Identification of spacewise and time dependent source terms in 1D heat conduction equation from temperature measurement at a final time

Inverse problems of identifying the unknown spacewise and time dependent heat sources F(x) and H(t) of the variable coefficient heat conduction equation u_t = (k(x)u_x)_x + F(x)H(t) from supplementary temperature measurement (u_T(x)?u(x, T_f)) at a given single instant of time T_f > 0, are investigated. For both inverse source problems, defined to be as ISPF and ISPH respectively, explicit formulas for the Fréchet gradients of corresponding cost functionals are derived. Fourier analysis of these problems shows that although ISPF has a unique solution, ISPH may not have a unique solution. The conjugate gradient method (CGM) with the explicit gradient formula for the cost functional J₁(F) is then applied for numerical solution of ISPF. New collocation algorithm, based on the piecewise linear approximation of the unknown source H(t), is proposed for the numerical solution of the integral equation corresponding to ISPH. The proposed two numerical algorithms are examined through numerical examples for reconstruction of continuous and discontinuous heat sources F(x) and H(t). Computational results, with noise free and noisy data, show efficiency and high accuracy of the proposed algorithms.     

Numerical calculation of local entropy generation in a methane–air burner

This study considers numerical simulation of the combustion of methane with air including 21% oxygen and 79% nitrogen in a burner and the numerical solution of the local entropy generation rate due to the high temperature and velocity gradients in the combustion chamber for various fuel flow rates (from 5 to 10 lpm). Swirling air flow is also used to burn the methane more efficiently. The effects of equivalence ratio (ϕ) and swirl number (S) on the combustion and entropy generation rate are investigated for different (consecutive) equivalence ratios (from 0.5 to 1.0) and swirl numbers (from 0 to 0.3). The numerical calculation of combustion is performed individually for these cases with the help of the Fluent CFD code. The volumetric entropy generation rate distributions and the other thermodynamic parameters are calculated numerically by using the results of the combustion calculations. The maximum values of the rates of reaction-1 and -2 decrease with the increase of ϕ. In the case of ϕ < 1, complete combustion occurs, and the combustion in the case of ϕ = 1 is very close to the complete combustion state. In the case of no swirl, the entropy generation rate decreases exponentially with the increase of ϕ in the cases of high Q_f, whereas they have quadratic profiles having their minimum values in cases of low Q_f. In terms of the entropy generation rates, the optimum equivalence ratios for Q_f = 5, 6, 7, >7 lpm in the case of S = 0 and Q_f = 10 lpm in the case of S = 0.3 are obtained as ϕ = 0.66, 0.8, 0.86, 1.0 and 0.92, respectively.     

High-pressure burning velocities measurements for centrally-ignited premixed methane/air flames interacting with intense near-isotropic turbulence at constant Reynolds numbers

This paper measures high-pressure turbulent burning velocities (S_T) of lean methane spherical flames at constant turbulent Reynolds numbers (Re_T ≡ u′L_I/ν), where u′ and L_I are the r.m.s. turbulent fluctuation velocity and the integral length scale of turbulence and ν is the kinematic viscosity of reactants. This is achieved by adopting a recently-built double-chamber, fan-stirred cruciform burner with perforated plates that can be used to generate intense near-isotropic turbulence with negligible mean velocities while controlling the product of u′L_I in proportion to the decreasing ν at elevated pressure (p) up to 1.2 MPa. Results show that when Re_T is fixed ranging from 6700 to 14,200, values of S_T decrease similarly as laminar burning velocities (S_L) with increasing p in minus exponential manners, revealing a global response of burning velocities to pressure. In general, the higher Re_T, the higher S_T/S_L at any fixed p. It is found that the curves of S_T/S_L as a function of u′/S_L all exhibit very strong bending under constant Re_T conditions. These results not only reveal that the important effect of Re_T on high-pressure S_T/S_L enhancement, but also suggest that recent findings related with the promotion effect of increasing pressure on S_T primarily due to the enhancement of flame instabilities via the thinner flame without any discussion on the influence of Re_T elevation at elevated pressure should be reconsidered. Moreover, we found that the modified values of S_T at mean progress variable $\overline{c}$ ≈ 0.5 show good agreements between Bunsen-type and spherical flames, suggesting that S_T determined at flame surfaces with $\overline{c}$ = 0.5 may be a better representative of itself regardless of the flame geometries. Finally, various general correlations of $S_{\text{T,}\overline{c}=0.5}$ are compared and discussed. It is found that the present scattering data under different p and Re_T conditions can be merged onto a single curve of ($S_{\text{T,}\overline{c}=0.5}$ ? S_L)/u′ = 0.14Da^0.47, where Da is the turbulent Damköhler number.     

