An LGEM to Identify Time-Dependent Heat Conductivity Function by an Extra Measurement of Temperature Gradient

We consider an inverse problem for estimating an unknown heat conductivity parameter α(t) in a heat conduction equation T_t(x,t) = α(t)T_xx(x,t) with the aid of an extra measurement of temperature gradient on boundary. Basing on an establishment of the one-step Lie-group elements G(r) and G(l) for the semi-discretization of heat conduction equation in time domain, we can derive algebraic equations from G(r) = G(l). The new method, namely the Lie-group estimation method (LGEM), is examined through numerical examples to convince that it is highly accurate and efficient; the maximum estimation error is smaller than 10^-5 for smooth parameter and for discontinuous and oscillatory parameter the accuracy is still in the order of 10^-2. Although the estimation is carried out under a large measurement noise, the LGEM is also stable.     

Molar Heat Capacity at Constant Volume of n-Butane at Temperatures from 141 to 342 K and at Pressures to 33 MPa

Molar heat capacities at constant volume (C _V) for normal butane are presented. Temperatures ranged from 141 to 342 K for pressures up to 33 MPa. Measurements were conducted on liquid in equilibrium with its vapor and on compressed liquid samples. The high purity of the samples was verified by chemical analysis. For the samples, calorimetric results were obtained for two-phase [C _v ⁽²⁾ ], saturated liquid (C or C _x ), and single-phase (C _V) molar heat capacities. The principal sources of uncertainty are the temperature rise measurement and the change-of-volume work adjustment. The expanded uncertainty (i.e., a coverage factor k=2 and thus a two-standard deviation estimate) for values of C _V is estimated to be 0.7%, for C _v ⁽²⁾ it is 0.5%, and for C it is 0.7%.     

Molar Heat Capacity at Constant Volume of Trifluoromethane (R23) from the Triple-Point Temperature to 342 K at Pressures to 33 MPa

Molar heat capacities at constant volume (C _v) of trifluoromethane (R23) have been measured with an adiabatic calorimeter. Temperatures ranged from the triple point to 342 K, and pressures up to 33.5 MPa. Measurements were conducted on the liquid in equilibrium with its vapor and on compressed liquid and gaseous samples. The samples were of high purity, as verified by chemical analysis. Calorimetric quantities are reported for the two-phase (C ⁽²⁾ _v), saturated-liquid (C or C_x), and single-phase (C _v) molar heat capacities. The C ⁽²⁾ _v data were used to estimate vapor pressures for values less than 100 kPa by applying a thermodynamic relationship between the two-phase internal energy U ⁽²⁾ and the temperature derivatives of the vapor pressure. The triple-point temperature and the enthalpy of fusion were also measured. The principal sources of uncertainty are the temperature rise measurement and the change-of-volume work adjustment. The expanded relative uncertainty (with a coverage factor k=2 and thus a two-standard deviation estimate) is estimated to be 0.7% for C _v, 0.5% for C ⁽²⁾ _v, and 0.7% for C .     

The influence of biaxiality of loading on the form of caustics in cracked plates

Summary The optical method of reflected caustics was applied up-to-now to problems of cracked plates under uniaxial loading. Only the problem of the biaxial tension of the plate has been considered for the particular case where the crack is transverse to the longitudinal axis of the plate which coincided with the loading axis. In this paper the influence of a biaxial loading of the plate on the form and orientation of the caustic was studied in connection with the orientation of the crack. New modified relations were given for the evaluation of the complex stress intensity factorK=K _I –iK _II in terms of the angle of the angular displacement of the caustic axis. For the accurate evaluation ofK _I andK _II nomograms of correction factors _y ^max , _x ^max and _x ^max were given in terms of the angle of inclination of the crack =(90–) and the biaxiality factork. Experimental evidence with PMMA internally cracked plates corroborated the results of theory.List of Symbols (z), (z) complex-stress function of Muskhelishvili - _xx , _yy , _xy crack tip stress referred to Cartesian coordinate system - r, polar coordinate system centered at crack tip - K _I ,K _II stress intensity factors for ModeI andII loading, respectively - angle of inclination of the crack - 90°- - k ratio of stresses at infinity - ₁, ₂ principal stresses at crack tip - a crack length - stress applied at infinity along the transverse boundaries of the plate - X _r,f ,Y _r,f parametric equations of the reflected caustics referred to the Cartesian systemOXY on the reference screen: (r) reflected caustics from rear face of the specimen and (f) reflected caustics from the front face of the specimen - r ₀ radius of the generatrix curve on the specimen around the crack tip (initial curve) - c _r,f optical constants of the material for reflections from the rear and front faces of the specimen respectively - _m magnification ratio of the optical set-up - z ₀ distance between the reference-screen and the middle plane of the specimen - z _i distance between the focus of the light beam and the middle plane of the specimen - d thickness of specimen - 2 for the reflected caustics from the rear face of the specimen and 1 for the reflected caustics from the front face of the specimen - C _r,f z ₀ dc _r,f / _m (2)^1/2 - v Poisson's ratio - E elastic modulus of the material - A (1+k)+(1–k) cos 2 - B (1–k) sin 2 - C 1+k ²+(1–k ²) cos 2 2 tan^–1 (B/A)=2 tan^–1 (K _II/K_I) - D _y ^max ,D _x ^max ,D _x ^min the maximum and the ninimum diameter of caustics along the axisOy andOx of the crack respectively - _y ^max , _x ^max , _x ^min the correction factors forD _y ^max ,D _x ^max andD _x ^min respectively - D _t ^max ,D _l ^max the maximum transverse and longitudinal diameters of the caustics respectively - _t ^max , _l ^max the correction factors forD _t ^max ,D _l ^max respectively With 13 Figures     

