Hydraulics: theoretical background 2. Water transport through pipes Free water surface = hydraulic grade line 2.1 Introduction One of the basic rules of nature is that water flows Weir from a high energy level to a low energy level. The high energy level can for instance be a storage tank Open at a high level as a water tower or a tank on a hill. It Open channel Open reservoir can also be the high pressure induced by pumps. reservoir For the drinking, sewerage and irrigation practise we Fig. 2.3 - Flow with a weir focus on the flow in open channel or partially filled pipes and flow through surcharged filled closed pipes. cause an out flow at the down stream end, but water These are the most common phenomena in the wa- can be stored in the channels causing a rise in water ter transport in these fields. level. In urban drainage systems a deliberate stor- age is wanted or even needed in the system, for in- stance by putting weirs at the down stream bound- ary. In open channel flow the storage capacity is an im- portant describing factor. The energy difference in pressurized flow in sur- charged and closed pipes is mostly induced by an energy input at the upstream boundary by pumps or by an elevated reservoir. Fig. 2.1 - Water tower Characteristic for water transport through partially filled pipes or open channels is that pressure at the fluid surface is atmospheric. Consequently open channel flow is always induced by surface fluid level difference between the upstream boundary and the Fig. 2.4 - Two reservoirs with pipes at the bottom downstream boundary. The upstream level may, for instance, be an open water surface or an inflow from a house manifold at a sewer system. A downstream flow can be an outflow over a weir. Flow is induced Pipe Pump Outflow by gravity. Fig. 2.5 - Pump and outflow Open channel flow allows for storage of water within the profile, through changes of water level. Upstream Closed and surcharged pipes don’t have significant inflow of water doesn’t necessarily instantaneously storage capabilities. The storage capability is for in- stance the stretching of the pipe and the compres- sion of the water. This causes high pressures be- Free water surface = hydraulic grade line cause of the stiffness of the pipe and the high compressibility of the water and is called water ham- mer and occurs when changes in flow velocity are relatively rapid. This type of flow is described sepa- Open rately. In closed pipe flow friction loss is the main Open channel Open describing factor for energy losses. reservoir reservoir In this chapter we first deal with pressurised flow, Fig. 2.2 - Two open reservoirs with open channel than with open channel flow and conclude with the 1 CT5550 - Water Transport phenomenon water hammer. dx 2.1 Pressurised flow Qin In pressurised flow through pipes the pressure at the A1 start of the pipe is higher than atmospheric. An open (high level) reservoir or a pump induces the pres- Qout sure at the beginning of the pipe (see figure XX’s). Fig. 2.7 -Mass balance Flow through the pipe will cause an energy loss due to friction loss and local losses caused by release of flow lines (entrance and decelaration losses). Schematically all the losses and levels are summa- Qtin¶=Qtout¶+¶¶Ax (eq. 2.3) rized static pressure and dynamic pressure in figure in words: ingoing mass equals outgoing mass plus XX. storage within the control volume. The storage is the changing of the cross section dA. Hdynamic Dividing the mass balance by dxdt gives Hstatic ¶¶AQ +=0 (eq. 2.4) ¶¶tx Momentum equation Fig. 2.6 - Two different energy levels Referring to Battjes (CT3310, chapter 2) the momen- tum equation is Mathematical description ¶Q¶¶æöQp2 QQ +ç÷+gAc+=f 0 (eq. 2.5) Two equations describe the flow through pipes: ¶t¶¶xèøAxAR 3 Continuity equation or mass balance 4 Motion equation or momentum balance with g : Gravitation 2.1.1 Mass balance/continuity equation R : Wet perimeter A control volume of pipe is considered with a cross cƒ : friction coefficient section A and a length dx The mass balance states that ingoing mass equals outgoing mass. Incoming The system of the continuity equation and the mass mass in a time frame dt is: balance are known as the equations of De Saint- Venant (1871) rrQindt= uAdt (eq. 2.1) The dimensionless coefficient cf can be expressed 2 in the Chézy coefficient as cƒ = g/C . With Q=uA the with momentum equation becomes 3 Qin : Incoming volume flow [m /s] r : Specific mass [kg/m3] u : Mean velocity [m/s] ¶u¶A¶¶¶uA2 pgA A+u+20Au++ugA+=2 uu A : Cross section of the pipe [m2] ¶t¶t¶¶¶xxxCR Outgoing mass is analogue with the ingoing mass (eq. 2.6) rrQoutdt= uAdt (eq. 2.2) 2.1.3 Rigid column approach Water transport through pipes is characterised with The mass balance over a time frame dt is slow changing boundary conditions. When the so- 2 Hydraulics: theoretical background ¶A Picture of de Saint-Venant o Elasticity of the pipe is negligible: = 0 ¶t o The fluid meets Newton’s criteria being that vis- Jean Cleade de cosity is constant and only dependent on tem- Saint-Venant (1797- perature. 