06_PressureDrop_web - ENU 4134 – Pressure Drop Models D Schubring Fall 2011 Learning Objectives 1-e-i Use empirical models(including those not

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Unformatted text preview: ENU 4134 – Pressure Drop Models D. Schubring Fall 2011 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 Remarks Course: In the next 5-6 weeks, the emphasis will shift from lecture-driven theoretical understanding to problem-driven empirical analysis. Topics covered: pressure drop (these notes, HW 3, Project 1); two-phase convective heat transfer (HW 4 and 5); and nuclear heat generation/transfer and SCA (HW 6, 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) 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 dP dz l (dP /dz )lfric (dP /dz )v fric (1) = fl De 2 Gm (1 − x )2 2ρl (2) = fv De 2 Gm x 2 2ρv (3) fric v 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 Rel dP dz 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 v fric Rev (4) (5) (6) (7) Lockhart-Martinelli Parameter (3) 0.2 1−x x 2 Xtt = Rev Rel 2 Xtt = x µl (1 − x )µv 2 Xtt = µl µv 0 .2 0.2 1−x x 2 ρv ρl 1−x x 1 .8 ρv ρl (8) 2 ρv ρl (9) (10) 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 ... where n = 0.2 when the McAdams correlation is used. (11) (12) 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 ): 1 C +2 X X = 1 + CX + X 2 φ2 = 1 + l (13) φ2 v (14) ... with C as a constant based on whether the fluid is laminar or turbulent. Assuming the latter: 20 1 +2 Xtt Xtt 2 = 1 + 20Xtt + Xtt φ2 = 1 + l (15) φ2 v (16) Coupling to Void Fraction After some tedious algebra: φ2 = (1 − α)−2 l (17) 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 (18) 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, or adjusting the switch-over v l criterion 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 (19) 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) dXtt /dx n = 0.2 for McAdams: (n − 2) (1 − x )n/2 x 3 dXtt = n/2 dx x (2x 5 − 4x 4 + 2x 3 ) (21) The d α/dz Term in L-M d α dXtt dx dα = dz dXtt dx dz =− (22) 2 (CXtt + 2) Xtt + CXtt + 1 4 3 2 2Xtt + 4CXtt + (2C 2 + 4) Xtt + 4CXtt + 2 (n − 2) (1 − x )n/2 x 3 x n/2 (2x 5 − 4x 4 + 2x 3 ) dx dz × (23) (24) Conclusion: not usually worth the pain for pure fluids with phase change. 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 GMSH − = 0.316Relo 0.25 = Gm Dh µl − = 0.316Rego0.25 = 2 Gm 2Dh ρl (25) (26) 2 Gm 2Dh ρg Gm Dh µg (27) (28) dP x3 dz fric ,go dP dP − dz fric ,go dz fric ,lo = GMSH (1 − x )1/3 + (29) dP + 2x dz fric ,lo (30) = 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) some purely empirical correlation you look up from a journal. If you need it, estimate d α/dz with finite difference Review of HEM HEM: 2 1G 2 fTP Dh 2ρm + Gm dx volfg + ρm g cos (θ) + dp dz m = − 2 ∂ vol dz 1 + Gm x ∂ p g (31) General SFM: − 2 dp 2 x ∂ volg 1 + Gm dz {α} ∂ p 2 +Gm 2 +Gm − 2 1 Gm + ρm g cos (θ) Dh 2ρl 2xvolg 2 (1 − x ) volf dx − {α} {1 − α} dz = flo φ2 lo x 2 volg (1 − x )2 volf + {α}2 {1 − α }2 d {α} dz Assume dx /dz = d α/dz = 0, incompressible gas, upflow: (32) Review of HEM HEM: 2 1 Gm dp = fTP + ρm g dz Dh 2ρm (33) 2 dp 1 Gm = flo φ2 + ρm g lo dz Dh 2ρl (34) − General SFM: − Phrase the HEM with φlo (as an SFM model): − dp dz φ2 ,HEM lo φ2 ,HEM lo = flo φ2 ,HEM lo = = ρl fTP ρm flo ρl µTP ρm µf 2 1 Gm + ρm g Dh 2ρl (35) (36) n (37) 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+ (38) (39) 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 (40) 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 many industrial steam-water applications, where ∆P << Pabs and constant heating is assumed – don’t use in core), the SFM pressure difference becomes: ∆P = G2 L flo × m× De 2ρl xout xout φ2 dx lo (41) 0 x 2 ρl (1 − xout )2 + out − 1 1 − αout αout ρv + 2 Gm ρl + Lρl g cos(θ) xout 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 (42) 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 (43) 1 1− 1− 0 r2 , r3 , and r4 tables still work in the limit xout = 1. ρv ρl α dx 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 geometry. lo 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 + 0.05 ln(10p )] (44) 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 = (1 − x )1.75 (1 − α)1.2 (45) For β < 0.9 and α > 0.5: 0.48 (1 − x ))1.75 (1 − α)n n = 1.9 + 0.0148p φ2 lo = (46) (47) p in MPa For β > 0.9: φ2 lo p in MPa = 0.0025p + 0.055 (1 − β )1.75 (1 − x )1.75 (48) ...
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This note was uploaded on 10/22/2011 for the course ENU 4134 taught by Professor Schubring during the Fall '11 term at University of Florida.

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