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Thermal stresses in radiant tubes due to axial, circumferential and radial temperature distributions
Applied Thermal Engineering 29 (2009) 1913–1920
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Thermal stresses in radiant tubes due to axial, circumferential and radial temperature distributions Mohammad A. Irfan a,*, Walter Chapman b,1 a b
Department of Mechanical Engineering, NWFP University of Engineering & Technology (NWFP UET), Peshawar, NWFP, Pakistan Design Engineer, North American Manufacturing Co., 4455 East 71st Street, Cleveland, OH 44105-5600, USA
a r t i c l e
i n f o
Article history: Received 21 September 2007 Accepted 26 August 2008 Available online 11 September 2008 Keywords: Thermal stresses Radiant tubes Temperature gradients
a b s t r a c t This paper presents an analysis of the thermal stresses in radiant tubes. The analytical analysis is verified using a finite element model. It was found that axial temperature gradients are not a source of thermal stresses as long as the temperature distribution is linear. Spikes in the axial temperature gradient are a source of high thermal stresses. Symmetric circumferential gradients generate thermal stresses, which are low as compared to the stress rupture value of radiant tubes. Radial temperature gradients create bi-axial stresses and can be a major source of thermal stress in radiant tubes. A local hot spot generates stresses, which can lead to failure of the tube. Ó 2009 Published by Elsevier Ltd.
1. Introduction Radiant tubes are commonly used in many industrial applications where isolation of the workload from the combustion environment is required. Melting, sintering and heat-treating are few of the applications where heat is transferred to the workload by radiation from the radiant tube. The radiant tube is internally heated with a flame produced by a burner. Service failures by cracking (stress rupture) of the tube have been the major hurdle in the life of radiant tubes. Failure of radiant tubes during services leads to shutdown of the furnace, leading to downtimes and huge losses. There is an increasing interest to investigate into the causes of failure of radiant tubes and hence suggest remedies providing longer service life of these tubes. There has been some previous work on failure probabilities of ceramic radiant tubes [1] (Segall et al., 1990) and development of radiant tube heating system for high temperature [2] (Nakagawa et al.). Analytical and Computation analysis of transient thermal stresses in elastic-plastic tubes was studied by various authors, Candella [3], Eraslan and Orcan [4,5]. However, there is a need for a comprehensive treatment on the causes of radiant tube failures. This paper takes an in-depth theoretical look at the thermal stresses generated due to axial, circumferential and radial temperature gradients. The results from mathematical models are further verified using Finite Element Analysis (F.E.A.). The results are valuable for analyzing and preventing radiant tube failures, and will help the industry in minimizing their losses. * Corresponding author. Tel.: +92 91 9216499; fax: +92 91 9216663. E-mail address: [email protected] (M.A. Irfan). 1 Tel.:+1 216 271 6000; fax: +1 216 641 7852. 1359-4311/$ - see front matter Ó 2009 Published by Elsevier Ltd. doi:10.1016/j.applthermaleng.2008.08.021
The outline of the paper is as follows. Section 2 presents a discussion on thermal stresses due to axial temperature distributions. Thermal stresses due to circumferential temperature gradients are discussed in Section 3. Stresses due to radial temperature gradients and a discussion of end effects are presented in Section 4. Section 5 looks into a combined radial and circumferential loading. A discussion on localized heating and hot spots is given in Section 6. Section 7 takes into consideration the effect of bends in the tube geometry. The paper is concluded in Section 8. 2. Thermal stresses due to axial temperature distributions Although Sauer [6] has developed some simple formulae for analysis of thermal stresses due to axial temperature distributions, Kent [7] has a more complete treatment, which has been employed in the following mathematical analysis. Consider a thin cylindrical shell of elastic material of mean radius c and of length L. Let the temperature rise be constant over any cross-section, but vary with the axial coordinate in a manner than can be expressed in the form of the following power series.
Tx ¼
1 x x2 x3 A þ B þ C 2 þ D 3 þ ...: 2 L L L
ð1Þ
where A, B, C, D, are constants. Substituting Eq. (1) into equilibrium equations [7], the stresses can be represented by:
Eac C 3D g 2 ðk 1Þ 3 ðbx þ wÞ 2 1l L bL Ea 3D bLC C ¼ h þ 2 w þ lrx 2 b2 L2 bL
rx ¼
ð2Þ
rt
ð3Þ
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M.A. Irfan, W. Chapman / Applied Thermal Engineering 29 (2009) 1913–1920
Nomenclature K L
a, b inner and outer radii of the cylinder radial distance ar c mean radius of cylindrical shell h thickness of cylindrical shell major and minor axis of an ellipse lb, la x length coordinate in axial direction A, B, C, D constants of Power Series coefficients of Fourier cosine series Ai, A0i E modulus of Elasticity I moment of Inertia of strip of unit width and depth
a g r, h k,w
l rt rx /
where,
b¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 3ð1 l Þ
ð4Þ
2
h c2
It is interesting to note that, no stresses result from temperature gradients in axial direction unless Eq. 1 includes at least the second power of x. This is evident from the fact that the equations representing the stresses have the coefficients C and D (coefficients A and B are not present). Lee [8] has developed a solution for thermal stresses of a thick walled cylinder under an arbitrary axial temperature gradient T(x). The solution satisfies the field equations of thermo elasticity and the boundary conditions (traction free) at the curved surfaces, but in general not at the end surfaces. The expressions for the radial and hoop stresses are reproduced below.
