Seeds

Chemical & Physical Processes
in Glass Melting
Quality of glass
melting process
Buchmayer

Ruud Beerkens
TNO Glass Group
Eindhoven, The Netherlands
Glass Service

ICG – EFONGA Spring School Montpellier 4-5 May 2009

1

Contents
• Overview of processes essential for glass melting
• Tools of analysis of industrial glass melting
• Melting-in of Raw materials
• Kinetics of Sand Dissolution
• Removal of Gas bubbles & Dissolved gases
• Evaporation processes
• Homogenisation


2


3

refractory corrosion
evaporation:
NaOH, KOH,
NOx and
HBO2,
heat transfer
water infiltration PbO, NaCl,
HF, SO2 etc.

flue gas
chemistry

melting kinetics foaming
fining and redox
refractory corrosion

deposition and
dust formation

1. Overview
Chemistry & Physics
of Glass
Melting Processes

emissions: Na2SO4, Na2B4O7 and PbO dust
HCl, HF, SO2, SO3 , SeO2, HBO2, H3BO3 etc.


4

Side-effects
•
•
•
•

Evaporation from melt
Refractory corrosion
Combustion & heating
Emissions from evaporation & combustion
Furnace

Raw materials mixed

Or

Homogeneous molten glass

Crucible
Melting-in of batch – chemical reactions/endothermic effects
Sand* grain dissolution
Removal of bubbles/gases
Dissolution of seed (fine bubbles) residue
Homogenisation
- Diffusion (slow)
- Velocity gradients – stretching of inhomogeneities

5

Parameters for processes in melting
Important parameters:
• Heat transfer

Temperature

viscosity, surface tension,
chemical activity, reaction kinetics, gas evolution..

• Flow characteristics – convection in melt, stirring
• Residence time: time-temperature history
• Exposure of melt to (reactive) atmosphere and refractory
lining

How to assess:
Temperatures and flows in glass melt ?

6

2. Tools: CFD Simulation Models Glass Furnaces
• Discretize total volume of furnace in small volume cells (> 1 million)
cells LxBxH: (1-10) x (1-10) x (1-10) cm
– GRID / mesh

• Input data process: pull, batch composition, fuel distribution, air
number
• Input furnace:
– Design
– Wall construction, including insulation

• Input glass: viscosity, heat conductivity, density, thermal
expansion, electric conductivity, solubility sand, solubility gases,…
• For each volume cell in tank & combustion chamber
– Energy conservation
– Momentum conservation
– Mass conservation (continuity) for melt and each chemical element
– Respect electro-neutrality

7

CFD Simulation Models
example: local conservation of energy
∂ (ρ c pT
∂t

) = − div (ρ c T v ) + div (λ
r

increase sensible
heat

p

convection
of heat

grad T ) + q

heat conduction

local
boosting or
cooling

Energy equation, conservation law for
energy in each volume element

8

Results of CFD simulation models
• Temperatures at all possible positions
– Combustion space
– Glass melt
– Refractory

• Glass melt and Combustion gas velocities
• Trajectories (particle tracing) in tank
– Thousands of different paths can be identified from charging end to throat or
spout

• Redox and dissolved gases
– Redox state of melt at each position (pO2 or Fe2+/Fe3+)

• Residence time distribution
– Minimum residence time is of importance for melting process

• Glass melt quality indices per trajectory
– Trajectory with minimum melting or fining index is decisive for glass

9

Application of CFD models
• For furnace design (lowest energy, highest glass quality)
–
–
–
–

Optimum depth of tank
Position bubblers or dam or burners
Size and design of throat
Design combustion chamber (LowNOx, less evaporation)

• For optimum process settings
–
–
–
–

Optimum fuel-boosting ratio
Temperature profile (energy distribution)
Bubbling rate
Creation of distinct spring zone to avoid short cut

• Time-transient (time dependent) for colour or pull change
– Optimize colour change process: reduce transition time

• Time-transient for process control (rMPC)
– Sensors give model continuous new information: model tracking
– Model continuously gives recommendation for input parameter changes to
follow optimum process path (low energy, high glass quality, constant T)

10

Geometry & Grid
for computational fluid dynamics (CFD)
analysis of glass furnace
Port Necks
crown

