CC BY
17Journal of Siberian Federal University. Engineering & Technologies 5 (2015 8) 669-674
УДК 532.528
Numerical Study Supercavitating of the Pump
Vladimir A. Kulagin*
Siberian Federal University 79 Svobodny, Krasnoyarsk, 660041, Russia
Received 14.02.2015, received in revised form 18.05.2015, accepted 22.06.2015
The problem of effective supercavitating (SC) pump is solved, and optimum load distribution along the radius of the blade is found taking into account clearance, degree of cavitation development, influence offinite number of blades, and centrifugal forces. Sufficient accuracy can be obtained using the equivalent flat SC-grid for design of any SC-mechanisms, applying the «grid effect» coefficient and substituting the skewed flow calculated for grids of flat plates with the infinite attached cavitation caverns. This article gives the universal design method and provides an example of SC-pump design.
Key words: Lagranian coefficient, base line vortex theory, supercavitating grids theory, equivalent grid, grid effect.
DOI: 10.17516/1999-494X-2015-8-5-669-674.
Численное исследование суперкавитирующего насоса
В.А. Кулагин
Сибирский федеральный университет Россия, 660041, Красноярск, Свободный, 79
Решена задача об эффективном СК-насосе и найдено оптимальное распределение нагрузки вдоль радиуса лопасти с учетом величины зазора, степени развития кавитации, влияния конечного числа лопастей и центробежных сил. Показано, что с достаточной точностью можно получить решение, используя эквивалентную плоскую СК-решетку для проектирования любых СК-механизмов, применяя коэффициент «эффекта решетки» и подставляя перекос потока, рассчитанный для решетки плоских пластин с бесконечными прикрепленными кавитационными кавернами. В статье изложен универсальный метод проектирования и представлен пример расчета конструкции СК-насоса.
Ключевые слова: переходной коэффициент, вихревая теория, базовая линия, теория суперкавитирующих профилей, эквивалентная решетка, эффект решетки.
© Siberian Federal University. All rights reserved Corresponding author E-mail address: v.a.kulagin@mail.ru
*
1. Introduction
The one effective application of the cavitation technology are supercavitating pumps (SC-pumps), which have advantages in variety of industrial processes [1, 2]. The basics for calculation of any cavitating equipment, working in the presence of developed cavitation, are their geometry parameters and hydraulic characteristics, i.e. the solution must contain the flow velocity and pressure distribution, and cavitation bubble's dimensions. For the given flow rate and net head, the design of the SC-pump's impeller requires calculation of its profile's geometry, the number of blades etc., as well as calculation of its rotation velocity, axial velocity etc.; but in addition, the design is complicated by generation of the cavity with required length, the number and size of the cavitation bubbles.
2. Load D istribution
Calculation algorithm of SC-pumps is based on the solution of a supercavitating flow around a blade's dlamnnt oe nouivolnnt fiat bldtei °rid [3]. In nhe first stage of the design of r totnay equipment, tint:; loest inatl aUccatiqu alono iht blddd's tgdii it ^aC^c^ifsiiEilLfdd,. aaft factor uudll gasute iloe adquireO oet lie ad with minimi odeygy losses.
Tha hesh k>rcl altocaüan nlong lhe tosis of a real S-iOpsmp'r tPa°o it daund by the hOTmula obtained by tht metlhm oe os^eosfc appooximatioos:
dCp = 4L(1 - obgß)^-df tg--?t-iE + i ' ()
2AP
where --p = —j - the siatic jpresnsi.ir^ o;s) erffice^(e;n't; H = #_<.<, — //_«, - the psetsure gane-ared by the
pto
SC-fc=p; o = (-1/=, = (Wo o.ioc - severse quality of t_e SC-jarofile sncl coeyparitiyn csnf^ the grid (fitst approximation); iiz = ^S^ii bVO z, ■■■ .1 - amesdment to tire limitad numbei oo dtalee; b n- Tr,yr'O - relaHo: radius of the blade; r, -R n -he curuect ami H^Hiku oudn sahius o0 lie hiadt eeopeytihaly; /t - the avgle ctJI^ only ol the rdattve with influence of t^l^is indncCvo speed ftg/d = n1 hg iee Vp1. lg; S— = /sc/CO0 —"tl:t^Er g^a^;; -¿^^c. — k^J?^snudbL
V = hOit^/tglo) ^ thb Lagraoglen coefficirot obtained taan tho solotton of variational L¡l>:l^o^les;пl, oompytud 0[ ttn formula:
L=_ZU - -_
1 + ; b£;i^iii[i 4-+b + i) l/if]' ;2 + 2b+ +1 bb + 1+
Vo + Was
X = tg ßiT = tgßU r =
cor — wfs
where =o f n0/<vr - the relative pace at the pump's inlet; a0, crr - tilt aaM and circumferential speed of fluid relative ln an uleamnt of ths blade alt the pump 's inlet ; whs, wC - lhe i^1 ond ciicumferectM speed al tise ^Ij^^^sm; (cm. pn. 1 By mCegcatinn ihcen /^^hj.^^ dC/df alonit the radii of hla blbde, we find the coeCficienC ol seatic presaure iin the fiaol apemoximneiaf.
