Heat Conduction Problems of Thermosensitive Solids under Complex Heat Exchange

To provide efficient investigations for engineering problems related to heating/cooling process in solids, the effect of thermosensitivity (the material characteristics depend on the temperature) should be taken into consideration when solving the heat conductivity problems (Carslaw & Jaeger, 1959; Noda, 1986; Nowinski, 1962; Podstrihach & Kolyano, 1972). It is important to construct the solutions to the aforementioned heat conduction problems in analytical form. This requirement is motivated, for instance, by the need to solve the thermoelasticity problems for thermosensitive bodies, for which the determined temperature is a kind of input data, and thus, is desired in analytical form. In general, the model of a thermosensitive body leads to a nonlinear heat conductivity problem. It is mentioned in (Carslaw & Jaeger, 1959) that the exact solutions of such problems can be determined when the temperature or heat flux is given on the surface by assuming the material to be “simply nonlinear” (thermal conductivity t  and volumetric volumetric heat capacity v c depend on the temperature, but the relation, called thermal


Introduction
To provide efficient investigations for engineering problems related to heating/cooling process in solids, the effect of thermosensitivity (the material characteristics depend on the temperature) should be taken into consideration when solving the heat conductivity problems (Carslaw & Jaeger, 1959;Noda, 1986;Nowinski, 1962;Podstrihach & Kolyano, 1972). It is important to construct the solutions to the aforementioned heat conduction problems in analytical form. This requirement is motivated, for instance, by the need to solve the thermoelasticity problems for thermosensitive bodies, for which the determined temperature is a kind of input data, and thus, is desired in analytical form.
In general, the model of a thermosensitive body leads to a nonlinear heat conductivity problem. It is mentioned in (Carslaw & Jaeger, 1959) that the exact solutions of such problems can be determined when the temperature or heat flux is given on the surface by assuming the material to be "simply nonlinear" (thermal conductivity t  and volumetric volumetric heat capacity v c depend on the temperature, but the relation, called thermal diffusivity tv ac   , is assumed to be constant). For construction of the solution in this case, it is sufficient to use the Kirchhoff's transformation to obtain the corresponding linear problem for the Kirchhoff's variable. This problem can be solved (Ditkin & Prudnikov, 1975;Galitsyn & Zhukovskii, 1976;Sneddon, 1951) by application of classical methods (separation of variables, integral transformations, etc.). The solutions to the heat conductivity problems for crystal bodies, whose thermal characteristics are proportional to the third power of the absolute temperature, can be constructed in a similar manner for the case of radiation heat exchange with environment.
In the case of complex heat exchange, the Kirchhoff transform makes the heat conductivity problem to be linear only in part. In the heat conductivity problem for the Kirchhoff's variable, the heat conduction equation is nonlinear due to dependence of the thermal diffusivity on the Kirchhoff's variable. The boundary condition of the complex heat exchange is also nonlinear due to a nonlinear expression of the temperature on the surface. Herein we discuss several approaches, developed by the authors for determining temperature distribution in thermosensitive bodies of classical shape under complex (convective, radiation or convective-radiation) heat exchange on the surface (Kushnir & Popovych, 2006, 2007; Kushnir & Protsiuk, 2009;Kushnir et al., 2001Kushnir et al., , 2008 At the initial moment of time, the temperature is uniformly distributed within the body: The key point of the solution method for the formulated non-linear heat conductivity problem (1)-(4), which is presented below, consists in the step-by-step linearization involving the Kirchhoff transformation along with linearization of the nonlinear term in the boundary conditions by means of the spline approximation.
Fo 0 p Here Let us apply the Kirchhoff's integral transformation (Carslaw & Jaeger, 1959;Noda, 1986;Podstrihach & Kolyano, 1972) to the problem (5) follows from the nonlinear heat conductivity equation (5). The boundary condition of convective-radiation heat exchange (6) can be partially linearized and presented as The boundary conditions (7) and initial condition (8)  Application of the Kirchhoff's variable allows us to linearize the nonlinear heat conductivity equation (5) and the second boundary condition (7) completely, whereas the convective-radiation heat exchange condition is linearized in a part. Due to the nonlinear expression   () a QT  , it is impossible to apply any classical method to solve the boundary problem (10)-(13). Therefore, it is necessary to linearize the boundary condition (11). In (Nedoseka, 1988;Podstrihach & Kolyano, 1972), the convective heat exchange condition has been considered. Therefore, the nonlinear expression () T  is simply replaced by  . As a result, the nonlinear convective heat exchange condition on  becomes linear. However, it has been shown in (Kushnir & Popovych, 2009;Popovych, 1993b;Popovych & Harmatiy, 1996) that this unsubstantiated linearization leads to the numerically or physically incorrect results. In our case, when we take into account the radiation constituent (which is nonlinear even for a non-thermosensitive material) and dependence of the heat transfer coefficient and emittance on the temperature, the considered substitution does not provide the complete linearization of the condition (11). Instead, the boundary condition (11) where () a i Q is expressed through () a i T by means of formula (15).
QT   is expressed as the first-order spline (17), then boundary condition (11) becomes linear Having solved the obtained linear problem (10), (12), (13), (16) or (10) The temperature can be computed with any given accuracy  for arbitrary segmentation of the time axis. However, the increasing of number of time-segments decreases the convergence of the proposed scheme. An appropriate choice of the initial moment of time can be done by means of the estimated 'a priory' time-dependence of the temperature on the surface xa  . We can also use the solution of corresponding boundary value problem for the body of the same shape with constant characteristics. Then the initial choice for values Fo i can be used as the appropriate one for the thermosensitive body.
The method of step-by-step linearization is applicable for determination of the temperature fields in thermosensitive plates, half-space, solid and hollow cylinders or spheres, space with cylindrical or spherical cavities, on the surfaces of which, the conditions of convective, radiation or convective-radiatio n h e a t e x c h a n g e m a y b e g i v e n . T h i s m e t h o d h a s b e e n efficiently used for solving the two-dimensional steady problem in thermosensitive body.