Burner port

tank

Deep Refiner

Batch Boosting electrodes

11

Example result CFD computation

Temperature contours


12

NOx
End-port fired furnace horizontal cross section at level of burners

Base case

4 inch higher crown


13

NOx
End-port fired furnace vertical cross section at 25 % from furnace length from port
NOx scaling in mole fraction

Base case

4 inch
higher crown
Burner port

Exit port (flue gas)
Lower NOx-concentration in exit


14

Glass melt path with lowest ‘temperature‘ index
canal

Temperatur indexpath i
e

T
= ∫
dt
η
doghouse

Temperature course of glass (melt) in typical float glass furnace with
minium temperature index
1800
1600
Temperature in oC

1400
1200
1000
800
600
400
200
0
0

2

4

6

8

10

12

14

Time in hours


15

Sensors (short overview)
• Combustion gases
• gas composition:
• laser optics
• electrochemical sensors (oxygen)

• Glass melt
• chemical composition
• LIBS = laser induced breakdown spectroscopy
emf
• redox / colour parameters
t/c
• Potentiometry
• Voltammetry

Type B
(mV)

(mV)

Alumina rod

Pt / Ni-NiO // ZrO2 // pO2 (glass) / Pt
EMF =

RT
pO 2 (glass)
⋅ ln
nF
pO 2 (ref.Ni/NiO)

Pt measuring
electrode


Zirconia cell

Ni/NiOreference mix
16

3. Melting-in of batch
• In glass furnaces (industrial)
– Kinetics determined by heat transfer through
batch blanket

• In small crucibles:
– Kinetics determined by contact between
different batch constituents and temperature


17

Return flow
for batch heating

Generation blisters
from refractory

Day
hopper

Refining
Bubble absorption

Hot spot &
evaporation

Conditioning of melt
Thermal homogeneity

Zone for sand
grain dissolution
Batch melting
• 40-60 minutes
• 80-90 % of net heat flux

Spring zone
& primary fining

Return flow
from working end


18

Scheme of melting process of batch blanket, charging velocity vg
(m/s)
combustion space
heat transfer
gas release

thickness

temperature
profile

reaction zone
figure 1b
glass melt layer

normal batch

Zipfel

glass
level

reaction zone

glassmelt
flow

heat
transferred

figure 1c


19

Detailed re-presentation of the batch melting process
in glass furnace
1500

o

C

Layer
glassmelt

melts

Dissolution
sand grains

loose batch

sand
grains
gas

melting
reactions
batch

melting
reactions

carbonates
(soda/lime)
gas

dissolution sand
grains

sand
grains

b. top of batch blanket

glass melt 1400
c.

o

C

bottom side of batch
blanket


20

Example:
Melting reactions of soda lime (dolomite) silica batch
• De-hydratation (100 oC physical bonded water & > 100 oC hydrates)
– Important for energy consumption: water evaporation is energy intensive

• Solid state reactions, formation of silicates, e.g.:
Carbonate route < 900 oC at fast heating rate
(melts at ±820 oC)

High amount of
heat required
(550-850 oC)

Na2CO3 + CaCO3

Na2Ca(CO3)2

Na2Ca(CO3)2 +2SiO2

Na2SiO3/CaSiO3 + 2CO2↑ reaction enhances > 820 oC

Na2CO3 + 2SiO2

Na2SiO3 + CO2↑

(790-850 oC)

• Formation of primary melt phases (alkali rich carbonates), e.g.:
Tm Na2CO3
Tm Na2Ca(CO3)2
Tm K2CO3

= 850 oC
= 820 oC
= 890 oC


21

Melting reactions of soda lime silica batch
limited kinetics may shift some reactions to higher temperatures
• Decomposition reactions of (Ca- and Mg-) carbonates:
heat required
CaCO3 + heat

CaO + CO2↑

MgCO3 + heat

MgO + CO2↑

MgCO3·CaCO3 + heat

MgO + CaCO3 + CO2↑

(910 oC at pressure 1 bar)
(540 oC at pressure of 1 bar)
(650 oC, 1 bar)

MgO still present up to 1150 oC.

