Modelingofheattransfercoefficient inthe furnace of CFB boilers by artificial neural network approach

《传热传质学》主要内容和专业词汇中英文对照Chapter 1 Thermodynamics and Heat Transfer主要内容1.Concepts:heat (thermal energy)、heat transfer、thermodynamics、total amount of heat transfer、heat transfer rate、heat flux、conduction、convection、radiation2.Equations:1) The first law of thermodynamics (conservation of energy principle)2) Heat balance equation: a) closed system; b) open system (steady-flow)3) Fourier’s law of heat conduction4) Newton’s law of cooling5) Stefan-Boltzmann law主要专业词汇heat transfer 传热、热传递、传热学thermodynamics热力学caloric 热素specific heat 比热mass flow rate 质量流率latent heat 潜热sensible heat 显热heat flux热流密度heat transfer rate热流量total amount of heat transfer总热量conduction导热convection对流radiation辐射thermal conductivity 热导率thermal diffusivity 热扩散率convection/combined heat transfer coefficient 对流/综合换热系数emissivity 发射率absorptivity 吸收率simultaneous heat transfer 复合换热Chapter 2 Heat Conduction Equation主要内容3.Concepts:temperature field、temperature gradient、heat generation、initial condition、boundary condition、steady\transient heat transfer、uniform\nonuniform temperature distribution4.Equations:1) Fourier’s law of heat conduction (§2-1)2) Heat conduction equation (in rectangular\cylindrical\spherical coordinates)(§2-2、§2-3)3) Boundary conditions: (§2-4)a)Specified temperature B. C.b) Specified heat flux B. C. (special case: insulation、thermal symmetry);c) Convection B.C.d) Radiation B.C.e) Interface B.C.4) Average thermal conductivity k ave(§2-7)5) Solution of one-dimensional, steady heat conduction in plane walls、cylinders andspheres (k =const)：a) no heat generation, specified temperature B.C.: (§2-5)T(x) or T(r)；Q(x) or Q(r), Q=constb) with heat generation, Specified temperature B.C. or Convection B.C. : (§2-6)∆T max=T o-T s= gs2/2nk ; q(x)=gx/n; T s=T + gs/nhcharacteristic length S, shape factor n:plane walls — s = L (half thickness), n = 1cylinders ——s = r o, n = 2spheres ——s = r o, n =35.Methods: Solve a heat transfer problem1) Mathematical formulation (differential equation & B.C.)2) General solution of equation3) Application of B.C.s4) Unique solution of the problemtemperature field\distribution 温度场\分布 temperature gradient 温度梯度 heat generation 热生成(热源) initial\boundary condition 初始\边界条件 transient heat transfer 瞬态(非稳态)传热 isothermal surface 等温面 Heat conduction differential equation 导热微分方程trial and error method 试算法 iterate 迭代 convergence 收敛Chapter 3 Steady Heat Conduction主要内容6. Concepts:multilayer\composite wall overall heat transfer coefficient Uthermal resistance R t thermal contact resistance R ccritical radius of insulation R crfin efficiency fin effectiveness7. Equations:✓ Multiplayer plane wall 、cylinders and spheres: 12totalT T Q UA T R ∞∞-==∆& ✓ Fin: fin equation ——0)()(=--∞T T hp dxdT kA dx d c1) Uniform cross-section:2) Varying cross-section: )(fin fin max fin,fin fin ∞-==T T hA Q Q b ηη&&thermal resistance热阻parallel 并联in series串联thermal contact resistance 接触热阻composite wall 复合壁面thermal grease 热脂cross-section 横截面temperature execess 过余温度hyperbolic 双曲线的exponent 指数fin 肋(翅)片fin base 肋基fin tip 肋端fin efficiency 肋效率fin effectiveness 肋片有效度Chapter 4 Transient Heat Conduction主要内容8.Concepts:lumped system analysis characteristic length (L c=V/A)Biot number (Bi=hL c /k) Fourier number ( τ = at/L)9.Equations:●Bi≤0, lumped system analys is (§4-1)●Bi>0, Heisler/Grober charts OR analytical expressions1-D：a) infinite large plane walls, long cylinders and spheres (§4-2)b) semi-infinite solids (§4-3)multidimensional: product solution (§4-4)主要专业词汇lumped system analysis 集总参数法characteristic length 特征长度（尺寸）dimension 量纲nondimensionalize 无量纲化dimensionless quantity 无量纲量semi-infinite solid 半无限大固体complementary error function 误差余函数series 级数production solution 乘积解Chapter 5 Numerical Methods in Heat Conduction主要内容10.Concepts:control volume (energy balance) method、finite difference method、discretization、node、space step、time step、mesh Biot number、mesh Fourier number、mirror image concept、explicit/implicit method、stability criterion (primary coefficients ≥0)Numerical error: 1) discretization/truncation error; 2) round-off error11.Methods:Numerical solution:1) Discretization in space and time (∆x, ∆t);2) Build all nodes’ finite difference formulations (including interior and bou ndary nodes);i.Finite difference methodii.Energy balance method (i.e. Control V olume method)3) Solution of nodal difference eqs. of heat conduction;a)Direct method: Gaussian Eliminationb)Iterative method: Gauss-Seidel iteration主要专业词汇control volume 控制容积finite difference有限差分Taylor series expression泰勒级数展开式mirror image concept 镜像法Elimination method 消元法direct/iterative method 直接/迭代方法explicit/implicit method 显式/隐式格式stability criterion 稳定性条件primary coefficients 主系数unconditionally 无条件地algebraic eq. 代数方程discretization/truncation error 离散/截断误差round-off error 舍入误差Chapter 6、7 Forced Convection and Natural Convection主要内容12.Concepts:Nu、Re、Gr、PrForce/natural convection、external/internal flow、velocity/thermal boundary layer flow regimes、laminar/turbulent flowhydrodynamic/thermal entry region、fully developed regionCritical Reynolds Number (Re c)、hydraulic diameter (D h)、film temperature (T f)、bulk mean fluid temperature (T b)logarithmic mean temperature difference ( ∆T ln)volume expansion coefficient (β = 1/T)effective thermal conductivity (K eff = K Nu)evaporation、boiling、condensation、nucleation sitepool/flow boiling、subcooled / satuated boilingnatural convection / uncleate / transition /film boilingburn out point13.Equations:Drag force ：F D = C f AρV2/2Heat transfer rate：Q = hA(T s-T∞)3．Typical Convection Phenomena:1) Forced convection：external flow——flow over flat plates (§6-4)——flow across cylinders and spheres (§6-5)internal flow——flow in tubes (§6-6)2) Natural convection：flow over surfaces (§7-2)flow inside enclosures (§7-3)3) Boiling Heat Transfer主要专业词汇Force/natural convection 自然/强制对流laminar/turbulent flow 层/湍流boundary layer 边界层laminar sublayer 层流底层buffer layer 缓冲层transition region 过渡区flow regimes 流态inertia/viscous force 惯性/粘性力shear stress 剪切应力friction/drag coefficient 摩擦/阻力系数friction factor 摩擦因子dynamic/kinematic viscous 动力/运动粘度wake 尾流stagnation point 滞止点flow separation 流体分离vortex 漩涡rotational motion 环流velocity fluctuation 速度脉动hydrodynamic 水动力学的hydraulic diameter 水力直径fully developed region 充分发展段volume flow rate 体积流量arithmetic/logarithmic mean temperature difference 算术/对数平均温差volume expansion coefficient 体积膨胀系数interferometer 干涉仪asymptotic渐近线的effective thermal conductivity 有效热导率evaporation 蒸发boiling 沸腾condensation 凝结pool boiling池内（大容器）沸腾flow boiling（管内）强制对流沸腾subcooled / satuated boiling 过冷/饱和沸腾nucleation site 汽化核心uncleate / transition /film boiling 核态/过渡/膜态沸腾burnout point烧毁点surface tension 表面张力vapor bubble 汽泡excess temperature 过热度thermodynamic equilibrium 热力平衡phase change/transformation 相变latent heat of vaporization 汽化潜热analogical method 类比法integral approach 积分近似法order of magnitude analysis 数量级分析法similarity principle 相似原理Chapter 9 Radiation Heat Transfer主要内容14.Concepts:black body、gray body、diffuse surface、emissive power (E)emissivity (ε)、absorptivity (α)、reflectivity (ρ)、transmissivity (τ)irradiation(G)、radiosity(J)、reradiating(adiabatic) surfaceview factor (F ij)、radiation network、space resistance、surface resistance radiation shieldgas radiation、transparent medium to radiation、absorbing and transmitting medium ws:Blackbody：(1) Planck’s distrib ution law(2) Stefan-Boltzmann’s law(3) Wien’s displacement lawGraybody：(4) Kirchhoff’s lawActual body：E (T) = ε E b(T) = ε σT4W/m2Gas：(5) Beer’s law3．Calculation:1) View factor：reciprocity/summation/superposition/symmetry Rulecrossed-strings method2) Radiation heat transfer：Radiation networkOpen system：between two surface (e.g. two large parallel plates)Enclosure：2-surface enclosure；3-surface enclosureRadiation shield主要专业词汇thermal radiation热辐射、quantum theory量子理论、index of refraction 折射系数electromagnetic wave/spectrum 电磁波/波谱、ultraviolet (UV) rays紫外线、infrared (IR) rays 红外线absorptivity 吸收率、reflectivity 反射率、transmissivity 透射率、emissivity (ε) 发射率(黑度)、specular/diffuse reflection 镜反射/漫反射irradiation (incident radiation) 投入辐射、radiosity 有效辐射spectral/directional/total emissive power单色/定向/总辐射力fraction of radiation energy 辐射能量份额(辐射比)、blackbody radiation function 黑体辐射函数view factor 辐射角系数、crossed-strings method交叉线法、reciprocity/summation/superposition/symmetry Rule相互/完整/和分/对称性net radiation heat transfer 净辐射热流量radiation network 辐射网络图、space/surface radiation resistance 空间/表面辐射热阻、reradiating surface重辐射面、adiabatic 绝热的radiation shield遮热板transparent medium to radiation辐射透热体、absorbing and transmitting medium吸收-透过性介质Chapter 10 Heat Exchangers主要内容16.Concepts:heat exchanger type---- double-pipe、compact、shell-and-tube、plate-and-frame、regenerative heat exchangerparallel/counter/cross/multipass flowoverall heat transfer coefficient (U) fouling factor (R f)heat capacity rate capacity rationlog mean temperature difference (ΔT lm)heat transfer effectiveness (ε)number of transfer units (NTU)17.Equations:1) heat balance eq.: Q = C h (T h,in - T h,out)=C c(T c,out - T c,in)2) heat transfer eq.: Q = UAΔT lm( LMTD method)or Q = εQ max = εC min (T h,in –T c,in) ( ε-NTU method) 3．Methods:1) LMTD Method：Select a heat exchangerKnown: C h、C c、3‘T’Predict: 1‘T’、Q、A2) ε-NTU Method：Evaluate the performance of a specified heat exchangerKnown: C h、C c、UA、T h,in、T c,inPredict: Q、T h,out、T c,out主要专业词汇double-pipe/compact/shell-and-tube/plate-and-frame/regenerative heat exchanger套管式/紧凑式/壳管式/板式/蓄热（再生）式换热器parallel/counter/cross/multipass flow 顺流/逆流/叉流/多程流area density 面积密度tube/shell pass 管程/壳程static/dynamic type 静/动态型baffle 挡板header 封头nozzle管嘴guide bar 导向杆porthole 孔口gasket 垫圈lateral 侧面的/横向的fouling factor 污垢因子heat capacity rate 水当量heat transfer effectiveness (ε) 传热有效度number of transfer units (NTU) 传热单元数《传热传质学》知识难点与重点Chapter 1 Thermodynamics and Heat Transfer第一章热力学与传热学1．传热学研究内容（温差=>传热）；Heat Transfer Research (Temperature Difference=> Heat Transfer)2.三种基本传热方式的机理和基本公式；The Mechanisms and Basic Formulas of Three Basic Modes of Heat Transfer.3.传热过程、传热方程式；Heat Transfer Process，Heat Transfer Equation4.导热系数、对流换热系数、传热系数的物理涵义、单位、基本数量级、影响因素和变化规律；Physical meanings ,units, fundamental orders,influencing factors and changes in laws of heat conduction coefficient，convection heat transfer coefficient，heat transfer coefficient.5.热阻与热流网络图；Thermal resistance and heat transfer network6,单位与单位制；Unit and system of unitsChapter 2 Heat Conduction Equation第二章导热方程式1.导热问题的求解目标（物体内部的温度场与热流场）；Determine Target of Heat Conduction(temperature field and heat field in the internal objects)2.温度场（稳态、非稳态、均匀、一维、二维、三维）；Temperature field （steady,transient,uniform,one-dimensional,two-dimensional,three-dimensional）3.等温面、等温线、热流线的性质及相互关系；Properties of isothermal surface, isotherm,heat flow and the relationship among them 4.方向导数、梯度的数学概念及相互关系；Mathematical concept of directional derivative , gradient and the relationship between them5.Fourier 定律；Fourier Law6.推导导热微分方程式的理论基础、简化假设及方程各项（内能、导热、内热源、导温系数、）的物理涵义；Theoretical bases of concluding heat conduction differential equation,simplified assumption and physical meanings of each term in the equation (Internal energy, heat conduction, internal heat source,temperature transfer coefficient, )7.定解条件【几何、物理、时间、边界（Ⅰ、Ⅱ、Ⅲ）】Conditions of determining the solution【geometry,physics,time,boundary(Ⅰ、Ⅱ、Ⅲ )】8.导热问题的求解方法（解析解、数值解）。

