分形理论及其在多孔介质和纳米流体热导率上的应用
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摘要
首先对分形几何和热传输性质的基本理论作了简要的综述,然后主要研究了弯曲度分形维数的分析解模型、两相和三相多孔介质有效热导率的一般模型和纳米流体有效热导率。
在本文第二部分,建立了弯曲度分形维数DT的分析解模型,该模型表示为孔隙率和介质微结构参数的函数,在该模型中没有经验常数。通过与实验作比较,证实了该模型的正确性。该模型的建立对于深刻理解多孔介质输运性质(渗透率、热传导和扩散)的机理有重要的指导意义。
在第三部分中,基于多孔介质具有统计自相似的特征和热-电类比模拟技术,本文建立了两相多孔介质热导率的一般模型。将谢尔宾斯地毯的边长选定为13,通过改变颗粒尺寸C (=3, 5, 7和9)的大小模拟孔隙率范围在0.14~0.80的多孔介质。多孔介质有效热导率可以表示为孔隙率(与阶数n相关),面积比,组成成分热导率之比和接触热阻的函数。本模型有效热导率的计算方法简单。与其他模型相比,本模型中参数较少,且每个参数都有明确的物理意义。在孔隙率为0.14~0.80的较大范围内,本模型的预测与其它模型和实验数据都吻合得很好。接着,将两相多孔介质热导率的一般模型推广到三相(未饱和)多孔介质热导率的一般模型,该分形几何模型也是基于电—热模拟方法和多孔介质统计自相似特性。在该模型中仍将谢尔宾斯地毯的边长选定为13,通过改变颗粒尺寸C (= 5, 7和9)的大小模拟孔隙率范围在0.14~0.60的多孔介质。本模型有效热导率的计算方法简单。三相(未饱和)多孔介质有效热导率可以表示为孔隙率(与阶数n相关)、面积比( Ant / A)、微结构尺寸(L和C)、组成成分热导率之比和饱和度的函数。本模型中参数较少,且每个参数都有明确的物理意义。在孔隙率0.14~0.60的范围内,本模型的预测值与实验数据吻合的很好。
本文第四部分对纳米流体导热系数进行了初步的理论研究。在考虑到纳米颗粒尺寸的分形分布和纳米颗粒在液体中做布朗运动导致对流换热的基础上,采用Monte Carlo方法和分形几何理论,本文建立了纳米流体有效热导率的分形模型。建立的模型表示成纳米颗粒热导率、基础液体的热导率、纳米颗粒尺寸、随机数、纳米颗粒体积份额和温度的函数。模型预测值与实验数据相比较,两者很好地相吻合。此外,在综合考虑液膜层和纳米颗粒凝聚的基础上提出了一个新的纳米流体等效热导率的理论模型。该模型等效热导率的大小与纳米流体的一些参数(颗粒和基础液体的热导率、纳米颗粒尺寸、纳米颗粒的体积份额和液膜层的厚度)紧密相关。将模型预测值与实验值进行比较,发现两者很好地相吻合,从而证实了本模型的有效性。
First, the fractal geometry theory and heat transfer in porous media are addressed. The focuses of this dissertation are involved in developing an analytical model for tortuous fractal dimension, a generalized model for the thermal conductivity of two-phase and three-phase porous media and the thermal conductivity of nanofluids.
In Chapter 2, an analytical expression for the fractal dimension for tortuous capillaries in porous media is derived and found to be a function of porosity and microstructures of porous media. There is no empirical constant in the proposed fractal dimension. The present model for the fractal dimension is verified by a comparison with the available analogy model. The present model may have the potential and significance on fractal analysis of transport properties (such as the permeability, dispersion, thermal and electrical properties) in porous media.
