Seetharaman S. Fundamentals of metallurgy
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7.10 Appendix: notation
a constant in equation 7.94c or radius of ellipsoidal martensite nuclei,
lattice parameter or chemical activity
A gaseous reactant or in the case of equation 7.41 a probability of
attachment
A
g
, A
p
external surface area of an individual grain and the pellet, respectively
b number of moles of solid B reacted per mole of gaseous reactant A
b burgers vector
B solid reactant
c, d stoichiometry coefficients or thickness of ellipsoidal martensite nuclei
C gaseous product or total molar concentration
C
i
molar concentration of gaseous species i
d
p
thickness of a slab or diameter of a cylinder or a sphere
D solid product or diffusivity
346 Fundamentals of metallurgy
D
ei
effective diffusion coefficient of gaseous species in the porous solid
D
g
effective diffusion coefficient of gaseous species in the product layer
around a grain
D
K
Knudsen diffusion coefficient
E Young's modulus or electrochemical potential
F
g
, F
p
shape factors for the grain and pellet, respectively ( 1, 2, or 3 for flat
plates, long cylinders, or spheres, respectively)
F Faraday's constant = 96587ëC/equivalents
F
ij
View factor for radiaton heat transfer between surfaces i and j
f fraction
gX conversion function defined by equations 7.75, 7.95 or 7.101
g gravity
G Gibbs free energy or activation energy
h convective heat transfer coefficient
ÿH heat of reaction
i current
i
0
exchange current
J flux (unit of species per area and time)
k reaction-rate constant, thermal conductivity or Bolzmann's constant
k
m
external-mass-transfer coefficient
K equilibrium constant or gradient energy
L diffusion length
m apparent rate constant, equation 7.97 or partition coefficient
M mobility
ÿ
_
m
A
molar rate of consumption of the gaseous reacta nt A per unit area of
solid
n
d
number of grains of solid reactant per unit volume of porous solid,
31 ÿ =4r
2
g
n
i
number of moles of species i per unit volume of the solid mixture
including the pore space, in the case of equation 7.41 the number of
atoms in a favorable position, or number of number of charge
equivalents in Faraday's law
N
i
molar flux vector for species i
N
a
Avogadro's number
N
i
mole fraction of species I
Nu Nusselt number, hd
p
/k
p
i
partial pressure of species i
p(X) conversion function defined by equations 7.76 or 7.79
P total pressure
Pr Prandtl number, C
p
=k
q heat flux vector
r
c
position of reaction interface in a non-porous particle
r
g
original radius of the grain
s strain
The kinetics of metallurgical reactions 347
R
i
rate of production of species i per unit volume of the pellet
R ideal gas constant
Re Reynolds number, du=
r radius
S
v
, S
v
o
surface area per unit volume of the solid (not pellet) at any time and its
initial value, respectively
s indicates strain energy or solid
Sc Schmidt number, =D
AB
Sh Sherwood number, k
m
d
p
/D
AB
Sh* modified Sherwood number defined by equation 7.74
t time
t
; t
g
; t
N
dimensionless times defined by eq uations 7.72, 7.92 and 7.99,
respectively
T temperature
u bulk velocity vector
u
i
velocity of species i
v
1
; v
2
net forward rates of reactions 7.109a and 7.109b, respectively, per unit
volume of the solid mixture includi ng the pore space
v velocity
V
g
, V
p
volume of a grain or the pellet, respectively
w local fraction conversion of solid reactant
x distance
x
i
mole fraction of fluid species i
X overall fractional conversion of solid reactant
Z volume of product solid formed from unit volume of reactant solid or
charge of an ion
Greek symbols
B
;
D
fractions of pellet volume initially occupied by solid reactants B and D,
respectively
i
transference number for the electrochemical half-cell reaction i
; denotation of phases
boundary layer thickness or thickness of deposited thin film
porosity or emissivity
volume change parameter defined by equation 7.81
number of gaseous products formed from one mole of gaseous reactant,
atomic vibration frequency or Poisson's ratio
indicates a parameter in equation 7.65 or overpotential
B
;
D
true molar densities of solids B and D, respective ly
^; ^
g
; ^
N
;
s
gas±solid reaction moduli defined by equations 7.93, 7.103, 7.100
