Let us consider an element of linear hydraulic circuit, of transverse area sw and length dl oriented in the direction of the flow w.
Let us suppose, for the moment, that the element is homogeneous and that the 2 phases move at the same celerity c.
dz is the elevation increase,dp is the pressure variation dpf is generally represented (4) as fraction df of the kinetic energy par kg c2/(2 vm)a and y so that the "kinetic balances", integrated over the part e to s of the circuit simplifies to (6).Picture euler_2 represents the node, delimited by the sections e and s.
wy and the internal "expansion source" s= (m1-m2)/δs effects are supposed to be concentrated in a thin, constant transverse surface area sw, entry volume, delimited by the sections e and ê of the node, so that oulet values ws and vms of the entry volume will apply to the whole remaining volume between the sections ê and s.It should be remembered that, in the course of the (m,h,b)-balances, the pressure, as well as the reference entry flows(wel, wea, wed for primary, wsan for secondary), are mean values, constant over δs .
The last m balances do, circuit_name have provided the nodal expansion sources s=(m1-m2)/dt.
The kinetic balances, enabled by the key-word euler recuperates those sources as such and aim now at updating the reference inlet flow of the flow branch containing the node .
Thus w1 actually represents the w assumed in the course of the last mass balances and w2 the searched new flows to be updated by means of the Euler balances.
l=(1,l9), the branch abc and the branch d.
wso outgoing o towards the expansion branch X of the pressu system is determined by closure of total primary system volume.wespr entering the spray is imposed by the pressu regulation and, therefore, considered here as a fixed value. as2 ys of the first node eliminates whith the inlet term -ae2 ye of the second, and if the branches are joined as a loop, like the recirculation loop of the sg those terms disappear, as well as the Δp terms.an from (10)f (11).
f depends on the node geometry and the distribution of singular and distributed friction factors.
f dependence with ws could be represented by a fitted correction.pe - ps across any node.vm are the last calculated values by the m,h-balances.
apzl is calculated (20) from pump head azpl [Accretion,Pressure,due to elevation Z,for loop L], and node geometrical elevations.
azpl is updated at bos (step, euler) it is advisable to put euler as the first keyword of the working sequence, in which case the vm values as well of the "sources sli" are those of preceding step. Note that, although the outlet o does belong to a branch, we have added a gravity contribution apzo for it, in order to insure that the sum of the az on closed loop abcod is zero.
w and wm.vm2) are those resulting from the last mass balances.(w2)2 is developed (23) about its current value w whose first guess is the bso w1, and the quadratic term is neglected.w2li are evaluated (24, 25) in terms of the reference branch inlet w2el and the nodal "sources" sli obtained from last m,h-balances.We thus obtain a linear system of l9+3 equations in searched flows w2al, w2ea, w2ed, and the common apOA ≡ pA -pO whose coefficients depend on the w1li, vm2li, sli, wli
After resolution, wli is updated as w2li, and the calculation is repeated until convergence of the reference flows, governed by the convergence criteria (?).
This method provides an exact solution of the original system of implicit, non-linear, kinetic balance equations.
Distribution of flow in the primary hydraulic network result from the initialization (ini, down-comer, bottom,..) of the primary components m,h,b-balances where the reference flows are imposed as the volumic (wvpl) or massic (wel) for the loops, and from wed or xwld for the dome bypass.
w2li - w1li dropped) the pressure loss apOA contributed by the various branches.apOA generally do not coincide because the input flows are design or measured values, whilst the hydraulic friction data are best calculated.cfrli, ... are operated by multipliers xcfrli, .. calculated in (ini, euler).azPA has been taken as reference and the other branches are adjusted. It would possibly be more sensible to impose apOA and adjust all friction factors on it.the primary components are initialized as usual starting from a with a cold enthalpy hsb= 1.195179e6, corresponding to a temperature of 545 K, which should be imposed as t1cn to the secondary because the system is initially isothermal (f2c=0).
Nevertheless ini, loop_1 adjusts t1cn to insure equilibrium initial state, so that any sensible guess value may be entered as input.
As indicated above, euler is last in the ini, call:2 sequence but is first in the working sequence S4.
zl%qgv) converges to 0. As already mentioned, the iterations must be over-relaxed to insure rapid convergence.a%apoa= pA - pO, and the branches friction multipliers. The loop_1 multiplier bl%xcv is 1 because that loop is the reference one.xcv_a is close to 1 which indicates the friction parameters of the loop and the abc branch are consistent with the input primary flow and pump head. On the other hand, the dome branch multiplier xcvd is far from 1, which means that the dome branch friction factor is not consistent with the imposed flow by-passed towards the dome!apoa ≡ pA - pO which pushes the flows through abc and apoa becomes also negative from sec=300 and represented as its opposite value.
q2c/3 (3 loops!) with the power qvgl1 lost in the sg.