Post on 13-Mar-2020
Chapter 2
1D heat conduction problems
2.1 1D heat conduction equation
When we consider one-dimensional heat conduction problems of a homogeneous isotropic solid, the Fourier equation simplifies to the form:
ππππππ‘ = π
π)πππ₯) + π (2.1)
If there is no heat generation, as is usually the case, such equation reduces to:
ππππ‘ = π
π)πππ₯) (2.2)
where π = ./0
. Furthermore, if the temperature distribution does not depend on time:
π)πππ₯) = 0 (2.3)
The stationary case of heat conduction in a one-dimension domain, like the one represented in figure 2.1, is particular simple to be solved.
Figure 2.1: the temperature within a solid media with prescribed Dirichelet boundary conditions
In fact, the general solution of the equation is in this case:
π π₯ = π΄π₯ + π΅ (2.4)
with A and B coefficients to be determined by imposing the boundary conditions.
Prescribed temperatures at both left and right surfaces:
π456 = π7
π458 = π9 (2.5)
we have:
π7 = π΅ and π9 = π΄π + π7 or ;<=;>8
= π΄ (2.6)
In conclusion:
π π₯ = π9 β π7
π π₯ + π7 (2.7)
Prescribed temperature at left surface ad assigned heat flux at the right surface.
Figure 2.2: the temperature within a solid media with prescribed temperature at left surface ad assigned heat flux at the right surface
πAB + πC β πDEF = 0 β΅ steady state
πAB = π458 , πC = π9 , πDEF = 0
π458 + π9 = 0
π456 = π7
π458 + π9 = 0 β βπππππ₯ 458
+ π9 = 0 β πππππ₯ 458
= π9
πππ = π9ππ₯
π[π]64 = π9[π₯]64
π π π₯ β π 0 = π9 π₯ β 0
π π π₯ β π7 = π9π₯
π π₯ = π9π π₯ + π7
Prescribed temperature at right side and assigned heat flux at left side
Figure 2.3: the temperature within a solid media with prescribed temperature at right side and assigned heat flux at left side
πAB + πC β πDEF = 0 β΅ steady state
πAB = 0 , πC = π7 , πDEF = π456
π7 β π456 = 0
π458 = π9
π456 = π7 β βπππππ₯ 456
= π7 β πππππ₯ 456
= βπ7
πππ = βπ7ππ₯
π[π]48 = βπ7[π₯]48
π π π β π π₯ = β π7 π β π₯
π π9 β π π₯ = β π7 π β π₯
π π₯ = π7π π β π₯ + π9
Convective and prescribed heat flux at both surfaces
Figure 2.4: the temperature distribution with convective and prescribed heat fluxes
Governing equation:
π)πππ₯) = 0 (2.8)
Boundary conditions:
πS4FT + π7 β π456 = 0 β β7 πV> β π + π7 + πππππ₯ = 0 ππ π₯ = 0 (2.9)
π458 + π9 β πS4FTT = 0 β βπππππ₯ + π9 β β9 π β πV< ππ π₯ = π (2.10)
where πV> and πV< are the temperatures of the surrounding media, π7 and π9 are surface heat generations per unit area per unit time, β7 and β9 are the heat transfer coefficients and subscripts a and b denote boundaries at π₯ = 0 and π₯ = π, respectively.
A general solution of eq. 2.8 is:
π π₯ = π΄π₯ + π΅ (2.11)
By substitution of the solution in eq. 2.9 and 2.10 we obtain:
ππ΄ + π7 = β7 π΄π₯ + π΅ β πβπ βππ΄ + π9 = β9 π΄π₯ + π΅ β πβπ
ππ΄ + π7 = β7 π΅ β πβπ πππ π₯ = 0 βππ΄ + π9 = β9 π΄π + π΅ β πβπ πππ π₯ = π
The coefficients A and B are:
π΄ =β7β9[ πβπ β πβπ +
π9β9β π7β7
]
π β7 + β9 + β7β9π
π΅ = πβπ +π7β7+πβ9 πβπ β πβπ + (
π9β9β π7β7
)
π β7 + β9 + β7β9π
If no surface heat generation π7 = π9 = 0
π π₯ = πV> + (πV< β πV> )β9(β7π₯ + π)
π β7 + β9 + β7β9π (2.12)
It is no possible to find a stationary solution by imposing an heat flux condition on both sides. In fact if ππ and ππ are not equal, the temperature response is not stationary since in presence of a variation of heat content, the temperature of the body should vary with time. If ππ = ππ the slope of the line is determined and the difference of temperature only can be obtained by no assumption on the temperature value. The case is analog to the mechanical response of an elastic spring free in the space at the end of which two forces with opposite sign are applied β rigid body motion
2.2 1D heat conduction: transient
Let us now consider a transient problem in which the temperature at x=0 is equal to Ta , the temperature at x=l is equal to zero and the initial condition is set as T=Ti(x). The governing equations read as follows
ππππ‘ =
ππππ)πππ₯) , π =
πππ (2.13)
π456 = π7 π458 = 0 πF56 = πA π₯ The solution is found by separation of variables, as we assume that the temperature can be expressed by the product of a function of position only f(x) and a function of time g(t)
( ) ( ) ( ), ,T x t f x g t= (2.14)
by substituting in the governing equation we obtain
2
2
( ) ( )( ) ( ) ,dg t d f xf x kg tdt dx
= (2.15)
that can also be written as
2
2
1 ( ) 1 ( ) ,( ) ( )dg t d f x
kg t dt f x dx= (2.16)
since the first member does not depend upon x, the second one does not depend upon t and the two members are equal, they can be set equal to a constant.
