Index
Note: Page numbers followed by “f” indicate figures and “t” indicate tables
A
Absorption, 254f Absorptivity, 254, 258f, 364 Advection, 104
Analogy between momentum, heat transfer, and mass transfer, 403
Chilton-Colburn analogy, 405, 408 Reynolds analogy, 405
ANN. See Artificial neural network (ANN) Apportioning of radiation falling on a surface, 254
opaque surface, 254
steady state, using energy balance, 254 Approximation using order of magnitude analysis,
121
continuity equation, 121, 125 energy equation, 125 momentum equation, 125 x-momentum equation, 121, 123 y-momentum equation, 121
Artificial intelligence, 327
Artificial neural network (ANN), 327–329 Average heat transfer coefficient, 108, 150, 375 Average Nusselt number, 136, 138
Azimuthal angle, 234
B
Bessel’s functions, 82
Binary diffusion coefficients, 399t Biot number, 68, 71
Black body, 235, 236
radiation functions (F-function) chart, 245t radiation law, 235
Body forces, 113 Boiling, 351
crisis, 353
Nusselt number, 360 Reynolds number, 360
Boundary conditions, 19
Boundary layer equations, approximate solution to, 128
flow over a flat plate, forced convection, 124, 136, 150
flow over a vertical plate, free convection, 124 integral energy equation, 130
from Leibniz rule, 129, 130 species transport equation, 130
Boussinesq approximation, 179, 189
Buoyancy force, 360
Burnout, 353
C
Center-line temperature charts, for long cylinder, 85f
Chen correlation, 367 Chilton-Colburn analogy, 149 Churchill-Chu correlation, 189 Clausius-Clapeyron equation, 355, 368 Coefficient of viscosity, 379, 388 Colburn equation, 163
Combined modes of heat transfer, 12 Concentration boundary layer, 401, 402f Condensation, 372
film condensation on vertical plate, 372 on horizontal tubes, 377, 377f
Condensate mass flow rate, 374 Conduction, 4, 104
across a gas layer between two infinitely long plates, 5f
in cylinder with insulation, and convection on outside, 25f
in horizontal direction, 6f Conservation laws, 110 Constitutive relationship, 114 Continuum, concept of, 12
Continuity equation, 121, 122, 125, 126, 129, 152, 153, 176, 180
Convecting tip, 55 Convection, 7, 104, 353
fundamental definition of, 105 mechanism of, 8
numerical methods, 319
Convective heat transfer, 8, 103, 105, 408 Convective mass transfer coefficient, 400 Convective mass transfer relations, 406
flow over a flat plate, 406 internal flow, 407
Cooling of ball bearings (heat treatment), 3 Cooling of hot ball bearings in cold fluid, 3f Coordinate systems, cylindrical and spherical, 18f Correction factor for gas emissivity, 287 Correlation constants, 287t, 288t
Cost function, 332
Counter flow heat exchanger, 207 effectiveness of, 216
417
418 Index
Critical heat flux, 353, 362, 369
Critical heat flux, flow boiling, 364
Critical heat flux, pool boiling, 352
Critical radius of insulation for cylinder, 25
Cross-flow heat exchanger, 211
D
Data-based models, 325, 327 Darcy friction factor, 378 Deep learning, 329
Deep neural networks (DNNs), 329
Departure from nucleate boiling (DNB), 353, 366 Diffuse surface, 250
Dirichlet boundary condition, 19 Dittus-Boelter equation, 163, 367
DNNs. See Deep neural networks (DNNs) Domain expertise, 346
Dryout in flow boiling, 369
E
Effectiveness-NTU method, 212 Electrical analogy, 22, 24 Electrical resistance network, 22f Electromagnetic theory, 9
Emissivities and absorptivities for a mixture of gases, calculation, 286
equations for, 287
Emissivity for different pressure conditions, 287 Energy balance, 375
Energy equation, 110, 115, 117, 118 with trial velocity, 130
Enthalpy chart (Grober’s chart) long cylinder, 86f
plane wall, 84f sphere, 88f
Equations and their classification
based on information propagation, 298 elliptic PDE, 298
hyperbolic PDE, 299 parabolic PDE, 299
based on linearity and order, 297 Error function, 76t