1.Solar RadiationThe output of sun is 2.8×1023KW. The energy reaching the earth is 1.5×1018KWH/year. When light travels from outer space to earth, solar energy is lost because of following reasons:Scattering:The rays collide with particles present in atmosphereAbsorption:Because of water vapor there is absorption.Cloud cover:The light rays are diffused because of clouds.Reflection:When the light rays hit the mountains present on the earth surface there is reflection. Climate:Latitude of the location, day (time in the year) also affects the amount of solar energy received by the place.The above mentioned factors determine the amount of power falling on the surface.Insolation:It is a quantity indicating the amount of incident solar power on a unit surface, commonly expressed in units of kW/m2. At the earth’s outer atmosphere, the solar insolation on a 1 m2 surface oriented normal to the sun’s rays is called SOLAR CONSTANT and its value is 1.37 kW/m2. Due to atmospheric effects, the peak solar insolation incident on a terrestrial surface oriented normal to the sun at noon on a clear day is on the order of 1 kW/m2. A solar insolation level of 1 kW/m2 is often called PEAK SUN. Solar insolation is denoted by ' I '.The graph shown gives the amount of power present in different wavelengths of radiation. It can be seen from the graph that 50% of solar energy is in the form of thermal energy. Solar PV captures the energy in visible region. Solar thermal captures energy in infrared region.WavelengthKW/m 2UltravioletVisibleInfrared0.2μm 0.4μm0.7μm4μmIrradiance:It is an amount of solar energy received on a unit surface expressed in units of kWh/m 2. Solar irradiance is essentially the solar insolation (power) integrated with respect to time. When solar irradiance data is represented on an average daily basis, the value is often called PEAK SUN HOURS (PSH) and can be thought of as the number of equivalent hours/day that solar insolation is at its peak level of 1 kW/m 2. The worldwide average daily value of solar irradiance on optimally oriented surfaces is approximately 5 kWh/m 2 or 5 PSH. Solar irradiance is denoted by ' H '.Now we know the definition of two basic terms commonly used in design of a photovoltaic systems. Of course, these terms are often used interchangeably. Hence, one has to be careful in looking at the unit that has been used. In designing a photovoltaic system, it is important to know the amount of insolation available to us at a given time so that the power can be captured using solar panels and convert it into electricity. Depending on the requirement, the size of the panel can be then designed.2. Radiation MeasurementWe know that the atmosphere is made up of ions and other particles including clouds. Hence, when the incident radiation passes through the atmosphere, some radiation penetrates and falls directly on to the panel, some radiation diffuses in atmosphere and travels to the panel and some radiation gets reflected from the surroundings of the panel and reaches the panel, the effect being called albedo effect. It becomes extremely important to know the amount of energy that has reached the panel through all the paths. There are several factors on which this energy is dependent. They are as follows: Latitude and longitude of the geographical location.Climatic conditions such as presence of clouds, water vapor etc. Time of the day.Time of the year. Angle of tilt. Collector design.Now, let us see how we make use of this information in calculating the solar energy available at the panel. The steps are as follows:1. Find the sun position with respect to the location. This is a function of latitude (φ), hour angle (ω) and declination angle (δ).),,(δϖφf n SunPositio =2. Find the available solar energy or irradiance with no atmosphere, H O . This is a function of sun position.)(n SunPositio f H O =3. Find the solar energy available on horizontal surface with atmospheric effects, H OA . This is a function of H O and clearness index K T .O T OA H K H =4. Find the actual solar energy available at the panel, H t . This is a function of H OA and the tilt factor R D .OA D t H R H =All the above mentioned steps can be written as an algorithm so that the moment available data is fed, the actual solar energy available at the panel can be calculated instantly. The algorithm would involve the following equations: βφ,Enter3651→=N ()⎟⎠⎞⎜⎝⎛−∗=365802sin 45.23N πδ Degrees, N = 1 on Jan 1st , N = 365 on Dec 31st()δφωtan tan cos 1⋅−=−sr ⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛+=365360cos 033.01N I I SC O KW/m2()()()δβφωωδβφπsin sin cos cos cos 24−+−∗=sr sr Oot I H KWh/m 2/day on atilted surface with no atmospheric effects. ()()()δφωωδφπsin sin cos cos cos 24sr sr Oo I H +∗=kWh/m 2/day(data fitting curve K T ⋅)⋅= - Clearness Index()⎟⎠⎞⎜⎝⎛−+⎟⎠⎞⎜⎝⎛++−=2cos 12cos 11βρβD D R D K K K R - Tilt Factorwhere ρ is the reflection factor which ranges between 0.2 to 0.7.kWh/m o D T t H R K H ∗∗=2/dayThis algorithm can be translated into any of the programming languages like C, C++ or MATLAB. Entering the known parameters, it becomes convenient to find out the solar energy available at any geographical location.Insolation at any locationWe need to develop an algorithm, which calculates insolation (H t ) in kWh/m 2 at any place, once we input the following parameters:a. Day of the year (N)b. Latitude of the location (φ)c. Tilt angle (β)d. Angle of declination (δ)e. Clearness Index (K T )f. Reflection co-efficient (varies from 0.2 to 0.7)Following flow chart gives an idea for developing the algorithm:We can see that once δ, N and φ are input, Extra Terrestrial Radiation (ETR) can bedetermined. Δ, the declination angle of the sun is assumed to be the same every year and δ = 0 in March 21st . Following Fourier series can be used to calculate δ:t B t B t B t A t A t A Ao 3sin 2sin sin 3cos 2cos cos 321321++++++=δ where)80(365360−=N t degreesA0, A1…B3 = ?ETR can be calculated from the following expression:()δφϖϖδφsin sin sin cos cos 24sr sr SC I k ETR +⋅⋅= kWh/m 2 where ⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛+=365360cos 033.01N kI SC = mean solar constant = 1.37 kW/m 2φ = latitude in degrees/radians δ = declination angle in degrees/radiansωsr =hour angle at sunrise in degrees/radians = cos -1(-tan φtan δ)The next parameter that needs to be known is K T , the clearness index. It is one of themost important and difficult factors to be determined since it depends on atmospheric conditions such as absorption, pressure, cloud-cover at the place etc., which are not constant at a given place. However, a model for K T could be developed based on the irradiance level (H) measured at different places and using the relationship K T = H/ETR. K T was initially modeled using linear polynomial regression and multiple regression techniques. Since the results obtained with these models were not very accurate, a model was developed using Fourier series techniques of curve fitting since K T is a periodic function of period one year.We have seen earlier that the irradiance in kWh/ m 2 can be calculated for any location by inputting latitude of the location, declination angle, day number of the year for a given tilt angle using algorithm. The plot of irradiance as a function of the year is shown in the following figure.We can see that the level varies with the day of the year. It may reach a peak at some day of the year and reach a bottom on some other day of the year. The peaks and valleys are the direct result of the amount of irradiance reaching the earth. This is a graph that gives us the irradiance level over one year period.3.Heat Transfer ConceptsThe important terminology one needs to know to understand the heat transfer mechanism is the following:a)Radiationb)Conductionc)Convectiond)Mass transportTo understand the meaning of each of these terms, let us take an example. Let us consider a typical flat plate solar collector that is used in solar water heater system and shown in the following figure.Water In Water Out Convection through AirIn the above figure it can be seen that the heat is transferred from Sun to the flat plate solar collector by radiation. Radiation is the process of heat transfer from source to the target directly. The plate gets heated and transfers part of the heat to the copper tubes carrying water by conduction. Conduction is a process of heat transfer between two metals or solids. Part of the heat from the plate gets lost due to convection of heat. Convection is a process of heat transfer from solids to the surroundings via fluids. The heat transferred to the copper tube gets eventually transferred to the water flowing through the tubes by mass transport. Mass transport is a process of heat transfer similar to convection with little difference. Convection is an uncontrolled process where as mass transport is a controlled process where the discharge rate of the fluid can be controlled. It is important to note at this point that the above-mentioned heat transfer processes are all dependent on the properties of materials.Modeling of Heat transfer system:To understand modeling concept, let us consider a spherical space as shown in the following figure:T a Most part of the spherical space is enclosed by a thermal insulating material such as thermo-foam. The balance part is covered by a thermal conducting material. Let a hot body at temperature T1 is placed at the middle of the space. The heat from the hot body flows outwards towards the boundary of the space by radiation and convection. Let the temperature at the inner surface of the conductive window of the space be T si. The heat flows through the thermal conductive window. Let the temperature at the outer surface of the window be T so. The heat further flows into the atmosphere by convection and radiation. Let the ambient temperature outside the surface be T a. It can be observed that T1>T si>T so>T a. The decrease of temperatures during the outflow is attributed to the loss of heat in the process of convection, radiation and conduction of heat. The following block diagram can represent the outward flow of heat from the hot body to ambient:aIn the block diagram, T1 is the temperature of the hot body, T si is the temperature at the inner surface of the thermal conductor window, T so is the temperature at the outer surface of the thermal conductor window, and T a is the ambient temperature. Also, Rθv1 is the thermal resistance of the convection path between hot body and the inner surface of the thermal conductor, Rθr1 is the thermal resistance of the radiation path between hot body and the inner surface of the thermal conductor, Rθc1 is the thermal resistance of the conduction path between inner and out surface of the thermal conductor window, Rθv2 is the thermal resistance of the convection path between outer surface of the thermalconductor and the ambient, and R θr2 is the thermal resistance of the radiation path between outer surface of the thermal conductor and the ambient.The heat flow from the hot body to the ambient is analogous to ohms law (v = I · r). That is effort = flow x resistance. With reference to the heat flow, effort is the temperature difference, flow is the power and resistance is the thermal resistance of the path. Hence, we can write T = P x R θ.Based on this analogy, let us find the following terms:Thermal Resistance of the convection path is given by111()si v v T T R P θθ−=, ºC/W Power transmitted through convection is given by111()si v v T T P R θθ−=, W Normalizing the power transmitted through convection by dividing it by unit area, we get11v v P T A r θθΔ== q Where r θv1 = R θv1 x A = Thermal Resistivity measured in ºC m 2/W In general we can writeTq r θΔ== h · ΔT Here h is called the Thermal Coefficient, one of the most important terms in heat transfer.ConductionConduction is a mode of heat transfer in which transfer of energy takes place from the more energetic to the less energetic particles of a substance due to interactions between the particles. Heat transfer processes by conduction can be quantified in terms of rate equation known as Fourier’s law. Let us consider a thin rectangular solid slab as shown in the following figure:qΔxIn the figure, q is the heat flux and is the rate of heat transfer in the x direction per unit area perpendicular to the direction of transfer. T1 is the temperature of the hotter side of the slab and T2 is the temperature of the colder side of the slab. We have seen earlier that the heat flux is proportional to the temperature difference (T1 – T2). That isq Α (T1 – T2).Hence, we can writeq = h ·ΔTwhere h is the thermal coefficient in W/˚C m2Now, let us define another constant called thermal conductivity, k, in terms of thermal coefficient and the thickness of the slab Δx. That isk = h ·Δx [(W/˚C m2) · m]; [W/˚C m]Hence, we can writek h x =Δ Substituting this in the expression for heat flux, we getTq k xΔ=⋅Δ Thermal conductivity, k, is a transport property and is the