Laboratory Location: 350 Mech E
Laboratory Supervisor: Ziliang (Liang) Guo (gzliang@me.umn.edu)
Objective:
The primary objective of this lab is the calculation permeability of a glass fiber mat using Darcy's Law coupled with empirical data. The empirical permeability is then to be compared with the permeability calculated from a model based on Cozeny-Karman equation. The permeability values of fiber preforms (such as the mat) are needed in RTM simulations where the fill patterns of complex geometry parts are observed, and process and mold design decisions are made.
Theory:
Resin transfer molding is a process for manufacturing continuous fiber reinforced composites. During impregnation, the fill pattern plays an important role on the quality of the end part, especially in relation to locations of air pockets, unwetted regions and proper placement of air vents. An important processing feature that affects the impregnation behavior is the permeability of the fibrous preform. Permeability is a directional property and it is a measure of how permeable a porous medium is (in RTM, the fibrous preform) to an impregnating fluid. It is a mathematical quantity defined as K in one-dimensional Darcy's Law:
where u is the superficial velocity (velocity of the incoming liquid in the absence of a porous medium), m is the viscosity of the impregnating liquid, DP is the pressure drop over a distance of L during impregnation.
When the impregnating flow is multi-dimensional (usually 2D in RTM), Darcy's Law can be extended to the below 2D form (provided principal permeability directions coincide with x and y coordinate directions):
and 
where u and v are the superficial velocities in x and y directions and Kxx and Kyy are the principal permeabilities in x and y directions.
The relative magnitudes of permeabilities are important in terms of flow front progression (fill pattern). In addition, the absolute permeability values affect the maximum pressure rise or the magnitude of the velocities in the mold during impregnation.
The permeability of a porous medium depends on a number of factors, mostly related to the porous architecture of the preform (capillary/spherical pores, specific surface area, porosity, etc.). The Carman-Kozeny relation is one of the most popular equations, used for calculating permeability of a random porous medium. For 1-D flow, it is:
where e is the porosity of the medium (e.g. 0.4 for 40 %), dg, average diameter of the granules and A, the Kozeny constant (can vary between 60 - 180, depending on the medium architecture). Note that this equation is used for low velocity flows - low Reynolds numbers, up to about 20. Reynolds number in porous media is defined as:
where r is the density of the impregnating liquid.
Figure 1. Porous media for Carman-Kozeny equation
For porous media composed of granules that are not spherical, the average granule diameter, dg, can be taken as
where Ag and Vg are the surface area and the volume of the granule, respectively.
Procedure:
A plexiglass two-part square cavity mold is built for this lab in which a three-layer glass fiber mat is to be impregnated for calculation of the permeability. The impregnating fluid is silicon oil, which has a Newtonian viscosity. The mold has a center gate.
Wear lab gloves - do not touch fibers without the gloves. Also keep your face away from the mat.
Observe/help the lab supervisor place the layers of the mat in the cavity and close the mold. Note ease/difficulty of handling the mat.
The lab supervisor will begin injecting the mold with silicon oil with the help of pressurized air. Note that the volumetric flow rate of the silicon oil is constant.
Record the following:
- the times the flow front crosses the two prespecified positions in the mold during impregnation (these positions are clearly marked on the mold), i.e. (t1,r1) and (t2,r2)
- the pressure value at the pressure gage (P1 at r1) when the flow front crosses the second prespecified position (r2). Note that the pressure at the flow front is always atmospheric (0 gage).
When the impregnation is over, assist the lab supervisor with clean-up (let her decide how much help you can give her).
Be sure to wash your hands with soap before you leave.
Calculations:
Determination of Permeability from the Experimental Data:
From the recorded data, you will calculate the permeability of the three-layer fiber mat. The impregnation pattern was circular so the results in x and y directions will not change. We will refer to either of these directions as r from now on.
For constant volumetric inlet flow rate Q, and constant fluid density, r, the superficial velocity at any r position is:
where h is the depth of the cavity. But u is also the local time derivative of r, i.e.
Thus, the previous equation becomes:
Integrating once:
The flow rate Q, and the constant can be determined by using the time-distance data recorded during the experiment: (t1,r1) and (t2,r2), i.e. two unknowns, two equations.
Once Q is known, so is the velocity, u at any r position.
The permeability will be determined from Darcy's law:
Integrating once with respect r,
The two unknowns in this equation are the permeability value, K and the constant.
When the flow front had crossed the second prepecified position in the mold (r2), the pressure at the gage was recorded (r1,P1). Using the two pressure values at r1 and r2, the two unknowns (and thus, the permeability) can be determined from the above equation.
Determination of Permeability from the Carman-Kozeny Relation:
For the Carman-Kozeny relation, the porosity and the granule diameter need to be calculated. The porosity value is the percentage of pores inside the cavity, i.e.
The fiber volume can be calculated from the fiberglass density and the areal mass of the layers. The total mass of the fibers in the cavity, mf, is calculated as:
where marea is the areal mass of a single fiberglass layer (mass/area) and Ai is the planar area of each cut layer that is to be placed in the mold cavity.
The total volume of the fibers in the cavity, Vf, is calculated as:
where rf is the density of the fibers.
For the calculation of the granule diameter, the surface area and the volume of the granules must be known. The glass mat is a continuous fiber medium. Therefore, the granules are cylinders in this case, with large aspect ratios (length to diameter). Assuming one very long fiber to fill the cavity and including only the lateral fiber surface for surface area calculation, the granule volume and surface area are:
and 
where df and Lf are the fiber diameter and length, respectively. The length of the hypothetical single fiber is:
Thus,
So the equivalent granule diameter is:
Reports:
Your report must be typed, and include the following sections: Objective, Experimental Procedure, Calculations and Discussion. The report for this lab session should not exceed 2 pages (excluding any experimental data sheet handed out during the lab).
In the Procedure section, you will give a brief outline of how the experiment was conducted.
In the Calculations section, you will include the recorded experimental data and detailed calculations for the two permeabilities.
In the Discussion section, you should mainly answer the following questions (but feel free to add any other observations, comments you might have):
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