entie next htp in calculation nf tee PC-punf) is to define llie spatict charaatarcctics octthe Wade gaip fo^ tli.::^ ;g:^;,e'i<s:iii iriic^iLiLi^!^., ii^lij^c^Wts ■'it^oi^lcl. (^nisiji^i^ iire^axioisi^aia craf^ Itlti^ C = Cy/Cx fop the eoad on the
aadiuo, obtainad in first oSage, a^il talring i^to accannt die conditions of stsength.
The steedy dealt with the SC-grid male of She folCowing pcofifssd flap pltte; baw wrap p^rcj^lle f = //£> el 0, 03' circle'e arch profile
-
third and fifth order (< 0 a,0e); profile with square load allocation (< 0 0,1)). Ranges of parameters for calculation were ahosen as Woliows: celative lengoa of lta caviOy Lc = J=/nc = 1,1 2 0 (lengtO to widah ration); reratiae curvcture of vcofitei fi = 0 -a 0,1; relativa grid ¿:_i'r-c = d.5-1- 1.5°; ^^em
angles gratings |3 r^ ^(0 - 75- ao^es of attack i = V — 20°
In Fig. 1, 2 and 3 vhcw the hodtotlygomir charcctaristics of SC-gridv fon flat pktei and curved profifer witd tt reclahgulcr disaribution r" pre scure aloog the ciht-^rcl e = o(ic,/o,--0,C:y). Here e = CXIC= 00 /(VT^iCyXx)1 Ic )= In/B= o= f/b/T,Cyip).
Design calculation of SC-pumps is based on the "base line" vortex theory, the theory of SC-grids, as well as on the method of equivalent SC-grid. The calculation is carried out in two phases. First determine the optimum load distribution along the radius of the blade on the desired characteristics of SC-pump: static pressure of HS, the flow Q, the rotational speed n, or the diameter blade's outer D (for a given D the unique value n can be evaluated among the several numbers n).
3. Profiling
Once the optimum distribution of circulation (load) is found, one proceed to the choice of structural elements of the blades suitable for this distribution, and profiling in accordance with the strength conditions and optimum for dynamic quality of profiles.
Fig. 1. Triangles of flow velocities, flowing on the blades of the SC-grid with an arbitrary gap
a b c
Fig. 2. Reverse quality factors £ = £(b/t, ¡C CJ for SC-grids of flat plates with -- 00, % -- xmin; where a - ¡0 = 15°; b - ¡0 = 30°; c - ¡0 = 60°
b/1=0,1
0,05
1,6 1,2 1,0 0,6 Will.' v/04
b/i=0,1
_l_I_l_
1,6 , 1,2
0 0,10
a
0,10
b
0,10
c
Fig. 3. Reverse quality factor e = e[b/t, fo, Cf for SC-grid made of curved profiles with the rectangular distribution of pressure along; the chord for fK oo , % Xmi.n , where a o f0 o 15° ; O - f° = 30°; c - ff0 = 6 0°
Optimal load distribution along the blade's rodius is found 0y the formula (1). Expressing the profile dcag by -lie multiplier (1 - Eyf, limit of thenumOer of Olade s 0>y KZi and calculating the Lograngion r from equat- e (2)) Oy the m-th-d of huccs-ive opproxim-iio ns, i.nd the distribution of load Oactoo olong the Olode's radius o;f a leo1 SC-pume oi).
On tln^ basis of the calculations one canbuold -iagoums to determioe the inductive dynamics X,- of optimal SCpemp with o- ;arl:^:]L"^:riaLit:"yT gep ond the <rea^re<3 so:]. cavitation -evelopmnnt for the given finite numO er a-Olade s foo the ntatect oalues of Sf, b jriiid ttrengon ^¡¡i^r^me^^r t) [f.
