10. PressureDrop_web

10. PressureDrop_web - ENU 4134 – Pressure Drop Models D...

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Unformatted text preview: ENU 4134 – Pressure Drop Models D. Schubring August 16, 2010 Learning Objectives 1-e-i Use empirical models (including those not based on the HEM or SFM) to estimate pressure drop 1-e-ii Develop a correlation for adiabatic, two-component two-phase pressure drop and compare to literature models (in project) 1-e-iii Identify issues related to experiments on two-phase flow (mostly in project) 1-e-iv Estimate design requirements and propose system for acquisition of pressure drop data (in project) 5-b Use correlations and/or models to analyze problems in nuclear thermal hydraulics 5-c 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 single-channel analysis (mostly in project) 5-d 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 5-6 weeks, the emphasis will shift from lecture-driven theoretical understanding to problem-driving empirical analysis. Topics covered: pressure drop (these notes, Project 1); critical flow (HW 4.5); two-phase 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 steam-water (most nuclear-relevant), refrigerants (de-emphasized in this course), and two-component flows (e.g., air-water). Most industrial applications are pure-fluid; most experiments are two-component (why is this?). Pressure Drop Models Outline Lockhart-Martinelli (mostly two-component) Muller-Steinhagen and Heck (two-component or refrigerants) ¨ Review of HEM (pure fluids, steam-water at high G ) Martinelli-Nelson (steam-water at low-to-moderate G ) Armand-Treschev (steam-water, BWR-type conditions) Wallis (annular – warning: difficult) Asali (annular – warning: difficult) The text also provides the models of Thom (similar to M-N), Baroczy (similar to M-N), Jones, and (one of the correlations by) Chisholm. These are no longer frequently used; they are not required for this course. Lockhart-Martinelli Model Assumptions: 1. The pressure (gradient) of the two fluids is equal at any axial position (i.e., pl (z ) = pv (z )) 2. Single-phase relations can be applied within each phase Lockhart-Martinelli 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). Lockhart-Martinelli Parameter (2) Most often, a turbulent smooth-tube 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 Lockhart-Martinelli 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 Lockhart-Martinelli Multiplier The L-M correlation is for the two-phase 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. Lockhart-Martinelli 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, two-component annular flow data can require both. Adjustments to Lockhart-Martinelli (Not in Text) Rather than developing a completely new correlation, some researchers have preferred to adjust L-M to their data. Adjustments have included: Using φ2 or φ2 exclusively v l Selecting a different single-phase friction factor Fitting the parameter(s) C Comments on Lockhart-Martinelli This is likely the most often cited two-phase pressure drop correlation. (To many chemical engineers, it is the only correlation). For comparing correlations, Lockhart-Martinelli 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 single-phase pressure fields – it ends up being a curve fit by virtue of φ2 (X 2 ). Integrating the L-M 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 L-M As a type of separated flow model, the L-M 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 11-95c: 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) Muller-Steinhagen and Heck – Motivation ¨ Consider, at a constant Gm , the function dP /dzfric (x ): There are two boundary conditions that must be satisfied: single-phase 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.) Two-phase multplier methods (i.e., SFM methods) will either have a discontinuity or violate a boundary condition. M¨ller-Steinhagen u and Heck is a purely empirical method to estimate frictional pressure gradient only – acceleration and gravity terms must come from somewhere else. Muller-Steinhagen 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). Muller-Steinhagen 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 two-component annular flow. No estimate of α is evident. Recommendation: use (1) drift flux model for bubbly/slug/churn, (2) M-N void fraction (Figure 11-17) for any steam-water regime, or (3) Rouhani-Axelsson for annular two-component (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 non-upflow: the friction term still holds. Martinelli-Nelson Model The Martinelli-Nelson model is designed to apply directly to steam-water systems. Assumptions: φ2 is a function of flow quality and fluid properties. Since lo two-phase 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 M-N lo Most often φ2 is looked up from a figure. The figure is often lo generated from the results of the Lockhart-Martinelli 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 L-M correlation): φ2 lo = 1 x x 1+ 0 20 1 +2 Xtt Xtt (1 − x )1.75 dx (38) φ2 in M-N lo Figure 11-15 in T&K α in M-N Figure 11-17 in T&K Integral Approach M-N is most useful when using an integral approach (computing ∆P rather than dP /dz ). When gas compressibility is ignored (reasonable for industrial steam-water 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 11-18 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 11-16 in T&K Graphical Correlation of r4 (gravity) Figure 11-20 in T&K, listed as for the Thom correlation (same value) Comments on Martinelli-Nelson Based on real data, φ2 is not independent of Gm or even geometry lo Gm . M-N 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 M-N model works best at moderate Gm (500-1000 kg m−2 s−1 ). This is somewhat below typical reactor Gm values, for which HEM is typically better Armand-Treschev 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 in-core BWR calculations. Two-Phase Multiplier for A-T 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 Flow-Specific 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 +: non-dimensional 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 low-pressure annular flow, τi ≈ τw , so the film velocity profile can be non-dimensionalized 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 non-dimensional 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 non-dimensionalizes 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.

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