A computational study and correlation of premixed isooctane air laminar reaction fronts diluted with EGR

The current work investigates the propagation of premixed laminar reaction fronts for mixtures of isooctane–air and recirculated combustion products (or EGR) under high pressure and temperature conditions. The work uses a transient one-dimensional flame simulation with a skeletal 215 species chemical kinetic mechanism to generate laminar burning velocity and front thickness predictions. The simulation was exercised over fuel–air equivalence ratios, unburned gas temperatures, pressures and EGR levels ranging from 0.1 to 1.0, 400 to 1000 K, 1 to 250 bar, and 0% to 60% (by mass) respectively, a range extending beyond that of previous researchers. Steady reaction fronts with burning velocities in excess of 5 cm/s could not be established under all of these conditions, especially when burned gas temperatures were below 1450 K and/or when characteristic reaction front propagation times were on the order of the unburned gas ignition delay. For a given pressure, T_u and T_b, the burning velocity of an EGR dilute mixture was found to be lower than that of an air dilute mixture, with the decrease in burning velocity attributed primarily to the reduced oxygen concentration’s effect on chemistry. Steady premixed laminar burning velocities were correlated using a modified two-equation form based on the asymptotic structure of a laminar flame, which produced an average error of 3.4% between the simulated and correlated laminar burning velocities, with a standard deviation of 4.3%, while additional correlations were constructed for reaction front thickness and adiabatic flame temperature. Correlations are presented based on a non-product equivalence ratio φ and a fraction of stoichiometric combustion products X_SCP. Conversion factors are provided to facilitate application to modern direct injection internal combustion engines with inherent charge stratification where the local global Φ is different from the global Φ of the residual gas.     

Optimal design of geometric parameters of double-layered microchannel heat sinks

This work uses an optimization procedure consisting of a simplified conjugate-gradient method and a three-dimensional fluid flow and heat transfer model to investigate the optimal geometric parameters of a double-layered microchannel heat sink (DL-MCHS). The overall thermal resistance R_T is the objective function to be minimized, and the number of channels N, channel width ratio β, lower channel aspect ratio α_l, and upper channel aspect ratio α_u are the search variables. For a given bottom area (10 × 10 mm) and heat flux (100 W/cm²), the optimal (minimum) thermal resistance of the double-layered microchannel heat sink is about R_T = 0.12 °C/m²W. The corresponding optimal geometric parameters are N = 73, β = 0.50, α_l = 3.52, and, α_u = 7.21 under a total pumping power of 0.1 W. These parameters reduce the overall thermal resistance by 52.8% compared to that yielded by an initial guess (N = 112, β = 0.37, α_l = 10.32, and α_u = 10.93). Furthermore, the optimal thermal resistance decreases rapidly with the pumping power and then tends to approach an constant value. As the pumping power increases, the optimal values of N, α_l, and α_u increase, whereas the optimal β value decreases. However, increasing the pumping power further is not always cost-effective for practical heat sink designs.     

A simple analytical model to study and control azimuthal instabilities in annular combustion chambers

This study describes a simple analytical method to compute the azimuthal modes appearing in annular combustion chambers and help analyzing experimental, acoustic and large eddy simulation (LES) data obtained in these combustion chambers. It is based on a one-dimensional zero Mach number formulation where N burners are connected to a single annular chamber. A manipulation of the corresponding acoustic equations in this configuration leads to a simple dispersion relation which can be solved by hand when the interaction indices of the flame transfer function are small and numerically when they are not. This simple tool is applied to multiple cases: (1) a single burner connected to an annular chamber (N = 1), (2) two burners connected to the chamber (N = 2), and (3) four burners (N = 4). In this case, the tool also allows to study passive control methods where two different types of burners are mixed to control the azimuthal mode. Finally, a complete helicopter chamber (N = 15) is studied. For all cases, the analytical results are compared to the predictions of a full three-dimensional Helmholtz solver and a very good agreement is found. These results show that building very simple analytical tools to study azimuthal modes in annular chambers is an interesting path to control them.     

