Thermal convection in a horizontal, eccentric annulus containing a saturated porous medium—an extended perturbation expansion

1984 ◽  
Vol 27 (12) ◽  
pp. 2277-2287 ◽  
Author(s):  
Haim H. Bau
1984 ◽  
Vol 106 (1) ◽  
pp. 166-175 ◽  
Author(s):  
H. H. Bau

A two-term regular perturbation expansion is presented for two-dimensional, steady-state thermal convection in a fully saturated porous medium bounded by two horizontal, eccentric cylinders. Both cylinders are impermeable to fluid motion and are maintained at different, uniform temperatures. The complicated boundary conditions are handled through the use of bicylindrical coordinates. Three geometrical configurations are considered: an eccentric annulus; a pipe buried in a semi-infinite medium; and two cylinders, one outside the other, imbedded in an infinite medium. Detailed results, however, are presented only for the first case. It is demonstrated that eccentric insulations may be more effective under certain conditions and therefore more economical than the currently used concentric ones.


1988 ◽  
Vol 187 ◽  
pp. 267-300 ◽  
Author(s):  
K. Himasekhar ◽  
Haim H. Bau

A saturated porous medium confined between two horizontal cylinders is considered. As a result of a temperature difference between the cylinders, thermal convection is induced in the medium. The flow structure is investigated in a parameter space (R, Ra) where R is the radii ratio and Ra is the Darcy-Rayleigh number. In particular, the cases of R = 2, 2½, 21/4 and 2½ are considered. The fluid motion is described by the two-dimensional Darcy-Oberbeck-Boussinesq's (DOB) equations, which we solve using regular perturbation expansion. Terms up to O(Ra60) are calculated to obtain a series presentation for the Nusselt number Nu in the form \[ Nu(Ra^2) = \sum_{s=0}^{30} N_sRa^{2s}. \] This series has a limited range of utility due to singularities of the function Nu(Ra). The singularities lie both on and off the real axis in the complex Ra plane. For R = 2, the nearest singularity lies off the real axis, has no physical significance, and unnecessarily limits the range of utility of the aforementioned series. For R = 2½, 2¼ and 21/8, the singularity nearest to the origin is real and indicates that the function Nu(Ra) is no longer unique beyond the singular point.Depending on the radii ratio, the loss of uniqueness may occur as a result of either (perfect) bifurcations or the appearance of isolated solutions (imperfect bifurcations). The structure of the multiple solutions is investigated by solving the DOB equations numerically. The nonlinear partial differential equations are converted into a truncated set of ordinary differential equations via projection. The steady-state problem is solved using Newton's technique. At each step the determinant of the Jacobian is evaluated. Bifurcation points are identified with singularities of the Jacobian. Linear stability analysis is used to determine the stability of various solution branches. The results we obtained from solving the DOB equations using perturbation expansion are compared with those we obtained from solving the nonlinear partial differential equations numerically and are found to agree well.


2021 ◽  
Author(s):  
Victorien Djotsa Nguimeya Ngninjio ◽  
Wang Bo ◽  
Christof Beyer ◽  
Sebastian Bauer

<p>Borehole thermal energy storage is a well-established technology for seasonal geological heat storage, where arrays of borehole heat exchangers (BHE) are installed in low permeability geological media dominated by conductive heat transfer. Increasing storage temperatures would increase storage capacities and rates and would thus allow for a better inclusion of BTES in the energy system. When using storage temperatures of up 90°C, however, highly permeable zones or intermediate layers may allow for thermally induced fluid migration and convective heat transport in the storage medium, which may increase heat losses from the storage and thus limit the thermal performance of the BTES system. Therefore, we present results from experimental work and subsequent numerical modelling aimed at quantifying thermally induced convection for a lab-scale BHE in a water saturated porous medium for a temperature range of 20°C to 70°C.</p><p>The experimental heat storage unit consists of a fully water saturated coarse sand within a cylindrical polypropylene barrel of 1.23 m height and 0.6 m radius and a vertical coaxial BHE, which is grouted by a thermally enhanced cement. The barrel is cooled from the outside using ventilators and laboratory air. A grid of 68 thermocouples is emplaced in the storage medium for monitoring the temperature distribution. For the stationary experiment, heat is transferred to the storage unit using a supply temperature of 70°C for 6 days until a steady state temperature distribution is achieved, followed by 3 days of heat recovery. The dynamic experiment begins with 3 days of heating with 70°C followed by 6 cycles of alternating heating at 70°C and cooling at approximately 18°C for 12 hours each.</p><p>The stationary experiment reveals a vertical temperature stratification, with temperatures increasing up to 48°C towards the top of the porous medium, as well as a horizontal temperature gradient along the top of the sand, while the lower part of the barrel and the outer wall remain at the laboratory temperature of approximately 18°C. This temperature distribution has stabilized after about 90 hours and represents a clear tilted thermal front, suggesting a significant contribution of induced thermal convection to the overall heat transport. The cyclic experiment shows a decrease of storage temperatures relative to the stationary experiment, with temperatures near to the BHE at the top of the porous lower by 2.5°C and 4.75°C, respectively, because the heating phase is not long enough to reach the stationary temperature distribution. This lower horizontal temperature gradient indicates a weakened thermal convection, however the thermal stratification is conserved. This shows that even under the cyclic loading conditions thermal convection may impair high temperature BTES operation and efficiency.</p><p>Numerical process simulation of coupled flow and heat transport accounting for variable density and the experimental boundary conditions reproduces the spatial and temporal temperature distribution of both experiments with good accuracy. This shows that induced thermal is causing the observed temperature distributions.</p>


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