Unsteady-state Heat Conduction Equation Research Articles

Effect of Pulsation and Acceleration of Liquid Metal Turbulent Flow Through a Horizontal Channel by Large Eddy Simulation

Abstract Pulsation and acceleration of liquid metal turbulent flow through a horizontal channel has been numerically studied using a large eddy simulation (LES) technique. The effect of inlet pulsation and acceleration on flow and heat transfer characteristics of low Prandtl number liquid metal flow have been investigated and reported here. Results have been presented for different Reynolds numbers, different amplitudes, and frequency with constant bottom wall thickness. The flow field is modeled as unsteady-state two-dimensional incompressible turbulent-forced convection flow. Turbulence is modeled using a LES technique. Two-dimensional unsteady-state heat conduction equation is solved to know the temperature distribution in the solid region. Finite difference method solver is developed for solving the governing equations using sixth-order accuracy of compact schemes. The average Nusselt number shows cyclic variation with respect to time in pulsation flows. The enhancement of heat transfer with pulsation at amplitude 0.4 and frequency 100 Hz is 6.51%. The rate of heat transfer increases in pulsation flow compared to quasi-steady flow. The inlet acceleration shows a significant effect on flow characteristics. The present results are compared with direct numerical simulation (DNS) results available in the literature and matching well with DNS data.

Study of Heat Conduction inside Rolling Calender Nip for Different Roll Temperatures

Calendering is a smoothening process at the final stage in textile industry, where fabric passes through the nips formed by two or more rolls in contact. The material used for making of rolls varies for different types of calenders. Depending on the quality of fabric required these rolls are hard or soft and can be heated to certain temperature, using induction process, hot water passage or heated oil passage inside the rolls. In rolling calendering process, fabric is pressed between two or more rolls, where nip is formed by the combination of alternate hard and soft rolls. In this paper effect of roll temperature and bonding time on fabric temperature has been discussed using single nip rolling calender, when fabric is inside the calender nip formed by hard and soft rolls having different temperatures using one dimensional unsteady state heat conduction equation.

A study on thermal calculation method for a plastic greenhouse with solar energy storage and heating

Plastic greenhouse has been widely used in agriculture and horticulture due to its prolonging period for crops growth. In this work, an approach that stores solar energy in the daytime and provides heat by earth-tube at night was proposed, and then applied to a plastic greenhouse to elevate the inside air temperature. A one-dimensional dynamic model was established to assist the design of the solar energy storage and heating system and to evaluate the system performance. Using the model developed in Matlab, the date-hour change patterns of characteristic temperatures in the plastic greenhouse were obtained, through calculating the heat gains of various surfaces and heat storage by hour from solar radiation and solving the unsteady-state heat conduction equation in the structure components of the greenhouse. The calculated results show good agreement with the measured data, indicating that the method is valid and can be applied to the design of solar energy storage and heating system as well as the thermal performance analysis of greenhouses.

A Mathematical Modeling of Heat Transfer in Continuous Casting Slab

A two-dimensional (2-D) heat transfer and solidification model has been established and applied to calculate the temperature distribution and solid shell thickness profile of a continuous casting slab in a steel plant. A finite difference method was used for the numerical simulation. For thermal analysis, the 2-D slice unsteady-state heat conduction equation with enthalpy convention was used. Meanwhile, non-linear material properties of specific heat and thermal conductivity as well as phase changes during solidification were considered in the model. The temperature distribution and solid shell thickness calculated by mathematical model agree with those predicted by industrial and experimental measurements. The model could also be used to predict the optimum process parameters on casting speed, heat removal rates and the water distribution of secondary cooling zone.

Heating process simulation for a body subjected to an intensive thermal effect on its surface

An approximate method of calculating the nonstationary temperature fields for the case of rapid and intensive heating of bodies is considered. The fast heating of a body is simulated by various boundary conditions at its heated surface. An approximate method is proposed in a one-dimensional formulation to solve the unsteady-state heat conduction equation with these boundary conditions. The applicability of this method in practice is confirmed by comparing the approximate temperature fields with the exact solution. An approximate temperature field is constructed for a particular case with consideration of a temperature dependence of thermal parameters, which allows one to describe the heat localization in a heated zone.