• Dissolution reactions of SiO2, e.g. (coarse limestone)
Na2O·2SiO2 + CO2↑ T > 790 oC
forms with SiO2 an eutectic melt
Or at further heating
fast Na2O·SiO2 formation (850 oC) –
limestone decomposes and:
2CaO + (SiO2 + Na2O·2SiO2 )eutectic melt
Na2O·2CaO·3SiO2 (> 900 oC)
Reactive calcination: Na2CO3 + 2SiO2

Silicate route: Silicate melt + SiO2
silica enriched melt T > 1000-1100 oC
Eutectic melt phases are formed above ±800-840 oC

22

Phase diagram for the system Na2O – SiO2 showing
eutectic sodium silicate melt phases

100 % SiO2

23

Scheme of melting reactions of soda lime glass batch
1080 oC: T s Na2SiO 3
910 oC : CaCO 3
CaO + CO 2 (gas)
o
850 C : T s Na2CO 3
820 oC : T s Na2Ca(CO 3)2
790 oC : T eut Na 2O.2SiO 2 + SiO 2
740 oC : T eut Na 2Ca(CO 3)2 + Na2CO 3
650 oC : MgCO 3.CaCO 3
MgO+CaCO 3+CO 2 (gas)
540 oC : MgCO 3 -> MgO + CO 2 (gas)

Dissolution of SiO 2, CaO,
MgO, Al2O 3 e.d. in melt phases
primary melts
decomposition
carbonates
solid state reactions
volatilisation of water

0

200

400

600

800
1000
temperature in oC

1200


1400

24

Overall chemical energy demand

-1

CaCO3(s) -> CaO(s) + CO2(g)

Chemical energy consumption rate [kJ·kgbatch ·K ]

MgCO3·CaCO3(s) -> MgO(s) + CO2(g) + CaCO3(s)

-1

10

8

Na2CO3(s) + SiO2(q) -> Na2O·SiO2(s) + CO2(g)
Na2CO3(s) -> Na2CO3(l)

6

Na2CO3(l) + SiO2(q) -> Na2O·SiO2(s) + CO2(g)
Na2O·SiO2(s) + SiO2(q) -> NS(l)
CaO(s) + melt

4

2

0
600

650

700

750

800

850

900

950

1000

-2

Temperature [°C]

Chemical enthalpy of batch reactions for float glass
from soda-sand-dolomite and limestone (positive: endothermic effects)

25

4. Dissolution of ‘refractory’ type raw material
in silicate melt
example: sand grains


26

Sand grain
Glass melt

CSiO2

Ce(T) = saturation level SiO2 in melt

100

Cb

Ce (T)

Cb

= bulk SiO2 level in melt
(depends on amount
dissolved sand)

Moving boundary
Diffusion of SiO2 in melt

27

Dissolution front

One-dimensional dissolution in multi-component liquid

Mass fraction SiO2 in saturated melt: we
Multi-component liquid

Dissolving material

w is mass fraction SiO2 in melt

a

x
dissolution

we
 ∂w 


 ∂x  a

w

ρ SiO2

 ∂w 
ρe ⋅ 

da
 ∂x  a
⋅
= − D⋅
dt
(1 − VA ⋅ ρ e ⋅ w e )


28

Mathematical description
(Ready & Cooper 1966)
-

Spherical symmetry – diffusion in 3 dimensions
Assuming constant diffusion coefficient in melt
Ideal solution, partial molar volume of SiO2 in melt is constant
Convection (term u) due to change of partial molar volume of SiO2 in sand versus in melt
Moving boundary: dissolving sand is partly staying in volume it came from

Mass flux (j) of dissolved SiO2

J
D
r
R
t
ρ
C
u

= mass flux of SiO2
= diffusion coefficient of SiO2 in silicate melt (m2/s)
= radial co-ordinate (distance from sand grain centre) (m)
= radius sand grain (m)
= time (s)
= density of melt (kg/m3)
= local SiO2 mass concentration (kg/m3)
= mass average velocity radial direction due to expansion by dissolution
(change in molar volume) (m/s)

29

Solution dissolution sand grain
without forced convection

Effect of moving boundary

a

=

actual grain size radius (m)

VA

=

partial specific volume of SiO2 in molten glass (m3/kg)

Cs

=

density of sand grain (kg/m3)

Ca

=

mass concentration SiO2 in saturated melt (kg/m3)


