This flow pattern is a transition regime between the slug flow and annular flow regimes.. When the flow rate is increased further, the entrained droplets congregate to form large lumps or wi
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upward One explanation for the instability is that the gravity and shear forces acting on the thin film of liquid of Taylorbubbles become similarin magnitude, such that the flow direction of the film oscillates between upward and downward
This flow pattern is a transition regime between the slug flow and annular flow regimes In small-diameter tubes, churn flow may not develop such that the flow passes directly from slug flow to annular flow
• Annular flow Here the bulk of the liquid flows as a thin film on the wall with the
gas as the continuous phase flowing up the centerof the tube, forming a liquid annulus with a gas core whose interface is disturbed by both large-magnitude waves and chaotic ripples Liquid may be entrained in the high-velocity gas core
as small droplets; the liquid fraction entrained may be similar to that in the film
This flow regime is quite stable and is often desirable for system operation and pipe flow
• Wispy annular flow When the flow rate is increased further, the entrained droplets
congregate to form large lumps or wisps of liquid in the central vapor core with
a very disturbed annular liquid film
• Mist flow When the flow rate is increased even further, the annular film becomes
very thin, such that the shear of the gas core on the interface is able to entrain all the liquid as droplets in the continuous gas phase (i.e., the inverse of the bubbly flow regime) The wall is intermittently wetted locally by impinging droplets
The droplets in the mist may be too small to be seen without special lighting and/ormagnification
Flow patterns in horizontal two-phase flows are influenced by the effect of gravity, which acts to stratify the liquid to the bottom and the gas to the top of the channel
Flow patterns encountered in co-current flow of gas and liquid in a horizontal tube are shown in Fig 9.10 The commonly identifiable flow patterns are:
• Bubbly flow The bubbles are dispersed in the continuous liquid with a higher
concentration in the upper half of the tube because of buoyancy effects However,
at high mass velocities, the bubbles tend to be dispersed uniformly in the tube as shearforces become dominant
• Stratified flow At low liquid and gas velocities, there is complete separation of
the two phases, with the gas in the top and the liquid in the bottom, separated by
an undisturbed horizontal interface
• Stratified–wavy flow With increasing gas velocity, waves form on the liquid–
gas interface traveling in the direction of the flow The amplitude of the waves depends on the relative velocity of the two phases, but their crests do not reach the top of the tube The waves have a tendency to wrap up around the sides of the tube, leaving thin films of liquid on the wall afterpassage of the wave
• Intermittent flow Further increasing the gas velocity, the waves grow in
magni-tude until they reach the top of the tube Thus, large amplimagni-tude waves wash the top of the tube intermittently, while slower-moving smaller-amplitude waves are often evident in between The large-amplitude waves contain a large amount of
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Figure 9.10 Flow pattens in horizontal flow
liquid and often have entrained bubbles The top wall is often wet continuously from the thin liquid film left behind by each large-amplitude wave
• Annular flow Similar to vertical upflow, at large gas flow rates the liquid forms
a continuous annular film around the perimeter of the tube, which tends to be noticably thickerat the bottom than the top The interface of the film is typically disturbed by small-amplitude waves, and droplets may be dispersed in the gas core At high gas fractions, the top of the tube eventually becomes dry first, with the flow reverting to the stratified–wavy flow regime
• Mist flow Similar to that occurring in vertical flow, all the liquid may become
entrained as small droplets in the high-velocity continuous gas phase
Intermittent flow is actually a composite of the plug and slug flow regimes These subcategories may be described as follows:
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• Plug flow This pattern is characterized by liquid plugs that are separated by
elongated gas bubbles The diameters of the bubbles are smaller than the tube, such that the liquid phase is continuous along the bottom of the tube Plug flow
may also be referred to as elongated bubble flow.
