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Piping Systems mass mass Energy Energy Conservati on Laws ; IN OUT IN OUT for CV & steady state Ein,i Eout, j Eout,k Ein,l if Ein , Eout on both side of Eq. i j k l J Energy m 2 Energy J e h head 2 ; H Head m mass N kg s Weight W pump P V2 L V2 V2 V2 h H g ; ; gz ; hp ; f ;K ; CfT ; etc. 2 m D 2 2 2 h in,i i hout , j hout,k j k VD QD 4 Q hin,l ; Re D A l circle VD shape CfT Power Q P Q ( h) m h ; f f (Re, ) ; K K D size,Re Piping Systems- Example 1-1 or (a) Type I (explicit) problem : Given: Li, Di, Qi; Find Hi Type II (implicit) problem : Given: Li, Di, Hi; Find Qi Type III (implicit) problem : Given: Li, Hi Qi; Find Di or (b) Example 1-1 (Continue) May be used to find any variable if the others are given, i.e. for any type of problem (I, II, III) Example 1-2 with MathCAD For inlet (0.78) & exit (1), p.18 For valve (55) & elbows (2*30), NOTE: K=CfT Given No pump Re & fT Solution Satisfying conservation of mass and energy equations we may solve or “guess and check” any piping problem …! Hardy-Cross Method & Program 0 Q b 0 • For every node in a pipe network: • Iteration # Eq.1 b Since a node pressure must be unique, then net pressure loss head around any loop must be zero: mustbe h f j 0 Eq.2 j • If we assume Qb0 to satisfy Eq. 1, the Eq. 2 will not be satisfied. • So we have to correct Qb0 for Qloop, so that Eq. 2 is satisfied, i.e.: Express all h f j h f j (Q) as function of Q . Since 0 h f j 0 , then j previous h next ( Q Q ) h (Q) fj fj next i Q dh fprevious j dQ Q ; h next 0 Qloop Q fj j Qi Qloop ...continu e until convergenc e Q L loop1 2 loop tol Hardy-Cross Method & Program (2) ...loops ...pipes Example 1-13 (cont.) Hardy-Cross subroutine ...guess (LX1) …while tolerance is not satisfied …correction (LX1) …new Qs (PX1)=(PXL)(LX1) …result (PX1) The results after Hardy-Cross iterations Example 1-13 Loop 1 Loop 2 This 3rd loop is not independent (no new pipe in it) 2 loops T 1 1 0 0 1 1 “Connection” matrix N 1 0 N 1 1 0 1 3 pipes Example 1-13 (cont.) " Connection " matrix : Example 1-13 (cont.) d’s L’s units & g Pump and reservoirs dhd/dQ derivative roughnesses Kin. viscosity Example 1-13 (cont.) Re & fT Laminar & turbulent f No minor losses Example 1-13 (cont.) Loss & device heads Derivative of h(Q) About constant Qi guesses from conservation of mass 2 loops 3 pipes “Connection” matrix N ...loops ...pipes Example 1-13 (cont.) Hardy-Cross subroutine ...guess …while tolerance is not satisfied Since assumed Q>0 …correction …new Qs …result The results after Hardy-Cross iterations