condução e conversao de calor app2

Prévia do material em texto

This appendix supports the thennal aspects of Chapters 2, 6 and subsequent chapters. A
more complete description of heat transport in solids is given in Carslaw and Jaeger (1959).
The basic law of heat conduction in an isotropic material is assumed; namely that the rate
q of heat transfer per unit area nonnal to an isothennal surface is proportional to the temper -
ature gradient in that direction and with K the thennal conductivity and T the temperature:
aT
an
(A2.1)q=-K
(A2.2a)
+~
(~ )2
dT dz
()2T
()Z2
= ~K d.xdydzdt
The heat accumulating due to convection, Hc ' isonv
Hconv = ~ uzpCT -UzPC~T+ -f dz~ ~dxdydt = -uzpC
aT
az
d.xdydzt
(A2.2b)
Conduction and convection of 
heat in solids 
352 Appendix 2 
Fig. A2.1 (a) A control volume for temperature analysis and (b) dependence of temperature on position and time for 
the example of Section A2.2.1 (IC= 10 mm2/s) 
Internal heat generation at a rate q* per unit volume causes an accumulation, Hint: 
Hint = q*dxdydzdt (A2.2~) 
Equating the sum of the terms (equations (A2.2a) to (A2.2~)) to the product of tempera- 
ture rise and heat capacity of the volume: 
aT a2T dK aT 2 aT 
p C - = K - + - - - u,pc - + q* (A2.3 a) 
at a z 2 dT ( az ) aZ 
The extension to three dimensions is straightforward: 
aT a2T a2T a2T dK JT 2 JT 2 aT 2 
pC-=K at ( - ax2 +T+F)+F( (T)+(x)+(+ 
-pc u, - JT + z i - aT + Uz- aT ) + q * 1 ax 
pC-=K(- a t ax2 +-+-) a Y 2 a z 2 -pc(Ux-+ ax icy--- aY + u, z aZ )+ 4* 
JY az 
(A2.3b) 
When thermal conductivity does not vary with temperature, equation (A2.3b) reduces 
to 
aT a2T d2T a2T aT aT 
(A2.4) 
Selected problems, with no convection 353
When itx = ity = itz = 0, and q* = 0 too, equation (A2.4) simplifies further, to
1 aT ~a2T a2T a2T
~--= -+-+- (A2.5)
1( at ax2 ay2 az2
where the diffusivity 1( equals K/pC. In this section, some solutions of equation (A2.5) are
presented that give physical insight into conditions relevant to machining.
A2.2.1 The semi-infinite solid z > 0: temperature due to an
instantaneous quantity of heat H per unit area into it over the
plane z = 0, at t = 0; ambient temperature T o
It may be checked by substitution that
41(t (A2.6)
is a solution of equation (A2.5). It has the property that, at t = 0, it is zero for all z > 0 and
is infinite at z = 0. For t > 0, dT/dz = 0 at z = 0 and
00
f pC(T -T Jdz = H (A2.7)
0
Equation (A2.6) thus describes the temperature rise caused by releasing a quantity of heat
H per unit area, at z = 0, instantaneously at ( = 0; and thereafter preventing flow of heat
across (insulating) the surface z = 0. Figure A2.l(b) shows for different times the dimen-
sionless temperature pC(T -T J/H for a material with 1( = 10 mm2/s, typical of metals. The
increasing extent of the heated region with time is clearly seen.
At every time, the temperature distribution has the property that 84.3% of the associ-
ated ~eat is contained within the ~ion z/V4-;t < 1. This res~lt is obtained by integrating
equation (A2.6) from z = ° to y 41(( .Values of the error function erf p, 
1
 
z2
 
C
(A2.8)
that results are tabulated in Carslaw and Jaegeill959). Physically, one can visualize the
temperature front as travelling a distance"" V 4/(t in time t. This is used in considering
temperature distributions due to moving heat sources (Section A2.3.2).
A2.2.2 The semi-infinite solid z > 0: temperature due to supply of heat
at a constant rate q per unit area over the plane z = 0, for t > 0;
ambient temperature T o
Heat dH = qdt' is released at z = O in the time interval t' tot' + dl'. The temperature rise
that this causes at z at a later time t is, from equation (A2.6)
354 Appendix 2
z
4Ir(t-t') (A2.9)
The total temperature is obtained by integrating with respect to t' from O to t. The temper-
ature at z = O will be found to be of interest. When q is independent of time
~V-;t
VIr K
(T- To) = --=- (A2.10)
The average temperature at z = 0, over the time interval ° to t, is 2/3rds of this. 
1
 
