boiling heat transfer-i introduction to pool boiling heat...
TRANSCRIPT
1
Kannan Iyer
Department of Mechanical Engineering
Indian Institute of Technology, Bombay
Introduction to Pool Boiling Heat Transfer
09:16 1/8
Boiling Heat Transfer-I
• Boiling is associated with transformation of liquid to
vapor by heating
• The formation of bubbles stir the fluid and breaks the
boundary layers thereby increasing the heat transfer
coefficient
• It differs from vaporization in the sense that it is
associated with the formation of bubbles
• The onset of bubble formation is called nucleation
• In equilibrium thermodynamics, boiling is assumed
to occur when water is heated to its saturated
temperature
09:16 2/53
Homogeneous Nucleation
• If the surface has mirror finish the onset of boiling is
considerably suppressed
• The bubbles are normally formed on the surface
scratches. The bubbles do not appear till the wall is
heated in excess of the saturation temperature, called
wall superheat.
• It has been seen that under clean surface conditions,
no boiling is seen up to 321 oC (~0.92 TC). According
to Blander and Katz (AIChE J, 21, 833-849, 1975) the
similar behaviour is seen in organic fluids (~0.89 TC).
• In industrial equipment that has sufficient scratches,
this homogeneous nucleation phenomenon is not of
much relevance
09:16 3/53Hetrogeneous Nucleation-I
• Bubbles generally originate from pits
• Air trapped in cavities help in nucleation
• Experiments indicate that removal air results in
suppression of nucleation
09:16 4/53
2
Hetrogeneous Nucleation-II
• If we assume the bubble to be of hemispherical
shape and perform a force balance between
surface tension and pressure, we can write
( ) σππ R2Rpp 2
lv=−
• For the bubble to grow LHS > RHS
• From thermodynamics, Clausius and Clayperon
equation is given as
fgsat
fg
sat vT
h
dT
dp=
σπR2
pv
pl
( )R
2pp lv
σ>− 1
2
09:16 5/53
• From Taylor series expansion, we can write
( )ambWsat
sat
ambsatWp)T(p
dp
dT)p(TT −+=
• Substituting for (dT/dp)sat we get,
( )ambWsat
fg
fgsat
ambsatWp)T(p
h
vT)p(TT −+=
• For bubble to grow, the condition for critical
radius can be invoked to eliminate ∆p and we can
write
Hetrogeneous Nucleation-II
R
2
h
vT)p(TT
fg
fgsat
ambsatW
σ=− 3
09:16 6/53
• For water at 1 bar, and for a 1 micron cavity, we can
compute the wall superheat by
substituting, Tsat= 373 K, ; σ= 0.059 N/m ;
hfg= 2.256 x 106J/kg ; vg= 1.672 m3/kg; R=10-6 m to
get TW-Tsat = 32 K
Hetrogeneous Nucleation-II
• Nucleation is significantly affected if there is a
temperature gradient normal to the wall
• In flow boiling, due to boundary layer presence,
there is a considerable variation
• Let us start with constant wall flux case and extend it
to the general case
09:16 7/53• The fluid temperature decreases as we
move away from the wall, the pressure
excess must correspond to the lowest
temperature, which is at the vertical tip of
the bubble
• Further, as the critical size of bubbles are of the order
10 microns, the temperature profile in this scale can
be assumed to be linear
• Labelling the vertical coordinate as y, we can write,
−=−=′′
y
TTk
dy
dTkq lW
ll
l 4
T
TW• This will ensure that at no surface of the
bubble there will be condensation
09:16 8/53
3
• For hemispherical bubble, y = Rcrit and we can write,
• For linear temperature profile in liquid, we can write,
T
V
y
• The slope will increase as heat flux is increased.
