They also provided a solution when the initial temperature distribution decays exponentially withx.. Constant Surface Heat Flux and Exponentially Decaying Energy Gener-ation When the sur
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ln
T s + T
T s − T
T s − T i
T s + T i
+ 2
arctanT
T s − arctan
T i
T s
= 4σA s T s3t
ρV c (3.283)
Equation (3.283) is useful, for example, in designing liquid droplet radiation systems for heat rejection on a permanent space station
Simultaneous Convective–Radiative Cooling In this case, the radiative term,
−σA s (T4−T4
s ) appears on the right-hand side ofeq (3.276) in addition to −hA s (T −
T∞) An exact solution for this case does not exist except when T∞ = T s = 0 For
this special case the exact solution is
1
3ln
1+σT3
i
/h(T /T i )3
(1 + σT3
i /h)(T /T i )3
= hA ρV c s t (3.284)
Temperature-Dependent Heat Transfer Coefficient For natural convection cooling, the heat transfer coefficient is a function of the temperature difference, and the functional relationship is
whereC and n are constants Using eq (3.285) in (3.276) and solving the resulting
differential equation gives
T − T∞
T i − T∞ =
1+nh ρV c i A s t
−1/n
(3.286) wheren = 0 and h i = C(T i − T∞)n.
Heat Capacity of the Coolant Pool Ifthe coolant pool has a finite heat capacity, the heat transfer to the coolant causesT∞to increase Denoting the properties ofthe hot body by subscript 1 and the properties ofthe coolant pool by subscript 2, the temperature–time histories as given by Bejan (1993) are
T1(t) = T1(0) − T1(0) − T2(0)
1+ ρ1V1c1/ρ2V2c2
T2(t) = T2(0) + T1(0) + T2(0)
1+ (ρ2V2c2/ρ1V1c1) (1 − e −nt ) (3.288)
wheret1(0) and T2(0) are the initial temperatures and
n = hA s ρ1V1c1+ ρ2V2c2
(ρ1V1c1)(ρ2V2c2) (3.289)
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q0⬙
T t T = T(0, ) = s i⫹at n ⫺k⫺⫺T = q⬙
X x=| 0 0 ⫺k⫺⫺ = h T ⫺T t
T
X x=| 0 [ a (0, )]
T x( ,0) =T i T x( ,0) =T i
h T, a
T s
Figure 3.37 Semi-infinite solid with (a) specified surface temperature, (b) specified surface heat flux, and (c) surface convection.
As indicated in Fig 3.37, the semi-infinite solid model envisions a solid with one identifiable surface and extending to infinity in all other directions The parabolic partial differential equation describing the one-dimensional transient conduction is
∂2T
Specified Surface Temperature Ifthe solid is initially at a temperatureT i, and iffor timet > 0 the surface at x = 0 is suddenly subjected to a specified temperature–
time variationf (t), the initial and boundary conditions can be written as
T (0, t) = f (t) = T i + at n/2 (3.291b)
wherea is a constant and n is a positive integer.
Using the Laplace transformation, the solution forT is obtained as
T = T i + aΓ1+n
2
(4t) n/2 i nerfc
x
2√
αt
(3.292)
whereΓ is the gamma function (Section 3.3.2) and i n erfc is the nth repeated integral
ofthe complementary error function (Section 3.3.1) The surface heat fluxq
0 is
q
0 = 2√n−1
αt (n−1)/2 kaΓ
1+n
2
i n−1erfc x
2√
αt
x=0
(3.293) Several special cases can be deduced from eqs (3.292) and (3.293)
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Case 1: f (t) = T0 This is the case ofconstant surface temperature which occurs whenn = 0 and a = T0− T i:
T (x, t) − T i
T0− T i = erfc
x
2√
αt
(3.294)
q
0 = k(T0− T i )
Case 2: f (t) = T i +at This is the case ofa linear variation ofsurface temperature
with time which occurs whenn = 2:
T (x, t) = T i + 4ati2erfc
x
2√
αt
(3.296)
q
Case 3: f (t) = T i + at1/2 This is the case ofa parabolic surface temperature
variation with time withn = 1.
