Unformatted text preview: ENU 4134 – Pressure Drop Models
D. Schubring August 16, 2010 Learning Objectives
1ei Use empirical models (including those not based on the HEM or SFM) to estimate pressure drop 1eii Develop a correlation for adiabatic, twocomponent twophase pressure drop and compare to literature models (in project) 1eiii Identify issues related to experiments on twophase ﬂow (mostly in project) 1eiv Estimate design requirements and propose system for acquisition of pressure drop data (in project) 5b Use correlations and/or models to analyze problems in nuclear thermal hydraulics 5c Use appropriate software (EES, TK Solver; scripting/compiled languages) to automate evaluation of correlations for a range of conditions or for more complex problems such as singlechannel analysis (mostly in project) 5d Identify assumptions used in development of models and critically evaluate the applicability of these assumptions for the TH conditions being modeled Notes
Course: In the next 56 weeks, the emphasis will shift from lecturedriven theoretical understanding to problemdriving empirical analysis. Topics covered: pressure drop (these notes, Project 1); critical ﬂow (HW 4.5); twophase convective heat transfer (HW 5 and 6); and nuclear heat generation/transfer and SCA (HW 7, Project 2). Pressure Drop Models Considered: The total number of published pressure drop models for straight pipe ﬂow easily extends into the 100’s. Diﬀerent correlations are advised for steamwater (most nuclearrelevant), refrigerants (deemphasized in this course), and twocomponent ﬂows (e.g., airwater). Most industrial applications are pureﬂuid; most experiments are twocomponent (why is this?). Pressure Drop Models Outline LockhartMartinelli (mostly twocomponent) MullerSteinhagen and Heck (twocomponent or refrigerants) ¨ Review of HEM (pure ﬂuids, steamwater at high G ) MartinelliNelson (steamwater at lowtomoderate G ) ArmandTreschev (steamwater, BWRtype conditions) Wallis (annular – warning: diﬃcult) Asali (annular – warning: diﬃcult) The text also provides the models of Thom (similar to MN), Baroczy (similar to MN), Jones, and (one of the correlations by) Chisholm. These are no longer frequently used; they are not required for this course. LockhartMartinelli Model Assumptions: 1. The pressure (gradient) of the two ﬂuids is equal at any axial position (i.e., pl (z ) = pv (z )) 2. Singlephase relations can be applied within each phase LockhartMartinelli Parameter X2 = (dP /dz )lfric (dP /dz )v fric fl De fv De
2 Gm (1 − x )2 2ρl 2 Gm x 2 2ρv (1) dP dz dP dz l =
fric v (2) (3) =
fric Note: These are not calculated by assuming all the ﬂow is as liquid (or gas). LockhartMartinelli Parameter (2)
Most often, a turbulent smoothtube relation is used to compute 2 X 2 . When this is done, Xtt is usually used as the symbol. The most frequent relation is the McAdams (seen below, n = 0.2) dP dz
l fric 2 Rel−0.2 Gm (1 − x )2 De 2ρl Gm (1 − x )De == µl 2 Re −0.2 Gm x 2 = 0.184 v De 2ρv Gm xDe == µv = 0.184 (4) (5) (6) (7) Rel dP dz
v fric Rev LockhartMartinelli Parameter (3) 2 Xtt 2 Xtt 2 Xtt = = = Rev Rel 0.2 1−x x
0.2 2 ρv ρl
2 (8) ρv ρl (9) (10) x µl (1 − x )µv µl µv
0 .2 1−x x
1 .8 1−x x ρv ρl LockhartMartinelli Multiplier The LM correlation is for the twophase multipliers, φ2 and φ2 v l (not φ2 and φ2 ). vo lo The conversions are as follows: φ2 lo φ2 vo = φ2 (1 − x )2−n l = φ2 x 2−n v (11) (12) ... where n = 0.2 when the McAdams correlation is used. LockhartMartinelli Correlations
