His dimensional reduced pressure correlation is αnb = 0.00417q0.7p0.69 whereαnb is the nucleate pool boiling coefficient in W/m2 · K, q the heat flux in W/m2, andpcrit the critical pressur
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Nu= αnb λL
σ
g(ρ L− ρG )
1/2
(9.42)
using the bubble departure diameter [which is the bracketed term in eq (9.42)] as the characteristic length, andαnbis the nucleate pool boiling heat transfer coefficient He defined the Reynolds numberusing the superficial velocity of the liquid as
h LGρ L
σ
g(ρ L− ρG )
1/2
ρL
where PrLis the liquid Prandt number For the lead constantC1, he introduced the empirical constantCsf to account forthe particularliquid–surface combination, so that
Nu=C1
sfRe
which he presented in the form
c pL ∆T
h LG = Csf
q
µL h LG
g(ρ L− ρG )
1/2n
[Pr]m+1 (9.45)
Thus, this gives∆T ∝ q nandαnbis obtainable from its definition (i.e.,αnb= q/∆T
where∆T = Tw − TsatandT wis the wall temperature) The exponents arem = 0.7
andn = 0.33 (thus equivalent to q ∝ ∆T3), except forwater, wherem = 0 Physical
properties are evaluated at the saturation temperature of the fluid Rohsenow provided
a list of values ofC sf for various surface–fluid combinations that has been extended
by Vachon et al (1967) in Table 9.2 Because this method requires a surface–fluid factor, it is inconvenient to use for general thermal design
TABLE 9.2 Values ofC sf for Rohsenow Correlation
n-Pentane on polished copper0.0154
Carbon tetrachloride on polished copper 0.0070
n-Pentane on lapped copper0.0049 n-Pentane on emery polished copper 0.0074
Wateron ground and polished stainless steel 0.0800 Wateron PTFE pitted stainless steel 0.0058 Wateron chemically etched stainless steel 0.0133 Wateron mechanically polished stainless steel 0.0132
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Reduced Pressure Correlation of Mostinski Mostinski (1963) applied the principle of corresponding states to the correlation of nucleate pool boiling data, arriving at a reduced pressure formulation without a surface–fluid parameter or fluid
physical properties His dimensional reduced pressure correlation is
αnb = 0.00417q0.7p0.69
whereαnb is the nucleate pool boiling coefficient in W/m2 · K, q the heat flux in
W/m2, andpcrit the critical pressure of the fluid in kN/m2 (i.e., in kPa) Pressure effects on nucleate boiling are correlated using the factorF P, determined from the expression
F P = 1.8p0.17
r + 4p1.2
wherep r is the reduced pressure, defined as p r = p/pcrit This correlation gives reasonable results for a wide range of fluids and reduced pressures
Physical Property Type of Correlation of Stephan and Abdelsalam
Stephan and Abdelsalam (1980) developed individual correlations for four classes
of fluids (water, organics, refrigerants, and cryogens), utilizing a statistical multi-ple regression technique These correlations used the physical properties of the fluid (evaluated at the saturation temperature) and are hence said to be physical property–
based correlations Their correlation applicable to organic fluids is
αnbd o
λL = 0.0546
ρG
ρL
1/2 qd o
λLTsat
0.67
h LG d2
o
a2
L
0.248
ρL− ρG
ρL
−4.33
(9.48)
The term at the left is a Nusselt number and the bubble departure diameterd o(meters)
is calculated with a Fritz type of equation:
d o = 0.0146β
2σ
g(ρ L− ρG )
1/2
(9.49)
Note that the contact angleβ is assigned a fixed value of 35° in this expression,
irrespective of the fluid, such that the lead constant becomes 0.511 In the expressions above,Tsatis the saturation temperature of the fluid in Kelvin anda L is the liquid thermal diffusivity in m2/s.