T (x, t) = T i + a√πt i erfc
x
2√
αt
(3.298)
q
0 = ka
2
π
Specified Surface Heat Flux For a constant surface heat fluxq
0, the boundary conditions ofeq (3.291b) is replaced by
− k ∂T (0, t)
and the solution is
T (x, t) = T i+2q0
√
αt
k i erfc
x
2√
αt
(3.301)
Surface Convection The surface convection boundary condition
− k ∂T (0, t)
∂x = h [T∞− T (0, t)] (3.302)
replaces eq (3.291b) and the solution is
T (x, t) − T i
T∞ − T i = erfc
x
2√
αt
− e (hx/k)+(h2αt/k2)erfc
h k
√
αt + x
2√
αt
(3.303)
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Constant Surface Heat Flux and Nonuniform Initial Temperature Zhuang
et al (1995) considered a nonuniform initial temperature of the form
wherea and b are constants to find a modified version ofeq (3.301) as
T (x, t) = a + bx + 2√αt
q
0
k + b
i erfc
x
2√
αt
(3.305) The surface temperature is obtained by puttingx = 0 in eq (3.305), which gives
T (0, t) = a + 2
αt
π
q
0
k + b
(3.306)
Zhuang et al (1995) found that the predictions from eqs (3.305) and (3.306) matched the experimental data obtained when a layer ofasphalt is heated by a radiant burner, producing a heat flux of41.785 kW/m2 They also provided a solution when the initial temperature distribution decays exponentially withx.
Constant Surface Heat Flux and Exponentially Decaying Energy Gener-ation When the surface of a solid receives energy from a laser source, the effect of this penetration ofenergy into the solid can be modeled by adding an exponentially decaying heat generation term, ˙q0e −ax /k (where a is the surface absorption
coeffi-cient), to the left side of eq (3.290) The solution for this case has been reported by Sahin (1992) and Blackwell (1990):
T = T i + (T0+ T i ) erfc
x
2√
αt
+ ˙q0
ka2
erfc
x
2√
αt
−1
2e a2αt+axerfc
x
2√
αt + a
√
αt
−1
2e a2αt−axerfc x
2a√αt − a
√
αt
+ e −ax e (a2αt−1)
(3.307)
Both Sahin (1992) and Blackwell (1990) have also solved this case for a convective boundary condition (h, T∞) atx = 0 Blackwell’s results show that for a given
absorption coefficienta, thermal properties of α and k, initial temperature T i, and surface heat generation ˙q0, the location ofthe maximum temperature moves deeper into the solid as time progresses Ifh is allowed to vary, for a given time the greater
value ofh provides the greater depth at where the maximum temperature occurs.
The fact that the maximum temperature occurs inside the solid provides a possible explanation for the “explosive removal of material” that has been observed to occur when the surface ofa solid is given an intense dose oflaser energy
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Consider one-dimensional transient conduction in a plane wall ofthickness 2L, a long
solid cylinder ofradiusr0, and a solid sphere ofradiusr0, each initially at a uniform temperatureT i At timet = 0, the exposed surface in each geometry is exposed to a
hot convective environment (h, T∞) The single parabolic partial differential equation describing the one-dimensional transient heating ofall three configurations can be written as
1
s n
∂
∂s
s n ∂T
∂s
= 1 α
∂T
wheres = x, n = 0 for a plane wall, s = r, n = 1 for a cylinder, and s = r, n = 2
for a sphere In the case of a plane wall,x is measured from the center plane The
initial and boundary conditions for eq (3.308) are
∂T (0, t)
k ∂T (L or r0, t)
∂s = h [T∞− T (L or r0, t)] (3.309c) According to Adebiyi (1995), the separation ofvariables method gives the solution forθ as
θ =
∞
n=1
2 Bi
λ2
n+ Bi2+ 2ν · Bi
RνJ−ν(λ n R)
J−ν(λ n ) e−λ
2
whereθ = (T∞− T )/(T∞− T i ), R = x/L for a plane wall, R = r/r0 for both cylinder and sphere, Bi= hL/k or hr0/k, τ = αt/L2orτ = αt/r2
0, ν = (1 − n)/2,
and theλnare the eigenvalues given by
λn J−(ν−1)(λ n ) = Bi · J−ν(λ n ) (3.311) The cumulative energy received over the timet is
Q = ρcV (T∞ − T i )
∞
n=1
2(1 + n) Bi2
1− e−λ 2
nτ
λ2
n
λ2
n+ Bi2+ 2ν · Bi (3.312)
Solutions for a plane wall, a long cylinder, and a sphere can be obtained from eqs
(3.310) and (3.312) by puttingn = 0, (ν = 1
2), n = 1(ν = 0), and n = 2(ν = −1
2),
respectively It may be noted that
J −1/2 (z) =
2
πz
1/2
cosz
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J1/2 (z) =
2
πz
1/2
sinz
J3/2 (z) =
2
πz
1/2
sinz
z − cos z
With these representations, eqs (3.310) and (3.312) reduce to the standard forms appearing in textbooks Graphical representations ofeqs (3.310) and (3.312) are
called Heisler charts.