30

Sand grain dissolution with convection by
glass melt velocity gradients & density
differences
ρ SiO2
h
we
ws
VA
a
t
ρs
ρSiO2
D

da
⋅
= − h ⋅ (w e ρ e − w sρ s ) /(1 − VA ⋅ w e ⋅ ρ e )
dt
= mass transfer coefficient SiO2 into melt (m/s)
= mass fraction SiO2 in saturated melt (depends on T, and glass) (kg/m3)
= mass fraction SiO2 in bulk melt (depends on dissolved sand)) (kg/m3)
= partial specific volume of SiO2 in molten glass (m3/kg)
= actual radius sand grain (m)
= time (s)
= density of melt (kg/m3)
= density sand grain (kg/m3)
= diffusion coefficient of SiO2 in silicate melt (m2/s)


31

Mass transfer coefficient




Sh
1
1

h=
⋅D ⋅  +
R

2
Sh
⋅D ⋅ t 
π⋅

2


h

=

D =

mass transfer coefficient (m/s)
diffusion coefficient of SiO2 in the molten glass, based on concentration
profiles given in mass fraction (D in m2/s)

R =

grain radius (m),

t

time (s),

=

Sh =

Sherwood number for mass transfer from spherical grain,
≈
2 + 0.89 {Re · Sc + (Gr · Sc)3/4}1/3 *
=
2 (no convection)
=
f (R2/3 , D-1/3, (grad v)1/3) (convection flow of the glass melt)
=
f (R3/4 , h-1/4 , D-1/4)

(free convection of surrounding melt relative to the sand grain: v = flow velocity of the
melt relative to the sand grain (m/s), η = viscosity (Pa.s)

32

30000
no convection
Dissoltion time in s

25000
20000
v-gradient
15000

-1

0.001 s

10000
v-gradient
5000

0.025 s

0
1350

-1

1450

1550

1650

1750

1850

T in K
Dissolution time required for complete dissolution of sand grains in
almost static and stirred soda-lime silica glass melts (forced convection
with velocity gradient grad v) at different temperatures. Initial size Ao=100 mm.

33

Dissolution of sand and alumina grains in static and convective sodalime-silica glass melt at 1700 K, moving boundary effect taken into
account (not on concentration profiles)
1.20E-04

1.00E-04

sand,
grad v=0

sand,
grad v= 0 steady state

sand,
-1

grad v=0.001 s

radius in m

8.00E-05

sand,
-1
grad v=0.01 s

6.00E-05

alumina,
-1
grad v=0.01 s

4.00E-05

alumina,
-1
grad v=0.001 s
sand,
steady state
-1
grad v=0.001 s

2.00E-05

0.00E+00
0

5000

10000

15000

20000

alumina,
grad v = 0

25000

30000

35000

time [s]


34

5. Fining Processes


35

Seeds after batch melting
Coarse sand

Seeds after batch melting
Fine sand


36

10 mm

Glass just after batch melting
- sample thickness ± 5 mm

37

0

to


8 mm

38

Bubbles & Seeds just after melting
•

Many small seeds and bubbles (Blisters) in glass melt
combustion space

(Mulfinger 1976 GTB)

heat transfer
gas release

– More than 100.000 per kg glass melt
thickness

reaction zone
figure 1b
glass melt layer

temperature
profile

glass
level

normal batch

– Most bubble diameters: 0.05 -0.4 mm

reaction zone

glassmelt
flow

•

heat
transferred

figure 1c

In most glass melts (using carbonates):

– bubbles in batch melting area: contain often mainly CO2
•

Large concentrations dissolved CO2 in melt

•

During sand grain dissolution in melt: generation
of fine CO2 seeds (Gispen)
from Glass Service

39

Fining
Objective of Fining:
Removal of bubbles and dissolved gases from the glass melt
Rising velocity of bubble:

v ascension
ρ =

Density of the glass melt [kg/m3]

η =

Viscosity of the melt [Pa·s]

R =

Bubble radius [m]

g =

Acceleration of gravity [m/s2]

c =

c ⋅ ρ ⋅ g ⋅R
=
η

2

Factor (e.g. Stokes c = 2/9)


40

Time to reach glass level at 1 meter [h]

