Practicality of Analytical Solutions for Heat Transfer in Circular Systems in the Infinite Region to Describe Temperatures Due to Disposal of Heat-Decaying Nuclear Waste

Abstract

Analytical solutions for temperatures in an infinite region bounded internally by a cylinder have proved to be useful for thermal analysis of heat-producing nuclear waste disposal scenarios where the thermal design criteria are peak temperatures. The practicality of an analytical solution for the temperature of the host rock used in forced-ventilation thermal analyses has been illustrated by a computational time of a few seconds. Prior to the use of an analytical temperature solution for the host rock, the computation time was on the order of hours. However, the published analytical temperature solution for the infinite region bounded internally by a cylinder with constant heat flux applied at the cylinder wall does not satisfy the boundary condition. This temperature solution is shown to be correct herein with respect to temperature predictions derived from a solution that does satisfy the boundary condition.

Keywords:

• Yucca Mountain
• nuclear waste
• thermal analysis
• analytical solutions

Nomenclature

$A_1$ =	=	defined parameter [Eq. (32)(32) $A_1\equiv k_2\sqrt{\epsilon }{J}_1\left(\sqrt{\epsilon }ua\right)J_0\left(ua\right)-k_1{J}_0\left(\sqrt{\epsilon }ua\right)J_1\left(ua\right)$(32) ]
$B_1$ =	=	defined parameter [Eq. (33)(33) $B_1\equiv k_2\sqrt{\epsilon }{Y}_1\left(\sqrt{\epsilon }ua\right)J_0\left(ua\right)-k_1{Y}_0\left(\sqrt{\epsilon }ua\right)J_1\left(ua\right).$(33) ]
$a$, $b$, $c$ =	=	inside radii of cylinders
$C$ =	=	defined parameter [Eq. (26)(26) $\begin{array}{rl}& C\equiv k_1\left\{J_0\left(\sqrt{\epsilon }ub\right)J_1\left(\sqrt{\epsilon }ub\right)+Y_0\left(\sqrt{\epsilon }ub\right)Y_1\left(\sqrt{\epsilon }ub\right)\right\}\left\{J_1\left(ub\right)Y_1\left(ua\right)-J_1\left(ua\right)Y_1\left(ub\right)\right\}\\ & -\sqrt{\epsilon }{k}_2\left\{J_1^2\left(\sqrt{\epsilon }ub\right)+Y_1^2\left(\sqrt{\epsilon }ub\right)\right\}\left\{J_0\left(ub\right)Y_1\left(ua\right)-J_1\left(ua\right)Y_0\left(ub\right)\right\}\end{array}$(26) ]
$D$ =	=	defined parameter [Eq. (27)(27) $D\equiv \frac{2k_1}{\pi \sqrt{\epsilon }ub}\left\{J_1\left(ub\right)Y_1\left(ua\right)-J_1\left(ua\right)Y_1\left(ub\right)\right\}.$(27) ]
$G_0$ =	=	temperature gradient (with respect to r) for a constant heat source
${\bar{G}}_0$ =	=	Laplace-transformation temperature gradient (with respect to r) for a constant heat source
$G_{\xi }$ =	=	temperature gradient (with respect to r) for an exponentially decaying heat source with decay constant $\xi$ used as $Q{e}^{-\xi t}$
${\bar{G}}_{\xi }$ =	=	Laplace-transformation temperature gradient (with respect to r) for an exponentially decaying heat source
$J_n\left(z\right)$ =	=	Bessel function of the first kind, order $n$
$k_i$ =	=	thermal conductivity of region i
$K_n(z)$ =	=	modified Bessel function of the second kind
