Gözenekli bir kanalda tam gelişmiş zorlamalı taşınımda akışkan ağdalılığının ısı transferi üzerine etkisi
Effects of viscous dissipation on fully developed forced convection in porous media
Yazar: CANSU UĞUR
Danışman: PROF. DR. AYŞE FİLİZ BAYTAŞ
İstanbul Teknik Üniversitesi / Enerji Enstitüsü / Enerji Bilim ve Teknoloji Ana Bilim Dalı
Derece: Yüksek Lisans, 2015
Konu: Enerji = Energy

Özet:
Gözenekli ortam gündelik hayatta her alanda karşımıza çıkmaktadır. Ciğerlerimiz, hücrelerimiz veya bir sünger gözenekli ortama verilebilecek en basit örneklerdir. Bir akışkanın içinden geçebildiği gözenekli ortamlarda enerji ve kütle geçişi pek çok mühendislik alanının ilgisini çekmektedir. Bugüne kadar gözenekli ortamdaki enerji denklemleri ve gözenekli ortam parametreleri üzerine pek çok araştırma yapılmıştır. Gözenekli ortamdaki akışkan akışı ve enerji geçişi ile ilgili hala hesaplanamayan parametreler bulunmakta ve bunlar üzerine halen pek çok araştırma yapılmaktadır.
Fransa'da bir hastaneye temiz su taşımak amacıyla 1856 yılında Henry Darcy tarafından başlatılan gözenekleri ortam çalışmaları, daha sonra gelen bilim adamlarının da katkılarıyla geliştirilmiş ve genel bir denkleme dönüşmüştür. Gözenekli ortamda kullanılan enerji denklemlerinin çoğunun çözümünde ağdalı akışkanın akış sırasında ağdalılık kaynaklı enerji kayıplarının sıcaklık üzerindeki etkisi ihmal edilmiştir. Pek çok çalışmada bu ağdalı kayıplar göz ardı edilmiş ve sıcaklık üzerindeki etkileri araştırılmamıştır. Bu çalışma ihmal edilen ağdalı kayıpların literatürde kullanılan bir model üzerinden incelenmesini içermektedir. Akışkanın modellemesi, gözenekli ortama sahip geçirgen olmayan iki paralel plaka duvarı, bu duvarlara uygulanan sabit ısı akısı ve tam gelişmiş akış şartları altında yapılmıştır. Akış tek yönlü ve zamana bağlı değişim olmadığı kabulu yapılmıştır. Gözenekli ortam genel momentum denklemi olan Brinkman Momentum Denklemini baz alan bu çalışmada ağdalı akışkanın akışı sırasında oluşan kayıpların sıcaklık üzerindeki etkileri incelenmiştir. Brinkman Momentum denklemi analitik olarak çözülmüş ve boyutsuz hale getirilerek, diferansiyel denklem çözümü yapılmıştır.
Diferansiyel çözüm sonucunda elde edilen boyutsuz hız denklemi enerji denkleminde kullanılmıştır. Enerji denkleminde yer alan ağdalı kayıplar teriminin çeşitli modeller baz alınarak değişimi incelenmiştir. Ağdalı kayıplar için Al-Hadhrami modeli ele alınmış değişen Brinkman ve Darcy sayıları ile gözenekli ortamdaki akışkan ve sabit cidar arasındaki sıcaklık farkları, değişen Brinkman sayısına bağlı olarak akışkanın sıcaklık artış ve düşüş profilleri tespit edilmiştir. Brinkman ve Darcy sayıları arasındaki etkileşim farklı parametreler için incelenmiştir. Nusselt sayısının enerji kayıplarını ifade etmedeki rolü, Nusselt'in Darcy ve Brinkman sayısı ile değişimi incelenmiştir. Maksimum sıcaklık farkı düşük Darcy sayılarında görülmüştür. Brinkman sayısının pozitif olması, akışkan sıcaklığının arttığını, negatif olmasının akışkan sıcaklığının azaldığını ifade ettiği görülmüştür. Darcy ve Brinkman sayılarından yola çıkarak akışkan sıcaklık değişimi tespit edilmiştir.

Abstract:
Porous media is a routine part of daily life. Porous materials has entered everywhere in daily life, in science and technology, in medicine and in natural structure of basic materials like rocks. Lungs, cells or a sponge can be given  as a few simple examples to porous media. Membrane technologies is the most popular industry which is built on porous media. The transport phenomena in porous medium has a important role in thermodynamics and fluid dynamics and it takes a lot of attention in recent years due to its vital role in different engineering applications. Flow of oil and gas in petroleum reservoirs, production planning of oil calculations are calculated based on those phenomena. Besides, nanotechnology and pebble-bed nuclear reactors and even the flow of liquids in biological systems are applied with basics rules of transport in porous media equations.
