Unsteady Surface Element Method Research Articles

A novel solution for vertical borehole heat exchangers in multilayered ground with the Robin boundary condition

The ground surface condition is crucial in the thermal analysis of borehole heat exchange (BHE) systems, as it directly influences the thermal regime within the subsurface. However, most existing analytical solutions assume a constant temperature at the ground surface equal to the undisturbed ground temperature, overlooking the impact of various anthropogenic and environmental factors that alter ground surface conditions. This study addresses this gap by proposing a new semi-analytical solution that incorporates the Robin boundary condition for BHEs in multilayered ground. The solution, developed using the unsteady surface element method, balances computational efficiency with analytical precision. The proposed solution was verified with an exact solution in a homogeneous body, showing a maximum error of approximately 0.2 %. Compared with prescribing a constant temperature at the ground surface, using a Robin boundary condition better captures the heat exchange between the ground and the surrounding environment. The results show that variations in the heat transfer coefficient at the ground surface can cause differences in the mean borehole wall temperature change of up to 7.45 % over about 31.7 years (109 s) for a 70-metre borehole. Meanwhile, a unit heat flux at the ground surface can raise the mean borehole wall temperature by 0.56 °C for a heat transfer coefficient of 0.25 Wm−2K−1 over 31.7 years. Additionally, the circulating fluid can transmit the impact of the ground surface condition to greater depths. The results indicate that even at the bottom of a 70-metre borehole, using a fixed temperature can underestimate the temperature change by up to 6.11 % over 31.7 years compared with using the Robin boundary condition. These findings emphasize the indispensability of the proposed solution for accurately predicting the thermal performance of BHEs.

Maximum Temperatures in Diamond Heat Spreaders Using the Surface Element Method

Diamond heat spreaders are very attractive for lowering the temperature of laser diodes and computer chip components. The use of diamonds as heat spreaders is very advantageous because its thermal conductivity is so large, about four times that of copper. The diamond heat spreader is mounted on a semi-infinite heat sink. The purposes of this paper are to present (a) a surface element method for the analysis of such composite systems, (b) a set of convenient algebraic equations for the maximum temperatures, (c) optimal geometry conditions and (d) some accurate numerical results. The analysis method is an adaptation of the unsteady surface element method but is different because the present problem is a steady-state one. The surface element method with one node gives a relatively simple algebraic solution, which contains all the important dimensionless groups. The one-node solution is simple, accurate, and has a form that can give deep insight into the effects of various parameters; for example, it premits derivation of the optimal geometry corresponding to the minimal temperature at the hot spot. The study shows that the optimal geometry of the thickness to radius aspect ratio of the diamond heat spreaders is about 0.4. In addition to the single-node analysis, a multinode analysis is developed and very accurate results are presented. These results show that the single-node analysis is generally within 2 percent.

Transient conjugated heat transfer from a rectangular hot film

A numerical calculation is carried out for the transient response of a flush-mounted hot-film sensor in a gas flow heated by a step change in the hot-film output flux. The numerical method used is called the unsteady surface element method. This calculation is part of a study of unsteady behavior of hot-film anemometers that involve substrate effects. The numerical results show that the early-time film temperature agrees well with an analytical solution, and the steady-state Nusselt number agrees with an existing numerical solution. The spatial average film temperature vs time is presented for several values of the rectangular hot-film aspect ratio. Nomenclature a = streamwise half-length of the hot film, m b = span wise half-length of the hot film, m K = thermal conductivity, W/(mK) — streamwise length of the yth surface element, m = spanwise length of the yth surface element, m M = number of time steps N = number of surface elements Np = flow parameter, Eq. (16) Nu = Nusselt number P = hot-film output flux, W/m2 Pe = Peclet number, p(2a)2/af q = heat flux, W/m2 qQ = hot-film output, W/m2 T = temperature Tav = spatial average hot-film temperature TQ = initial temperature T- = (T - TQ)l(qQalKs) t+ = ast/a2 x = streamwise coordinate x+ = x/a Xj = center of the yth surface element y = spanwise coordinate y+ = yla yj = center of the ;th surface element z = coordinate normal to the interface a = thermal diffusivity, m2/s /3 = velocity gradient, s1 F( ) = gamma function Ai/r = influence function, Eq. (13) TJ = dummy space variable in the spanwise direction A = dummy time variable //,( ) = unit step function f = dummy space variable in the streamwise direction $ = fundamental solution, defined in Appendix A

Conjugated heat transfer from a strip heater with the unsteady surface element method

The unsteady surface element method is extended to conjugated heat transfer from a strip heater where the unsteadiness is caused by a step change in the heater output flux. The geometry is a model of a flush-mounted, hot-film anemometer for air flow. The numerical results for the heater temperature agree with the analytical solution for early times and the steady-state results for the Nusselt number agree with an existing numerical solution.

Multinode Unsteady Surface Element Method With Application to Contact Conductance Problem

The unsteady surface element method is a powerful numerical technique for solution of linear transient two- and three-dimensional heat transfer problems. Its development originated with the need of solving certain transient problems for which similar or dissimilar bodies are attached one to the other over a part of their surface boundaries. In this paper a multinode unsteady surface element (MUSE) method for two arbitrary geometries contacting over part of their surface boundaries is developed and formulated. The method starts with Duhamel’s integral (for arbitrary time and space variable boundary conditions) which is then approximated numerically in a piecewise manner over time and the boundaries of interest. To demonstrate the capability of the method, it is applied to the problem of two semi-infinite bodies initially at two different temperatures suddenly brought into perfect contact over a small circular region. The results show excellent agreement between the MUSE solution and the other existing solutions.

Simplified Laplace Transform Inversion for Unsteady Surface Element Method for Transient Conduction

The unsteady surface element method is an analytical procedure for solving certain types of transient heat conduction problems. The method is intended for thermally contacting bodies of similar or dissimilar geometries such as occur in the contact conductance, fin, and temperature measurement problems. The method utilizes Duhamel's integral. The two simplified procedures in this paper utilize an approximate inverse Laplace transform relation given by Schapery to obtain functional forms of the solutions. In certain cases the approximate inverse Laplace relations simplify the solution greatly and produce acceptable accuracy. Tauberian, series, and numerical tests define these successful cases. The results presented are useful for the intrinsic thermocouple problem and as background for more general heat conduction problems.