Heat Transfer in Porous Media - Literature review Example

Literature Review Table of Contents Literature Review- Heat Transfer In Porous Media 3 3References 9 Literature Review- Heat Transfer In Porous MediaLiterature on heat transfer in porous media stresses the importance of understanding the phenomenon because of the way practical, industrial activities, for instance, such as the composite manufacturing process for polymers, requires an understanding of the mechanics of the way molds are filled in the interim processes for resins, which require modeling, and which require the use of suitable modeling equations as a result (Languri and Ganji n.d., pp. 631-636). The Darcy model is often discussed as a convective model that is relevant for understanding heat transfer in porous media in particular, and is relevant in discussions that attempt to gain insights into the nature of the phenomenon as it applies to various scientific disciplines. This literature review therefore puts some focus on Darcy’s law and explores aspects of Darcy models that relate to the identification and rationale for identification of Pre-Darcy regions. This literature review also defines porous media, first and foremost as material volumes that are characterized or made up of solid matrices, with a hollow or void that is connected through (Khaled and Vafai 2003, pp. 4989-5003; Vafai and Tien 1981, pp. 195-203). Moreover, the proceeding discussion is put into the context of viewing heat transfer in porous media in terms of laminar flows within regions defined as being Darcy parts. The literature states that such laminar flows are governed by Darcy’s law, and therefore this relationship adds relevance to a more in-depth look at Darcy’s law as discussed above. To be more specific, Darcy’s law as it pertains to laminar flows asserts that there is a proportional relationship between fluid velocity on the one hand and the gradient of pressure on the other, given constant measures of resistance (Seguin, Montillet and Comiti 1998, pp. 3571-3761; Holditch and Morse 1976, pp. 1169-1171). When we talk of laminar flows, we talk about smooth and regular fluid paths, as opposed to flows that are turbulent, which are characterized by non-regular, fluctuating, and mixing paths. Streamline flows refer to the same reality of laminar flows, and in such flows, there is a regularity and constancy in the flow properties of a fluid along every point in that fluid, including the pressure and the speed/velocity of the fluid. Laminae refer to layers of water, very thin, that are imagined as constitutive of fluid flows along a horizontal path for instance. The laminae are imagined to all slide from one another to create flow, in the same way that a card deck flows horizontally when stacked and then made to slide (Encyclopedia Britannica 2015). In a more general and pragmatic sense, we define Darcy’s law here as that equation that governs the way fluid flows in a porous medium, as when water flows through porous rock media for instance. Darcy’ law captures the essence of the dynamics of fluid flow through porous media in terms of the differences in pressures between two points where the fluid flows, as well as the impedance characteristics of the porous medium through which the fluid flows. This latter characteristic is defined as the medium’s level of permeability. Pressure is defined as local pressure that goes above the pressure in the fluid from its inherent hydrostatic characteristics, and is something that goes up in proportion to depth, because of gravitational effects. Darcy’s law defines a relationship that is proportional, as discussed above, between drops in pressure on the one hand and the rate of discharge of a fluid passing through a porous medium. In equation form, the flow rate of a fluid is equal to the product of one, the conductivity of the fluid or K, the area of the column in cross section A, and the gradient for hydraulics, or dh/dl (GWPC & IOGCC 2015). K in the previous sentence also refers to the medium’s measure of permeability, and the literature states that permeability is tied to the geometric configuration of a medium (Khaled and Vafai 2003, p. 4994). Still stated another way , Darcy’s law says that there is a direct proportional relationship between changes in elevation in media on the one hand and the fluid flow rate in porous media on the other, as well as an inverse proportion relationship to how far the two points in the medium are separated. With regard to this latter aspect of Darcy’s law, one can expect laminar flows through two points in a porous medium to be of a faster flow rate when the two points in the medium are closer to each other than when they are farther apart. On the other hand, where the elevation change is steep, so