Determination of thermal conductivity and emissivity of electromagnetically levitated high-temperature droplet based on the periodic laser-heating method: Theory

Recently, a novel method of measuring the thermophysical properties, especially thermal conductivity, of high-temperature molten materials using the electromagnetic levitation technique has been developed by [H. Fukuyama, H. Kobatake, I. Minato, K. Takahashi, T. Tsukada, S. Awaji, Establishment of noncontact AC calorimetry of high-temperature melts using solid platinum spheres as a reference, in: Proceedings of 16th Symposium on Thermophysical Properties, CD-ROM, 2006, p. 937; H. Kobatake, H. Fukuyama, I. Minato, T. Tsukada, S. Awaji, Noncontact AC calorimetry of liquid silicon with suppressing convections in a static magnetic field, in: Proceedings of 16th Symposium on Thermophysical Properties, CD-ROM, 2006, p. 625], where the method was based on periodic laser-heating, and a static magnetic field was superimposed to suppress convection in an electromagnetically levitated droplet. In the present work, the periodic laser-heating method was modeled to estimate the thermal conductivity and emissivity of the electromagnetically levitated droplet using a measured parameter, i.e., the phase lag between the modulated light and the temperature variations detected by a pyrometer, Δ ϕ s, at various frequencies of the modulated light ω. Here, the unsteady-state heat conduction equation for the droplet accompanying radiative heat transfer to the ambient was simplified and transformed to steady-state linear equations. The experimental relation between Δ ϕ s and ω was fitted by the mathematical model proposed here to estimate simultaneously the thermal conductivity and emissivity of molten silicon. Also, the numerical simulations for unsteady thermal field in the electromagnetically levitated droplet which was periodically laser-heated were carried out to demonstrate the validity of the proposed simplified model, and then to investigate the sensitivity of the thermophysical properties to the relation between Δ ϕ s and ω.

The Temperature Uniformity During Air Drying of a Colloidal Liquid Droplet

ABSTRACT The mathematical analysis for drying of individual droplets containing dissolved or suspended solids has been given significant importance due to the increasing popularity of the spray drying operation in the production of various chemicals, ceramics, drugs, food, and dairy powders as well as nanoparticles. The physical and biological qualities of the final products primarily depend on the history experienced by the droplet within the dryer. It is, therefore, desirable to ‘estimate’ the droplet's behavior and various characteristics such as moisture content and temperature profiles accurately. In the literature, several models have been presented to estimate moisture and temperature profiles inside a particle considering various assumptions. One common assumption is the uniform temperature distribution within the droplets being dried. The present article has presented an estimation procedure to evaluate the temperature distribution within a porous skim milk droplet to determine whether the uniform temperature distribution assumption is reasonable. Here, the surface-center temperature differences were estimated by considering the one-dimensional, unsteady-state heat conduction equation for a spherical droplet. Shrinkage of droplets was taken into consideration during modeling. A new concept of the Biot number has also been applied in the current article to assist in the determination of the rate-limiting process.

Temperature dependent electrical conductivities of fruit purees during ohmic heating

Ohmic heating takes its name from Ohm’s law; the food material switched between electrodes has a role of resistance in the circuit. In this study, the apricot and peach purees were heated on a laboratory scale static ohmic heater by applying voltage gradients in the range of 20–70 V/cm. The voltage gradient was statistically significant on the ohmic heating rates for both purees ( P < 0.05). The linear temperature dependent electrical conductivity relations were obtained. Bubbling was observed above 60 °C especially at high voltage gradients. The ohmic heating system performance coefficients were in the range of 0.49–1.00. The unsteady-state heat conduction equation for negligible internal resistance was solved with an ohmic heating generation term by the finite difference technique. The predictions of the mathematical model using obtained electrical conductivity equations were found to be very accurate.