A thermodynamic property formulation for ethylene from the freezing line to 450 K at pressures to 260 MPa

A new thermodynamic property formulation based upon a fundamental equation explicit in Helmholtz energy of the form A=A(, T) for ethylene from the freezing line to 450 K at pressures to 260 MPa is presented. A vapor pressure equation, equations for the saturated liquid and vapor densities as functions of temperature, and an equation for the ideal-gas heat capacity are also included. The fundamental equation was selected from a comprehensive function of 100 terms on the basis of a statistical analysis of the quality of the fit. The coefficients of the fundamental equation were determined by a weighted least-squares fit to selected P--T data, saturated liquid and saturated vapor density data to define the phase equilibrium criteria for coexistence, C _v data, velocity of sound data, and second virial coefficients. The fundamental equation and the derivative functions for calculating internal energy, enthalpy, entropy, isochoric heat capacity (C _v), isobaric heat capacity (C _p), and velocity of sound are included. The fundamental equation reported here may be used to calculate pressures and densities with an uncertainty of ±0.1%, heat capacities within ±3 %, and velocity of sound values within ±1 %, except in the region near the critical point. The fundamental equation is not intended for use near the critical point. This formulation is proposed as part of a new international standard for thermodynamic properties of ethylene.Paper presented at the Ninth Symposium on Thermophysical Properties, June 24–27, 1985, Boulder, Colorado, U.S.A.     

Molar heat capacity at constant volume of difluoromethane (R32) and pentafluoroethane (R125) from the triple-point temperature to 345 K at pressures to 35 MPa

Molar heat capacities at constant volume (C _v) of dill uoromethane (R32) and pentalluoroethane (R125) were measured with an adiabatic calorimeter. Temperatures ranged from their triple points to 345 K, and pressures up to 35 MPa. Measurements were conducted on the liquid in equilibrium with its vapor and on compressed liquid samples. The samples were of a high purity, verified by chemical analysis of each fluid. For the samples, calorimetric results were obtained for two-phase (C _v ⁽²⁾ ), saturated liquid (C orC _x ), and singlephase (C _v) molar heat capacities. TheC data were used to estimate vapor pressures for values less than 0.3 MPa by applying a thermodynamic relationship between the saturated liquid heat capacity and the temperature derivatives of the vapor pressure. The triple-point temperature (T _tr) and the enthalpy of fusion (_fus H) were also measured for each substance. The principal sources of uncertainty are the temperature rise measurement and the change-ofvolume work adjustment. The expanded uncertainty (at the two-sigma level) forC _v is estimated to be 0.7%, forC _v ⁽²⁾ it is 0.5%, and forC it is 0.7%.     

Transient mixed convective heat transfer with melting effect from the vertical plate in a liquid saturated porous medium

This paper aims to numerically analyze the melting effect on transient mixed convective heat transfer from a vertical plate in a liquid saturated porous medium. The Darcy’s flow with the Boussinesq approximation and energy equations for this study is transformed into the non-dimensional equations by using pseudo-similarity coordinate (ζ) and dimensionless time (ξ). The resulting the boundary value problem is solved by the method of lines (MOLs) with central finite difference and Newton’s iteration to completely obtain the solutions for the whole transient mixed convective heat transfer from transient heat conduction to steady mixed convection in porous medium with melting effect in the presence of aiding and opposing external flows. As shown in the results, it is both found that the rate of dynamic heat transfer is reduced with increasing melting strength and the response time from transient heat conduction to steady mixed convection in porous medium for aiding external flow is shorter than that for opposing external flow in the presence of melting effect. Additionally, the melting effect may be reduced with increasing mixed convective strength Gr/Re^1/2 during transient mixed convective heat transfer in porous medium with aiding external flow.     

