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IntroductionIntroduction

Zach Frye, University of Wisconsin-Eau Claire

Faculty Advisors: Mohamed Elgindi, and John Drost, Department of Mathematics

Funded by: University of Wisconsin-Eau Claire Faculty/Student Research Collaboration Differential Tuition Grants

Zach Frye, University of Wisconsin-Eau Claire

Faculty Advisors: Mohamed Elgindi, and John Drost, Department of Mathematics

Funded by: University of Wisconsin-Eau Claire Faculty/Student Research Collaboration Differential Tuition Grants

An inclined membrane trough can carry fluids without the use of pumps. Also membrane troughs are easy to transport, create, and cost little to manufacture. This research examines the dynamic inclined membrane trough that carries a fluid. We are interested in the flow rate, which varies for various trough shapes and inclinations. In particular, the following question is posed. What would be the maximum flow possible through an inclined membrane trough of given lateral perimeter?

Approximations of the Cross Section of a Membrane Trough that Produces Optimum Flow Approximations of the Cross Section of a Membrane Trough that Produces Optimum Flow

Membrane Trough ModelMembrane Trough Model

Flow Rate Flow Rate The flow rate of the inclined membrane trough is a measure of the longitudinal velocity through the cross sectional shape. The velocity can be determined by solving the following elliptic equation governing longitudinal parallel flow.

where w = normalized longitudinal velocity.

The boundary conditions for the elliptical differential equation are (1) a velocity on the membrane surface of zero , where is the boundary of the shape; and (2) a shear on the surface of zero.

To calculate the flow rate the velocity must be integrated over the cross sectional area of the trough:

Cross section of the membrane trough

Results & DiscussionResults & Discussion

Membrane trough model

Perturbation SolutionPerturbation Solution

The software Matlab uses the finite element method (FEM) to determine the velocity of the above elliptic differential equation with respect to the given boundary conditions, then to integrate for the flow rate as indicated above. Figure (c) shows the FEM mesh used, for the case when the

trough is almost flattened (a = 0.9). The maximum flow rate was found to occur when a = 0.651. Figure (d) shows the flow rate as a function of the trough width a.

The trough is supported by two rigid bars A and B in figure (a). The trough is inextensible so the material of the membrane does not affect the calculation. Since the trough is symmetric longitudinally we only examine half of the cross section of the trough. Furthermore, each variable is divided by half the lateral perimeter length, thus normalizing each variable. The membrane trough is described as follows: (figure (b))

The local curvature is governed by the static pressure difference which is:

The Cartesian coordinates (x ,y) are related to the angle theta by the kinematic relations:

The boundary conditions are then:

Given the half width opening a, the previous equations can be solved for the unknowns , x, y, h, . Using the Matlab numerical software, the boundary value problem can be solved resulting in the shape of the trough.

Flow Rate v. Trough Width

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.2 0.4 0.6 0.8 1 1.2

Parameter a (Half Through Width)

Flo

w R

ate

When a = 1, the trough is flat, and therefore, the solution of the boundary value problem is

known. We use perturbation techniques to approximate the solution for the shape when a 1. We derive analytical formulas for the solution as power series expansions with respect to the parameter = h <<1. Our formulas are valued to the second order of . The analytical approximation is useful in examining the qualitative behavior of the solution set as well in validating the numerical approximations obtained. To achieve this the algebraic tools from the software Maple was used. For instance, the perturbation expansion for is given by:

where,

)()( yhs

),cos()( sx )sin()( sy

0)0()0()0( yx,)1( ax hy )1(

= lateral perimeter length= width of the trough= local angle of inclination= arc length from the bottom= Depth= normalized tension= proportionality constant

12 w

wdydxFR

| 0w

ReferencesWang, Chang-Yi. “Optimum Membrane Trough”. Journal of the Engineering Mechanics Division. P:206-211. April 22, 1981.

Transportation of sanitary water is a global problem, and the use of membrane troughs that deliver optimum flow is one solution. This research also leads to the question of what if the membrane trough was closed to create a tube, what characteristics would give that vessel optimum flow? This new investigation would apply to blood vessels and may enhance our understanding of blood flow in the human body.

(c) Trough shape when a = 0.9

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