Reduced Pressure Correlation of Cooper with Surface Roughness
Cooper (1984) proposed the following reduced pressure expression for the nucleate pool boiling heat transfer coefficient:
αnb= 55p0.12−0.4343 ln Rp
r (−log10p r ) −0.55 M −0.5 q0.67 (9.50)
Note that this is a dimensional correlation in whichαnbis in W/m2· K, the heat flux
q is in W/m2, andM is the molecularweight and R p the mean surface roughness
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in micrometers (Rp is set to 1.0µm for undefined surfaces) Increasing the surface
roughness has the effect of increasing the nucleate boiling heat transfer coefficient
SinceR pmay be affected by fouling oroxidation of the surface, it is common to use his standard value of 1.0µm for R p Although he recommends multiplyingαnbby
a factor of 1.7 for horizontal copper cylinders, the equation more accurately predicts boiling of the new generation of refrigerants on copper tubes without applying this factor, and that is the approach recommended here The correlation covers a database
of reduced pressures from 0.001 to 0.9 and molecular weights from 2 to 200 and is highly recommended for general use
Fluid-Specific Correlation of Gorenflo Gorenflo (1993) proposed a reduced pressure type of correlation that utilizes a fluid-specific heat transfer coefficientα0, defined for each fluid at the fixed reference conditions ofp r0 = 0.1, Rp0 = 0.4 µm,
andq0 = 20,000 W/m2 His values forα0are given in Table 9.3 for various fluids
The general expression for the nucleate boiling heat transfer coefficientαnbat other conditions is
αnb= α0F PF (q/q0) n (R p /R p0 )0.133 (9.51)
where the pressure correction factorF P F is
F PF = 1.2p0.27
r + 2.5p r+ p r
andp ris the reduced pressure The exponentn on the heat flux ratio is also a function
of reduced pressure:
n = 0.9 − 0.3p0.3
The value ofn decreases with increasing reduced pressure, typical of experimental
data Surface roughness is included in the last term of eq (9.51), whereR p is in micrometers (set to 0.4µm when unknown) The method above is forall fluids listed
except water and helium For water, the corresponding equations are
F PF = 1.73p0.27
r +
6.1 + 0.68
1− pr
p2
n = 0.9 − 0.3p0.15
This method is applicable for0.0005 ≤ pr ≤ 0.95 using the values of α0in the list
For other fluids, experimental values can be input at the standard reference conditions cited in the table, oranothercorrelation can be used to estimateα0 This method is accurate over a very wide range of heat flux and pressure and is probably the most reliable of those presented However, this approach is not extendable to boiling of mixtures, which is of interest in numerous industrial processes
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TABLE 9.3 Values of α 0 in W/m 2· K at pr0 = 0.1, q0 = 20,000 W/m 2 , andR p0 = 0.4 µm, with pcrit in bar
aAt triple point.
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9.5.3 Departure from Nucleate Pool Boiling (or Critical Heat Flux)
The maximum attainable heat flux in the nucleate boiling regime is the DNB or CHF point shown in Section 9.2 The maximum in the nucleate boiling curve heat flux is reached when a hydrodynamic instability destabilizes the vapor jets rising from the heater surface Zuber (1959) was first to demonstrate that this process is governed by the Taylor and Helmholtz instabilities His model was refined by Lienhard and Dhir (1973a) for an infinite flat heated surface facing upward and has been extended to othergeometries
Taylor instability governs the collapse of an infinite, horizontal planar interface of
liquid above a vapororgas, and the Taylorwavelength is that which predominates
at the interface during the collapse Applied to the DNB phenomenon, vapor jets are formed above a large, flat horizontal heater surface as illustrated in Fig 9.8
For jets in a rotated square array as shown, their in-line spacing between jets is the two-dimensional Taylorwavelengthλd2, but sinceλd2 =√2λd1, the characteristic dimension can be considered to beλd1, which is equal to
λd1
(ρ
L− ρG )g
σ
1/2
The Helmholtz instability is that which causes a planarliquid interface to go unstable
when a vapor or gas flowing parallel to the interface reaches some critical velocity
Presently, it is the rising vapor jet that creates the instability The critical velocity of the vaporu Gis
u G=
2πσ
where λH is the Helmholtz wavelength of a disturbance in the jet wall Setting
λd1= λH and performing an energy balance, the expression forqDNBis obtained:
qDNB= ρG h LG
2πσ
ρG
1
2π√3
g(ρ L− ρG )
σ
π
or
qDNB= 0.149ρ1/2G h LG4
This expression is valid for flat infinite heaters facing upward, providing good agree-ment with experiagree-mental results for large flat heaters with vertical sidewalls to prevent lateral liquid flow as long as the diameters or widths of the heaters are larger than
3λd1 Using dimensional analysis, Kutateladze (1948) had already arrived at nearly the same result:
qDNB= Cρ1/2
G h LG4
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Figure 9.8 Vaporjets on a horizontal heaterat DNB showing the Taylorwavelengths
where he found the empirical factorC to be 0.131 based on a comparison to
experi-mental data Zuber’s original analysis yielded a nearly identical value ofC = π/24 =
0.1309, while the Lienhard and Dhir solution gives C = 0.149, which is 15% higher
Lienhard and Dhir (1973a,b) extended this theory to finite surfaces using the expression
qDNB
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TABLE 9.4 qDNB Ratios for Finite Bodies
Geometry
qDNB
diameter
L≥ 2.7
Small flat heater1.14Aheater/λ2
d1 Width or
diameter
0.07 ≤ L≤ 0.2
Horizontal cylinder 0.89 + 2.27e −3.44√R
RadiusR R≥ 0.15
Large horizontal cylinder 0.90 RadiusR R≥ 1.2
Small horizontal cylinder 0.94/(R)1/4 RadiusR 0.15 ≤ R≤ 1.2
Small sphere 1.734/(R)1/2 RadiusR R≤ 4.26
Small horizontal ribbon oriented verticallya
1.18/(H)1/4 Height of
sideH
0.15 ≤ H≤ 2.96
Small horizontal ribbon oriented verticallyb
1.4/(H)1/4 Height of
sideH
0.15 ≤ H≤ 5.86
Any large finite body 0.9 LengthL AboutL≥ 4
Small slendercylinderof any cross section
1.4/(P)1/4 Transverse
perimeterP
0.15 ≤ P≤ 5.86
Small bluff body Constant/(L)1/2 LengthL AboutL≥ 4
Source: Adapted from Lienhard (1981).
