Analytical Solution of Transient Heat Conduction through a Hollow Cylindrical Thermal Insulation Material of a Temperature Dependant Thermal Conductivity

Dr. Mishaal Abdulameer Abdulkareem Lecturer,Mechanical Engineering Department College of Engineering, Al-Mustansiriyah University, Baghdad, Iraq dr.mishal04@gmail.com ABSTRUCT The one-dimensional, cylindrical coordinate, non-linear partial differential equation of transient heat conduction through a hollow cylindrical thermal insulation material of a thermal conductivity temperature dependent property proposed by an available empirical function is solved analytically using Kirchhoff’s transformation. It is assumed that this insulating material is initially at a uniform temperature. Then, it is suddenly subjected at its inner radius with a step change in temperature. Four thermal insulation materials were selected. An identical analytical solution was achieved when comparing the results of temperature distribution with available analytical solution for the same four case studies that assume a constant thermal conductivity. It is found that the characteristics of the thermal insulation material and the pressure value between its particles have a major effect on the rate of heat transfer and temperature profile.


INTRODUCTION
The performance of air separation plants, storage tanks, transfer lines and transport vessels for cryogenic liquids and liquefied hydrocarbons depends majorly on the characteristics of its thermal insulating materials.Most of these thermal insulations operate under atmospheric or medium vacuum pressure, and it use either Perlite (a loose granulated material of volcanic glass origin heated at 850-900 C o to vaporize the high water content that is trapped in its structure and allowing its volume to be porous and expanded up to 16 7 − times its original volume), or mineral fibers in the form of shells or mats (Verschoor and Greebler, 1952), (Kropschot and Burges, 1962) and (Kaganer, 1969).For specific applications, when only a small space is available for thermal insulation, a system of multilayered foils is used in a high vacuum.It is preferred in the field of transfer lines of liquid hydrogen and liquid helium as well as components in the space technology and in the field of physical basic experimental research (Hoffman, 2006).A common property of all cryogenic thermal insulating materials is that it operates under high temperature deference between atmospheric air and cryogenic fluid temperatures.Therefore, filling it inside a vacuumed leak tight annular space separating the atmosphere from cryogenic fluid vessels is necessary to avoid the drop in its efficiency due to the penetration and freeze of water vapor and carbon dioxide.
The sudden filling of an empty cryogenic liquid storage tank initially at atmospheric temperature with a cryogenic liquid at its saturation temperature will initiate a sudden high temperature difference between the terminals of the annular space containing the thermal insulating material.This high temperature difference is behind the dependence of thermal conductivity of the thermal insulation material on its temperature.In addition, it will initiate a potential for the evaporation of cryogenic liquid due to the transient heat transfer inside the cryogenic liquid storage tank.This energy loss is of a great economic interest especially when the size of cryogenic liquid storage tank is relatively big.(Zivkovic et al, 2010) have used the PAK-T software package, which is based on the finite element method using the Galerkin approach to solve the non-linear transient twodimensional heat conduction through an insulation wall of tank for transportation of liquid aluminum.The objective was to optimize, under certain boundary conditions, the thickness of the insulation material which its thermal properties is a temperature dependent.
(Singh, Jain and Rizwan-Uddin, 2008) presented an analytical double-series solution for transient heat conduction in polar coordinates (2-D cylindrical) for multi-layer domain in the radial direction with spatially non-uniform but time-independent volumetric heat sources.Inhomogeneous boundary conditions of the third kind are applied in the direction perpendicular to the layers.Only homogeneous boundary conditions of the first or second kind are applicable on θ = constant surfaces.(Amiri, Kayhani and Norouzi, 2012) have investigated analytically the unsteady heat conduction in composite fiber winded cylindrical shape laminates.This solution is valid for the most generalized boundary conditions that combine the effects of conduction, convection and radiation both inside and outside the cylindrical composite laminates.The Laplace transformation has been used to change the problem domain from time into frequency.An appropriate Fourier transformation has been derived using the Sturm-Liouville theorem.Due to the difficulty of applying the inverse Laplace transformation, the Meromorphic function method is utilized to find the transient temperature distribution in laminate.
In this paper, the one-  .To validate the results, the temperature distribution will be compared with an available analytical solution for the same four case studies that assume a constant thermal conductivity.A summery table will present the general analytical solution for the history of temperature profiles and heat transfer rates of any size and type of thermal insulation material that is subjected at ( )