characteristic of the slab material.Now, if A is the area of the slab then the heat rate by conduction, P is the product of heat flux and the area. That isT P q A k A xΔ=⋅=⋅⋅Δ Let us summarize the units of various constants we have considered so far. a) Thermal resistance R θ = ˚C/W or ˚K/Wb) Thermal resistivity r θ = ˚C· m 2/W or ˚K· m 2/W c) Thermal co-efficient h = W/˚C· m 2 or W/˚K· m 2 d) Thermal conductivity k = W/˚C· m or W/˚K· mAlso, we can write expression for all the above-mentioned constants in terms of thermalconductivity as follows:a) Thermal conductivity = kb) Thermal co-efficient kh x =Δc) Thermal resistivity xr k θΔ=d) Thermal resistance xR k AθΔ=⋅Thermal conductivity is an important property of material and plays an important role in the conduction process. The k values for different materials are given in the followingtable:Material k value in W/˚C/mCopper 385 Aluminum 211 Steel 47.6 Glass 1.05 Brick 0.6Concrete 1.7 Asbestos 0.32 Polyurethane 0.025 Polystyrene 0.035 Still air0.026Problem:Calculate Thermal resistance of the path ifa) The slab is made up of glass having area of 1 m 2 and thickness of 5 mm. b) The slab is made up of brick having area of 1 m 2 and thickness of 220 mm. Solution:We know that the expression for thermal resistance is given by xR k AθΔ=⋅ a) Δx = 5 mm = 5 x 10-3 m; k = 1.05; A = 1 m 2 35101.051x R θ−= = 0.004762 ˚C/Wb) Δx = 220 mm = 220 x 10-3 m ; k = 0.6 ; A = 1 m 23220100.61x R θ−= = 0.366667 ˚C/WA typical example of heat transfer by conduction is the case of heat dissipation insemiconductor devices such as junction transistors (BJT and FET). Any device that has a voltage across it and current through it simultaneously dissipates power. This power is equal to the product of the voltage and the current. We know that the power is the rate at which energy is converted from one form to another. In semiconductor devices, the energy changes from electrical energy to heat energy.Now, heat energy can be compared to current and temperature to voltage. When current flows in a semiconductor device, a voltage is produced and when heat energy flows in any substance, the temperature rises. The thermal resistance measured in ˚C/W of the substance determines how much the temperature rises when heat energy flows in it.The maximum operating temperature of a semiconductor junction is about 200 ˚C. Depending upon the type of case material, any given semiconductor device may have a lower maximum operating temperature. If this maximum temperature is exceeded, the device gets destroyed. An important part of the design process for large signal semiconductor circuit is making sure that the junction does not rise to a destructive temperature. To safeguard this, heat sinks are frequently used with semiconductor devices. Thermal resistance of the heat sink is the one that controls the temperature. We have seen earlier that the thermal resistance R θ can be given by the following expression:xR k AθΔ=⋅Here, the thermal resistance depends on the thickness of the heat sink material, thermal conductivity of the material and the area of the heat sink material. Normally, the thickness of the material is almost standardized and the thermal conductivity is the property of the material. That leaves the flexibility of varying only the surface area of the heat sink to vary the thermal resistance of the heat sink. To design heat sink with appropriate surface area, we need to know the maximum allowable temperature for the junction and the ambient temperature. The temperature of a piece of the material depends upon both the temperature rise caused by heat energy flowing in a thermal resistance and the temperature of the surrounding air. Stated mathematically:A Diss Temperature T P R t θ=+max ,maxJ At Diss T T R P θ−=Following figure shows how a semiconductor device is mounted on to a heat sink.The thermal resistance of the whole unit is made up of the following three parts:1. R θJC is the thermal resistance from junction of the device to case of thedevice. This specifies how many degrees hotter than the case the semiconductor junction will become for each watt dissipated.2. R θCS is the thermal resistance from case to heat sink. It specifies how many degrees hotter than the heat sink the case of the device will become for each watt dissipated.3. R θSA is the thermal resistance from heat sink to ambient. It specifies how many degrees hotter than the surrounding air the heat sink will become for each watt dissipated.Since all three thermal resistances appear in series with the flow of heat energy, the total thermal resistance is the sum of the three as shown in the following figure:P Tt JC CS SA R R R R θθθθ=++Substituting for Rot in the above equation we get the following:max ,maxJ AJC CS SA Diss T T R R R P θθθ−++=Normally, R θJC is provided in the data sheet of any device and R θCS can be assumed to be 1 ˚C/W. Hence, the thermal resistance of the heat sink can be calculated by rearranging the terms as follows:max ,maxJ ASA JC CS Diss T T R R R P θθ−=−−θConvectionConvection is a mode of heat transfer between a solid and a fluid when there is a temperature difference between a fluid and a solid. The convection heat transfer mode is comprised of following two mechanisms:•Energy transfer due to random molecular motion (diffusion).•Energy transferred by the bulk motion of the fluid.Such motion in the presence of a temperature gradient contributes to heat transfer.Let us consider a fluid flow over a heated surface as shown in the following figure:Heated SurfaceT S T FFluidΔxThe interaction between the fluid and the heated surface results in the development of a region in the fluid through which the velocity varies from zero at the surface to a finite value associated with the flow. This region of the fluid is known as the velocity boundary layer. Also, if the surface and flow temperature differ, there will be a region of the fluid through which the temperature varies from T S at the surface to T F in the outer flow. This layer is called the thermal boundary layer. If T S > T F, convection heat transfer will occur between the surface and the outer flow.The convection heat transfer mode is sustained both by random molecular motion and by the bulk motion of the fluid within the boundary layer. The contribution due to random molecular motion dominates near the surface where the fluid velocity is low. The contribution due to bulk fluid motion originates from the fact that the boundary layer grows as the flow progresses in the x direction. In effect, the heat that is conducted into this layer is swept downstream and is eventually transferred to the fluid outside the boundary layer.Convection heat transfer may be classified into the following types:1.Free Convection: Here the flow is induced by buoyancy forces, which arise fromdensity differences, caused by temperature variations in the fluid.2. Forced Convection: Here the flow is caused by external means such as fan, pump or atmospheric winds.Let us derive expressions for convective thermal resistance, convective thermal resistivity, convective thermal conductivity and convective thermal coefficient. We have seen in earlier that the expression for power (heat flow) is given by the following:TP k A xΔ=⋅⋅Δ, W; wherek = Thermal conductivityA = Cross sectional area perpendicular to the heat flowΔT = T S – T F = Temperature difference between surface and outer flow Δx = Thickness of the fluidHere most of the parameters are measurable except the fluid thickness, by. It can vary with surroundings. Hence, an additional parameter is added which is measurable and the fluid thickness is taken as a fraction of the measurable parameter as given by the following expression:X T P k A x XΔ=⋅⋅⋅Δ; W; whereX = Characteristic dimensionThe ratio of Characteristic dimension to the fluid thickness is called as the Nusselt number, a dimensionless quantity represented by N . HenceTP k A XΔ=⋅Ν⋅⋅; WRearranging the terms, we can get the expression for heat flow rate/unit area (heat density) and convective thermal resistance asPT q k A XΔ==⋅Ν⋅; W/m 2v T XR P k θΔ==⋅Ν⋅A; ˚C/WNow, we know that the thermal resistivity is the product of thermal resistance and area.v X Xr A k A k θ=⋅=⋅Ν⋅⋅Ν; ˚Cm 2/WFrom the above expression, we can write the expression for convective thermal coefficient (reciprocal of thermal resistivity) asv k h Xθ⋅Ν=; W/˚C/m 2Substituting for convective thermal coefficient in the expression for heat density and heat flow rate we get the following expressions:v q h T θ=⋅Δ; W/m 2v P h T A θ=⋅Δ⋅; WDetermination of Nusselt Number:Nusselt Number is a dimensionless number that can be determined experimentally only. In convection mode of heat transfer, we need to know the following for determining the Nusselt Number:• Speed of the fluid flow • Property of the fluid • Geometry of the solidNusselt Number for different classes of convection is different. They are called1. Rayliegh Number for free convection. It is represented by A and is given by the following expression:3g X Tβδν⋅⋅⋅ΔΑ=⋅; whereg = Acceleration due to gravity = 9.81 sec/m 2 β = Coefficient of thermal expansion X = Characteristic Dimension δ = Thermal diffusivity υ = Kinemetic viscosity of fluid2. Reynolds Number for forced convection. It is represented by R and is given by the following expression: u XR ν⋅=; whereu = mean velocity of flow X = Characteristic dimension υ = Kinemetic viscosity of fluidWe can see that the Nusselt number is a function of both Rayliegh Number and Reynolds Number, that isN = f ( A , R )The boundary layer in convection mode of heat transfer can be any of the following:• Laminar : In the laminar boundary layer, fluid motion is highly ordered and it ispossible to identify streamlines along which particles move. Since the velocity component u is in the direction normal to the surface, it can contribute significantly to the transfer of momentum and energy through the boundary layer.• Turbulent : In the turbulent boundary layer, fluid motion is highly irregular and is characterized by velocity fluctuations. These fluctuations enhance the transfer of momentum and energy and hence increase surface friction as well as convection transfer rates.The boundary layer can be classified into laminar or turbulent based on the Rayliegh Number and Reynolds Number.For Rayliegh Number:a) A ≥ 105: Turbulent flowb)103≤ A ≤ 105: Laminar flowc) A < 103: Free convection is not possibleFor Reynolds Number:a)R ≥ 2300 :Turbulent flowb)R < 2300 : Laminar flowLet us find expressions for Nusselt number for various shapes and for Laminar & Turbulent flows in terms of either Rayliegh Number or Reynolds Number for free convection and forced convection respectively.Free Convection - Rayliegh Number:Horizontal flat plate of length ‘a’ and breadth ‘b’ as shown in the following figure:Characteristic Dimension:()2a b X+ =For Laminar Flow:•102 < A < 105•N= 0.54 A 0.25 For Turbulent Flow:•A > 105•N = 0.14 A 0.33Circular plate with diameter ‘d’ as shown in the following figure:=Characteristic Dimension: X dFor Laminar Flow:•102 < A < 105•N = 0.54 A0.25For Turbulent Flow:•A > 105•N = 0.14 A 0.33Horizontal cylinder with diameter ‘d’ as shown in the following figure:dCharacteristic Dimension X = dFor Laminar Flow:•104 < A < 109•N = 0.47 A0.25For Turbulent Flow:•A > 109•N = 0.10 A 0.33Vertical cylinder with length ‘l’ as shown in the following figure:lCharacteristic dimension X = lFor Laminar Flow:•104 < A < 109•N = 0.56 A0.25For Turbulent Flow:•109 < A < 1012•N = 0.20 A 0.4Parallel plates at an angle ‘θ’ as shown in the following figure:Characteristic Dimension X = Distance between parallel platesFor θ < 50˚For Laminar Flow:•Not possibleFor Turbulent Flow:•A > 105•N = 0.062 A0.33Forced Convection – Reynolds NumberFlat plate of length ‘a’ or Circular plate of diameter ‘a’ as shown in the following figure:Characteristic Dimension X = aFor Laminar Flow:•R < 5 x 105•N = 0.664 x R0.5 x (υ/δ) 0.33For Turbulent Flow:•R > 5 x 105•N = 0.37 x R0.8 x (υ/δ) 0.33Vertical cylinder with length ‘l’ and diameter ‘D’ where the flow is over the cylinder as shown in the following figure:DCharacteristic Dimension X = Diameter, DFor Laminar Flow:•0.1 < R < 1000•N = (0.35 + 0.56 R 0.52) x (υ/δ) 0.3For Turbulent Flow:•1000 < R < 5 x 105•N = 0.26 R0.6 x (υ/δ) 0.3Horizontal cylinder with diameter ‘d’ where the flow is into the cylinder as shown in the following figure:Characteristic Dimension X = Length, LFor Laminar Flow:•R < 2300•N = 1.86 (R x υ/δ x D/L) 0.33For Turbulent Flow:•R > 2300•N = 0.027 x R0.8 x (υ/δ) 0.33Application of Convection PrinciplesProblem 1:Consider two conducting square plates measuring 1 m on each side, placed parallel to each other, separated by a distance of 3 cm. Let air is the medium of their separation. Let the vertical sides formed by the plates are thermally insulated so that no heat flows through them. Let the bottom plate is at 70 ˚C and the top plate is 45 ˚C. Calculate the heat flow in watts from the bottom plate to the top plate by convection through the air separating the plates.Solution:Following figure explains the problem.3 cmWe have seen earlier that the expression for the power flow P is given by the following expression:N kP A TX⋅=⋅⋅Δ, W, whereA = Area of the surface perpendicular to the heat flow = 1 m2N = Nusselt numberk = Thermal conductivity of the fluid (air) = 0.028 W/m˚KX = Characteristic dimension = Vertical distance between the plates for parallel plates. ΔT = Temperature difference between the plates。