If frictional pressure gradient is known, φ2 and φ2 can be v l calculated from the data and then correlated. Lockhart and Martinelli proposed (with X as the positive square root of X 2 ): φ2 = 1 + l φ2 v 1 C +2 X X = 1 + CX + X 2 (13) (14) ... with C as a constant based on whether the ﬂuid is laminar or turbulent. Assuming the latter: φ2 = 1 + l φ2 v 20 1 +2 Xtt Xtt 2 = 1 + 20Xtt + Xtt (15) (16) Coupling to Void Fraction After some tedious algebra: φ2 = (1 − α)−2 l (17) (18) This estimate of α are then used in gravitation and acceleration parts. Note that the estimate of α requires φl , even if dp /dz takes φv . Which Multiplier? (Not in Text)
The two multipliers are not consistent for the same ﬂow. This stems from the empirical correlations for φ2 and φ2 . v l The transition is based on the liquid Reynolds number: Rel == ρl {jl }De Gm (1 − x )De = µl µl (19) When this is larger than 4000, φ2 is to be used. Otherwise, φ2 . v l At typical BWR conditions (ρl = 740 kg m−3 , µl = 9.5 × 10−5 kg m−1 s−1 , De ≈ 0.01 m), this requires {jl } > 0.05 m s−1 (or Gl >37 kg m−2 s−1 ). So, for reactor applications, φ2 is almost always the appropriate l choice. Low pressure, twocomponent annular ﬂow data can require both. Adjustments to LockhartMartinelli (Not in Text) Rather than developing a completely new correlation, some researchers have preferred to adjust LM to their data. Adjustments have included: Using φ2 or φ2 exclusively v l Selecting a diﬀerent singlephase friction factor Fitting the parameter(s) C Comments on LockhartMartinelli
This is likely the most often cited twophase pressure drop correlation. (To many chemical engineers, it is the only correlation). For comparing correlations, LockhartMartinelli is the ”Mendoza Line” of models – any new proposal that cannot perform better is rapidly discarded. While the correlation started with assumptions regarding physics – mechanical equilibrium and the applicability of singlephase pressure ﬁelds – it ends up being a curve ﬁt by virtue of φ2 (X 2 ). Integrating the LM correlation (total pressure loss in channel during boiling) can be a challenge as the function is non linear in x and branches about Rel . Typically, it is handled numerically (ﬁnite volume approach). The d α/dz Term in LM
As a type of separated ﬂow model, the LM correlation must be able to produce estimates of dx /dz (presumably through an energy balance) and d α/dz (below): dα d α dx d α dXtt dx = = dz dx dz dXtt dx dz d α/dXtt from Equation 1195c:
2 dα (CXtt + 2) Xtt + CXtt + 1 =− 4 3 2 dXtt 2Xtt + 4CXtt + (2C 2 + 4) Xtt + 4CXtt + 2 (20) (21) dXtt /dx n = 0.2 for McAdams: dXtt (n − 2) (1 − x )n/2 x 3 = n/2 dx x (2x 5 − 4x 4 + 2x 3 ) Conclusion: not usually worth the pain for pure ﬂuids. (22) MullerSteinhagen and Heck – Motivation ¨
Consider, at a constant Gm , the function dP /dzfric (x ): There are two boundary conditions that must be satisﬁed: singlephase expressions at x = 0 and x = 1. The function has been observed to be continuous and reasonably smooth. There is a single peak in the function, at a quality of x = 0.9 to x = 0.95, depending on ﬂuid properties, Gm , and geometry. (Enrichment: this is clearly an annular ﬂow. Indeed, this peak occurs at roughly the same ﬂow quality as the critical ﬁlm ﬂow rate.) Twophase multplier methods (i.e., SFM methods) will either have a discontinuity or violate a boundary condition. M¨llerSteinhagen u and Heck is a purely empirical method to estimate frictional pressure gradient only – acceleration and gravity terms must come from somewhere else. MullerSteinhagen and Heck Correlation ¨
dP dz fric ,lo Relo dP dz fric ,go Rego dP dz fric ,MH GMH
2 Gm 2Dh ρl − = 0.316Relo 0.25 (23) (24) = Gm Dh µl
2 Gm 2Dh ρg − = 0.316Rego0.25 (25) (26) = Gm Dh µg dP x3 dz fric ,go dP dP − dz fric ,go dz fric ,lo = GMH (1 − x )1/3 + = dP + 2x dz fric ,lo (27) (28) Note: GMSH is not a mass a ﬂux, but a pressure gradient (Pa/m). MullerSteinhagen and Heck Comments ¨ For a wide range of ﬂuids, this is observed to provide a good estimate when it is integrated from x = 0 to x = 1 and a fair estimate at any given x . It is a strong performer in twocomponent annular ﬂow. No estimate of α is evident. Recommendation: use (1) drift ﬂux model for bubbly/slug/churn, (2) MN void fraction (Figure 1117) for any steamwater regime, or (3) RouhaniAxelsson for annular twocomponent (as in project). If you need it, estimate d α/dz with ﬁnite diﬀerence Review of HEM