Method of linearizing parameters
The method of step-by-step linearization makes it possible to determine the solutions to the two-dimensional heat conductivity problems in thermosensitive bodies with simple nonlinearity, when the nonlinear term in the condition of complex heat exchange for the Kirchhoff's variable depends on one (spatial or time) variable only. In this section, we consider an efficient method for solving the steady-state and transient heat conductivity problems of arbitrary dimension those describe the propagation of heat in thermosensitive bodies with simple nonlinearity under the convective heat exchange with environment. Let the body occupies region D with surface S. The surface (whole or a part) is subjected to the convective heat exchange with the environment of temperature p t . From the moment of time 0   , the heat sources (,,,) Wxyz are acting in the body. The temperature in the body shall be determined from the following heat conduction equation: is obtained, where 000 ,, Xx lYy lZz l   are dimensionless coordinates; is the dimensionless function of heat sources. As a result, the initial problem is partially linearized, meanwhile the condition (27) remains nonlinear. The latter conditions have been obtained from the conditions of convective heat exchange due to nonlinear expression () T  on the surface S . For solving the problem (26)-(28) by using an analytical method, it is necessary to linearize this condition. Let us prove the possibility of such linearization. Consider the simplest case of linear dependence of heat conductivity coefficient on the temperature: where k is a constant. From the equation (9), the formula From the physical standpoint, the square root is chosen to be positive. After substitution of the equation (31) into the boundary condition (27), the last one takes the form Be decomposing the square root in (32) into the series and restricting this series with two terms, the boundary condition www.intechopen.com is obtained. The solution of equation (26) with boundary conditions (28), (33) is an approximate solution to the boundary value problem (26), (28), (32). To determine the exact solution, the equation (26) is to be solved under initial condition (28) and the following linear boundary condition For an arbitrary value of  , the solution (35) meets the equation (26) and the initial condition (28). In order the solution (35) to satisfy the nonlinear conditions (32) and (34), the parameter  is to be the solution of the equation After some transformations, this equation can be given as This equation holds for every moment of time Fo . After the paramenter  is found, we substitute it into (35). In such manner, the expression for Kirchhoff's variable is obtained. The temperature in the body is then calculated by means of the relation (31). Note that the boundary condition (34) can be represented as where Bi Bi (1  ) where ,