For some configurations it is possible to construct multidimensional transient conduc-tion soluconduc-tions as the product ofone-dimensional results given in Secconduc-tions 3.8.2 and 3.8.3 Figure 3.38 is an example ofa two-dimensional transient conduction situation
in which the two-dimensional transient temperature distribution in a semi-infinite plane wall is the product ofthe one-dimensional transient temperature distribution in
an infinitely long plane wall and the one-dimensional transient temperature distribu-tion in a semi-infinite solid Several other examples ofproduct soludistribu-tions are given by Bejan (1993)
Explicit Method For two-dimensional transient conduction in Cartesian coordi-nates, the governing partial differential equation is
T x x t( , , )1 2 ⫺T⬁ T x t( , )2 ⫺T⬁ T x t( , )1 ⫺T⬁
T ,h⬁
T ,h⬁
T ,h⬁ T ,h⬁ T ,h⬁ T ,h⬁
semi-infinite plane wall
infinitely long plane wall
semi-infinite solid
x1
x1
x2
x2
Figure 3.38 Product solution for a two-dimensional transient conduction problem
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∂2T
∂x2 +∂ ∂y2T2 = 1α∂T ∂t (3.313) which assumes no internal heat generation and constant thermal properties Approx-imating the second-order derivatives inx and y and the first-order derivative in t by
for various nodes (see Fig 3.33) can be expressed using the Fourier modulus, Fo=
α ∆t/(∆x)2, Bi = h ∆x/k, and t = p ∆t.
• Internal node: With Fo ≤ 1
4,
T i,j p+1= FoT1p +1,j + T i−1,j p + T i,j+1 p + T i,j−1 p + (1 − 4Fo)T i,j p (3.314)
• Node at interior corner with convection: With Fo(3 + Bi) ≤ 3
4,
T i,j p+1=2
3Fo
T i+1,j p + 2T i−1,j p + 2T i,j+1 p + T i,j−1 p + 2BiT∞
+1− 4Fo −4
3Bi· FoT i,j p (3.315)
• Node on a plane surface with convection: With Fo(2 + Bi) ≤ 1
2,
T i,j p+1 = Fo2T i−1,j p + T i,j+1 p + T i,j−1 p + 2Bi · T∞
+ (1 − 4Fo − 2Bi · Fo) T i,j p (3.316)
• Node at exterior corner with convection: With Fo(1 + Bi) ≤ 1
4,
T i,j p+1= 2FoT i−1,j p + T i,j−1 p + 2Bi · T∞
+ (1 − 4Fo − 4Bi · Fo) T i,j p = 0 (3.317)
• Node on a plane surface with uniform heat flux: With Fo ≤ 1
4,
T i,j p+1 = (1 − 4Fo)T i,j p + Fo2T i−1,j p + T i,j+1 p + T i,j−1 p + 2Fo · q∆x
k (3.318)
The choice of∆x and ∆t must satisfy the stability constraints, introducing each
ofthe approximations given by eqs (3.314)–(3.318) to ensure a solution free of numerically induced oscillations Once the approximations have been written for each node on the grid, the numerical computation is begun witht = 0(p = 0), for which
the node temperatures are known from the initial conditions prescribed Because eqs
(3.313)–(3.318) are explicit, node temperatures att = ∆t(p = 1) can be determined
from a knowledge of the node temperatures the preceding time,t = 0(p = 0) This
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“marching out” in time type ofcomputation permits the transient response ofthe solid to be determined in a straightforward manner However, the computational time necessary to cover the entire transient response is excessive because extremely small values of∆t are needed to meet the stability constraints.