STATEMENT OF THE PROBLEM
Consider a cylindrical storage tank of liquefied cryogenic fluid is thermally insulated with a hollow cylindrical super insulating material of temperature dependent thermal conductivity proposed by an available

ANALYTICAL SOLUTION
To solve this problem, the following assumptions are considered: 1.
The temperature is a function of ( ) To linearize the non-linear partial differential equation in system (1), a corrected Kirchhoff's transformation, (Arpaci, 1966), in accordance to the zero lower limit of the absolute temperature scale, as shown in Fig. 2, is used as follows: Substituting eq.(2) into system (1) and rearranging yields; Substitute eq.(4) into system (3) and rearranging, yields; Substitute eq. ( 6) into system (5) and rearranging yields; To solve system (7), assume; Substituting eq. ( 11) into system (7) and using the method of separation of variables, then rearranging, yields; Substitute eq. ( 10), ( 12) and ( 13) into eq.( 8), yields; To estimate the value of ( ) λ , substitute the boundary condition of system (7) into eq.( 14) and rearranging yields; The values of ( ) n λ represent the roots of eq. ( 15), where ( ) Substitute the initial condition of system (7) into eq.( 14), and use eq.( 16), then rearranging, yields; Multiply eq. ( 17) by  1 , R R with the assumption that the integral of the infinite sum is equivalent to the sum of the integration (Arpaci, 1966), we have; All the terms on the right hand side of eq. ( 18) equals zero, except when ( ) The denominator of eq. ( 19) is estimated as follows, (Carslaw and Jaeger, 1959); To estimate the nominator of eq. ( 19), the following are used, (Carslaw and Jaeger, 1959); ) Substitute eq. ( 20) and ( 21) into eq.( 19), and rearranging yields; Substitute eq. ( 22) into eq.( 14), and rearranging yields; Substitute eq. ( 4) into (6), and rearranging yields; Hence, equate eq. ( 23) and eq.( 24) leads to the general analytical close form solution, and it is given as follows; To normalize eq. ( 25), assume the following: Hence, substituting eq. ( 26) into eq.( 25) and rearranging, yields the general analytical close form solution of unsteady temperature distribution through the cylindrical thermal insulation material, as follows: The unsteady heat transfer through the cylindrical thermal insulation material is given as follows:- Differentiating eq. ( 25) and rearranging, yields: Where: Substituting eq. ( 30) and eq.( 31) into ( 29), yields:- To normalize eq. ( 32), substitute eq. ( 26) into eq.( 32) and rearranging, yields:- And the heat transfer at the inner surface of the thermal insulation material ( ) The general analytical close form solution of unsteady heat transfer through the cylindrical thermal insulation material is given as follows:

Temperature profile:
Substitute the value of ( ) m k into eq.( 28) and rearranging, yields;

Heat transfer: Mishaal Abdulameer Abdulkareem Analytical Solution of Transient Heat Conduction Through a Hollow Cylindrical Thermal Insulation Material of a Temperature Dependant Thermal Conductivity
Substitute the value of ( ) m k into eq.( 33) and rearranging, yields; ( )

Heat transfer:
Substitute the value of ( ) 33) and rearranging, yields;

THERMAL CONDUCTIVITY
The dependence of thermal conductivity on temperature is suggested by the empirical function ( ) , (Hoffman, 2006).The values of ( ) , for the four selected thermal insulation materials are given in Table 1.This empirical function is valid in a temperature rang of ( ) . The thermal conductivity of the first insulation material, Perlite in air at ( ) Pa 5 10 atmospheric pressure, is linearly dependent on temperature, (Perlite Institute Thermal Data Sheet, 1970).In the literatures (Christiansen and Hollingworth, 1958) and (Christiansen, Hollingworth and Marsh, 1959), measurements of the thermal conductivity of micro fine Fiberglass mats are published.This material consists of fibers type 'AA' of Owens-Corning Fiberglass Corporation.The behavior of the thermal conductivity and its mean value for each of these thermal insulating materials for a temperature range of ( ) is shown in Fig. 3.

SPECIFIC HEAT CAPACITY
Each of the four selected thermal insulation materials is originally made from Quartz glass.Therefore, the temperature dependence of specific heat capacity of Quartz

Journal of Engineering Volume 19 November 2013 Number 11
1491 glass for a temperature range of ( ) (Corruccini and Gniewek, 1960).In order to evaluate the mean value of specific heat capacity ( ) m C of Quartz glass, its relation with temperature is represented using a 4 th Degree polynomial fit as shown in Table 3 and it is plotted with its mean value in Fig. 4.

