Effective Thermal Conductivity Equation Research Articles

Effect of radiation on the effective thermal conductivity of encapsulated capsules containing high-temperature phase change materials

Abstract The use of packed beds containing encapsulated capsules can markedly improve the efficiency of latent heat thermal energy storage systems. The capsule effective thermal conductivity is a crucial parameter for modelling the melting process within the capsule and to investigate the thermal performance of the packed bed. This study simulated the effect of radiation on the melting process of encapsulated high-temperature molten salt using a numerical model. Results showed that radiative processes inside the molten salt could accelerate the melting process. When comparing simulations that incorporated radiation compared to those that did not, melting time differed by as much as 18% because the radiation changed the temperature distribution of the liquid molten salt and enhanced heat transfer. The effect of radiation on the molten salt melting process was then qualitatively analyzed using dimensionless numbers. When the ratios of conductive and radiative heat transfer N were 0.91, 1.62, and 2.24, the effect of radiation became less significant and the time required to complete the melting process was reduced by 25%, 19%, and 7%, respectively. An equation for effective thermal conductivity considering radiative heat transfer in encapsulated capsules was derived, which was valid within a limited range.

Thermal Conduction Simulation Based on Reconstructed Digital Rocks with Respect to Fractures

Effective thermal conductivity (ETC), as a necessary parameter in the thermal properties of rock, is affected by the pore structure and the thermal conduction conditions. To evaluate the effect of fractures and saturated fluids on sandstone’s thermal conductivity, we simulated thermal conduction along three orthogonal (X, Y, and Z) directions under air- and water-saturated conditions on reconstructed digital rocks with different fractures. The results show that the temperature distribution is separated by the fracture. The significant difference between the thermal conductivities of solid and fluid is the primary factor influencing the temperature distribution, and the thermal conduction mainly depends on the solid phase. A nonlinear reduction of ETC is observed with increasing fracture length and angle. Only when the values of the fracture length and angle are large, a negative effect of fracture aperture on the ETC is apparent. Based on the partial least squares (PLS) regression method, the fluid thermal conductivity shows the greatest positive influence on the ETC value. The fracture length and angle are two other factors significantly influencing the ETC, while the impact of fracture aperture may be ignored. We obtained a predictive equation of ETC which considers the related parameters of digital rocks, including the fracture length, fracture aperture, angle between the fracture and the heat flux direction, porosity, and the thermal conductivity of saturated fluid.

Modeling of the temperature field in a porous thermal barrier coating

Abstract Thermal barrier coatings (TBCs) are advanced materials systems with low thermal conductivity. One of the reasons for the low thermal conductivity in TBCs is that they contain porous structures created by a network of micro-voids. In the present investigation, experimental and analytical studies of heat transfer in TBCs having different levels of porosity were performed. The ceramic coatings were analyzed using scanning electron microscopy to calculate the level of porosity and micro-pore size distribution. A two-dimensional FE model was then developed, where a stochastic method was used to define randomly distributed porous structures equivalent to porosities of 1%, 3%, and 5%. The results showed that the heat flux and temperature gradient were affected by the interactions between neighboring micro-pores and micro-pores/ceramic coatings, and that the effect of the micro-pores was limited to a small area (2–2.5 times the micro-pore radius). Based on the obtained results, a set of effective thermal conductivity equations are proposed which more clearly describe the heat transfer process in a porous TBC structure. Two different equivalent thermal resistance models were used to study the heat transfer process under low porosity ( 3%) conditions.

Study on AZ91D chip for injection molding (2nd Report, Effect of dimension on effective thermal conductivity after consolidation)

The purpose of this study is to propose an effective model to estimate the effective thermal conductivity of compacts made from magnesium alloy particles, considering the effects of particle size, temperature and pressure. In order to investigate the influence of the particle size, six kinds of particles were prepared in which the cross sectional area was different. The test particles made from AZ91D were poured into a compression vessel and then a given pressure was applied to the piston of vessel for six hours under constant temperature condition. After compression, effective thermal conductivity of compacts were measured by steady state comparative-longitudinal heat flow method. The numerical calculation considering two-dimensional steady heat conduction was also conducted by using the actual cross section of compacts. Test results showed that the effective thermal conductivity of compacts were increased with increase of applied pressure and temperature elevation. Calculation results also revealed that by considering the actual cross section of compacts, the effective thermal conductivity could be well estimated when filling fraction is lower than 0.8. However, when filling fraction is exceed 0.8, the effect of thermal contact resistance between particles on thermal conductivity is not negligible. The experimental equation of effective thermal conductivity of compacts was determined to discuss the effects of cross sectional area of particles.