HEM:
1G 2 fTP Dh 2ρm + Gm dx volfg + ρm g cos (θ) + dp dz m = − 2 ∂ vol dz 1 + Gm x ∂ p g
2 (29) General SFM: −
2 dp 2 x ∂ volg 1 + Gm dz {α} ∂ p 2 +Gm 2 +Gm = flo φ2 lo 2 1 Gm + ρm g cos (θ) Dh 2ρl 2xvolg 2 (1 − x ) volf dx − {α} {1 − α} dz − x 2 volg (1 − x )2 volf + {α}2 {1 − α }2 d {α} dz (30) Assume dx /dz = d α/dz = 0, incompressible gas, upﬂow: Review of HEM
HEM: − General SFM: −
2 1 Gm dp = fTP + ρm g dz Dh 2ρm (31) 2 dp 1 Gm = flo φ2 + ρm g lo dz Dh 2ρl (32) Phrase the HEM with φlo (as an SFM model): − dp dz = flo φ2 ,HEM lo = = ρl fTP ρm flo ρl µTP ρm µf
2 1 Gm + ρm g Dh 2ρl (33) (34) φ2 ,HEM lo φ2 ,HEM lo n (35) In the event that dx /dz = 0, or compressible gas, or nonupﬂow: the friction term still holds. MartinelliNelson Model The MartinelliNelson model is designed to apply directly to steamwater systems. Assumptions: φ2 is a function of ﬂow quality and ﬂuid properties. Since lo twophase ﬂow occurs along the saturation line, P and T are coupled so that φ2 = φ2 (P , x ) only. lo lo φ2 need not be considered. vo Computation of φ2 in MN lo Most often φ2 is looked up from a ﬁgure. The ﬁgure is often lo generated from the results of the LockhartMartinelli correlation (except with n = 0.25 instead of n = 0.2 – listed ﬁrst) or the analytical expression of Jones (listed second): φ2 lo φ2 lo 1 20 + 2 (1 − x )1.75 Xtt Xtt ρf − 1 x 0.824 + 1 = 1.2 ρg = 1+ (36) (37) Although these look much diﬀerent than one another, they produce similar results. For constant heating, an average multiplier can be computed (such as that from the LM correlation): φ2 lo = 1 x
x 1+
0 20 1 +2 Xtt Xtt (1 − x )1.75 dx (38) φ2 in MN lo
Figure 1115 in T&K α in MN
Figure 1117 in T&K Integral Approach
MN is most useful when using an integral approach (computing ∆P rather than dP /dz ). When gas compressibility is ignored (reasonable for industrial steamwater applications, for which ∆P << Pabs and constant heating is assumed, the SFM pressure diﬀerence becomes: ∆P = + + G2 L flo × m× De 2ρl xout
2 Gm ρl xout 0 φ2 dx lo (39) (1 − xout )2 x 2 ρl + out − 1 1 − αout αout ρv
xout Lρl g cos(θ) xout 1− 1−
0 ρv ρl α dx This is sometimes written as: ∆P =
2 flo Gm L G2 (r3 ) + m (r2 ) + Lρl g cos(θ)(r4 ) 2De ρ l ρl (40) Integral Approach (2) The ﬁrst part of the acceleration term appears to produce garbage (0/0) when xout = 1, but it actually goes to 0. We’ll skip the analytical x (α) and related calculus and provide the result that, in this limit: ∆P = +
1 G2 flo × m ×L φ2 dx lo De 2ρl 0 2 Gm ρl − 1 + Lρl g cos(θ) ρl ρv (41)
1 1− 1−
0 ρv ρl α dx r2 , r3 , and r4 tables still work in the limit xout = 1. Graphical Correlation of r2 (acceleration)
Figure 1118 in T&K, listed as for the Thom correlation (same value) – can calculate from equation; don’t really need a chart Graphical Correlation of r3 (friction)
Figure 1116 in T&K Graphical Correlation of r4 (gravity)
Figure 1120 in T&K, listed as for the Thom correlation (same value) Comments on MartinelliNelson Based on real data, φ2 is not independent of Gm or even geometry lo Gm . MN assumed a separated ﬂow in their model (which in vertical ﬂow implies annular ﬂow). Since dispersed ﬂows (e.g., bubbly) occur and regime transitions are functions of Gm and De (in general), the implicit regime identiﬁcation is incorrect. The MN model works best at moderate Gm (5001000 kg m−2 s−1 ). This is somewhat below typical reactor Gm values, for which HEM is typically better ArmandTreschev Correlation Correlation for α (old news): α = β [0.833 + 0.005 ln(10p )] (42) p in MPa This relation for α is enough to compute ρm and the gravitational part. The accelerational part comes from the dx /dz term (given or energy balance) and the d α/dz term. It is fairly strong for incore BWR calculations. TwoPhase Multiplier for AT