does the fluid flow accelerate between two points (Farlex 2015; Preziosi and Farina n.d.). The literature states that in boundaries of media, and in media that are characterized by as being highly porous, the boundaries have effects on heat transfer as well as in fluid flow rates, and so too do forces that are tied to inertia impact flow rates and heat transfer rates. In these places and contexts, Darcy’s law is said to generally break down, because of the simple fact that those conditions are not factored into the Darcy equation. Darcy’s law has no provisions for boundary conditions for the bounds of the solid media, as well as for the impact of forces of inertia on fluid flow rates. Darcy’s law’s limitations are said to be more relevant in modern contexts, where industrial and scientific uses of media that are characterized by high levels of porosity are growing with the passage of time. Various patches have been attempted to make Darcy’s law relevant in such conditions as in the boundaries of media and to consider the effects of inertia, but the observations are generally that those patches that have been attempted via the introduction of relevant inertia and boundary variables into the Darcy equation lack empirical foundations, and therefore are suspect in terms of their applicability in all kinds of contexts. Other attempts at covering up the shortcomings of Darcy’s law in such contexts resort to averaging techniques that in some way account for the missing impact of inertia and the physical bounds of media, but do not resolve the conundrums that occur at the physical bounds and that occur due to the way Darcy’s equation excludes inertial effects in a fundamental sense. Taking off from this line of thinking, one sees the Darcy region as that region where the Darcy-established relationships are valid, and the boundaries of the media constitute regions where the Darcy law collapses and becomes non-applicable (Vafai and Tien 1981, p. 195-196). On the other hand, non-Darcy regions and pre-Darcy regions have more specific meanings in the literature, with non-Darcy flows for instance being defined as high rates of fluid flow resulting in the breakdown of Darcy’s linearly-established relationships. Non-Darcy regions in this latter context are those regions where the flow rates are such that they go beyond the bounds of the regimes of laminar flows, and is interesting to note that in particular physical scenarios, such as in oil wells, such non-Darcy regions are observed at the bounds of the wells (Orodu, Makinde and Orodu 2012). One way to define pre-Darcy flow is to define it in the context of Darcy and non-Darcy regimes. In a medium, the Darcy region is that region where Darcy’s law applies, and the non-Darcy region is that medium region that dominates after the end of the Darcy region. The pre-Darcy region is that region before the medium area where Darcy’s law operates (Orodu, Makinde and Orodu 2012, p. 1939). Elsewhere in the literature, pre-Darcy flows are defined as those regions in a medium defined by very low flow rates of fluids or very low velocities of fluid flows, and discussions in the literature point to some practical applications of knowledge of pre-Darcy flows in oil reservoirs in generating larger yields of oil recoveries from oil fields, for instance. This is a definition of the pre-Darcy region as those regions where the Darcy equation does not hold true, and where the fluid flow rates are not high enough for Darcy’s law to operate. Differentiating pre-Darcy regions from post-Darcy regions, one can say that the differences are with regard to fluid flow rates at the bounds before and after the Darcy equations are relevant, with linear relationships in the equation being operative, and the pre-Darcy region simply as that region before the Darcy region operates, characterized by flow rates that lie below the threshold for Darcy’s law to become applicable (Gavin, Longmuir 2004). Laminar flow through a porous medium, moreover, has been described in the literature as being amenable to characterization via numbers that have no dimensions, including the Reynolds number, the porosity of the medium that is porous, and the coefficient of friction. From this perspective, the flow rate of the fluid by volume as well as the drop in pressure are characterizations of the parameters of the laminar flow process (Hovekamp 2002, pp. 1-7). Reynolds number, which is the fluid velocity times the pipe’s internal diameter divided by the viscosity of the fluid from an absolute point of view, determines whether a flow s laminar or turbulent, with Reynolds\s numbers below 2,300 characterizing laminar flow and Reynolds\s numbers above 4,000 characterizing turbulent flows (PipeFlow Software 2015). The friction coefficient