.9: Thermal conductivity of gaseous chlorine 9A.10: Thermal conductivity of chlorine-air mixtures 9A.11: Thermal conductivity of quartz sand 9A.12: Calculation of molecular diameters from transport properties 9C.1: Enskog theory for dense gases

10A.1: Heat loss from an insulated pipe 10A.2: Heat loss from a rectangular fin 10A.3: Maximum temperature in a lubricant 10A.4: Current-carrying capacity of a wire 10A.5: Free convection velocity 10A.6: Insulating power of a wall 10A.7: Viscous heating in a ball-point pen 10B.1: Heat conduction from a sphere to a stagnant fluid 10B.2: Viscous heating in slit flow 10B.3: Heat conduction in a nuclear fuel rod assembly 10B.4: Heat conduction in an annulus 10B.5: Viscous heat generation in a polymer melt 10B.6: Insulation thickness for a furnace wall 10B.7: Forced-convection heat transfer in flow between parallel plates 10B.8: Electrical heating of a pipe 10B.9: Plug flow with forced-convection heat transfer 10B.10: Free convection in an annulus of finite height 10B.11: Free convection with temperature-dependent viscosity 10B.12: Heat conduction with temperature-dependent thermal conductivity 10B.13: Flow reactor with exponentially temperature-dependent source 10B.14: Evaporation loss from an oxygen tank 10B.15: Radial temperature gradients in an annular chemical reactor 10B.16: Temperature distribution in a hot-wire anemometer 10B.17: Non-Newtonian flow with forced-convection heat transfer 10B.18: Reactor temperature profiles with axial heat flux 10C.1: Heating of an electric wire with temperature-dependent electrical and thermal conductivity 10C.2: Viscous heating with temperature-dependent viscosity and thermal conductivity 10C.3: Viscous heating in a cone-and-plate viscometer 10D.1: Heat loss from a circular fin 10D.2: Duct flow with constant wall heat flux and arbitrary velocity distribution

11A.1: Temperature in a friction bearing 11A.2: Viscosity variation and velocity gradients in a nonisothermal film 11A.3: Transpiration cooling 11A.4: Free-convection heat loss from a vertical surface 11A.5: Velocity, temperature, and pressure changes in a shock wave 11A.6: Adiabatic frictionless compression of an ideal gas 11A.7: Effect of free convection on the insulating value of a horizontal air space 11B.1: Adiabatic frictionless processes in an ideal gas 11B.2: Viscous heating in laminar tube flow (asymptotic solutions) 11B.3: Velocity distribution in a nonisothermal film 11B.4: Heat conduction in a spherical shell 11B.5: Axial heat conduction in a wire 11B.6: Transpiration cooling in a planar system 11B.7: Reduction of evaporation losses by transpiration 11B.8: Temperature distribution in an embedded sphere 11B.9: Heat flow in a solid bounded by two conical surfaces 11B.10: Freezing of a spherical drop 11B.11: Temperature rise in a spherical catalyst pellet 11B.12: Stability of an exothermic reaction system 11B.13: Laminar annular flow with constant wall heat flux 11B.14: Unsteady-state heating of a sphere 11B.15: Dimensionless variables for free convection 11C.1: The speed of propagation of sound waves 11C.2: Free convection in a slot 11C.3: Tangential annular flow of a highly viscous liquid 11C.4: Heat conduction with variable thermal conductivity 11C.5: Effective thermal conductivity of a solid with spherical inclusions 11C.6: Interfacial boundary conditions 11C.7: Effect of surface-tension gradients on a falling film 11D.1: Equation of change for entropy 11D.2: Viscous heating in laminar tube flow 11D.3: Derivation of the energy equation using integral theorems