F
Fanning friction factor, 155 Fick’s law of diffusion, 397 Film boiling, 363
expression for combined effects of convection and radiation, 364
expression for film boiling on, 363 flat surface, 363
horizontal cylinder, 363 sphere, 363
radiation heat transfer, 364
Film condensation, on horizontal tubes, 377 Film condensation, on vertical plate, 372, 373f
average heat transfer coefficient, 375 condensate mass flow rate, 374
film Reynolds number, 375 Laminar-wavy, 376
local heat transfer coefficient, 374 turbulent, 376
Fin, effectiveness of, 53 Fin efficiency, 53
Fin heat transfer, 48 analysis of, 48
Finite difference method, 300
create a mesh or grid, 303, 305, 308, 310 discretize governing equations, 303, 305, 308,
310
mathematical modelling, 303, 304, 308, 310 postprocess to find desired quantities, 304, 307,
309, 311
practical considerations in engineering problems, 320
solve the set of equations, 304, 306, 309, 311 First law of thermodynamics, 1
Flow boiling, 364
in microchannels, 371 regimes, 364
Flow over a cylinder, 137
correlation for Nusselt number, 138 comprehensive equation, 138
Flow over a sphere, 138 average Nusselt number, 138
Fluid flow, and heat transfer over a flat plate, 8f Forced convection, 7, 121
over a flat plate, 104f
Fourier’s law of heat conduction, 9, 105 Free convection, 173
Fully developed flow, 157
analytical solution, Nusselt number for, 161 bulk mean temperature, 162
for constant surface heat flux, 161 convective heat transfer, 157
correlation for turbulent flow inside tubes and ducts, 163
internal flow, 159
from Newton’s law of cooling, 158, 162 variation of heat transfer coefficient and, 158f variation of surface temperature and mean
temperature, 160f F-function chart, 245
G
Gas radiation, 278
Gauss-Seidel method, 304, 308, 309, 311 Governing equations, and quest for analytical
solution
average or mean velocity calculation, 154 continuity equation, 152
energy equation, 156 Fanning friction factor, 155
fully developed conditions, 157 mean temperature, 156 momentum equations, 153
x –momentum, 153 r –momentum, 153
Newton’s law of cooling, 157 noncircular ducts, 156 thermal considerations, 156
Gradient descent, 332–334
Grashof number, 181, 182, 404, 405 Gray surface, 251
Grober’s chart, 93
H
Heat and mass transfer, simultaneous, 408 Heat conduction equation, 65
Heat exchangers, 3, 199, 204f analysis, 203, 204f
energy balance, 203 based on compactness, 202
based on direction of fluid flow counterflow, 199
cross flow, 200 parallel flow, 199
based on mechanical design concentric tube heat exchanger, 200 multipass heat exchanger, 201 shell and tube heat exchanger, 201
based on nature of heat exchange process direct contact-type heat exchanger, 199 recuperator type of heat exchanger, 199 regenerator type of heat exchanger, 199
based on physical state of working fluid condenser, 201
evaporator, 202 design of, 229
Heat exchanger effectiveness, 203, 214 counter low, 207
parallel flow, 205
relations for shell and tube, and cross-flow, 210, 211
Heat exchangers, with phase change, 208 Heat flux, 9, 17, 327, 353, 366f
Index 419
determination of, 280 Heat transfer, 1, 6, 403
and its applications, 3
in cylindrical coordinates, 23 modes of, 4
in plane wall, 20 rate of, 137
through the cylinder, 24
with thickness of insulation, 26f
Heat transfer coefficient, 8t, 65, 106, 166, 205 methods of estimating, 109
Newton’s law vs. fundamental definition, 106 Heat transfer in flows across a bank of tubes, 139
average Nusselt number, 139 generic expression, 140
Heisler’s charts, 82, 90, 92 Heterogeneous nucleation, 356 Homogeneous nucleation, 356
I
Incident radiation, 236, 254, 256 Incompressible flow, 121, 157 Insulated tip, 51
Integral method for fluids with Pr < 1, 134 Integral momentum, solution to, 130 Internal flows, 151