Energy conserving dissipative particle dynamics study of phonon heat transport in thin films

Phonon heat transport in thin films is solved by using dissipative particle dynamics with energy conserving (eDPD), and the size effect on thermal conductivity is analyzed. When the thickness of a film is comparable with the phonon mean free path, boundary scattering becomes dominant. To incorporate both the phonon–phonon scattering effect and phonon-boundary scattering effect, we adopt a particle–particle collision flux model and a particle–wall collision flux model in eDPD. Using these models, we investigated the heat transport between two parallel and infinite plates for 0.1<Kn<10 where the heat transport is both ballistic and diffusive. The temperature jump is observed at the boundaries and the temperature profiles are in good agreement with the solution to the equation of phonon radiative transport (EPRT). Moreover, we vary the thickness of the film and calculate the corresponding effective thermal conductivity to investigate the size influence. The present results are compared with the simple analytical solution based on Matthiessen’s rule and the effective thermal conductivity equation derived from the EPRT and it is found that the effective thermal conductivity can be predicted correctly by eDPD.

Effective Thermal Conductivity of Open Cell Polyurethane Foam Based on the Fractal Theory

Based on the fractal theory, the geometric structure inside an open cell polyurethane foam, which is widely used as adiabatic material, is illustrated. A simplified cell fractal model is created. In the model, the method of calculating the equivalent thermal conductivity of the porous foam is described and the fractal dimension is calculated. The mathematical formulas for the fractal equivalent thermal conductivity combined with gas and solid phase, for heat radiation equivalent thermal conductivity and for the total thermal conductivity, are deduced. However, the total effective heat flux is the summation of the heat conduction by the solid phase and the gas in pores, the radiation, and the convection between gas and solid phase. Fractal mathematical equation of effective thermal conductivity is derived with fractal dimension and vacancy porosity in the cell body. The calculated results have good agreement with the experimental data, and the difference is less than 5%. The main influencing factors are summarized. The research work is useful for the enhancement of adiabatic performance of foam materials and development of new materials.

New theoretical equation for effective thermal conductivity of two-phase composite materials

A new theoretical equation that describes the thermal conductivity of a two-phase composite is proposed. The Cheng–Vachon equation has been modified, and a new parameter, i.e. the modified volume fraction, is introduced. The modified volume fraction is a function of the physical volume fraction and thermal conductivity ratio of constituent phases. It was found that the new equation can describe the thermal conductivity of different two-phase composites, such as Cu/solder, Al/water, Al/air and cellular concrete, very accurately.

Effective thermal conductivity of fluid-saturated rocks: Experiment and modeling

Effective thermal conductivity (ETC) of dry, gas-, oil-, and water-saturated rocks with various porosities has been measured over a temperature range from 273K to 523K at atmospheric pressure with a steady-state guarded parallel-plate apparatus. The expanded uncertainty of thermal conductivity and temperature measurements at the 95% confidence level with a coverage factor of k=2 were estimated to be 4% and 30mK, respectively. This uncertainty in ETC measurement does not include the uncertainty due to contact thermal resistance and radiative conductivity. The temperature coefficients, (∂lnλ/∂T)P, for fluid-saturated rocks were calculated by using the measured ETC. We interpreted measured ETC data for fluid-saturated rocks using various theoretical models in order to check their accuracy, predictive capability, and applicability. The effect of saturating fluids, structure (size, shape, and distribution of the pores), porosity, and mineralogical composition on temperature and porosity dependences of the ETC of fluid-saturated rocks was discussed. A new simple equation for ETC of fluid-saturated rocks which takes into account structure of porous media has been proposed. Using the Hofmiester model and measured thermal conductivities of dry rock materials, the values of thermodynamic properties (density, thermal expansion coefficient, enthalpy, and heat capacity) were predicted.

A New Theoretical Equation for Effective Thermal Conductivity of Cellular Concrete

A new theoretical equation that represents the thermal conductivity of two-phase composite has been proposed. The Cheng-Vachon equation has been modified by introducing a new parameter that is the corrected porosity. It was found that the new equation can describe the thermal conductivity of cellular concrete very accurately. Development of the equation is helpful to understanding heat transfer mechanism and improving thermal property of cellular concrete.

