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A closed-form solution for axisymmetric conduction in a finite functionally graded cylinder
International Communications in Heat and Mass Transfer 108 (2019) 104280
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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
A closed-form solution for axisymmetric conduction in a finite functionally graded cylinder
T
A. Amiri Deloueia, , A. Emamianb, S. Karimnejadb, , H. Sajjadia ⁎
a b
⁎
Department of Mechanical Engineering, University of Bojnord, Bojnord 945 3155111, Iran Mechanical Engineering Department, Shahrood University of Technology, Shahrood 361 9995161, Iran
ARTICLE INFO
ABSTRACT
Keywords: Functionally graded material Heat transfer Analytical solution Cylinder General inhomogeneous boundary condition
An exact general analytical solution is derived for heat conduction problem in an axisymmetric cylinder made of functionally graded material whose thermal conductivity differs in two directions. It is assumed that the thermal conductivity coefficients in radial and longitudinal directions are the power functions of radius. This is accomplished by taking advantage of Sturm-Liouville theory to get a suitable Fourier transformation. The general inhomogeneous thermal boundary conditions are adopted in both radial and longitudinal directions. The obtained solution is adequately verified against available analytical results. Also, two illustrative cases are considered to guarantee the solution's accuracy from a practical perspective. Particularly, the importance of the material constants on temperature distribution in both radial and longitudinal directions is uncovered. The results show that the present exact solution satisfies well the energy equation and arbitrary complex boundary conditions.
1. Introduction Functionally graded material (FGM) is a non-homogeneous composite material in which material properties could change smoothly in a certain direction. The premise behind the FGM was to gradually substitute changing interface with the sharp interface which exists in conventional composite materials [1–4]. Depending on the area of use, FGM structure can change stepwise or continuously [5,6]. Thus, these types of structures permit materials with different and even incompatible features to be combined and used properly [5,7]. For instance, the most prevalent type of FGM which is achieved through the combination of ceramic and metal, smoothly integrates the metal's fracture toughness with ceramic's high thermal resistance. In other words, this assembly is using certain properties of each constituent to serve the desired purpose. As known, thermal studies are frequently encountered in a variety of uses [8–12]. Having had high strength- and stiffness-to-density ratio, thermal resistance, and flexible designability make FGM dominate commodity market, biomaterials, optics, and engineering industries [6,13,14]. Being greatly heat resistive material encouraged engineers to utilize FGM in high-temperature environments. To ensure the satisfactory performance of such materials, especially during service condition, indepth investigations on the heat transfer problem are of critical importance. This can be regarded as an aid in the process of designing, ⁎
anticipating material characteristic, and also analyzing composition profile. Slurry pipe, heat pipe, roller engine components, shaft, and building materials are some of the most common practical applications of FGM [5,15]. Thus, detailed studies regarding cylindrical geometry are absolutely needed. So far, heat conduction solutions for conventional composite structures have been extended widely [16–21]. In case of cylindrical composite pin fin, Bahadur and Bar-Cohen [22] derived a closed form analytical solution under orthotropic condition. Kayhani et al. [23] analytically studied the steady-state temperature distribution for a cylinder made of composite laminate. The same group, in 2012, extended their research to the transient situation [24]. They hired Meromorphic function to obtain the unsteady temperature distribution. Li and Lai [25] considered a composite hollow cylinder under a set of boundary conditions and analytically studied heat conduction. It is revealed that analytical solutions with designating proper values for the coefficients of boundary conditions would result in a certain reported solution. They have illustrated that in the case of constant temperature boundary condition, their solution is an extended version of Jaeger's solution [26]. For a hollow composite cylinder, Torabi and Zhang [27] implementing a numerical-analytical scheme and explored temperature distribution along with entropy generation rate. Under the convective boundary condition and constant temperature boundary condition they presented a complete analytical solution.
Corresponding authors. E-mail addresses: [email protected] (A. Amiri Delouei), [email protected] (S. Karimnejad).
https://doi.org/10.1016/j.icheatmasstransfer.2019.104280
0735-1933/ © 2019 Elsevier Ltd. All rights reserved.
International Communications in Heat and Mass Transfer 108 (2019) 104280
A. Amiri Delouei, et al.
Nomenclature
m q′′ R r, θ, z r0 r1 T I ρ ϕn λ
The coefficient constant An a1, a2, c1, c2 Constant coefficient (W/m2K) The constant coefficient n order Bn, Cn b1, b2, d1, d2 Constant coefficient (W/mK) cp Specific heat capacity (m2/s2K) f1, f2 Arbitrary function of r h Convective coefficient of heat transfer (W/m2K) kr, kz The heat conductivity coefficients in r and z directions kr0, kz0 The conduction coefficients in r and z directions The conduction coefficients ratio k∗
Material constant coefficients Heat flux (W/m2) Independent function of r Coordinate system Inner radius (m) Outer radius (m) Temperature distribution (K) Independent function of z Density (kg/m3) Eigenfunction Separation constant
cylindrical geometry. In the following study, the Strum-Liouville theorem is used to derive the desired Fourier transformation. The Fourier eigenfunction and eigenvalue will be achieved by solving a set of equations and using cylinder's boundary conditions in z direction. Unlike most former studies, the thermal properties of the cylinder vary in two directions. The new proposed analytical solution is properly validated using the known data. Afterward, two cases which can be attributed to the actual problems are solved and temperature distribution under the effect of various parameters is discussed. Analytically satisfying the combination of the thermal boundary conditions is a difficult process which the obtained solution is capable of fulfilling it.
Several scholars motivated by the extensive use and role that functionally graded materials (FGMs) are able to fulfill and possess in comparison with composite materials. It is pointed out that controlling of thermal stresses using FGMs in structural components is possible. Analyzing FGMs under thermal loading is rather hard, therefore, only a small number of research papers concerning this matter can be found [28]. Sladek et al. [29] proposed a computational method to study the heat conduction problem for FGMs. They applied the method to a hollow cylinder in which its material properties changes exponentially. Haghighi et al. [30] developed a numerical method based on finite element method and differential quadrature approach to tackling the two-dimensional heat transfer problems of FGMs. It is shown that the method is accurate and is not cumbersome. The non-uniform convective-radiative boundary conditions are applied to the boundaries and investigated. Although numerical approaches [31,32] play an essential role in understanding the behavior of functionally graded materials, their applications are somewhat limited and results are not relatively solid. For instance, the boundary integral equation methods [29,33] which have been employed widely, need a closed form or computable notation of the partial differential equation (PDE). However, some advancements regarding this issue have been made. Gray et al. [34] further extended boundary integral analysis applications to exponentially graded materials and verified the results with exact solutions. On the other hand, the analytical solution is an attractive field of study whose outcomes are most accurate and reliable [35,36]. Utilizing analytical approaches, deep insight into heat conduction problems is practicable. Also, they have been broadly used as a verification test to numerical methods. Analytical studies regarding the FGM cylinders are quite scarce and limited studies generally lack complex thermal boundary conditions [37] or are done for specific grading patterns [38]. Tarn and Wang [39] studied heat conduction end effects in FGM circular cylinder in which inhomogeneity changes radially and derived an exact solution. Hiring matrix algebra and eigenfunction expansion they calculated the decay length. An analytical research on hyperbolic heat conduction for FGM cylinder is conducted by Babaei and Chen [40]. They solved one dimensional (radial) heat conduction problem in a long FGM cylinder. Hosseini et al. [41] presented an analytical solution for a hollow cylinder made of FGM in which its properties distributed through the thickness. The obtained solution for the axisymmetric shell is restricted to constant thermal boundary conditions. The leading objective of the present study is to acquire a two-dimensional (r, z) steady-state analytical solution for heat conduction in a finite axisymmetric thick hollow FGM cylinder. Moreover, the solution is derived under the most general inhomogeneous linear thermal boundary conditions both in radial and longitudinal directions. The central advantage of the implied general thermal boundary conditions is the capability of handling a wide variety of steady-state problems in
2. Problem description and solution method In this section, the mathematical model which is used for this solution is described. A finite length (L) thick cylindrical vessel made of functionally graded material with inner and outer radii equal to r0 and r1 is considered. The general thermal boundary conditions are considered both in radial and longitudinal directions. The material properties are different functions of radius which is because of the fact that FGM layers are graded radially. 2.1. The energy equation and general thermal boundary conditions For the analysis, a cylindrical coordinate system (r, θ, z) is implemented as illustrated in Fig. 1. The three-dimensional heat transfer in an axisymmetric FGM cylinder is as [42]:
Fig. 1. The geometry and coordinate system of the present problem and variation of materials properties from inner radius to the outer.