Davis and Anderson postulated tangency of the liquid
profile as the condition for nucleation
y
2
h
vT)p(TT
fg
fgsat
ambsatV
σ=− 5
2
fg
fgsatV
y
2
h
vT
dy
dT σ−=⇒ 6
l
l
k
q
dy
dT ′′−= 7
09:16 9/53
• Equating the slope at the point of nucleation, we get
q
k2
h
vTy l
fg
fgsat2
′′=⇒
σ
• At the point of tangency, the values of temperatures of
vapour and liquid also have to be same. Eqs. (4) and
(5) can be rewritten as,
8
l
Wlk
yqTT
′′−=
y
2
h
vT)p(TT
fg
fgsat
ambsatV
σ+=
y
2
h
vT
k
yq)p(TT
fg
fgsat
l
ambsatW
σ+
′′=−
+
′′=−
fg
fgsat
l
2
ambsatWh
2vT
k
yq
y
1)p(TT
σ
09:16 10/53
• Substituting for y from Eq. (8), we get
+
′′
=−fg
fgsat
fg
fgsat
5.0
l
fg
fgsat
ambsatWh
2vT
h
vT2
q
k2
h
vT
1)p(TT
σσ
σ
( )
′′=−
lfg
fgsat2
ambsatWk
q
h
vT8)p(TT
σ9
( )
=
−
−
lfg
fgsat
bulkW
2
ambsatW
k
h
h
vT8
TT
)p(TT σ10
( )
′′=
′′=−
lfg
fgsat
2
fg
fgsat
fg
fgsatl
2
ambsatWk
q
h
vT8
h
vT2
h
2vT
1
k
q)p(TT
σσ
σ
4
09:16 11/53
• Eq. (9) is used for the case of heat flux specified
cases, whereas Eq.(10) can be used for wall
temperature specified cases
• Frost and Dzakowic (ASME 67-HT-61(Ht. Tr. Conf.)
showed that
( )2
llfg
fgsat2
ambsatWpr
1
k
q
h
vT8)p(TT
′′=−
σ
09:16 12/53
4
Introduction-I09:16
• We have understood that boiling starts with
nucleation of bubbles in crevices
• We have also established criterion for the onset of
bubble both in the presence of temperature gradient
as well is in its absence.
• We shall now look at the growth of these bubbles as
they have a say in predicting the heat transfer during
nucleate boiling.
• The exact solution demands solution of three
dimensional Navier-Stokes equations with complex
boundary conditions as the interfaces have to be
tracked.
13/53 09:16
• We shall look at crude modelling by making several
assumptions to get some analytical solutions that are
illustrative
• The two extremes that are normally addressed in
bubble growth are:
• Inertia controlled growth in the initial stages
• Heat transfer controlled growth in the latter parts
of the growth
• We shall look at them now
Introduction-II14/53
09:16
• Rayleigh derived this equation by using the first law
of thermodynamics
• The assumptions are:
• The pressure inside the bubble is constant at
psat(TW)
• The control volume is the infinite shell of water
with bubble interface on one side (p = constant)
• One-dimensional spherically symmetric
incompressible flow conditions exist
Inertia Controlled regime
( ) 0Vrrr
1r
2
2=
∂
∂
• Continuity equation implies
15/53 09:16
( ) CVrr
2 =⇒
At r = RB, Vr = dRB/dt = BR&
2
B
2
Br
r
RRV
&
=⇒
• The kinetic energy of the liquid from r=RB to ∞ can
be written as
2
2
B
2
B
R
f
22
r
R
f
2
r
RR
2
1drr4
2
Vdrr4
BB
= ∫∫
∞∞ &
ρπρπ
16/53
5
09:16
2
B
3
Bf
B
2
B
4
Bf
R
2
2
B
4
BfRR2
R
1RR2dr
r
1RR2
B
&&& πρπρπρ === ∫∞
• Invoking first law, where the change of KE should be
equal to work done (under no heat transfer), we can
write,
( )( )∞−−= p)T(pRR3
4RR2 Wsat
3
0
3
B
2
B
3
Bf ππρ &
• Differentiating the above expression, we get
( ) ( )∞−=+ p)T(pRR4RR3RRR2R2WsatB
2
BB
2
B
2
BBB
3
Bf&&&&&& ππρ
• Simplifying, we get,
f
Wsat2
BBB ρ
p)T(pR
2
3RR
∞-=+ &&& 1
17/53 09:16
