Fluid Thermal Expansion Research Articles

Well Shut-In Pressure Determination Method for Deepwater Drilling Considering Fluid-Solid-Heat Coupling

Blowout is one of the most serious safety threats in deepwater drilling. Considering the characteristics of gas invasion in complex formations, gas migration and distribution, and dynamic changes in temperature inside a wellbore, a deepwater well-closing pressure determination method considering thermal-fluid-solid coupling was proposed. The model was verified using actual data, and the average error in the increase in casing pressure during the closing process was found to be 5.42%. The shut-in pressure of oil and gas wells under a transient shut-in process was analyzed. The results showed that the fluid thermal expansion caused by temperature recovery had a significant impact on the change in wellhead backpressure after well closure. Furthermore, the time required for the wellbore pressure to recover to the formation pressure varies nonlinearly with factors such as geothermal gradients, pit gains, bottom-hole pressure, and gas production indices. A pressure calculation chart was developed for deepwater drilling. The shut-in time for deepwater drilling should not be less than 15 min, and in cases where the gas production index is small and the bottomhole pressure difference is large, the shut-in time should exceed 90 min. The results can provide theoretical guidance and technical support for well shut-in processes during deepwater drilling.

Outcomes of Uniform as well as Non-Uniform Temperature Profiles on the Onset of Double Diffusive Magneto-Darcy-Rayleigh-Benard Convection in a Two Layer set up in the presence of Local Thermal Non-Equilibrium

Outcomes of uniform as well as non-uniform temperature profiles that is linear, parabolic, inverted parabolic, Piecewise Linear Profile Heated from Below (PLHB), Piecewise Linear Profile Cooled from Above (PLCA) and Step Function (SF) temperature profile on the onset of Double Diffusive Magneto-Darcy-Rayleigh-Benard (DDMDRB) Convection in a two layer set up in the presence of Local Thermal Non-Equilibrium with incompressible fluid horizontally surrounded by adiabatic rigid boundaries. Analytical solution to the attained problem is accomplished by means of regular perturbation technique. The consequences of altering the parameters namely fluid phase thermal expansion ratio, solid phase thermal expansion ratio, solid phase thermal diffusivity ratio, inter-phase diffusivity ratio, solute Rayleigh number, Chandrasekhar number, thermal ratio and porous parameter have been elucidated graphically.

Impact of Magnetic Field on the Onset of Double Diffusive Rayleigh-Benard Convection Governed by Local Thermal Non-Equilibrium in a Double Layered System

Onset of Double Diffusive Rayleigh-Benard-Magneto (DDRBM) convection is studied for a double layered system composed of incompressible fluid confined by adiabatic rigid peripheries under the governance of Local Thermal Non-Equilibrium (LTNE). Result of the acquired problem is obtained analytically through the mode of Regular Perturbation. Consequence of physical factors such as solid phase thermal expansion ratio, solid phase thermal diffusivity ratio and inter-phase thermal diffusivity ratio that favours LTNE are being analysed. The outcome of altering the constraints namely fluid phase thermal expansion ratio, solute Rayleigh number, Chandrasekhar number, thermal ratio, concentration ratio, fluid phase solute diffusivity ratio, solute diffusivity ratio in fluid layer and porous layer subject to LTNE set-up are paralleled with that of LTE set-up with diagrammatic representation.

Seal Qualification and Single-Trip Lockdown Sleeve Reduce Risk

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper OTC 29943, “Seal Qualification and Single-Trip Lockdown Sleeve Reduce Risk Through Innovative and Advanced Technology,” by Scott Patrick Ellisor, Rodrigo Albani Queiroz, and Carlos Eduardo Martins Gaban, Dril-Quip, et al., prepared for the 2019 Offshore Technology Conference Brasil, Rio de Janeiro, 29-31 October. The paper has not been peer reviewed. Copyright 2019 Offshore Technology Conference. Reproduced by permission. Traditionally, a wellhead system requires a drilling lockdown sleeve (LDS) during drilling operations to ensure that the seal assembly is rigid and prevents dynamic movement. By eliminating the LDS, however, the operator will save multiple trips into the well. Because of a seal-assembly design outlined in the complete paper, development of a single-trip LDS was possible for wellhead systems requiring extremely high lockdown requirements, allowing the user to save trips required by legacy designs. New Subsea Challenges In high-pressure/high-temperature subsea wells, annular pressure buildup, caused by thermal expansion of fluid, endangers well integrity. This phenomenon is caused by the increase of pressure during the production phase, temperature differential during the drilling phase, and fluid parameters. This type of annuli pressure buildup causes uplift as high as 3,200,000 lbf. The number of times the system needs to be shut down could be as many as eight per year for a 25-year well life; the life of the field during the production phase, therefore, could see as many as 200 total cycles. Saving Rig Time by Implementing New Technologies The technologies presented in the complete paper are the single-trip LDS and the dynamic casing hanger seal assembly. The single-trip LDS is used if an extremely high lockdown capacity is required. When this LDS is used, the need to run a lead impression tool (LIT) before running the lockdown sleeve is eliminated. Thus, an extra tool on the rig floor is eliminated, as well as a trip into the well. By eliminating the LIT run, between 5 and 10 hours could be saved, depending on water depth. Similar to the way in which LDS saves operators both operating expense and capital expenditure, the dynamic casing hanger seal assembly does as well. The dynamic casing hanger seal assembly has a higher lockdown capacity than the industry-standard seal assembly, which allows this seal to be run downhole without installation of an LDS. Because of this design, running the casing hanger seal assembly could save between three and five trips into the well, reducing overall risk.

