BOOKCOMP, Inc. — John Wiley & Sons / Page 1297 / 2nd Proofs / Heat Transfer Handbook / Bejan NOMENCLATURE 1297 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1297], (67) Lines: 1842 to 1991 ——— -1.37268pt PgVar ——— Long Page PgEnds: T E X [1297], (67) Least Squares If N consecutive measurements of the output y(i), 1 = 1 ···N , are made at successive sampling periods, these can be concatenated in a measured output vectorY as Y = [y(n + 1)y(n +2) ··· y(n + N)] T (17.109) These will be compared with their respective predictions Y by the ARMA process model, in which the corresponding augmented vectors Φ can be assembled in ma- trix Φ: Φ = [Φ T (n) Φ T (n +1) ···Φ T (n +N − 1)] T (17.110) where the elementary vectors Φ also contain the previously measured outputs y(i), i = 1 ···n and inputs Q(i), i = 1 ···n. Thus the ARMA model of the thermalsystem can be written as Y = ΦΘ. The least squares method minimizes the quadratic index of the output deviations ε =Y − Y (i.e., J = ε T ε) setting the parameters Θ to Θ = (Φ T Φ) −1 Y = Φ † Y (17.111) where Φ † is the pseudoinverse of Φ. NOMENCLATURE Roman Letter Symbols a grinding cut depth, m ARMA output coefficient, dimensionless A state matrix, dimensionless A 1 preexponential frequency factor in cure kinetics model, s −1 A c ,A o ,A 1 cross-sectional area, m 2 b chip width, m grinding width, m saturation point, dimensionless ARMA input coefficient, dimensionless B input matrix, dimensionless B 1 ,B 2 ,B 3 parameters defined in eqs. (17.26)–(17.28), dimensionless Bi Biot number, dimensionless ˆc degree of crystallization, dimensionless c specific heat capacity, kJ/kg · K saturation level, dimensionless C output matrix, dimensionless C A instantaneous resin concentration in the resin–catalyst mixture at any time, t, kg/m 3 C A0 initial resin concentration in the resin–catalyst mixture, kg/m 3 d delay, dimensionless BOOKCOMP, Inc. — John Wiley & Sons / Page 1298 / 2nd Proofs / Heat Transfer Handbook / Bejan 1298 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1298], (68) Lines: 1991 to 1991 ——— 0.00563pt PgVar ——— Normal Page PgEnds: T E X [1298], (68) d s grinding wheel diameter, m D diameter, m differential, dimensionless period, dimensionless D direct matrix, dimensionless D b degree of bonding, dimensionless D h degree of healing, dimensionless D ic degree of intimate contact, dimensionless e error, dimensionless E activation energy for viscosity in the chemorheological model, kJ/kmol error (Laplace transform), dimensionless expectation, dimensionless E 1 ,E 2 activation energies in the cure kinetics models, kJ/kmol f transfer function, dimensionless F,F forces associated with cutting, N g controller transfer function, dimensionless G transfer function, dimensionless G discrete state matrix, dimensionless h heat transfer coefficient, W/m 2 · K depth of cut, m thermoplastic tow thickness, m sensor transfer function, dimensionless discrete-time index, dimensionless H plate thickness, m H detectability matrix, dimensionless discrete input matrix, dimensionless H T total heat of crystallization, kJ/kg H u theoretical ultimate heat of crystallization, kJ/kg ∆H c heat of crystallization, kJ/kg ∆H R heat of the cure reaction, kJ/kg i discrete time index, dimensionless I identity matrix, dimensionless J quadratic performance index, dimensionless k thermal conductivity, W/m · K discrete time index, dimensionless k o Kozeny constant, dimensionless K gain, dimensionless K controller matrix, dimensionless K o modified Bessel function of the first kind, of order zero, dimensionless K 10 ,K 20 preexponential frequency factors in the cure kinetics models, s −1 l input number, dimensionless length of heated region, m BOOKCOMP, Inc. — John Wiley & Sons / Page 1299 / 2nd Proofs / Heat Transfer Handbook / Bejan NOMENCLATURE 1299 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1299], (69) Lines: 1991 to 1991 ——— 0.73116pt PgVar ——— Normal Page PgEnds: T E X [1299], (69) l c frictional contact length, m l s shear length, m L length, m thickness, m characteristic length, m L observer matrix, dimensionless m, n exponents in the cure kinetics models, dimensionless output number and state number, dimensionless m 1 ,m 2 Pe ± Pe 2 + 4Bi /2, dimensionless N describing function, dimensionless sample number, dimensionless p stiffness of a fiber network, Pa P perimeter, m grinding power, W pressure, Pa P pdrs matrix, dimensionless Pe P ´ eclet number, dimensionless q heat flux, W/m 2 Q heat source strength, W Q(t) heat release during cure per unit mass of resin-catalyst sample, kJ/kg Q state penalty matrix, dimensionless r,x, y, z spatial coordinates, m r f fiber radius, m R fraction of shear energy removed by the chip, dimensionless void radius, m universal gas constant, kJ/kmol · K R, X,Y coordinates, dimensionless R resultant force, N input penalty matrix, dimensionless s Laplace variable, dimensionless S radius of a resin shell surrounding a void, m S stabilizability matrix, dimensionless t time, s T temperature, K sampling period, s u grinding energy, J/m 3 integration variable, dimensionless v velocity, m/s void fraction, dimensionless output noise, dimensionless excitation, dimensionless v a maximum fiber volume fraction, dimensionless v f fiber volume fraction in the composite, dimensionless BOOKCOMP, Inc. — John Wiley & Sons / Page 1300 / 2nd Proofs / Heat Transfer Handbook / Bejan 1300 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1300], (70) Lines: 1991 to 2068 ——— 12.20792pt PgVar ——— Normal Page PgEnds: T E X [1300], (70) v I initial fiber volume fraction, dimensionless V velocity, m/s Lyapunov function, dimensionless w state noise, dimensionless W D deformation work per unit volume, J/m 3 y parameter in the degradation kinetics model, dimensionless output, dimensionless y output measurement, dimensionless Y measured output, dimensionless zZ-transform variable, dimensionless Greek Letter Symbols α thermal diffusivity, m 2 /s degree of degradation, dimensionless β friction angle shape constant, dimensionless γ characteristic thickness, m [= A c /P ] adaptation gain, dimensionless γ o rake angle, deg ε fraction of grinding power entering the workpiece as heat, dimensionless degree of cure, dimensionless [= (C A0 − C A )/C AO ] state deviation, dimensionless θ temperature, dimensionless Θ parameters, dimensionless κ permeability, m 2 λ constant in the chemorheological model, dimensionless eigenvalue (observer), dimensionless µ coefficient of friction, dimensionless dynamic viscosity, Pa · s eigenvalue (controller), dimensionless ρ density, kg/m 3 σ interfacial bond strength, Pa τ time, dimensionless time (Lagrangian), s τ s flow stress, N/m 2 φ porosity, dimensionless characteristic polynomial, dimensionless φ o shear plane angle, deg ϕ resin volume fraction in the fiber–resin mixture, dimensionless [= 1 −v f ] Φ augmented state, dimensionless ω angular frequency, s −1 BOOKCOMP, Inc. — John Wiley & Sons / Page 1301 / 2nd Proofs / Heat Transfer Handbook / Bejan REFERENCES 1301 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 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