Implicit Method In the implicit method, the second derivatives in x and y are approximated by central differences but with the use oftemperatures at a subsequent
time,p + 1, rather than the current time, p, while the derivative in t is replaced by
a backward difference instead of a forward difference Such approximations lead to
the following equations:
• Internal node:
(1 + 4Fo)T i,j p+1− FoT i+1,j p+1 + T i−1,j p+1 + T i,j+1 p+1 + T i,j−1 p+1= T i,j p (3.319)
• Node at interior corner with convection:
(1 + 4Fo)1+1
3Bi T i,j p+1−2
3Fo
T i+1,j p+1 + 2T i−1,j p+1 + 2T i,j+1 p+1 + T i,j−1 p+1= T i,j p +4
3Fo· Bi · T∞ (3.320)
• Node on a plane surface with convection:
[1+ 2Fo(2 + Bi)] T i,j p+1− Fo2T i−1,j p+1 + T i,j+1 p+1 + T i,j−1 p+1
• Node at exterior corner with convection:
1+ 4Fo(1 + Bi)T i,j p+1− 2FoT i−1,j p+1 + T i,j−1 p+1= T i,j p + 4Bi · Fo · T∞ (3.322)
• Node on a plane surface with uniform heat flux:
(1 + 4Fo)T i,j p+1+ Fo2T i−1,j p+1 + T i,j+1 p+1 + T i,j−1 p+1= T i,j p +2Fo· q∆x
k (3.323)
The implicit method is unconditionally stable and therefore permits the use of higher values of∆t, thereby reducing the computational time However, at each time
t, the implicit method requires that the node equations be solved simultaneously rather than sequentially.
Other Methods Several improvements ofthe explicit and implicit methods have been advocated in the numerical heat transfer literature These include the three-time-level scheme ofDufort and Fankel, the Crank–Nicholson method, and alternating
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direction explicit methods For a discussion ofthese methods as well as stability analysis, the reader should consult Pletcher et al (1988)
Examples ofperiodic conduction are the penetration ofatmospheric temperature cycles into the ground, heat transfer through the walls of internal combustion engines, and electronic components under cyclic operation The periodicity may appear in the differential equation or in a boundary condition or both The complete solution to
a periodic heat conduction problem consists ofa transient component that decays
to zero with time and a steady oscillatory component that persists It is the steady oscillatory component that is ofprime interest in most engineering applications In this section we present several important solutions
Environment
Revisit the lumped thermal capacity model described in Section 3.8.1 and consider
a scenario in which the convective environmental temperatureT∞ oscillates sinu-soidally, that is,
wherea is the amplitude ofoscillation, ω = 2πf the angular frequency, f the
frequency in hertz, andT ∞,mthe mean temperature ofthe environment
The method ofcomplex combination described by Arpaci (1966), Myers (1998), Poulikakos (1994), and Aziz and Lunardini (1994) gives the steady periodic solution as
θ = √ 1
where
θ = T − T a ∞,m B = ρV c hAω τ =ρV c hAt β = arctan B (3.326)
A comparison ofeq (3.325) with the dimensionless environmental temperature vari-ation (sinBτ) shows that the temperature ofthe body oscillates with the same
fre-quency as that ofthe environment but with a phase lag ofβ As the frequency of
oscillation increases, the phase angleβ = arctan B increases, but the amplitude of
oscillation 1/(1 + B2)1/2decreases.
Consider the semi-infinite solid described in Section 3.8.2 and let the surface temper-ature be ofthe form
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In this case, eqs (3.291) are still applicable, although the initial condition ofeq
(3.291a) becomes irrelevant for the steady periodic solution, which is
T (x, t) = T i + ae−[(ω/2α)1/2 x] cosωt − ω
2α
1/2
x
(3.328)
Three conclusions can be drawn from this result First, the temperatures at all lo-cations oscillate with the same frequency as the thermal disturbance at the surface
Second, the amplitude ofoscillation decays exponentially withx This makes the
so-lution applicable to the finite thickness plane wall Third, the amplitude ofoscillation decays exponentially with the square root ofthe frequencyω Thus, higher-frequency
disturbances damp out more rapidly than those at lower frequencies This explains why daily oscillations ofambient temperature do not penetrate as deeply into the ground as annual and millenial oscillations The surface heat flux variation follows directly from eq (3.328):
q(0, t) = −k ∂T (0, t) ∂x = kaω
α
1/2
cos
ωt −π
4
(3.329) and this shows thatq(0, t) leads T (0, t) by π/4 radians.
In this case, the boundary condition ofeq (3.327) is replaced with
q(0, t) = −k ∂T
∂x (0, t) = q0cosωt (3.330) and the solution takes the form
T (x, t) = T i+q0
k
α ω
1/2
e−[ω/2α)1/2 x] cos
ωt − ω
2α
1/2
x − π
4
(3.331)
It is interesting to note that the phase angle increases as the depthx increases with the
minimum phase angle ofπ/4 occurring at the surface (x = 0) A practical situation
in which eq (3.331) becomes useful is in predicting the steady temperature variations induced by frictional heating between two reciprocating parts in contact in a machine
This application has been described by Poulikakos (1994)
The surface boundary condition in this case is