Conduction and convection of
heat in solids
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 temperature gradient in that direction and with K the thennal conductivity and T the temperature:
aT
q=-K
=
)2
(~
()2T
+~
~K
()Z2
(A2.1)
an
(A2.2a)
d.xdydzdt
dT
dz
The heat accumulating due to convection, H conv '
is
aT
Hconv
= ~ uzpCT
-UzPC~T+
-f
dz~ ~dxdydt
= -uzpC
d.xdydzt
az
(A2.2b)
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 temperature rise and heat capacity of the volume:
aT
a2T
dK
az2
dT
pC-=K-+-
at
aT
( az )
aT
2
- - u,pc -+ q*
(A2.3a)
aZ
The extension to three dimensions is straightforward:
aT
pC-=K
at
(
a2T
a2T
a2T
dK
JT
2
JT
2
aT
2
+T+F)+
(T)+(x)+(+
F(
ax2
-pc
1
u, JT + z i - aT + Uz- aT ) + q *
JY
ax
az
(A2.3b)
When thermal conductivity does not vary with temperature, equation (A2.3b) reduces
to
aT
pC-=K(at
a2T
ax2
d2T
+-+-)
aY2
a2T
az2
aT
-pc(Ux-+
ax
aT
icy--aY + u,
z
aZ )+
4*
(A2.4)
Selected problems, with no convection
353
When itx = ity = itz = 0, and q* = 0 too, equation (A2.4) simplifies further, to
1
--=
1(
aT
~a2T
a2T
a2T
at
-+-+ax2
ay2
az2
~
(A2.5)
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-infinitesolid z > 0: temperaturedue to an
instantaneousquantity of heat H per unit area into it over the
plane z = 0, at t = 0; ambient temperature To
It may be checked
by substitution
that
1
z2
41(t
C
(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
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 dimensionless 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 associated ~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,
(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-infinitesolid z > 0: temperaturedue to supply of heat
at a constantrate q per unit areaover the plane z = 0, for t > 0;
ambient temperature To
Heat dH = qdt' is released at z = O in the time interval t' tot'
that this causes at z at a later time t is, from equation (A2.6)
+ dl'. The temperature rise
354 Appendix 2
2
z
-
1
4Ir(t-t')
(A2.9)
The total temperature is obtained by integrating with respect to t' from O to t. The temperature at z = O will be found to be of interest. When q is independent of time
2
(T- To) = --=-
~V-;t
K
VIr
(A2.10)
The average temperature at z = 0, over the time interval ° to t, is 2/3rds of this.
A2.2.3 The semi-infinite solid z > 0: temperaturedue to an
instantaneousquantity of heat H releasedinto it at the point
x = y = z = 0, at t = 0; ambient temperature To
In this case of three-dimensional
heat flow, the equivalent to equation (A2.6) is
H
T-To=4pC
1
-=-:
e
(7t1Ct)3/2
x'+y'+z'
4Kt
(A2.11)
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 To
Heat flows into the solid over the surface area shown in Figure (A2.2a). In the time interval 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
(b)
z'
(c)
Fig. A2.2 Someproblemsrelevantto machining:(a)surfaceheatingof a stationarysemi-infinitesolid;(b) an infinite
solidmovingperpendicular
to a planeheatsource;(c)a semi-infinitesolidmovingtangentiallyto the planeof a surface
heatsource
b
2qa
(T-
T O)max
a
nK
(T-
+
sinh-]
=
T O)av = (i-
T O)max
b .a
Slnh-l b
a
~r/~+~\/1
31rKl \ b
a I \
+~\'/~~-~1
a2 I
a2
b J
(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 xdirection.
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 .2b); am ient
temperature To
i
g
In the steady state, the form of equation (A2.4) (with q* = 0) to be satisfied is
a2T
lc--
az2
. JT
-'Z
(A2.15)
aZ
The temperature distribution
( T - To) =
4
~
, ~ 2 0 ; (T-To)=-
PCU,
4
e
U,Z
, 220
(A2.16)
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 the
heat 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 dimensionless group (qalK),
(A2.17b)
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
(A2.18)
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 properties 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:
m
/(1) = L /e(T)
e=l
(A2.19)
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 variation 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 quantities q*, q and h, and the elements of [H]e depend mainly on the conduction and convection terms of Ie(1). Assembly of all the element equations to create a global equation
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 righthand rule whereby the first three nodes are listed in an anticlockwise manner when viewed
from the fourth one. Node i is at (xi,yi, zi) and so on for the other nodes. Temperature Te
anywhere in the element is related to the nodal temperatures { T ) = (Ti Tk Tl)Tby
5
(A2.21)
T e = [Ni
Nj Nk Nl]{ T ) = [N]{ T }
where [N] is known as the element's shape function.
N.=
'
where
a 1. =
1
~
6Ve
(ai+ bix + ciy + d,z)
12 :2 I,
Fig. A2.3 A tetrahedral finite element
x. y j
zj
1
bi=-l]
;21
Yj
zj
Numerical (finite element) methods 359
and
(A2.22)
This may be checked by showing that, at the nodes, T e 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 function of the surface’s nodal temperatures:
T = [N,”’N;]
{ T ] = [N’][ T ]
(A2.23)
where
N.’ =
1
~
(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. Aikjis the area of the element’s triangular face: it may also be written in global coordinates as
(A2.25)
A2.4.2 Tetrahedral element thermal stiffness 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, =
bibi + cici + didi
bibj + cicj + d.d.
‘ I
bibk+ tick + didk
bib, + cicl + didl
bjbi + cjci + didi
b.b.
+ CJ. CJ . + d.d.
J J
J J
bjbk+ cjck + djdk
bibl + cjcl + djdl
b,bi + ckci + dkdi b,bi + clci + d,di
bkbj + ckcj+ dkdj blbj + clcj + dldj
bkbk+ ckck+ dkdk blbk+ clck + dldk
bkbl+ ckc, + dkdl b,b, + clc, + dldl
1
360 Appendix 2
uxbi + uYci+ uzdi
uxbi+ u,ci + u,di
uxbi + U,ci + uzdi
uxbi+ uYci+ uzdi
2 1
+A[
hA,' 1 2
12
1 1
0 0
1
1
2
0
0
0
0
0
]
uX b.J + uYc.J + uzdj
uX bJ. + uY c.J + uzdj
U b. + U C. + U d.
X J
YJ
Z J
uxbj+ Uycj + uzdj
uxbk+ UYck+ uzdk
uxbk+ uYck + uzdk
uxbk+ uYck+ uzdk
uxbk+ UYc k + uzdk
uxbl+ uYcl+ uzdl
Uxbl+ U,cl + Uzdl
Uxb,+ UYcl+ u,d,
uxb,+ U,cl + u,d,
1
(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 interface. 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)
(A2.28b)
where
zs =
F, cos @ - FT sin @
sin @;
zf =
F, sin a + FT cos a
V
fd
cos a
v, =cos(@- a) 'work;
sin @
v, = cos(@
-
a)
'work
1
(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.
A2.4.4 Extension to transient conditions
The functional, equation (A2.1S), 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
2
1
1
1
1
2
1
1
1
1
2
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
-a~
at
1 = (I -8)
Jay
aT
aT
1
+8
at
(A2.31a)
-at
-J
n+l
or
Tn+l
,!i;-
~av=
(A2.31b)
111
~~
where (Jis 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 equations (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,
an equation is created for temperatures at time tn+l in terms of temperatures at time tn: in
global assembled form
[C]
(A2.32)
~t
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 determine temperature distributions in orthogonal machining. Proc. Inst. Mech. Eng. Lond. 188,
627-638.