Two-dimensional thermomechanical analysis of continuous casting process

In this work the two-dimensional analysis of continuous casting of low carbon steel was presented. The interaction between moved ingot, copper mould and transport rolls was modeling. The influence of liquid steel ferrostatic pressure and coupled between temperature and deformation fields were taken into consideration.For thermal analysis (with phase change), the two-dimensional unsteady-state heat conduction equation with enthalpy convention was used. For non-linear mechanical analysis the elasto-plastic material model with bilinear isotropic hardening was used. The Von Mises yield criterion was determined by the yield stress and the hardening modulus.For numerical solution the commercial system ANSYS®, which based on finite elements method was applied. The temperature distributions and thickness of solid shell were computed. Additionally, inside the solid shell, the distribution of stresses caused by thermal stresses and influence of mould and transport rolls were calculated.

The moving boundary problems from melting and freezing to drying and frying of food

It was possible to describe large number of processes such as melting, solidification, microwave thawing, spray-drying, and freeze-drying by a single theoretical model. All these applications and some others have one thing in common, that is the formation of a moving interface during the process of heat and mass transfer. This interface may be defined by a sharp boundary or by a region, depending on the application considered. The unsteady-state heat conduction equation was found to describe well the transport of heat in these processes. Two different numerical solutions have been developed, one for the sharp interface and the other for materials that undergo phase transformation within a range of temperatures. In spite of the large variation in the applications considered in this paper, the agreement was good between the theoretical predictions and our experiments, and some of the measurements available in the literatures.

Time evaluation for the steady state condition of the weight before the measurements

The temperature difference between the weight and its surrounding is an important parameter in the evaluation of the waiting time required by the weights to reach a steady state condition before starting a measurement, because the thermal gradients cause the thermal convection effect on the mass during the weighing process. In this study, investigation of both theoretical and experimental applications were carried out to evaluate the steady state condition of the weight before the calibration. Two kinds of unsteady state heat conduction equation were introduced, theoretical calculation results were compared to experimental results, and comparisons were very satisfactory for a 20 g weight. Then theoretical calculations were extended for different initial temperatures and for different weights. It was realized that the steady state condition of the waiting time was less dependent on initial temperature, but greatly dependent on the sizes of the weights.

Determination of the thermal conductivities of building and insulating materials by the probe method

A simple and inexpensive electrical circuit based on the transient probe method has been developed for the determination of thermal conductivities of building and insulating materials namely limebrick, gypsum, rockwool and polystrene foam. The solution of the unsteady state heat conduction equation has been approximated by considering certain simplifying assumptions which have been justified by experimental observations. The overall accuracy of the thermal conductivity measurements is estimated to be within ± 3%.

PREDICTION of MASS‐AVERAGE and SURFACE TEMPERATURES, and the TEMPERATURE PROFILES AT the COMPLETION of FREEZING FOR SHAPES INVOLVING ONE‐DIMENSIONAL HEAT TRANSFER

ABSTRACTA model for predicting the temperature profiles of simple‐shaped foodstuffs at the end of freezing was developed. It was shown that with appropriate selection of effective thermal diffusivity and initial temperature data, the standard solution of the unidimensional unsteady state heat conduction equation can be used in predicting the average and surface temperatures of infinite slabs at the end of freezing operation. For the calculation of average and surface temperatures in infinite cylinders and spheres, unsteady state solutions were corrected by an empirical factor that was derived from temperature profiles predicted by an accurate finite difference scheme. the temperature profiles calculated from the proposed model were compared with the predicted results obtained from a numerical model. Mean absolute errors between the predictions of the proposed model and the numerical model were 0.54C and 0.46C for average and surface temperatures, respectively.