2
 
-
 
2
 
1
A2.2.3 The semi-infinite solid z > 0: temperature due to an
instantaneous quantity of heat H released into it at the point
x = y = z = 0, at t = 0; ambient temperature T o
In this case of three-dimensional heat flow, the equivalent to equation (A2.6) is
x'+y'+z'
4Kt
H
T-To=- -=-: e (A2.11)
4pC (7t1Ct)3/2
Equation (A2.ll) is a building block for determining the temperature caused by heating
over a finite area of an otherwise insulated surface, which is considered next.
A2.2.4 The semi-infinite solid z > 0: uniform heating rate q per unit area
for t > 0, over the rectangle -a < x < a, -b < y < b at z = 0;
ambient temperature T o
Heat flows into the solid over the surface area shown in Figure (A2.2a). In the time inter-
val t' to t' + dt', the quantity of heat dH that enters through the area dA = dx'dy' at (x', y')
is qdAdt'. From equation (A2.1l) the contribution of this to the temperature at any point
(x, y, z) in the solid at time t is
(X-x')2+(y-y')2+Z;
.
4K(t-t') (A2.12)
Integrating over time first, in the limit as t and t' approach infinity (the steady state),
(A2.13)
Details of the integration over area are given by Loewen and Shaw (1954). At the surface
Z = 0, the maximum temperature (at x = y = 0) and average temperature over the heat
source are respectively
Selected problems, with convection 355
q
z'(b) (c)
Fig. A2.2 Some problems relevant to machining: (a) surface heating of a stationary semi-infinite solid; (b) an infinite
solid moving perpendicular to a plane heat source; (c) a semi-infinite solid moving tangentially to the plane of a surface
heat source
2qa
nK
b
a
b
a
.aSlnh-l -
b
(T- T O)max = sinh-] +
~r/~+~\/1 +~\'/~~-~1
31rKl \ b a I \ a2 I a2 b J
(T- T O)av = (i- T O)max
(A2.14)
Figures A2.2(b) and (c) show two classes of moving heat source problem. In Figure
A2.2(b) heating occurs over the plane z = 0, and the solid moves with velocity
itz through the source. In Figure A2.2(c), heating also occurs over the plane z = 0, but
the solid moves tangentially past the source, in this case with a velocity itx in the x-
direction.
356 Appendix 2 
A2.3.1 The infinite solid with velocity uz: stead heatin at rate 9 per 
unit area over the plane z 0 (Figure A i g .2b); am ient 
temperature To 
In the steady state, the form of equation (A2.4) (with q* = 0) to be satisfied is 
a2T . JT 
lc-- (A2.15) 
az2 - ' Z aZ 
The temperature distribution 
U,Z 4 4 - 
( T - To) = ~ , ~ 2 0 ; (T -To)=- e , 2 2 0 (A2.16) 
PCU, PCU, 
satisfies this. For z > 0, the temperature gradient is zero: all heat transfer is by convection. 
For z = - 0, aT/& = q/K: from equation (A2. l), all the heating rate q is conducted towards 
-z. It is eventually swept back by convection towards + z . 
A2.3.2 Semi-infinite solid z > 0, velocity: Ox steady heating rate per 
unit area over the rectangle -a < x < a, -b < y e b, z = 0 ( ? igure 
A2.2(c)); ambient temperature To 
Two extremes exist, depending on the ratio of the time 2alux, for an element of the solid 
to pass the heat source of width 2a to the time a 2 k for heat to conduct the distance 2a 
(Section A2.2.1). This ratio, equal to 21cl(zixa), is the inverse of the more widely known 
Peclet number P,. 
When the ratio is large (P, << l), the temperature field in the solid is dominated by 
conduction and is no different from that in a stationary solid, see SectionA2.2.4. Equations 
(A2.14) give maximum and average temperatures at the surface within the area of the heat 
source. When bla = 1 and 5, for example, 
b 
-- - 1 : f T - T ) = 
- = 5: (T - Tolmax _.__ 
a 
(A2.17a) 
At the other extreme (P, >> I), convection dominates the temperature field. Beneath theheat source, aTl& >> aTlax or aTlay; heat conduction occurs mainly in the z-direction and 
temperatures may be found from Section A2.2.2. At z = 0, the temperature variation from 
x = - a to x = + a is given by equation (A2.10), with the heating time t from 0 to 2ulux. 
Maximum and average temperatures are, after rearrangement to introduce the dimension- 
less group (qalK), 
(A2.17 b) 
Numerical (finite element) methods 357
Because these results are derived from a linear heat flow approximation, they depend only
on the dimension a and not on the ratio b/a, in contrast to p e « I conditions.