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International Communications in Heat and Mass Transfer 108 (2019) 104280
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1 T rkr r r r
1 r2
+
k
T
+
z
kz
T z
= cp
DT , Dt
The separation of variables (SOV) technique is utilized to find the exact solution. For this purpose the temperature distribution is defines as: T(r, z) = ℜ(r)ℑ(z). Using SOV method to solve Eq. (4) and applying homogeneous thermal boundary conditions, the following equations in the longitudinal, z, direction are obtained:
(1)
where T is temperature distribution. Also, cp and ρ indicate specific heat capacity and density, correspondingly. For steady-state conductive heat transfer in a finite axisymmetric FGM cylinder, the Eq. 1 can be reduced to:
1 T rkr (r ) r r r
T + k z (r ) z z
= 0,
2I (z )
z2
(2)
where kr and kz are the heat conductivity coefficients in r and z directions, respectively. It is supposed that the thermal conductivity coefficients, kr and kz, are the power functions of r as:
k r (r ) = k r 0 r m ,
(3a)
k z (r ) = k z 0 r m .
(3b)
r2
+
1 T 1 (m + 1) + r r k
2T
z2
= 0,
kr 0 . kz0
a2 T (r , L) + b2
(6b)
c1 T (r0 , z ) + d1
T (r0 , z ) = g1 (z ), r
(6c)
c2 T (r1 , z ) + d2
T (r1 , z ) = g2 (z ). r
s ( z ) f (z ) b
s (z )
F (f ) n=1
n (z ),
a2 b1 )
4 n An2
b1
(11)
n cos( n z ).
n cos( n L)
+ (a2 a1 + b1 b2
2 n)
sin(
n L)
(12)
= 0.
L
f (z ) a1 sin(
n z)
b1
n cos( n z ) dz ,
(13)
0
( a12 + (b1
F (f ) =
(6d)
+ (b1
n)
2)
sin(2
2 n ) )L
n L)
2a1 b1
+ 2a1 b1
n cos(2 n L) + 2 n
. (14)
n
2 n An
2
a2 cos(
n L)
a1 b2
n sin( n z )
f2 (r ) + f1 (r )
2 nF
(f ), (15)
2.3. Analytical solution The following relations are obtained by applying the constructed Fourier transformation to Eq. (4) and the boundary conditions in the radial, r, direction (Eqs. (6c) and (6d)): k
2U
r2
+
1 U (m + 1) r r
2 nU
=
2 n An
2
a2 cos(
n L)
a1 b2
n sin( n z )
f2 (r )
f1 (r )
(16)
(7)
where s(z) is the weighting function. Also, ϕn(z) is the eigenfunction obtained from the solution of the homogeneous equation with homogeneous thermal boundary conditions in the z direction. The inverse Fourier transformation is defined as [23]:
f (z ) =
n z)
From the definition of the Fourier transformation, the second derivative with respect to z is given by
,
a
= a1 sin(
(a12
n (z ) dz
2 n (z ) dz
(10b)
An =
Concerning the current inhomogeneous thermal boundary conditions, it is appropriate to employ Sturm-Liouville theory to get a suitable Fourier transformation for the arbitrary function f(z) [23]:
a
I (L ) = 0. z
where
(6a)
2.2. Construction of Fourier transformation
F (f ) =
a2 I (L) + b2
F (f ) =
where the inner and outer radii are r0 and r1,respectively. L is the length of cylinder. The constants a1, a2, c1 and c2 have the same dimension as the convection coefficient. The constants b1, b2, d1 and d2 have the same dimension as the conduction coefficient, as well.
b
(10a)
The weighting function is constant based on Sturm-Liouville theory and the homogeneous boundary conditions in the z direction. A suitable Fourier transformation for this problem is obtained (F) by substituting these relations into the Sturm-Liouville equation (Eq. (7)):
The general linear thermal boundary conditions with f1(r), f2(r), g1(z) and g2(z) being arbitrary functions are as follows [23,25]:
T (r , L ) = f2 (r ), z
I (0) = 0, z
(a1 b2
(5)
T (r , 0) = f1 (r ), z
(9)
By applying the boundary conditions in the z direction (Eq. (10)), the following equation in the form of trigonometric is obtained for the eigenvalues
(4)
a1 T (r , 0) + b1
= 0,
a1 I (0) + b1
n (z )
where
k =
2 I (z )
Having solved Eq. (9) and using boundary conditions introduced by Eqs. (10), the eigenfunction of this problem is obtained as
In the above-noted equations, kr0, kz0, and m are the constant coefficients. The values of k and m are defined based on the material properties of the functionally graded cylinder. Introducing Eq. (3) into Eq. (2), the following equation is found: 2T
+
c1 U (r0 , n) + d1
U (r0 , n) = G1 (n), r
(17a)
c2 U (r1 , n) + d2
U (r1 , n) = G2 (n) r
(17b)
where
U (r , n) = F (T (r , z )),
(18a)
Gi (n) = F (gi (z )) i = 1, 2. ,
(18b)
If the right-hand side of Eq. (16) is equal to ℏ(r, n), the general solution of this equation will be as
(8)
3
International Communications in Heat and Mass Transfer 108 (2019) 104280
A. Amiri Delouei, et al. m 2
U (r , n ) = r
(B I
n m 2
)
( n r ) + Cn K m ( n r ) + w (r , n), 2
conductivity ratio in z direction is considered to be 20 W/mK as [22]. The cooling rate versus 1/k⁎ for three values of convection coefficients (h = 100, 500, and 1000 W/m2K) is plotted in Fig. 2. From the figures, it is apparent that results match very well with the previous study [22], and the obtained solution using the aforementioned procedure is able to accurately predict cooling rate in various situations.
(19)
where I and K are modified Bessel functions of the first and second kinds, respectively. n
2 n
=
1 2
k
,
(20a)
3.2. Two applied cases
and w(r, n) is the non-homogeneous solution for Eq. (16). Its general form is given as r1
w (r , n) = I m ( n r ) × 2
r0
In order to examine the solution's reliability from the practical perspective, two industrial illustrative cases are assessed in this section and it is tried to mimic the real practical situation. Thus, different combinations of thermal boundary conditions ranging from convection, constant temperature to constant and varied heat flux are imposed on the examples. Steady-state heat conduction problem for an axisymmetric finite hollow cylindrical FGM vessel is investigated and temperature distribution under the impact of various parameters is discussed. As mentioned earlier, the cylindrical geometry is one of the most encountered structures in real-world applications such as heat pipe, cooling pipe, roller engine components, pressure vessel, and shaft. Two cases are described successively. The first case is a FGM hollow cylinder of inner radius r0 = 0.1 m, outer radius r1 = 0.2 m, and length L = 0.45 m. The geometry and the boundary conditions are demonstrated in Fig. 3. As can be seen, the cylinder's inner surface is experiencing constant temperature. Moreover, the outer surface of the body is partially subjected to longitudinal and radial varied heat flux along with convection current. Additionally, another illustrative industrial example with constant heat flux, convection, and the varied temperature is considered. The geometry and the boundary conditions of the second case are shown in Fig. 4. The geometrical parameters are as follows: Inner and outer radii are equal to 0.1 m and 0.3 m, respectively and the length is 0.3 m. Temperature distribution in both radial and longitudinal directions at different values of material constant, m, which can be customized regarding material features, for both cases are displayed in Figs. 5 and 6. This parameter ranges from 1 to 4 and the heat conductivity ratio (k⁎) is maintained fixed at 1. For the first case, longitudinal and radial distributions of temperature are demonstrated at r = 0.15 m and z = L/ 6, correspondingly. As expected, with an increase in values of m, temperature distribution rises in two directions. It is seen that from the inner radius to the outer, temperature increases monotonically. For the second case, temperature distribution curves in the radial and longitudinal directions are plotted at z = L/4 and r = 0.15 m, respectively (Fig. 6). Contrary to the longitudinal temperature variation, in the radial direction, the temperature reaches a virtually constant plateau as radius upturns. As it is observed in the figures, material constant strongly affects the temperature distribution in two directions. Contours
r1
r × K m ( n r ) × (r , n ) dr + K m ( n r ) × 2 2
r0
r × I m ( n r ) × (r , n) dr , 2
(20b) Lastly, by applying the boundary conditions to the inner and outer surface of the cylinder in the direction of r, the unknown coefficient will be achieved. Introducing the Eq. (19) into Eqs. (17a) and (17b), the following relations will be obtained
Bn
nm
+ Cn
nm
= F1, nm,
(21)
Bn
nm
+ Cn
nm
= F2, nm,
(22)
Where αnm, βnm, χnm, δnm, F1, nm and F2, nm are the constant coefficients. For the sake of brevity they are shown in the appendix section. The unknown coefficients of Bn and Cn can be achieved as follows
Bn =
F1, nm
nm
nm
nm
Cn =
F1, nm
nm
nm
nm
F2, nm
nm
, n = 1, 2, …,
nm nm
F2, nm
nm
, n = 1, 2, ….