Steam jet ejector cooling powered by waste or solar heat

A small scale steam jet ejector experimental setup was designed and manufactured. This ejector setup consists of an open loop configuration and the boiler operate in the temperature range of T_b = 85–140 °C. The typical evaporator liquid temperatures range from T_e = 5 °C to 10 °C while the typical water-cooled condenser pressure ranges from P_c = 1.70 kPa to 5.63 kPa (T_c = 15–35 °C). The boiler is powered by two 4 kW electric elements while a 3 kW electric element simulates the cooling load in the evaporator. The electric elements are controlled by means of variacs.Primary nozzles with throat diameters of 2.5 mm, 3.0 mm and 3.5 mm are tested while the secondary ejector throat diameter remains unchanged at 18 mm. These primary nozzles allow the boiler to operate in the temperature range of T_b = 85–110 °C. When the nozzle throat diameter is increased, the minimum boiler temperature decreases. A primary nozzle with a 3.5 mm throat diameter was tested at a boiler temperature of T_b = 95 °C, an evaporator temperature of T_e = 10 °C and a critical condenser pressure of P_crit = 2.67 kPa (22.6 °C). The system's COP is 0.253.In a case study the experimental data of a solar powered steam jet ejector air conditioner is investigated. Solar powered steam ejector air conditioning systems are technical and economical viable when compared to conventional vapour compression air conditioners. Such a system can either utilise flat plate or evacuated tube solar thermal collectors depending on the type of solar energy available.     

Influence of fuel fraction gradient on triple flame velocity in plain and axis-symmetrical channels

Dynamics of laminar triple flame investigated numerically for the different mixture degrees. One-step methane–air chemistry adequate to reach and lean mixture combustion was accepted. Velocity of triple flame is determined as a function of methane concentration logarithm gradients μ = d(ln Y₁)/dx (characterizing mixing degree). It is found that maximum velocity of the triple flames correspond to the value of the methane concentration logarithm gradients μ ≈ 1000 m^?1 for plain and μ ≈ 2000 m^?1 for axis-symmetrical channels. The maximum velocity of triple flame in plain and axis-symmetrical channels in the case of non-gradient incoming gas flow is about twice bigger than normal laminar flame velocity S_f ≈ 2.1S_l.     

Spray water cooling heat transfer at high temperatures and liquid mass fluxes

Spray water cooling is an important technology used in industry for the cooling of materials from temperatures up to 1800 K. The heat transfer coefficient in the so-called steady film boiling regime is known to be a function of the water impact density. Below a specific surface temperature T_L, the heat transfer coefficient shows a strong dependence on temperature (Leidenfrost effect). These findings are the results of complex self-organizing two-phase boiling heat transfer phenomena.The heat transfer coefficient was measured by an automated cooling test stand (instationary method) under clean (non-oxidizing) surface conditions. Compared to the common thought, an additional temperature dependency in the high temperature regime was found. The heat transfer from the material to the outflowing spray water is explained by a simple model of the two-phase flow region. From the experimental data, an analytic correlation for the dependence of the heat transfer coefficient α as an analytic function of water impact density V_S and temperature ΔT is provided.For water temperatures around 291 K, surface temperatures between 473 and 1373 K, i.e. ΔT > 180 K and water impact densities between V_S = 3 and 30 kg/(m² s) the heat transfer coefficient α was measured. The spray was produced with full cone nozzles (v_d ≈ 13–15 m/s, d_d ≈ 300–400 μm).     