HH oH
Tt t t  is temperature of the body with constant characteristics.
By subtraction equations of the problem (26) The boundary value problem (41) temperature for the thermosensitive body is replaced with the surface temperature for the body with constant characteristics (whose thermal diffusivity is equal to the thermal diffusivity of thermosensitive body and the heat conductivity coefficient is equal to the reference value of the heat conductivity coefficient 0 Thus, if the surface temperature () s T  of the thermosensitive body in the condition (27) is equal to the corresponding temperature of the body with constant characteristics, then the boundary value problem for the Kirchhoff's variable  should be solved with the condition (33). Then the solution of this problem presents the difference of the temperature in the same-shape body with constant characteristics and the initial temperature: As it was mentioned above, the substitution of () in the case of linear dependence of the heat conductivity coefficient on the temperature is equivalent to keeping only two terms in the series, into which the square root in expression for the temperature through the Kirhoff's variable has been decomposed. This linearization does not guarantee a sufficient solution approximation. To overcome this difficulty, we consider the boundary value problem for the variable  with the linear condition (37) instead of the nonlinear condition (27), which involves an additional parameter  . Having solved the obtained linear problem, the Kirhoff's variable  is found as a function of the coordinates and parameter  . The parameter  should be chosen in the way to satisfy the nonlinear condition (27) with any given accuracy. Thus for determination of the temperature field in the body with simple nonlinearity for arbitrary temperature dependence of heat conductivity coefficient under convective heat exchange between the surface and environment, the corresponding solution of the nonlinear heat conductivity problem can be determined by following the proposed algorithm of the method of linearized parameters: -to present the problem in dimensionless form; -to linearize the problem in part by using integral Kihhoff transformation; -to linearize the problem completely by linearizing the nonlinear condition on

The method of linearizing parameters for the steady-state heat conduction problems in piecewise-homogeneous thermosensitive bodies
Determination of the temperature fields in piecewise-homogeneous bodies subjected to intensive thermal loadings is an initial stage that precedes the determination of steady-state or transient thermal stresses in the mentioned bodies. Let us assume that the elements of piecewise-homogeneous body are in the ideal thermal contact and the limiting surface is under the condition of complex heat exchange with environment. Mathematical model for determination of the temperature fields in such structures leads to the coupled problem for a set of nonlinear heat conduction equations with temperature-dependent material characteristics in the coupled elements. By making use of the Kirhoff's integral transformation for each element by assuming the thermal conductivity to be constants, the problem can be partially linearized. The nonlinearities remain due to the thermal contact conditions on the interfaces and the conditions of complex heat exchange on the surfaces. To obtain an analytical solution to the coupled problem for the Kirchhoff's variable, it is necessary to linearize this problem. The possible ways of such a linearization and, thus, determination of the general solution to the heat conduction problems in piecewisehomogeneous bodies are considered below in this section.