nm nm
(23a) (23b)
It should be noted that for solid cylinders ( T (0, z ) ), the coefficient Cn r should be set to zero. At the end, by means of the inverse Fourier transformation (Eq. (8)) to Eq. (19), the temperature distribution will be achieved as: T (r , z ) = n=1
U (r , n ) × n (z ) m
= n=1
r 2 Bn I m ( n r ) + Cn K m ( n r ) + w (r , n) × (a1 sin( n z ) 2
2
b1 n cos( n z )).
(24) 3. Results and discussion First and foremost, in this section, the solution's validity will be checked with those of Bahadur and Bar-Cohen [22]. Afterward, two industrial illustrative cases with considering a combination of thermal boundary conditions are discussed. Using the current solution and further analyzing the impact of material constants on the distribution of temperature will provide the engineers with a better understanding of the heat exchange in FGMs. 3.1. Verification test The correctness of the presented exact analytical solutions with general inhomogeneous thermal boundary condition is investigated in this sub-section. Because the same studies are few, to verify the ongoing problem it is required to simplify the current solution to the published results. To do so, the material properties, boundary conditions, and geometry (i.e., r = 0.45 cm and L = 5 cm) are considered as those of Bahadur and Bar-Cohen [22]. In reference [22] an analytical solution for orthotropic cylindrical pin fin is obtained and in particular, watercooled fins are studied. To compare, the value of m is set to zero to fulfill the orthotropic thermal conductivity situation. Also, the
Fig. 2. Comparison of cooling rate variations with conductivity ratio at three values of convection coefficients.
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A. Amiri Delouei, et al.
Fig. 3. Schematic illustration of the geometry and boundary conditions of the first case.
of temperature distribution in r and θ directions at k∗ = 1 for case 1 and 2 are illustrated in Figs. 7 and 8 to make a clear observation of the influence of material constant (m). In order for further illustration, Table 1 shows temperature distribution in radial direction for both case 1 and 2. As can be found out, as for case 1, with an increase in the values of m, temperature increases.
To have a more detailed assessment of the radial temperature distribution at different longitudinal locations, four sections from L/6 to L in z direction for both cases are considered and depicted in Figs. 9 and 10. For cases 1 and 2, the temperature variation is approximately the same from L/6 to L/2 and afterward at the end of the cylinder temperature slightly drops. For the second case as it is illustrated in Fig. 10,
Fig. 4. Schematic illustration of the geometry and boundary conditions of the second case.
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Fig. 7. Contours of temperature distribution in r and θ directions at k∗ = 1 and different values of “m” ranging from 1 to 4 for case 1.
Fig. 5. Temperature distribution in a) longitudinal (r = 0.15 m) and b) radial (z = L/6) direction at different values of “m” ranging from 1 to 4 at k∗ = 1for the first case.
Fig. 8. Contours of temperature distribution in r and θ directions at k∗ = 1 and different values of “m” ranging from 1 to 4 for case 2. Table 1 Temperature distribution in radial direction at different values of “m” ranging from 1 to 4 at “k∗ ” = 1 for the first case (z = L/6) and the second case (z = L/ 4). Case 1
Case 2
r
m=1
m=2
m=3
m=4
r
m=1
m=2
m=3
m=4
0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2
300 312 322 330 338 345 352 358 364 370 376
300 317 330 340 348 356 362 367 373 377 382
300 323 338 349 358 365 371 376 380 384 387
300 328 346 358 367 373 378 382 385 388 391
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
455 415 388 369 354 344 335 329 324 321 318
455 400 369 349 336 328 322 318 316 314 314
455 384 350 331 320 313 310 307 306 306 305
455 368 332 314 306 301 299 298 297 296 295
Fig. 6. Temperature distribution in a) longitudinal (r = 0.15 m) and b) radial (z = L/4) direction at different values of “m” ranging from 1 to 4 at k∗ = 1 for the second case. Fig. 9. Radial distribution of temperature at different longitudinal locations at k∗ = 1 for case 1.
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International Communications in Heat and Mass Transfer 108 (2019) 104280
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Fig. 10. Radial distribution of temperature at different longitudinal locations at k∗ = 1 for case 2.
Fig. 13. Temperature distribution in a) longitudinal (r = 0.2 m) and b) radial (z = L/6) direction at different values of “k∗” at m = 1 for the first case.
Fig. 11. Longitudinal distribution of temperature at different radial sections at k∗ = 1 for case 1.
Fig. 12. Longitudinal distribution of temperature at different radial sections at k∗ = 1 for case 2.
it can be concluded that all cases reach a constant value at the same radial location. To see the similar results in various radial sections through z, refer to Figs. 11 and 12. With the growth of radius from the inner wall to the outer wall, the gap among different temperature variations steadily narrows for both cases. To further study the temperature distribution, here, the effect of heat conductivity ratio (k∗= kr0/kz0) for both cases are studied (Figs. 13 to 16). Four different values of k∗(=0.3, 0.6, 1, 5) are considered while the value of m is kept the same (=1). It can be seen from figures that for both case 1 and case 2, in the same circumstances but different values of k∗, the temperature distribution is different. Fig. 13 is concerned with the effect of heat conductivity ratio on temperature distribution in two
Fig. 14. Temperature distribution in a) longitudinal (r = 0.15 m) and b) radial (z = L/4) direction at different values of “k∗” at m = 1 for the second case.
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Similarly, for the second case, Fig. 14a and b demonstrate longitudinal (at r = 0.15 m) and radial (at z = L/4) temperature distribution, respectively. It is evident from Fig. 14b that even with the growth of heat conductivity, the temperature distribution is reduced, which demonstrates the importance of thermal environments. Also, in this case, the gap among different lines marginally increases. For cases 1 and 2, contours of temperature distribution in two directions are shown in Figs. 15 and 16. One can conclude that even in the small values of k∗, the temperature distribution in either direction changes noticeably. As known, a change in material properties can considerably alter the patterns of temperature distribution in the structure. k* is one of the important factors which contributes to this matter. As in Figs. 15 and 16, in relatively small values of k* (less than 1), the variations are noticeable. In other words, has a tremendous impact on the rate of cooling. On the other hand, for larger values of m (larger than 1), the variations can be ignored. From the figures, all lines of temperature distribution for both cases are reasonable and properly satisfy imposed thermal boundary conditions. Further, the variations of temperature distribution in two directions are very beneficial in designing flexible and favorable FGMs. In Tables 2 and 3, for case 1 and 2, temperature distribution versus z at different values of m, radius, and k∗, are shown. It is seen that for both of which with a rise in z, temperature drops.
Fig. 15. Contours of temperature distribution in r and θ directions at m = 1 for case1.
4. Conclusions Two-dimensional heat transfer in FGMs plays an important role in designing and optimization of structures for multi-functional areas relevant to various practical applications. This paper analytically tackled the heat conduction problem in a finite length FGM cylindrical vessel under general boundary condition. Having employed inhomogeneous thermal boundary conditions, the Sturm-Liouville theory is used to acquire the analytical solution for two-dimensional heat conduction. At first, the solution is validated with published data, and then two practical cases with various mixtures of boundary conditions are solved. Temperature distribution under the influence of various parameters at different fragments of the FGM cylinder is reported. The steady-state exact analytical solution that presented in this paper is believed to be beneficial for the design of FGM vessels for different operating conditions.