12A.1: Unsteady-state heat conduction in an iron sphere 12A.2: Comparison of the two slab solutions for short times 12A.3: Bonding with a thermosetting adhesive 12A.4: Quenching of a steel billet 12A.5: Measurement of thermal diffusivity from amplitude of temperature oscillations 12A.6: Forced convection from a sphere in creeping flow 12B.1: Measurement of thermal diffusivity in an unsteady-state experiment 12B.2: Two-dimensional forced convection with a line heat source 12B.3: Heating of a wall (constant wall heat flux) 12B.4: Heat transfer from a wall to a falling film (short contact time limit) 12B.5: Temperature in a slab with heat production 12B.6: Forced convection in slow flow across a cylinder 12B.7: Timetable for roasting turkey 12B.8: Use of asymptotic boundary layer solution 12B.9: Non-Newtonian heat transfer with constant wall heat flux (asymptotic solution for small axial distances) 12C.1: Product solutions for unsteady heat conduction in solids 12C.2: Heating of a semi-infinite slab with variable thermal conductivity 12C.3: Heat conduction with phase change (the Neumann-Stefan problem) 12C.4: Viscous heating in oscillatory flow 12C.5: Solar heat penetration 12C.6: Heat transfer in a falling non-Newtonian film 12D.1: Unsteady-state heating of a slab (Laplace transform method) 12D.2: The Graetz-Nusselt problem 12D.3: The Graetz-Nusselt problem (asymptotic solution for large z) 12D.4: The Graetz-Nusselt problem (asymptotic solution for small z) 12D.5: The Graetz problem for flow between parallel plates 12D.6: The constant wall heat flux problem for parallel plates 12D.7: Asymptotic solution for small z for laminar tube flow with constant heat flux 12D.8: Forced conduction heat transfer from a flat plate (thermal boundary layer extends beyond the momentum boundary layer)

13B.1: Wall heat flux for turbulent flow in tubes (approximate) 13B.2: Wall heat flux for turbulent flow in tubes 13C.1: Wall heat flux for turbulent flow between two parallel plates 13D.1: The temperature profile for turbulent flow in tubes

14A.1: Average heat transfer coefficients 14A.2: Heat transfer in laminar tube flow 14A.3: Effect of flow rate on exit temperature from a heat exchanger 14A.4: Local heat transfer coefficient for turbulent forced convection in a tube 14A.5: Heat transfer from condensing vapors 14A.6: Forced-convection heat transfer from an isolated sphere 14A.7: Free convection heat transfer from an isolated sphere 14A.8: Heat loss by free convection from a horizontal pipe immersed in a liquid 14A.9: The ice-fisherman on Lake Mendota 14B.1: Limiting local Nusselt number for plug flow with constant heat flux 14B.2: Local overall heat transfer coefficient 14B.3: The hot-wire anemometer 14B.4: Dimensional analysis 14B.5: Relation between h loc and h ln 14B.6: Heat loss by free convection from a pipe 14C.1: The Nusselt expression for film condensation heat transfer coefficients 14C.2: Heat transfer correlations for agitated tanks 14D.1: Heat transfer from an oblate ellipsoid of revolution

15A.1: Heat transfer in double-pipe heat exchangers 15A.2: Adiabatic flow of natural gas in a pipeline 15A.3: Mixing of two ideal-gas streams 15A.4: Flow through a Venturi tube 15A.5: Free batch expansion of a compressible fluid 15A.6: Heating of air in a tube 15A.7: Operation of a simple double-pipe heat exchanger 15B.1: Performance of a double-pipe heat exchanger with variable overall heat transfer coefficient 15B.2: Pressure drop in turbulent flow in a slightly converging tube 15B.3: Steady flow of ideal gases in ducts of constant cross section 15B.4: The Mach number in the mixing of two fluid streams 15B.5: Limiting discharge rates for Venturi meters 15B.6: Flow of a compressible fluid through a convergent-divergent nozzle 15B.7: Transient thermal behavior of a chromatographic device 15B.8: Continuous heating of a slurry in an agitated tank 15C.1: Parallel-counterflow heat exchangers 15C.2: Discharge of air from a large tank 15C.3: Stagnation temperature 15D.1: The macroscopic entropy balance 15D.2: Derivation of the macroscopic energy balance 15D.3: Operation of a heat-exchange device 15D.4: Discharge of a gas from a moving tank 15D.5: The classical Bernoulli equation