衡量换热器效果的指标英文回答：Measuring the effectiveness of a heat exchanger involves several key indicators that help evaluate its performance. These indicators provide insights into the efficiency, thermal performance, and overall effectiveness of the heat exchanger. Two commonly used indicators are the heat transfer coefficient and the effectiveness.The heat transfer coefficient, also known as theoverall heat transfer coefficient (U-value), represents the rate at which heat is transferred through the heat exchanger. It is a measure of the heat transfer efficiency and is typically expressed in watts per square meter per degree Celsius (W/m²·°C). A higher heat transfer coefficient indicates better heat transfer performance.For example, let's say we have two heat exchangers, A and B. Heat exchanger A has a heat transfer coefficient of300 W/m²·°C, while heat exchanger B has a heat transfer coefficient of 200 W/m²·°C. This means that heat exchanger A is more efficient at transferring heat compared to heat exchanger B.The effectiveness of a heat exchanger is another important indicator. It measures how effectively the heat exchanger is able to transfer heat between two fluids. The effectiveness is defined as the ratio of the actual heat transfer to the maximum possible heat transfer. It is expressed as a percentage, with higher values indicating better performance.For instance, let's consider two heat exchangers, C and D. Heat exchanger C has an effectiveness of 80%, while heat exchanger D has an effectiveness of 60%. This means that heat exchanger C is more effective at transferring heat between the fluids compared to heat exchanger D.In addition to these indicators, other factors such as pressure drop, fouling, and cost-effectiveness should also be considered when evaluating the effectiveness of a heatexchanger. Pressure drop refers to the decrease in fluid pressure as it flows through the heat exchanger. Lower pressure drop is generally desirable as it reduces the energy required for fluid circulation.Fouling is another factor to consider, as it refers to the accumulation of unwanted deposits on the heat transfer surfaces. Fouling reduces the heat transfer efficiency and can lead to increased energy consumption and maintenance costs. Regular cleaning and maintenance can help mitigate fouling effects.Lastly, cost-effectiveness is an important consideration. It involves assessing the initial investment cost of the heat exchanger, as well as the operating and maintenance costs over its lifespan. A cost-effective heat exchanger should provide good performance while minimizing overall costs.中文回答：衡量换热器效果的指标有几个关键指标，可以帮助评估其性能。

A rc hi v eo fSI DMathematical Modeling of Heat Transfer for Steel Continuous Casting ProcessM. Alizadeh 1*, H. Edris 2 and A. Shafyei 31, 2, 3- Department of Materials Engineering, Isfahan University of Technology, Isfahan,8415683111, IranAbstractHeat transfer mechanisms and the solidification process are simulated for a continuous casting machine and the geometric shape of the liquid pool is predicted considering different conditions. A heat transfer and solidification model is described for the continuous casting of steel slabs. The model has been established on the basis of the technical conditions of the slab caster in the continuous casting unit of Mobarakeh Steel Company. This model involves a two-dimensional (2-D) transient energy equation. The governing equation was solved using the finite-volume procedure. The boundary conditions of the mold, water spray cooling, and air cooling regions have been defined. The mathematical model is able to predict the shell thickness, temperature distribution in the mold and shell, and the interfacial gap between shell and mold. The modeling results were verified by the measured slab surface temperatures and a reasonable agreement was achieved.Keywords: Heat transfer, Solidification, Continuous casting, Mathematical modeling.IntroductionA continuous casting process setup is shown inFigure 1. Different regions of cooling in the continuous casting machine are shown in this figure.Since heat transfer plays an important role incontinuous casting, it is desirable to have aquantitative understanding of the heat flow processeswhich will permit the prediction of the shell profileand temperature distribution as a function of thecasting variables.Fig. 1. Schematic of continuous casting process.∗Corresponding author:Tel: +98- 311- 3912750-1 Fax: +98- 311- 3912752 E-mail: alizadeh@ma.iut.ac.irAddress: Dept. of Materials Engineering, Isfahan University of Technology, Isfahan, 8415683111, IranIn the copper mold heat is transported to the cooling water. Below the mold, in the so-called secondary cooling zone, cooling is performed by water sprays, by contact with water cooled supportrolls, and in the lower part of the machine, by radiation. The mold must be able to extract sufficient heat from the incoming molten steel to solidify a shell which can support the liquid pool at the mold’s exit, and thereby minimize breakouts. It is also important for the mold to be able to remove the heat uniformly to avoid the formation of thin regions in the shell which could rupture and lead to breakouts or surface cracks. Therefore, the mold provides support for the newly solidified steel shell at a time when the shell is weak and sensitive to bulging due to ferrostatic pressure 1).Heat transfer in the mold region is controlled by: the convection of liquid superheat to the shell surface, latent heat evolution in the mushy zone, heat conduction through the solid shell, the thickness and other properties of the interface between the shell and the mold, heat conduction through the copper mold, and heat convection to the mold-cooling water 2).The mechanism of heat transfer from the surface of the strand to the cooling water is complex. It involves resistances in the layer of flux or resolidified mold powder, in the air gap, the mold wall, and in the mold/cooling water interface.1. Ph.D. Sstudent, Department of Materials Engineering, Isfahan University of Technology2. Associate Professor, Department of Materials Engineering, Isfahan University of Technology3. Assistant Professor, Department of Materials Engineering, Isfahan University of TechnologyA rc hi v eo fSI DInternational Journal of ISSI, Vol.3 (2006), No. 2Table 1: Comparison of thermal conductivities of materials present in the continuous casting mold [3].MaterialsTemperature(o C) Thermal conductivity(W m -1 K -1)Steel St 37 1200 29 Copper 30-130 385 Casting flux 1000-1300 0.5 to 1.2 Water 25 0.62Radiation conductivity of gas gap 1000 0.043Fig. 2. Different regions and simulation domain in continuous casting process.The thermal conductivities of different layers are compared in Table 1. As shown in this table, the air gap has the largest resistance to heat flow, while the other parts have a comparatively small resistance. Therefore, the pattern of heat removal in the mold is dependent largely upon the dynamics of gap formation. The air gap or contact resistances can be generated by the shrinkage of the steel shell away from the mold walls, especially after the flux is completely solid and unable to flow into the gaps. More researches were performed to show that the air gap was usually created in the lowest one-third of the mold length 3).In the upper part of the secondary cooling zone, the strand is usually sprayed by water emerging from nozzles arranged in the spaces between the rolls. The rate at which heat is extracted from the strand surface by water sprays has been measured by many researchers. These researchers have shown that under normal continuous casting conditions, in which shell surface temperatures range between 700 and 1200 o C, surface temperature has a little effect on the heat transfer coefficient. All studies agree that, in the stated temperature range, the spray water flux has the most effect on the heat transfer coefficient. Moreover, the temperature of sprayed water does not have a large influence on the heat transfer coefficient 4).In the lower part of the secondary cooling zone, heat transfer is preferred mainly by radiation and by roll contact. Therefore, it should be considered that the oxide scales generated on the surface of the strand could cause a thermal resistance in this zone 5).In this study, the mathematical model has been established on the basis of the technical conditions of the slab caster in the continuous casting unit of Mobarakeh Steel Company. In this model, steel heat capacity and steel thermal conductivity were considered as functions of steel temperature and chemical composition. Considering these functions, the governing equation is a non-linear equation. In this study the equation is solved in non-linear state. This model is also capable of predicting the temperature distribution, including the solidus and liquidus isotherm which defines the solid shell and mushy zone, respectively, as a function of section size, pouring temperature, steel composition, casting speed, mold length and spray conditions.Mathematical modelingFigure 2 shows the different regions of the continuous casting machine and the model considered for physical simulation of the caster. A typical method of modeling the strand thermal condition is shown in this figure. The mathematical model is applied to slices of strand that start at the meniscus and travel through the machine at the casting speed. New slices are generated periodically. A sufficient number of slices exist in each cooling zone to give an accurate representation of the thermal condition in each zone. In this model, only a quarter of the strand is considered due to the symmetry of the heat flow conditions (Figure 3).A rc hi v eo fSI DInternational Journal of ISSI, Vol.3 (2006), No. 2Fig. 3. A symmetric quarter of strand width section with physical coordinate.AssumptionsThe following assumptions are made to simplify the mathematical model 6):-Conduction can take place only in the transverse directions.