Fig. 16. Contours of temperature distribution in r and θ directions at m = 1 for case2.
directions for the first case. Fig. 13a shows the longitudinal temperature distribution at r = 0.15 m and Fig. 13b is responsible for radial temperature distribution at z = L/4. As can be found out, an increase in the k∗ in r and z directions makes the heat flux transfer and convection more effective. The distribution of temperature is substantially affected by the larger values of k∗ and also higher temperature is seen close to the outer surface.
Table 2 Temperature distribution in longitudinal direction at different values of “m” (r = 0.15 m), radial sections (k∗ = 1) and k∗ (r = 0.2 m) for the first case. z
m=1
m=2
m=3
m=4
(r₀ + r₁)/3
(r₀ + r₁)/2
(r₀ + r₁)/1.5
k* = 0.3
k* = 0.6
k* = 1
0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275 0.3 0.325 0.35 0.375 0.4 0.425 0.45
345 345 345 343 342 340 338 335 332 328 324 320 315 310 304 298 292 285 278
356 355 355 354 352 350 348 345 342 338 334 329 324 319 313 307 301 294 286
365 365 364 363 361 359 357 354 351 347 343 338 333 327 322 315 309 301 294
373 373 372 371 370 367 365 362 358 355 350 346 340 335 329 322 315 308 301
300 301 300 299 298 296 294 292 289 286 282 279 274 270 265 260 254 249 242
345 345 345 343 342 340 338 335 332 328 324 320 315 310 304 298 292 285 278
376 376 375 374 372 370 367 364 361 357 353 348 343 337 331 325 318 310 303
359 359 358 357 355 353 351 348 345 341 337 332 327 322 316 310 303 296 289
371 371 370 369 367 365 363 360 356 352 348 343 338 333 327 320 314 306 299
376 376 375 374 372 370 367 364 361 357 353 348 343 337 331 325 318 310 303
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Table 3 Temperature distribution in longitudinal direction at different values of “m” (r = 0.15 m), radial sections (k∗ = 1) and k∗ (r = 0.15 m) for the second case. z
m=1
m=2
m=3
m=4
(r₀ + r₁)/2
2(r₀ + r₁)/3
3(r₀ + r₁)/4
k* = 0.3
k* = 0.6
k* = 1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
377 377 376 375 373 371 368 365 361 357 353 348 342 336 330 323
391 389 388 387 385 383 381 378 374 371 366 362 357 351 345 339
402 402 401 400 399 396 394 391 387 384 380 376 371 366 360 354
410 410 409 408 407 405 403 400 397 394 390 386 381 377 371 366
338 338 338 336 335 333 331 328 324 321 317 312 307 302 296 290
313 313 312 311 310 308 306 303 300 297 293 289 284 279 274 268
306 306 305 304 303 301 299 296 293 290 286 282 278 273 268 262
356 355 355 354 352 350 347 344 341 337 333 328 323 317 311 305
368 367 367 365 364 362 359 356 352 348 344 339 334 328 322 315
377 377 376 375 373 371 368 365 361 357 353 348 342 336 330 323
Appendix A. Appendix nm
nm
nm nm
= c1 r0
m 2 Im 2
( n r0 )
=
m c1 r0 2 K m 2
=
m c2 r1 2 I m 2
=
m c2 r1 2 K m 2
( n r0)
( n r1) ( n r1)
d1 n r0
m 2 I m + 1 ( n r0 ), 2
(A1)
m d1 n r0 2 Kn + 1 ( n r0),
(A2)
m d2 n r1 2 In + 1 ( n r1),
(A3)
m d2 n r1 2 Kn + 1 ( n r1),
(A4)
F1, nm = G1 (n)
c1 w (r0 , n)
d1
w (r0 , n) , r
(A5)
F2, nm = G2 (n)
c2 w (r1 , n)
d2
w (r1 , n) . r
(A6)
[14] B.V. Sankar, An elasticity solution for functionally graded beams, Compos. Sci. Technol. 61 (2001) 689–696. [15] F. Kuznik, D. David, K. Johannes, J.-J. Roux, A review on phase change materials integrated in building walls, Renew. Sust. Energ. Rev. 15 (2011) 379–391. [16] M.M. Shahmardan, M. Norouzi, M.H. Kayhani, A.A. Delouei, An exact analytical solution for convective heat transfer in rectangular ducts, J. Zhejiang Univ. Sci. A 13 (2012) 768–781. [17] A.A. Delouei, M. Norouzi, Exact analytical solution for unsteady heat conduction in fiber-reinforced spherical composites under the general boundary conditions, J. Heat Transf. 137 (2015) 101701. [18] A.A. Masumi, G.H. Rahimi, G.H. Liaghat, The use of the layerwise theory in heat transfer analysis of metal composite vessel by DQM, Int. J. Therm. Sci. 132 (2018) 14–25. [19] I. Dülk, T. Kovácsházy, A method for computing the analytical solution of the steady-state heat equation in multilayered media, J. Heat Transf. 136 (2014) 091303. [20] Y. Chang, R. Tsou, Heat conduction in an anisotropic medium homogeneous in cylindrical regions—unsteady state, J. Heat Transf. 99 (1977) 41–46. [21] A. Aziz, W. Khan, Classical and minimum entropy generation analyses for steady state conduction with temperature dependent thermal conductivity and asymmetric thermal boundary conditions: regular and functionally graded materials, Energy 36 (2011) 6195–6207. [22] R. Bahadur, A. Bar-Cohen, Orthotropic thermal conductivity effect on cylindrical pin fin heat transfer, ASME 2005 Pacific Rim Technical Conference and Exhibition on Integration and Packaging of MEMS, NEMS, and Electronic Systems Collocated with the ASME 2005 Heat Transfer Summer Conference, 2005, pp. 245–252. [23] M. Kayhani, M. Norouzi, A.A. Delouei, A general analytical solution for heat conduction in cylindrical multilayer composite laminates, Int. J. Therm. Sci. 52 (2012) 73–82. [24] A.A. Delouei, M. Kayhani, M. Norouzi, Exact analytical solution of unsteady axisymmetric conductive heat transfer in cylindrical orthotropic composite laminates, Int. J. Heat Mass Transf. 55 (2012) 4427–4436. [25] M. Li, A.C. Lai, Analytical solution to heat conduction in finite hollow composite cylinders with a general boundary condition, Int. J. Heat Mass Transf. 60 (2013) 549–556. [26] J. Jaeger, XXVI. Heat conduction in composite circular cylinders, London Edinburgh Dublin Philos. Mag. J. Sci. 32 (1941) 324–335. [27] M. Torabi, K. Zhang, Temperature distribution and classical entropy generation
References [1] M. Norouzi, A.A. Delouei, M. Seilsepour, A general exact solution for heat conduction in multilayer spherical composite laminates, Compos. Struct. 106 (2013) 288–295. [2] P. Erasenthiran, V.E. Beal, Functionally graded materials, Rapid Manuf. Ind. Revolut. Digi. Age (2006) 103–124. [3] Y. Miyamoto, W. Kaysser, B. Rabin, A. Kawasaki, R.G. Ford, Functionally Graded Materials: Design, Processing and Applications, vol. 5, Springer Science & Business Media, 2013. [4] M. Krumova, C. Klingshirn, F. Haupert, K. Friedrich, Microhardness studies on functionally graded polymer composites, Compos. Sci. Technol. 61 (2001) 557–563. [5] R.M. Mahamood, E.T. Akinlabi, Types of functionally graded materials and their areas of application, Functionally Graded Materials, Springer, 2017, pp. 9–21 ed:. [6] V. Birman, L.W. Byrd, Modeling and analysis of functionally graded materials and structures, Appl. Mech. Rev. 60 (2007) 195–216. [7] A. Markworth, K. Ramesh, W. Parks, Modelling studies applied to functionally graded materials, J. Mater. Sci. 30 (1995) 2183–2193. [8] A. Wakif, Z. Boulahia, R. Sehaqui, Numerical study of the onset of convection in a Newtonian nanofluid layer with spatially uniform and non uniform internal heating, J. Nanofluids 6 (2017) 136–148. [9] A. Wakif, Z. Boulahia, R. Sehaqui, Numerical analysis of the onset of longitudinal convective rolls in a porous medium saturated by an electrically conducting nanofluid in the presence of an external magnetic field, Results Phys. 7 (2017) 2134–2152. [10] A. Wakif, Z. Boulahia, S. Mishra, M.M. Rashidi, R. Sehaqui, Influence of a uniform transverse magnetic field on the thermo-hydrodynamic stability in water-based nanofluids with metallic nanoparticles using the generalized Buongiorno's mathematical model, Eur. Phys. J. Plus 133 (2018) 181. [11] S. Karimnejad, A.A. Delouei, M. Nazari, M. Shahmardan, M. Rashidi, S. Wongwises, Immersed boundary—thermal lattice Boltzmann method for the moving simulation of non-isothermal elliptical particles, J. Therm. Anal. Calorim. (2019) 1–15. [12] M. Norouzi, M. Davoodi, O.A. Bég, A.A. Joneidi, Analysis of the effect of normal stress differences on heat transfer in creeping viscoelastic dean flow, Int. J. Therm. Sci. 69 (2013) 61–69. [13] D. Jha, T. Kant, R. Singh, A critical review of recent research on functionally graded plates, Compos. Struct. 96 (2013) 833–849.