• Using Clayperon’s Equation,
gsat
fg
vT
h
dT
dp=
• Using linearized expansion, we can write
( )( )gsat
fg
satW
Wsat
vT
h
pTT
p)T(p=
−
−
∞
∞
( )( )∞∞ -=-⇒ pTT
vT
hpp
satW
gsat
fg
v 2
18/53
09:16
• Eliminating the pressure difference from Eqs.(1) and
(2), we can write
( )( )∞−=
+ pTT
vT
vhR
2
3RR satW
gsat
ffg2
BBB&&&
• At very low bubble radius, the first term is negligible
and hence we can write,
( )( )∞−= pTTvT
vhR
2
3satW
gsat
ffg2
B&
( )( )∞−= pTTvT
vh
3
2R
satW
gsat
ffg
B&
• Since RHS is a constant, bubble grows linearly with time
19/53
Heat transfer Control
• Before we look at heat transfer control, let us look at
heat transfer in semi-infinite plate
0
h, T∞
• The governing equation
0 ≤ x ≤ ∞; 0 ≤ t2
2
x
T
t
T
α
1
∂
∂=
∂
∂
• Boundary conditions
T(0,x) = Ti ; T(t,0) = Ts ; T(t,x→∞) = Ti
09:16
• We will link it to bubble dynamics a bit later
20/53
6
• Mathematically, problems that have boundary at
infinity are often solved by a method called similarity
solution
1-D Transient in a Semi-Infinite Plate-II
• In this method, a new variable, called similarity
variable is introduced
• There are systematic ways by which this can be
derived, but often involves some qualitative
arguments
• Thus the governing equation will be transformed into
an ODE from PDE
• This variable is chosen such that T becomes a
function of only this variable
09:16 21/53
5.0)tα4(
1
ηd
dT
x
η
ηd
dT
x
T=
∂
∂=
∂
∂• Using chain rule
1-D Transient in a Semi-Infinite Plate-III
( ) 5.0t4
x
αη =
• In this course we shall give you the form of the
variable. Note that it will be a combination of x and t
)tα4(
1
ηd
Td2
2
=
t2
η
ηd
dT
)tα4(t2
x
ηd
dT
)t(
5.0
)α4(
x
ηd
dT
t
η
ηd
dT
t
T5.05.15.0
-=
-=
-=
∂
∂=
∂
∂
( ) 5.0tα4
1
x
η=
∂
∂⇒
( ) 5.15.0t
5.0
α4
x
t
η,
-=
∂
∂
09:16
5.02
2
5.02
2
)tα4(
1
x
η
ηd
Td
)tα4(
1
ηd
dT
xx
T
xx
T
∂
∂=
∂
∂=
∂
∂
∂
∂=
∂
∂
22/53
1-D Transient in a Semi-Infinite Plate-IV
• Substituting for the partial derivatives in the heat
equation, we get
)tα4(
1
ηd
Td
t2
η
ηd
dT
α
12
2
=-
η2ηd
dT
ηd
Td2
2
-=⇒
ηd
dTη2
ηd
Td2
2
-=• Thus we get an ODE in η
• The boundary conditions
T(t,0) = Ts T(η = 0) = Ts
T(0,x) = Ti ; T(t,x→∞) = Ti T(η = ∞) = Ti
09:16 23/53
1-D Transient in a Semi-Infinite Plate-V
ηd
dT
ηd
TdT
ηd
dT2
2 +
+ =⇒=
• To get the solution, we make the transformation
• The equation in the new variables can be written as
+
+
-= Tη2ηd
dT
ηd
dTη2
ηd
Td2
2
-=• The governing equation is
ηdη2T
dT-=⇒ +
+
Cη)Tln(2 +-=+
• Integration gives2η
1eCT-+ =⇒
2η
1eCηd
dT-=⇒ ∫=∫=-∫ =⇒ --
η
0
u
1
η
0
η
1S
T
T
dueCηdeCTTdT22
S
09:16
u is a
dummy
variable
24/53
7
Error Function
∫ -η
0
udue
2
• The integral occurs very frequently in physics
• Though not integrable in explicit form, it has been
integrated with series expansion and tables have been
constructed under what is called Error Function
• It turns out that 2
πdue
0
u2
=∫∞
-
• Hence erf(∞)=1, erf(0) = 0
∫= -x
0
udue
π
2)x(erf
2
erf(x) is tabulated in
in many books
x=3 is as good as ∞
Erf(3) = 0.99998
• A complimentary error function
is also defined as
erfc(x) = 1 – erf(x)
09:16 25/53
∫∫=∞
∞
+-∞
∞
dxdyeI)yx( 22
Consider ∫∫= -∞ π2
0
)r(
0
θrdrde2