Local thermal non-equilibrium effects on thermal pressurization in saturated porous media considering thermo-osmosis and thermal-filtration

This paper presents new fully-coupled analytical thermo-poroelastic solutions, introducing four new components to the traditional Local Thermal Non-Equilibrium (LTNE) theory: thermo-osmosis; thermal-filtration; heat sinks due to thermal expansion of the pore fluid and the solid phase; and vertical confinement. Thermo-osmosis, thermal-filtration, and fluid dilation are found to have very different effects in case of LTNE versus local thermal equilibrium (LTE), resulting in a fundamentally different thermo-hydraulic-mechanical (THM) response. The aforementioned non-traditional thermal processes cause generation of notably higher pore pressures in the porous medium that cannot be otherwise predicted using current solutions. Fluid temperatures are more sensitive to these non-traditional thermal processes, compared to the solid phase temperatures. Thermo-osmosis is found to have more significant effects under LTNE compared to LTE, and to be the dominant mechanism affecting solid temperatures, pore pressures and stresses. Thermal-filtration and heat sink due to fluid thermal expansion are the dominant mechanisms affecting fluid temperatures. The vertical confinement and formation anisotropy is shown to notably impact the THM response of a porous layer to heat source or sink. The impact of the aforementioned four novel components are found to be enhanced under an impermeable hydraulic boundary compared to a permeable boundary condition.

Simulation of an Adaptive Fluid-Membrane Piezoelectric Lens

In this paper, we present a finite-element simulation of an adaptive piezoelectric fluid-membrane lens for which we modelled the fluid-structure interaction and resulting membrane deformation in COMSOL Multiphysics®. Our model shows the explicit coupling of the piezoelectric physics with the fluid dynamics physics to simulate the interaction between the piezoelectric and the fluid forces that contribute to the deformation of a flexible membrane in the adaptive lens. Furthermore, the simulation model is extended to describe the membrane deformation by additional fluid forces from the fluid thermal expansion. Subsequently, the simulation model is used to study the refractive power of the adaptive lens as a function of internal fluid pressure and analyze the effect of the fluid thermal expansion on the refractive power. Finally, the simulation results of the refractive power are compared to the experimental results at different actuation levels and temperatures validating the coupled COMSOL model very well. This is explicitly proven by explaining an observed positive drift of the refractive power at higher temperatures.

Design and application of foamed spacer to mitigate annular pressure induced by fluid thermal expansion

Annular pressure build-up (APB) caused by fluid thermal expansion is a severe risk in deepwater wells. Foamed spacers can alleviate APB by carrying nitrogen into a sealed annulus. To properly design a foamed spacer, a prediction model based on the nonideal gas assumption is proposed for obtaining the optimal nitrogen quantity. The proposed model includes the pressure increment caused by nitrogen migration and fluid thermal expansion. A laboratory experiment is carried out to validate the proposed model. Traditional methods based on the ideal gas equation are compared with the proposed model; the results show that the error of the proposed model is less than that of the traditional model. A case study shows that more than 5% nitrogen should be used in the annulus. But, when the injected nitrogen quantity is greater than 20%, the annular pressure hardly decreases with the increase of nitrogen. For widespread applications, formulations of dimensionless pressure increases are developed using the Pi theorem, and the dimensionless graphs are presented. The pressure increase at different initial conditions can be obtained. By referring to the dimensionless graphs, engineers could determine the migration pressure caused by gas dissociation and the nitrogen quantity to mitigate the annular pressure increase.