ð1 mÞ rr ¼ 2aE
(
mT 00
"
2
2
r 2 ða2 þ b Þ þ
16ð1 þ mÞ
a2 b r2
#)
þ higher order terms
ð1 mÞ rt ¼ 2aE
(
mT 00
16ð1 þ mÞ
ð5Þ
"
2
2
3r2 ða2 þ b Þ þ
þ higher order terms
a2 b r2
#)
ð6Þ
Yang and Lee [9] present a series solution for a thick walled cylinder subjected to a temperature distribution which varies both radially and axially. A typical thermal gradient as experimentally found in radiant tubes [10] was applied to Kent’s model [7] depicted by Eqs. 2 and 3, and Lee’s model [8] as given by Eqs. 5 and 6. Both the models gave similar results showing negligible stresses due to axial thermal gradients present in Radiant Tubes. This is probably due to the absence of modeling bends and end conditions in the above equations, which are further investigated in Section 7. To further understand the deformation shapes and resulting stresses due to linear and non-linear (such as sudden rises in) temperature gradients a simple axi-symmetric analysis was carried out in ANSYS. A linear temperature gradient was applied to a tube. The temperature was varied form 24 °C (75°F) to 135 °C (275°F) in linear fashion. The stress field due to the linear temperature gradient was found to be essentially zero as discussed earlier. Fig. 1 shows a simulation of the spike in temperature profile of recuperative burners [10]. The spike is 24 °C (75 °F) higher than its neighboring sections, whereas the rest of the tube the temperature gradient is applied in 7 °C (20 °F) step rises. Fig. 2 shows the axial stress in the tube due to a spike in the temperature gradient. It can be seen that appreciable tensile stresses of the order of 17.24 MPa (2.5 kpsi) are created which are enough to cause the failure of the tube by stress rupture. Note that the creep stress for Super 22HÒ [11] at
modulus of foundation = Eh/c2 length of cylindrical shell coefficient of thermal expansion distance from neutral axis polar coordinates function of x Poisson’s ratio tangential (hoop) stress axial (longitudinal) stress Airy’s stress function
1800 °F is 22 MPa (3.2 kpsi) and the stress for rupture is 2 MPa (2.9 kpsi) for 14 months. When the rate or degree of deformation is the limiting factor, the design stress is based on creep stress. When fracture is the limiting factor, stress to rupture values should be used in design. The above analysis reiterates the fact that linear temperatures gradients do not create any stresses whereas spikes (non-linear temperature rise) in the temperature profile can be detrimental to tube life. The extreme case of sudden rise in temperature of a small section is discussed in Section 6 as localized heating. 3. Thermal stresses due to circumferential temperature distributions From experimental measurements, radiant tubes have been shown to have a circumferential temperature gradient. The difference between the maximum and minimum temperature is around 9 °C (15 °F). It is worth investigating to know whether this difference would create a notable stress in the tube. It is known that there are no stresses generated if a cylinder is uniformly heated to a high temperature. It just expands in the axial and radial direction. The question is what happens if there is a temperature gradient varying with h (but not varying with radius). First an analytical approach is taken. This is a case of an asymmetrical temperature distribution, i.e. the temperature distribution varies as a function of h – the angular position. Two general cases are presented here. First case deals with the temperature fields, which do not result in the creation of thermal stresses. Second case deals with symmetric temperature fields, which can lead to thermal stresses. Using the Airy’s Stress function /, the compatibility equation [7] can be written as,
r4 / þ Ear2 T ¼ 0
ð7Þ
Where / accounts for the non-temperature dependent stresses (e.g. due to external forces), a is the coefficient of linear expansion, and T is the Temperature field. The necessary condition for thermal stresses to be zero is that the Laplacian of Temperature field should be zero,
r2 T ¼ 0
ð8Þ
o2 T 1 oT 1 o2 T þ ¼0 þ or 2 r or r 2 oh2
ð9Þ
where r and h are the polar coordinates. So temperature fields which have Laplacian (r2T) as zero will not generate any thermal stresses. A typical example is
Tðr; hÞ ¼ T 0
r cosðhÞ r0
ð10Þ
M.A. Irfan, W. Chapman / Applied Thermal Engineering 29 (2009) 1913–1920
1915
Fig. 1. A linear temperature gradient ranging form 24 C (75 F) to 135 C (275 F) along the tube with a spike (sudden non-linear rise) in temperature at one section. The spike is 42 C (75 F) higher than the neighboring temperature.
Fig. 2. Axial stress due to the spike (sudden non-linear rise) in temperature at one section. It can clearly be seen that a spike can be a considerable source of stress 16 MPa (2367 psi) whereas linear temperature distributions do not create any stress.
The second case deals with a particular case of symmetric temperature distribution, which results in the generation of thermal stresses. The circumferential temperature distribution in radiant tubes can closely be approximated to be symmetrical about the vertical plane and thus be an even function of h. In general we can represent the circumferential temperature distribution by an even Fourier series.
First a mathematical model was built using Goodier’s formulation [12]. Consider thick walled cylinder where the temperature varies around the circumference and through the thickness, but not along the length. Let h be the angular coordinate. Then the inside T1 and outside T2 temperatures are functions of h only and can be developed in Fourier series (leaving the odd functions of sinh),
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M.A. Irfan, W. Chapman / Applied Thermal Engineering 29 (2009) 1913–1920
T 1 ðhÞ ¼ Ao þ A1 cosðhÞ þ A2 cosð2hÞ þ . . .
ð11Þ
T 2 ðhÞ ¼ A0o þ A01 cosðhÞ þ A02 cosð2hÞ þ . . .
ð12Þ
Timoshenko [13] has further shown that the higher harmonics (cos2h etc.) do not contribute to displacements and stresses. We can clearly see that the Laplacian (r2T) is not equal to zero. Thus the above symmetric circumferential temperature field will generate thermal stresses. The hoop stresses are given as [12],
! 2 2 a2 b a2 þ b rh ¼ jr cos h 4 þ 2 3 r r 2 2 Ea A1 A01 a b where j ¼ þ 2ð1 mÞ a b b4 a4
ð13Þ ð14Þ
where a, b are the inner and outer radii of the cylinder. A computer program was written based on the above equations and the results were compared with the F.E.A. model. Circumferential temperature gradients were applied to a section with a difference of 330 °C (625 F) between the maximum and minimum temperatures. The hottest spot was assumed to be at the bottom and the coolest spot was assumed to be at the top of the section under consideration. The resulting hoop stress field at the outer shell shown in Fig. 3. The results were found consistent with the mathematical analysis carried out using the formulae given earlier. It takes almost a 330 °C (625 °F) circumferential temperature difference to create stresses of the order of 20 MPa (3 kpsi). In reality the circumferential temperature difference in radiant tubes is not more than 15 °C (60 °F). It is interesting to recall that from previous experiments the difference in the maximum and minimum temperature measured across the circumference was never more than 9 °C (15 F). Taking into account the error of measurement of K type thermocouples (+/ 13 C, +/ 24 F) the maximum temperature difference could have been 17 C (63 F). This confirms that circumferential temperature gradients are not a major source of thermal stress. It is also interesting to note that as the thickness of the tube is reduced, there is a corresponding decrease in the magnitude of
thermal stress generated. Fig. 4 shows an extreme case where the thickness of the tube was reduced to 0.254 mm (0.01 inch) as opposed to 8.5 mm (0.334 in.) resulting in negligible thermal stresses. It is worth mentioning that secondary thermal stresses can also exist due to T00 (h). 4. Radial temperature gradients and end effects Having discussed axial and circumferential temperature gradients and their effects on thermal stress, next the thermal stresses created due to radial temperature gradients are evaluated. A mathematical model was built using the formulation of Roark [14].
