A borehole temperature during drilling in a fractured rock formation

Drilling in brittle crystalline rocks is often accompanied by a fluid loss through the finite number of the major fractures intercepting the borehole. These fractures affect the flow regime and temperature distributions in the borehole and rock formation. In this study, the problem of borehole temperature variation during drilling of the fractured rock is analyzed analytically by applying the approximate generalized integral-balance method. The model accounts for different flow regimes in the borehole, for different drilling velocities, for different locations of the major fractures intersecting the borehole, and for the thermal history of the borehole exploitation, which may include a finite number of circulation and shut-in periods. Normally the temperature fields in the well and surrounding rocks are calculated numerically by the finite difference and finite element methods or analytically, utilizing the Laplace-transform method. The formulae obtained by the Laplace-transform method are usually complex and require tedious numerical evaluations. Moreover, in the previous research the heat interactions of circulating fluid with the rock formation were treated assuming constant bore-face temperatures. In the present study the temperature field in the formation disturbed by the heat flow from the borehole is modeled by the heat conduction equation. The thermal interaction of the circulating fluid with the formation is approximated by utilizing the Newton law of cooling at the bore-face. The discrete sinks of fluid on the bore-face model the fluid loss in the borehole through the fractures. The heat conduction problem in the rock is solved analytically by the heat balance integral method. It can be proved theoretically that the approximate solution found by this method is accurate enough to model thermal interactions between the borehole fluid and the surrounding rocks. Due to its simplicity and accuracy, the derived solution is convenient for the geophysical practitioners and can be readily used, for instance, for predicting the equilibrium formation temperatures.     

Non-Fourier heat conduction in a finite medium subjected to arbitrary periodic surface disturbance

The non-Fourier transient heat conduction in a finite medium under arbitrary periodic surface thermal disturbance is investigated analytically. In order to obtain the desired temperature field from the known solution for non-Fourier heat conduction under a harmonic disturbance, the principle of superposition along with the Fourier series representation of an arbitrary periodic function is employed. The developed method can be applied for more realistic periodic boundary conditions occurred in nature and technology.     

Numerical analysis of unsteady heat conduction in regular solid bodies comprising natural convection to nearby fluids

The present paper addresses unsteady, unidirectional heat conduction in regular solid bodies (vertical plate, horizontal cylinder, and sphere) that exchange heat by natural convection with a neighboring fluid. From thermal physics, natural convection constitutes a worst-case scenario for forced convection cooling. Under the premises of natural convection heat transfer, the unsteady, 1-dimensional heat conduction equation consists in a linear parabolic partial differential equation with a dominant natural convection boundary condition represented by the mean convective coefficient that depends upon temperature. As expected, the nonlinear unsteady, unidirectional heat conduction problem is complex and does not admit an exact, analytical solution. Instead, the nonlinear unsteady, unidirectional heat conduction problem forcibly necessitates approximate numerical treatment with the finite difference method. The computed dimensionless center, surface, and mean temperatures varying with dimensionless time are obtained numerically and are graphed for 3 solids: iron, aluminum, copper exposed to 3 fluids: air, water, oil; the 6 media are used in numerous engineering applications.     

APPLICATION OF THE NETWORK METHOD TO HEAT CONDUCTION PROCESSES WITH POLYNOMIAL AND POTENTIAL-EXPONENTIALLY VARYING THERMAL PROPERTIES

Nonlinear transient heat conduction in a finite slab with potential-exponential temperature-dependent specific heat and thermal conductivity is investigated numerically by using the network method. A general network model for this process is proposed, whatever the exponent of the temperature-dependent functions may be, including initial and boundary conditions. With this network model and using the electrical circuit simulation program PSPICE, time-dependent temperature and heat flux profiles at any location can be obtained. This approach allows us to solve this conduction problem by a general, efficient, and relatively simple method. To show the accuracy of the network method, a comparison is made of the present results and those obtained by other methods for a particular case.     

Heat Transfer in a Horizontal Cylinder With Eccentric Surfaces

Heat transfer in horizontal cylinders exposed to free convection and radiation is of importance in many industries. Usually this problem is treated by adopting a concentric geometry, disregarding that the external surface temperature is not uniform. If an eccentric geometry is used, the external surface temperature should have a larger variation, changing the flow around the cylinder and the heat transfer coefficient, either improving or reducing the heat transfer. A numerical analysis is presented of the heat transfer in a horizontal cylinder with an internal isothermal surface eccentric to the external surface that is exposed to air free convection and radiation. The conduction problem was solved analytically and integrated numerically, while the free convection was solved by the PHOENICS software. The parameters analyzed were the ratio of radius, the ratio between the material and air thermal conductivities, the Rayleigh number, the emissivity of the outer surface, and the eccentricity between the external and inner surfaces. The parameters of a proposed equation to estimate the total heat of an eccentric arrangement in terms of the total heat of the corresponding concentric arrangement and the ratio between the convective and conductive thermal resistances were determined for given ratios of radius and eccentricities.     