16A.1: Approximation of a black body by a hole in a sphere 16A.2: Efficiency of a solar engine 16A.3: Radiant heating requirement 16A.4: Steady-state temperature of a roof 16A.5: Radiation errors in temperature measurements 16A.6: Surface temperatures on the Earth's moon 16B.1: Reference temperature for effective emissivity 16B.2: Radiation across an annular gap 16B.3: Multiple radiation shields 16B.4: Radiation and conduction through absorbing media 16B.5: Cooling of a black body in vacuo 16B.6: Heat loss from an insulated pipe 16C.1: Integration of the view-factor integral for a pair of disks 16D.1: Heat loss from a wire carrying an electric current

17A.1: Prediction of a low-density binary diffusivity 17A.2: Extrapolation of binary diffusivity to a very high temperature 17A.3: Self-diffusion in liquid mercury 17A.4: Schmidt numbers for binary gas mixtures at low density 17A.5: Estimation of diffusivity for a binary mixture at high density 17A.6: Diffusivity and Schmidt number for chlorine-air mixtures 17A.7: The Schmidt number for self-diffusion 17A.8: Correction of high-density diffusivity for temperature 17A.9: Prediction of critical cD AB values 17A.10: Estimation of liquid diffusivities 17B.1: Interrelation of composition variables in mixtures 17B.2: Relations among fluxes in multicomponent systems 17B.3: Relations between fluxes in binary systems 17B.4: Equivalence of various forms of Fick's law for binary mixtures 17C.1: Mass flux with respect to volume average velocity 17C.2: Mass flux with respect to the solvent velocity 17C.3: Determination of Lennard-Jones potential parameters from diffusivity data of a binary gas mixture

18A.1: Evaporation Rate 18A.2: Sublimation of small iodine spheres in still air 18A.3: Estimating the error in calculating the absorption rate 18A.4: Chlorine absorption in a falling film 18A.5: Measurement of diffusivity by the point-source method 18A.6: Determination of diffusivity for ether-air system 18A.7: Mass flux from a circulating bubble 18B.1: Diffusion through a stagnant film---alternate derivation 18B.2: Error in neglecting the convection term in evaporation 18B.3: Effect of mass transfer rate on the concentration profiles 18B.4: Absorption with chemical reaction 18B.5: Absorption of chlorine by cyclohexene 18B.6: Two-bulb experiment for measuring gas diffusivity---quasi-steady-state analysis 18B.7: Diffusion from a suspended droplet 18B.8: Method for separating helium from natural gas 18B.9: Rate of leaching 18B.10: Constant-evaporating mixtures 18B.11: Diffusion with fast second-order reaction 18B.12: A sectioned-cell experiment for measuring gas-phase diffusivity 18B.13: Tarnishing of metal surfaces 18B.14: Effectiveness factors for thin disks 18B.15: Diffusion and heterogeneous reaction in a slender cylindrical tube with a closed end 18B.16: Effect of temperature and pressure on evaporation rate 18B.17: Reaction rates in large and small particles 18B.18: Evaporation rate for small mole fraction of the volatile liquid 18B.19: Oxygen uptake by a bacterial aggregate 18C.1: Diffusion from a point source in a moving stream 18C.2: Diffusion and reaction in a partially impregnated catalyst 18C.3: Absorption rate in a falling film 18C.4: Estimation of the required length of an isothermal reactor 18C.5: Steady-state evaporation 18D.1: Effectiveness factors for long cylinders 18D.2: Gas absorption in a falling film with chemical reaction

19A.1: Dehumidification of air 19B.1: Steady-state evaporation 19B.2: Gas absorption with chemical reaction 19B.3: Concentration-dependent diffusivity 19B.4: Oxidation of silicon 19B.5: The Maxwell-Stefan equations for multicomponent gas mixtures 19B.6: Diffusion and chemical reaction in a liquid 19B.7: Various forms of the species continuity equation 19C.1: Alternate form of the binary diffusion equation 19D.1: Derivation of the equation of continuity 19D.2: Derivation of the equation of change for temperature for a multicomponent system 19D.3: Gas separation by atmolysis or "sweep diffusion" 19D.4: Steady-state diffusion from a rotating disk