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Let us adopt the method of linearizing parameters to solution of the steady-state heat conduction problems for coupled bodies of simple shape, for instance, n -layer thermosensitive cylindrical pipe. The pipe is of inner and outer radii 0 rr  and n rr  , respectively, with constant temperatures b t and H t on the inner and outer surfaces. The layers of different temperature-dependent heat conduction coefficients are in the ideal thermal contact. The cylindrical coordinate system ,, rz  is chosen with z -axis coinciding with the axis of pipe. The temperature field in this pipe can be determined from the set of heat conduction equations () , where the constituents with the indices "0" are dimensional constants and the asterisked terms are dimensionless functions, 0 t is the reference temperature. In the dimensionless form, the problem (45) Consider the heat conduction coefficients in the form of linear dependence on the temperature is obtained from the problem (48)-(50). Here The initially nonlinear heat conduction problem is partially linearized due to application of the Kirchhoff's variables. However, the conditions for temperature, that reflects the temperature equalities of the neighbouring layers, remain nonlinear (the first group of conditions (54)). By integrating the set of equations (52) with boundary conditions (53) and contact conditions (54), the set of transcendent equation can be obtained for determination of constant of integration. This set can be solved numerically. The efficiency of numerical methods depends on the appropriate initial approximation. Unfortunately, it is very complicated to determine the definition domain for the solution of this set of equations and thus to present a constructive algorithm for determination of the initial approximation. The possible way around this problem is to decompose the square root in the first conditions (54) into series by holding only two terms. Then, instead of mentioned conditions, the following approximated conditions are obtained: Application of the conditions (55), instead of exact ones, separates the interfacial conditions. This fact allows us to consider the boundary problem (52)-(54) replacing the conditions (54) by the following ones: where i  are unknown constants (linearizing parameters). By substitution we obtain 1 11 , a t = , 1 , 1 where нн (1 ) ; Besides the initial data, the solution (66) contains n arbitrary constants i  and satisfies the equation (52), boundary conditions (53) and the second group of the contact conditions (54).

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The linearized parameters i  will be selected to satisfy the first group of the conditions (54).
By assuming that one of the linearizing parameters i  , for instance, is equal to zero, the for determination of the rest 1 n  linearizing parameters. The solution should be found in a neighborhood of zero. From the set (67), we determine the values of linearization parameters and thus the Kirchhoff's variables. Then the temperature in layers is For example, we consider the two-layer pipe (2 ) n  . The Kirchhoff's variables for this case are expressed as where (2) (1) 00 tt  is equal to zero, and 2  is denoted as  . The value of  shall be obtained from the equation  To simplify the explanation of the linearized parameters method for solving the heat conductivity problem in the coupling thermal sensitive bodies, the constant temperatures on bounded surfaces of piecewise-homogeneous bodies were considered. If the conditions of www.intechopen.com convective heat exchange are given, then the final linearization of the obtained nonlinear conditions on Kirchhoff's variables may be fulfilled using the method of linearizing parameters. The method of linearizing parameters can be successfully used for solution of the transient heat conduction problems.

Determination of the temperature fields by means of the step-by-step linearization method
To illustrate the step-by-step linearization method, consider the solution of the centrosymmetrical transient heat conduction problem. Let us consider the thermosensitive hollow sphere of inner radius 1 r and outer radius 2 r . The sphere is subjected to the uniform temperature distribution p t and, from the moment of time 0 with boundary and initial conditions www.intechopen.com is obtained, where The is asymmetric unit function (H. Korn & T. Korn, 1977;Podstrihach et al., 1984), Fo (j) i are the points of segmentation of the time axis (0;Fo). After substitution of the expression (82) into the boundary conditions (79), the boundary value problem (78)-(80) becomes linear. For its solving, the Laplace integral transformation can be used (Ditkin & Prudnikov, 1975). As a result, the Laplace transforms of the Kirhoff's variables are determined as 1 (1) The inverse Laplace transformation can be found by means of the Vashchenko-Zakharchenko expansion theorem of and shift theorem (Lykov, 1967). As a result, the following expression for Kirchhoff's variable where n  are roots of the characteristic equation Let us provide the numerical implementation of the proposed solution method to determine the time-variation of the temperature on the surface 1   of solid sphere exposed to the condition of convective heat exchange. We assume c t = 300ºС (573 K) and this value is also chosen to be the reference temperature; the initial temperature is p t  20ºC (293 K . The results of computation are shown in Figure 1. In Figure 2, the dependence of the temperature on the radial coordinate at the moment of time Fo 0,1  is shown for some values of the Biot number. The solid lines correspond to the solution of the heat conduction problem, obtained by using the step-by-step method, i.e., when the Kirchhoff's variable is computed by the formula (94). The dash-dot line corresponds to the solution of the problem when the boundary condition is linearized by changing () T  for  . In this case, the Kirchhoff's variable is calculated by formula (98). The dashed line presents the solution of corresponding linear problem when thermal characteristics are constant. In the considered case, neglecting the temperature dependence in thermal properties leads to the increasing of the temperature values. In the same time, the unsubstantiated linearization of boundary condition increases the temperature and leads to physically improper results. As it follows from the figures, at some moments of time, the temperature on surface of sphere is greater than the temperature of heating environment. The authors (Nedoseka, 1988;Podstrihach & Kolyano, 1972) did not give much attention to this matter because mainly they considered the temperature fields in thermosensitive bodies due to the internal heat sources. In this case increasing of the temperature is unbounded.