Coupled drying and devolatilization of wet coal in fluidized beds

A single particle model is proposed for the coupled drying and devolatilization (under inert conditions) of coals in fluidized beds. The model assumes that heat transfer to and through the particle is the rate-controlling mechanism for drying. It is also assumed to be the rate-controlling mechanism for devolatilization in conjunction with the kinetics of overall coal decomposition. The drying is assumed to take place from a moving boundary (the drying front) and an unsteady-state heat conduction equation (in spherical coordinates) with a convective boundary condition is solved using approximate analytical techniques. The resultant temperature profile in the dried shell is used in a numerical integration over the volume of particle, using the non-isothermal coal decomposition kinetic expression proposed by Anthony et al. (1975) Proc. 15 th Symp. Combustion, to determine the devolatilization characteristics. After the drying is complete, the remaining possible volatile evolution at any particular bed temperature is obtained from a similar integration with the temperature profile determined from the analytical solution for the unsteady-state heat conduction equation with a convective boundary condition and a non-uniform initial temperature condition. Results of parametric analyses are presented along with the comparison of model predictions with data reported in the literature.

Analysis and simulation of a solar water pump for lift irrigation

A solar water pump for lift irrigation, which was shown to be economically viable, was proposed by Rao and Rao [5]. A “modified pump” is suggested, which is suitable for village water supply. The thermodynamic analysis of the pumps is presented. Though the solar water pump is intended to be operated with flat-plate collectors, it is analysed whether the pump could be run more efficiently when coupled with concentrating collectors. The analysis is also applicable for bellow actuated solar water pumps. Preliminary experimental studies showed that the heat losses are 2–3 times the theoretical energy requirement and the losses to the water tank shell accounted for a major part of the total heat losses. To reduce these losses, it is proposed that the inner surface of the water tank shell be lined with a resin bonded cork insulation. A method to evaluate the heat losses to the shell with insulation by solving the unsteady state heat conduction equation for a composite cylindrical body with time varying convective boundary conditions is presented. The heat losses are reduced to 5–15 per cent of the theoretical energy requirement with the use of the internal insulation. The major problem encountered in operating the pump was the inadequate condensation of the working fluid, which resulted in failure of the suction of water into the water tank. To ensure proper suction of water, the conditions to be maintained are analysed. The presence of water vapor and air in the spent vapor has to be taken into consideration while designing the condenser. An algorithm to evaluate the year round performance of the water-cooled pump for any location, given the lift and collector area, is presented. The cost of the water-cooled pump is compared with the costs of other types of solar water pumps and it is shown that the pump under consideration costs several times less than the other ones. The conditions at which the pump will be economical vis-a-vis diesel and electrical pumps are presented.