By setting the constant equal to -s2 we obtain two separate equations
2( ) ( ) 0,dg t ks g tdt
+ = (2.17)
And
22
2
( ) ( ) 0.d f x s f xdx
+ = (2.18)
The general solution of the first equation can be easily obtained by searching solution of the kind π π‘ = πbF and by finding the characteristic equation
2 0,ksΞ± + = (2.19)
that leads to the general solution
π π‘ = πc for π ) = 0, π π‘ = π)π=eCfF for π ) β 0 (2.20)
The general solution of the second equation can be sought in the same way or set directly as
π π₯ = πiπ₯ + πj for π ) = 0
π π₯ = πk sin π π₯ + πn cos π π₯ for π ) β 0 (2.21)
In conclusion the general solution of the original equation, that should be valid for arbitrary values of s can be written as
π π₯, π‘ = π=eCfF π΄ sin π π₯ + π΅ cos π π₯ + πΆπ₯ + π· (2.22) with
2 5 2 6 1 3 1 4, , , .A c c B c c C c c D c c= = = = (2.23)
We now impose the satisfaction of boundary conditions by substituting then in the latter expression:
π 0, π‘ = π΅π=eCfF + π· = π7,
π π, π‘ = π=eCfF π΄ π ππ π π + π΅ cos π π + πΆπ + π· = 0 (2.24)
In order to satisfy the first equation for every value of t since π=eCfF is never equal to zero it is necessary that
0, .aB D T= = (2.25)
For the second equation we have now 2
sin 0ks taAe sl Cl Tβ + + = (2.26)
that can only be satisfied for
sin π π = 0 and πΆ = β;>8
(2.27)
The values of k for which sin π π = 0 are
1,2,3...nns nlΟ= = (2.28)
they are the eigenvalues of our problem.
The temperature can then be expanded in an infinite series of form
2
1( , ) 1 sin ,nks t
a n nn
xT x t T A e s xl
ββ
=
β β= β +β ββ β β (2.29)
when An are unknown coefficients still to be determined. We now impose the initial condition π π₯, 0 = πA(π₯) by substitution in the preceding equation we obtain
πA π₯ β π7 1 βπ₯π = π΄B sin π Bπ₯
V
B5c
(2.30)
In order to determine the coefficients An corresponding to initial condition we can profit of the properties of the sinusoidal function. In fact by multiplying both sides of the equation by sin π uπ₯ and integrating it from 0 to l (that is by finding the value of the scalar product between the terms at the first and the second member of the equation and the generic sinusoidal function sin π uπ₯ ) we obtain:
π ππ π Bπ₯ sin π uπ₯ππ₯ = 0 for π β π 8
6
π ππ) π Bπ₯ ππ₯ = π2 for π = π
8
6
(2.31)
And, for the coefficients An
0
2 ( ) 1 sin .l
n i a nxA T x T s xdx
l lβ‘ β€β β= β ββ ββ’ β₯β β β£ β¦β« (2.32)
In conclusion the expression of the temperature can be written as
2
01
2( , ) 1 { ( ) 1 sin }sin .nl ks t
a i a n nn
x xT x t T T x T s xdx s xel l l
ββ
=
β‘ β€β β β β= β + β ββ β β ββ’ β₯β β β β β£ β¦β β« (2.33)
From the form of the solution it is clear that the first term represents the stationary response that will eventually reached after a transient. The second term is on the contrary a time-dependent one, with the tendency of decaying to zero as time increases. In the case that different boundary conditions are imposed, say on both sides x=0,l imposed heat flux by convection, as expressed below
Figure 2.5: The temperature profile within a solid media which separates two semi-infinite fluid media. πV> and πV< correspond, respectively, to the temperature of the external media for x<0 and for x>l.