• Slug flow At higher gas velocities, bubbles are entrained in the liquid slugs, and
the elongated bubbles become similarin size to the channel height The liquid slugs can also be characterized as large-amplitude waves
9.7.2 Flow Pattern Maps for Vertical Flows
For vertical upflow, Fig 9.13 (shown later) shows the typical regimes that would
be encountered from inlet to outlet of an evaporator tube It is beyond our scope in this chapter to present flow pattern maps for vertical flows A flow pattern map is
a diagram utilized to delineate the transitions between the flow patterns, typically plotted on log-log axes using dimensionless parameters to represent the liquid and gas velocities Hewitt and Roberts (1969) and Fair (1960) are widely quoted flow pattern maps for vertical upflows
9.7.3 Flow Pattern Maps for Horizontal Flows
For evaporation, Fig 9.14 (shown later) depicts the typical sectional views of the flow structure For condensation, the flow regimes are similar with the exception that the top tube wall is not dry in stratified types of flow, but instead, is coated with
a thin condensing film of condensate The most widely quoted flow pattern maps for predicting the transition between two-phase flow regimes for adiabatic flow in horizontal tubes are those of Baker (1954) and Taitel and Dukler (1976), whose descriptions are available in numerous books and publications Their transition curves should be considered as zones similarto that between laminarand turbulent flow
Forsmall-diametertubes typical of heat exchangers, Kattan et al (1998a) pro-posed a modification of the Steiner(1993) map, which itself is a modified Taitel–
Duklermap, and included a method forpredicting the onset of dryout at the top of the tube in evaporating annular flows This flow pattern map is presented here as it
is used in Section 9.9 forpredicting local flow boiling coefficients based on the local flow pattern The flow regime transition boundaries of the Kattan–Thome–Favrat flow pattern map are depicted in Fig 9.11 (bubbly flow is at very high mass velocities and
is not shown) This map provides the transition boundaries on a linear–linear graph with mass velocity plotted versus gas orvaporfraction forthe particularfluid and flow channel, which is much easierto use than the log-log format of othermaps
The transition boundary curve between annular and intermittent flows to stratified–
wavy flow is
˙mwavy=
16A3
Gd gd iρLρG
χ2π2
1− (2h Ld − 1)2 0.5 π
2
25h2
Ld
(1 − χ) −F1 (q)
We Fr
−F2 (q)
0.5
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0 100 200
500
300
600
400 700
Vapor quality
s)
R-134a Dia = 12mm Tsat= 10°C
Stratified wavy
Intermittent
Annular
Mist flow
Stratified [Strat]
[IA]
[Mist]
[Wavy]
Figure 9.11 Kattan–Thome–Favrat flow pattern map illustrating flow regime transition boundaries
The high-vapor-quality portion of this curve depends on the ratio of the Froude number(FrL) to the Webernumber(WeL), where FrL is the ratio of the inertia to the surface tension forces and WeLis the ratio of inertia to gravity forces The mass
velocity threshold for the transition from annular flow to mist flow is
˙mmist =
7680A2
Gdgd iρLρG
χ2π2ξP h
Fr We
L
0.5
(9.71)
Evaluating the expression above for the minimum mass velocity of the mist flow transition gives the value ofχmin, which forχ > χminis
The transition between stratified–wavy flow and fully stratified flow is given by the
expression
˙mstrat =
(226.3)2A
Ld A2
GdρG (ρ L− ρG )µ L g
χ2(1 − χ)π3
1/3
(9.73)
The transition threshold into bubbly flow is
˙mbubbly=
256A Gd A2
Ld d1.25
i ρL (ρ L− ρG )g
0.3164(1 − χ)1.75π2P idµ0.25
L
1/1.75
(9.74)
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In the equations above, the ratio of We to Fr is
We Fr
L= gd i2ρL
and the friction factor is
ξP h=
1.138 + 2 log π
1.5A Ld
−2
(9.76)
The nondimensional empirical exponentsF1(q) and F2(q) in the ˙mwavy boundary equation include the effect of heat flux on the onset of dryout of the annular film: the transition of annular flow into annular flow with partial dryout, the latter classified as stratified–wavy flow by the map They are
F1(q) = 646.0
q
qDNB
2
+ 64.8
q
qDNB
(9.77a)
F2(q) = 18.8
q
qDNB
The Kutateladze (1948) correlation for the heat flux of departure from nucleate boil-ing,qDNB, is used to normalize the local heat flux:
qDNB= 0.131ρ1/2
G h LG
g(ρ L− ρG )σ 1/4 (9.78)
The vertical boundary between intermittent flow and annular flow is assumed to occur
at a fixed value of the Martinelli parameterX tt, equal to 0.34, whereX ttis defined as
X tt=
1− χ χ
0.875ρ
G
ρL
0.5µ
L
µG
0.125
(9.79) Solving forχ, the threshold line of the intermittent-to-annular flow transition at χIAis
χIA = 0.2914
ρG
ρL
−1/1.75
µL
µG
−1/7 + 1
−1
(9.80)
Figure 9.12 defines the geometrical dimensions of the flow, whereP Lis the wetted perimeter of the tube,P G the dry perimeter in contact with only vapor,h the height