rh ¼
2r 2i ro 1 ln 2 2 ro ri r r o 2ð1 tÞ ln i
DT aE
ð15Þ
ri
rh ¼ raxial at the outer radius
ð16Þ
It was found out that a temperature difference of 9 °C (15 °F) would generate tensile longitudinal and hoop stresses of the order of 10.34 MPa (1.5 kpsi) at the outer radius. The outer skin of the tube is under bi-axial tension. Primary cracking would occur in a direction perpendicular to maximum principal stress (which would be the longitudinal stress and the bending stress) and secondary cracking could occur in the direction of hoop stress. It is interesting to note that the stress for rupture is 20 MPa (2.9 kpsi) for 14 months at 982 C (1800 F) for the tube material Super 22 H. Similar conclusions were mentioned by Segall et al. [1], who argued that the stress field is tri-axial, however the radial stresses can be neglected as they are small and compressive in nature and do not contribute to crack propagation. It is worthwhile mentioning here that the mathematical formulation is true only for infinitely long cylinders and for regions which are away from the ends of the cylinder. This does raise a concern over what happens at the ends. There are two cases to consider in the case of radial temperature gradients. First is the case
Fig. 3. Hoop stress in the outer shell due to a cosine temperature distribution around the circumference of the tube. The hottest potion of the tube is under tension of about 20 MPa (2880 psi).
M.A. Irfan, W. Chapman / Applied Thermal Engineering 29 (2009) 1913–1920
1917
Fig. 4. Hoop stress in the outer shell due to a cosine temperature distribution around the circumference of the tube. The thickness of tube has been reduced to 0.254 mm (0.01 in.) as opposed to 8.5 mm (0.334 in.) as presented in Fig. 3. Hoop stresses can be reduced dramatically to 0.861 MPa (124.909 psi) by reducing the thickness of the tube.
for the free end. For thin walled tubes where the hoop stress away from the ends can be approximated as,
rh ¼
aEDT 2ð1 mÞ
ð17Þ
It has been shown [12] that the tangential (hoop) stress at the free end can be represented by,
aEDT rh ¼ 2ð1 mÞ
! pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m2 pffiffiffi m þ 1 3
ð18Þ
6. Localized heating and hot spots
Assuming m = 0.3, the above expression reduces to,
rh ¼ 1:25
aEDT 2ð1 mÞ
The combined effect of radial and circumferential temperature gradients was also investigated. The value of tensile stress due to a radial temperature difference of 9 °C (15 °F) is added to the value of tensile stress due to circumferential temperature gradient. The superimposed tensile stresses can be detrimental to the life of a radiant tube. It is interesting to note that there is a cross-section where the combined stress goes to zero. On a similar note the combined effects of radial and axial gradients can be quite detrimental to the life of radiant tubes.
ð19Þ
which says that the hoop stress at the ends is 25% more than the stress calculated for an infinite cylinder. The increase in stress comes from the bending moment at the free ends due to the outward deflection of the tube. On the other hand if the deflection of the end is prevented by rigid supports, the tangential stress at the end can be reduced by approximately 43% as compared to the stress in an infinite cylinder. It might be noted that the above analysis is for steady state conditions. Transient conditions can generate more severe stresses. 5. Radial and circumferential temperature fields In practice radial and circumferential temperature gradients exist simultaneously and the stresses created due to each of those have to be superimposed [12]. This basically means superimposing the results of Sections 3 and 4. Recall in Section 3 it was shown that there would be a maximum tensile stress at the outer shell corresponding to the region of highest temperature. Section 4 illustrated that for the case of a temperature difference of 9 °C (15 °F) between the inside and outside of the cylinder (the inside being hotter) a tensile stress of 10.34 MPa (1.5 kpsi) is created on the outer boundary.
A leading cause of failure of radiant tubes is localized heating and creation of hot spots. In this section a few different mathematical formulations are discussed, each dealing with a different shape of the hot spot. Also a Finite Element Analysis using commercial software is presented. The first formulation deals with the heating of a small arc. If it is assumed that the shell is heated uniformly over a small arc of circumferential length k along the length of its outer surface the temperature distribution may be approximated by T2, constant over k, with the rest of the outer surface and the whole inner surface kept constant at T1. It may be shown that the maximum circumferential stress is given by the following [7].
rh ¼
3 ðT 2 T 1 Þ k 4 ð1 mÞ R
ð20Þ
A program was written to evaluate stresses due to different hot spot temperature gradients. It was found out that a temperature difference of 10 C (50 F) should lead to a stress of about 27.6 MPa (4 kpsi). The mathematical formulation is only true for very small circumferential elements. Goodier [12] expresses the stresses due to a small circular spot at temperature T1, in a plate with temperature T0, as
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M.A. Irfan, W. Chapman / Applied Thermal Engineering 29 (2009) 1913–1920
1 2
rr ¼ rh ¼ EaðT 1 T 0 Þ inside the hot spot
ð21Þ
9 a2 ¼ 12 EaðT 1 T 0 Þ r2r = outside the hot spot a2 ¼ 1 EaðT T Þ r ;
ð22Þ
rr rh
2
1
0 r2
The above formulation leads to a compressive stress of 26.2 MPa (3.8 kpsi) inside the hot spot and a tensile stress of 26.2 MPa (3.8 kpsi) at the outer boundary of the hot spot. It is interesting to note that for a hot spot of any shape, the greatest stress depends on the shape but not on the size. For an elliptical hot spot the hoop stress is represented by
rh ¼ EaðT 1 T 0 Þ= 1 þ
lb la
ð23Þ
For a (lb/la) (major axis/minor axis) ratio of 2 the hoop stress comes out to be 17.24 MPa (2.5 kpsi.) Fig. 5 shows the simulation of a hot spot in F.E.A. The hot spot was kept 10 °C (50 °F) higher than the surrounding material. The hot spot expands as a ‘‘blister” on the surface of the tube. Looking at the outer shell, the hot spot is under compression in the axial direction and is under tension in the hoop direction. The maximum hoop stress is of the order of 13.79 MPa (2 kpsi) as shown in Fig. 6, which is comparable to hoop stress of the elliptical hot spot model. A more comprehensive mathematical analysis has been developed by Segal and Hellmann [15]. It is interesting to note the pitfall in applying such relationships in FEA, because of the method of interpolation will significantly affect the higher order differentials of temperature gradient. Hence it is hard to accurately predict the stresses in hot spots using the above FEA analysis. 7. Effect of bends in radiant tube An F.E.A. model of the radiant W-tube was built in ANSYS. To study the exclusive effect of temperature field, the effect of gravity
as well as the effect of internal and external support was neglected. A linear temperature gradient was applied across a W-tube. The tube grows outward in the axial direction. As the bend grows, it extends down as well as in the axial direction. The straight portion of the tube was found to be stress free, however stresses of the order of 0.7 MPa (0.1 kpsi) to 1.4 MPa (0.2 kpsi) appear around the bends. The effect can be clearly illustrated by looking at the Von-Mises (equivalent) stress as shown in Fig. 7. It might be worthwhile mentioning two possible explanations for this behavior around the bends. First the mathematical argument of linear temperature gradients producing no stress as discussed in Section 2 is true for straight tubes only. Second, temperature is applied to the F.E.A. model at the nodes. Each set of nodes across the circumference is assigned a particular temperature and a step jump of temperature is given to the next set of nodes. The distance between the nodes at the top and bottom of the tube leg is the same across the straight portion of the tube, and hence a truly uniform temperature gradient can be applied. However at the bends, the nodes are situated much closer together at the inner radius than at the outer radius. Thus a truly linear temperature gradient does not exist at the bends.