-Forced convective heat flow in the liquid pool is considered by defining an effective liquid thermal conductivity as:l eff K K ⋅=7(1) -The density of steel is constant, but specific heatcapacity and heat conductivity of steel are functions of temperature and chemical composition and therefore not constant.Model formulationThe energy conservation equation can be written as 7):(2)()()()()⎟⎠⎞⎜⎝⎛∂∂∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂+⎟⎠⎞⎜⎝⎛∂∂∂∂=∂∂+∂∂+∂∂+∂∂z T k z y T k y x T k x wH vH y uH x H t effeffeffρρρρ To simplify the equation, a transformation as wt z −=ζ is used. Therefore this is: (3)()ls ls s sH H H t y x H H q tHy H y x H x tH +==+∂∂−⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂+⎟⎠⎞⎜⎝⎛∂∂∂∂=∂∂;,,,;0ζρααIn order to solve the governing equation, it is necessary to transform the physical domain into a computational domain. In general, this sort of transformation is used, and leads to a uniformly spaced grid in the computational domain but the points in physical domain may be unequally spaced. The original partial differential equation is transformed from physical coordinates (x, y ) to computational coordinates (ξ,η) by applying the chain rule of partial derivatives.;JS H H J H t x y s y x s s +⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂=⎟⎠⎞⎜⎝⎛∂∂ξηηαηηξξαξ (4);tH S l ∂∂−= y x J ηξ= In the above equations "S " is a term for heat source due to the metal phase transformation (liquid to solid). In order to establish the region of phase change, the latent heat contribution is specified as a function of temperature i.e. 8):f l l L f H ⋅=(5) Where L f is the latent heat of the phase changeand the liquid fraction (f l ) is computed by:⎪⎪⎭⎪⎪⎬⎫⎪⎪⎩⎪⎪⎨⎧≤≥≥−−≥=.......01sol sol liq sol liq sol liq l T T when T T T when T T T T T T when f (6)A typical 2D cartesian control volume is shownin Figure 4. This C.V contains a central node (P) with four neighborhood points (E, W, N, and S). The integral form of equation (4) is obtained on the control volume by the finite volume method 9,10): a P H sP = a E H sE + a W H sW + a N H sN + a S H sS + b (7) a P = a W + a E + a S + a N + a P oa E =ey x e ⎟⎟⎠⎞⎜⎜⎝⎛ΔΔηξξηα a W =wyx w ⎟⎟⎠⎞⎜⎜⎝⎛ΔΔηξξηα a N =nx y n ⎟⎟⎠⎞⎜⎜⎝⎛ΔΔξηηξα a S =sx ys ⎟⎟⎠⎞⎜⎜⎝⎛ΔΔξηηξαa P o =PJ t ⋅ΔΔ⋅Δηξ b = a P o (H sP o + H lP o - H lP ) + o Pq J ρηξΔ⋅Δ (8)Fig. 4. A typical control volume and the notation used for a Cartesian 2D grid [9].To approximate the variable values on the surface of the control volume, the Quadratic Upwind Interpolation (QUICK) algorithm is used 9). In the QUICK scheme, the variable profile between two points approximated by a parabola instead of aArc hi v eo fSI DInternational Journal of ISSI, Vol.3 (2006), No. 2straight line (Figure 5), on a uniform Cartesian grid leads to 9):W E P e φφφφ818386−+= (9)Fig. 5. Approximation of gradients at cell faces [9].Boundary conditionsTo solve the above equation, the boundary conditions are needed for different regions include the mold, water spray cooling, and air cooling. Figure 6 shows some machine cooling layouts while the technical information belonging to each zone is shown in Table 2. A general form of the boundary condition can be expressed through an equation (10), in which the heat transfer coefficient, h , is estimated for different cooling zones.()water b b T T h n T k −=∂∂−| (10)To determine the temperature of the boundaries, the discretion of the bnT |∂∂ on boundary points isrequired. According to the QUICK scheme b nT |∂∂could be calculated by the following correlation [10]: nT T T nT W P b bΔ+−=∂∂398(11)Therefore, as seen in equation (10), finding the heat transfer coefficient for different regions such as: mold, water spray, and air cooling is necessary.Fig. 6. Secondary cooling of the slab caster.In the mold, several thermal resistance layers exist between the steel shell surface and the recirculation water. All the thermal resistances in the mold are shown in Figure 7. The effective thermal resistance of the water channel is estimated from the water channel heat transfer coefficient, thermal conductivity and the thickness of scale deposits on the surface of the cooling-water channel 11):⎟⎟⎠⎞⎜⎜⎝⎛+=w scale scale water h k d r 1 (12)Fig. 7. Thermal resistances existing between the shell surface and water channel in the mold.Table 2. Secondary cooling zones variables.No. Zone Length zone(m)SegmentNumber of spray nozzles Water flow rate(m 3/sec)Number rollin zoneRoll radius (m)1 0.439 - -2 0.220 - -3 0.303 - 30 0.003975 - -4 0.925 0 38 0.004967 5 0.140 5 1.470 0 38 0.0048426 0.200 6 1.475 1 10 0.004858 5 0.2507 1.725 2 10 0.003975 5 0.3008 1.725 3 10 0.003733 5 0.3009 3.950 4,5 20 0.005667 10 0.350 10 5.200 Roll 37-47 22 0.006483 11 0.380,0.440 11 9.400 Roll 48-63 - Air cooling 16 0.440A rc hi v eo fSI DInternational Journal of ISSI, Vol.3 (2006), No. 2 The heat transfer coefficient between the water and the side walls of the water channel (h w ) is calculated assuming a turbulent flow through an equivalent-diameter pipe (D ) using the empirical correlation of Sleicher and Reusse 11):()21Pr Re 015.05c c water w Dkh += (13)()Pr 6.0215.0333.0;Pr 424.088.0−+=+−=e c cOther thermal resistances shown in Fig 7 can be calculated by the following correlations [11]:;;slag slag slag moldmold mold k d r k dr == ;air air air k d r = (14)Which d slag can be found from the powder consumption per mass of product (M slag (kg/ton)): ()N W N W M d slagsteel slag slag +×××=2ρρ (15) Also, d air includes a gap due to shrinkage of the steel shell, which can be calculated by the thermal linear expansion relation of steel using following correlation for each control volume: T l l Δ⋅⋅=Δλ (16) Where, λ is linear thermal expansion coefficient of steel and l is length of each control volume. Moreover, the heat transfer coefficient due to radiation is calculated by: ()()212221T T T T h rad ++=εσ (17) Heat transfer mechanisms in the spray cooling zones below the mold are defined in Figure 8. The heat extraction due to the water sprays is a function of the water flux. The relationship between the rate of heat extraction by the water sprays and the spray variables has been established in a number ofexperimental studies. One of the most widely usedrelations has been presented by Nozaki's 12): ();0075.013925.055.0water water spray T Q h ×−××= (18)Fig. 8. Heat transfer mechanisms in the secondarycooling zones [12].Due to the high temperature of the strand surfaceand the exposure of water to the surface, an oxidescale is produced on the surface of the strand. Despite the low thickness, the scale can have animportant role in the heat transfer control. Therefore,the effective heat transfer coefficient should be considered as 12):sprayscsceff h k h 11+=δ (19)It has already been mentioned that the cooling ofthe strand in the lower part of the secondary cooling zone is mainly done by radiation 12). Therefore, the equation for the heat transfer coefficient is given as follows:()().2.2am s am s rad T T T T h ++=εσ (20) Besides the radiation, heat transfer is alsoachieved by natural convections, but this part is rather small and can be neglected in comparison to radiation cooling.The symmetrical boundary condition has been considered for midplanes as follows:0.,00.,0=∂∂−==∂∂−=yTk y xTk x (21)Computation and verificationThe algebraic equation of the boundary conditions has been solved with a Tridiagonal Matrix Algorithm (TDMA) solver 13). As seen in equations(7) and (8), to solve the algebraic equation, it is necessary to know the latent enthalpy in a new time step (H l ). To update the amount of the latent enthalpy an iterative solution is used for each time step:11++−+=k sP k sP k lP k lP H H H H (22)As seen in the above equation, by using the sensitive enthalpy that has been obtained by solving the energy equation (H sP k+1), latent enthalpy could be updated in the k+1th inner iteration in each time stepto achieve a certain convergence. The calculation mentioned has been programmed in the FORTRAN language. The mathematic simulation starts by setting the initial steel temperature at the pouring temperature. Input parameters, in Table 3, in the standard cases are verified by the measured temperatures on the shell surface of the strand.Equilibrium lever-rule calculations are performed on a Fe-C phase diagram in order to calculate steel phase fractions. By this means, phase field lines are specified as simple linear functions of carbon equivalent content. The carbon of the steel is applied as the carbon equivalent content that is calculated by the following correlation 14): wt%CE=wt%C+0.04(wt%Mn)-0.14(wt%Si)-0.04(wt%Cr)-0.1(wt%Mo)-0.24(wt%Ti)+0.1(wt%Cu) For a 0.16%C, 1.3%Mn, 0.5%Si, 0.05%Cr, 0.03%Mo and 0.01%Ti plain carbon steel, the carbon equivalent percentage calculated as 0.135 and also the equilibrium phase diagram model calculates T liq.=1528 o C, T sol = 1494 o C. The solid fraction-A rc hi v eo fSI DInternational Journal of ISSI, Vol.3 (2006), No. 2temperature curve in the mushy zone obtained from the model is shown in Figure 9. As seen in this figure the relation between temperature and solid fraction of steel in the mushy zone is non-linear.Solidified shell thickness is one of the most important calculated parameters in the model, and the influence of the grid spacing on this parameter should be considered. The influence of the gridspacing on the solidified shell thickness, in exit point of the mold, is shown in Table 4. The thickness amounts in width and narrow sides are presented in this table. It is clear from Table 4 that when grid spacing is reduced, solidified thicknesses are changed and they are stable in a narrow limit with reduction of grid spacing lower than a certain limit. It can be concluded that the solidified shell thickness is independent of the mesh size with the reduction of grid spacing lower than a certain limit.The mold zone is a complex and important area in continuous casting machine. The solid shell growth in this zone is complicated and the results of this study are compared to those of some other researchers. The comparison between the results of the model in this study and experimentalmeasurement by some other researchers is shown in Figure 10. The figure shows the variation of growth of the solid shell thickness on the ingot for low carbon steel (0.06% C). It is clear that the results from numerical solution in this study have a very good compatibility with those of three 3-D model of Thomas and experimental measurement of Alberny and co-worker.Table 3. Input data for standard conditions.Carbon equivalent content, CE pct 0.132 pctSteel density, ρ7500 kg/m 3 Steel emissivity, ε0.8 Mold copper plates thickness 0.043×0.030 m ×m Total mold length 0.704 m Mold copper plates width 2.220×0.215 m ×m Scale thickness on the surface of mold cold face 0.00001 m Mold conductivity, k mold 315 W/mK Mold powder conductivity, k slag 1.27 W/mK Air conductivity, k air 0.083 W/mKMold powder density, ρslag0.650 kg/m 3 Mold powder consumption rate, M slag 0.8 kg/ton steel Casting speed, V c 0.0167 m/secPour temperature, T in 1546 oCLiquidus temperature, T liq. 1528.6 oCSolidus temperature, T sol. 1494 oC Working mold length 0.659 m Slab geometry, W ×N 1.250×0.203 m ×m Scale conductivity on the surface of slab, k sc 0.5 W/mK Scale conductivity on the surface of mold, k scale 1.0 W/mK Water channel geometry, large & small plates 25×5×29 & 22×5×26mm 3 Average cooling water temperature in the mold 28 oC Water flow rate entering the mold small plate, 0.0061 m 3/sec Water flow rate entering the mold large plate, 0.0553 m 3/sec Latent heat of the steel phase change, L f 272140 J/kgWater conductivity , k water0.615 W/mKSteel specific heat capacity [11], C p ()()()()⎪⎪⎪⎪⎪⎪⎭⎪⎪⎪⎪⎪⎪⎬⎫⎪⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪⎨⎧≥=≤=≤+=≤=−==+=+=liqp liq sol p sol op p o p p op o p T T C T T T C T T C T C T C T C T C T C T C T C T C T C 7877721100*334.02681100850648850750*766.338497507001431700500*836.0268500*376.0456p p p p p p p p p p J/kgK Steel conductivity, k steelf l *k liq +(1-f l )*k sol W/mK Solid steel conductivity, k sol 33.0 W/mK Effective molten steel conductivity, k liq 7*43.0 W/mKScale thickness on the surface of slab , δsc0.001 mA rc hi v eo fSI DInternational Journal of ISSI, Vol.3 (2006), No. 2Table 4. Effect of mesh size on the shell thickness at mold exit.Grid system #nx ×n y Shell thickness at middle of wide face (m)Shell thickness at middle of narrow face (m)25×10 0.0145 0.0111 25×15 0.0135 0.0110 50×15 0.0136 0.0118 50×20 0.0133 0.0119 100×200.0136 0.0117 100×300.0131 0.0117Fig. 9. Phase fraction variation with temperature in mushy zone.Fig. 10. Comparison between model results and other references.Results and discussionThe calculated surface temperatures of a slab, for the Table 3 conditions as a function of the distance below the meniscus, are presented in Figure 11. This