Fining
Time to reach glass surface (1 meter)
250
1400 OC
200
1450 OC

1350 OC

150
100
1500 OC
50
0
0

100

200

300

400

500

Bubble diameter [µm]

41

III. Fining gases and
other dissolved
gases diffuse
strongly into bubble
II. start of fining:
gases diffuse into
bubble

I. static bubble

Reaction in melt: release of fining gases
Pgases melt > pt (pt is pressure in bubble)

42

Two fining steps
•

First step: primary fining
– High temperatures
– Bubble agglomeration and bubble size growth
– Dissolved gases diffuse from melt in to bubbles (like
bubbles in soda drinks)
– Ascension to glass melt surface

• Second step: Secondary fining/Refining (secondary fining)
– Dissolution of (small) remaining bubbles
• Only effective if bubble contains gases (CO2, O2, SO2+O2)
that dissolve in cooling melts
• Glass melt should be lean in dissolved gases


43

Mechanism sulfate primary fining
Fining process in glass melt
pSO2 · pO21/2 ·aNa2O

K=

pSO2· pO21/2

K’ =
aNa2SO4

[SO3]

Increasing temperatures lead to increasing K-values →
extra oxygen gas & SO2 gas release:
- oxygen & SO2 molecules diffuse into growing bubbles
- bubble ascension increases (vascension~R2)
- sulfate retention decreases

44

Fining (primary)
– Fining agents added to the batch to enhance the
rising velocity of bubbles
– Often used fining agent: Sodium sulphate
Fining reaction: T > TFining onset
Na2SO4 ⇔ Na2O + SO2 (gas) +1/2 O2 (gas)

Cm CO2

pSO2 ⋅ pO2
K =
[SO3 ]
'

CO2

Stripping of CO2
and N2 from melt

Cm N2

SO2
O2

N2
Dilution of N2 & CO2 in bubble by fining gases


45

Multicomponent diffusion of gases in bubbles





d
Shi Di
1
1

⋅ ⋅ Rg ⋅ T ⋅ (Csi − Cii) ⋅  +
(4πR3 ⋅ pt ) /(3Rg ⋅ T) = 4πR2 ⋅ Σi
R

dt
2 pt
Shi
⋅ Di ⋅ t 
π⋅

2



[

]

Shi = 1+ (1+ 2·v·R/Di )1/3


46

Fining/Refining: degassing & removal of bubbles
• Mostly applied fining agents in glass industry: Na2SO4 & Sb2O5
– Na2SO4 (m) → SO2 (g) + 0.5 O2 (g) + Na2O (m)
– Sb2O5 (m) → Sb2O3 (m) + O2 (g)
• Na2SO4 added in concentrations 0.1 – 1 wt. % to batches of:
– Soda lime glass for container, float and tableware
– E-borosilicate glass for fibres
• Na2SO4 partly decomposes during batch melting & releasing SO2 in
early melting stages
• Dissociation temperature of Na2SO4 in melt:
– Between 1350 – 1480 ºC, depending on redox state
– Between 1100-1350 oC (reduced batches) Na2SO4+Na2S
reactions forming SO2 and or S2 gas.

47

Fe2+/Fetotal

Sulfur retention (wt.%SO3)

0.6
0.5

80

70

60

Sulfur only in Sulfur in
form of S2form of SO42-, S2-

40

25

15

%

Sulfur only in
form of SO42-

(probably also SO32-?)

0.4
0.3

1400 oC

0.2
0.1

1500 oC

0.0
-8

-7

-30

-6
-5
-4
-3
Log pO2 in the melt at 1400°C (bar)

-20

-10

0

+10

-2

-1

+20

redox number

48

Mass
Spectrometer
Synthetic gas

30 mm


49

High temperature test facility


50

1. Primary fining
– Removal of bubbles by rising of bubbles to melt
surface
– Bubble growth under influence of fining agents
– Stripping of dissolved gases by growing of gas bubbles
(dilution)

Fining


51

Enhanced Sulfate Fining by
Dissolved Water in Melt
O2
N2

CO2
O2

H2O
H2O

Oxy-case

SO2

SO2
O2
N2

CO2
O2
SO2

H2O
Air case
SO2

In oxygen-fired glass furnace:
peH2O = 0.25-0.40 bar
Fining only if:
peSO2 + peO2 > 0.70 - 0.75 bar
In air-fired furnace:
peH2O = 0.10-0.15 bar
Fining only if :
peSO2 + peO2 > 0.9 bar