For β < 0.9 and α < 0.5: φ2 lo For β < 0.9 and α > 0.5: φ2 lo 0.48 (1 − x ))1.75 (1 − α)n n = 1.9 + 0.0148p = (44) (45) = (1 − x )1.75 (1 − α)1.2 (43) p in MPa For β > 0.9: φ2 lo p in MPa = 0.0025p + 0.055 (1 − β )1.75 (1 − x )1.75 (46) Annular FlowSpeciﬁc Models (Film Roughness Models)
Force balances: τi ρcore g D − 2δ 4 τw =− D − 2δ 4 1−
2 ρcore jg ,core Pabs dP − dz (47) − RD (UD − UE ) = τi − D − 2δ D 1 dP + ρl g 4 dz D 2 − (D − 2δ )2 D (48) Need a correlation for dP /dz (total, not just frictional), τw or (usually) τi + appropriate entrainment information/models/assumptions. Wallis Model for τi fWallis fWallis ,modiﬁed fWallis ,modiﬁed τi = 0.005 1 + 300 = fBlasius δ D δ D δ D (49) (50) (51) (52) 1 + 300 − = 0.079Resg0.25 1 + 300 ρg (jg ,core − vliquid ,interface )2 =f 2 Simple Approach to Wallis Correlation
Assume that entrainment is negligible. ρcore = ρg , RD = 0 (droplet deposition mass ﬂux). Further assume that vliquid ,interface = 0 and jg ,core = jg . Solve the following equations simultaneously (TK Solver, EES, etc.): τi τi τw =
− 0.079Resg0.25 δ 1 + 300 D
2 ρcore jg ,g Pabs 2 ρ g jg 2 (53) (54) =− = τi − D − 2δ 4 1− dP D − 2δ − ρg g dz 4 D − 2δ D 1 dP + ρl g 4 dz D 2 − (D − 2δ )2 D (55) Need δ from experimental data. More Sophisticated Use of Wallis – Liquid Film Flow
y – distance from wall, u – general velocity, u – friction velocity. Subscript +: nondimensional wall coordinate u = τ ρl u (y ) u yu ν δu ν (56) (57) (58) (59) u+ = y+ = δ+ = Universal velocity proﬁle (“Universal” than to “proﬁle”.) + y + 5.0 ln y + − 3.05 u= 2.5 ln y + + 5.5 applies more to “velocity” if y + < 5 if 5 < y + < 30 if 30 < y + (60) More Sophisticated Use of Wallis – Liquid Film Flow 2
The value of vliquid ,interface is then solved iteratively with τ , etc. In lowpressure annular ﬂow, τi ≈ τw , so the ﬁlm velocity proﬁle can be nondimensionalized with either. A dimensionless ﬁlm ﬂow rate, mL,ﬁlm , is deﬁned by: ˙+ mL,ﬁlm = ˙+ mL,ﬁlm ˙ π D µl the ﬁlm height: if δ + < 5 if 5 < δ + < 30 if 30 < δ + (62) (61) By integrating the UVP, this can be related to 2 0.5 (δ + ) mL,ﬁlm = ˙+ −64 + 3δ + + 2.5δ + ln (δ + ) 12.05 − 8.05δ + + 5δ + ln (δ + ) More Sophisticated Use of Wallis – Entrainment
Whatever doesn’t ﬂow in the ﬁlm must be entrained, allowing calculation of ρcore : = ml − ml ,ﬁlm ˙ ˙ = (mg + ml ,Ent ) ˙ ˙ ml ,Ent ˙ mg ˙ + ρg ρl
−1 ml ,Ent ˙ ρcore (63) (64) Ishii and collaborators have suggested the following model for droplet deposition rate, RD : RD
− = 0.022 (ρcore − ρg ) Usg Reg 0.25 ρg ρcore − ρg 0.26 (65) Take UD as Usg (droplets deposit at the core velocity) and UE as vliquid ,interface (droplets are entrained at the top of the liquid ﬁlm). These equations (obviously!) require a computer solver. Need to use Equations 47 and 48, not the simpliﬁed versions. An Alternate τi Correlation – Asali Asali further couples the τi correlation into the friction factor, using δ + as the nondimensional ﬁlm thickness parameter rather than δ/D : τi ,As φHN
− − = 0.046KEsg Reg 0.2 1 + 0.45Reg 0.2 φHN δ + − 4 (66) = µl µg ρg ρl 0.5 (67) Normally, Asali nondimensionalizes by τi (which is a bit faster with most implementations), but the eﬀect is usually small. ...
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This note was uploaded on 02/07/2011 for the course ENU 4134 taught by Professor Schubring during the Spring '11 term at University of Florida.
 Spring '11
 Schubring

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