THERMAL DIFFUSIVITY
The behavior of thermal diffusivity and its mean value for each of the four selected thermal insulation materials for a temperature range of ( ) is shown in Fig. 5.

RESULTS
Consider four empty cylindrical storage tanks of liquid nitrogen for instance.Each of these tanks is thermally insulated with a hollow cylindrical super insulating material of outside radius ( ) and initially at ( )  2006).The values of ( ) , are given in Table 1.Then, each of these tanks is suddenly filled with saturated liquid nitrogen at ( ) Therefore, each of the four selected thermal insulation materials is subjected at ( ) with a constant temperature of ( ) To plot the transient temperature distribution and the rate of heat transfer of the insulating material, it is necessary to estimate the roots ( ) λ .The values of these roots are given in Table 4, (Cole et. al., 2011).
Figure 6 shows the profiles of temperature distribution for each of the four selected thermal insulation materials for two case studies.The first considers a constant mean value of thermal conductivity ( ) for the temperature range ( ) . The general analytical close form solution of the first case study is identical to the solution of (Carslaw and Jaeger, 1959) using the same assumptions.Therefore, the validation of the analytical solution is accomplished perfectly.The second case study is for a temperature dependent thermal conductivity proposed by an available empirical function ( ) , (Hoffman, 2006).Each profile of the second case study was converged after a maximum of 5 iterations for a temperature residual value of ( ) . Table 5 shows the summery of the present work analytical solutions for both case studies.
Figure 7 shows the heat transfer profiles for each of the four selected thermal insulation materials.It is clear that the thermal insulation material that has the lowest value of thermal conductivity has the lowest heat transfer value.Table 5 shows the summery of the present work analytical solutions.
For both Figures 6 and 7, it is clear that the characteristics of the thermal insulation material and the pressure value between its particles have a major effect on the rate of heat transfer and consequently the temperature profile.For instance, the dominant heat transfer mode when choosing Perlite at ( ) Pa 5 10 atmospheric pressure is by heat conduction of the interstitial gas between the particles, whereas the heat transfer by radiation is negligible.When the pressure within a thermal insulation material is lowered to a value, the percentage of heat transfer by heat conduction of the interstitial gas between the particles becomes negligibly small when compared with the percentage heat transfer by radiation and conduction over the bulk material.The gas pressure, at which this is reached, depends on the characteristic diameter of the thermal insulation material.For Perlite with a characteristic mean particle diameter of ( ) , a gas pressure of ( ) is sufficient, for Microglass spheres with a characteristic mean particle diameter of ( ) and for micro fine Fiberglass with a mean fiber diameter of ( ) respectively.Figures 8 and 9 shows the time history of temperature profiles and rate of heat transfer for each of the four selected thermal insulation materials for a temperature dependent thermal conductivity proposed by an available empirical function ( ) , (Hoffman, 2006).
The general analytical solution for the history of temperature profiles and heat transfer rates of any size and type of thermal insulation material that is subjected to the assumptions that are listed in article (3) is given in Table 5 and is plotted in Figures 10 and 11 respectively.

CONCLUSIONS
It is found that the characteristics of the thermal insulation material and the pressure value between its particles have a major effect on the rate of heat transfer and temperature profile.The dominant mode of heat transfer when choosing a specific thermal insulation material at atmospheric pressure is by heat conduction of the interstitial gas between its particles, whereas the heat transfer by radiation is negligible.When the pressure within a thermal insulation material is lowered to a vacuum level, the percentage of heat transfer by heat conduction of the interstitial gas between its particles becomes negligibly small when compared with the percentage heat transfer by radiation and conduction over the bulk material.On the other hand, the optimum selection of thermal insulating material for a specific cylindrical storage tank of liquefied cryogenic fluid is that with a minimum heat leakage (minimum boil off rate of cryogenic fluid), a minimum amount of insulating material (minimum cost) and a maximum storage capacity of the storage tank (minimum thickness of the thermal insulating material).This optimum selection is accomplished when choosing the micro fine Fiberglass mats when compared with the three other thermal insulating materials.
insulating material is initially at a uniform temperature ( ) i T .Then, it is suddenly subjected at It is required to find the transient temperature distribution of the insulating material and the rate of heat transfer. 2