In thf sec-od staig^ of <o;iJ.cu:lat^oii chose stnuctural riaments of Ijlrdas ^h^t provide found circulation and diohoibution of^ tbe induced spee-s. Usion tire thino-am uZ ZZukovfkiy for liftinreg force on element of the Olafe d) = pvUlrft, an- the following i;x]i)]-^si^i]3n
dy = ^ — bdr
wri-e down tlie fquatien foir coupling of the blade ¡and the flow:
1 — i
2
r =-C-Vib. (3)
Substitutinf (3) inio tfe exxpunejssiion for the static head pressure lightly loaded SC-pump
R
Q H^v^coj r^ r
rs
oOtain pvln b
dH*=-^HzC-r^rdr- (4)
Taking into consideration rhat
= ^r - wis =VZ~ wS; 1 cos/1 sin/
- on -
Hs 8,18
f-0C //y / .
i0? /8f,oHr-
0i6
H/D=2
Fig. 4 Universal cliart orhydrodynamic characteristics for SC-pumps working under developed cavitation with Hs = fid), "5=^ = f-H/f,b/T,x,Kf
then from (3) derive the formula to determine the hydro-mechanical parameter of SC-pump's blade: C --1 s[hPi dCP
^ T 2 f2KtKf 1 -f Was df
It shoulst be ranted that in the charta od hydfodynamic characteristics of SC-grids (see Fig. 2, 3), one crn add the curves of equal valuei of the parameter Crn/T. Thus, defining for each radius the vrlue Pi and parameter CybCTfor a set of SCC-geids, one cam choose the grid thai has the lowest inverse quality
a •
Then, uting r method of the equivalent SC-grid [1], for known b/T, P,■, LK, find a factor which takes into account the influence of the "grid effect" without the skewed flow Jp = JP fi8T >P> Lk) ■ Sfmilar calculations for grids of curved profiles with gibitrary length of the cavern thow what />> - the fa.ctor influencing the "grid effect" (lor grid density b/T > 0.6). But the "grid effect practically does nof depends i n the curvature / profile' s torm and ths cavern le ngth LK. The vo rtnx bpte line the t) ry cao Ire used to find the stream skew tor,- = Afr neat the impeller. By Cp and/p —-ermine the retup angle nf the bladr on ensh radius <r>:
c = e(Cc
hyimp ta r n
' sap
where the designation "imp" - impeller; "sap" - standaalone profile. Writing a, = p - fr, gives cy(ncy\
v=i\ni) =*
¡J \ 'sap
Therefore H/D = nnx%qr. For convenience, the calculations of design write down in a table form. Based on the dercaibed methods, celcnlation of ihe generic diagrams of hydrodynamia chf rarteristics of SC-impellrrs can r>e done .
The Fig. 4 illustrates one rnat diagram of supercavilartng regimer for SC-pump's impellers (b/r = 1; x = 0,25; K£ = 0,96; profile is a wedge-shaped flai plate) chat are used for eimultaneous pumping and processing of the working fluids [1].
4. Conclusion
In its current state, this method allows design of rotary equipment, which simultaneously pumps the medium, and uses effects of cavitation on the pumped medium (SC-pump). The highly turbulent cavitation bubbles collapse, observed under the developed cavitation condition, have numerous applications [1, 4]. For the given conditions and working fluids, designer should be aware of the negative and the positive consequences of cavitation effects, especially for the extremal case - supercavitation.
References
[1] Ивченко В.М., Кулагин В.А., Немчин А.Ф. Кавитационная технология: монография / ред. акад. Г.В. Логвинович. Красноярск: Изд-во КГУ, 1990. 200 с.
[2] Демиденко Н.Д., Кулагин В.А., Шокин Ю.И. Моделирование и вычислительные технологии распределенных систем: монография / ред. чл.-корр. РАН А.М. Федотов. Новосибирск: Наука, 2012. 424 с.
[3] Кулагин В.А. Вильченко А.П., Кулагина Т.А. Моделирование двухфазных суперкавитационных потоков / ред. В.И. Быков. Красноярск: КГТУ, 2001. 187 с.
[4] Kulagin V.A., Likhachev D.S., Feng-Chen Li. Modeling supercavitating flow in supercavitating pumps // Submitted to International Conference on Pumps and Fans (ICPF 2015), Hangzhou, China. 2015.