Numerical Simulation of Heat and Mass Transfer during Nanosecond Laser Chemical Vapor Deposition on a Particle Surface

Chemical vapor deposition of TiN on a spherical particle surface subjected to nanosecond laser heating is simulated numerically here. A thermal model is developed to describe the heat transfer within the particle, chemical reaction on the particle surface, and mass transfer in the gases. The heat conduction and mass diffusion equations are discretized using the finite volume method with fully implicit schemes, and solved with tridiagonal matrix method. Temporal distributions of particle surface temperature and deposited film thickness resulting from multiple-pulse irradiation are analyzed for a wide range of parameters including laser fluence, pulse repetition frequency, pulse width, initial particle temperature, particle radius, and total pressure in the reaction chamber.     

Finite line-source model for borehole heat exchangers: effect of vertical temperature variations

A solution to the three-dimensional finite line-source (FLS) model for borehole heat exchangers (BHEs) that takes into account the prevailing geothermal gradient and allows arbitrary ground surface temperature changes is presented. Analytical expressions for the average ground temperature are derived by integrating the exact solution over the line-source depth. A self-consistent procedure to evaluate the in situ thermal response test (TRT) data is outlined. The effective thermal conductivity and the effective borehole thermal resistance can be determined by fitting the TRT data to the time-series expansion obtained for the average temperature.     

A note on the determination of the thermal properties of a material in a transient nonlinear heat conduction problem

The determination of the temperature dependent thermal properties and the temperature distribution inside a heat conducting material when heat flux boundary conditions are prescribed is investigated. Assuming that the material has a known constant thermal diffusivity then the heat conduction problem is linearised by employing the Kirchhoff transformation and additional measurements of the temperature at an arbitrary space location are imposed in order to render a unique solution. The dependence of the thermal conductivity with the temperature is obtained as the sum of an infinite series, whilst the temperature solution is obtained implicitly and is then calculated numerically. The characteristics of the solutions with respect to the spatial position where the sensor is located is also discussed.     

Computer simulation of borehole ground heat exchangers for geothermal heat pump systems

Computer simulation of borehole ground heat exchangers used in geothermal heat pump systems was conducted using three-dimensional implicit finite difference method with rectangular coordinate system. Each borehole was approximated by a square column circumscribed by the borehole radius. Borehole loading profile calculated numerically based on the prescribed borehole temperature profile under quasi-steady state conditions was used to determine the ground temperature and the borehole temperature profile. The two coupled solutions were solved iteratively at each time step. The simulated ground temperature was calibrated using a cylindrical source model by adjusting the grid spacing and adopting a load factor of 1.047 in the difference equation. With constant load applied to a single borehole, neither the borehole temperature nor the borehole loading was constant along the borehole. The ground temperature profiles were not similar at different distances from the borehole. This meant that a single finite difference scheme was not sufficient to estimate the performance of a borefield by superposition. The entire borefield should be discretized simultaneously. Comparison was made between the present method and the finite line source model with superposition. The discrepancies between the results from the two methods increased with the scale of borefield. The introduction of time schedule revealed a discrepancy between the load applied to the ground heat exchanger and that transferred from the borehole to the ground, which was usually assumed to be the same when using analytical models. Hence, in designing a large borefield, the present method should give more precise results in dynamic simulation.     

ROUGH SIZE ESTIMATION OF A THERMALLY FRACTURED ZONE IN AN INFINITE HOT ROCK MASS

Abstract

Concerning the extraction of geothermal energy from a deep thermal reservoir by the downhole coaxial heat exchanger with a thermally insulated inner pipe proposed by Morita et al, we obtained rough estimates of a size of the fractured zone induced by thermal stresses due to injecting cold water into the hot rock mass through the pipe. We assumed complete spherical symmetry of the temperature and stress fields. At the rough estimation, we considered three typical or extreme cases. (1) The fracturing affects neither the loading capacity of a fractured rock mass nor the temperature distribution within the formation. (2) The fractured zone completely loses its loading capacity and is fully invaded by the borehole water. No disturbance of the fracturing makes any difference in the temperature. (3) The rock formation is assumed to have an appropriately increased fictitious conduction substituted for the heat transfer enhanced by the expected convection within the fractured zone in order to discuss the effects of an occurrence of heat convection within the fractured zone on the temperature and stress distributions and the fractured zone size. As a result, the size of the zone has been estimated to be about ten or more times the borehole radius.     

