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# Tp3 siphonic roof drainage systems gutters(dr)

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### Tp3 siphonic roof drainage systems gutters(dr)

1. 1. Siphonic roof drainage systems with Gutters Marc Buitenhuis Hydraulic research engineer Akatherm International BV, Panningen, The Netherlands 02-03-2009AbstractOn the European continent it is common to install siphonic roof drainage systems directly onflat roofs. In other parts of the world, like the UK and Australia for example, the rain water iscollected in gutters first.In this article the theory behind the flow in gutters is described. 1. Introduction: Flow in gutters q qOn the European continent it is common toinstall siphonic roof drainage systemsdirectly on flat roofs. In other parts of theworld, like the UK and Australia for vexample, the rain water is collected in D v + dvgutters first. βUsually these gutters are installed under a A A + dAsmall inclination so that the water will flow Wtowards the roof outlet, driven by gravity.The siphonic drainage system will add asuction force to this through its working Mass continuityprinciple.The flow will also experience counter The mass flowing out of this controlforces: besides the friction force there is an volume must equal the incoming massopposing hydrostatic pressure resulting flow, which consists of the mass flowfrom the additional amount of water upstream of the control volume plus thesupplied by the runoff from the roof along source term of the runoff from the roof:the gutter length. In this article the equations that describe Q = A( x ) ⋅ v( x ) + q = A( x + ∂x ) ⋅ v ( x + ∂x ) =gutter flow will be presented. [ A( x ) + ∂A( x )] ⋅ [v( x ) + ∂v( x )] => q = A( x ) ⋅ ∂v ( x ) + ∂A( x ) ⋅ v( x ) + ∂A( x ) ⋅ ∂v ( x ) 2. Theoretical backgroundA control volume can be defined for which Momentum equationthe mass and momentum continuityequations can be derived. Similarly the momentum forces acting on the control volume can be constructed.
2. 2. The inertia forces of the flow in the length Falling water termdirection of the gutter are constructedsimilarly as the mass continuity: The water dropping from the roof in to the M = −ρ ( x ) ⋅ v( x ) + ρ ( x +∂ ) ⋅ A A 2 x gutter will add its inertia to the controlv ( x +∂ ) 2 = 2 ρ ( x ) ⋅ v ( x ) ⋅ ∂ ( x ) + x A v volume system. In the length direction this accounts for:ρ A( x ) ⋅ v( x ) 2 +O ( ∂ ) 2 ∂ x Frunoff = ρqv f ⋅ ∂x ⋅ sin (θ ) withFrictional Force v f = 2 g ( h0 + x sin (θ ) )The frictional forces acting on the flow are Hydrostatic pressure termpresent on the wall boundaries only.  ∂p  As mentioned above there will be a net τF friction = − w  p + ∂x  2  static pressure force acting on the controlThe wall shear stress can be expressed by: volume caused by the additional amount of ρv 2 water supplied by the runoff.τw = f with For an arbitrary cross sectional area with 2 1 the form function L(y) describing the 7.4n 2 width of the gutter at depth y the staticf = 1 / 3 ; n = 0.0382ε 6 RH pressure term can be described by: Fp = ρg ∫ L( y ) ⋅ y ⋅ dy Dand RH/ 3 = A / p 1 0or rewritten: Cρv 2τ w = 1 / 3 with C = 3.7n 2 RH q q v v + dv β The net hydrostatic pressure force in the τw g length direction is then described by: dFp d  D  L( y ) ydy ∂x dx  ∫0 ∂x = ρg  dx Gravitational Force or:The gravitational force has a component in dFp d  D dDthe length direction when the gutter is ∂x = ρg  ∫0 L( y ) ydy   ∂xinstalled under inclination: dx dD   dx  ∂A Fgravitation = ρg  A + ∂x sin (θ ) ≈ Total Momentum equation  2   ∂A  Taking all first order terms and dividing byρg  A + ∂x ⋅ S0  2  ρ∂x the following equation results:when θ is small and S0 is the slope.