of the completely stratified liquid layer, andP i the length of the phase interface
Similarly,A LandA Gare the corresponding cross-sectional areas Normalizing with the tube internal diameterd i, six dimensionless variables are obtained:
h Ld = h
d i P Ld =
P L
d i P Gd =
P G
d i P id =
P i
d i
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A Ld = A L
d2
i
A Gd =A G
d2
i
(9.81) Forh Ld ≤ 0.5:
P Ld = 8(h Ld )0.5 − 2 [h Ld (1 − h Ld )]0.5
A Ld =
12 [h Ld (1 − h Ld )]0.5 + 8(h Ld )0.5
h Ld
4 − A Ld (9.82) Forh Ld > 0.5:
P Gd= 8(1 − h Ld )0.5 − 2 [h Ld (1 − h Ld )]0.5
A Gd=
12 [h Ld (1 − h Ld )]0.5 + 8(1 − h Ld )0.5
(1 − h Ld )
4 − A Gd
(9.83) For0≤ h Ld ≤ 1:
P id = 2 [h Ld (1 − h Ld )]0.5 (9.84)
Sinceh is unknown, an iterative method utilizing the following equation is necessary
to calculate the reference liquid levelh Ld:
X2
tt = P Gdπ+ P id
1/4 π2
64A2
Gd
P
Gd + P id
A Gd +
P id
A Ld
π
P Ld
1/4
64A3
Ld
π2P Ld (9.85)
Once the reference liquid levelh Ld is known, the dimensionless variables are calcu-lated from eqs (9.82)–(9.84) and the transition curves for the new flow pattern map are determined with eqs (9.70)–(9.80)
Figure 9.12 Cross-sectional and peripheral fractions in a circular tube
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This map was developed from a database for five refrigerants: two single-com-ponent fluids (134a and 123), two near-azeotropic mixtures (402A and R-404A), and one azeotropic mixture (R-502) The test conditions covered the following range of variables: mass flow rates from 100 to 500 kg/m2· s, vaporqualities from
4 to 100%, heat fluxes from 440 to 36,500 W/m2, saturation pressures from 0.112
to 0.888 MPa, Weber numbers from 1.1 to 234.5, and liquid Froude numbers from 0.037 to 1.36 The Kattan–Thome–Favrat flow pattern map correctly identified 96.2%
of these flow pattern data
Z¨urcher et al (1997) obtained additional two-phase flow pattern observations for the zeotropic refrigerant mixture R-407C at an inlet saturation pressure of 0.645 MPa, and the map accurately identified these new flow pattern data Z¨urcher et al (1999) also obtained two-phase flow pattern data for ammonia with a 14–mm bore sight glass formass velocities from 20 to 140 kg/m2· s, vaporqualities from 1 to 99% and heat
fluxes from 5000 to 58,000 W/m2, all taken at a saturation temperature of 4°C and saturation pressure of 0.497 MPa Thus, the mass velocity range in the database was extended from 100 kg/m2· s down to 20 kg/m2· s In particular, it was observed that
the transition curve ˙mstrat was too low and eq (9.73) was corrected empirically by adding+20χ as follows:
˙mstrat=
(226.3)2A Ld A2
GdρG (ρ L− ρG )µ L g
χ2(1 − χ)π3
1/3 + 20χ (9.86)
where ˙mstratis in kg/m2· s The transition from stratified–wavy flow to annular flow
at high vaporqualities was, instead, observed to be too high, and hence an additional empirical term with an exponential factor modifying the boundary at high vapor qualities was added to eq (9.70) to take this into account as
˙mwavy (new) = ˙mwavy − 75e −(χ2−0.97)2/χ(1−χ) (9.87)
where the mass velocity is in kg/m2· s The movement of these boundaries has an
effect on the dry angle calculationθdryin the Kattan et al (1998c) flow boiling heat transfermodel and shifts the onset of dryout to slightly highervaporqualities, which
is in agreement with the ammonia heat transfer test data
To utilize this map, the following parameters are required: vapor quality (χ), mass
velocity (˙m), tube internal diameter (d i), heat flux (q), liquid density (ρ L), vapor density (ρG), liquid dynamic viscosity (µL), vapordynamic viscosity (µG), surface tension (σ), and latent heat of vaporization (h LG), all in SI units The local flow pattern
is identified by the following procedure:
1 Solve eq (9.85) iteratively with eqs (9.79), (9.82), (9.83), and (9.84)
2 Evaluate eq (9.81)
3 Evaluate eqs (9.75)–(9.78)
4 Evaluate eqs (9.70), (9.71) or(9.72), (9.73), (9.74), and (9.80)
5 Compare these values to the given values ofχ and ˙m to identify the flow pattern.
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Note that eq (9.87) should be used in place of eqs (9.70) and (9.86) should be used in place of eq (9.80) to utilize the most updated version The map is thus specific to the fluid properties, flow conditions (heat flux), and tube internal diameter input into the equations The map can be programmed into any computer language, evaluating the transition curves in incremental steps of 0.01 in vapor quality to obtain
a tabular set of threshold boundary points, which can then displayed as a complete map with ˙m versus χ as coordinates.
9.8FLOW BOILING IN VERTICAL TUBES
Convective evaporation in a vertical tube is depicted in Fig 9.13 At the inlet, the liquid enters subcooled As the liquid heats up, the wall temperature rises until it
Figure 9.13 Flow patterns during evaporation in a vertical tube with a uniform heat flux
(From Collier and Thome, 1994.)