Predicting effects of blood flow rate and size of vessels in a vasculature on hyperthermia treatments using computer simulation

BackgroundPennes Bio Heat Transfer Equation (PBHTE) has been widely used to approximate the overall temperature distribution in tissue using a perfusion parameter term in the equation during hyperthermia treatment. In the similar modeling, effective thermal conductivity (Keff) model uses thermal conductivity as a parameter to predict temperatures. However the equations do not describe the thermal contribution of blood vessels. A countercurrent vascular network model which represents a more fundamental approach to modeling temperatures in tissue than do the generally used approximate equations such as the Pennes BHTE or effective thermal conductivity equations was presented in 1996. This type of model is capable of calculating the blood temperature in vessels and describing a vasculature in the tissue regions.MethodsIn this paper, a countercurrent blood vessel network (CBVN) model for calculating tissue temperatures has been developed for studying hyperthermia cancer treatment. We use a systematic approach to reveal the impact of a vasculature of blood vessels against a single vessel which most studies have presented. A vasculature illustrates branching vessels at the periphery of the tumor volume. The general trends present in this vascular model are similar to those shown for physiological systems in Green and Whitmore. The 3-D temperature distributions are obtained by solving the conduction equation in the tissue and the convective energy equation with specified Nusselt number in the vessels.ResultsThis paper investigates effects of size of blood vessels in the CBVN model on total absorbed power in the treated region and blood flow rates (or perfusion rate) in the CBVN on temperature distributions during hyperthermia cancer treatment. Also, the same optimized power distribution during hyperthermia treatment is used to illustrate the differences between PBHTE and CBVN models. Keff (effective thermal conductivity model) delivers the same difference as compared to the CBVN model. The optimization used here is adjusting power based on the local temperature in the treated region in an attempt to reach the ideal therapeutic temperature of 43°C. The scheme can be used (or adapted) in a non-invasive power supply application such as high-intensity focused ultrasound (HIFU). Results show that, for low perfusion rates in CBVN model vessels, impacts on tissue temperature becomes insignificant. Uniform temperature in the treated region is obtained.ConclusionTherefore, any method that could decrease or prevent blood flow rates into the tumorous region is recommended as a pre-process to hyperthermia cancer treatment. Second, the size of vessels in vasculatures does not significantly affect on total power consumption during hyperthermia therapy when the total blood flow rate is constant. It is about 0.8% decreasing in total optimized absorbed power in the heated region as γ (the ratio of diameters of successive vessel generations) increases from 0.6 to 0.7, or from 0.7 to 0.8, or from 0.8 to 0.9. Last, in hyperthermia treatments, when the heated region consists of thermally significant vessels, much of absorbed power is required to heat the region and (provided that finer spatial power deposition exists) to heat vessels which could lead to higher blood temperatures than tissue temperatures when modeled them using PBHTE.

A new heat transfer model of inorganic particulate-filled polymer composites

Thermal conductivity is an important parameter for characterization of thermal properties of materials. Various complicated factors affect the thermal conductivity of inorganic particulate-filled polymer composites. The heat transfer process and mechanisms in an inorganic particulate-filled polymer composite were analyzed in this article. A new theoretical model of heat transfer in these composites was established based on the law of minimal thermal resistance and the equal law of the specific equivalent thermal conductivity, and an relevant equation of effective thermal conductivity (Keff) for describing a relationship between Keff and filler volume fraction as well as other thermal parameters were derived based on this model. The values of Keff of aluminum powder-filled phenol–aldehyde composites and graphite powder-filled phenol–aldehyde composites were estimated by using this equation, and the calculations were compared with the experimental measured data from these composites with filler volume fraction from 0 to 50% in temperature range of 50–60 °C and the predictions by Maxwell–Eucken equation and Russell equation. The results showed that the predictions of the Keff by this equation were closer to the measured data of these composites than the other equations proposed in literature.