Three-Dimensional Thermoelastic Analysis of Rectangular Plates with Variable Thickness Subjected to Thermomechanical Loads

Three-dimensional thermoelastic analysis of simply supported rectangular plates with variable thickness and subjected to thermomechanical loads is investigated. An approximate analytical solution method is proposed. The temperature field and the displacements are represented by a double harmonic series, respectively. First, the heat conduction equation is solved analytically to obtain the nonlinear temperature series with unknown coefficients. Then the three-dimensional equilibrium differential equations are analytically solved to obtain the displacement component series with unknown coefficients. Thus the stress component series can be determined. The unknown coefficients in the temperature series and stress component series are approximately determined by using the upper surface and lower surface conditions of the plate. With the proposed procedure, the solutions satisfy the governing differential equations, the loading conditions, and the simply supported boundary conditions. The proposed solution method shows a good convergence and the results agree well with those obtained from a commercial finite element software, ANSYS. Two examples are used to demonstrate the effectiveness of the proposed solution method. The simultaneous effects of temperature change and applied mechanical load on the behavior of the plate are examined.     

Homotopy Perturbation Method for Thermal Stresses in an Annular Fin with Variable Thermal Conductivity

An approximate analytical solution method for thermal stresses in an annular fin with variable thermal conductivity is presented. Homotopy perturbation method (HPM) is employed to estimate the non-dimensional temperature field by solving nonlinear heat conduction equation. The closed-form solutions for the thermal stresses are formulated using the classical thermoelasticity theory coupled with HPM solution for temperature field. The plane state of stress conditions are considered in this study. The effects of thermal parameters such as variable thermal conductivity parameter (β), thermogeometric parameter (K), and the non-dimensional coefficient of thermal expansion (χ) on the temperature field and stress field are studied. The results for temperature field and stress field obtained from HPM-based solution are found to be in very close agreement with the results available in literature. Furthermore, the HPM solution is found to be very efficient and handles nonlinear heat transfer equation with greater convenience.     

Approximate Analytic Estimate of Axial Fluid Conduction in Laminar Forced Convection Tube Flows with Zero-to-Uniform Step Wall Heat Fluxes

The influence of axial fluid conduction on low Péclet number flows in the thermal entrance region of long circular tubes is investigated in this theoretical study. The convective heat transport of viscous fluids relates to a specific condition under which the first part of the tube ( x h 0) is insulated and the second part of the tube ( x > 0) receives a heat flux of uniform intensity. A conjugate one-dimensional lumped model produces a solution of compressed algebraic form that is able to deliver dependable mean bulk temperatures that are in perfect agreement with those obtained numerically by the formal conjugate two-dimensional distributed model. As a by-product of the succession of algebraic calculations within the platform of the lumped model, the critical Péclet number Pe cr has been easily quantified. This number is a figure-of-merit of remarkable importance in the modeling of forced heat convection in tubes because it establishes the threshold between two contrasting situations: one embracing axial fluid conduction (finite Pe) and the other implicating negligible axial fluid conduction (Pe M X ). In addition, the local wall temperatures were calculated with an approximate engineering procedure, showing good agreement with those determined numerically by the formal conjugate two-dimensional distributed model.     

Forced convective heat transfer in a microtube including rarefaction,viscous dissipation and axial conduction effects

Subsonic gas convective heat transfer in a microtube with a constant cross-sectional area and uniform wall temperature is investigated both analytically and numerically. First, the effect of rarefaction on heat transfer characteristics, at a distance from the inlet where Nu becomes constant, is analytically investigated for two cases: (i) including and (ii) neglecting the viscous dissipation effect. An exact solution for Nu in fully developed flow is presented for the case without viscous dissipation, while a closed-form solution for the asymptotic Nu is also provided for the case with viscous dissipation. Next, a numerical model is employed to investigate the simultaneous effects of rarefaction, viscous dissipation, and axial conduction for developing hydrodynamic and temperature conditions. The Nusselt number is substantially affected by viscous dissipation, rarefaction and axial conduction.     

Periodic heat conduction in a solid homogeneous finite cylinder

Analytic solution of the steady periodic, non-necessarily harmonic, heat conduction in a homogeneous cylinder of finite length and radius is given in term of Fourier transform of the fluctuating temperature field. The solutions are found for quite general boundary conditions (first, second and third kind on each surface) with the sole restriction of uniformity on the lateral surface and radial symmetry on the bases. The thermal quadrupole formalism is used to obtain a compact form of the solution that can be, with some exception, straightforwardly extended to multi-slab composite cylinders. The limiting cases of infinite thickness and infinite radius are also considered and solved.     