The porous medium has to have two basic properties to be able to confirm that material is eligable to be defined as porous media. The material must contain relatively small void spaces, called pores distributed randomly or quite homogeneously  in a solid matrix and cores generally contain some fluid such as water or oil or a mixture of them like in petroleum reservoirs. The second property must be the penetration, all pores must be connected with each other and allow the fluid to transport through pores. Gas or liquids should be able to transport one face of the material and emerge on other side. Sandstone, limestone, dolomites, a spoon  of sugar, bread, ceramics are examples of porous media which we encountered in every part of daily life. Solid matrix of porous medium can be consolidated such as concrete or solid matrix can exist in non-consolidated form like sand filters.  The structure and network of pores determines the macroscopic properties of porous media. Permeability, porosity and turtuosity are the parameters defined  based on the pore and matrix properties. When the porosity and turtuosity are the characteristics of porous media, permeability is the fluid tranport property of the porous medium. Porosity is defined as the ratio of pore volume to bulk volume of matrix. Imbibition, mercury or gas injections methods are used to determine the effective porosity value. Pores can be distributed homogeneouly in matrix, porosity can be accepted a constant value for homogeneous medium. However, most of the matrixes contain randomly distributed pores and porosity change depending upon location inside the matrix. Porosity is dependent on type of the medium and porosity value is defined between zero and one.If there is no connection between pores and it is a solid and no void material the value of porosity is defined as zero. If the material has no solid matrix inside then porosity value will be defined as one. Each void in control volume can be connected more than one pore or it can be connected only one pore. Porosity may be constant for a homogeneous medium but in general, most of the matrixes have a heterogenenous media and porosity changes depending upon the location and direction of considered control volume.  
Porosity is the most important feature of porous medium and affect all physical parameters in media. Turtuosity is not a physical parameter depends upon the pore diameter, porosity and channel shape. Turtuosity can be imagined as the path the fluid flows through inside the porous media. The type of fluid flows through the channel also affect the turtuosity. Turtuosity is difficult to measure experimentally. In the simplest way, turtuosity is explained as a bundle of capillary tubes through the porous media and for this case, it is calculated as the ratio of pore length to porosius medium thickness. Turtuosity is dependent on type of material and flow conditions and also the structure of porous medium. It can be changed according to flow direction. Permeability is the capability of porous medium to transport the fluid through the void spaces. Permeability depends on the pore distribution and porous media geometry. Permeability first defined by Darcy in 1856. The measurement of permeability can be achieved under isotropic conditions by using both liquid and gas fluids. Some liquids may affect pore structure and change porosity value so they affect also permeability. Type of liquid is important for permeability value. We need to consider a control volume like we did flow in a channel in fluid dynamics to be able to define mass, momentum and energy equations. We need to define a relatively small control volume which represents properties of all matrix. However, our control volume both includes solid matrix and fluid. Those assumption enable us to define basic equations for flow in porous media. Forced convective heat and mass transfer of a fluid in porous media is the subject of  various engineering applications because of its significant role in transport processes such as petroleum reservoirs or nuclear reactors.
Calculation of parameters of porous media and energy equation in porous media has received much attention in recent years from a variety of engineering disciplines. The researches on calculations of new parameters is still in progress related to energy transport and fluid flow and flow in porous media becomes the subject of a variety of studies. Porous media studies was started by Henry Darcy in 1856 to transport clean water to a nearby hospital by using porous media. Those studies were developed by scientists after Darcy and obtained a general equation.  There are a large number of analytical studies on forced convection in porous media in the literature, however very few of them take into consideration the effects of viscous dissipation on temperature distribution in porous media. Viscous dissipation effect on temperature distribution is neglected in the most of the studies. In present study, effects of viscous dissipation on temperature distributions is analyzed by an analytical model in the literature. The modelling of fluid is taken into consideration under steady state operation condition and unidirectional flow in a parallel plate channel filled with a porous medium height of channel 2H and constant heat flux qw is applied to impermeable wall. The effects of viscous dissipation on temperature profile has been analyzed by utilizing Brinkman Momentum Equation as general enery moment equation.  Brinkman momentum equation is solved analytically and dimensionless form is obtained for differential equation solution. Dimensionless velocity profile is obtained by solving differential equation and velocity profile later is used in energy equation. 
Viscous dissipation terms in energy equation is calculated by taking into consideration Al-Hadhrami model. Temperature difference between the impermeable wall and fluid is modelled by Al-Hadhrami method by changing Darcy and Brinkman parameters. Fluid temperature increase and decrease profiles are determined  depending upon the change in Brinkman number.  In large Darcy number, flow profiles starts to behave as Poiseuille flow. Change of temperature difference between wall and fluid is determined for lower Darcy number related to a constant Brinkman number and vice versa. The role of Nusselt number to indicate the energy loss is indicated. Nusselt number changes depending upon Brinkman and Darcy number is investigated. Maximum temperature difference is determined in low Darcy number. The positive form of Brinkman number shows the increase in fluid temperature and the negative form of Brinkman number shows the decrease in fluid temperature. Temperature distribution is determined by studying the relation between Darcy and Brinkman numbers.