A self-adaptive LGSM to recover initial condition or heat source of one-dimensional heat conduction equation by using only minimal boundary thermal data

The present study is concerned with the recovery of an unknown initial condition for a one-dimensional heat conduction equation by using only the usual two boundary conditions of the direct problem for heat equation. The algorithm assumes a function for the unknown initial condition and derives an inverse problem for estimating a spatially-dependent heat source F(x) in T_t(x, t) = T_xx(x, t) + F(x). A self-adaptive Lie-group shooting method, namely a Lie-group adaptive method (LGAM), is developed to find F(x), and then by integrations or by solving a linear system we can extract the information for unknown initial condition. The new method possesses twofold advantages in that no a priori information of unknown functions is required and no extra data are needed. The accuracy and efficiency of present method are confirmed by comparing the estimated results with some exact solutions.     

Analysis of entropy generation on mixed convection in square enclosures for various horizontal or vertical moving wall(s)

Entropy generation during the mixed convection process have been studied in a square enclosure for various moving horizontal (cases 1a–1d) or vertical wall(s) (cases 2a–2c) where the bottom wall of the cavity is isothermally hot, side walls are cold, and the top wall is adiabatic. Simulations have been performed for Prandtl number Pr = 0.026 and 7.2, Reynolds number Re = 10 − 100, and Grashof number Gr = 10³ − 10⁵. Results show that, in the case of the horizontally moving wall(s) (cases 1a–1d), the overall heat transfer rate $\left(\overline{N{u}_b}\right)$ and total entropy generation (S_total) are identical for cases 1a–1d and the cup-mixing temperature (θ_cup) is high for case 1b at Pr = 0.026, Re = 100, and Gr = 10⁵. Similarly, in the case of the vertically moving wall(s) (cases 2a–2c), $\overline{N{u}_b}$ and S_total are identical for cases 2a–2c with the maximum θ_cup occurring for the case 2a. At Pr = 7.2, Gr = 10⁵, and Re = 10, case 1a and case 1c are preferable for horizontally moving wall(s) and either of case 2a–2c is preferable for vertically moving wall(s). At Pr = 7.2, Gr = 10⁵, and Re = 100, case 1d may be preferable for the horizontally moving wall(s) and case 2a may be preferable for the vertically moving wall(s).     

Wide-optical bandgap with improved conductivity p-μc-Si:Ox:H films prepared by Cat-CVD

Boron-doped hydrogenated microcrystalline silicon oxide (p-μc-Si:O_x:H) films have been deposited using catalytic chemical vapor deposition (Cat-CVD). The single-coiled tungsten catalyst temperature (T_fil) was varied from 1850 to 2100 °C and films were deposited on glass substrates at the temperatures (T_sub) of 100–300 °C. Different catalyst-to-substrate distances of 3–5 cm and deposition pressures from 0.1 to 0.6 Torr were considered.Optical and electrical characterizations have been made for the deposited samples. The sample transmittance measurement shows an optical-bandgap (Eg_opt) variation from 1.74 to 2.10 eV as a function of the catalyst and substrate temperatures. One of the best window materials was obtained at T_sub=100 °C and T_fil=2050 °C, with Eg_opt=2.10 eV, dark conductivity of 3.0×10^?3 S cm^?1 and 0.3 nm s^?1 deposition rate.     

Streamwise interaction of burning drops

The spatial distribution of drops and their interactions are influential parameters in spray combustion. Most available researches on this subject were about lateral spacing or were performed in micro-gravity. Studies about upstream/downstream convective interaction of burning drops are scarce. In this study, drop strings of different spacing were investigated in a high-temperature oxidizing environment for their flame transition, flame width variation and drop evaporation rate. The flame transition showed that along the flow direction, the drop flame initially located ahead of the drop, became a spherical envelope flame, then moved behind the drop, and finally burned as a wake flame. It was found that a drop string with an initial drop spacing (S_i) of 2.5 or 5 was surrounded by a bulk flame tube, exhibited local group burning and soot layer. In addition, for S_i = 2.5, spacing instability and collision merging of the burning drops occurred; the wake flame stretched away from the drop could attach to and stabilize on the rear drop. In the experiment, for all cases, most of drops in the string were not surrounded by the flame. For S_i < 30, the drop evaporation rate was lower than that of a single drop. For 30 < S_i < 75, the drop evaporation rate was higher than that of a single drop. The interaction of drops diminished if S_i was more than 75.     