3. 3. (v ∂A + 2 Av∂v ) = ∂( Av ) = 2 2 The maximum depth will be found when ∂x ∂x the nominator is 0: 2 Cv 2 Cp gA3 S0 v f A S0 2 A− p + gAS 0 + qv f S 0 − − 2 2 − + = 0 => RH/ 3 1 RH/ 3 1 q x qx 2 x d  D dD Cp 2 A  gA3 v f A  2  L( y ) ydy  dD  ∫0g  + = 2 2 +  S0  dx RH/ 3 1 x qx 2  q x   q⋅x From this equation the optimum slope ofwith v = the gutter can be determined. A(v ∂A + 2 Av∂v 2 = ) ( ∂ Av 2 = ) The above theory neglects one effect and ∂x ∂x that is the effect of the suction of the dA ∂D siphonic system. Thus the above equationv2 + 2qv dD ∂x is the worst case depth profile.Substitution in the equation above and 3. Conclusions ∂Dderiving from it gives: ∂x In this article the theory of the flow in 2 Cp gA3 S0 v f A S0 2 A gutters has been described. The theory − 2 2 − + neglects the effect of suction by the∂D R1 / 3 q x qx 2 x = H siphonic system and thus the equations∂x dA A g d2 L( y ) ydy D dD q 2 x 2 dD ∫0 − predict a worst case depth profile. 4. Referenceswhich defines the depth profile along thegutter. 1. Gwilym T. Still, Flows in Gutters &When the denominator becomes 0 the Downpipes, 2001, University of Warwick, school of engineeringequation goes to infinity. In practice theflow will then go from supercritical tosubcritical and form a hydraulic jump ( vFr = = 1 with h=hydraulic ghdepth=A/surface width).Also when x tends to zero (far end ofgutter) the solution tends to infinity. So itis necessary to take an approximation forthe depth at this point.When the friction or hydraulic pressureterms increase the gradient of the depthprofile increases. When the slope or fallingwater velocity increases the gradientdecreases, i.e. the water is flowing awayeasier. From this the conclusions can bedrawn that decreasing the friction orincreasing the slope helps to minimize thewaterlevel in the gutter. The slope will belimited practically.
4. 4. (v ∂A + 2 Av∂v ) = ∂( Av ) = 2 2 The maximum depth will be found when ∂x ∂x the nominator is 0: 2 Cv 2 Cp gA3 S0 v f A S0 2 A− p + gAS 0 + qv f S 0 − − 2 2 − + = 0 => RH/ 3 1 RH/ 3 1 q x qx 2 x d  D dD Cp 2 A  gA3 v f A  2  L( y ) ydy  dD  ∫0g  + = 2 2 +  S0  dx RH/ 3 1 x qx 2  q x   q⋅x From this equation the optimum slope ofwith v = the gutter can be determined. A(v ∂A + 2 Av∂v 2 = ) ( ∂ Av 2 = ) The above theory neglects one effect and ∂x ∂x that is the effect of the suction of the dA ∂D siphonic system. Thus the above equationv2 + 2qv dD ∂x is the worst case depth profile.Substitution in the equation above and 3. Conclusions ∂Dderiving from it gives: ∂x In this article the theory of the flow in 2 Cp gA3 S0 v f A S0 2 A gutters has been described. The theory − 2 2 − + neglects the effect of suction by the∂D R1 / 3 q x qx 2 x = H siphonic system and thus the equations∂x dA A g d2 L( y ) ydy D dD q 2 x 2 dD ∫0 − predict a worst case depth profile. 4. Referenceswhich defines the depth profile along thegutter. 1. Gwilym T. Still, Flows in Gutters &When the denominator becomes 0 the Downpipes, 2001, University of Warwick, school of engineeringequation goes to infinity. In practice theflow will then go from supercritical tosubcritical and form a hydraulic jump ( vFr = = 1 with h=hydraulic ghdepth=A/surface width).Also when x tends to zero (far end ofgutter) the solution tends to infinity. So itis necessary to take an approximation forthe depth at this point.When the friction or hydraulic pressureterms increase the gradient of the depthprofile increases. When the slope or fallingwater velocity increases the gradientdecreases, i.e. the water is flowing awayeasier. From this the conclusions can bedrawn that decreasing the friction orincreasing the slope helps to minimize thewaterlevel in the gutter. The slope will belimited practically.