Effectiveness of mixing coaxial flows swirled in opposite directions

The mixing of coaxial turbulent flows swirled in opposite directions is experimentally studied. The effectiveness of this mixing is compared with mixing after an agitating grid.Notation z, r, cylindrical coordinate system - r₁, r₂ inside and outside radius of annular channel - H=r₂–r₁ radial gap in annular channel - y=(r–r₁)/H dimensionless radial coordinate - V(v_z, v_r, v) mean velocity vector - v, v_z pulsative components of velocity in the direction of the mean velocity vector and in the axial direction - P_*, P total and static pressure - , loss coefficients Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 41, No. 3, pp. 407–413, September, 1981.     

The fourth-dimention concept in the finite element analysis of transient heat transfer problems

This study presents the finite element analysis for the transient heat transfer problems by introducing the Fourth-dimension concept. Time is treated as an additional dimension in the solution domain thereby Increasing the number of dimensions by one. For instance, a three-dimensional transient problem can be considered as a four-dimensional problem in the x, y, z, t domain and a two-dimensional transient problem can be considered as a three-dimensional steady-state problem in the x, y, t domain, respectively. The variational principle of the finite element method and the techniques existing for steady-state problems can be directly utilized. Numerical calculations were performed for heat conduction problems, laminar-turbulent convective heat transfer problems, and radiative heat transfer problems.     

Isochoric Heat Capacity Measurements for 2,2-Dichloro-1,1,1-Trifluoroethane (R123) at Temperatures from 167 to 341 K and 1-Chloro-1,2,2,2-Tetrafluoroethane (R124) from 94 to 341 K at Pressures to 35 MPa

Molar heat capacities at a constant volume (C _v) of 2,2-dichloro-1,1,1-trifluoroethane (R123) and 1-chloro-1,2,2,2-tetrafluoroethane (R124) were measured with an adiabatic calorimeter. Temperatures ranged from 167 K for R123 and from 94 K for R124 to 341 K, and pressures were up to 33 MPa. Measurements were conducted on the liquid in equilibrium with its vapor and on compressed liquid samples. The samples were of a high purity, verified by chemical analysis of each fluid. For the samples, calorimetric results were obtained for two-phase (C ⁽²⁾ _v), saturated liquid (C or C _x ), and single-phase (C _v) molar heat capacities. The C data were used to estimate vapor pressures for values less than 100 kPa by applying a thermodynamic relationship between the saturated liquid heat capacity and the temperature derivatives of the vapor pressure. Due to the tendency of both R123 and R124 to subcool, the triple-point temperature (T _tr) and the enthalpy of fusion ( _fus H) could not be measured. The principal sources of uncertainty are the temperature rise measurement and the change-of-volume work adjustment. The expanded uncertainty (at the 2 level) for C _v is estimated to be 0.7%, for C ⁽²⁾ _v it is 0.5%, and for C it is 0.7%.     

Temperature-enthalpy approach to the modelling of self-propagating combustion synthesis of materials

A new temperature-enthalpy approach has been proposed to model self-propagating combustion synthesis of advanced materials. This approach includes the effect of phase change which might take place during a combustion process. The effect of compact porosity is also modelled based on the conduction, convection and radiation in the local scale. Various parametric studies are made to analyse numerically the effects of activation energy, non-reacting phase content, porosity, Biot number, etc. The model predictions of the combustion pattern are in close agreement with those observed in experiments.Nomenclature c Concentration (wt %) - B _i Biot number =hL/k - f Fractional value - c _p Specific heat (J kg^–1 K) - h Heat-transfer coefficient (W m^–2 K) - L Height of material,m - Q Heat of reaction (J kg^–1) - H _SL ^* Latent heat of fusion (J kg^–1) - H _SE ^* Latent heat of fusion at eutectic (J kg^–1) - k Thermal conductivity (W m^–1 K) - k Equilibrium partition coefficient - Reaction kinetic function - t Time (s) - Non-dimensional time - T Temperature (K) - T ₀ Initial temperature (K) - Non-dimensional temperature - H Enthalpy (J kg^–1) - Kinetic function - Non-dimensional enthalpy - v _f Volume fraction of non-reactive phase - V Volume (m₃) - k ₀ Pre-exponential constant to reaction rate (s^–1) - z Cartesian co-ordinate - z^* Non-dimensional co-ordinate - Non-dimensional reacted fraction - Density (kg m^–3) - A non-dimensional temperature - Pore surface emissivity - Planck's constant - i Initial state - r Reacted state - l, L Liquid state - s Solid state - E Eutectic - M Melting point of pure material - P Centre of control volume - s Southern side of central volume - S Southern control volume - n Northern side of central volume - N Northern control volume - * Non-dimensional term - n New time level - o Old time level - m Iteration level     