In Chapter 3, a generalized model for the thermal conductivity of two phase porous media is derived based on the self-similarity existing in porous media and thermal-electrical analogy technique. In this model, the geometry model, Sierpinski carpet, is chosen to approximate the statistically self-similar porous media. The Sierpinski carpets with the same side length L=13 and different cutout sizes C (=3, 5, 7 and 9) and the different fractal dimensions are adopted to model the real porous media in a wide range of porosities, 0.14~0.80. The present model for the thermal conductivity of porous media is found to be a function of the porosity (related to stage n of Sierpinski carpet), the ratio Ant / A of areas, the ratioβof component thermal conductivities, contact thermal resistance t + and microstructures L and C. This model has the least parameters, Ant / A and t +, compared to the other models and every parameter in this model has clear physical meaning. The model predictions are compared with the existing experimental data and other models’, and the results show that the present model presents a good agreement with the existing experimental data in a wide range of porosities, 0.14~0.80. Furthremore, the proposed thermal conductivity model for two-phase porous media is then extended to analyze unsaturated/three-phase porous media. The Sierpinski carpets with side length L = 13 and cutout sizes C = 5, 7 and 9, respectively, depending on the porosity concerned. The recursive formulae are presented and expressed as a function of porosity, ratio of areas, ratio of component thermal conductivities, and saturation. The model predictions are compared with those of available experimental data, and good agreement between them in a wider range of porosities 0.14~0.60 is obtained.
The effective thermal conductivity of nanofluids is studied in Chapter 4. First, the Monte Carlo simulations combined with the fractal geometry theory are performed. Based on the fractal character of nanoparticles in nanofluids, the probability model for nanoparticle’s sizes and the effective thermal conductivity model are derived, in which the effect of the micro-convection due to the Brownian motion of nanoparticles in the fluids is taken into account. The proposed model is expressed as a function of the thermal conductivities of the base fluid and the nanoparticles, the volume fraction, fractal dimension for particles, the size of nanoparticles and the temperature, as well as random number. The predictions by the present Monte Carlo simulations are shown in good accord with the existing experimental data. Then, by taking into account the nanolayer and nanoparticles’aggregation, a new model for the effective thermal conductivities of nanofluids is proposed. This model is expressed as a function of the thickness of nanolayer, the nanoparticle size, the nanoparticle volume fraction and thermal conductivities of suspended nanoparticles and base fluid. The theoretical predictions on the effective thermal conductivities of nanofluids are shown to be in good agreement with the available experimental data. The validity of the proposed model is thus verified.
在本文第二部分,建立了弯曲度分形维数DT的分析解模型,该模型表示为孔隙率和介质微结构参数的函数,在该模型中没有经验常数。通过与实验作比较,证实了该模型的正确性。该模型的建立对于深刻理解多孔介质输运性质(渗透率、热传导和扩散)的机理有重要的指导意义。
在第三部分中,基于多孔介质具有统计自相似的特征和热-电类比模拟技术,本文建立了两相多孔介质热导率的一般模型。将谢尔宾斯地毯的边长选定为13,通过改变颗粒尺寸C (=3, 5, 7和9)的大小模拟孔隙率范围在0.14~0.80的多孔介质。多孔介质有效热导率可以表示为孔隙率(与阶数n相关),面积比,组成成分热导率之比和接触热阻的函数。本模型有效热导率的计算方法简单。与其他模型相比,本模型中参数较少,且每个参数都有明确的物理意义。在孔隙率为0.14~0.80的较大范围内,本模型的预测与其它模型和实验数据都吻合得很好。接着,将两相多孔介质热导率的一般模型推广到三相(未饱和)多孔介质热导率的一般模型,该分形几何模型也是基于电—热模拟方法和多孔介质统计自相似特性。在该模型中仍将谢尔宾斯地毯的边长选定为13,通过改变颗粒尺寸C (= 5, 7和9)的大小模拟孔隙率范围在0.14~0.60的多孔介质。本模型有效热导率的计算方法简单。三相(未饱和)多孔介质有效热导率可以表示为孔隙率(与阶数n相关)、面积比( Ant / A)、微结构尺寸(L和C)、组成成分热导率之比和饱和度的函数。本模型中参数较少,且每个参数都有明确的物理意义。在孔隙率0.14~0.60的范围内,本模型的预测值与实验数据吻合的很好。
本文第四部分对纳米流体导热系数进行了初步的理论研究。在考虑到纳米颗粒尺寸的分形分布和纳米颗粒在液体中做布朗运动导致对流换热的基础上,采用Monte Carlo方法和分形几何理论,本文建立了纳米流体有效热导率的分形模型。建立的模型表示成纳米颗粒热导率、基础液体的热导率、纳米颗粒尺寸、随机数、纳米颗粒体积份额和温度的函数。模型预测值与实验数据相比较,两者很好地相吻合。此外,在综合考虑液膜层和纳米颗粒凝聚的基础上提出了一个新的纳米流体等效热导率的理论模型。该模型等效热导率的大小与纳米流体的一些参数(颗粒和基础液体的热导率、纳米颗粒尺寸、纳米颗粒的体积份额和液膜层的厚度)紧密相关。将模型预测值与实验值进行比较,发现两者很好地相吻合,从而证实了本模型的有效性。
First, the fractal geometry theory and heat transfer in porous media are addressed. The focuses of this dissertation are involved in developing an analytical model for tortuous fractal dimension, a generalized model for the thermal conductivity of two-phase and three-phase porous media and the thermal conductivity of nanofluids.