A new structural model of effective thermal conductivity for heterogeneous materials with co-continuous phases

A new structural model for a heterogeneous material with multiple continuous phases is proposed. The corresponding equation for effective thermal conductivity was derived using three methods. The new model is substantially different from the conventional five fundamental structural models (Series, Parallel, two forms of Maxwell–Eucken, Effective Medium Theory). The model has two applications. First, as a new fundamental structural model to produce composite models using the combinatory rules previously proposed by J.F. Wang, J.K. Carson, M.F. North, D. J. Cleland, A new approach to modelling the effective thermal conductivity of heterogeneous materials, International Journal of Heat and Mass Transfer, 49 (17–18) (2006) 3075–3083. Second, to narrow the bounds of the effective thermal conductivity for heterogeneous materials where the physical structure can be characterised into general classes.

Heat transfer in polymer composites filled with inorganic hollow micro-spheres: A theoretical model

The heat transfer mechanisms in inorganic hollow micro-spheres filled polymer composites are analyzed in the present paper. This heat transfer includes mainly three mechanisms: (1) thermal conduction between solid and gas; (2) thermal radiation between the hollow micro-sphere surfaces; and (3) natural thermal convection of the gas in the micro-hollow spheres. A theoretical model of heat transfer in polymer/inorganic hollow micro-sphere composites is established based on the law of minimal thermal resistance and the equal law of the specific equivalent thermal conductivity, and a corresponding equation of effective thermal conductivity is derived. The effective thermal conductivity ( k eff ) of hollow glass bead-filled polypropylene composites is estimated by using this equation, and is compared with the numerical simulations by means of a finite element method. The results show that the variation of the theoretical estimations of k eff are similar to the numerical simulations at lower filler volume fraction ( φ f ⩽20%). Moreover, k eff decreases linearly with increasing φ f , and reduces somewhat with increase of filler size.

Local heating of human skin by millimeter waves: Effect of blood flow

We investigated the influence of blood perfusion on local heating of the forearm and middle finger skin following 42.25 GHz exposure with an open ended waveguide (WG) and with a YAV mm wave therapeutic device. Both sources had bell-shaped distributions of the incident power density (IPD) with peak intensities of 208 and 55 mW/cm(2), respectively. Blood perfusion was changed in two ways: by blood flow occlusion and by externally applied vasodilator (nonivamide/nicoboxil) cream to the skin. For thermal modeling, we used the bioheat transfer equation (BHTE) and the hybrid bioheat equation (HBHE) which combines the BHTE and the scalar effective thermal conductivity equation (ETCE). Under normal conditions with the 208 mW/cm(2) exposure, the cutaneous temperature elevation (DeltaT) in the finger (2.5 +/- 0.3 degrees C) having higher blood flow was notably smaller than the cutaneous DeltaT in the forearm (4.7 +/- 0.4 degrees C). However, heating of the forearm and finger skin with blood flow occluded was the same, indicating that the thermal conductivity of tissue in the absence of blood flow at both locations was also the same. The BHTE accurately predicted local hyperthermia in the forearm only at low blood flow. The HBHE made accurate predictions at both low and high perfusion rates. The relationship between blood flow and the effective thermal conductivity (k(eff)) was found to be linear. The heat dissipating effect of higher perfusion was mostly due to an apparent increase in k(eff). It was shown that mm wave exposure could result in steady state heating of tissue layers located much deeper than the penetration depth (0.56 mm). The surface DeltaT and heat penetration into tissue increased with enlarging the irradiating beam area and with increasing exposure duration. Thus, mm waves at sufficient intensities could thermally affect thermo-sensitive structures located in the skin and underlying tissue.

Effect of effective tissue conductivity on thermal dose distributions of living tissue with directional blood flow during thermal therapy

This study proposes a modified transient bioheat transfer equation based on combing the porous medium property and the scalar effective thermal conductivity equation in order to include the directional effect of blood flow. By applying the porous medium model to describe the collective behavior of the heat transfer in living tissue with many small blood vessels, an analytical solution can be obtained by Green's function. Simulation results quantitatively show that the blood perfusion rate, the averaged blood velocity, the porosity and the heating period are the crucial factors determining the distribution of thermal dose for thermal therapies. In addition, longer heating schemes induce dependencies of both the blood perfusion and the enhanced thermal conductivity on increased temperature.

Thermal conductivity and limiting oxygen index of basic rubber blend/aluminium hydroxide particulate composite

Measurements of the effective thermal conductivity and the limiting oxygen index of a basic rubber blend/aluminium hydroxide particulate composite have been presented and interpreted in the frame of the percolation theory. The experimental data of the thermal conductivity were fitted by the equation for effective thermal conductivity and from this fitting, the threshold percolation was obtained. From the fitting of the experimental data of the limiting oxygen index, the interval in which the threshold percolation lies was obtained. It was shown that the threshold percolation obtained from measurement of the effective thermal conductivity also lies in that interval.