Put r2 = t ( ) πe2
π2θd
2
dteI
0
tπ2
0
t
0
=-=∫∫=⇒∞--
∞
Note that dxeIwhere,I4dyedxeI0
x
1
2
1
yx222
∫==∫∫=∞
-∞
∞
-∞
∞
-
From above2
πIorπI4
1
2
1==
Integration 09:16 26/53
1-D Transient in a Semi-Infinite Plate-VI
S
η
0
u
1 TdueCT2
+∫= -• The solution
S1 T)η(erf2
πC +=
• The Boundary condition, T(η=∞) = Ti implies
s1i T)(erf2
πCT +∞=
s1iT
2
πCT +=⇒
( )π
2TTC si1 -=⇒
( )ssi T)η(erfTTT +-=∴
( ) θ)η(erfTT
TT
si
s==
-
-⇒
η
θ
00
10.99
1.82
09:16 27/53
1-D Transient in a Semi-Infinite Plate-VII
• For η = 1.82, θ=0.99; This implies that η = 1.82
is for all practical purposes is ∞
( )82.1
t4
x82.1
5.0=
α⇒=η ( ) 5.0
t64.3x α=⇒
• The above numbers can be interpreted in the
following manner
• x > 3.64 (αt)0.5 can be considered as infinitely thick
• Similarly for t < x2/(13.25 α), the plate can be
considered infinite
• Now we will turn our attention to heat transferred
09:16 28/53
8
=∂
∂-=′′
=0xx
Tkq
1-D Transient in a Semi-Infinite Plate-VIII
2η
1 eCηd
dT-=• We had shown that
1
0η
Cηd
dT=⇒
=
( )π
2TT si -=
( ) ( )5.0
is
5.0
si
)tπα(
TTk
π
2
)tα4(
TTkq
-=
--=′′∴
5.0
0η)tα4(
1
ηd
dTk
=
-
5.0)t4(
1
d
dT
x
T
αη=
∂
∂
09:16 29/53 09:16
• The heat transferred into the
bubble can be viewed similar
to semi-infinite plate as
shown
• The infinite pool can be
viewed as the semi-infinite
region
• Rate of vaporization can be
estimated by using the energy
balance as
T
dt
dRRπρ4rRπρ
3
4
dt
d
dt
dmRπ4q B2
Bg
3
Bg
g2
B===
h
′′
fg
30/53
09:16
( )
gfggfg h
-=
h
′′=
ρ)tπα(
TTk
ρ
q
dt
dR5.0
is
l
B
( )tπ
αJa
ρ)tπα(
TTcρ
cρ
k
dt
dR l
5.0
l
ispl
pl
lB=
h
-=
gfg
l
l
Where ( )
gfg
l
h
-=
ρ
TTcρJa
ispl
Integration of Eq. (1) gives
1
cπ
tαJa2R
l
B +=
31/53 09:16
• Using BC of RB = 0 at t = 0, implies c = 0
π
tαJa2R
l
B =
• More complex analysis gives,
π
tαJa32R
l
B =⇒tπ
αJa3R
l
B =⇒ &
tπ
αJaR
l
B =⇒ &
• Bubble grows as square root of time in the later half,
• Though RB and its time derivative are functions of
time, its product does not depend on time. This was
exploited by Forster and Zuber to arrive at a
correlation for heat transfer
32/53
9
Pool Boiling Heat Transfer-I
Nucleate boiling Film boilingTransition
Leidenfrost pointSingle-phase
Critical heat flux
Boiling curve at 1 atm
09:16 33/53
Pool Boiling Heat Transfer-II
• Free convection region ∆TSat < 5 oC (single phase)
• Vapor formed at the free surface
• Onset of nucleation ∆Tsat ~ 5 oC
• Bubbles nucleate, grow and detach from the surface
• Increase of wall superheat leads to more vigorous
nucleation and rapid increase in heat transfer
• As the superheat is increased, the vapor formation
become vigorous, it blankets the surface and the heat
transfer decreases. This turn around point is called
the Critical Heat Flux or Boiling crisis
09:16 34/53
Pool Boiling Heat Transfer-III• As the superheat is increased, more blanketing causes
the heat transfer to drop, till the entire heated surface
is blanketed
• Now the radiation heat transfer also starts playing a
role and eventually, the heat transfer starts increasing
due to increase convection and radiation heat transfer
• The second turnaround point is called Leidenfrost
point or rewetting point.