Conclusion
In this chapter, the formulations of non-linear heat conduction problems for the bodies with temperature-dependent characteristics (thermosensitive bodies) are given. The efficient analytico-numerical methods for solution of the formulated problems are developed. Particularly, the step-by-step linearization method is proposed for solution of onedimensional transient problems of heat conduction, which describe the temperature fields in thermosensitive structure members of simple nonlinearity under complex (convective, radiation or convective-radiation) heat exchange boundary conditions. The coefficient of heat exchange and emissivity of the surface, that is under heat exchange with environment, are also dependent on the temperature. The method provides: reduction of the heat conduction problem to the corresponding dimensionless problem; -partial linearization of the obtained problem by means of the Kirchhoff's transform; -complete linearization of the nonlinear condition on the Kirchhoff's variable  , that has been obtained from the condition of complex heat exchange due to approximation of the nonlinear term by specially constructed spline of zero or first order; construction of the solution to the linearized boundary value problem for  by means of the appropriate analytical method; determination of the temperature in question by means of the inverse Kirchhoff's transform; -determination of the unknown parameters of spline-approximation, those remain in the expression for the temperature, by means of the collocation method. The method is verified by the solutions of transient heat conduction problems for thermosensitive solid and hollow spheres subjected to heating (cooling) due to the heat exchange over the limiting surface. This method can be efficiently used fro solution of twodimensional steady-state heat conduction problems. The efficient method of linearizing parameters is proposed for determination of the temperature fields in structure members with simple nonlinearity due to convective heat exchange through the limiting surfaces for an arbitrary dependence of the heat conduction coefficient on the temperature. The main feature of this method consists in the fact that the complete linearization of the nonlinear condition for the Kirchhoff's variable  (obtained form the condition of convective heat exchange) is achieved by substitution of the nonlinear www.intechopen.com term () T  by (1 ) p T     with unknown parameter  . This parameter can be found by satisfaction of the nonlinear condition for  with required accuracy.
The method of linearizing parameters is adopted to solution of the nonlinear steady-state and transient heat conduction problems for contacting thermosensitive bodies of simple geometrical shape under conditions of the ideal thermal contact at the interfaces and complex heat exchange on the limiting surfaces. Its approbation is provided for the n-layer cylindrical pipe under given temperatures on its inner and outer surfaces. It these surfaces are subjected to the convective heat exchange, then the complete linearization of the obtained nonlinear conditions for the Kirchhoff's variable  can be done by means of the method of linearizing parameters. The content of this book covers several up-to-date approaches in the heat conduction theory such as inverse heat conduction problems, non-linear and non-classic heat conduction equations, coupled thermal and electromagnetic or mechanical effects and numerical methods for solving heat conduction equations as well.

Acknowledgment
The book is comprised of 14 chapters divided into four sections. In the first section inverse heat conduction problems are discuss. The first two chapters of the second section are devoted to construction of analytical solutions of nonlinear heat conduction problems. In the last two chapters of this section wavelike solutions are attained.The third section is devoted to combined effects of heat conduction and electromagnetic interactions in plasmas or in pyroelectric material elastic deformations and hydrodynamics. Two chapters in the last section are dedicated to numerical methods for solving heat conduction problems.