Thermoelastic study of a functionally graded annular fin with variable thermal parameters using semiexact solution

Abstract

In this paper, the thermoelastic behavior of a functionally graded material (FGM) annular fin is investigated. The material properties of the annular fin are assumed to vary radially. The heat transfer coefficient and internal heat generation are considered to be functions of temperature. A closed form solution of nonlinear heat transfer equation for the FGM fin is obtained using the homotopy perturbation method (HPM) which leads to nonuniform temperature distributions within the fin. The temperature field is then coupled with the classical theory of elasticity and the associated thermal stresses are derived analytically. For the correctness of the present closed form solution for the stress field, the results are compared with the ANSYS-based finite element method (FEM) solution. The present HPM-based closed form solution of the stress field exhibits a good agreement with the FEM results. The effect of various thermal parameters such as the thermogeometric parameter, conduction-radiation parameter, internal heat generation parameter, coefficient of variation of thermal conductivity, and the coefficient of thermal expansion on the thermal stresses are discussed. The results are presented in both nondimensional and dimensional form. The dimensional stress analysis discloses the suitability of FGM as the fin material in practical applications.     

Thermo-Elastic Analysis of a Cracked Half-Plane Under a Thermal Shock Impact Using the Hyperbolic Heat Conduction Theory

The transient thermal stresses around a crack in a thermo-elastic half-plane are obtained under a thermal shock using the hyperbolic heat conduction theory. Fourier, Laplace transforms and singular integral equations are applied to solve the temperature and thermal stress fields consecutively. The integral equations are solved numerically and the asymptotic fields around the crack tip are obtained. Numerical results show that the hyperbolic heat conduction have significant influence on the dynamic temperature and stress field. It is suggested that to design materials and structures against fracture under thermal loading, the hyperbolic model is more appropriate than the Fourier heat conduction model.     

TRANSIENT THERMOELASTIC ANALYSIS OF AN ANNULAR FIN WITH COUPLING EFFECT AND VARIABLE HEAT TRANSFER COEFFICIENT

A hybrid numerical method of the Laplace transformation and the finite difference method is applied to solve the transient thermoelastic problem of an annular fin, in which the thermomechanical coupling effect is taken into account in the governing equation of heat conduction and the heat transfer coefficient is a function of the radius of the fin. The general solutions of the governing equations are first solved in the transform domain. Then the inversion to the real domain is completed via the method of matrix similarity transformation and Fourier series technique. The transient distributions of temperature increment and thermal stresses of the fin in the real domain are calculated numerically. The presented method is more efficient in computing time and is applicable to other types of boundary conditions.     

A least squares method for a longitudinal fin with temperature dependent internal heat generation and thermal conductivity

Approximate but highly accurate solutions for the temperature distribution, fin efficiency, and optimum fin parameter for a constant area longitudinal fin with temperature dependent internal heat generation and thermal conductivity are derived analytically. The method of least squares recently used by the authors is applied to treat the two nonlinearities, one associated with the temperature dependent internal heat generation and the other due to temperature dependent thermal conductivity. The solution is built from the classical solution for a fin with uniform internal heat generation and constant thermal conductivity. The results are presented graphically and compared with the direct numerical solutions. The analytical solutions retain their accuracy (within 1% of the numerical solution) even when there is a 60% increase in thermal conductivity and internal heat generation at the base temperature from their corresponding values at the sink temperature. The present solution is simple (involves hyperbolic functions only) compared with the fairly complex approximate solutions based on the homotopy perturbation method, variational iteration method, and the double series regular perturbation method and offers high accuracy. The simple analytical expressions for the temperature distribution, the fin efficiency and the optimum fin parameter are convenient for use by engineers dealing with the design and analysis of heat generating fins operating with a large temperature difference between the base and the environment.     

Fully developed forced convection of the Phan-Thien-Tanner fluid in ducts with a constant wall temperature

Two problems of laminar-forced convection in pipes and channels, under fully developed conditions, are solved for an imposed constant temperature at the wall, with fluids obeying the simplified Phan-Thien-Tanner (SPTT) model. The fluid properties are taken as constants and axial conduction is negligible. The first case represents the asymptotic behaviour of the Graetz problem for the SPTT fluid, i.e., equilibrium between axial convection and radial conduction of thermal energy with negligible viscous dissipation. The solution is given by an analytical expression but it is only approximate (within 0.3%) as it was obtained with an algebraic method based on successive approximations. The second problem has an exact analytical solution representing the equilibrium between viscous dissipation and radial heat conduction, with negligible axial convection and a constant wall temperature.