and 7.73, respectively
interfacial tension or a phase
! factor in equation 7.51 that depends on the area of the critically sized
cluster and atom vibration frequency
348 Fundamentals of metallurgy
spacing between ledges or wavelength of the sinu soidal concentration
fluctuation
indicates time of nucleation
shear modulus or chemical potential
Subscripts
A, C gaseous species A and C, respectively
b bulk stream value
B, D solid species B and D, respectively
e effective or environment
eff effective
g grain
i species i, based on initial conditions or interface
1, 2, 3 . . . species in a system
o starting value
! indicates a process going from one state to another
m indicates molar quantity or melt
s solid
b bulk
i interface
Superscripts
* when associated with D indicates radioactive tracer diffusion constant ,
when associated with G indicates an activation energy and when
associated with r or C indicates a critical radius or concentration
respectively
0 reference state
p pellet
s surface or solid
T total
The kinetics of metallurgical reactions 349
8.1 Introduction
8.1.1 Specific heat
The quantity of heat required to raise the temperature of a unit mass by one
degree is called the specific heat C and the molar heat capacity represents the
heat capacity of one mole of a substance. The numerical values are given in
units of calories/mol/ëC or in calories/gm/ëC. The specific heat can be measured
under constant pressure (C
p
) or constant volume (C
v
). In metallic materials it is
C
p
which is of greater relevance as it can be directly measured experimentally
and C
v
can be estimated from C
p
using the relation
C
p
ÿ C
v
2
VT
8:1
where
! Coefficient of volume expansion
1
V
dV
dT
P
!Compressibility coefficient ÿ
1
V
dV
dP
T
V!Molar volume
The molar heat capacity of a binary alloy is estimated from the heat capacities of
the constituent elements, as per the Neumann±Kopp rule.
Thus,
C
p
X
1
C
p1
X
2
C
p2
... 8:2
where X
1
, X
2
are the atom fraction of the constituents.
When the variation of enthalpy (Q E PV ) with temperature (T )is
known, C
p
dQ =dTis determined at a given temperature by graphically or
numerically determining the slope of the Q ÿ T plot. By the same argument
given the specific heat variation with temperature, the enthalpy could be
determined by integration (either graphical or numerical).
8
Thermoanalytical methods in metals processing
O N M O H A N T Y , The Tata Iron and Steel Company, India
8.1.2 Specific heat of crystals and phase transition
As a material is heated, a variety of disorders can occur; each involves
absorption of thermal energy and therefore, a contribution to the specific heat of
the material. When the temperature of a body is raised, the amplitude of
vibration of the atoms would increase, the frequency remaining constant. The
amount of vibrational energy absorbed when the temperature is raised by one
degree is known as vibrational heat capacity and when expressed for unit mass
of the material, it is known as vibrational specific heat. The variation of
vibrational specific heat with temperature has been investigated first by Einstein
and then modified by Debye in order to comply with the observed data at
different temperatures.
1
The Debye expression thus becomes:
C
v
3R
12T
3
3
Z
=T
0
x
3
dx
e
x
ÿ 1
ÿ
3=T
e
=T
ÿ 1
" #
8:3
where
Debye characteristic temperature
h
m
, x
h
kT
! Bollzman's const.;
m
! maximum frequency of vibration
The above expression is similar to Einstein's expression but yields better results
at lower temperatures compared to the Einstein expression; the latter assuming a
constant frequency (
m
) of vibration for all atoms. Th e Debye characteristic
temperatures are given in Table 8.1.
Beyond the Debye±Einstein characteristic temperature (i.e. T= 1:0) the
C
v
per gm atom for all metals tend to assume a limiting value of ~6 cal/gm atom
and at 0ëK it goes to zero. The value of h= (the form being same in both
the Einstein and Debye approaches) depends on and plays a crucial role in
relating to the elastic properties of the materials. The frequency of vibration ()
is given by
1
2
ya
m
r
8:4
where, y ! elastic const. of the material, a ! atomic spacing, m ! atomic
mass.