8. Conclusions 1. It was shown that linear temperature gradients in the axial direction do not produce thermal stresses in the straight portion of the tube. However a spike in the temperature gradient 24 °C (75 °F) higher than the neighboring temperature field can cause thermal stresses of the order of 17.25 MPa (2.5 kpsi). 2. Circumferential temperature gradients are not a major source of thermal stress. It would require a temperature difference of almost 329 °C (625 °F), varying as a cosine distribution across the circumference to create stresses of the order of 20 MPa (3 kpsi.)
Fig. 5. Simulation of a hot spot. The tube has a temperature range of 24 C (75 F) to 52 C (125 F). The temperature of the hot spot is 28 C (50 F) higher than the uniform surrounding temperature.
M.A. Irfan, W. Chapman / Applied Thermal Engineering 29 (2009) 1913–1920
1919
Fig. 6. Hoop stress field in the outer shell due to the hot spot. Note that the expanding ‘‘blister” creates a tensile hoop stress field with a maximum stress of 13.4 MPa (1946 psi) on the outer shell of the tube.
Fig. 7. Von Mises (equivalent) stress in the bends due to a linear temperature gradient. The maximum stress is about 1.8 MPa (263 psi.)
3. The most critical temperature gradient is the radial temperature gradient. A temperature difference of 11 °C (20 °F) between the inner and outer radius of the tube can create a bi-axial tensile stress of about 13.79 MPa (2 kpsi) at the outer radius of the tube.
4. A combination of radial and circumferential temperature gradient adds up the tensile stresses at the outer radius. This increases the propensity of failure initiation at the outer radius of the tube.
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M.A. Irfan, W. Chapman / Applied Thermal Engineering 29 (2009) 1913–1920
5. Hot spots are a major source of thermal stress. A hot spot with temperature 10 °C (50 °F) higher than its surrounding material can easily generate a ‘‘blister” with a tensile hoop stress of the order of 13.79 MPa (2 kpsi.) 6. The bends carry a higher stress level than the straight portion of the tube.
References [1] A.E. Segall, J.R. Hellmann, P. Strzepa, Experimental and analytical evaluation of the mechanical performance of a gas-fired ceramic radiant tube at steadystate, J. Tes. Eval. 18 (4) (1990) 250–255. [2] T. Nakagawa, M. Obashi, K. Nuta, T. Naito, Y. Abe, N. Ohmori, Development of radiant tube heating system for high-temperature and for high-combustion load, Mizushima Works, Kawasaki Steel Corporation and Otto Corporation, 1996. [3] A. Cardella, Analytical methodology and boundary problem for computing temperature and thermal stresses in tubes, Heat Technol. (Pisa, Italy) 20 (1) (2002) 61–67. [4] Y. Orcan, A.N. Eraslan, Thermal stresses in elastic–plastic tubes with temperature-dependent mechanical and thermal properties, J. Thermal Stresses 24 (11) (2001) 1097–1113.
[5] A.N. Eraslan, Y. Orcan, Computation of transient thermal stresses in elasticplastic tubes: effect of coupling and temperature-dependent physical properties, J. Thermal Stresses 25 (6) (2002) 559–572 (14). [6] G. Sauer, Simple formulae for the approximate computation of axial stresses in pipes due to thermal stratification, Int. J. Press. Vessels Piping 69 (1996) 213– 223. [7] C.H. Kent, F. Ark, Thermal stress in thin-walled cylinders, Trans. of ASME, APM53-13, (1931) 167–180. [8] C.W. Lee, Thermoelastic stresses in thick-walled cylinders under axial temperature gradient, Trans. ASME Ser. E 88 (2) (1996) 467–469. [9] K.W. Yang, C.W. Lee, Thermal Stresses in thick-walled circular cylinders under axisymmetric temperature distribution, ASME J. Eng. Ind. (1971) 969–975. [10] D. Quinn, Industrial radiant tube temperature uniformity and its effects on tube life and production, Technical Report, North American Manufacturing Co., Cleveland OH, 1998. [11] Product Data Sheet for Super 22HÒ Radiant Tube Material, Duraloy Technologies, Scottdale, PA, 2006, http://www.duraloy.com. [12] J.N. Goodier, Thermal stresses and deformation, J. Appl. Mech. 24 (3) (1957) 467–474. [13] S. Timoshenko, Strength of Materials Part II: Advanced Theory and Problems, Robert E. Kreiger Publishing Co., NY, 1976. [14] W.C. Young, Roark’s Formulas for Stress and Strain, 6th ed., McGraw Hill, 1989. [15] A.E. Segall, J.R. Hellmann, Analysis of Gas-Fired Ceramic Radiant Tubes During Transient Heating: II - Thermoelastic Stress Analysis, ASTM J. Test. Eval. 20 (1) (1992) 25–32.