Pair explosion in an exponential atmosphere

The problem of the interaction of two coaxial explosions in a barometric atmosphere is solved numerically based on the complete system of Navier-Stokes equations. Basic regularities that occur in the interference of two spherical shock waves of different intensities are studied. The last stage of the processes, when shock wave processes become unimportant and convection plays a dominant part, is investigated.Notation t time - r, z cylindrical coordinates - v=(u, ) velocity - density - p pressure - T temperature - ,k dynamic viscosity and thermal conductivity - V(t) calculation region - f(t), _± (t boundaries of the calculation region - z ₁ ,z ₂ altitudes of the centers of the lower and upper explosions - R ₁ ,R ₂ initial radii of the regions involved in the explosions - altitude of the homogeneous atmosphere - g acceleration due to gravity - adiabatic exponent - , , parameters - M Mach number - Re Reynolds number - Pr Prandtl number - c _p ,c _v specific heats Department of Theoretical Problems, Russian Academy of Sciences, Moscow. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 66, No. 6, pp. 657–661, June, 1994.     

Study on Isochoric Specific Heat Capacity of Liquid R-410A and HFE-347pcf2

The isochoric heat capacity (c _v ) of R-410A [a mixture of 49.81 mass% difluoromethane (HFC-32) + 50.19 mass% pentafluoroethane (HFC-125)] and 1,1,2,2-tetrafluoroethyl-2,2,2-trifluoroethylether (HFE-347pcf2) was measured at temperatures from 277 K to 400 K and at pressures up to 30 MPa. The reported density measurements for R-410A and HFE-347pcf2 are in the single-phase region and cover a density range above 0.92 g·cm^?3 and 1.33 g·cm^?3, respectively. The measured data of R-410A are compared with data reported by other researchers. Also, the measured data of R-410A are examined with an available equation of state. As a result, it is found that the present c _v data for R-410A agree well with those by other researchers and the calculated values with the equation of state in the measurement range except near the critical isochore.     

Static fatigue and creep resistance of a commercial sialon

The static fatigue and creep resistances of a commercial sialon material were evaluated by elevated temperature flexural stress rupture testing in air. The material is a ceramic alloy with a solid solution sialon phase, Si_6-z Al _z O _z N_8-z , with a substitution levelz less than 1. The utilization of yttria sintering aids and post-sintering heat treatments leads to a fully crystalline grain-boundary phase. Creep and static fatigue resistances were excellent at temperatures up to 1200° C.     

Frequency Response of AC Susceptibility in Melt Textured YBCO Superconductors

We have reviewed the scaling relations for frequency dependent AC susceptibility proposed by Lee. et al. and other contributions. Based on vortex glass analysis, we derived a scaling equation for the peak temperature of the imaginary part and ac magnetic field frequency f. The peak temperature T_p is found to obey: T_p = Cf^1/(v(z–1))+T_g, where C is a constant, and T_g is the vortex glass temperature. The scaling relationship was applied to our melt textured YBCO superconductors, which was found in good agreement with the experimental data.     

Influence of microstructure on strength and fracture toughness ofβ-Sialon

-Sialon (Si_6–z Al _z O _z N_8–z ) withz = 0.5 was fabricated by hot-pressing of a spray dried mixture of –Si₃N₄ and aluminium-isopropoxide solution. Phase composition, flexural strength and microstructure of a sintered body were investigated. Phases identified by XRD were -Sialon and a small amount of O-Sialon. The flexural strength (three-point bending) was about 1500 MN m^–2. This value, about three times higher than that of -Sialon fabricated from -Si₃N₄ and -Al₂O₃ powder, was mainly due to the homogeneous microstructure without large defects such like clusters of large grains. -Sialon was heat treated at 2000 °C for 2 h in 4 M Pa N₂ to develop elongated -Sialon grains with high aspect ratio. Microstructure, flexural strength and fracture toughness (K _Ic) of it were investigated. Both strength andK _Ic were lower than those of hot-pressed sample, even though an elongated microstructure was achieved. This fact showed that the toughening of -Sialon with elongated grains could not be achieved without grain boundary phase which resulted in a crack deflection.     