figure shows the calculated surface temperatures at the centers of the wide and narrow faces and at the corners of the slab caster. The central areas of the wide faces are cooled one-dimensionally, whereas the slab corners are subject to 2-D cooling. The slab corners can therefore become significantly colder than other parts of the inner wide face. At the beginning of straightening, the slab corner temperature is 230 o C less than the temperature at the center of the wide face. Control of corner cooling is critical for much of continuously cast products. The slab corners tend to have meniscus marks, which act as stress risers. A combination of temperature, stress risers and a low ductility region in the 700-900 o C temperature range during the straightening process often leads to cracks in the corners. One way to increase the corner temperature is to widen the strip of unsprayed strand at the corner. However, as the non-sprayed strip is widened, a hot spot will develop between the colder corner and the sprayed area. Thus, the design must be in a way to ensure that it does not cause other quality problems. Also, as seen in Figure 11, the intensity of heat transfer of mist spray cooling is less than cooling in the lower part of the mold for all of the curves, because of this, the model predicts a 200 o C reheating of the slab surfaceon leaving the mold. A similar situation also exists in water spray cooling and air cooling regions on the surface of the wide face slab.Fig. 11. Predicted surface temperature of strand. Figure 12 shows the solidified shell thickness profiles of both the narrow and wide faces of the slab. This figure shows a sudden change of slope at the beginning of the solidified shell growth curve. It is indicated that the rate of solidified shell growth is clearly high in the mold region. Figure 13 shows a set of data of the local heat flux density along the length of the mold [15]. There is a maximum of heat flux density somewhat below the meniscus. Downward, the heat flux density usually decreases. If the air gapA rc hi v eIDInternational Journal of ISSI, Vol.3 (2006), No. 2between the mold and strand approximately has uniform thickness, or it increases uniformly in the downward direction, the heat flux density along the length of the mold continuously decreases.Fig. 12. Predicted solidified shell thickness.Fig. 13. Heat flux density distribution along the length of the mold.Fig. 14. Effect of casting speed on the pool depth.The casting speed is the most effective parameter in changing the position of the solidified shell thickness profiles. The relation of the "metallurgical length" (maximum length of the liquid pool) with the casting speed is shown in Figure 14. Increase in casting speed decreases the holding time of the slab in the secondary cooling zones and increases the length of the liquid core. Therefore, casting speed is the most important factor in controlling mold heat extraction. The solidified shell thickness as a function of the casting speed of bothwide face and narrow face of the strand are shown in Figure 15. Since, a lower casting speed provides more time for the heat to be extracted from the shell, the shell thickness increases. Moreover, the shell thickness in the initial solidification stage decreases at the high casting speed, as shown in Figure 15, which often easily causes the breakout of strand. Therefore, to prevent this unfavorable defect, the casting speed is limited. It should be mentioned that steel composition and slab width in comparison to casting speed has no significant effect on the shell thickness. As seen in this figure, solidified shell thickness for wide face is higher than it is for narrow face. This is because of the different air gap sizes between mold powder and mold wall in both wide and narrow faces. The air gap size along the length of the mold is predicted by the model for both faces in Figure 16. As seen in this figure, total shrinkage value in narrow face is more than wide face as expected because width size of slab is much larger than thickness size in cross section of strand. Furthermore, this phenomenon could change Fig. 15. Effect of casting speed on shell thickness.Fig. 16. air gap size along the mold.Heat transfer in the mold is governed by these three resistances: the casting-mold interface, the mold wall, and the mold-cooling water interface. Although, thermal resistance due to the air gapA rc hi v e o fSI DInternational Journal of ISSI, Vol.3 (2006), No. 2 should also be considered while the amount of air gap thermal resistance is usually quite large compared to the other resistances especially for the lower portion of the mold. Temperatures of three points: slag layer/mold wall interface (T 1), mold wall (T 2), and water channel wall (T 3) as shown in Figure 7 are predicted by the model. Since, the heat flux for steady state conditions will be constant and independent of distance: (23)mold rad airradair slag s rad airrad air slags slags total water s r rr r r r T T rr r r r T T r T T r T T q +⎟⎟⎠⎞⎜⎜⎝⎛+⋅+−=⎟⎟⎠⎞⎜⎜⎝⎛+⋅+−=−=−=′321The above relation represents three equations for the three unknowns, T 1, T 2, and T 3. The results of the model predictions are shown in Figure 17(a) and Figure 17(b). As seen in these figures, the existence of the air gap between the shell surface and mold wall causes the T 1 to increase both in wide and narrow faces of strand. Since the air gap thickness in the narrow face is larger than the wide face, T 1 in the narrow face is much higher than in the wide face. Therefore, the pattern of heat removal in the mold is dependent largely upon the dynamics of gap formation. It causes temperature difference to exist in solidified shell and has a strong influence on transverse cracks generation in the mold region.Fig. 17. Predicted temperatures for T 1, T 2 and T 3 points in Fig 7 -(a) wide face and (b) narrow face (T 1=slag layer/mold wall interface temperature, T 2=mold wall temperature, and T 3=water channel wall temperature).The effect of the amount of superheat temperature of the molten steel, entering the mold, on the solidified shell thickness is shown in Figure 18. At first, with increasing the superheat temperature, the superheat flux will increase and therefore, the solidification rate decreases. This, in turn, reduces the thickness of the steel shell. On the other hand, by increasing the superheat temperature, the driving force for the heat transfer will increase. Therefore, as seen in Figure 18, the influence of the superheat temperature is insignificant to shell growth, especially in the wide face of the strand. Low superheat, near liquidus temperature, is beneficial to continuous casting. Internal quality is improved as the equated zone is made significantly larger. Therefore, a more desirable structure with greater resistance to halfway cracks is produced. Centerline segregation and porosity are also reduced or eliminated.Fig. 18. Effect of steel superheat temperature on the shell thickness at mold exit.Conclusions1- A finite volume heat transfer and solidification model has been formulated to predict the temperature field and liquid pool position in the continuous casting process under different conditions. This has been verified with the temperature measurement of slab surface.2- Casting speed is the most effective parameter on mold heat removal. Therefore, it is the most important factor in controlling solidified shell thickness and slab temperature.3- Since, the air gap size in narrow face of mold is higher than the wide face, the breakout of strand often occurs in the narrow face.4- Air gap existing in the casting-mold interface causes a large thermal resistance for heat transfer from the solidified shell to the mold. Therefore, it has a strong influence on product quality and casting problems, especially for the narrow face of strand. 5- High superheat temperature may cause breakouts at the mold exit, especially for the narrow face, so it should be exactly kept at a low level.。

金属大变形过程中温度场的Abaqus模拟马璇【摘要】Based on the Abaqus,the finite element modeling and simulation of the large deformation of metal was to analyze the temperaturefield.Considering the deformation heat,the friction heat and the contact heat,the mathematical model of temperature field and the mathematical model of water-cooled and air-cooled were validated.To describe the distribution of the temperature,the parameter of high temperature was used.The results indicate that the temperature of plate core change little while surface temperature change sharply.At the same time,it is important to consider the heat exchange between the surface and the surrounding.The quantity of heat was concerned with the temperature and the coefficient of heat transfer.The deformation heat has a significant impact on the distribution of temperature field.%基于 Abaqus有限元仿真软件，在考虑到变形热、摩擦热和接触热等因素的情况下，建立了轧件内部温度场的数学模型和轧件空冷及水冷的数学模型，并利用实验获得的金属高温物性参数，对金属的大变形过程进行了有限元建模和仿真，模拟了金属变形过程中纵剖面的温度场分布及变化规律。

Modeling of Heat Transfer in Nanoporous Silica - Influence of MoistureR. Coquard1, D. Quenard21. Centre Thermique de Lyon (CETHIL), UMR CNRS 5008, Domaine Scientifique de la Doua, INSA de Lyon, Bâtiment Sadi Carnot, 9 rue de la physique, 69621 Villeurbanne CEDEX, France. Phone: 04 72 43 84 74 Email: remi.coquard1@insa-lyon.fr2. Centre Scientifique et Technique du Bâtiment (CSTB), 24 rue Joseph FOURIER, 38400 Saint Martin d’Hères, France. Phone : 04 76 76 25 46 Email : quenard@cstb.frAbstractThis work is interested in the modelling of the thermal behaviour of nanoporous silica. These highly porous materials are suitable for the manufacturing of Vacuum Insulation Panels given that the exceptionally small size of their pores permit to reduce drastically the conductive heat transfer when a primary vacuum is applied. Nanoporous silicas have been widely studied experimentally but only few studies have already dealt with the theoretical modelling of their thermal characteristics from their nanometric structure. That is the reason why we have developed a model of heat transfer based on a simplified representation of their structural morphology which permits us to analyse the contributions of each heat transfer modes (gaseous conduction, solid conduction, radiation) to the total heat transfer.The coupled radiation-conduction heat transfer is treated using a simultaneous resolution of the Energy Equation and the Radiative Transfer Equation. The effective conductivity of the medium is computed by means of a Finite Element Method while its monochromatic radiative properties are calculated assuming that only silica particles are present and that they scatter radiation independently. The model can take into account the presence of condensed water at the surface of silica particles as well as the possible contact area between neighbouring particles. The results of the model permits us to investigate the evolution of the equivalent thermal conductivity of the nanoporous silica with different parameters related to their structural characteristics (density, particle size, contact area between particles …) or to the external conditions (pressure, water content …). They notably indicate in which conditions the silica should be used in order to maximize their insulating performances. The results of the model are compared with experimental results previously published and show good agreement. In particular, the bad influence of the presence of condensed water and of the contact area between touching particles on the conductive heat transfer are clearly confirmed theoretically. This demonstrates the important role played by the adsorbed water on the thermal performances of this very absorbent material. The study also reveals that, for materials with common densities containing silica particles alone, the radiative heat transfer is significant and noticeably degrades the insulating performances of the material. Thus, the addition of opaque particles to the solid matrix is indispensable.1Nomenclaturea : constant parameter proper to the gas consideredA cube : characteristic length of the cubic lattice in mc0 : light speed in