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analyses in an asymmetric cooling composite hollow cylinder with temperaturedependent thermal conductivity and internal heat generation, Energy 73 (2014) 484–496. F. De Monte, An analytic approach to the unsteady heat conduction processes in one-dimensional composite media, Int. J. Heat Mass Transf. 45 (2002) 1333–1343. J. Sladek, V. Sladek, C. Zhang, Transient heat conduction analysis in functionally graded materials by the meshless local boundary integral equation method, Comput. Mater. Sci. 28 (2003) 494–504. M.G. Haghighi, M. Eghtesad, P. Malekzadeh, Coupled DQ–FE methods for two dimensional transient heat transfer analysis of functionally graded material, Energy Convers. Manag. 49 (2008) 995–1001. H. Wang, An effective approach for transient thermal analysis in a functionally graded hollow cylinder, Int. J. Heat Mass Transf. 67 (2013) 499–505. K.-Z. Chen, X.-A. Feng, Computer-aided design method for the components made of heterogeneous materials, Comput. Aided Des. 35 (2003) 453–466. M. Bonnet, Boundary integral equation methods for elastic and plastic problems, Encyclopedia of Computational Mechanics Second Edition, 2017, pp. 1–33. L. Gray, T. Kaplan, J. Richardson, G.H. Paulino, Green's functions and boundary integral analysis for exponentially graded materials: heat conduction, J. Appl. Mech. 70 (2003) 543–549. A. Wakif, Z. Boulahia, R. Sehaqui, A semi-analytical analysis of electro-thermo-
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hydrodynamic stability in dielectric nanofluids using Buongiorno's mathematical model together with more realistic boundary conditions, Results Phys. 9 (2018) 1438–1454. M. Norouzi, A. Emamian, M. Davoodi, An analytical and experimental study on dynamics of a circulating Boger drop translating through Newtonian fluids at inertia regime, J. Non-Newtonian Fluid Mech. 267 (9) (2019) 1–13. S.M. Hosseini, M.H. Abolbashari, A unified formulation for the analysis of temperature field in a thick hollow cylinder made of functionally graded materials with various grading patterns, Heat Tran. Eng. 33 (2012) 261–271. M. Jabbari, A.H. Mohazzab, A. Bahtui, One-dimensional moving heat source in a hollow FGM cylinder, J. Press. Vessel. Technol. 131 (2009) 021202. J.-Q. Tarn, Y.-M. Wang, End effects of heat conduction in circular cylinders of functionally graded materials and laminated composites, Int. J. Heat Mass Transf. 47 (2004) 5741–5747. M.H. Babaei, Z. Chen, Transient hyperbolic heat conduction in a functionally graded hollow cylinder, J. Thermophys. Heat Transf. 24 (2010) 325–330. S.M. Hosseini, M. Akhlaghi, M. Shakeri, Transient heat conduction in functionally graded thick hollow cylinders by analytical method, Heat Mass Transf. 43 (2007) 669–675. M. Asgari, M. Akhlaghi, Transient heat conduction in two-dimensional functionally graded hollow cylinder with finite length, Heat Mass Transf. 45 (2009) 1383–1392.
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
A closed-form solution for axisymmetric conduction in a finite functionally graded cylinder
T
A. Amiri Deloueia, , A. Emamianb, S. Karimnejadb, , H. Sajjadia ⁎
a b
⁎
Department of Mechanical Engineering, University of Bojnord, Bojnord 945 3155111, Iran Mechanical Engineering Department, Shahrood University of Technology, Shahrood 361 9995161, Iran
ARTICLE INFO
ABSTRACT
Keywords: Functionally graded material Heat transfer Analytical solution Cylinder General inhomogeneous boundary condition
An exact general analytical solution is derived for heat conduction problem in an axisymmetric cylinder made of functionally graded material whose thermal conductivity differs in two directions. It is assumed that the thermal conductivity coefficients in radial and longitudinal directions are the power functions of radius. This is accomplished by taking advantage of Sturm-Liouville theory to get a suitable Fourier transformation. The general inhomogeneous thermal boundary conditions are adopted in both radial and longitudinal directions. The obtained solution is adequately verified against available analytical results. Also, two illustrative cases are considered to guarantee the solution's accuracy from a practical perspective. Particularly, the importance of the material constants on temperature distribution in both radial and longitudinal directions is uncovered. The results show that the present exact solution satisfies well the energy equation and arbitrary complex boundary conditions.
1. Introduction Functionally graded material (FGM) is a non-homogeneous composite material in which material properties could change smoothly in a certain direction. The premise behind the FGM was to gradually substitute changing interface with the sharp interface which exists in conventional composite materials [1–4]. Depending on the area of use, FGM structure can change stepwise or continuously [5,6]. Thus, these types of structures permit materials with different and even incompatible features to be combined and used properly [5,7]. For instance, the most prevalent type of FGM which is achieved through the combination of ceramic and metal, smoothly integrates the metal's fracture toughness with ceramic's high thermal resistance. In other words, this assembly is using certain properties of each constituent to serve the desired purpose. As known, thermal studies are frequently encountered in a variety of uses [8–12]. Having had high strength- and stiffness-to-density ratio, thermal resistance, and flexible designability make FGM dominate commodity market, biomaterials, optics, and engineering industries [6,13,14]. Being greatly heat resistive material encouraged engineers to utilize FGM in high-temperature environments. To ensure the satisfactory performance of such materials, especially during service condition, indepth investigations on the heat transfer problem are of critical importance. This can be regarded as an aid in the process of designing, ⁎
anticipating material characteristic, and also analyzing composition profile. Slurry pipe, heat pipe, roller engine components, shaft, and building materials are some of the most common practical applications of FGM [5,15]. Thus, detailed studies regarding cylindrical geometry are absolutely needed. So far, heat conduction solutions for conventional composite structures have been extended widely [16–21]. In case of cylindrical composite pin fin, Bahadur and Bar-Cohen [22] derived a closed form analytical solution under orthotropic condition. Kayhani et al. [23] analytically studied the steady-state temperature distribution for a cylinder made of composite laminate. The same group, in 2012, extended their research to the transient situation [24]. They hired Meromorphic function to obtain the unsteady temperature distribution. Li and Lai [25] considered a composite hollow cylinder under a set of boundary conditions and analytically studied heat conduction. It is revealed that analytical solutions with designating proper values for the coefficients of boundary conditions would result in a certain reported solution. They have illustrated that in the case of constant temperature boundary condition, their solution is an extended version of Jaeger's solution [26]. For a hollow composite cylinder, Torabi and Zhang [27] implementing a numerical-analytical scheme and explored temperature distribution along with entropy generation rate. Under the convective boundary condition and constant temperature boundary condition they presented a complete analytical solution.
Corresponding authors. E-mail addresses: [email protected] (A. Amiri Delouei), [email protected] (S. Karimnejad).
https://doi.org/10.1016/j.icheatmasstransfer.2019.104280
0735-1933/ © 2019 Elsevier Ltd. All rights reserved.