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Thermal stresses in radiant tubes due to axial, circumferential and radial temperature distributions Mohammad A. Irfan a,*, Walter Chapman b,1 a b
Department of Mechanical Engineering, NWFP University of Engineering & Technology (NWFP UET), Peshawar, NWFP, Pakistan Design Engineer, North American Manufacturing Co., 4455 East 71st Street, Cleveland, OH 44105-5600, USA
a r t i c l e
i n f o
Article history: Received 21 September 2007 Accepted 26 August 2008 Available online 11 September 2008 Keywords: Thermal stresses Radiant tubes Temperature gradients
a b s t r a c t This paper presents an analysis of the thermal stresses in radiant tubes. The analytical analysis is verified using a finite element model. It was found that axial temperature gradients are not a source of thermal stresses as long as the temperature distribution is linear. Spikes in the axial temperature gradient are a source of high thermal stresses. Symmetric circumferential gradients generate thermal stresses, which are low as compared to the stress rupture value of radiant tubes. Radial temperature gradients create bi-axial stresses and can be a major source of thermal stress in radiant tubes. A local hot spot generates stresses, which can lead to failure of the tube. Ó 2009 Published by Elsevier Ltd.
1. Introduction Radiant tubes are commonly used in many industrial applications where isolation of the workload from the combustion environment is required. Melting, sintering and heat-treating are few of the applications where heat is transferred to the workload by radiation from the radiant tube. The radiant tube is internally heated with a flame produced by a burner. Service failures by cracking (stress rupture) of the tube have been the major hurdle in the life of radiant tubes. Failure of radiant tubes during services leads to shutdown of the furnace, leading to downtimes and huge losses. There is an increasing interest to investigate into the causes of failure of radiant tubes and hence suggest remedies providing longer service life of these tubes. There has been some previous work on failure probabilities of ceramic radiant tubes [1] (Segall et al., 1990) and development of radiant tube heating system for high temperature [2] (Nakagawa et al.). Analytical and Computation analysis of transient thermal stresses in elastic-plastic tubes was studied by various authors, Candella [3], Eraslan and Orcan [4,5]. However, there is a need for a comprehensive treatment on the causes of radiant tube failures. This paper takes an in-depth theoretical look at the thermal stresses generated due to axial, circumferential and radial temperature gradients. The results from mathematical models are further verified using Finite Element Analysis (F.E.A.). The results are valuable for analyzing and preventing radiant tube failures, and will help the industry in minimizing their losses. * Corresponding author. Tel.: +92 91 9216499; fax: +92 91 9216663. E-mail address: [email protected] (M.A. Irfan). 1 Tel.:+1 216 271 6000; fax: +1 216 641 7852. 1359-4311/$ - see front matter Ó 2009 Published by Elsevier Ltd. doi:10.1016/j.applthermaleng.2008.08.021
The outline of the paper is as follows. Section 2 presents a discussion on thermal stresses due to axial temperature distributions. Thermal stresses due to circumferential temperature gradients are discussed in Section 3. Stresses due to radial temperature gradients and a discussion of end effects are presented in Section 4. Section 5 looks into a combined radial and circumferential loading. A discussion on localized heating and hot spots is given in Section 6. Section 7 takes into consideration the effect of bends in the tube geometry. The paper is concluded in Section 8. 2. Thermal stresses due to axial temperature distributions Although Sauer [6] has developed some simple formulae for analysis of thermal stresses due to axial temperature distributions, Kent [7] has a more complete treatment, which has been employed in the following mathematical analysis. Consider a thin cylindrical shell of elastic material of mean radius c and of length L. Let the temperature rise be constant over any cross-section, but vary with the axial coordinate in a manner than can be expressed in the form of the following power series.
Tx ¼
1 x x2 x3 A þ B þ C 2 þ D 3 þ ...: 2 L L L
ð1Þ
where A, B, C, D, are constants. Substituting Eq. (1) into equilibrium equations [7], the stresses can be represented by:
Eac C 3D g 2 ðk 1Þ 3 ðbx þ wÞ 2 1l L bL Ea 3D bLC C ¼ h þ 2 w þ lrx 2 b2 L2 bL
rx ¼
ð2Þ
rt
ð3Þ
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M.A. Irfan, W. Chapman / Applied Thermal Engineering 29 (2009) 1913–1920
Nomenclature K L
a, b inner and outer radii of the cylinder radial distance ar c mean radius of cylindrical shell h thickness of cylindrical shell major and minor axis of an ellipse lb, la x length coordinate in axial direction A, B, C, D constants of Power Series coefficients of Fourier cosine series Ai, A0i E modulus of Elasticity I moment of Inertia of strip of unit width and depth
a g r, h k,w
l rt rx /
where,
b¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 3ð1 l Þ
ð4Þ
2
h c2
It is interesting to note that, no stresses result from temperature gradients in axial direction unless Eq. 1 includes at least the second power of x. This is evident from the fact that the equations representing the stresses have the coefficients C and D (coefficients A and B are not present). Lee [8] has developed a solution for thermal stresses of a thick walled cylinder under an arbitrary axial temperature gradient T(x). The solution satisfies the field equations of thermo elasticity and the boundary conditions (traction free) at the curved surfaces, but in general not at the end surfaces. The expressions for the radial and hoop stresses are reproduced below.