πS4FT = β7 πV> β π
πS4FTT = β9 π β πV< (2.34)
with
πS4FT + π7 β π = 0 , π7 = 0 β β7 πV> β π + πππππ₯ = 0 on π₯ = 0 (2.35)
π + π9 β πS4FTT = 0 , π9 = 0 β βπππππ₯ β β9 π β πV< = 0 on π₯ = π (2.36)
It is possible to use the same expression of the solution obtained before:
π π₯, π‘ = π=eCfF π΄ π ππ π π₯ + π΅ cos π π₯ + πΆπ₯ + π· (2.37)
For obtaining the solution in this case similar steps can be followed to the ones used for the previous set of boundary conditions. The solution in this case will result as follows
π π₯, π‘ = πV> + πV< β πV>
β9 β7π₯ + ππ β7 + β9 + β7β9π
+
+2π)π B) + β9) β7 sin π Bπ₯ + ππ B cos π Bπ₯
π π)π B) + β7) π)π B) + β9) + π β7 + β9 π)π B) + β7β9
V
B5c
π=eCfF..
πA π₯ β πV> + πV< β πV>
β9 β7π₯ + ππ β7 + β9 + β7β9π
β7 sin π Bπ₯8
6
+ ππ B cos π Bπ₯ ππ₯,
(2.38)
with sn being any positive root of the transcendental equation
( )2 2tan .a b
a b
s h hsl
s h hλλ
+=
β (2.39)
The stationary part of solution
π π₯, π‘ β β = πV> + πV< β πV>
β9 β7π₯ + ππ β7 + β9 + β7β9π
(2.40)
can be further examined. In fact, the boundary condition in x=0 reads
π7 = πππππ₯ = β7 π β πV> (2.41)
if the conductivity of the material is low or the convention thermal coefficient is high, the temperature on the wall reaches the temperature of the fluid πV> . In fact, with low Ξ» and high ha
we have .{>βͺ 1 and
πβ7ππππ₯ = (π β πV>) (2.42)
with the first term of the equation equal to zero and π = πV>. This results can also obtained from the above expression of the stationary part of the solution in presence of convective conditions on both sides, that can be written as
π π₯ = πV> + πV< β πV>
β9 β7π₯ + ππ β7 + β9 + β7β9π
= πV> + πV< β πV>
β7β9 π₯ + πβ7
π β7 + β9 + β7β9π
= πV> + πV< β πV>
π₯ + πβ7
π 1β9+ 1β7
+ π
(2.43)
and, considering the same assumption .{>βͺ 1 and .
{<βͺ 1 we finally have
π π₯ = πV> + πV< β πV>48 . (2.44)
That is the temperature of the media at both sides of the solid can be directly assumed as the wall temperature. It is also interesting to note that, also with entering fluxes at both sides qa and qb, the temperature of the body T in the stationary response cannot have temperatures higher than the highest of the media by which the body is surrounded in perfect agreement with the physics of heat transfer. It is also obvious that, also in the presence of an entering flux, the body cannot increase indefinitely its temperature that cannot increase higher than the one from which the flux is generated.
Figure 2.6: The temperature within a wall with prescribed Dirichlet and Neumann boundary conditions
The 1D thermal behaviour just described can be applied to the case of an indefinite plate of thickness l with the side at isothermal condition where either the temperature, also variable with
the time t, or the heat flux can be applied. The same results are also valid for a one-dimensional solid (like a bar or a cable) with uniform section, provided that the lateral surface of the bar is isolated, that is no heat flux is present through it.
2.3 Honeycomb panel
Figure 2.7: Structure of a honeycomb sandwich panel: assembled view (A), and exploded view (with the two face sheets B, and the honeycomb core C). Ribbons run along the x direction, and are glued side
by side in counter-phase along the y direction as detailed.
Honeycomb panels (Figure 2.7) are structural elements with great stiffness-to-mass ratio, widely used in aerospace vehicles. Heat transfer through honeycomb panels is non-isotropic and difficult to predict. If the effect of the cover faces is taken aside, and convection and radiation within the honeycomb cells can be neglected in comparison with conduction along the ribbons (what is the actual case in aluminium honeycombs), heat transfer across each of the dimensions is:
π4 = ππΉ4π΄4βπ4π4
with πΉ4 =32πΏπ
ποΏ½ = ππΉοΏ½π΄οΏ½βποΏ½ποΏ½
with πΉοΏ½ =πΏπ
ποΏ½ = ππΉοΏ½π΄οΏ½βποΏ½ποΏ½ with πΉοΏ½ =
83πΏπ
(2.45)
where F is the factor modifying solid body conduction (the effective conductive area divided by the plate cross-section area), which is proportional to ribbon thickness, Ξ΄, divided by cell size, s (distance between opposite sides in the hexagonal cell, not hexagon side, a, in Figure 2.7; π =3π), and depends on the direction considered: x is along the ribbons (which are glued side by
side), y is perpendicular to the sides, and z is perpendicular to the panel. For instance, for the rectangular unit cell pointed out in Figure 2.7, of cross-section area 3as, the solid area is 8aΞ΄, and the quotient is Fz=(8/3)Ξ΄/s.