Since y 10
11
dynes/cm
2
, a 10
ÿ8
cm, m 10
ÿ23
gm, comes out to be of
the order of 10
12
±10
13
per second; substituting this in the relation h= gives
Table 8.1 Debye characteristic temperatures () of selected metals
Material (diamond) ! Na Al Cu Ag Fe Pb Be C
Debye temp. () ! 150 385 315 215 420 88 1000 2000
Thermoanalytical methods in metals processing 351
the characteristic temperatures in the order of 50±500ëK; which is in agreement
with the observed value. Further, it may be seen that increases with increasing
strength of binding (increasing `y') and with decreasing mass. Consequently,
strongly bound crystals of light atoms, e.g. diamond, have high characteristic
temperatures, and weakly bound crystal of heavy atoms, e.g. lead, have low
characteristic temperature s. It has alr eady been shown that the quant ity
experimentally measured is C
p
and the C
v
is derived from this using equation 8.1.
The free energy change at any given temperature is given by the general
expression
F E
o
ÿ
Z
T
0
Z
T
0
C
p
T
dT
dT 8:5
where E
o
! internal energy, C
p
! specific heat. The second term in equation
8.5 essentially represents entropy(s), that is @F=@T , at constant pressure.
The expression shows that as the temperature increases, the free energy drops
and it decreases as the specific heat increases. Thus, if there are two phases, A
and B, and A has lower E
o
and lower specific heat, then A would be more stable
at lower temperat ure but, at sufficiently higher temperature, the phase B will be
stabler. A large specific heat is associated with a low characteristic temperature
and hence with a low vibrational frequency. The vibrational frequency decreases
as the strength of the binding of the lattice decrease. The more weakly bound
phase B in the above situation (that has higher E
o
) has a larger vibrational
specific heat; hence higher entropy. This phase would therefore be more stable
at higher temperatures. Barrett discovered that Li, stable at room temperature in
the bcc form has an fcc structure at liquid air temperature.
2
The polymorphism
in iron is somewhat complex; here bcc structure (±iron) is replaced by the fcc
structure (±iron) above 910ëC ; but the bcc phase re appears (as ±iron) at
temperatures above 1400ëC. Here, the magnetic change (ferro to para on
heating) and the electronic specific heat (that becomes prominent in transition
metals such as iron) are responsible for the observed phase changes.
1
A discontinuous change in the heat content takes place during melting (at
constant temperature) and also during several allotropic/structural transforma-
tions. Here, the entropy change dQ=dT would be infinite. Transformation at
Curie temperature also shows a similar change. The order±disorder transforma-
tion in ± brass exemplifies another type of transformation that takes place over
a temperature; here C
p
at transformation is large, but finite.
The Q ÿ T and dQ=dT ÿ T variations are show n in Fig. 8.1(a) and (b) for
first and second types of transformations, respectively.
8.1.3 Estimation of thermal effects
As has been shown, the heat content can increase abruptly at melting and this
increment is equal to the latent heat. Usually, the specific heat of liquid is higher
352 Fundamentals of metallurgy
than that of solid and therefore, a supercool ed liquid would manifest higher C
p
.
The latent heat also depends on the temperature of transformation and as
supercooling increases the latent heat decreases. The exte nt of heat effects
would decide the accuracy required and therefore the experimental technique to
be used. The solid state phase transformations for instance, are associated with
low heat effects demanding more sensitive equipme nt for monitoring the
transformations. From the foregoing it will be seen that if there is a structural
change taking place during heating or cooling, it will be accompanied by
discontinuities or abnormal curvature in a time±temperature curve. Under
suitable conditions these curves can be analyzed to identify the temperature or
the temperature range associated with a transformat ion.
Direct and inverse thermal methods
The simplest technique is to place a specimen in a furnace which is designed to
have a negligible temperature gradient in a zone that is larger than the dimension
of the specimen. The specimen could be self-supported if the investigation is all
in the solid state or else, need a crucible when solid±liquid transformation is
under study. The furnace is to be heated or cooled linearly and the temperature
of the specimen is recorded at suitable time intervals. For measuring temperature
a calibrated thermocouple or a pyrometer is used; the thermocouple needs to be
sheathed, if liquid alloys are used. The rate of heating or cooling should be so
adjusted as to avoid a pronounced temperature gradient across the specimen. A
recommended rate is 1±2 ëC/min for a specimen of about 20 mm diameter, this is
T
Q
Q
(a)
(b)
dQ
dT
dQ
dT
8.1 Variation Q-T and (dQ/dT) ÿ T for (a) isothermal and (b) non-isothermal
transformations.