1886) graduated at the Ecole Polytech- The continuity equation then transforms in nique in 1816. He had ¶Q a fascinating career = 0 which becomes: as a civil engineer ¶x and mathematics ¶Q¶uA¶u¶¶Au teacher at the Ecole =0®=0®+Au=00®= des Ponts et Chaus- ¶x¶x¶x¶¶xx (eq. 2.7) sées where he suc- ceeded Coriolis. Seven years after Navier’s death, And the momentum balance than becomes: Saint-Venant re-derived Navier’s equations for a viscous flow, considering the internal viscous ¶¶upgA stresses, and eschewing completely Navier’s mo- A+gA+=2 uu 0 (eq. 2.8) lecular approach. That 1843 paper was the first to ¶¶txCR properly identify the coefficient of viscosity and its Q role as a multiplying factor for the velocity gradi- substituting u = and dividing by gA gives ents in the flow. He further identified those prod- A ucts as viscous stresses acting within the fluid because of friction. Saint-Venant got it right and 1 ¶¶QpQQ ++=0 (eq. 2.9) recorded it. Why his name never became associ- gA¶¶txCAR22 ated with those equations is a mystery. certainly it is a miscarriage of technical attribution. Considering a piece of pipe with length L and inte- It should be remarked that Stokes, like de Saint- grating the equations over this pipe length gives Venant, correctly derived the Navier-Stokes equa- tions but he published the results two years after decSaint-Venant. x=L¶¶pQx==l1 xl QQ dx=-dx-Þdx In 1868 de Saint-Venant was elected to succeed òòò22 x=0¶¶txx==00gAtCAR Poncelet in the mechanics section of the Académie QQ des Sciences. By this time he was 71 years old, 1 ¶Q p21-p=--LL22 but he continued his research and lived for a fur- gA¶tCAR ther 18 years after this time. At age 86 he trans- (eq. 2.10) lated (with A Flamant) Clebsch’s work on elastic- ity into French and published it as Theorie de ¶Q In stationary flow, the term becomes zero or l’élasticité des corps solides and Saint-Venant ¶t negligible: flow will only slowly change over time. added notes to the text which he wrote himself. For the roughness of the pipe wall in this equation the Chézy-coefficient is used. Often this is replaced called rigid column simplification is applied the pre- 8g by the Darcy-Weissbach friction coefficient l = . sumptions are made: C2 Different formulas are applied to calculate the fric- o Uniform and stationary flow tion coefficient l, which is referred to in the next para- o Prismatic pipe: The cross section of the pipe graph. doesn’t change over the length of the pipe re- Combined with the substitution of the Darcy- ¶A Weissbach friction coefficient the socalled Darcy- sulting in = 0 ¶x Weissbach equation remains: o Water is incompressible 3 CT5550 - Water Transport 8LLQQ l p-p==l 0,0826 QQ 12 p 2gDD55 (eq. 2.11) Another popular representation of the Darcy Weiss- bach formula is: lLu2 D=H (eq. 2.12) Dg2 Fig. 2.8 - Moody diagram 2.3 Friction coefficients and local losses In Battjes (CT2100, chapter 12.4) an extensive elabo- Local losses ration on different friction parameters is given, both Local losses are caused by sudden deceleration of theoretically and experimentally determined. The flows combined with release of flow lines from the most used formulas are those of Manning, Chézy pipe wall. Examples are given in figure 2.9. and White Colebrook. White and Colebrook (1937) performed experiments In fact local losses are separately addressed because to asses how ë varies in the transition from smooth this is a violation of the assumption that the pipes to rough conditions as a function of the Reynolds ¶A are prismatic = 0 . number at a constant relative roughness. They found ¶x that l in technical rough pipes much smoother var- ies than in experiments of Nikuradse. Colebrook de- Local losses depend on the velocity in the pipe and veloped an expression with which l can be are expressed in an analogue formula as the Darcy deterimend as a function of the relative roughness Weissbach equation. in Nikuradse’s coefficient k/D and the Reynolds number: u2 D=H x (eq.