ð1 mÞ rr ¼ 2aE
(
mT 00
"
2
2
r 2 ða2 þ b Þ þ
16ð1 þ mÞ
a2 b r2
#)
þ higher order terms
ð1 mÞ rt ¼ 2aE
(
mT 00
16ð1 þ mÞ
ð5Þ
"
2
2
3r2 ða2 þ b Þ þ
þ higher order terms
a2 b r2
#)
ð6Þ
Yang and Lee [9] present a series solution for a thick walled cylinder subjected to a temperature distribution which varies both radially and axially. A typical thermal gradient as experimentally found in radiant tubes [10] was applied to Kent’s model [7] depicted by Eqs. 2 and 3, and Lee’s model [8] as given by Eqs. 5 and 6. Both the models gave similar results showing negligible stresses due to axial thermal gradients present in Radiant Tubes. This is probably due to the absence of modeling bends and end conditions in the above equations, which are further investigated in Section 7. To further understand the deformation shapes and resulting stresses due to linear and non-linear (such as sudden rises in) temperature gradients a simple axi-symmetric analysis was carried out in ANSYS. A linear temperature gradient was applied to a tube. The temperature was varied form 24 °C (75°F) to 135 °C (275°F) in linear fashion. The stress field due to the linear temperature gradient was found to be essentially zero as discussed earlier. Fig. 1 shows a simulation of the spike in temperature profile of recuperative burners [10]. The spike is 24 °C (75 °F) higher than its neighboring sections, whereas the rest of the tube the temperature gradient is applied in 7 °C (20 °F) step rises. Fig. 2 shows the axial stress in the tube due to a spike in the temperature gradient. It can be seen that appreciable tensile stresses of the order of 17.24 MPa (2.5 kpsi) are created which are enough to cause the failure of the tube by stress rupture. Note that the creep stress for Super 22HÒ [11] at
modulus of foundation = Eh/c2 length of cylindrical shell coefficient of thermal expansion distance from neutral axis polar coordinates function of x Poisson’s ratio tangential (hoop) stress axial (longitudinal) stress Airy’s stress function
1800 °F is 22 MPa (3.2 kpsi) and the stress for rupture is 2 MPa (2.9 kpsi) for 14 months. When the rate or degree of deformation is the limiting factor, the design stress is based on creep stress. When fracture is the limiting factor, stress to rupture values should be used in design. The above analysis reiterates the fact that linear temperatures gradients do not create any stresses whereas spikes (non-linear temperature rise) in the temperature profile can be detrimental to tube life. The extreme case of sudden rise in temperature of a small section is discussed in Section 6 as localized heating. 3. Thermal stresses due to circumferential temperature distributions From experimental measurements, radiant tubes have been shown to have a circumferential temperature gradient. The difference between the maximum and minimum temperature is around 9 °C (15 °F). It is worth investigating to know whether this difference would create a notable stress in the tube. It is known that there are no stresses generated if a cylinder is uniformly heated to a high temperature. It just expands in the axial and radial direction. The question is what happens if there is a temperature gradient varying with h (but not varying with radius). First an analytical approach is taken. This is a case of an asymmetrical temperature distribution, i.e. the temperature distribution varies as a function of h – the angular position. Two general cases are presented here. First case deals with the temperature fields, which do not result in the creation of thermal stresses. Second case deals with symmetric temperature fields, which can lead to thermal stresses. Using the Airy’s Stress function /, the compatibility equation [7] can be written as,
r4 / þ Ear2 T ¼ 0
ð7Þ
Where / accounts for the non-temperature dependent stresses (e.g. due to external forces), a is the coefficient of linear expansion, and T is the Temperature field. The necessary condition for thermal stresses to be zero is that the Laplacian of Temperature field should be zero,
r2 T ¼ 0
ð8Þ
o2 T 1 oT 1 o2 T þ ¼0 þ or 2 r or r 2 oh2
ð9Þ
where r and h are the polar coordinates. So temperature fields which have Laplacian (r2T) as zero will not generate any thermal stresses. A typical example is
Tðr; hÞ ¼ T 0
r cosðhÞ r0
ð10Þ
M.A. Irfan, W. Chapman / Applied Thermal Engineering 29 (2009) 1913–1920
1915
Fig. 1. A linear temperature gradient ranging form 24 C (75 F) to 135 C (275 F) along the tube with a spike (sudden non-linear rise) in temperature at one section. The spike is 42 C (75 F) higher than the neighboring temperature.
Fig. 2. Axial stress due to the spike (sudden non-linear rise) in temperature at one section. It can clearly be seen that a spike can be a considerable source of stress 16 MPa (2367 psi) whereas linear temperature distributions do not create any stress.
The second case deals with a particular case of symmetric temperature distribution, which results in the generation of thermal stresses. The circumferential temperature distribution in radiant tubes can closely be approximated to be symmetrical about the vertical plane and thus be an even function of h. In general we can represent the circumferential temperature distribution by an even Fourier series.
First a mathematical model was built using Goodier’s formulation [12]. Consider thick walled cylinder where the temperature varies around the circumference and through the thickness, but not along the length. Let h be the angular coordinate. Then the inside T1 and outside T2 temperatures are functions of h only and can be developed in Fourier series (leaving the odd functions of sinh),
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M.A. Irfan, W. Chapman / Applied Thermal Engineering 29 (2009) 1913–1920
T 1 ðhÞ ¼ Ao þ A1 cosðhÞ þ A2 cosð2hÞ þ . . .
ð11Þ
T 2 ðhÞ ¼ A0o þ A01 cosðhÞ þ A02 cosð2hÞ þ . . .
ð12Þ
Timoshenko [13] has further shown that the higher harmonics (cos2h etc.) do not contribute to displacements and stresses. We can clearly see that the Laplacian (r2T) is not equal to zero. Thus the above symmetric circumferential temperature field will generate thermal stresses. The hoop stresses are given as [12],
! 2 2 a2 b a2 þ b rh ¼ jr cos h 4 þ 2 3 r r 2 2 Ea A1 A01 a b where j ¼ þ 2ð1 mÞ a b b4 a4
ð13Þ ð14Þ
where a, b are the inner and outer radii of the cylinder. A computer program was written based on the above equations and the results were compared with the F.E.A. model. Circumferential temperature gradients were applied to a section with a difference of 330 °C (625 F) between the maximum and minimum temperatures. The hottest spot was assumed to be at the bottom and the coolest spot was assumed to be at the top of the section under consideration. The resulting hoop stress field at the outer shell shown in Fig. 3. The results were found consistent with the mathematical analysis carried out using the formulae given earlier. It takes almost a 330 °C (625 °F) circumferential temperature difference to create stresses of the order of 20 MPa (3 kpsi). In reality the circumferential temperature difference in radiant tubes is not more than 15 °C (60 °F). It is interesting to recall that from previous experiments the difference in the maximum and minimum temperature measured across the circumference was never more than 9 °C (15 F). Taking into account the error of measurement of K type thermocouples (+/ 13 C, +/ 24 F) the maximum temperature difference could have been 17 C (63 F). This confirms that circumferential temperature gradients are not a major source of thermal stress. It is also interesting to note that as the thickness of the tube is reduced, there is a corresponding decrease in the magnitude of
thermal stress generated. Fig. 4 shows an extreme case where the thickness of the tube was reduced to 0.254 mm (0.01 inch) as opposed to 8.5 mm (0.334 in.) resulting in negligible thermal stresses. It is worth mentioning that secondary thermal stresses can also exist due to T00 (h). 4. Radial temperature gradients and end effects Having discussed axial and circumferential temperature gradients and their effects on thermal stress, next the thermal stresses created due to radial temperature gradients are evaluated. A mathematical model was built using the formulation of Roark [14].