International Communications in Heat and Mass Transfer 108 (2019) 104280
A. Amiri Delouei, et al.
Nomenclature
m q′′ R r, θ, z r0 r1 T I ρ ϕn λ
The coefficient constant An a1, a2, c1, c2 Constant coefficient (W/m2K) The constant coefficient n order Bn, Cn b1, b2, d1, d2 Constant coefficient (W/mK) cp Specific heat capacity (m2/s2K) f1, f2 Arbitrary function of r h Convective coefficient of heat transfer (W/m2K) kr, kz The heat conductivity coefficients in r and z directions kr0, kz0 The conduction coefficients in r and z directions The conduction coefficients ratio k∗
Material constant coefficients Heat flux (W/m2) Independent function of r Coordinate system Inner radius (m) Outer radius (m) Temperature distribution (K) Independent function of z Density (kg/m3) Eigenfunction Separation constant
cylindrical geometry. In the following study, the Strum-Liouville theorem is used to derive the desired Fourier transformation. The Fourier eigenfunction and eigenvalue will be achieved by solving a set of equations and using cylinder's boundary conditions in z direction. Unlike most former studies, the thermal properties of the cylinder vary in two directions. The new proposed analytical solution is properly validated using the known data. Afterward, two cases which can be attributed to the actual problems are solved and temperature distribution under the effect of various parameters is discussed. Analytically satisfying the combination of the thermal boundary conditions is a difficult process which the obtained solution is capable of fulfilling it.
Several scholars motivated by the extensive use and role that functionally graded materials (FGMs) are able to fulfill and possess in comparison with composite materials. It is pointed out that controlling of thermal stresses using FGMs in structural components is possible. Analyzing FGMs under thermal loading is rather hard, therefore, only a small number of research papers concerning this matter can be found [28]. Sladek et al. [29] proposed a computational method to study the heat conduction problem for FGMs. They applied the method to a hollow cylinder in which its material properties changes exponentially. Haghighi et al. [30] developed a numerical method based on finite element method and differential quadrature approach to tackling the two-dimensional heat transfer problems of FGMs. It is shown that the method is accurate and is not cumbersome. The non-uniform convective-radiative boundary conditions are applied to the boundaries and investigated. Although numerical approaches [31,32] play an essential role in understanding the behavior of functionally graded materials, their applications are somewhat limited and results are not relatively solid. For instance, the boundary integral equation methods [29,33] which have been employed widely, need a closed form or computable notation of the partial differential equation (PDE). However, some advancements regarding this issue have been made. Gray et al. [34] further extended boundary integral analysis applications to exponentially graded materials and verified the results with exact solutions. On the other hand, the analytical solution is an attractive field of study whose outcomes are most accurate and reliable [35,36]. Utilizing analytical approaches, deep insight into heat conduction problems is practicable. Also, they have been broadly used as a verification test to numerical methods. Analytical studies regarding the FGM cylinders are quite scarce and limited studies generally lack complex thermal boundary conditions [37] or are done for specific grading patterns [38]. Tarn and Wang [39] studied heat conduction end effects in FGM circular cylinder in which inhomogeneity changes radially and derived an exact solution. Hiring matrix algebra and eigenfunction expansion they calculated the decay length. An analytical research on hyperbolic heat conduction for FGM cylinder is conducted by Babaei and Chen [40]. They solved one dimensional (radial) heat conduction problem in a long FGM cylinder. Hosseini et al. [41] presented an analytical solution for a hollow cylinder made of FGM in which its properties distributed through the thickness. The obtained solution for the axisymmetric shell is restricted to constant thermal boundary conditions. The leading objective of the present study is to acquire a two-dimensional (r, z) steady-state analytical solution for heat conduction in a finite axisymmetric thick hollow FGM cylinder. Moreover, the solution is derived under the most general inhomogeneous linear thermal boundary conditions both in radial and longitudinal directions. The central advantage of the implied general thermal boundary conditions is the capability of handling a wide variety of steady-state problems in
2. Problem description and solution method In this section, the mathematical model which is used for this solution is described. A finite length (L) thick cylindrical vessel made of functionally graded material with inner and outer radii equal to r0 and r1 is considered. The general thermal boundary conditions are considered both in radial and longitudinal directions. The material properties are different functions of radius which is because of the fact that FGM layers are graded radially. 2.1. The energy equation and general thermal boundary conditions For the analysis, a cylindrical coordinate system (r, θ, z) is implemented as illustrated in Fig. 1. The three-dimensional heat transfer in an axisymmetric FGM cylinder is as [42]:
Fig. 1. The geometry and coordinate system of the present problem and variation of materials properties from inner radius to the outer.
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International Communications in Heat and Mass Transfer 108 (2019) 104280
A. Amiri Delouei, et al.
1 T rkr r r r
1 r2
+
k
T
+
z
kz
T z
= cp
DT , Dt
The separation of variables (SOV) technique is utilized to find the exact solution. For this purpose the temperature distribution is defines as: T(r, z) = ℜ(r)ℑ(z). Using SOV method to solve Eq. (4) and applying homogeneous thermal boundary conditions, the following equations in the longitudinal, z, direction are obtained:
(1)
where T is temperature distribution. Also, cp and ρ indicate specific heat capacity and density, correspondingly. For steady-state conductive heat transfer in a finite axisymmetric FGM cylinder, the Eq. 1 can be reduced to:
1 T rkr (r ) r r r
T + k z (r ) z z
= 0,
2I (z )
z2
(2)
where kr and kz are the heat conductivity coefficients in r and z directions, respectively. It is supposed that the thermal conductivity coefficients, kr and kz, are the power functions of r as:
k r (r ) = k r 0 r m ,
(3a)
k z (r ) = k z 0 r m .
(3b)
r2
+
1 T 1 (m + 1) + r r k
2T
z2
= 0,
kr 0 . kz0
a2 T (r , L) + b2
(6b)
c1 T (r0 , z ) + d1
T (r0 , z ) = g1 (z ), r
(6c)
c2 T (r1 , z ) + d2
T (r1 , z ) = g2 (z ). r
s ( z ) f (z ) b
s (z )
F (f ) n=1
n (z ),
a2 b1 )
4 n An2
b1
(11)
n cos( n z ).
n cos( n L)
+ (a2 a1 + b1 b2
2 n)
sin(
n L)
(12)
= 0.
L
f (z ) a1 sin(
n z)
b1
n cos( n z ) dz ,
(13)
0
( a12 + (b1
F (f ) =
(6d)
+ (b1
n)
2)
sin(2
2 n ) )L
n L)
2a1 b1
+ 2a1 b1
n cos(2 n L) + 2 n
. (14)
n
2 n An
2
a2 cos(
n L)
a1 b2
n sin( n z )
f2 (r ) + f1 (r )
2 nF
(f ), (15)
2.3. Analytical solution The following relations are obtained by applying the constructed Fourier transformation to Eq. (4) and the boundary conditions in the radial, r, direction (Eqs. (6c) and (6d)): k
2U
r2
+
1 U (m + 1) r r
2 nU
=
2 n An
2
a2 cos(
n L)
a1 b2
n sin( n z )
f2 (r )
f1 (r )
(16)
(7)
where s(z) is the weighting function. Also, ϕn(z) is the eigenfunction obtained from the solution of the homogeneous equation with homogeneous thermal boundary conditions in the z direction. The inverse Fourier transformation is defined as [23]:
f (z ) =
n z)
From the definition of the Fourier transformation, the second derivative with respect to z is given by
,
a
= a1 sin(
(a12
n (z ) dz
2 n (z ) dz
(10b)
An =
Concerning the current inhomogeneous thermal boundary conditions, it is appropriate to employ Sturm-Liouville theory to get a suitable Fourier transformation for the arbitrary function f(z) [23]:
a
I (L ) = 0. z
where
(6a)
2.2. Construction of Fourier transformation
F (f ) =
a2 I (L) + b2
F (f ) =
where the inner and outer radii are r0 and r1,respectively. L is the length of cylinder. The constants a1, a2, c1 and c2 have the same dimension as the convection coefficient. The constants b1, b2, d1 and d2 have the same dimension as the conduction coefficient, as well.