52

Evolution of fining gas: water lean & rich melts

50

water vapor pressure

40

0 bar

30

0.20 bar

20

0.60 bar

3

m /batch

Volume of gas in

Gas evolution during sulfate fining of soda lime glass
melt - effect of water vapor level 60

10
0
1300

1400

1500

1600
o

Temperature in C


53

Stripping of dissolved gases from melt


54

Pressure in melt before and during fining & cooling

Total internal pressure in melt
bar

[S]initial= 0.3 mass% SO3 300 mgr water/kg [Fe2+]/ [Fetotal] = 20 %

1.2
1.0
0.8
0.6
0.4
0.2
0.0
1200

1300

1400

1500

1600

o

Temperature in C


55

Partial pressure in float glass melt during heating

Partial pressure in float glass melt
in bar

[S]initial = 0.3 mass% SO3 300 mgr water/kg [Fe2+]/ [Fetotal] = 20 %
1.E+00

SO2

1.E-01
H2 O
1.E-02
O2
1.E-03
CO2
1.E-04
N2
1.E-05
1250

1350

1450

1550
o

Temperature in C


56

O2
fining gas

CO2

mol/m3

mol/m3

Modeling dissolved gas distribution in glass melt tank

57

2. Secondary fining (refining)
– Re-absorption of residual gases during controlled cooling
• Chemical solubility SO2 and O2 increases with
decreasing temperature: gases will be re-absorbed
during cooling.
• Physical solubility of dissolved gases increases slightly
with decreasing temperature: these gases will also be reabsorbed during cooling

Refining

58

Fining at low pressure

- Same amount of gas needs large volume
- Low partial pressures in bubble will stimulate gas diffusion
from melt into bubble


59

6. Evaporation processes


60

Multi-component glasses
• Different oxides
• Oxides in glass with high chemical activity or vapour
pressure:
– React at glass melt surface with combustion gases
– Evaporate from glass melt surface
– Show depletion at surface layer
INCONGRUENT EVAPORATION


61

Evaporation mechanisms
•

Direct evaporation of volatile glass components from the surface of the melt,
e.g. volatilization of PbO from lead crystal melts;

•

Evaporation of components by reactions in the melt itself, forming volatile
compounds; generally such volatile compounds exhibit high activity coefficients
and weak bonding with other glass melt species.
–

•

An example: formation of alkali borates in alkali borosilicate melts,
subsequently evaporation of alkali meta-borates/tetra-borates takes place,
Na2O(melt) + B2O3 (melt) ⇔ 2NaBO2 (melt) ⇒ 2 NaBO2 (vapor)
Evaporation by reactions of certain glass melt components with gas
species at the surface of the melt.
The evaporation rate & vapor pressure depends on the composition of
the gas atmosphere above the melt.
B2O3 (glass melt) + H2O

⇒ 2HBO2 (vapor)

Na2O (glass melt) + H2O (gas)

⇒ 2NaOH (vapor)

Na2O(glass melt) + CO (gas)

⇒ 2Na (vapor) + CO2


62

Kinetics of evaporation
The volatilization rate is often determined by:
• mass transport of the relevant species from the
melt interior (bulk) to the surface;
• the vapor pressures of the volatile components at the
surface of the melt, dependent on the glass composition,
temperature and gas atmosphere;
• the mass transfer of evaporated species from the surface
of the melt into the main gas stream above the melt.

63

Reactive evaporation
For reactive evaporation of a component j in the melt reacting with
a gas k and forming gaseous species i with saturation pressure
pi*, according to reaction:
n·j (melt) + m·k (gas)
Reaction equilibrium: pi*q

q·i (gas)

= K · ajn · pkm

The values of K (equilibrium constant, assuming chemical
equilibrium at the glass melt surface) and aj (activity of component j
in the molten glass at the surface) can be determined
experimentally or by thermodynamic modeling


64

Static melt and static atmosphere above the
melt (interface x = 0)
Diffusion in melt of reacting glass melt component j:

dCj/dt = Dm,j·δCj2/δx2

Time dependency: Cjsurface(t).