Experimental evaluation of two simple thermal models using transient temperature analysis

Thermal models are used to predict temperature distributions of heated tissues during thermal therapies. Recent interest in short duration high temperature therapeutic procedures necessitates the accurate modelling of transient temperature profiles in heated tissues. Blood flow plays an important role in tissue heat transfer and the resultant temperature distribution. This work examines the transient predictions of two simple mathematical models of heat transfer by blood flow (the bioheat transfer equation model and the effective thermal conductivity equation model) and compares their predictions to measured transient temperature data. Large differences between the two models are predicted in the tissue temperature distribution as a function of blood flow for a short heat pulse. In the experiments a hot water needle, above ambient, delivered a 20 s heating pulse to an excised fixed porcine kidney that was used as a flow model. Temperature profiles of a thermocouple that primarily traversed the kidney cortex were examined. Kidney locations with large vessels were avoided in the temperature profile analysis by examination of the vessel geometry using high resolution computed tomography angiography and the detection of the characteristic large vessel localized cooling or heating patterns in steady-state temperature profiles. It was found that for regions without large vessels, predictions of the Pennes bioheat transfer equation were in much better agreement with the experimental data when compared to predictions of the scalar effective thermal conductivity equation model. For example, at a location mm away from the source, the measured delay time was s compared to predictions of 9.4 s and 5.4 s of the BHTE and ETCE models, respectively. However, for the majority of measured locations, localized cooling and heating effects were detected close to large vessels when the kidney was perfused. Finally, it is shown that increasing flow in regions without large vessels minimally perturbs temperature profiles for short exposure times; regions with large vessels still have a significant effect.

Ultrasonic heating with waveguide interstitial applicator array

Ultrasound interstitial applicators can be used for heating tumors near air and bone interfaces, where use of noninvasive ultrasound methods becomes difficult. We describe in this paper an ultrasonic waveguide three-applicator array for interstitial thermotherapy. We first discuss the temperature distribution and the pattern of heat deposition using the effective thermal conductivity equation. This equation is used then for finite element analysis of temperature modeling. We discuss heating in porcine brain tissue and present results with the array in a large volume tissue phantom. Simulation results agreed well with the experiment. The average difference between measured and simulation temperatures was <0.4/spl deg/C, less than the precision of temperature determination. Further modeling of temperature distribution in the human brain was based on the above simulations. We show that the arrays can be useful for thermotherapy in tissues with an appropriate choice of physical parameters of the applicators.

A generic tissue convective energy balance equation: Part I--theory and derivation.

A new equation for calculating temperatures in living tissues, the tissue convective energy balance equation (TCEBE), is derived using only a few assumptions. The resulting equation is basic, general and applicable to any tissue. The (unsolved) TCEBE is used: (a) to relate both Pennes' BHTE perfusion-related parameter (W) and the effective thermal conductivity equation's perfusion-related parameter (keff) to the true capillary perfusion Pcap, and (b) to show that both W and keff are defined, nonphysiological variables, which are only related to Pcap in a problem-dependent manner. Finally, the derivation of the relationship between W and Pcap provides a complete derivation of Pennes' BHTE, something that has not been previously done.

Blood flow cooling and ultrasonic lesion formation.

This article examines lesion formation using focused ultrasound and demonstrates how blood flow may affect lesion dimensions using a theoretical model. The effects of blood flow on temperature distributions during ultrasonic lesioning are examined for both regional cooling by the microvasculature and localized cooling due to thermally significant vessels. Regional cooling was critically assessed using two models: the Pennes bioheat transfer equation and the scalar effective thermal conductivity equation. Localized cooling was modeled by adding an advective term in the heat diffusion equation in regions enclosed by thermally significant vessels. A finite difference approach was used to solve the basic equations of heat transfer in perfused tissues in cylindrical coordinates. The extent of the lesioned tissue was determined by the accumulated thermal dose at each location. The size of the lesion was then calculated from the boundaries of the thermal isodose curves generated by the simulations. The results were compared to published in vivo lesion data in rat liver. It was shown that even for short ultrasound exposure times (approximately 8 s), blood flow may play an important role in the thermal dose distribution.