In Chapter 2, an analytical expression for the fractal dimension for tortuous capillaries in porous media is derived and found to be a function of porosity and microstructures of porous media. There is no empirical constant in the proposed fractal dimension. The present model for the fractal dimension is verified by a comparison with the available analogy model. The present model may have the potential and significance on fractal analysis of transport properties (such as the permeability, dispersion, thermal and electrical properties) in porous media.
In Chapter 3, a generalized model for the thermal conductivity of two phase porous media is derived based on the self-similarity existing in porous media and thermal-electrical analogy technique. In this model, the geometry model, Sierpinski carpet, is chosen to approximate the statistically self-similar porous media. The Sierpinski carpets with the same side length L=13 and different cutout sizes C (=3, 5, 7 and 9) and the different fractal dimensions are adopted to model the real porous media in a wide range of porosities, 0.14~0.80. The present model for the thermal conductivity of porous media is found to be a function of the porosity (related to stage n of Sierpinski carpet), the ratio Ant / A of areas, the ratioβof component thermal conductivities, contact thermal resistance t + and microstructures L and C. This model has the least parameters, Ant / A and t +, compared to the other models and every parameter in this model has clear physical meaning. The model predictions are compared with the existing experimental data and other models’, and the results show that the present model presents a good agreement with the existing experimental data in a wide range of porosities, 0.14~0.80. Furthremore, the proposed thermal conductivity model for two-phase porous media is then extended to analyze unsaturated/three-phase porous media. The Sierpinski carpets with side length L = 13 and cutout sizes C = 5, 7 and 9, respectively, depending on the porosity concerned. The recursive formulae are presented and expressed as a function of porosity, ratio of areas, ratio of component thermal conductivities, and saturation. The model predictions are compared with those of available experimental data, and good agreement between them in a wider range of porosities 0.14~0.60 is obtained.
The effective thermal conductivity of nanofluids is studied in Chapter 4. First, the Monte Carlo simulations combined with the fractal geometry theory are performed. Based on the fractal character of nanoparticles in nanofluids, the probability model for nanoparticle’s sizes and the effective thermal conductivity model are derived, in which the effect of the micro-convection due to the Brownian motion of nanoparticles in the fluids is taken into account. The proposed model is expressed as a function of the thermal conductivities of the base fluid and the nanoparticles, the volume fraction, fractal dimension for particles, the size of nanoparticles and the temperature, as well as random number. The predictions by the present Monte Carlo simulations are shown in good accord with the existing experimental data. Then, by taking into account the nanolayer and nanoparticles’aggregation, a new model for the effective thermal conductivities of nanofluids is proposed. This model is expressed as a function of the thickness of nanolayer, the nanoparticle size, the nanoparticle volume fraction and thermal conductivities of suspended nanoparticles and base fluid. The theoretical predictions on the effective thermal conductivities of nanofluids are shown to be in good agreement with the available experimental data. The validity of the proposed model is thus verified.
引文
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