on the other hand relates to the Reynolds number, in a simple equation 64 divided by the Reynolds number. On the other hand, the derivation of pressure drops in laminar flow shows us that pressure drops occur in direct proportion to the flow velocity when those flows are laminar. The Reynolds number therefore establishes whether first, a flow is laminar or not, and two, whether or not one can establish a direct proportion relationship between pressure loss on the one hand and the flow velocity on the other. Laminar flows establish this direct proportion relationship between the pressure drops and the flow velocity. To integrate, medium porosity impacts the Reynolds number, which in turn relates to the coefficient of friction. The pressure drop in turn relates to the flow velocity when the flow is laminar, and laminar flows are determined to be in existence when the Reynolds number is below 2,300. The link between the factors discussed above and heat transfer is that, as explained in the beginning of this literature review, heat transfer in porous media can be discussed as laminar flows in porous media (Sleigh 2009; Bengston 2010). 1 2 3 References Bengston, H. (2010). Pipe Flow Calculation 3: The Friction & Frictional Head Loss. Bright Hub Engineering [online]. Available at: http://www.brighthubengineering.com/hydraulics-civil-engineering/55227-pipe-flow-calculations-3-the-friction-factor-and-frictional-head-loss/ [accessed 4/21/2015] Encyclopedia Britannica (2015). Laminar Flow. Britannica.com. [online]. Available at: http://www.britannica.com/EBchecked/topic/328742/laminar-flow [accessed 4/21/2015] Farlex (2015). Darcy’s Law. The Free Dictionary. [online]. Available at: http://www.thefreedictionary.com/Darcy%27s+law [accessed 4/21/2015] Gavin, Longmuir. (2004). Pre-Darcy Flow: A Missing Piece of the Improved Oil Recovery Puzzle? SPE/DOE Symposium on Improved Oil Recovery. [online]. Available at: https://www.onepetro.org/conference-paper/SPE-89433-MS [accessed 4/21/2015]. GWPC & IOGCC (2015). Fluid Flow in the Subsurface (Darcy’s Law). FracFocus. [online]. Available at: https://fracfocus.org/groundwater-protection/fluid-flow-subsurface-darcys-law [accessed 4/21/2015]. Holditch, S. and Morse, R. (1976). The Effects of Non-Darcy Flow on the Behavior of Hydraulically Fractured Gas Wells. Journal of Petroleum Technology [online]. Available at: http://www.codexpublishinginc.com/portals/66/spe%205586_non-darcy%20flow.pdf [accessed 4/21/2015] Hovekamp, T. (2002). Experimental and Numerical Investigation of Porous Media Flow with regard to the Emulsion Process. Swiss Federal Institute of Technology Zurich. [online]. Available at: http://e-collection.library.ethz.ch/eserv/eth:26305/eth-26305-02.pdf [accessed 4/21/2015]. Khaled, A. and Vafai, K. (2003). The role of porous media in modeling flow and heat transfer in biological tissues. International Journal of Heat and Mass Transfer 46. [online]. Available at: http://vafai.engr.ucr.edu/Publications/new/PDF%20Papers/khaled-6.pdf [accessed 4/21/2015] Languri, E. and Ganji, D. (n.d.). Heat Transfer in Porous Media. IntechOpen. [online]. Available at: http://cdn.intechopen.com/pdfs-wm/13435.pdf [accessed 4/21/2015]. Orodu, O., Makinde, F. and Orodu, K. (2012). Experimental Study of Darcy and non-Darcy Flow in Porous Media. International Journal of Engineering and Technology 2 (12). [online]. Available at:http://www.iet-journals.org/archive/2012/dec_vol_2_no_12/99422135248353.pdf [accessed 4/21/2015]. PipeFlow Software (2010). Reynold’s Numbers. PipeFlow.com. [online]. Available at: http://www.pipeflow.com/pipe-pressure-drop-calculations/reynolds-numbers [accessed 4/21/2015]. Preziosi, L. and Farina, A. (n.d.). On Darcy’s Law for Growing Porous Media. Politecnico di Torino. [online]. Available at: http://calvino.polito.it/~preziosi/pubs/growing.pdf [accessed 4/21/2015]. Seguin, D., Montillet, A. and Comiti, J. (1998). Experimental characterisation of flow regimes in various porous media- I: Limit of laminar flow regime. Chemical Engineering Science 53 (21). [online]. Available at: www.researchgate.net/profile/A_Montillet/publication/244116127_Experimental_characterisation_of_flow_regimes_in_various_porous_mediaI_Limit_of_laminar_flow_regime/links/02e7e53c7898ce4d9a000000.pdf [accessed 4/21/2015]. Sleigh, A. (2009). Laminar and Turbulent Flows. University of Leeds School of Engineering CIVE1400. [online]. Available at: http://www.efm.leeds.ac.uk/CIVE/FluidsLevel1/Unit00/index.html [accessed 4/21/2015]. Vafai, K. and Tien, C. (1981). Boundary and Inertia Effects on Flow and Heat Transfer in Porous Media. International Journal of Heat and Mass Transfer 24. [online]. Available at: hhttp://vafai.engr.ucr.edu/Publications/new/PDF%20Papers/Tien-2.pdf [accessed 4/21/2015] Read More