Hydrogen uptake characteristics of mischmetal based alloy

Hydrogen storage properties of Mm_39.2Ni_42.1Mn_4.9Al_1.25Co_10.2Fe_2.35 alloy have been systematically studied in the present work. An attempt is made to relate the content of hydrogen with change in resistance. It is found that the resistance of material increases with the increase in value of H/M due to hydrogen absorption. Pressure composition (P-C-T) isotherm using water displacement method has been investigated in the temperature and pressure ranges of 308 ≤ T ≤ 338 K and 0.5 ≤ P ≤ 10 bar, respectively. The P-C isotherms show the presence of two single α and β regions one mixed α + β phase. The maximum H (wt%) was found to be around 1.53 at 308 K and around 6 bar. Since enthalpy is an index of thermochemical stability of metal hydride the thermo dynamical parameters viz., the relative partial molar enthalpy (ΔH) and relative partial molar entropy (ΔS) of dissolved hydrogen have been calculated by plotting the Van’t Hoff plot. The variation of ΔH and ΔS with the hydrogen concentration confirm the phase boundaries.     

Dual mixed convection flows in a vertical channel

The classical problem of the fully developed mixed convection flow with frictional heat generation in a vertical channel bounded by isothermal plane walls having the same temperature is revisited in this paper. The existence of dual solutions of the local balance equations is pointed out. They are either columnar upflows or cellular down–up–down flows. Below a maximum value Ξ_max of the governing parameter Ξ = Ge Pr Re (the product of the Gebhart, Prandtl and Reynolds numbers), for any given Ξ a pair of different solutions occurs. The value Ξ_max corresponds to a maximum value of the Reynolds number above which no laminar solution can be found. At this maximum value, the two solution branches bifurcate from each other. In the neighborhood of the bifurcation point Ξ_max even small perturbations can cause transitions from one flow regime to the other. In the paper, the mechanical and thermal characteristics of the dual flow regimes are discussed in detail both analytically and numerically.     

Influence of natural convection on the melting of ice block surrounded by water on all sides

The ice block at initial temperature T_is = 0 °C is fixed at the center of a long, prismatic enclosure with isothermal vertical walls and insulated horizontal walls. The enclosure is completely filled with water at initial temperature T_il = 0 °C. Six numerical simulations were performed by varying vertical wall temperatures from T_W = 2 to 12 °C (range of Rayleigh number from 4.22 × 10⁶ to 2.28 × 10⁷). In the case of T_W > 8 °C the ice melts faster from above and for T_W < 8 °C from below. In the case of T_W = 8 °C, two vortices are separated by nearly vertical 4 °C isotherm and the average Nusselt number remains constant during the convection dominated regime.     

The effect of cycle boundary conditions and adsorbent grain size on the water sorption dynamics in adsorption chillers

The aim of this work was an experimental study of the temporal evolution of isobaric adsorption uptake (release) for simplest configuration of an adsorbent-heat exchanger unit, namely, a monolayer of loose adsorbent grains located on a metal plate. The study was performed by a large temperature jump method at four various boundary conditions of an adsorptive heat transformation cycle typical for air-conditioning application driven by low temperature heat: T_e = 5 and 10 °C, T_c = 30 and 35 °C and T_HS = 80 °C. The size of the Fuji silica grains was varied from 0.2 to 1.8 mm to investigate its effect on water sorption dynamics. For each boundary set and grain size the experimental kinetic curve could be described by an exponential function up to 80–90% of the equilibrium conversion. Desorption runs are found to be faster than appropriate adsorption runs by a factor of 2.2–3.5, hence, for optimal durations of the isobaric ad- and desorption phases of the chilling cycle should be selected accordingly. The size R of the adsorbent grains was found to be a powerful tool to manage the dynamics of isobaric water ad-/desorption. For large grains the characteristic time was strongly dependent on the grain size and proportional to R². Much less important appeared to be an impact of the boundary conditions which variation just weakly affected the dimensionless kinetic curves for the four tested cycles. The maximal specific cooling/heating power was proportional to the maximal temperature difference ΔT and the contact area S between the layer and the metal plate, and can exceed 10 kW/kg. The heat transfer coefficient α estimated from this power was as large as 100–250 W/(m² K) that much exceeds the value commonly used to describe the cycle dynamics.     