$P$ =	=	defined parameter [Eq. (38)(38) $P\equiv k_3\sqrt{{\epsilon }_3}\left\{J_1\left(ua\right)Y_0\left(ub\right)-J_0\left(ub\right)Y_1\left(ua\right)\right\},$(38) ]
$p$ =	=	Laplace-transformation variable, also $\lambda$
$Q$ =	=	thermal flux or power density
$q_i$ =	=	defined variable $q_i^2=\lambda /{\kappa }_i$ for region i
$R$ =	=	defined parameter [Eq. (39)(39) $R\equiv k_2\left\{J_1\left(ub\right)Y_1\left(ua\right)-J_1\left(ua\right)Y_1\left(ub\right)\right\},$(39) ]
$r_i$ =	=	radial position in region i
$t$ =	=	time
$u$ =	=	contour integration (with respect to) variable, defined from $\lambda ={\kappa }_i{u}^2{e}^{\pm i\pi }$, $i\left[\equiv \right]\sqrt{-1}$
$v_i$ =	=	temperature of region i (used to avoid confusion with time t)
${\bar{v}}_i$ =	=	Laplace-transformation temperature of region i
$X$ =	=	defined parameter [Eq. (40)(40) $\begin{array}{rl}& X\equiv +k_1{k}_3\sqrt{{\epsilon }_1}\sqrt{{\epsilon }_3}{J}_1\left(\sqrt{{\epsilon }_1}ua\right)Y_1\left(\sqrt{{\epsilon }_3}ub\right)\left\{J_0\left(ua\right)Y_0\left(ub\right)-J_0\left(ub\right)Y_0\left(ua\right)\right\}\\ & +k_2{k}_3\sqrt{{\epsilon }_3}{J}_0\left(\sqrt{{\epsilon }_1}ua\right)Y_1\left(\sqrt{{\epsilon }_3}ub\right)\left\{J_0\left(ub\right)Y_1\left(ua\right)-J_1\left(ua\right)Y_0\left(ub\right)\right\}\\ & +k_1{k}_2\sqrt{{\epsilon }_1}{J}_1\left(\sqrt{{\epsilon }_1}ua\right)Y_0\left(\sqrt{{\epsilon }_3}ub\right)\left\{J_1\left(ub\right)Y_0\left(ua\right)-J_0\left(ua\right)Y_1\left(ub\right)\right\}\\ & +k_2^2{J}_0\left(\sqrt{{\epsilon }_1}ua\right)Y_0\left(\sqrt{{\epsilon }_3}ub\right)\left\{J_1\left(ua\right)Y_1\left(ub\right)-J_1\left(ub\right)Y_1\left(ua\right)\right\},& (40)\end{array}$(40) ]
$Y$ =	=	defined parameter (no subscript) [Eq. (41)(41) $\begin{array}{rl}& Y\equiv +k_1{k}_3\sqrt{{\epsilon }_1}\sqrt{{\epsilon }_3}{J}_1\left(\sqrt{{\epsilon }_1}ua\right)J_1\left(\sqrt{{\epsilon }_3}ub\right)\left\{J_0\left(ub\right)Y_0\left(ua\right)-J_0\left(ua\right)Y_0\left(ub\right)\right\}\\ & +k_2{k}_3\sqrt{{\epsilon }_3}{J}_0\left(\sqrt{{\epsilon }_1}ua\right)J_1\left(\sqrt{{\epsilon }_3}ub\right)\left\{J_1\left(ua\right)Y_0\left(ub\right)-J_0\left(ub\right)Y_1\left(ua\right)\right\}\\ & +k_1{k}_2\sqrt{{\epsilon }_1}{J}_0\left(\sqrt{{\epsilon }_3}ub\right)J_1\left(\sqrt{{\epsilon }_1}ua\right)\left\{J_0\left(ua\right)Y_1\left(ub\right)-J_1\left(ub\right)Y_0\left(ua\right)\right\}\\ & +k_2^2{J}_0\left(\sqrt{{\epsilon }_1}ua\right)J_0\left(\sqrt{{\epsilon }_3}ub\right)\left\{J_1\left(ub\right)Y_1\left(ua\right)-J_1\left(ua\right)Y_1\left(ub\right)\right\}.& (41)\end{array}$(41) ]
$Y_n\left(z\right)$ =	=	Bessel function of the second kind, order $n$

Greek

${\epsilon }_i$ =	=	ratio of thermal diffusivities as defined in specific text
${\kappa }_i$ =	=	thermal diffusivity of region i
$\lambda$ =	=	Laplace transformation variable, also $p$
$\xi$ =	=	exponential decay constant used as $e^{-\xi t}$

Subscripts

$0$ =	=	temperature gradient for a constant heat flux
$i$ =	=	region
$n$ =	=	Bessel function order
$\xi$ =	=	gradient for an exponentially decaying heat flux

Acknowledgments

No funding was received for this work.