boundary layer thickness, for forced convection, 152
with constant heat flux, 159
with constant wall mass concentration, 154 with constant wall mass flux, 156, 163 with constant wall temperature, 159
Irradiation, 256, 259, 272, 276, 285 Isothermal heat transfer, 1
K
Kirchoff’s law, 255
L
Laplace equation, 18
Law of conservation of mass, 110 Learning process, 333f
data collection, 331
feedback for finding optimum parameters, 332
forward prediction pass, 332 initializing parameters, 331 learning optimum parameters, 331
mathematical formulation, 330 selecting hypothesis function, 330
Learning rate, 333
420 Index
Leckner correlations, 286 Leibniz rule, 183 Leidenfrost point, 353 Lewis number, 403 Linear regression, 333
direct optimization, 334
learning or iterative optimization, 334 to minimize mean squared error or cost
function, 334 Liquid superheat, 356
LMTD (logarithmic mean temperature difference), 164, 207, 209, 210, 223, 224, 226, 228, 229
vs.effectiveness-NTU methods of heat exchanger analysis, 223
Local heat transfer coefficient, 109 Lockhart-Martinelli parameter, 368 Long fin, 55
Loss of coolant accident (LOCA), 174 Lumped capacitance method, 68
M
Machine learning, 327 common algorithms in, 328 reinforcement learning, 328 supervised learning, 328 unsupervised learning, 328
Mass transfer, 403 Mean beam lengths, 283t Mean squared error, 332 Mean temperature, 156
Mid-plane temperatures charts, for plane wall, 83f Midpoint temperature charts, for sphere, 87f Mixed convection, 105
Modified Reynolds analogy, 149 Molecular interaction, 4 Momentum equations, 112 Momentum transfer, 403
N
Natural convection, 104, 105, 173, 360 boundary layer equations, 176 dimensionless numbers governing, 179 empirical correlations for, 189
from heated sphere, 190
from horizontal cylinders, 190 from other geometries, 191 from vertical cylinders, 190
governing equations continuity equation, 176 energy equation, 176 x-momentum equation, 176
y-momentum equation, 176 nondimensional numbers, 176 over a flat plate, 175f, 175
Natural convection over heated plate, 105f Navier-Stokes equations, 115
Net radiative heat transfer from a surface, 256 Neumann boundary condition, 19
Neural networks, in heat transfer, 329, 337 applications in heat transfer, 347 backpropagation, 344
bias unit, 338
engineering problems, practical considerations in, 346
hidden layers, 341 input layer, 338 learning paradigm, 329 learning process, 330 linear model, 338f linear regression, 333 loss function, 339 modifications
nonlinearity, 339 activation function, 339 sigmoid function, 339
neurons, 338
optimization in neural networks. See Gradient descent
output layer, 338 regression task, 330
universal approximation theorem, 342 weighted sum, 338
weights, 338
Newton’s law of cooling, 7, 157, 162 Newton's law of viscosity, 373 Newtonian fluid, 114
Noncircular ducts, 156
hydraulic diameter, equation for, 156 Nondimensionalization of governing equations
boundary layer energy equation, 126 continuity equation, 126
energy equation, 126
local skin friction coefficient, 127 mean or average Nusselt number, 128 momentum equation, 126 x-momentum equation, 126
Nucleate boiling, 360. See also Boiling Nucleation, 355, 356f
Numerical methods, 296
finite difference method (FDM), 296 finite element method (FEM), 296 finite volume method (FVM), 296
Nusselt number, 136, 149, 181, 182, 189, 333, 404
O
Ohm’s law, 22
One-dimensional, steady state heat conduction, with heat generation, 41
plane wall with heat generation, 41 Onset of nucleate boiling (ONB), 353 Opaque surface, 254, 256, 256f Optically thin limit, 281
Ordinary differential equation (ODE), 74, 78, 295, 325
Orthogonal functions, 80
P
Parallel-flow heat exchanger, 205 effectiveness of, 214
Parallel-flow vs. counterflow heat exchangers, 219 Partial differential equation (PDE), 74, 295, 325 Partial pressure, 286