5. 5. (v ∂A + 2 Av∂v ) = ∂( Av ) = 2 2 The maximum depth will be found when ∂x ∂x the nominator is 0: 2 Cv 2 Cp gA3 S0 v f A S0 2 A− p + gAS 0 + qv f S 0 − − 2 2 − + = 0 => RH/ 3 1 RH/ 3 1 q x qx 2 x d  D dD Cp 2 A  gA3 v f A  2  L( y ) ydy  dD  ∫0g  + = 2 2 +  S0  dx RH/ 3 1 x qx 2  q x   q⋅x From this equation the optimum slope ofwith v = the gutter can be determined. A(v ∂A + 2 Av∂v 2 = ) ( ∂ Av 2 = ) The above theory neglects one effect and ∂x ∂x that is the effect of the suction of the dA ∂D siphonic system. Thus the above equationv2 + 2qv dD ∂x is the worst case depth profile.Substitution in the equation above and 3. Conclusions ∂Dderiving from it gives: ∂x In this article the theory of the flow in 2 Cp gA3 S0 v f A S0 2 A gutters has been described. The theory − 2 2 − + neglects the effect of suction by the∂D R1 / 3 q x qx 2 x = H siphonic system and thus the equations∂x dA A g d2 L( y ) ydy D dD q 2 x 2 dD ∫0 − predict a worst case depth profile. 4. Referenceswhich defines the depth profile along thegutter. 1. Gwilym T. Still, Flows in Gutters &When the denominator becomes 0 the Downpipes, 2001, University of Warwick, school of engineeringequation goes to infinity. In practice theflow will then go from supercritical tosubcritical and form a hydraulic jump ( vFr = = 1 with h=hydraulic ghdepth=A/surface width).Also when x tends to zero (far end ofgutter) the solution tends to infinity. So itis necessary to take an approximation forthe depth at this point.When the friction or hydraulic pressureterms increase the gradient of the depthprofile increases. When the slope or fallingwater velocity increases the gradientdecreases, i.e. the water is flowing awayeasier. From this the conclusions can bedrawn that decreasing the friction orincreasing the slope helps to minimize thewaterlevel in the gutter. The slope will belimited practically.
6. 6. (v ∂A + 2 Av∂v ) = ∂( Av ) = 2 2 The maximum depth will be found when ∂x ∂x the nominator is 0: 2 Cv 2 Cp gA3 S0 v f A S0 2 A− p + gAS 0 + qv f S 0 − − 2 2 − + = 0 => RH/ 3 1 RH/ 3 1 q x qx 2 x d  D dD Cp 2 A  gA3 v f A  2  L( y ) ydy  dD  ∫0g  + = 2 2 +  S0  dx RH/ 3 1 x qx 2  q x   q⋅x From this equation the optimum slope ofwith v = the gutter can be determined. A(v ∂A + 2 Av∂v 2 = ) ( ∂ Av 2 = ) The above theory neglects one effect and ∂x ∂x that is the effect of the suction of the dA ∂D siphonic system. Thus the above equationv2 + 2qv dD ∂x is the worst case depth profile.Substitution in the equation above and 3. Conclusions ∂Dderiving from it gives: ∂x In this article the theory of the flow in 2 Cp gA3 S0 v f A S0 2 A gutters has been described. The theory − 2 2 − + neglects the effect of suction by the∂D R1 / 3 q x qx 2 x = H siphonic system and thus the equations∂x dA A g d2 L( y ) ydy D dD q 2 x 2 dD ∫0 − predict a worst case depth profile. 4. Referenceswhich defines the depth profile along thegutter. 1. Gwilym T. Still, Flows in Gutters &When the denominator becomes 0 the Downpipes, 2001, University of Warwick, school of engineeringequation goes to infinity. In practice theflow will then go from supercritical tosubcritical and form a hydraulic jump ( vFr = = 1 with h=hydraulic ghdepth=A/surface width).Also when x tends to zero (far end ofgutter) the solution tends to infinity. So itis necessary to take an approximation forthe depth at this point.When the friction or hydraulic pressureterms increase the gradient of the depthprofile increases. When the slope or fallingwater velocity increases the gradientdecreases, i.e. the water is flowing awayeasier. From this the conclusions can bedrawn that decreasing the friction orincreasing the slope helps to minimize thewaterlevel in the gutter. The slope will belimited practically.