ASTM - Intensive Quenching Systems - Engineering and Design 2010 - N I Kobasko, M A Aronov, J A Powell, G E Totten

CHAPTER 2 n TRANSIENT NUCLEATE BOILING AND SELF-REGULATED THERMAL PROCESSES 31
where:
q cr1 is the first critical heat flux density (W/m 2 ); and
q cr2 is the second critical heat flux density (W/m 2 ).
This dependence is very important and widely used for
the computation of temperature fields for steel quenching in
liquid media.
In most cases, steel parts are heated up to the austenization
temperature of the steel, higher than A C3 temperature. In
this case, the temperature field is uniform, since all points
within the part are at the furnace temperature T 0 . Upon
immersion into the cold quenchant, the boundary liquid layer
is formed. When compared to the duration of quenching process,
the formation of the boundary layer is very short, and its
duration in many cases can be neglected. What is most important
is not the time for formation of the boundary liquid boiling
layer, but the process of further cooling that can proceed
with or without film boiling. These processes are completely
different, and their development depends on events during the
initial period of time. Therefore, the study of the processes
occurring at the initial period of time are extremely important.
Assume that there is no film boiling and the main cooling
process is nucleate boiling followed by single-phase convection.
The mathematical model for computation of the
temperature fields for steel quenching when there is no film
boiling and nucleate boiling prevails has the form:
Cr @T þ divðk grad TÞ ¼0;
@r ð23Þ
@T
@r þ bm k ðT
T SÞ m
Tðr; 0Þ ¼T 0 :
¼ 0;
S
ð24Þ
ð25Þ
Boundary conditions in Eq 24 can be also obtained
using approximate equations (see Table 5). At the establishment
of single-phase convection, these boundary conditions
are the normal boundary conditions of the third kind:
@T
@r þ a conv
k ðT T mÞ
¼ 0:
S
ð26Þ
Initial conditions include temperature distribution of the
cross-section after nucleate boiling, which can be shown as:
Tðr; s nb Þ¼uðrÞ;
ð27Þ
where:
a conv is the heat transfer coefficient during convection;
r is a radius of a cylinder or a sphere, and r ¼ x for a plate;
s nb is time of transient nucleate boiling;
/(r) is the temperature field of the cross-section at the time
of completion of nucleate boiling; and
T m is bulk temperature.
It should be noted that T m < T S , where T S is a saturation
temperature of the quenchant in a boundary liquid layer.
The time of transition from nucleate boiling to the
single-phase convection is determined from:
q nb ffi q conv ;
ð28Þ
where:
q nb is the heat flux density at the conclusion of nucleate
boiling; and
q conv is heat flux density at the beginning of convection.
For the approximate analytical solution of this problem,
a method of variational calculus will be used [8]. The process
of cooling is divided into two stages. The first stage is an
irregular thermal process. Minimizing functions are built on
the basis of laws of calculus of variations. The solution
found will satisfy Eqs 23 and 24. The method of solving this
problem is described below [9–12].
2.4 ANALYTICAL SOLUTION TO THERMAL
PROBLEMS RELATED TO STEEL QUENCHING
2.4.1 Statement of the Problem
In the process of heating and cooling of steel parts (heat
treatment technology), new structures and mechanical properties
of material are formed [1,13]. The selection of optimal
conditions for heat treatment of metals is impossible without
the computation of temperature fields, which are necessary
to predict structural and thermal stresses. When there is a
wide range of different parts being heat-treated, the use of
numerical methods is even more difficult because of the
time and expense involved, especially for parts with a complex
configuration. In these cases, approximate solutions are
often utilized. Therefore, it is important to develop approximate
analytical methods to facilitate the calculations of temperature
fields encountered during quenching.
The mathematical model describing the heat transfer
during quenching is [8]:
C V ðTÞT s ¼ div½kðTÞr ! TŠþq V ðTÞ; s > 0; ð29Þ
Tðx; 0Þ ¼f ðxÞ; x 2 D; ð30Þ
kðTÞ @T
@ n ! þ FðTÞ
¼ 0; x 2 S; ð31Þ
S
where:
C V (T) is volumetric heat capacity;
k(T) is thermal conductivity;
!
n is outer normal;
S is the surface surrounding area D;
q V (T) is the density of internal heat sources; and
F(T) is the heat flux density at the surface.
The system of equations 29–31 describes both heating and
cooling processes. During heating, F(T) is a total heat flux density
due to processes of radiation, convection, and heat transfer
and during cooling, processes of boiling and convection.
2.4.2 Statement of the Variational Problem
The variational problem equivalent to problem of Eqs 29–31
is a universal approach [9,10] that is used in nonequilibrium
thermodynamics for the description of quasi-linear processes
of heat transfer. For the heat transfer processes discussed
here, the function will be:
I < T > ¼
Z s
Z
ds
0 V
Z T
3
f
Z s
0
Z
þ
V
(
½CVðTÞT s
q V ðTÞŠ
)
kðTÞdT 1 !
2 k2 ðTÞðrTÞ 2 dV
Z
ds
dV
Z T0
f
ds
Z T
f
FðTÞkðTÞdT
C V ðT 0 Þ½T 0 f ðxÞŠkðT 0 ÞdT 0 ! min ð32Þ