Dm,j is the inter-diffusion coefficient of the volatile component j in the melt.

at t = 0
at t > 0
at t > 0

-∞ < x < 0
x
−∞
x =0

Cj = Cjbulk
Cj = Cjbulk
Cj = Cjsurface(t)

For the vapor i in a static gas phase with partial vapor pressure pi,
the diffusion process in the gas phase can be described in a similar way:

δ(pi/RgT)/δt = Dg,i ·δ2(pi/RgT) /δx2
Dg,i is the diffusion coefficient of the vapor I in the gas phase.

at t = 0
at t > 0
at t > 0

0<x<∞
x
∞
x =0

pi = pi,gasbulk
pi = pigasbulk
pi = pi*(t)


65

-3

Na2 O concentration [mol.m ]

4970
4968
4966
4964
4962

5 sec.

4960

50 sec.
250 sec.

4958
4956

DNa2O=3.3 10-11 m 2.s -1

4954
0

0.01

0.02

0.03

0.04

0.05

0.06

distance from surface [mm]

Calculated time dependent- Na2O concentration profiles in static melt
Situation: static conditions in semi-infinite gas phase with 0.55 bar vapor pressure
and semi-infinite soda-lime-silica melt
(13 wt% Na2O, 10 wt% CaO, 5 wt% MgO, 72 wt% SiO2).
Dg,NaOH = 2.7 10-4 m2·s-1 , Dm,Na2O = 3.3 10-11 m2·s-1

66

Evaporation in gas flow above molten glass
pb,i
main gas stream
velocity, vg

diffusion of gas i in
gas boundary layer, Dgi

surface reaction:
n·j (melt)+ m·k (gas) ⇒ q·i (gas) p*i
glass melt surface
C j-profile
Example:
Na2O (m) + H2O(g) ⇔ 2NaOH (g)

Transport of component
j in the melt, Dm,j


67

Evaporation in gas flow
Average evaporation rate (over length Lg of gas flow above
melt ) of component i (formed by reaction of glass
compound j) into (turbulent) gas phase:
Qm,j =(ni/qi)·A·vg0.8·ρg0.47·µgas-0.47·Dg0.667·Lg-0.2· Rg-1·T-1·B·Cj,x=0(t)

The proportionality parameter B depends on the furnace
atmosphere composition and the chemical activity of the
volatile component in the melt.
For NaOH-evaporation, the B value depends on the water
vapor pressure in the furnace atmosphere, B ∼ pH2O0.5

68

Mass transfer equations
Average evaporation rate (rate of loss of glass component j)
over length Lg from leading edge:
Qm,j = -Dm,j·(δCj/δx)x=0 = α·Cj,x=0(t)
α = (ni/qi) · A· vg0.8·ρg 0.47·µgas-0.47·Dg0.667·Lg-0.2· Rg-1·T-1· B
Turbulent flow of gas
v = velocity, g refers to gas phase, Rg is universal gas constant, T in K,
B ratio between vapour pressure i and surface concentration component j
A = between 0.03 and 0.04 for turbulent gas flow (Re > 300000 or for disturbed flows)


69

Solution in flowing gas and static melt
For kd defined as α/DmNa2O the solution for a single component j
Evaporating from a static melt in flowing gas phase
Assuming complete depletion at surface for t

∞

MQm.j is the total evaporation mass loss per unit surface area between
time 0 and τ

MQm.j = (Cj,0/kd)·{exp(kd2·DmNa2O·τ)·erfc[kd·(DmNa2O·τ)0.5] -1
+ 2kd·(DmNa2O·τ/π)0.5}
Cj,x=0(t) = Cj,0· exp(kd2·DmNa2O·τ)·erfc[kd·(DmNa2O·τ)0.5]
Cj,0 = bulk concentration compound j at t=0

70

5000
-3

melt [mol.m ]

Na2O concentration in

5500

4500

stagnant gas
4000

Lg= 2 m, v= 2 m.s-1
Lg= 2 m, v = 5 m.s-1

3500

Lg= 0.5 m, v= 2 m.s-1
3000
0

0.5

1

1.5

2

2.5

distance from surface [mm]