Heat and mass transfer bibliography—Soviet works

s of Papers and Communications of the 4th AllUnion Conference on Heat and Mass Transfer, Minsk ($972). S. F. CHISTYAKOV and D. V. RADUN, Thermal Me~ure~~nt~ and Apparatuses. Vysshaya Shkola, Moscow (1972). V. A. FEDOSEEV et al. (Eds.), Physics of Aerodispersed Systems. vyp. 5, (Collected Papers). Izd-vo Kievsk. Universiteta, Kiev (1971). Heat and Mass Transfer. Vol. 2, part 1, Heat and Mass Transfer involuing Physicaf-Chemical Trunsformations. Minsk (1972). Heat and Mass Trcmsfer. Vol. 2, part 2, Heat and Mass Transfer Inuoluing Physical-Chemical Transformations, Minsk (1972). Heat nnd Mass Transfer. Vol. 3.~ Heat and Mass Transfer in Rheological Systems. Minsk (1972). Heat and Mass Transfer. Vol. 4, Heat and Mass Transfer in Technological Processes and Apparatuses of Chemical Indllstry. Minsk (1972). Heat and Mass Transfer. Vol. 5, part 1, Heat and Mass Transfer in Dispersed Systems. Kiev (1972). Heat and Mass Transfer. Vol. 5, part 2, Heat and Mass Transfer in Dispersed Systems. Kiev (1972). Heat and Mass Transfer. Vol. 6, Heat and Mass Transfer in Capillary-Porous Bodies and Drying Processes. Minsk (1972). Heat and Mass Transfer. Vol. 7, Tr~sferab~e Substance Properties. Minsk (1972). Heat and Mass Trunsfer, Vol. 8, General Problems of Heat and Mass Transfer Theory. Minsk (1972). Heat and Mass Transfer. Vol. 9, part 1. Minsk (1972). L. A. KOZDOBA. Electrical Simulation. of Heat and Mass Transfer Phenomena. Energiya, Moscow (1972). A. V. LUIKOV (Ed.), Conve~fion in Channels (Collected Papers}. Minsk (1971). A. V. LUIKOV (Ed.), Drying and Evaporation at Production of Mineral Fertilizers (Collected Papers). Moscow (1971). A. I. BEXTINOV, B. L. ALIEVSKII, A. G. SHERSTYUK and T. S. DVORYADKINA, A temperature field in a disk-type armature of acyclic electrical machines, Izuest. AN SSSR. Energetiku i Transport No. 2, 131 (1972). V. N. ELISEEV, An unsteady-state temperature field of a semitransparent envelope with heat sources distributed irregularly, &vest. Vuzoo. Mushinostr. No. 12, 52 (1971). V. I. FEDOROV, On heat transfer mechanisms in electron melts, Teplofz. Vysok. Temper. 10(2), 290 (1972). A. I. GORBAN, V. A. PRYADKO~~~ A. 1. LUKIN, A simplified calculation method for unsteady-state heat fluxes ao cording to temperature oscillations in a wall, Trudy Nikol~le~sk. Korablestroit. In-ta, 42, 79 (1971). Yu. P. GUPALO, Yu. S. RYAZANT~EV and A. T. CHALYUK, On a temperature field, rising in motion of a reacting sphere at small finite P&let and Reynolds numbers, Zhurn. Prikl. Mekhan. i Tekhn. Fiziki No. 2, 59 (1972). V. A. GUREVICH, On a pulse shape of a periodical point heat source on a surface of a massive body, Fizika i Khimj.~a Ubr~botki Ma~erialou No. 2, 19 (1972). A. P. IVANOV, Unsteady-state thermal conditions in scattering media, Zhurn. Prikl. Spektroskopii 16(4), 709 (1972). V. K. IVANOV, A problem of quasiinversion for a heat conduction equation in uniform metrics, D@erenrs. Uravneniya 8(4). 65.2 (1972). E. Z. KOROL, Functions of a heat conduction equation source for a circular ring, ~aLlchn. Trudy In-m Mekhaniki MGU No. 8, 83 (1971). M. I. LESHCHINSKII, Calculation of an electric heater of a plate for a vulcanizing press with a uniform temperature field, Kuuchuk i Rezina No. 5, 23 (1972). A. D. LEZHNEV, Some particular case for solving an unsteady-state heat conduction equation with a movable boundary, Teplofiz. Vysok. Temper. 10(2), 430 (1972). B. A. LINEV, Solution of a problem of unsteady-state heat conduction with mixed boundary conditions, Nuuchn. Trudy In-ta Mekhaniki MGU No. 8, 75 (1971). E. A. LLJCIINIKGV, Study of a temperature field of a semiinfinite body from a movable elliptic heat source. Sborn. HEAT CONDUCTJON Nauchn. Trudoo Perm. Politekhn’ln-ta No. 84, 44 (1971). A. V. BASHKATOV. V. S. POSTNIKOV, F. N. RYZHKOV and R. M. PETRICHENKO and Yu. K. PROKHOROV, Calculation A. A. UGLOV, Determination of heat fields in welding by of the temperature state for ICE pistons, Energoa vibrating electronic beam, Fizika i Khimiyu Obrabotki mashjnostr. No. 3, 19 (1972). V. L. RVACHW and A. P. SLESARENKO, On solution of Materialov No. 2, 23 (1972). mixed boundary-value problems in different media for