vacuum (≈ 2.99776.108 m/s)C ext, C sca et C abs : extinction, scattering and absorbing cross sections in m²d c : contact diameter between neighboring grains in md eau : diameter of the water meniscus in md grain : diameter of the silica grains in mh : Planck constant (≈ 6.626.10-34 J.s)Iλ : Monochromatic radiant intensity (W/m²/Str/µm)I0λ : Monochromatic radiant intensity of the black-body at temperature T (W/m²/Str/µm) k : thermal conductivity in W/m/KK : Boltzmann constant (≈ 1.3805.10-23 J/K)L g : mean free path of the gas molecules (m)~: complex refractive index=ikn−nn~: equivalent refractive indexeqN grain :Number of grains per volume unit (m-3)P : Pressure in PaP max : Maximal strain on the material in PaPλ(θ) : Monochromatic scattering phase functionq c: Conductive heat flux density (W/m²)q r : Radiative heat flux density (W/m²)Q k : thermal power passing through the horizontal slab z k in WR : Radius of the particles in mt p : Characteristic size of the pores in mT : temperature in KT cold, T hot : temperature of the hot and cold boundaries in KTh water : thickness of the condensed water on the sides of the silica grains in mx, y, z : cartesian coordinatesGreek symbolsβλ, σλ, et κλ : monochromatic extinction, scattering and absorption coefficient in m-1γ : convergence parameterε : porosityλ : radiation wavelength in µmµ =cosθ : directing cosine of the directionρ : density in kg/m3σSB: Stefan-Boltzmann constant (≈ 5.67.10-8 W/m²/K4)θ : angle between incident and scattering directions in radωλ = σλ/βλ : monochromatic scattering albedo (-)ξ : convergence criterionSubscriptsair : of airch : relative to the chain of silica grainside: relative to the lateral boundaries of the computation volumewater : of watereff: effectiveeq : equivalentg : gasbound. : relative to the horizontal hot and cold boundaries of the computation volume i,j,k : relative to the discretised nodes of coordinates x i, y j, z kfree : free gassi : of silicax, y, z : relative to the point of coordinates x, y, zλ : at radiation wavelength λ21 IntroductionUsual building thermal insulators (cellular or fibrous materials) have now reached a limit in their thermal performances and it appears that an equivalent thermal conductivity of 0.03 W/m/K is an almost incompressible value. Thus, the improvement of the thermal insulation required to reduce the expenditure of energy of the building sector could only be achieved by the development of new thermal insulators able to reach lower equivalent thermal conductivities. One of the most promising materials are the Vacuum Insulating Panels made of nanoporous media under air vacuum. The principle of action of these new insulators is based on the “Knudsen effect”. By this technique, it is theoretically possible to reach equivalent conductivities lower than the thermal conductivity of free air. The Knudsen effect is already used in several products commercialized by industrial groups such as POREXTHERM or CABOT-CORP.Several studies have already been interested in the thermal properties of this kind of materials. Rochais, Domingues and Enguehard [1] have developed a model enabling to study the transient or steady-state conductive heat transfer through a fractal representation of a nanoporous material in order to compute the thermal conductivity and diffusivity of these materials. At the same time, Enguehard [2] have modeled the radiative heat transfer in these highly porous materials by developing a simplified model based on the Rosseland approximation and the Mie Theory under the assumption of independent scattering. The author pointed out the fact that, for the density considered, thermal radiation contributes significantly to the total heat transfer. Then, he suggested to introduce micrometric particles into the nanometric network in order to reduce the radiative contribution. The theoretical results showed that a substantial decrease of this contribution could be achieved by introducing a relatively small amount of particle with suitable sizes (diameter close to 1µm).Quenard and Sallée [3] have also conducted a detailed experimental study on the thermophysical properties of these materials. The samples tested, with porosities greater than 90%, are essentially composed of fumed silica associated with a very small amount of fibers insuring the mechanical consistency. Some opaque particles are also added to the material to make it more opaque to thermal radiation. The authors measured the porosities, the specific areas and the pore size distribution. They have also been interested in the evolution of their equivalent thermal conductivity with pressure, temperature and water content using the hot-wire method which enable a simple, fast and non-intrusive measurement in a wide range of experimental conditions. The experimental results showed that, at room temperature and atmospheric pressure, the equivalent conductivity of these materials is close to 20 mW/m/K. This conductivity could be reduced noticeably to approximately 6 mW/m/K by applying a primary vacuum (P=1-10 hPa). Moreover, the measurements reveal that the equivalent conductivity increases significantly with the temperature which certainly implies that a non negligible part of the heat transfer is due to thermal radiation. Finally, Quenard and Sallee clearly pointed out the very bad influence of water adsorption on the thermal performances of nanoporous silica whose thermal conductivity increases linearly with the moisture content at a rate of approximately 1.5 to 2 mW/m/K per mass percentage of adsorbed waterResearchers from the Bavarian Center for Energy (ZAE Bayern) [4, 5, 6, 7] have also looked into the characterization of opacified nanoporous medium under partial air vacuum. They have shown that for the opacified silica aerogel, equivalent conductivities of only several mW/m/K could be reached at ambient temperature [4]. For fumed, precipitated or pyrogenic silica powders [5, 6], the addition of opacifiant particles allows to reach equivalent conductivities close to 3 mW/m/K under partial air vacuum. However, they also remarked that the mechanical strains undergone by the material reduces the insulating abilities of the material significantly. Finally, the ZAE Bayern also studied the influence of condensed water on the thermal properties of fumed silica [7] by measuring their conductivities for different moisture content using a guarded hot-plate apparatus. The increase of the thermal conductivity is noticeable and is almost proportional to the water content.One can notice that, at the moment, the thermal properties of nanoporous materials have mainly been studied from an experimental point of view. The relation between their nanometric structure and the evolution of their thermal characteristics with the different parameters (density, pressure, water content …) remain relatively badly known. Thus, we have developed a simple model representing the silica grain network constituting the porous structure which allowed us to estimate the thermal conductivity of these insulators from their structural characteristics (density, solid conductivity, grains diameter, contact area between grains …). We assumed that the macroscopic heat conduction law are still valid at the nano-grain scale. Moreover, we have also modeled the presence of condensed water to predict the influence on the equivalent conductivity. Finally, given that radiative heat transfer could be significant, we also computed the radiative properties of non-opacified34 nanoporous silica. The knowledge of these properties and of the effective conductivity of the nanoporous silica permits to compute their equivalent conductivity.2 Structure of the nanoporous silicaThe use of nanoporous materials proves to be particularly convenient for the manufacturing of V.I.P. given that the very small size of the pores allows to benefit from the Knudsen effect without requiring high vacuum. Silica aerogels, the most common nanoporous material, are constituted of pyrogenic or precipitated silicium micrograins ( amorphous SiO 2). These micro-grains are almost spherical and their characteristic size is approximately 10 nm. Fumed silica could also be considered for this application. In this case, the structure is still constituted of silica grains with very small diameters but they are obtained from the combustion of silica tetrachlorure (SiCl 4) in aqueous atmosphere. For these two kinds of nanoporous silica, the structure of the grain network is random and the chains have random orientations. Thus, the properties of the material could be considered as isotropic. The scheme and the photograph of Fig. 1 illustrate the structure of the network obtained by considering that all the grains have a uniform size.In order to simplify the theoretical morphology of nanoporous silica, We have modeled the grain network by considering that all the identical grains are spread on the arêtes of cubes abutted sides by sides. The grains are mechanically linked to their neighbors. The illustration of this cubical network is given on Fig. 2.The porosity ε and the average size of the pores t p constituting the nanoporous silica only depend on the diameter of the silica grains d grain (radius r grain ) and on the length of the arête A cube of the cubical network. We have computed these two parameters for different grain diameter and different length of arête by generating the entire grain network from a unique representative cube incorporating several grains on its arêtes. The porosity of the material is estimated by generating a great number of points randomly distributed in the elementary cube and by calculating the proportion of points belonging to an air pore. To match the porosity of usual nanoporous silicas which is comprised between 0.9 and 0.95, our representation corresponds to a ratio grain cube d A / comprised between 7 and 10.As regards the average size of the pores, it is very difficult to define it given that the nanometric structure does not form closed pores. However, we managed to express a characteristic length of the pores. Actually, this characteristic size represents the average distance that an air molecule have to travel after impacting a silica grain to reach the silica matrix again. To compute this average distance, we randomly generate a point into the air pores and a direction of propagation of the molecule. Then, a ray-tracing procedure determines the distance between the two silica grains encountered in this direction on both sides of the point. For a great number of rays, it is possible to estimate the average size of the pores. According to our geometric model, for grain diameters close to 10 nm, the characteristic size of the pores is several hundreds of nanometers for usual silica (porosity between 0.9 and 0.95). This average size is the same order of magnitude as the mean free path of air molecules and allows to benefit from the KNUDSEN effect which reduces the thermal conductivity of air. Indeed, according to Griesinger et al. [8], the conductivity k g of the gas can be estimated by :Figure 1: Simplified representation of the random structure of nanoporous silicas5p g libre g g t L a k k /...21+= (1)For air, under atmospheric pressure: k air,free = 0.0262 W/m/K ; a = 1.5 and L g ≈ 67 nm.Thus, according to our model, when the diameter of the grain is about 10 nm, the decrease of thermal conductivity of air due to the Knudsen effect for usual silicas (ε between 0.9 and 0.95) is comprised between 12 and 15 mW/m/K at ambient pressure and is already very interesting.3 Modelling of the heat transfer in nanoporous silicaThe principle of action of all traditional thermal insulators is to prevent the convective heat transfer due to the temperature gradient and to limit the conductive heat transfer through the solid matrix. Thus, the heat transfer is primary due to the conduction through the gas (generally air). In nanoporous materials, the thermal conductivity of air could even be reduced by the Knudsen effect. Actually, when the insulators is highly porous, another heat transfer mode due to the propagation of thermal radiation occurs. In the case of 1-D steady-state heat transfer between two horizontal boundaries at different temperatures, the total heat flux can be computed by solving the Energy Equation (EE) and the Radiative Transfer Equation (RTE). To solve these equations, it is necessary to know the effective thermal conductivity and the infrared radiative properties of the nanoporous silica.3.1Modelling of the conductive heat transfer 3.1.1 Thermal conductivity without condensed waterIn order to compute the effective conductivity of dry nanoporous silica composed of a solid and a gas phase, we used a model based on the Finite Element Method (FEM) applied to the grain network. This model assumes that the heat transfer by conduction at the grain scale follow the macroscopic laws represented by the Fourier equation :T k q c ∂−= (2)Although the grain diameter is not vastly greater than the size of the silica molecules, we will consider, in the next, that the Fourier law is suitable. From this hypothesis, we have modeled the thermal conductivity of a chain of silica grain in punctual contact, noted k ch . This conductivity can be calculated by analyzing the temperature field in a representative cube containing a grain subjected to a temperature gradient. It is necessary to solve the differential equation of heat conduction in steady-state by the Finite Element Method. The FEM has already been used by Druma et al. [9], Quintard et al. [10] and Bakker [11] for conductive heat transfer calculations in materials made of several phases. The principle is to simulate an experimental steady-state apparatus measuring the conductivity and to solve the steady-state energy equation 0=∇∇).