b
(10a)
The weighting function is constant based on Sturm-Liouville theory and the homogeneous boundary conditions in the z direction. A suitable Fourier transformation for this problem is obtained (F) by substituting these relations into the Sturm-Liouville equation (Eq. (7)):
The general linear thermal boundary conditions with f1(r), f2(r), g1(z) and g2(z) being arbitrary functions are as follows [23,25]:
T (r , L ) = f2 (r ), z
I (0) = 0, z
(a1 b2
(5)
T (r , 0) = f1 (r ), z
(9)
By applying the boundary conditions in the z direction (Eq. (10)), the following equation in the form of trigonometric is obtained for the eigenvalues
(4)
a1 T (r , 0) + b1
= 0,
a1 I (0) + b1
n (z )
where
k =
2 I (z )
Having solved Eq. (9) and using boundary conditions introduced by Eqs. (10), the eigenfunction of this problem is obtained as
In the above-noted equations, kr0, kz0, and m are the constant coefficients. The values of k and m are defined based on the material properties of the functionally graded cylinder. Introducing Eq. (3) into Eq. (2), the following equation is found: 2T
+
c1 U (r0 , n) + d1
U (r0 , n) = G1 (n), r
(17a)
c2 U (r1 , n) + d2
U (r1 , n) = G2 (n) r
(17b)
where
U (r , n) = F (T (r , z )),
(18a)
Gi (n) = F (gi (z )) i = 1, 2. ,
(18b)
If the right-hand side of Eq. (16) is equal to ℏ(r, n), the general solution of this equation will be as
(8)
3
International Communications in Heat and Mass Transfer 108 (2019) 104280
A. Amiri Delouei, et al. m 2
U (r , n ) = r
(B I
n m 2
)
( n r ) + Cn K m ( n r ) + w (r , n), 2
conductivity ratio in z direction is considered to be 20 W/mK as [22]. The cooling rate versus 1/k⁎ for three values of convection coefficients (h = 100, 500, and 1000 W/m2K) is plotted in Fig. 2. From the figures, it is apparent that results match very well with the previous study [22], and the obtained solution using the aforementioned procedure is able to accurately predict cooling rate in various situations.
(19)
where I and K are modified Bessel functions of the first and second kinds, respectively. n
2 n
=
1 2
k
,
(20a)
3.2. Two applied cases
and w(r, n) is the non-homogeneous solution for Eq. (16). Its general form is given as r1
w (r , n) = I m ( n r ) × 2
r0
In order to examine the solution's reliability from the practical perspective, two industrial illustrative cases are assessed in this section and it is tried to mimic the real practical situation. Thus, different combinations of thermal boundary conditions ranging from convection, constant temperature to constant and varied heat flux are imposed on the examples. Steady-state heat conduction problem for an axisymmetric finite hollow cylindrical FGM vessel is investigated and temperature distribution under the impact of various parameters is discussed. As mentioned earlier, the cylindrical geometry is one of the most encountered structures in real-world applications such as heat pipe, cooling pipe, roller engine components, pressure vessel, and shaft. Two cases are described successively. The first case is a FGM hollow cylinder of inner radius r0 = 0.1 m, outer radius r1 = 0.2 m, and length L = 0.45 m. The geometry and the boundary conditions are demonstrated in Fig. 3. As can be seen, the cylinder's inner surface is experiencing constant temperature. Moreover, the outer surface of the body is partially subjected to longitudinal and radial varied heat flux along with convection current. Additionally, another illustrative industrial example with constant heat flux, convection, and the varied temperature is considered. The geometry and the boundary conditions of the second case are shown in Fig. 4. The geometrical parameters are as follows: Inner and outer radii are equal to 0.1 m and 0.3 m, respectively and the length is 0.3 m. Temperature distribution in both radial and longitudinal directions at different values of material constant, m, which can be customized regarding material features, for both cases are displayed in Figs. 5 and 6. This parameter ranges from 1 to 4 and the heat conductivity ratio (k⁎) is maintained fixed at 1. For the first case, longitudinal and radial distributions of temperature are demonstrated at r = 0.15 m and z = L/ 6, correspondingly. As expected, with an increase in values of m, temperature distribution rises in two directions. It is seen that from the inner radius to the outer, temperature increases monotonically. For the second case, temperature distribution curves in the radial and longitudinal directions are plotted at z = L/4 and r = 0.15 m, respectively (Fig. 6). Contrary to the longitudinal temperature variation, in the radial direction, the temperature reaches a virtually constant plateau as radius upturns. As it is observed in the figures, material constant strongly affects the temperature distribution in two directions. Contours
r1
r × K m ( n r ) × (r , n ) dr + K m ( n r ) × 2 2
r0
r × I m ( n r ) × (r , n) dr , 2
(20b) Lastly, by applying the boundary conditions to the inner and outer surface of the cylinder in the direction of r, the unknown coefficient will be achieved. Introducing the Eq. (19) into Eqs. (17a) and (17b), the following relations will be obtained
Bn
nm
+ Cn
nm
= F1, nm,
(21)
Bn
nm
+ Cn
nm
= F2, nm,
(22)
Where αnm, βnm, χnm, δnm, F1, nm and F2, nm are the constant coefficients. For the sake of brevity they are shown in the appendix section. The unknown coefficients of Bn and Cn can be achieved as follows
Bn =
F1, nm
nm
nm
nm
Cn =
F1, nm
nm
nm
nm
F2, nm
nm
, n = 1, 2, …,
nm nm
F2, nm
nm
, n = 1, 2, ….
nm nm
(23a) (23b)
It should be noted that for solid cylinders ( T (0, z ) ), the coefficient Cn r should be set to zero. At the end, by means of the inverse Fourier transformation (Eq. (8)) to Eq. (19), the temperature distribution will be achieved as: T (r , z ) = n=1
U (r , n ) × n (z ) m
= n=1
r 2 Bn I m ( n r ) + Cn K m ( n r ) + w (r , n) × (a1 sin( n z ) 2
2
b1 n cos( n z )).
(24) 3. Results and discussion First and foremost, in this section, the solution's validity will be checked with those of Bahadur and Bar-Cohen [22]. Afterward, two industrial illustrative cases with considering a combination of thermal boundary conditions are discussed. Using the current solution and further analyzing the impact of material constants on the distribution of temperature will provide the engineers with a better understanding of the heat exchange in FGMs. 3.1. Verification test The correctness of the presented exact analytical solutions with general inhomogeneous thermal boundary condition is investigated in this sub-section. Because the same studies are few, to verify the ongoing problem it is required to simplify the current solution to the published results. To do so, the material properties, boundary conditions, and geometry (i.e., r = 0.45 cm and L = 5 cm) are considered as those of Bahadur and Bar-Cohen [22]. In reference [22] an analytical solution for orthotropic cylindrical pin fin is obtained and in particular, watercooled fins are studied. To compare, the value of m is set to zero to fulfill the orthotropic thermal conductivity situation. Also, the
Fig. 2. Comparison of cooling rate variations with conductivity ratio at three values of convection coefficients.
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International Communications in Heat and Mass Transfer 108 (2019) 104280
A. Amiri Delouei, et al.
Fig. 3. Schematic illustration of the geometry and boundary conditions of the first case.
of temperature distribution in r and θ directions at k∗ = 1 for case 1 and 2 are illustrated in Figs. 7 and 8 to make a clear observation of the influence of material constant (m). In order for further illustration, Table 1 shows temperature distribution in radial direction for both case 1 and 2. As can be found out, as for case 1, with an increase in the values of m, temperature increases.
To have a more detailed assessment of the radial temperature distribution at different longitudinal locations, four sections from L/6 to L in z direction for both cases are considered and depicted in Figs. 9 and 10. For cases 1 and 2, the temperature variation is approximately the same from L/6 to L/2 and afterward at the end of the cylinder temperature slightly drops. For the second case as it is illustrated in Fig. 10,
Fig. 4. Schematic illustration of the geometry and boundary conditions of the second case.
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International Communications in Heat and Mass Transfer 108 (2019) 104280
A. Amiri Delouei, et al.
Fig. 7. Contours of temperature distribution in r and θ directions at k∗ = 1 and different values of “m” ranging from 1 to 4 for case 1.
Fig. 5. Temperature distribution in a) longitudinal (r = 0.15 m) and b) radial (z = L/6) direction at different values of “m” ranging from 1 to 4 at k∗ = 1for the first case.