aHeated on both sides.
bOne side insulated.
whereqDNB/qDNB,Z is a constant or a geometrical expression with a characteristic
dimension of the particular surface, and eq (9.60) withC = π/24 gives the value of
qDNB,Z The dimensionless length scale is
L= 2π√3 L
whereλd1is given by eq (9.56) andL may be a length L, a perimeter P , a height
H, ora radius R, which give the values of L, P, H, andR, respectively, according
to the type of geometry listed in Table 9.4 The methods are accurate to within about 20% fora wide range of fluids
9.5.4 Film Boiling and Transition Boiling
Film boiling bears a strong similarity to falling film condensation, where the rising va-porfilm is analogous to the falling liquid film Recognizing this fact, Bromley (1950) used the Nusselt equation for film condensation on horizontal cylinders to predict film boiling on the same geometry He did this simply by changing the thermal con-ductivity and the kinematic viscosity from liquid to vapor properties and introducing
an empirical lead constant of 0.62 ratherthan 0.728 Thus, fora cylinderof diameter
D, the film boiling heat transfer coefficient α f bis
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αf b = 0.62
ρG (ρ L− ρG )gh
LGλ3
G
Dµ G (T w − Tsat )
1/4
(9.63)
The vapor properties are evaluated at the film temperature (Tsat+ ∆T /2), the
liq-uid properties at the saturation temperatureTsat, and the latent heat is corrected for sensible heating effects of the vaporas
h
LG = h LG
1+ 0.34
c pG ∆T
h LG
(9.64)
Berenson (1960) proposed a film boiling model for film boiling on cylinders, incor-porating the Taylor hydrodynamic wave instability theory into the Bromley model, arriving at
αf b= 0.425
ρG (ρ L− ρG )gh
LGλ3
G
µG (T w − Tsat )σ/g(ρ L− ρG ) 1/2
1/4
(9.65)
where the physical properties are evaluated as above but 0.5 is used in eq (9.64) rather than 0.34 For film boiling on spheres, Lienhard (1981) has similarly shown that
αf b= 0.67
ρ
G (ρ L− ρG )gh
LGλ3
G
Dµ G (T w − Tsat )
1/4
(9.66)
At large superheats, thermal radiation may be important, depending on the emissivity
of the heated surface,εw Bromley (1950) proposed combining the contributions of film boiling and thermal radiation on cylinders whenαrad < α f bas follows:
αtotal= αf b+3
where the radiation heat transfer coefficientαrad from the heater to the surrounding liquid orvessel is
αrad= qrad
T w − Tsat =
εwσSB(T4
w − T4 sat)
andσSBis the Stephan–Boltzmann constant (σSB = 5.67×10−8W/m2· K4) Apply-ing this analogy to vertical plates is less tenable because in film boilApply-ing the liquid–
vaporinterface becomes Helmholtz unstable nearthe leading edge Leonard et al
(1976) showed that replacing the diameterD in eq (9.63) with the one-dimensional
Taylorinstability wavelengthλd1 given by eq (9.56) gives satisfactory results for film boiling on relatively tall vertical plates
The minimum heat flux at which film boiling can be maintained is again controlled
by the Taylorinstability of the interface (i.e., point MFB in Section 9.2) Zuber(1959) supposed that the minimum is reached when the vapor generation rate becomes too
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small to sustain Taylor wave action on the free interface for stable generation of bubbles Fora horizontal cylinder,qMFBis then
qMFB= Cρ G h LG4
σg(ρ L− ρG )
whereC is an arbitrary empirical constant, which Berenson (1960) identified
exper-imentally to be equal to 0.09
Transition boiling occurs between the heat fluxes at the DNB and MFB points on surfaces in which the surface temperature is the independent variable For example, the transition boiling portion of the pool boiling curve may be obtained by quenching
a hot sphere If the Biot number of the sphere is small, an energy balance on the sensible heat removed from the body can be used to determine the transient surface heat flux during that portion of the cooling process Thus, measuring the temperatures
of the sphere and the liquid, the transition heat transfer coefficient can be obtained
However, tests show that the transition boiling curve obtained by cooling may differ significantly from that obtained by heating