(T K r r r (3)Figure 2 : Theoretical representation of the cubical grain network proposed for the modelling of thethermal properties of nanoporous silicaElementarycube dgrain6 The resolution is based on a spatial discretization in cubical elements of the volume modeled. Each volume is made of silica or air and a node is placed at its center. The thermal conductivity of the solid node is k Si = 1.4 W/m/K and that of air volume is set to k air = 0.0001 W/m/K. We voluntarily set the thermal conductivity of air to a very small value given that air is confined in a very narrow space comprised between two neighboring grains and is subjected to a strong Knudsen effectOnce the effective conductivity of the chain of grain has been computed, the conductivity of the entire network forming the nanoporous silica is calculated assuming that it is actually made of two continuous phases with different conductivities : one phase corresponding to the grain in contact forming the skeleton of the network and one phase corresponding to air. We used a numeric model based on the FEM similar to the previous model to compute the effective conductivity of the entire network of silica grain. The model used only differs from the previous one by the geometry of the discretised volume which now corresponds to a cubical volume whose arête are occupied by the chains of grain. The thermal conductivity of the silica chain used for the computation has been computed in the previous step: k ch . On the other hand, the thermal conductivity of air used is not the same as that of the previous computation given that air is no more confined between the silica grains but in the entire network. Then, the mean free path of the air molecules is noticeably greater than previously and the thermal conductivity of air could no more be neglected although the Knudsen effect is present. The thermal conductivity used is given by Eq. (1)3.1.2 Conductivity in the presence of condensed water and contact between grains Due to their high specific area and the hydrophilic character of silica, the risk of water condensation in nanoporous silica is major. Thus, it is necessary to take into account the influence of the presence of condensed water on its thermal properties. In addition, the contact between neighboring grains is not punctual since the mechanical strain could lead to a deformation of the grains. A model enabling the simulation of these two phenomenon has been developed.The ideal structure of the nanoporous silica is then modified to take into account the condensed water and the contact diameter d c between two touching grains. We assumed that water can condense according to two means :- by forming a meniscus with diameter d water located at the junction between two grains,- by forming a continuous water layer with thickness Th water at the surface of the grainsThe cube representing the chain of grain composing the nanoporous silica could be illustrated by the scheme of Fig. 3.The effective conductivity of the chain of grain in the presence of condensed water could be computed using a Finite Element Method similar to that used previously but in which, the spatial discretization is modified. In the new configuration, three different phases are considered : a solid phase (conductivity k Si ), a liquid phase (conductivity k water = 0.6 W/m/K) and a gas phase ( k air = 0.0001 W/m/K). The proportions of each phase depend on the values of d grain , d water , Th water and d c .3.1.3Evolution of the effective conductivity with the different parametersFigure 3 : Illustration of the cube representing the chain of silica grains in the presence ofcondensed water and of a non-punctual contactRepresentative cube containing a silica graincondensed water7Influence of the densityThe density of nanoporous silica only depends on their porosity : 2).1(SiO ρερ−= (4) with ρSiO2 = 2500 kg/m 3 the density of silicaIn order to analyze the relative influence of these two effects, we have conducted the computations without the presence of condensed water and assuming punctual contacts between grains fro different densities and different grain diameters. Two air pressures have been modeled : atmospheric pressure (1000 hPa) and a primary vacuum (P=10 hPa). The results are depicted on Fig. 4 where we have also represented the equivalent thermal conductivities measured by Quenard et al.[3] on opacified fumed silicas and by Caps et al. [5] on opacified pyrogenic silicas. In both cases, theauthors have shown that due to the presence of opacifiers, the radiative heat transfer at ambient temperature could be neglected and that the equivalent conductivity is sensibly equal to the effective conductivity.The results of Fig. 4 show that, for relatively weak densities, the theoretical effective conductivity decreases with the density when the silica grain are sufficiently small. This can be explained by the fact that, in this range of densities, the diminution of the conductivity of air due to the Knudsen effect when the silica gets denser (decrease of the average size of the pores) is more important than the increase of the thermal conduction through the solid phase. On the other hand, when the silica grains are relatively large (> several tens of nm), the decline of thermal conductivity of air is practically imperceptible and the augmentation of solid conduction due to the densification is predominant. This causes an increase of the effective conductivity with the density.The evolution observed when the grains are sufficiently small is quite noteworthy given that for the other porous materials classically used for thermal insulation, the effective conductivity generally tends to increase with the density.One can also notice that for common densities and grain diameters of approximately 10 nm, the effective conductivity at atmospheric pressure is lower than the thermal conductivity of free air. The interest of the Knudsen effect is then clearly apparent even without partial vacuum. At the same time, when a partial air vacuum is applied, the theoretical effective conductivity reaches particularly low values lower than 10 mW/m/K.All these theoretical conclusions agree well with the experimental results of Quenard and Sallee [3] who measured an equivalent thermal conductivity at atmospheric pressure slightly lower than the thermal conductivity of free air. However, it seems that our theoretical results slightly overestimates the conductivity of nanoporous silica for the two pressures considered.Influence of the grain diameterThe size of the grain forming the nanoporous matrix affects directly the average pore size t p . Then, very small grains are required in order to maximize the Knudsen effect. The influence of the parameter d grain on the effective conductivity is illustrated on Fig. 4. One can notice that silica grains100 200300400500600 700density in kg/m 3 k eff inW/m/K Figure 4 : Theoretical evolution of the effective conductivity of nanoporous silicas with their densities at atmospheric pressure and under a partial air vacuum (P=10 hPa) for different grain diameters8 with diameters lower than several tens of nm are required so that the Knudsen effect is perceptible at atmospheric pressure. Above this limit, microporous silicas are not suitable to reach very low thermal conductivities. However, for lower pressures, we notice that when the grains are small enough, their size no more affects the effective conductivity since the conductivity of confined air is then extremely low and almost negligible when compared with the conductivity of the silica skeleton.Influence of air pressureWe have seen that the air pressure have an influence on the mean free path of air molecules and thus on the strength of the Knudsen effect. To illustrate this more clearly, we have depicted on Fig. 5 the variations of the effective conductivity with pressure for different porosities and assuming that the grain diameter is d grain = 10 nm. We have also illustrated on this figure the equivalent conductivities measured at different air pressures by the CSTB (Quenard et Sallée [3]) and the ZAE Bayern (Caps et al. [5]) on opacified fumed silicas.The influence of air pressure is significant. Then, when an advanced air vacuum in the order of 10-100 hPa is applied in common nanoporous silicas (porosities between 0.9 and 0.95, d grain ≈ 10 nm), very weak thermal conductivities lower than 10 mW/m/K can be obtained theoretically. But, if we reduce the pressure under 10 hPa, the improvement of the conductivity is less and less pronounced. Very low pressures are not necessarily interesting particularly if we consider that the mechanical strains on the structure are then increased. This results are in very good agreement with the measurements given by Quenard and Sallee [3] and Caps et al. [5].Influence of the contact between grainsOwing to the mechanical strain undergone by the nanoporous silicas during the manufacturing process or utilization, the contact between grains is not punctual. Actually, there is a contact area between the two touching grains that is likely to enhance the conductive heat transfer in the solid phase. This effect has been underlined by Caps and Fricke [6] who proposed a law of variation of the conductivity in the form max P A k c =.We have evaluated from a theoretical point of view the influence of this non-punctual contact on the effective conductivity of the nanoporous silica by assuming that contact area is circular. We varied the contact diameter d c . The computations were conducted at atmospheric pressure for nanoporous silica with densities 0.9455 and 0.968 with a grain diameter d grain = 10 nm. We specify that, in each case considered, the porosity of the silica remains unchanged when d c varies. The variations observed are thus entirely due to the variation of the contact. They are illustrated on Fig. 6As expected, the contact area between the grains leads to an increase of k eff . The conduction of heat by solid-solid contact is enhanced. However, one can notice that this influence is relatively limited as long as the deformation of the grains is weak compared with the grain diameter (d c /d grain < 0.2). But, it becomes significant when the deformation is more important.Pressure in hPak eff inW/m/KFigure 5 : Theoretical evolution of the effective conductivity of nanoporous silica with air pressure for different densities and d grain = 10 nm0,010,02 0,03 0,040,1 1 10101001000。