Analysis on Experimental Investigation and Mathematical Modeling of Incompressible Flow Through Ceramic Foam Filters
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IN metallurgy, ceramic foam filters are used to remove solid particles and inclusions from molten metal.[1–7] In order to predict the pressure gradient necessary to achieve the desired flow rate, it is essential to determine the permeability coefficients of the filter.[2,3] To obtain the coefficients (Darcy and Non-Darcy coefficients), permeametry studies[8–10] are needed. Here, the permeametry is based on incompressible liquid flow through a porous media. In deriving the coefficients, the pressure drop DP [Pa] along the height of the porous media L [m] is determined as a function of the superficial velocity Vs [m/s]. Subsequently, the Darcy and NonDarcy permeability coefficients k1 [m2] and k2 [m] can be determined using the Forchheimer equation for incompressible fluids [Eq. 1],[11] when the fluid density q (kg/ m3) and fluid dynamic viscosity l [PaÆs] are known, DP lVs qV2s ¼ þ : L k1 k2
½1
SHAHIN AKBARNEJAD, PhD Student, LAGE TORD INGEMAR JONSSON, Researcher, and P¨AR G¨ORAN J¨ONSSON, Professor, are with the Department of Materials Science and Engineering, Royal Institute of Technology (KTH), Stockholm, Sweden. Contact e-mail: [email protected] MARK WILLIAM KENNEDY, Postdoctoral Fellow, and RAGNHILD ELIZABETH AUNE, Professor, are with Department of Materials Science and Engineering, Norwegian University of Science and Technology (NTNU), Trondheim, Norway. Manuscript submitted February 15, 2016. Article published online May 31, 2016. METALLURGICAL AND MATERIALS TRANSACTIONS B
It has been mentioned in the literature that fluid bypassing, from the gap between the filter and the filter holder during permeametry, can result in an underestimation of the pressure gradient.[2,9,10,12–22] Therefore, it is of vital importance to adequately seal the filter prior to experimental measurements being taken. In addition to the Forchheimer equation [Eq. 1], it is also very common to use the Brinkman equation [Eq. 2] or extended versions of the equation, i.e., the Brinkman–Forchheimer equation, to investigate incompressible fluid flow through a packed bed.[23–25] Brinkman defines the equation [Eq. 2] where P is the pressure, g the fluid viscosity, k the permeability of the porous mass, v is rate of flow through a surface element of unit area, and g0 Dv is the divergence of the stress tensor.[23] g rP ¼ v þ g0 Dv ½2 k here g0 may not be the same as g the fluid viscosity. Brinkman emphasizes that the precise nature of the formula depends on the assumption made for g0 and as an approximation suggests a formula [Eq. 3] where V0 is the total volume of the particles and V the total volume of the column. The equation [Eq. 2] satisfactorily agreed with experiment data only at V0/V values less than 0.6, while using Eq. [3] to estimate g0 . V0 0 g ¼ g 1 þ 2:5 ½3 V Since 1947, when the Brinkman equation [Eq. 2] was introduced, several complementary studies have been performed. Thus, extended versions of the Brinkman VOLUME 47B, AUGUST 2016—2229
equation, i.e., the Brinkman–Forchheimer equation, have been introduced.[24–26] Due to similarities between packe
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