A more detailed analysis (Carslaw and Jaeger, 1959) shows equations (A2.16) and
(A2.l7) to be reasonable approximations as long as Uxa/(2K) < 0.3 or > 3 respectively.
Applying them at Uxa/(2K) = 1 leads to an error of ~20%.
Steady state (aT/at = 0) solutions of equation (A2.4), with boundary conditions
T = Ts on surfaces ST of specified temperature,
KaT/an = 0 on thermally insulated surfaces Sqo'
KaT/an = -h(T-To) on surfaces Sh with heat transfer (heat transfer coefficient h),
KaT/an = -q on surfaces Sq with heat generation q per unit area.
may be found throughout a volume V by a variational method (Hiraoka and Tanaka, 1968).
A temperature distribution satisfying these conditions minimizes the functional
h
(T2 -2T oT)dS
(A2.19)
m
/(1) = L /e(T)
e=l
where Ie(1) means equation (A2.18) applied to an element and m is the total number of
elements. If an element's internal and surface temperature variations with position can be
written in terms of its nodal temperatures and coordinates, Ie(1) can be evaluated. Its vari-
ation tSle with respect to changes in nodal temperatures can also be evaluated and set to
zero, to produce an element thermal stiffness equation of the form
[H]e{T} = {F}e (A2.20a)
where the elements of the nodal F-vector depend on the heat generation and loss quanti-
ties q*, q and h, and the elements of [H]e depend mainly on the conduction and convec-
tion terms of Ie(1). Assembly of all the element equations to create a global equation
"' (A2.18)
where the temperature gradients aT/ax, aT/ay, at/az, are not varied in the minimization
process. The functional does not take into account possible variations of thermal proper-
ties with temperature, nor radiative heat loss conditions.
Equation (A2.18) is the basis of a finite element temperature calculation method if its
volume and surface integrations, which extend over the whole analytical region, are
regarded as the sum of integrations over finite elements:
358 Appendix 2 
[HI{TI = {FI (A2.20b) 
and its solution, completes the finite element calculation. The procedure is particularly 
simple if four-node tetrahedra are chosen for the elements, as then temperature variations 
are linear within an element and temperature gradients are constant. Thermal properties 
varying with temperature can also be considered, by allowing each tetrahedron to have 
different thermal properties. In two-dimensional problems, an equally simple procedure 
may be developed for three-node triangular elements (Tay et al., 1974; Childs et al., 
1988). 
A2.4.1 Temperature variations within four-node tetrahedra 
Figure A2.3 shows a tetrahedron with its four nodes i, j , k, 1, ordered according to a right- 
hand rule whereby the first three nodes are listed in an anticlockwise manner when viewed 
from the fourth one. Node i is at (xi, y i , zi) and so on for the other nodes. Temperature Te 
anywhere in the element is related to the nodal temperatures { T ) = (Ti 5 Tk Tl)T by 
Te = [Ni Nj Nk Nl] { T) = [N] { T } (A2.21) 
where [N] is known as the element's shape function. 
1 
N . = ~ (ai + bix + ciy + d,z) 
' 6Ve 
where 
x. y j zj 1 Yj zj 
a . 1 = 12 : 2 I , b i = - l ] ; 21 
Fig. A2.3 A tetrahedral finite element 
Numerical (finite element) methods 359 
and 
(A2.22) 
This may be checked by showing that, at the nodes, Te takes the nodal values. Nj, Nk and 
N , are similarly obtained by cyclic permutation of the subscripts in the order i, j , k, 1. V, is 
the volume of the tetrahedron. 
In the same way, temperature T s over the surface ikj may be expressed as a linear func- 
tion of the surface’s nodal temperatures: 
T = [N,”’N;] { T ] = [N’] [ T ] (A2.23) 
where 
1 
N.’ = ~ (u] + bix‘ + ciy’) 
2Aikj 
and 
The other coefficients are obtained by cyclic interchange of the subscripts in the order i, k, 
j . x’, y’ are local coordinates defined on the plane ikj. Aikj is the area of the element’s trian- 
gular face: it may also be written in global coordinates as 
(A2.25) 
A2.4.2 Tetrahedral element thermal stiff ness equation 
Equation (A2.21), after differentiation with respect to x, y and z , and equation (A2.23) are 
substituted into Ze(T) of equation A2.19. The variation of Ze(7) with respect to Ti, T., Tk and T, 
