Asymptotic expansions for the ground heat transfer due to a borehole heat exchanger with a Neumann boundary condition

Abstract

We model the heat transfer of a borehole heat exchanger considering a finite-line heat source inside a semi-infinite medium with constant thermal conductivity and geothermal gradient. As a novelty with respect to other models, periodic heat flux changes are allowed on the ground surface. From this model, we calculate analytic expressions for the average ground temperature by integrating the exact solution along the line-source. Also, asymptotic formulas for the intermediate and long timescales are derived.

Appendices

Equivalence of Eqns. (32) and (33)

According to (31) and (28)–(29), rewrite (32) as

$$\begin{aligned} v_\mathrm{d}\left( r,z,t\right)= & {} \frac{Q_{z}}{8\pi \lambda }\int _{r^{2}/4kt}^{\infty }\frac{\mathrm {e}^{-u}}{u} \left. \mathrm {erf}\left( \frac{w}{r}\sqrt{u}\right) \right| _{w=z-L}^{z+L}\mathrm {d}u \\= & {} \frac{Q_{z}}{8\pi \lambda }\int _{0}^{t}\exp \left( -\frac{r^{2}}{4k\left( t-t^{\prime }\right) }\right) \left. \mathrm {erf}\left( \frac{w}{2\sqrt{ k\left( t-t^{\prime }\right) }}\right) \right| _{w=z-L}^{z+L}\frac{ \mathrm {d}z^{\prime }}{t-t^{\prime }} \\= & {} \frac{q_{L,t}}{8}\int _{0}^{t}\frac{\mathrm {d}z^{\prime }}{\left[ \pi k\left( t-t^{\prime }\right) \right] ^{3/2}}\int _{-L}^{L}\exp \left( -\frac{ r^{2}+\left( z-z^{\prime }\right) ^{2}}{4k\left( t-t^{\prime }\right) } \right) \mathrm {d}z^{\prime }, \end{aligned}$$

where we have taken into account that $q_{L,t}/k=Q_{z}/\lambda$. Exchanging the order of integration, we calculate the inner integral with the substitution $w^{2}=(r^{2}+( z-z^{\prime }) ^{2})/ (4k(t-t^{\prime }))$, arriving at

$$\begin{aligned} v_\mathrm{d}\left( r,z,t\right)= & {} \frac{q_{L,t}}{8}\int _{-L}^{L}\mathrm {d}z^{\prime }\int _{0}^{t}\frac{\mathrm {d}z^{\prime }}{ \left[ \pi k\left( t-t^{\prime }\right) \right] ^{3/2}}\exp \left( -\frac{ r^{2}+\left( z-z^{\prime }\right) ^{2}}{4k\left( t-t^{\prime }\right) } \right) \\= & {} \frac{Q_{z}}{4\pi \lambda }\int _{-L}^{L}\mathrm {erfc}\left( \sqrt{\frac{ r^{2}+\left( z-z^{\prime }\right) ^{2}}{4k\left( t-t^{\prime }\right) }} \right) \frac{\mathrm {d}z^{\prime }}{\sqrt{r^{2}+\left( z-z^{\prime }\right) ^{2}}}, \end{aligned}$$

where we have applied the definition of the complementary error function [18, Eq. 7.2.2]. Finally, split the integration interval as $\left( -L,L\right) =\left( -L,0\right) \cup \left( 0,L\right)$, and perform in the first interval the substitution $z^{\prime }\rightarrow -z^{\prime }$ to obtain (33), grouping terms.

Dirichlet boundary condition

Following [5], we consider a sinusoidal temperature oscillation on the surface, thus we have the following Dirichlet boundary condition,

$$\begin{aligned} T\left( r,0,t\right) =T_{0}+T_\mathrm{s}\sin \omega t. \end{aligned}$$

(75)

The temperature components $v_{0}\left( z,t\right)$, $v_\mathrm{s}\left( z,t\right)$ and $v_\mathrm{d}\left( r,z,t\right)$ of the time-dependent temperature field $T\left( r,z,t\right)$ satisfying the boundary-value problem (1), (2) and (75) is [5, 6],

$$\begin{aligned} v_{0}\left( z,t\right) +v_\mathrm{s}\left( z,t\right) =T_{0}+k_{\text {geo}}z+\frac{ T_\mathrm{s}}{2}\mathrm {Im}\left( \mathrm {e}^{\mathrm {i}\omega t}\left[ A^{+}\left( z,t\right) +A^{-}\left( z,t\right) \right] \right) , \end{aligned}$$

(76)

where $A^{\pm }\left( z,t\right)$ has been defined in (12), and

$$\begin{aligned} v_\mathrm{d}\left( r,z,t\right) =\frac{Q_{z}}{4\pi \lambda }\int _{r/\left( 2\sqrt{kt }\right) }^{\infty }\frac{\mathrm {e}^{-u^{2}}}{u}\left[ 2\mathrm {erf}\left( \frac{z}{r }u\right) -\mathrm {erf}\left( \frac{L+z}{r}u\right) +\mathrm {erf}\left( \frac{L-z}{r}u\right) \right] \mathrm {d}u. \end{aligned}$$

(77)