Fig. 8. Contours of temperature distribution in r and θ directions at k∗ = 1 and different values of “m” ranging from 1 to 4 for case 2. Table 1 Temperature distribution in radial direction at different values of “m” ranging from 1 to 4 at “k∗ ” = 1 for the first case (z = L/6) and the second case (z = L/ 4). Case 1
Case 2
r
m=1
m=2
m=3
m=4
r
m=1
m=2
m=3
m=4
0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2
300 312 322 330 338 345 352 358 364 370 376
300 317 330 340 348 356 362 367 373 377 382
300 323 338 349 358 365 371 376 380 384 387
300 328 346 358 367 373 378 382 385 388 391
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
455 415 388 369 354 344 335 329 324 321 318
455 400 369 349 336 328 322 318 316 314 314
455 384 350 331 320 313 310 307 306 306 305
455 368 332 314 306 301 299 298 297 296 295
Fig. 6. Temperature distribution in a) longitudinal (r = 0.15 m) and b) radial (z = L/4) direction at different values of “m” ranging from 1 to 4 at k∗ = 1 for the second case. Fig. 9. Radial distribution of temperature at different longitudinal locations at k∗ = 1 for case 1.
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Fig. 10. Radial distribution of temperature at different longitudinal locations at k∗ = 1 for case 2.
Fig. 13. Temperature distribution in a) longitudinal (r = 0.2 m) and b) radial (z = L/6) direction at different values of “k∗” at m = 1 for the first case.
Fig. 11. Longitudinal distribution of temperature at different radial sections at k∗ = 1 for case 1.
Fig. 12. Longitudinal distribution of temperature at different radial sections at k∗ = 1 for case 2.
it can be concluded that all cases reach a constant value at the same radial location. To see the similar results in various radial sections through z, refer to Figs. 11 and 12. With the growth of radius from the inner wall to the outer wall, the gap among different temperature variations steadily narrows for both cases. To further study the temperature distribution, here, the effect of heat conductivity ratio (k∗= kr0/kz0) for both cases are studied (Figs. 13 to 16). Four different values of k∗(=0.3, 0.6, 1, 5) are considered while the value of m is kept the same (=1). It can be seen from figures that for both case 1 and case 2, in the same circumstances but different values of k∗, the temperature distribution is different. Fig. 13 is concerned with the effect of heat conductivity ratio on temperature distribution in two
Fig. 14. Temperature distribution in a) longitudinal (r = 0.15 m) and b) radial (z = L/4) direction at different values of “k∗” at m = 1 for the second case.
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Similarly, for the second case, Fig. 14a and b demonstrate longitudinal (at r = 0.15 m) and radial (at z = L/4) temperature distribution, respectively. It is evident from Fig. 14b that even with the growth of heat conductivity, the temperature distribution is reduced, which demonstrates the importance of thermal environments. Also, in this case, the gap among different lines marginally increases. For cases 1 and 2, contours of temperature distribution in two directions are shown in Figs. 15 and 16. One can conclude that even in the small values of k∗, the temperature distribution in either direction changes noticeably. As known, a change in material properties can considerably alter the patterns of temperature distribution in the structure. k* is one of the important factors which contributes to this matter. As in Figs. 15 and 16, in relatively small values of k* (less than 1), the variations are noticeable. In other words, has a tremendous impact on the rate of cooling. On the other hand, for larger values of m (larger than 1), the variations can be ignored. From the figures, all lines of temperature distribution for both cases are reasonable and properly satisfy imposed thermal boundary conditions. Further, the variations of temperature distribution in two directions are very beneficial in designing flexible and favorable FGMs. In Tables 2 and 3, for case 1 and 2, temperature distribution versus z at different values of m, radius, and k∗, are shown. It is seen that for both of which with a rise in z, temperature drops.
Fig. 15. Contours of temperature distribution in r and θ directions at m = 1 for case1.
4. Conclusions Two-dimensional heat transfer in FGMs plays an important role in designing and optimization of structures for multi-functional areas relevant to various practical applications. This paper analytically tackled the heat conduction problem in a finite length FGM cylindrical vessel under general boundary condition. Having employed inhomogeneous thermal boundary conditions, the Sturm-Liouville theory is used to acquire the analytical solution for two-dimensional heat conduction. At first, the solution is validated with published data, and then two practical cases with various mixtures of boundary conditions are solved. Temperature distribution under the influence of various parameters at different fragments of the FGM cylinder is reported. The steady-state exact analytical solution that presented in this paper is believed to be beneficial for the design of FGM vessels for different operating conditions.
Fig. 16. Contours of temperature distribution in r and θ directions at m = 1 for case2.
directions for the first case. Fig. 13a shows the longitudinal temperature distribution at r = 0.15 m and Fig. 13b is responsible for radial temperature distribution at z = L/4. As can be found out, an increase in the k∗ in r and z directions makes the heat flux transfer and convection more effective. The distribution of temperature is substantially affected by the larger values of k∗ and also higher temperature is seen close to the outer surface.
Table 2 Temperature distribution in longitudinal direction at different values of “m” (r = 0.15 m), radial sections (k∗ = 1) and k∗ (r = 0.2 m) for the first case. z
m=1
m=2
m=3
m=4
(r₀ + r₁)/3
(r₀ + r₁)/2
(r₀ + r₁)/1.5
k* = 0.3
k* = 0.6
k* = 1
0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275 0.3 0.325 0.35 0.375 0.4 0.425 0.45
345 345 345 343 342 340 338 335 332 328 324 320 315 310 304 298 292 285 278
356 355 355 354 352 350 348 345 342 338 334 329 324 319 313 307 301 294 286
365 365 364 363 361 359 357 354 351 347 343 338 333 327 322 315 309 301 294
373 373 372 371 370 367 365 362 358 355 350 346 340 335 329 322 315 308 301
300 301 300 299 298 296 294 292 289 286 282 279 274 270 265 260 254 249 242
345 345 345 343 342 340 338 335 332 328 324 320 315 310 304 298 292 285 278
376 376 375 374 372 370 367 364 361 357 353 348 343 337 331 325 318 310 303
359 359 358 357 355 353 351 348 345 341 337 332 327 322 316 310 303 296 289
371 371 370 369 367 365 363 360 356 352 348 343 338 333 327 320 314 306 299
376 376 375 374 372 370 367 364 361 357 353 348 343 337 331 325 318 310 303
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Table 3 Temperature distribution in longitudinal direction at different values of “m” (r = 0.15 m), radial sections (k∗ = 1) and k∗ (r = 0.15 m) for the second case. z
m=1
m=2
m=3
m=4
(r₀ + r₁)/2
2(r₀ + r₁)/3
3(r₀ + r₁)/4
k* = 0.3
k* = 0.6
k* = 1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
377 377 376 375 373 371 368 365 361 357 353 348 342 336 330 323
391 389 388 387 385 383 381 378 374 371 366 362 357 351 345 339
402 402 401 400 399 396 394 391 387 384 380 376 371 366 360 354
410 410 409 408 407 405 403 400 397 394 390 386 381 377 371 366
338 338 338 336 335 333 331 328 324 321 317 312 307 302 296 290
313 313 312 311 310 308 306 303 300 297 293 289 284 279 274 268
306 306 305 304 303 301 299 296 293 290 286 282 278 273 268 262
356 355 355 354 352 350 347 344 341 337 333 328 323 317 311 305
368 367 367 365 364 362 359 356 352 348 344 339 334 328 322 315
377 377 376 375 373 371 368 365 361 357 353 348 342 336 330 323
Appendix A. Appendix nm
nm
nm nm
= c1 r0
m 2 Im 2
( n r0 )
=
m c1 r0 2 K m 2
=
m c2 r1 2 I m 2
=
m c2 r1 2 K m 2
( n r0)
( n r1) ( n r1)
d1 n r0
m 2 I m + 1 ( n r0 ), 2
(A1)
m d1 n r0 2 Kn + 1 ( n r0),
(A2)
m d2 n r1 2 In + 1 ( n r1),
(A3)
m d2 n r1 2 Kn + 1 ( n r1),
(A4)
F1, nm = G1 (n)
c1 w (r0 , n)
d1
w (r0 , n) , r
(A5)
F2, nm = G2 (n)
c2 w (r1 , n)
d2
w (r1 , n) . r
(A6)
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