rh ¼
2r 2i ro 1 ln 2 2 ro ri r r o 2ð1 tÞ ln i
DT aE
ð15Þ
ri
rh ¼ raxial at the outer radius
ð16Þ
It was found out that a temperature difference of 9 °C (15 °F) would generate tensile longitudinal and hoop stresses of the order of 10.34 MPa (1.5 kpsi) at the outer radius. The outer skin of the tube is under bi-axial tension. Primary cracking would occur in a direction perpendicular to maximum principal stress (which would be the longitudinal stress and the bending stress) and secondary cracking could occur in the direction of hoop stress. It is interesting to note that the stress for rupture is 20 MPa (2.9 kpsi) for 14 months at 982 C (1800 F) for the tube material Super 22 H. Similar conclusions were mentioned by Segall et al. [1], who argued that the stress field is tri-axial, however the radial stresses can be neglected as they are small and compressive in nature and do not contribute to crack propagation. It is worthwhile mentioning here that the mathematical formulation is true only for infinitely long cylinders and for regions which are away from the ends of the cylinder. This does raise a concern over what happens at the ends. There are two cases to consider in the case of radial temperature gradients. First is the case
Fig. 3. Hoop stress in the outer shell due to a cosine temperature distribution around the circumference of the tube. The hottest potion of the tube is under tension of about 20 MPa (2880 psi).
M.A. Irfan, W. Chapman / Applied Thermal Engineering 29 (2009) 1913–1920
1917
Fig. 4. Hoop stress in the outer shell due to a cosine temperature distribution around the circumference of the tube. The thickness of tube has been reduced to 0.254 mm (0.01 in.) as opposed to 8.5 mm (0.334 in.) as presented in Fig. 3. Hoop stresses can be reduced dramatically to 0.861 MPa (124.909 psi) by reducing the thickness of the tube.
for the free end. For thin walled tubes where the hoop stress away from the ends can be approximated as,
rh ¼
aEDT 2ð1 mÞ
ð17Þ
It has been shown [12] that the tangential (hoop) stress at the free end can be represented by,
aEDT rh ¼ 2ð1 mÞ
! pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 m2 pffiffiffi m þ 1 3
ð18Þ
6. Localized heating and hot spots
Assuming m = 0.3, the above expression reduces to,
rh ¼ 1:25
aEDT 2ð1 mÞ
The combined effect of radial and circumferential temperature gradients was also investigated. The value of tensile stress due to a radial temperature difference of 9 °C (15 °F) is added to the value of tensile stress due to circumferential temperature gradient. The superimposed tensile stresses can be detrimental to the life of a radiant tube. It is interesting to note that there is a cross-section where the combined stress goes to zero. On a similar note the combined effects of radial and axial gradients can be quite detrimental to the life of radiant tubes.
ð19Þ
which says that the hoop stress at the ends is 25% more than the stress calculated for an infinite cylinder. The increase in stress comes from the bending moment at the free ends due to the outward deflection of the tube. On the other hand if the deflection of the end is prevented by rigid supports, the tangential stress at the end can be reduced by approximately 43% as compared to the stress in an infinite cylinder. It might be noted that the above analysis is for steady state conditions. Transient conditions can generate more severe stresses. 5. Radial and circumferential temperature fields In practice radial and circumferential temperature gradients exist simultaneously and the stresses created due to each of those have to be superimposed [12]. This basically means superimposing the results of Sections 3 and 4. Recall in Section 3 it was shown that there would be a maximum tensile stress at the outer shell corresponding to the region of highest temperature. Section 4 illustrated that for the case of a temperature difference of 9 °C (15 °F) between the inside and outside of the cylinder (the inside being hotter) a tensile stress of 10.34 MPa (1.5 kpsi) is created on the outer boundary.
A leading cause of failure of radiant tubes is localized heating and creation of hot spots. In this section a few different mathematical formulations are discussed, each dealing with a different shape of the hot spot. Also a Finite Element Analysis using commercial software is presented. The first formulation deals with the heating of a small arc. If it is assumed that the shell is heated uniformly over a small arc of circumferential length k along the length of its outer surface the temperature distribution may be approximated by T2, constant over k, with the rest of the outer surface and the whole inner surface kept constant at T1. It may be shown that the maximum circumferential stress is given by the following [7].
rh ¼
3 ðT 2 T 1 Þ k 4 ð1 mÞ R
ð20Þ
A program was written to evaluate stresses due to different hot spot temperature gradients. It was found out that a temperature difference of 10 C (50 F) should lead to a stress of about 27.6 MPa (4 kpsi). The mathematical formulation is only true for very small circumferential elements. Goodier [12] expresses the stresses due to a small circular spot at temperature T1, in a plate with temperature T0, as
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M.A. Irfan, W. Chapman / Applied Thermal Engineering 29 (2009) 1913–1920
1 2
rr ¼ rh ¼ EaðT 1 T 0 Þ inside the hot spot
ð21Þ
9 a2 ¼ 12 EaðT 1 T 0 Þ r2r = outside the hot spot a2 ¼ 1 EaðT T Þ r ;
ð22Þ
rr rh
2
1
0 r2
The above formulation leads to a compressive stress of 26.2 MPa (3.8 kpsi) inside the hot spot and a tensile stress of 26.2 MPa (3.8 kpsi) at the outer boundary of the hot spot. It is interesting to note that for a hot spot of any shape, the greatest stress depends on the shape but not on the size. For an elliptical hot spot the hoop stress is represented by
rh ¼ EaðT 1 T 0 Þ= 1 þ
lb la
ð23Þ
For a (lb/la) (major axis/minor axis) ratio of 2 the hoop stress comes out to be 17.24 MPa (2.5 kpsi.) Fig. 5 shows the simulation of a hot spot in F.E.A. The hot spot was kept 10 °C (50 °F) higher than the surrounding material. The hot spot expands as a ‘‘blister” on the surface of the tube. Looking at the outer shell, the hot spot is under compression in the axial direction and is under tension in the hoop direction. The maximum hoop stress is of the order of 13.79 MPa (2 kpsi) as shown in Fig. 6, which is comparable to hoop stress of the elliptical hot spot model. A more comprehensive mathematical analysis has been developed by Segal and Hellmann [15]. It is interesting to note the pitfall in applying such relationships in FEA, because of the method of interpolation will significantly affect the higher order differentials of temperature gradient. Hence it is hard to accurately predict the stresses in hot spots using the above FEA analysis. 7. Effect of bends in radiant tube An F.E.A. model of the radiant W-tube was built in ANSYS. To study the exclusive effect of temperature field, the effect of gravity
as well as the effect of internal and external support was neglected. A linear temperature gradient was applied across a W-tube. The tube grows outward in the axial direction. As the bend grows, it extends down as well as in the axial direction. The straight portion of the tube was found to be stress free, however stresses of the order of 0.7 MPa (0.1 kpsi) to 1.4 MPa (0.2 kpsi) appear around the bends. The effect can be clearly illustrated by looking at the Von-Mises (equivalent) stress as shown in Fig. 7. It might be worthwhile mentioning two possible explanations for this behavior around the bends. First the mathematical argument of linear temperature gradients producing no stress as discussed in Section 2 is true for straight tubes only. Second, temperature is applied to the F.E.A. model at the nodes. Each set of nodes across the circumference is assigned a particular temperature and a step jump of temperature is given to the next set of nodes. The distance between the nodes at the top and bottom of the tube leg is the same across the straight portion of the tube, and hence a truly uniform temperature gradient can be applied. However at the bends, the nodes are situated much closer together at the inner radius than at the outer radius. Thus a truly linear temperature gradient does not exist at the bends.