20A.1: Measurement of diffusivity by unsteady-state evaporation 20A.2: Absorption of oxygen from a growing bubble 20A.3: Rate of evaporation of n -octane 20A.4: Effect of bubble size on interfacial composition 20A.5: Absorption with rapid second-order reaction 20A.6: Rapid forced-convection mass transfer into a laminar boundary layer 20A.7: Slow forced-convection mass tranfer into a laminar boundary layer 20B.1: Extension of the Arnold problem to account for interphase transfer of both species 20B.2: Extension of the Arnold problem to nonisothermal diffusion 20B.3: Stoichiometric boundary condition for rapid irreversible reaction 20B.4: Taylor dispersion in slit flow 20B.5: Diffusion from an instantaneous point source 20B.6: Unsteady diffusion with first-order chemical reaction 20B.7: Simultaneous momentum, heat, and mass transfer: alternate boundary conditions 20B.8: Absorption from a pulsating bubble 20B.9: Verification of the solution of the Taylor-dispersion equation 20C.1: Order-of-magnitude analysis of gas absorption from a growing bubble 20C.2: Effect of surface curvature on absorption from a growing bubble 20C.3: Absorption with chemical reaction in a semi-infinite medium 20C.4: Design of fluid control circuits 20C.5: Dissociation of a gas caused by a temperature gradient 20D.1: Two-bulb experiment for measuring gas diffusivities-analytical solution 20D.2: Unsteady-state interphase diffusion 20D.3: Critical size of an autocatalytic system 20D.4: Dispersion of a broad pulse in steady, laminar axial flow in a tube 20D.5: Velocity divergence in interfacially embedded coordinates

21A.1: Determination of eddy diffusivity 21A.2: Heat and mass transfer analogy 21B.1: Wall mass flux for turbulent flow with no chemical reactions 21B.2: Alternate expressions for the turbulent mass flux 21B.3: An asymptotic expression for the turbulent mass flux 21B.4: Deposition of silver from a turbulent stream 21B.5: Mixing-length expression for the velocity profile

22A.1: Prediction of mass transfer coefficients in closed channels 22A.2: Calculation of gas composition from psychrometric data 22A.3: Calculating the inlet air temperature for drying in a fixed bed 22A.4: Rate of drying of granular solids in a fixed bed 22B.1: Evaporation of a freely falling drop 22B.2: Effect of radiation on psychrometric measurements 22B.3: Film theory with variable transport properties 22B.4: An evaporative ice maker 22B.5: Oxygen stripping 22B.6: Controlling diffusional resistance 22B.7: Determination of diffusivity 22B.8: Marangoni effects in condensation of vapors 22B.9: Film model for spheres 22B.10: Film model for cylinders 22C.1: Calculation of ultrafiltration rates

23A.1: Expansion of a gas mixture: very slow reaction rate 23A.2: Height of a packed-tower absorber 23B.1: Effective average driving forces in a gas absorber 23B.2: Expansion of a gas mixture: very fast reaction rate 23B.3: Startup of a chemical reactor 23B.4: Irreversible first-order reaction in a continuous reactor 23B.5: Mass and enthalpy balances in an adiabatic splitter 23B.6: Flow distribution in an ideal cascade 23B.7: Isotope separation and the value function 23C.1: Irreversible second-order reaction in an agitated tank 23C.2: Protein purification 23C.3: Physical significance of the zeroth and first moments 23C.4: Analogy between the unsteady operation of an adsorption column and a cross-flow heat exchanger 23D.1: Unsteady-state operation of a packed column 23D.2: Additivity of the lower moments 23D.3: Start-up of a chemical reactor 23D.4: Transient behavior of N reactors in series

24A.1: Thermal diffusion 24A.2: Ultracentrifugation of proteins 24A.3: Ionic diffusivities 24B.1: The dimensions of the Lorentz force 24B.2: Junction potentials 24B.3: Donnan exclusion 24B.4: Osmotic pressure 24B.5: Permeability of a perfectly selective filtration membrane 24B.6: Model insensitivity 24C.1: Expressions for the mass flux 24C.2: Differential centrifugation 24C.3: Transport characteristics of sodium chloride 24C.4: Departures from electroneutrality 24C.5: Dielectrophoretic driving forces 24C.6: Effects of small inclusions in a dielectric medium 24C.7: Frictionally induced selective filtration 24C.8: Thermodynamically induced selective filtration 24C.9: Facilitated transport 24D.1: Entropy flux and entropy production