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Unformatted text preview: ENU 4134 – Pressure Drop Models
D. Schubring Fall 2011 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 Remarks
Course:
In the next 56 weeks, the emphasis will shift from
lecturedriven theoretical understanding to problemdriven
empirical analysis.
Topics covered: pressure drop (these notes, HW 3, Project 1);
twophase 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 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)
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
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). 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 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) LockhartMartinelli 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) 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 ... where n = 0.2 when the McAdams correlation is used. (11)
(12) 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 ):
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, 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, or adjusting the switchover
v
l
criterion
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 (19) 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) 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 LM 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. 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
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). 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) 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 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+ (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 LM correlation):
φ2
lo = 1
x x 1+
0 20
1
+2
Xtt
Xtt (1 − x )1.75 dx (40) 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 many industrial steamwater 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 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 geometry.
lo
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 + 0.05 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 incore BWR calculations. TwoPhase Multiplier for AT
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.
 Fall '11
 Schubring

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