Peclet number, 135 Phase-change heat transfer, 12
Physics and data in heat transfer, 326 Physics-based model, 325
Physics-informed neural networks (PINNs), 348 Physics vs. data methods, 325
Planck’s distribution, 237 Plane wall problem, 11f Pool boiling, 352
critical heat flux, 362 curve, 353, 353f, 354f film boiling, 363 nucleate boiling, 360
temperature profile in liquid, 352f Positive work, 1
Prandtl number, 181, 182, 186, 367, 403, 404 Prevost’s law, 233
Problem of convection, 108
Q
Quanta, 233
Quantum theory, 9
R
Radiation, 233 absorbed, 255 classification of, 234t
concepts and definitions in, 234 electromagnetic theory, 233 incident, 236, 254, 255 quantum theory, 233
reflected, 254, 272 transmitted, 254
Index 421
Radiation heat transfer between surfaces, 261 Radiation heat transfer coefficient, 10
Radiation view factor, and its determination, 262 Radiative transfer equation (RTE), 279 Radiosity, 256, 261, 272, 275, 285, 286 Radiosity-irradiation method, 272 Rayleigh-Jeans distribution, 240
Rayleigh number, 182
Real surfaces, properties of, 248 diffuse surface, 250 emissivity, 248
gray surface, 251 Rectangular fin, 54 Reflection, 254f, 254 Reflectivity, 254, 255, 258f Regression algorithm, 333 Re-radiating surface, 275 Reynolds analogy, 147 Reynolds number, 136, 140
Robin boundary condition (or) mixed condition, 19 Rohsenow correlation, 361
RTE. See Radiative transfer equation (RTE)
S
Saturated water and steam properties table, 381t Schmidt number, 403, 404
Semi-infinite approximation, 72 Separation of variables, method of, 77 Shell and tube heat exchanger, 210 Sherwood number, 404
Silent boiling, 353 Single-phase convection, 352 Solar radiation, 278, 278f, 280 Solid angle, 235
Solid conductor, 4
Spectral directional absorptivity, 255 diffuse and gray surface, 255
Spectral directional intensity, 236 Stanton number, 149, 404 Steady state conduction, 28
composite wall, 28 parallel connection, 30
series-parallel connection, 31 thermal contact resistance, 35
composite cylinder, 35 composite sphere, 38 cylindrical annulus, 23f plane wall, 20 spherical shell, 26
Stefan-Boltzmann law, 239 Stokes’ hypothesis, 117 Subcooling, 369, 372, 392
422 Index
Superheat, 3, 351, 353, 356, 360, 368
Surface forces, 112
Surface tension force, 360
Surface wettability, 358
T
Taylor series, 300
and finite difference formulae, 300 first derivative, 301
order of accuracy, 314 second derivative, 302 Temperature distribution chart
long cylinder, 86f plane wall, 84f sphere, 88f
Temperature profiles, 130, 148 Thermal boundary layer, 401, 402f Thermal boundary layer thickness, 357
Thermal conductivity, 6, 7t, 17, 22, 71, 106 Thermal diffusivity, 17
Thermal radiation, 9, 233 Thermal resistance, 22, 26f
Three-dimensional conduction equation, 15 Three-dimensional control volume, 16f Total pressure drop, 380
for case of evaporation with constant wall heat flux condition, 380
Transient conduction, 66f Transmission processes, 254f Transmissivity, 254 Turbulent flow, 144
average velocity, 145
eddy viscosity of momentum, 146
eddy viscosity of heat, 146 governing equations, 145
total shear stress, and total heat flux, 146 turbulent Prandtl number, 147
Two-dimensional, steady state systems, analysis of, 94
Two-phase heat transfer, 351 Two-phase pressure drop, 378
acceleration pressure gradient, 379 frictional pressure gradient, 378 gravitational head pressure gradient, 379 mean two-phase viscosity, 379
U
Universal black body curve, 243, 244f Unsteady conduction, 312
unsteady conduction equation, 312
V
Variable area fins, 59
Velocity boundary layer, 401, 402f Velocity profile, 374
View factor, 262–264, 266, 269f, 271, 285, 293 View factor algebra, 264
W
Wavelength, 233
Wien’s displacement law, 238
Wien’s distribution, 240
Z
Zenith angle, 234