Thermoanalytical methods in metals processing 353
expected to produce a gradient not exceeding 0.1 ëC between the outer to the
centre while studying S±L transformations.
The method of inverse rate cooling woul d sometimes show more sensitive
points of transition. Here, the time taken to heat or cool through a small uniform
temperature interval is noted and plotted against the corresponding temperature .
However, the inverse rate method does not give any additional information and
the accuracy may go down especially when isothermal transformation with
supercooling are studied, in comparison to the normal cooling curves.
The characteristics would be complicated in case of non-isothermal
transformations particularly when there are overlapping reactions; for example,
in the steels, during heating or cooling. For analyzing these curves, the
fundamental knowledge on specific heat would be usef ul. In Fig. 8.2 the specific
heat curve for carbon±steel (a) specific heat; (b) a somewhat ideal direct heating
curve; and (c) an inverse heating curve are shown.
3
In the real life situation, the heating and cooling curves appear different in
form. While one makes an allowance for rounding-off the discontinuities,
reference is usually made to cooling curves. The magnitude of the upper
discontinuity is more pronounced in an experimental cooling curve due to
undercooling. Owing to gradual changes in specific heat on heating, it is difficult
to detect the upper change point during heating. Since the actual specific heat
change is small on heating, it is replaced in practice by the irregularity shown as
a dotted line in Fig. 8.2(c). From the above discussion it would be evident that
care needs to be taken to distinguish betwee n fluctuations due to internal
structural changes in the specimen.
Time dt/dT
0.5
0.4
0.3
0.2
0.1
0
840
800
760
720
700 740 780 820 860
Temperature ( C)
Specific heat
T C
o
o
8
8
(a) (b) (c)
8.2 Specific heat and heating curves for a 0.2% carbon-steel (a) specific heat
(b) direct (c) inverse rate.
354 Fundamentals of metallurgy
Differential thermal analysis (DTA)
Small changes in thermal effects are emphasized through this technique. Here,
the difference in temperature (T) between a standard and a specimen is plotted
against the temperature of the specimen. The standard has to be a material that
undergoes no transformation in the temperature range of investigation. The
specimen and standard could be placed side by side, or the standard may be in
the form of a thick-walled hollow cylinder, surrounding the specimen, care taken
to ensure that the specimen and the standard do not touch each other. The latter
method is considered very sensitive.
One may record the differential heating or cooling curves, these are of
importance in solid state transformations.
8.1.4 Introduction to specific thermoanalytical methods
Thermal analysis as defined by the international confederation for thermal
analysis (ICTA) and accepted by the IUPAC and ASTM refers to a group of
techniques in which a physical property of a substance, or its reaction products is
measured as a function of temperature while the substance is subjected to a
controlled temperature programme. Therefore one could cover a number of
techniques here. A comprehensive table has been provided by Gallagher ,
4
simplified in Table 8.2.
As would be evident, a wide spectrum of methods can come under
thermoanalytical methods as long as the measurement is conducted under a
controlled temperature environment. Keeping in view the small space available
and the level of treatment being rather basic, four general techniques are dealt
with in this chapter, namely: (i) thermogravimetry (TG); (ii) differential thermal
analysis (DTA) and differential scanning calorimetry (DSC); and (iii) evolved
gas analysis (EGA). It is also known that two or more techniques are to be
Table 8.2 Principal thermoanalytical methods
Property Technique Acronym
Mass Thermogravimetry TG
Apparent mass Thermomagnetometry TM
(change by induced
magnetic field)
Volatiles Evolved gas detection EGD
Evolved gas analysis EGA
Radioactive decay Emanation thermal analysis ETA
Temperature Differential thermal analysis DTA
Heat or heat flux Differential scanning calorimetry DSC
Dimension Thermodilatometry DA
Thermoanalytical methods in metals processing 355