8. Conclusions 1. It was shown that linear temperature gradients in the axial direction do not produce thermal stresses in the straight portion of the tube. However a spike in the temperature gradient 24 °C (75 °F) higher than the neighboring temperature field can cause thermal stresses of the order of 17.25 MPa (2.5 kpsi). 2. Circumferential temperature gradients are not a major source of thermal stress. It would require a temperature difference of almost 329 °C (625 °F), varying as a cosine distribution across the circumference to create stresses of the order of 20 MPa (3 kpsi.)
Fig. 5. Simulation of a hot spot. The tube has a temperature range of 24 C (75 F) to 52 C (125 F). The temperature of the hot spot is 28 C (50 F) higher than the uniform surrounding temperature.
M.A. Irfan, W. Chapman / Applied Thermal Engineering 29 (2009) 1913–1920
1919
Fig. 6. Hoop stress field in the outer shell due to the hot spot. Note that the expanding ‘‘blister” creates a tensile hoop stress field with a maximum stress of 13.4 MPa (1946 psi) on the outer shell of the tube.
Fig. 7. Von Mises (equivalent) stress in the bends due to a linear temperature gradient. The maximum stress is about 1.8 MPa (263 psi.)
3. The most critical temperature gradient is the radial temperature gradient. A temperature difference of 11 °C (20 °F) between the inner and outer radius of the tube can create a bi-axial tensile stress of about 13.79 MPa (2 kpsi) at the outer radius of the tube.
4. A combination of radial and circumferential temperature gradient adds up the tensile stresses at the outer radius. This increases the propensity of failure initiation at the outer radius of the tube.
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M.A. Irfan, W. Chapman / Applied Thermal Engineering 29 (2009) 1913–1920
5. Hot spots are a major source of thermal stress. A hot spot with temperature 10 °C (50 °F) higher than its surrounding material can easily generate a ‘‘blister” with a tensile hoop stress of the order of 13.79 MPa (2 kpsi.) 6. The bends carry a higher stress level than the straight portion of the tube.
References [1] A.E. Segall, J.R. Hellmann, P. Strzepa, Experimental and analytical evaluation of the mechanical performance of a gas-fired ceramic radiant tube at steadystate, J. Tes. Eval. 18 (4) (1990) 250–255. [2] T. Nakagawa, M. Obashi, K. Nuta, T. Naito, Y. Abe, N. Ohmori, Development of radiant tube heating system for high-temperature and for high-combustion load, Mizushima Works, Kawasaki Steel Corporation and Otto Corporation, 1996. [3] A. Cardella, Analytical methodology and boundary problem for computing temperature and thermal stresses in tubes, Heat Technol. (Pisa, Italy) 20 (1) (2002) 61–67. [4] Y. Orcan, A.N. Eraslan, Thermal stresses in elastic–plastic tubes with temperature-dependent mechanical and thermal properties, J. Thermal Stresses 24 (11) (2001) 1097–1113.
[5] A.N. Eraslan, Y. Orcan, Computation of transient thermal stresses in elasticplastic tubes: effect of coupling and temperature-dependent physical properties, J. Thermal Stresses 25 (6) (2002) 559–572 (14). [6] G. Sauer, Simple formulae for the approximate computation of axial stresses in pipes due to thermal stratification, Int. J. Press. Vessels Piping 69 (1996) 213– 223. [7] C.H. Kent, F. Ark, Thermal stress in thin-walled cylinders, Trans. of ASME, APM53-13, (1931) 167–180. [8] C.W. Lee, Thermoelastic stresses in thick-walled cylinders under axial temperature gradient, Trans. ASME Ser. E 88 (2) (1996) 467–469. [9] K.W. Yang, C.W. Lee, Thermal Stresses in thick-walled circular cylinders under axisymmetric temperature distribution, ASME J. Eng. Ind. (1971) 969–975. [10] D. Quinn, Industrial radiant tube temperature uniformity and its effects on tube life and production, Technical Report, North American Manufacturing Co., Cleveland OH, 1998. [11] Product Data Sheet for Super 22HÒ Radiant Tube Material, Duraloy Technologies, Scottdale, PA, 2006, http://www.duraloy.com. [12] J.N. Goodier, Thermal stresses and deformation, J. Appl. Mech. 24 (3) (1957) 467–474. [13] S. Timoshenko, Strength of Materials Part II: Advanced Theory and Problems, Robert E. Kreiger Publishing Co., NY, 1976. [14] W.C. Young, Roark’s Formulas for Stress and Strain, 6th ed., McGraw Hill, 1989. [15] A.E. Segall, J.R. Hellmann, Analysis of Gas-Fired Ceramic Radiant Tubes During Transient Heating: II - Thermoelastic Stress Analysis, ASTM J. Test. Eval. 20 (1) (1992) 25–32.