The heat transfer process in the transition regime can be thought of as a combi-nation of nucleate and film boiling, occurring eitherside by side orone afteranother
in rapid succession at the same location on the heater surface The endpoints of this regime are given by the values ofqDNBandqMFB Very approximate values of the transition boiling heat transfer coefficient can be obtained by linear interpolation be-tween theirrespective values atqDNBandqMFB For more details on transition boiling, referto Witte and Lienhard (1982) orDhirand Liaw (1987)
9.6 INTRODUCTION TO FLOW BOILING
For evaporation under forced-flow conditions, heat transfer includes both a convective contribution and a nucleate boiling contribution, whose relative importance depends
on the specific conditions The process of flow boiling is most commonly used inside vertical tubes, in horizontal tubes, in annuli, and on the outside of horizontal tube bundles The local flow boiling heat transfer coefficient is primarily a function of vapor quality, mass velocity, heat flux, flow channel geometry and orientation, two-phase flow pattern, and fluid properties Because the flow boiling coefficient is a function of vapor quality, these calculations are typically done locally in thermal design methods Flow boiling is the most typical of industrial applications; pool boiling is sometimes applied to cooling of electronic parts In the next five sections we present a review of two-phase flow patterns (Section 9.7), flow boiling inside vertical tubes (Section 9.8), flow boiling in horizontal tubes (Section 9.9), boiling on tube bundles (Section 9.10) and post-dryout heat transfer inside tubes (Section 9.11)
9.7 TWO-PHASE FLOW PATTERNS
Flow boiling heat transfer is closely related to the two-phase flow structure of the evaporating fluid Commonly observed flow structures are defined as two-phase flow
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patterns that have particular identifying characteristics Analogous to criteria for delineating laminar flow from turbulent flow in single-phase flow, two-phase flow pattern maps are used to predict the transition from one type of two-phase flow pattern
to another and hence to identify which flow pattern is occurring at the particular local conditions under consideration In this section, first the flow patterns themselves are described for internal tube flows, then a flow pattern map and its flow regime transition equations are presented for horizontal tubes For a more comprehensive treatment of two-phase flow transitions, refer to Barnea and Taitel (1986)
9.7.1 Flow Patterns in Vertical and Horizontal Tubes
Flow patterns encountered in co-current upflow of gas and liquid in a vertical tube are shown in Fig 9.9 The commonly identifiable flow patterns are:
• Bubbly flow In this regime, the gas is dispersed in the form of discrete bubbles
in the continuous liquid phase The shapes and sizes of the bubbles may vary widely, but they are notably smaller than the pipe diameter
• Slug flow Increasing the gas fraction, bubbles collide and coalesce to form larger
bubbles similar in size to the pipe diameter These have a characteristic hemi-spherical nose with a blunt tail end, similar to a bullet, and are referred to as
Taylor bubbles Successive bubbles are separated by a liquid slug, which may
include smallerentrained bubbles These bullet-shaped bubbles have a thin film
of liquid between them and the channel walls, which may flow downward due to the force of gravity, even though the net flow of liquid is upward
• Churn flow Further increasing the velocity, the flow becomes unstable and the
liquid travels up and down in an oscillatory fashion, although the net flow is
Figure 9.9 Flow patterns in vertical upflow
... transition heat transfer coefficient can be obtainedHowever, tests show that the transition boiling curve obtained by cooling may differ significantly from that obtained by heating
The heat. .. tube bundles (Section 9.10) and post-dryout heat transfer inside tubes (Section 9.11)
9.7 TWO-PHASE FLOW PATTERNS
Flow boiling heat transfer is closely related to the two-phase... on the heater surface The endpoints of this regime are given by the values ofqDNBandqMFB Very approximate values of the transition boiling heat transfer