Modeling and ecodesigning crossflow ventilation fans with Mathematica
aa r X i v : . [ phy s i c s . f l u - dyn ] J u l Modeling and ecodesigning crossflowventilation fans with Mathematica
Gianluca Argentini
Research & Development Dept., Riello Burners - [email protected]@gmail.com
Abstract
The efficiency of a simple model of crossflow fan is maximized when the geom-etry depends on a design parameter. The flow field is numerically computedusing a Galerkin method for solving a Poisson partial differential equation.
Keywords incompressible flow, stream function, differential problem, Galerkin method,impeller, fan.
A crossflow, or tangential, fan ([1]) is a ventilating structure where air is movedfrom an inlet zone to an outlet one using an impeller that forces the air to followtrajectories tangentially to the ideal circumferences concentrical to impeller it-self, so that fluid goes across fan on planes that are normal to impeller rotationaxes. The inlet and the outlet zone are physically separated by a fixed object,called vortex wall (see Fig.1).Crossflow fans have the important property to get relatively high values offlow rate with small geometrical dimensions. Two typical engineering questionsare how to design their housing and where to place the vortex wall between theinlet and the outlet zone ([5]). In this work, we try to get an answer to thesecond question using a simple two-dimensional geometry in the
Mathematica numerical environment.
We consider a squared housing D with edge of length L . The impeller is ideallyrepresented by a circle of radius R , with R < L . In a cartesian coordinate1igure 1: A crossflow fan schema.system x, y , the left bottom vertex of the housing is the point ( − L/ , − L/ s , with − L/ < s < L/ s, L/
2] is the inlet, the segment [ − L/ , s ] is the outlet, so thatthe fluid particles have a counter-clokwise movement into the fan.Figure 2: The geometry of fan. Now we describe the mathematical model used for simulating the air-flow intothe fan. We consider a zero-divergence velocity field ( u, v ) in the 2D geometry,so that a scalar stream function Ψ exists such that (see [6]) u = ∂ y Ψ , v = − ∂ x Ψ (1)2he expression ∂ x v − ∂ y u = − ∆Ψ is called vorticity ([3]). If the case ofa simple counter-clokwise rotating field centered at the origin, we have u = − wy, v = wx ([6]), so that − ∆Ψ = 2 w , where w is the rotation angular speed.In the case of our crossflow fan, we can also considered for fluid particles anegative horizontal velocity component in the upper half (0 ≤ y ≤ L/
2) of thesquare and a positive one in the bottom half ( − L/ ≤ y ≤ s = 0,a simple way for describe this field ( u f , v f ) is to consider the formulas u f = − Asin ( πy/L ) , v f = 0 (2)where A is a constant (see Fig.3). - - - - Figure 3: The u f profile for L = 4 . A = 1.If s = 0, we use the correction u f = − Asin [ π ( s + y ) /L ]. Therefore, inthe case of an estimated field ( − wy + u f , wx ), the vorticity is − ∆Ψ = 2 w − πA/Lcos [ π ( s + y ) /L ]. Note that, if s = 0 and y small, using the approximation cosα ≈ w = 2 w cos [ π ( s + y ) /L ], therefore we choose A = 1 (weare interested on comparison between geometries, not on absolute value of flowrate) and obtain the following simple expression for the vorticity of our crossflowfan: − ∆Ψ = (2 w − π/L ) cos [ π ( s + y ) /L ] (3)Previous formula is a differential equation of Poisson type ([9]) for the un-known function Ψ. On the three edges of the square that are walls for the fanwe can impose the usual boundary conditionsΨ( − L/ , y ) = Ψ( x, − L/
2) = Ψ( x, L/
2) = 0 (4)because the normal component of flow along these edges is null (see [6]).3
Numerical resolution of Poisson equation If s = 0, it is possible to try a symbolic resolution of previous differentialproblem, being the equation linear and nothing that a particular solution hasthe form Ψ = Bcos [ π ( s + y ) /L ], with B constant. For our purposes, we applya version of the numerical method of Galerkin ([8]), in the following fashion.We seek for a solution in the formΨ( x, y ) = N X i,j =1 c ij φ ij ( x, y ) (5)where c ij are constants to be computed and φ ij are the linearly independentfunctions φ ij = ( x + L/ y + L/ i ( y − L/ j (6)which satisfy the boundary conditions (4). The choice of these particular test-functions is justified by the fact that the flow is mainly determined by theposition of the upper and bottom wall. Let f ( y ) = 2 w − πA/Lcos [ π ( s + y ) /L ].Then the Poisson equation can be approximated as − N X i,j =1 c ij ∆ φ ij = f (7)Multiply the two members by φ nm and integrate over the square D : − N X i,j =1 c ij Z D ( φ nm ∆ φ ij ) = Z D ( f φ nm ) (8)This is a system of N linear algebraic equations in the unknowns c ij which canbe solved by usual numerical methods.We have computed the coefficients c ij in the case M = 4 using the methodsbuilt-in into the software Mathematica , ver. 7.0, for numerical integration andnumerical resolution of linear systems. The geometrical and physical valueswhich have been used for computations are impeller radius R = 2.0 cm androtation rpm = 2000 ( w = 2 πrpm/ L = 2( R + 0.2) cm, position of vortexwall pvw such that − . ≤ pvw ≤ +1 . pvw = +1.5 is shown in Fig.4. In the last years European Union has emanated some Directives about energyrelated machines, with the purpose of regulating their design and utilizationfor ecological optimizations ([2]). In particular, some studies has revealed theimportance of optimize the design of ventilation fans for having high energeticefficiency ([7]). 4igure 4: The velocity field for pvw =1.5 cm.Suppose that the rotation speed of the impeller is constant for all the possiblevalues of flow rate. This is only a first quite good approximation. Then, theefficiency is determined by the product of the flow rate Q and the pressure head ∆ p of the fan. The pressure head is usually considered as the difference of theflow pressure at outlet and the flow pressure at inlet. The flow rate, defined asthe dot product of the velocity field and the normal to the outlet surface, canbe computed in our case with the formula Q = Z pvw − L/ u ( L/ , y ) dy (9)Pressure p and velocity field v are related by Navier-Stokes equations ρ ( ∇ v ) v = −∇ p , where ρ = 0.001 g/cm is the value of air density, which we suppose tobe constant along the flow. Once we have numerically computed v , we canestimate numerically the pressure along the right vertical side of the fan casingby a path line integral R ( ∇ p ) · d l (see [10]), so that if a is the ordinate of a pointe.g. in the outlet segment [ − L/ , pvw ], we have p ( a ) = Z pvwa ∂ y p ( L/ , y ) dy (10)For our purpose, we define the pressure head of the crossflow fan has the differ-ence of the pressure at the middle point at outlet and the pressure at the middlepoint at inlet: ∆ p = p [1 / pvw − L/ − p [1 / pvw + L/ Q ∆ p for values of pvw in the interval[ − . , . .
1, the profile of the efficiency is shown in Fig.5. - - - pvwQ D p Figure 5: Graph of the product Q ∆ p .The maximum is obtained for pvw = 0.6 cm, an information which were noteasily guessed whitout a numerical analysis. From Fig.6, we can note that, withthis geometrical configuration, the speed at outlet is almost the double of thespeed at inlet. - - - (cid:144) s Figure 6: Profiles of u (black) and v (red) along the right side of the fan, in thecase pvw = 0.6 cm . References [1] F.Bleier,
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