Local concentration profile in soda-lime silica melt after 7200 seconds exposure time,
calculated for NaOH-evaporation from static melt in static or flowing gas phases,
(Lg= downstream distance from leading edge ).
Temperature = 1500 oC, pH2O = 0.55 bar. Dm,Na2O = 3.3 10-11 m2·s-1
Glass composition (mass %): SiO2 =72, Na2O =13, MgO = 5, CaO = 10

71

parameter is temperature:
5000

1723K
1773 K
1823 K

4000

-3

[mol.m ]

Na2O-surface concentration

6000

1873 K
3000
2000
1000
0
0

5000

10000

15000 20000
time [s]

25000

30000

Change in Na2O-surface concentration soda-lime-silica melt at different
temperatures in flowing gas (5 ms-1), 1 meters downstream.
pH2O in gas = 0.55 bar & Na2O in glass = 13 mass%.
Dm,Na2O= 8 10-10 exp(-5655/T) in upper graph

72

Experimental – Set up to study
(reactive) evaporation from molten glass
Thermocouples
Platinum funnel

Porous
plate

Gases IN:

Platinu
m gas
samplin
g probe

Platinum
coating (30 cm)

Platinum boat

N2, H2O,
O2

melt


73

Mass transfer in gas phase during
transpiration – evaporation test
Shtranspiration

0 .5


2


1 .5
=  C1 + C 2 ⋅ Re d ⋅ Sc + 
 ⋅ C3 ⋅ (Re d ⋅ Sc ) 


 1 + 22 ⋅ Sc 



h g ,i =

Q g ,i =

Vessel with
liquid of melt

1

3

Shtranspiration ⋅ D g ,i
d

hg ,i
R ⋅T

(

⋅ pi (t ) * − pi bulk

)

2.5E-03

x=0

pi*(t) (e.g. p*NaOH or p*NaBO2)
can be derived from evaporation
(transpiration experiments)
From measured Qg,i and Sherwood
relations derived with model liquids

Water evaporation rate
QH2O (moles s -1 m-2)

2.0E-03

1.5E-03
Measurerments

1.0E-03

CFD model
Empirical equation (2.19)

5.0E-04

0.0E+00
0

100

200
Reynolds number
Re (-)


300

400

74

1.E-06

Modeled aNa2O (-)

SiO2:Na2O:CaO = 74:16:10 mol
1.E-07

Na2O.2SiO2
y = 0.9823x
R2 = 0.9283

1.E-08

1.E-09
1.E-09

1.E-08

1.E-07

1.E-06

Measured aNa2O (-)

Na2O activity at glass melt surface determined by transpiration test

measuring p*NaOH: Na2O + H2O

2 NaOH


75

Derivation chemical activity of volatile glass
component at surface of melt
K = exp(-∆G/RT) = p*NaOH2/aNa2O.pH2O
From thermodynamic tables:
∆G = GfNa2O+GfH2O- 2GfNaOH(g)
p*NaOH is measured from QNaOH and pH2O is
controlled
aNa2O (surface) can be determined
K is calculated by standard Gibbs free energy values of
products & reactants of reaction

76

Non static melt & non static gas phase
free convection by density gradients
Gas flow
Low Na2O

Mid Na2O

High Na2O

Float glass melt with Na2O concentration differences

77

6. Homogeneity of glass
Micro-mixing: transfer from high to low chemical activity by diffusion

Cm
- π 2 . D.t
= A exp
Co
Lo 2
Macro-mixing: elongation of in-homogeneities exposed to velocity gradient in melt

C(x,t)

dC
d 2C
= D⋅ 2
dt
dx

t=0
Cm (t)

Slow diffusion processes
t = t1

t = t2
Co

Lo

78

Macro-mixing
Reduction of diffusion distance, Lo
velocity v + dv

velocity v

y
In the case, t dv/dy >> 1:

L=

L0
dv
⋅
t
dy

For macro-mixing in combination with diffusion (by approximation):

Cm
π 2 . D . t 3 .( dv / dy ) 2
= A .exp −
Co
L2
0
A

= proportionality factor dependent on the shape of the cord

79

Macro versus only Micro mixing
• Small velocity gradients (> 0.01 m/s per m) enhance
homogenisation process with factor 20 to 100
• Velocity gradients by:
– Stirring
– Bubbling
– Temperature gradients
free convection


80

Thank you for your attention
What does a number tell us without the proper unit?


81

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