is established by differentiation and set equal to zero. [HI, and { F ] , (equation (Ai.20a)) are 
[HI, = 
1 bibi + cici + didi bibj + cicj + d.d. bib, + cicl + didl bjbi + cjci + didi b.b. + C . C . + d.d. bibl + cjcl + djdl b,bi + ckci + dkdi bkbj + ckcj + dkdj bkbl + ckc, + dkdl b,bi + clci + d,di blbj + clcj + dldj b,b, + clc, + dldl ‘ I J J J J J J bibk + tick + didk bjbk + cjck + djdk bkbk + ckck + dkdk blbk + clck + dldk 
360 Appendix 2 
1 uxbi + uYci + uzdi u b. + u c. + uzdj uxbk + UYck + uzdk U b. + U C. + U d. uxbj + Uycj + uzdj uxbl + uYcl + uzdl X J Y J uxbi + u,ci + u,di u b. + u c. + uzdj uxbk + u c + uzdk Uxbl + U,cl + Uzdl uxbi + U,ci + uzdi uxbk + uYck + uzdk Uxb, + UYcl + u,d, X J Y J Y k X J Y J Z J uxbi + uYci + uzdi 
2 1 1 0 
uxbk + U Y k c + uzdk uxb, + U,cl + u,d, 
+A[ h A , ' 1 2 1 0 ] 
12 1 1 2 0 
0 0 0 0 (A2.26) 
and 
Global assembly of equations (A2.20a), with coefficients equations (A2.26) and 
(A2.27), to form equation (A2.20b), or similarly in two-dimensions, forms the thermal part 
of closely coupled steady state thermal-plastic finite element calculations. 
A2.4.3 Approximate finite element analysis 
Finite element calculations can be applied to the shear-plane cutting model shown in 
Figure A2.4. There are no internal volume heat sources, q*, in this approximation, but 
internal surface sources q, and qf on the primary shear plane and at the chip/tool inter- 
face. If experimental measurements of cutting forces, shear plane angle and chip/tool 
contact length have been carried out, q, and the average value of qf can be determined as 
follows: 
(A2.28a) 
(A2.2 8 b) 
where 
(A2.29) 
In general, q, is assumed to be uniform over the primary shear plane, but qf may take on a 
range of distributions, for example triangular as shown in Figure A2.4. 
1 F, cos @ - FT sin @ F, sin a + FT cos a zs = sin @; zf = fd V cos a sin @ v, = 'work; v, = 'work cos(@ - a) cos(@ - a) 
A2.4.4 Extension to transient conditions 
The functional, equation (A2.1 S), supports transient temperature calculation if the q* term 
is replaced by (q* - pC&?/&). Then the finite element equation (A2.20a) becomes 
Numerical (finite element) methods 361 
/ICv, 
[CI, = ~ 
20 
2 1 1 1 
1 1 2 1 
1 1 1 2 
1 2 1 1 
362 Appendix 2
Over a time interval At, separating two instants tn and tn+l' the average values of nodal
rates of change of temperature can be written in two ways
aT
at
aT 1
-at -J n+l
(A2.31a)-a~ 1 = (I -8)
at Jay
+8
or
~~
Tn+l (A2.31b)
111
,!i;- ~av=
where (J is a fraction varying between O and 1 which allows the weight given to the initial
and final values of the rates of change of temperature to be varied. After multiplying equa-
tions (A2.31) by [C], substituting [C] { aT/at} terms in equation (A2.3la) for ( {F}-[H] {T} )
terms from equation (A2.30), equating equations (A2.3la) and (A2.31 b ), and rearranging,
anequation is created for temperatures at time tn+l in terms of temperatures at time tn: in
global assembled form
[C]
~t
(A2.32)
This is a standard result in finite element texts (for example Huebner and Thomton,
1982). Time stepping calculations are stable for (J ~ 0.5. Giving equal weight to the start
and end rates of change of temperature ( (J = 0.5) is known as the Crank-Nicolson method
(after its originators) and gives good results in metal cutting transient heating calculations.
Carslaw, H. S. and Jaeger, J. C. (1959) Conduction of Heat in Solids, 2nd edn. Oxford: Clarendon
Press.
Childs, T. H. C., Maekawa, K. and Maulik, P. (1988) Effects of coolant on temperature distribution
in metal machining. Mat. Sci. and Technol. 4, 1006-1019.
Hiraoka, M. and Tanaka, K. ( 1968) A variational principle for transport phenomena. Memoirs of the
Faculty of Engineering, Kyoto University 30, 235-263.
Huebner, K. H. and Thomton, E. A. (1982) The Finite Element Methodfor Engineers, 2nd edn. New
York: Wiley.
Loewen, E. G. and Shaw, M. C. (1954) On the analysis of cutting tool temperatures. Trans. ASME
76, 217-231.
Tay, A. 0., Stevenson, M. G. and de Vahl Davis, G. (1974) Using the finite element method to deter-
mine temperature distributions in orthogonal machining. Proc. Inst. Mech. Eng. Lond. 188,
627-638.