Pool boiling of R-123/oil mixtures on enhanced tubes having different pore sizes

The effect of enhanced geometry (pore diameter, gap width) is investigated on the pool boiling of R-123/oil mixture for the enhanced tubes having pores with connecting gaps. Tubes having different pore diameters (and corresponding gap widths) are specially made. Significant heat transfer degradation by oil is observed for the present enhanced tubes. At 5% oil concentration, the degradation is 26–49% for T_sat = 4.4 °C. The degradation increases 50–67% for T_sat = 26.7 °C. The heat transfer degradation is significant even with small amount of oil (20–38% degradation at 1% oil concentration for T_sat = 4.4 °C), probably due to the accumulation of oil in sub-tunnels. The pore size (or gap width) has a significant effect on the heat transfer degradation. The maximum degradation is observed for d_p = 0.20 mm tube at T_sat = 4.4 °C, and d_p = 0.23 mm tube at T_sat = 26.7 °C. The minimum degradation is observed for d_p = 0.27 mm tube for both saturation temperatures. It appears that the oil removal is facilitated for the larger pore diameter (along with larger gap) tube. The highest heat transfer coefficient with oil is obtained for d_p = 0.23 mm tube, which yielded the highest heat transfer coefficient for pure R-123. The optimum tube significantly (more than 3 times) outperforms the smooth tube even with oil. The heat transfer degradation increases as the heat flux decreases.     

Sintering and electrical properties of Ce0.8Y0.2O1.9 powders prepared by citric acid-nitrate low-temperature combustion process

Ce_0.8Y_0.2O_1.9 nanopowders were prepared using a citric acid-nitrate low-temperature combustion process. The effect of pH value of the solutions on the ionization of citric acid and the chelating of metal ions was studied. It was found that when the pH value of the solutions was bigger than 6, the citric acid was completely ionized and the stable complexes were formed between rare earth metal ions and the citric acid. Then the stable gels were obtained. The effect of the amount of oxidants (Φ) on the combustion fashions and combustion reaction time of the gels, the properties of the as-synthesized powders, the sintering behavior of the as-synthesized powders and the conductivity of the sintered pellets was also investigated. The results showed that the green density, sintered density and the ionic conductivity of the specimens increased with Φ. When Φ was equal to 1.5, the powders with good dispersion and compressibility were obtained, and the green density of the powders was 52.5%. The relative density of the sintered sample was over 95% at 1350 °C for 4 h and the conductivity of the sintered specimen was 0.034 S cm⁻¹ at 700 °C.     

Optoelectronic properties of chemically deposited Bi2S3 thin films and the photovoltaic performance of Bi2S3/P3OT solar cells

Thin films of bismuth sulfide (Bi₂S₃), prepared on conductive tin-doped indium oxide (ITO)-glass substrates by chemical deposition showed a variation of optical band gap with thickness: 1.8 eV for a 50 nm film (deposited at 40 °C for 30 min) to 1.5 eV for a 200 nm film deposited for 2 h. The electronegativity for Bi₂S₃ compound is 5.3 eV, as estimated from the ionization energy and electron affinity of elemental Bi and S, and thus the electron affinity of chemically deposited Bi₂S₃ film was deduced to be 4.5 eV. In the energy level analysis of ITO/Bi₂S₃/P3OT/Au structure, Bi₂S₃ was established as an electron acceptor. To produce ITO/Bi₂S₃/P3OT/Au solar cell structures, poly3-octylthiophene (P3OT), prepared in the laboratory was dissolved in toluene and was drop-cast on the Bi₂S₃ film and a gold film was thermally evaporated. Under 100 mW/cm² tungsten-halogen irradiation incident from the ITO-side, the cell using a Bi₂S₃ film with thickness of 50 nm has the highest open circuit voltage (V_oc) of 440 mV and short-circuit current density (J_sc) of 0.022 mA/cm². The addition of a CdS thin film (90 nm) between ITO and B₂S₃ would increase V_oc to 480 mV, and J_sc to 0.035 mA/cm². The role of surface morphology and optoelectronic properties of the Bi₂S₃ film in the photovoltaic performance of the junction is discussed.