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surpasses the saturation temperature of the liquid, at which point subcooled boiling begins, where bubbles nucleate and grow in the thermal boundary layer and condense
in the subcooled core Farther up the tube, the liquid bulk reaches its saturation temperature and the convective boiling process passes through the bubbly flow, churn flow, and finally, into the annular flow regime At the dryout point, the annular liquid film is depleted completely, eitherby evaporation orby entrainment into the vapor core, and the wall temperature rises significantly in order to dissipate the applied heat flux The process proceeds in the post-dryout (mist or drop flow) regime First, saturated wet wall convective boiling is described, and post-dryout heat transfer is then described in Section 9.11
In the forced-convective evaporation regime the heat transfer coefficient is less dependent on heat flux than in nucleate pool boiling, while its dependence on the local vapor quality appears as a new and important parameter Both the nucleate and convective heat transfer mechanisms must be taken into account to predict heat transfer data in the convective boiling regime Their relative importance varies from dominance of nucleate boiling at low vaporqualities and high heat fluxes to the dominance of convection at relatively high vapor qualities and low heat fluxes
9.8.1 Chen Correlation
The Chen (1963) correlation was the first to attain widespread acclaim and has served
as the starting point for the development of most other flow boiling correlations since
Superposition of the nucleate boiling and convective boiling mechanisms is assumed, such that the local two-phase flow boiling coefficientαtpis obtained as
whereαnbis the nucleate boiling contribution andαcbis the convective contribution, which Chen presented as
αtp= αFZS + α L F (9.89) The Forster and Zuber (1955) correlation is used to calculate the pool boiling heat transfer coefficientαFZas
αFZ= 0.00122
λ0.79
L c0.45
pL ρ0.49 L
σ0.5µ0.29
L h0.24
LGρ0.24 G
∆T0.24
sat ∆p0.75
while forliquid-only heat transfer, coefficientαLis obtained with the Dittus–Boelter correlation:
αL = 0.023Re0.8
L · Pr0.4 L
λ
L
d i
(9.91) The liquid Reynolds numberReL above is based on the fraction of liquid flowing alone in the channel [i.e., using ˙m(1 − χ) to calculate Re L] In the Forster–Zuber
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equation, the wall superheat∆Tsatand pressure difference∆psat are the active pa-rameters: the pressure difference refers to that obtained with the fluid’s vapor pres-sure curve evaluated at the wall and saturation temperatures, respectively, and∆psat
is in N/m2 Thus, this approach leads to an iterative calculation when the heat flux is specified since the wall temperature is not known beforehand
Forced flow creates a sharper temperature gradient at the wall relative to that in nucleate pool boiling, which has an adverse effect on bubble nucleation Thus, nucle-ation is partially suppressed, which Chen accounted for by introducing a nuclenucle-ation suppression factorS The convective boiling contribution α cbis a product ofαLtimes
a two-phase multiplierF , which enhances this heat transfer mode The suppression
factorS, two-phase multiplier F , Martinelli parameter X tt, and two-phase Reynolds numberRetpused in his method are calculated as follows:
1+ 0.00000253Re1.17
tp
(9.92)
F =
1
X tt + 0.213
0.736
(9.93)
X tt=
1− χ χ
0.9
ρG
ρL
0.5
µL
µG
0.1
(9.94)
Note, however, that when 1/X tt ≤ 0.1, F is set equal to 1.0 The Chen correlation is
applicable over the entire evaporation range in which the heated wall remains wet
9.8.2 Shah Correlation
Shah (1982) proposed a method for implementing his chart method Similar to Chen,
he included two distinct mechanisms: nucleate boiling and convective boiling
How-ever, instead of adding these two contributions, his method chooses the larger of
the nucleate boiling coefficientαnband the convective boiling coefficientαcb In his
method, the first step is to calculate the dimensionless parameterN, which forvertical
tubes at all values of the liquid Froude number FrLis given as
whereC0is determined from
C0=
1− χ χ
0.8
ρG
ρL
0.5
(9.97)
ForN > 1.0, the values of α nbandαcbare calculated from the following expressions
and the largervalue is chosen as the local heat transfercoefficientαtp The value
of the liquid-only convective heat transfer coefficientαL used in these expressions
... the largervalue is chosen as the local heat transfercoefficientαtp The valueof the liquid-only convective heat transfer coefficientαL... a product ofαLtimes
a two-phase multiplierF , which enhances this heat transfer mode The suppression
factorS, two-phase multiplier F , Martinelli parameter...
sat ∆p0.75
while forliquid-only heat transfer, coefficientαLis obtained with the Dittus–Boelter correlation:
αL