Seebeck coefficient of V2O5-R2O3-TeO2 (R = Sb or Bi) glasses

The thermoelectric power of glasses in the systems V₂O₅-Sb₂O₃-TeO₂ and V₂O₅-Bi₂O₃-TeO₂ was measured at temperatures in the range 373–473 K. The glasses in both systems were found to be n-type semiconductors. The Seebeck coefficient, Q, at 473 K was determined as –192 to –151 VK^–1 for V₂O₅-Sb₂O₃-TeO₂ glasses, and –391 to –202 VK^–1 for V₂O₅-Bi₂O₃-TeO₂ glasses. For these glasses in both systems, Heikes' formula was satisfied adequately for the relationship between Q and In [C _v/(1-C_v)] (C _v = V⁴⁺/V_total, C _v is the ratio of the concentration of reduced vanadium ions), and discussions confirmed small polaron hopping conduction of the glasses in both systems. Mackenzie's formula relating to Q and V⁵⁺/V⁴⁺ was also applicable to the glasses in both systems, and it was concluded that the dominant factor determining Q was C _v.     

Frequency Response of AC Susceptibility in Melt Textured YBCO Superconductors

We have reviewed the scaling relations for frequency dependent AC susceptibility proposed by Lee. et al. and other contributions. Based on vortex glass analysis, we derived a scaling equation for the peak temperature of the imaginary part and ac magnetic field frequency f. The peak temperature T_p is found to obey: T_p = Cf^1/(v(z–1))+T_g, where C is a constant, and T_g is the vortex glass temperature. The scaling relationship was applied to our melt textured YBCO superconductors, which was found in good agreement with the experimental data.     

Recent advances in the numerical analysis of heat pipes

Numerical simulation of heat pipes has progressed significantly in recent years. The state-of-the-art has been advanced in steady state, continuum transient, and frozen startup simulation for high, moderate, and low temperature heat pipes of conventional cylindrical and nonconventional geometries such as wing leading edges and spacecraft nosecaps. This review summarizes these advancements and discusses the important results.List of symbols A cross-sectional area of the vapor channel, m² - c specific heat, J/(kg-K) - C _p specific heat at constant pressure, J/(kg-K) - C _v specific heat at constant volume, J/(kg-K) - D vapor space diameter, m - D _v coefficient of self-diffusion, m²/s - G vapor mass flux, kg/(m²-s) - h convective heat transfer coefficient, W/(m²-K) - h _fg latent heat of evaporation, J/kg - H latent heat due to melting or freezing, J/kg - k thermal conductivity, W/(m-K) - Kn Knudsen number, /D - L total length of the heat pipe, m - L _a length of the adiabatic section, m - L _c length of the condenser, m - M molecular weight, kg/kmol - Ma Mach number, - m _i mass flux at the liquid-vapor interface, kg/(m²-s) - P pressure, N/m² - q heat flux, W/m² - Q heat input at the active evaporator, W - Q _o heat output at the condenser, W - r radial coordinate, m - R gas constant, J/(kg-K) - R _o outer pipe wall radius, m - R _u universal gas constant, J/(kmol-K) - R _v vapor space radius, m - t time, s - T temperature, K - T _i,c interfacial temperature on the continuum vapor flow side, K - T _i,r interfacial temperature on the rarefied vapor flow side, K - T _rf reference (saturation) temperature, K - T _tr transition vapor temperature, K - v radial velocity, m/s Funding for this work was provided by a joint effort of the NASA Lewis Research Center and the Thermal Technology Center of Wright Laboratory under contract No. F33615-88-C-2820     

Determination of the local angular radiation coefficients in certain two-body systems

A method is shown and formulas are derived by which local angular radiation coefficients can be determined in certain two-body systems where the configuration is arbitrary but one of the bodies is either a cylinder or a rectangular plate.Notation _int radiation vector of body 1 - _E ^int intrinsic radiation intensity of body 1 - _x, _y, _z components of the geometrical radiation vector along rectangular coordinates - r₀=x²+z² shortest distance from point M(x, y, z) to linear radiator - ₀ , ₀ ^' angles subtending the two segments of the linear radiator from point M(x, y, z) on area element 2 of irradiated surface - l length of the cylinders - x, y, z space coordinates of point M Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 22, No. 6, pp. 1080–1088, June, 1972.