Example
Evaluate the mean core-panel values of density and thermal conductivity through-the-thickness for a core made of aluminium foil with Ο=2700 kg/m3, Ξ»=150 W/(mΒ·K), thickness Ξ΄=30 ΞΌm and s=3 mm cell pattern.
Solution.
Fz = (8/3)(Ξ΄/s) = (8/3)(0.03/3) = 0.027
Οhoneycomb = ΟΒ· Fz = 2700Β·0.027=73 kg/ m3
λz honeycomb = λ· Fz = 150·0.027 = 4 W/(m·K).
2.4 Standard test method for thermal properties measurement
There are several standards to measure the thermal properties of a specimen. Here two methods based on ASTM C518 and C117.
The first standard provides a measure by means of a heat flux apparatus. The heat flow meter apparatus is used widely because it is relatively simple in concept, rapid, and applicable to a wide range of test specimens. The precision and bias of the heat flow meter apparatus can be excellent provided calibration is carried out within the range of heat flows expected. This means calibration shall be carried out with similar types of materials, of similar thermal conductivities, at similar thicknesses, mean temperatures, and temperature gradients, as expected for the test specimens. The heat flow meter apparatus establishes steady state one-dimensional heat flux through a test specimen between two parallel plates at constant but different temperatures. Fourierβs law of heat conduction is used to calculate thermal conductivity, and thermal resistivity or thermal resistance and thermal conductance.
Figure 2.8: Apparatus with one heat flux transducer and one specimen.
Figure 2.9: Apparatus with one heat flux transducer and two specimens.
Figure 2.10: Apparatus with two heat flux transducers and one specimen.
The general features of a heat flow meter apparatus with the specimen or the specimens installed are shown in Figures 2.8, 2.9 and 2.10. A heat flow meter apparatus consists of two isothermal plate assemblies, one or more heat flux transducers and equipment to control the environmental conditions when needed. The two plate assemblies should provide isothermal surfaces in contact with either side of the test specimen. The assemblies consist of heat source or sink, a high conductivity surface, means to measure surface temperature, and means of support. A heat flux transducer may be attached to one, both, or neither plate assembly, depending upon the design. In all cases, the area defined by the sensor of the heat flux transducer is called the metering area and the remainder of the plate is the guard area. The two plate assemblies provide isothermal surfaces in contact with either side of the test specimen. The surfaces of the plate assemblies in contact with the specimen(s) shall be instrumented with precision temperature sensors such as thermocouples, platinum resistance thermometers (RTD), and thermistors.
When only one specimen is used, the thermal conductivity is calculated as follows:
π = π π πΏΞπ (2.46)
where π is the calibration factor of the heat flux transducer (W/m2)/V.
The uncertainties in S, q, L, and Ξπ (πΏπ, πΏπ, πΏπΏ and πΏΞπ) can be used to form the uncertainty πΏπ by the usual error propagation formula where the total uncertainty is calculated from the square root of the sums of the squares of the individual standard deviations.
πΏππ
)
=πΏππ
)
+πΏππ
)
+πΏπΏπΏ
)
+πΏΞπΞπ
)
(2.47)
When two specimens (a and b) are used, the average thermal conductivity is calculated as follows:
π7οΏ½S = π π2
πΏ7 + πΏ9Ξπ7 + Ξπ9
(2.48)
Another method to evaluate the thermal properties of a material is the standard ASTM C117. A general arrangement of the mechanical components of the apparatus is illustrated in Figure 2.11. This consists of a hot surface assembly comprised of a metered section and a primary guard, two cold surface assemblies, and secondary guarding in the form of edge insulation, a temperature-controlled secondary guard(s), and often an environmental chamber. Some of the components illustrated in Figure 2.11 are omitted in systems designed for ambient conditions, although a controlled laboratory environment is still required; edge insulation and the secondary guard are typically used only at temperatures that are more than Β±10Β°πΆ from ambient. At ambient conditions, the environmental chamber is recommended to help eliminate the effects of air movement within the laboratory and to help ensure that a dry environment is maintained.
Figure 2.11: General arrangement of the mechanical components of the guarded-hot-plate apparatus.
The purpose of the hot surface assembly is to produce a steady-state, one-dimensional heat flux through the specimens. The purpose of the edge insulation, secondary guard, and environmental chamber is to restrict heat losses from the outer edge of the primary guard. The cold surface assemblies are isothermal heat sinks for removing the energy generated by the heating units; the cold surface assemblies are adjusted so they are at the same temperature.
Figure 2.12: Illustration of idealized heat flow in a guarded-hot-plate apparatus.