• The heat transfer beyond this point is called film
boiling
• We shall briefly look at some details
09:16 35/53
Pool Boiling Heat Transfer-IV
– Liquid motion is strongly influenced by
nucleation of bubbles at the surface.
– ‘h’ rapidly increases with wall superheat
– Heat transfer is principally due to contact
of liquid with the surface (single-phase
convection) and not due to vaporization.
• Nucleate Boiling (Wall super heat < 30 oC at 1 atm)
� Isolated bubbles Region 5 < ∆TSat < 10 oC
� Jets and Columns Region (10 < ∆TSat < 30 oC at 1 atm)
– Increasing nucleation density causes
bubbles to coalesce to form jets and slugs
– Liquid wetting impaired
– ‘h’ starts decreasing with increase in
superheat
09:16 36/53
10
Pool Boiling Heat Transfer-V• Critical Heat Flux (Wall super heat ~30 oC at 1 atm)
� Heat transfer by conduction and
radiation across vapor blanket
� Usually not a preferred mode of
cooling but can occur during the ECCS
injection in an uncovered core
• Film Boiling (Wall super heat >120 oC at 1 atm)
� Typically 1MW/m2 at 1 atm and increases with pressure
� If the wall pumps heat flux, there is a potential for the
wall to melt as the heat transfer coefficient is very low
here due to vapor blanketing.
09:16 37/53
• Transition Boiling ( 30 oC < ∆Tsat < 120 oC at 1 atm)
� Called Unstable film boiling
� Surface conditions oscillate between nucleate and film
boiling.
Pool Boiling Heat Transfer-VI
• Boiling Heat Transfer Correlations
� Different models exist and there is no single view on this
aspect.
� We shall just list some correlations for application
purposes
09:16 38/53
09:16
• Taking cue from turbulent convection heat transfer
promoted by the movement of eddies, nucleate
boiling heat transfer can also be represented by the
classic Dittus-Boelter type equation
mn PrReANu =⇒
• To properly account for bubble scales, Rohsenow
used departure bubble diameter as the length scale
)ρρ(g
σ2θL
gl -∝⇒ b
θ
Ref: Von Carey, Liquid-Vapor Phase-Change Phenomenon
Models for Nucleate Boiling-I39/53 09:16
• If one performed an energy balance on the heated
surface, one can write a bubble velocity scale as
qGh fg′′= qhρV fggb
′′=⇒
fgg
b hρ
qV
′′=⇒
• Experiments suggest that subcooling had little
influence on the heat transfer coefficient. Hence the
Tsat rather than TB is used as the temperature scale
-
′′=⇒
satW TT
qh
Models for Nucleate Boiling-II40/53
11
09:16
• As the Reynolds number has to be that of liquid that
cools the surface
l
b
µ
GLR =e⇒ l
• From the previous slide,
qGh fg′′=
• As this mass flux can be equally viewed from either
the liquid or the vapour standpoint, Rel is defined as
l
bbg
µ
LVρR =e⇒ l
Models for Nucleate Boiling-III41/53 09:16
l
b
k
hLN =u
.PrReANu1r1 s--=
• Defining Nu to be Rohsenow used
• Plugging in the expressions for each non-dimensional
number as stated, he arrived at the corrrelation
( )
r/1
fg
satpl
l
5.0
vl
r/1
sffglh
T∆cPr
ρρg
σ
)C(
1
hµ
q
-=
′′s/r-
• In the above equation,
7.1s,33.0r,A
θCCsf ===
Models for Nucleate Boiling-IV42/53
09:16
• Csf is an empirical constant found by experiment and
is tabulated in handbooks. Typically its value is
0.013. Some have recommended value of s=1
• An alternate route has been followed by Forster and
Zuber
• They used scales derived from bubble dynamics25.0
2
c
2
B
c
2
Bf
cscale R
R
R/σ2
RρRD =
&
l
scale
lW k
D
)TT(
qNu
-
′′=
• Similarly, Re was scaled as
l
BBf
µ
RRρRe
&
=
Models for Nucleate Boiling-V43/53 09:16
• By substituting for the bubble Radius and Bubble
velocity as derived earlier, we can write
1
l
2 PrJaπRe-=
• RC from our previous derivation can be written as
))T(pT(p(
ρ2R
lsatWsat
c -)=
• Using the above definitions, Forster and Zuber fitted
experimental data into the form
mnPrReANu =⇒ A= 0.0015, n=0.62, m = 0.33
Models for Nucleate Boiling-VI44/53
12
09:16
• The above equation when substituted with
appropriate scales reduce to the following equation
24.0
g
29.0
l
24.0
fg
5.0
79.0
l
49.0
l
45.0
pl
75.0
sat
24.0
sat
lW ρµhσ
kρcp∆T∆00122.0
TT
q=
-
′′
• There are other correlations. At times different
correlations are off by as much as 100% and one
needs to be careful when sizing surface area for HX.