To calculate the average value of (76) along the borehole heat exchanger, we use the integral given in (22), arriving at

$$\begin{aligned} \left\langle v_{0}\left( z,t\right) +v_\mathrm{s}\left( z,t\right) \right\rangle =T_{0}+k_{\text {geo}}\frac{L}{2}+\frac{T_\mathrm{s}d_\mathrm{p}}{4L}\mathrm {Im}\left( \left( 1-\mathrm {i}\right) \mathrm {e}^{\mathrm {i}\omega t}\left[ A^{+}\left( z,t\right) +A^{-}\left( z,t\right) +2\,\mathrm {erfc}\left( \sqrt{\mathrm {i}\omega t}\right) \right] \right) , \end{aligned}$$

(78)

Also, in [5] we found

$$\begin{aligned} \left\langle v_\mathrm{d}\left( r,z,t\right) \right\rangle =\frac{Q_{z}}{4\pi \lambda }\int _{r/\sqrt{4kt}}^{\infty }\frac{\mathrm {e}^{-u^{2}}}{u} \left[ 4\mathrm {erf}\left( \frac{Lu}{r}\right) -2\mathrm {erf}\left( \frac{ 2Lu}{r}\right) -\frac{\left( 3+\mathrm {e}^{-4L^{2}u^{2}/r^{2}}-4\,\mathrm {e}^{-L^{2}u^{2}/r^{2}}\right) r}{\sqrt{\pi }Lu} \right] \mathrm {d}u, \end{aligned}$$

(79)

where with the aid of the tabulated integral [25, Eq. 3.461(5)],

$$\begin{aligned} \int _{x}^{\infty }\frac{\mathrm {e}^{-az^{2}}}{z^{2}}\mathrm {d}z=\frac{\mathrm {e}^{-ax^{2}}}{x}-\sqrt{ \pi a}\ \mathrm {erfc}\left( \sqrt{a}x\right) ,\quad \mathrm {Re}\,a>0, \quad x>0, \end{aligned}$$

we may rewrite (79) as

$$\begin{aligned} \left\langle v_\mathrm{d}\left( r,z,t\right) \right\rangle= & {} \frac{Q_{z}}{4\pi \lambda }\left\{ \int _{r/\sqrt{4kt}}^{\infty }\left[ 4 \mathrm {erf}\left( \frac{Lu}{r}\right) -2\mathrm {erf}\left( \frac{2Lu}{r} \right) \right] \frac{\mathrm {e}^{-u^{2}}}{u}\mathrm {d}u\right. \nonumber \\&-\,\frac{2\sqrt{kt}}{\sqrt{\pi }L}\exp \left( -\frac{4L^{2}+r^{2}}{4kt} \right) \left[ 1-4\exp \left( \frac{3L^{2}}{4kt}\right) +3\exp \left( \frac{ L^{2}}{kt}\right) \right] +\frac{3r}{L}\mathrm {erfc}\left( \frac{r}{2\sqrt{kt }}\right) \nonumber \\&\left. -\, 4\sqrt{1+\frac{r^{2}}{L^{2}}}\mathrm {erfc}\left( \frac{1}{2}\sqrt{ \frac{r^{2}+L^{2}}{kt}}\right) +\sqrt{4+\frac{r^{2}}{L^{2}}}\mathrm {erfc} \left( \frac{1}{2}\sqrt{\frac{r^{2}+4L^{2}}{kt}}\right) \right\} . \end{aligned}$$

(80)

It is worth noting that (78) and (80) are not given explicitly in the literature.

Regarding the asymptotic behavior of (78), we apply to (78) the result given in (24), thereby

$$\begin{aligned} \left\langle v_{0}\left( z,t\right) +v_\mathrm{s}\left( z,t\right) \right\rangle \approx T_{0}+k_{\text {geo}}\frac{L}{2}+\frac{T_\mathrm{s}d_\mathrm{p}}{\sqrt{2}L}\left[ \sin \left( \omega t-\frac{\pi }{4}\right) +\mathrm {e}^{-L/d_\mathrm{p}}\sin \left( \omega t- \frac{L}{d_\mathrm{p}}-\frac{\pi }{4}\right) \right] , t\rightarrow \infty {.} \end{aligned}$$

(81)

When $d_\mathrm{p}\ll L$, we can neglect the last term in (81). It is worth noting that for the latter case, we found a typo in [5].

Finally, according to [5], the asymptotic formula of (79) for the long timescale $t\gg L^{2}/4k$ is

$$\begin{aligned} \left\langle v_\mathrm{d}\left( r,z,t\right) \right\rangle\approx & {} \frac{Q_{z}}{ 4\pi \lambda }\left\{ 4\sinh ^{-1}\left( \frac{L}{r}\right) -2\sinh ^{-1}\left( \frac{2L}{r}\right) +3\frac{r}{L}-4\sqrt{1+\frac{r^{2}}{L^{2}}} \right. \nonumber \\&\qquad \quad +\,\left. \sqrt{4+\frac{r^{2}}{L^{2}}}-\frac{L^{3}}{12\sqrt{\pi }\left( kt\right) ^{3/2}}\left( 1-\frac{3\left( L^{2}+r^{2}\right) }{20kt}\right) \right\} , \end{aligned}$$

(82)

and for the intermediate timescale $r^{2}/k\ll t\ll L^{2}/k$ is

$$\begin{aligned} \left\langle v_\mathrm{d}\left( r,z,t\right) \right\rangle \approx \frac{Q_{z}}{ 4\pi \lambda }\left\{ -\gamma +\log \left( \frac{4kt}{r^{2}}\right) +\frac{3r }{L}-\frac{6}{L}\sqrt{\frac{kt}{\pi }}+\frac{r^{2}}{4kt}-\frac{3r^{4}}{2L^{3} \sqrt{\pi kt}}\right\} . \end{aligned}$$

(83)