Models for Nucleate Boiling-VII45/53 09:16
• Critical heat flux is the limiting factor in the
operation of heat flux driven systems (nuclear)
• Kuteteladze, based on his Russian work gave an
empirical relation
( ) 25.0
gffgg σ)ρρ(ghρ131.0q -=′′
• Based on the work of Zuber, Lienhard, we can
construct the following physics
• CHF is governed by Rayleigh-Taylor as well as
Kelvin-Helmholtz Instabilities,
Models for CHF-I46/53
09:16
• Based on inviscid flow assumption, one can show
that Taylor wave length to be
( ))ρρ(g
σ3π2λ
gf
T -=
• Similarly, one can derive the Helmholtz
wave length to be under the assumption that
ug>>ul, ρl>> ρg
2
gg
H uρ
πσ2λ =
Models for CHF-II47/53 09:16
• Performing energy balance, we get
Tλ
fgg
2
T hmqλ &=′′
2
ggg d4
πuρm =&
• Assuming d = λT/2, and from the
above two equations, we can write
fgg
g hρ
q
π
16u
′′=
• Also,Hg
g λρ
πσ2u =
Models for CHF-III48/53
13
09:16
• Substituting for ug from the last equation into the one
above, assuming λH= λT, and substituting the
expression for λT, Lienhard derived
( ) 25.0
gffgg σ)ρρ(ghρ149.0q -=′′
( ) 25.0
gffgg σ)ρρ(ghρ131.0q -=′′
• This is same as Kuteteladze’s experimental fit.
• Lienhard’s model is assumed to be a better fit than
Zuber’s. Many more corrections are available
Models for CHF-IV
• Earlier using similar arguments and using p/d =
Zuber derived,
6
49/53 09:16
• For the minimum critical heat flux at the
Leidenfrost’s point, correlations are available
25.0
2
gf
gf
fgg )ρρ(
σ)ρρ(ghρ09.0q
-
-=′′
Berenson,
1961
• Several others are available. See Collier and Thome,
Carey, Tong and Tang, etc.
Models for CHF-V50/53
09:16
• Similar to Nusselt Condensation, we can derive
4/1
satwgg
3
fgvlg
x )TT(µk
gxh)ρρ(ρ707.0Nu
-
-=
• We can derive the average h as done before
• During film boiling radiation heat transfer
also plays an important role
4/1
satwg
3
gfgvlg
x x)TT(µ
gkh)ρρ(ρ707.0h
-
-=
Film Boiling-I51/53 09:16
( ) 4/1
erfaceintwall
satw
2
sat
2
wg
rad
1ε
1
ε
1
)TT)(TT(σρh
+
++=
• Since radiation and convection is present
simultaneously, they get coupled.
• Explicit for of effective heat transfer coefficient can
be obtained by using the following relations
radiationconvectionoverallh75.0hh +=
For hrad < hconv
Film Boiling-II52/53
14
09:16
+
++=
conv
radconv
rad
radconvoverall
h
h62.2
1
h
h25.075.0hhh
For 0 < hrad / hconv < 1
• Convection has to be modified for turbulent
conditions and relations are available
Film Boiling-III53/53