A three-dimensional thermo-hydro-mechanical coupled model for enhanced geothermal systems (EGS) embedded with discrete fracture networks

Fractures play an important role in the geothermal energy recovery from enhanced geothermal systems (EGS) involving interacted multi-physical fields. A strongly coupled thermo-hydro-mechanical (THM) model is proposed to simulate the process of the long-term geothermal production in three-dimensional geothermal reservoirs containing arbitrary discrete fracture networks (DFNs). By introducing a strong discontinuity concept into the DFN model, the aperture variation of each fracture induced by the fluid pressure, external stresses and thermal expansion in the period of production can be captured. Non-isothermal fluid flow in DFN and the local thermal non-equilibrium (LTNE) between fluid and rock matrix are formulated and coupled with the fracture deformation model. Verification is carried out against an analytical solution, followed by a sensitivity and convergence analysis concerning time step and mesh size. This approach is then applied to Habanero EGS project in Australia to evaluate the geothermal productivity and efficiency for a period of 20 years with different injection and production pressures. The results demonstrate that the proposed model is robust and effective to simulate the THM coupled process in 3D fractured reservoirs.

In-Situ Upgrading Process Offers Potential for Heavy-Oil Recovery

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 190695, “Simulations of In-Situ Upgrading Process: Interpretation of Laboratory Experiments and Study of Field-Scale Test,” by A. Perez-Perez, SPE, M. Mujica, and I. Bogdanov, SPE, Computational Hydrocarbon Laboratory for Optimized Energy Efficiency, and A. Brisset, SPE, and O. Garnier, SPE, Total, prepared for the 2018 SPE Improved Oil Recovery Conference, Tulsa, 14–18 April. The paper has not been peer reviewed. In-situ upgrading (IU) is a promising method of improved viscous- and heavy-oil recovery. The IU process involves a reservoir being exposed to temperatures greater than 300°C long enough to promote a series of chemical reactions. In this work, the authors developed a numerical model of IU on the basis of laboratory experiences and validated results, applying the model to an IU test published in the literature. Simulation results for the cores submitted have shown that oil production and oil-sample quality were well-predicted by the model. Introduction In this paper, results of a previously published experiment and those of an unpublished experiment (Experiment B) are modeled and discussed. To accomplish this, an in-house code is developed for coupling the kinetic model with the thermodynamic description, a process detailed in the complete paper. This code was able to represent both the reaction dynamics of bitumen-fraction decomposition and the phase distribution and production of pseudocomponents. The resulting model was used to simulate an IU field-scale test (using data from Shell’s Viking pilot). Experimental Simulation A 2D homogeneous radial model was used to simulate two IU experiments. Elemental gridblock size was 5 mm in both directions. The model size was 5×30 cells (Experiment A) and 5×138 cells (Experiment B). A total of 30 pseudocomponents was provided in the simulation model. A sensitivity analysis was performed to identify the effect of some parameters in the IU model on the basis of experimental conditions in the experiments. This analysis was focused on less-certain properties related to the core and physical properties of oil pseudocomponents. Experimental and Simulation Results. In the process of model development and adaptation, more effort has been dedicated to specify the thermodynamic model than the kinetic model. The final model is capable of successfully predicting liquid production at various test scales and conditions. To explain the mismatch observed between simulation and experimental results of solid generation during IU, the authors performed estimations of solid mass that would be generated under the IU conditions of both experiments in a closed system. The resulting amount of generated solids is the highest possible because the initial hydrocarbons were all retained inside a core. The estimated amount of pyrobitumen for Experiment A was 0.022 kg. This value is still lower than that reported for laboratory conditions (0.024 kg). With regard to Experiment B, taking into account that 35% of the original oil was evacuated by thermal expansion of fluids, the estimated mass of pyrobitumen was approximately 0.070 kg (for closed volume). This value is significantly lower than that reported by the laboratory for this experiment (0.090 kg). The most probable reason is a natural difference in the physicochemical conditions for measurements in kinetic tests and IU experiments.

Oscillatory instability in a weakly non-Boussinesq fluid layer with a deformable surface

We investigate the oscillatory convective instability in a liquid layer with a deformable free surface. We provide linear stability analysis within a non-Boussinesq approach as our aim is to examine the influence of deformability of the free surface on the Rayleigh–Bénard–Marangoni instability. Within this approach, fluid is assumed to be isothermally incompressible; the density variations are accounted for in the continuity equation and in the buoyancy and inertial terms of the momentum equations. The numerical results show significant differences in stability behavior as compared to results obtained using the Boussinesq approach. Moreover, a novel oscillatory mode of instability is revealed for the zero Marangoni number and zero gravity. This result could not be obtained within the framework of the conventional Boussinesq approximation. Thorough investigation of the novel oscillatory mode let us propose the mechanism of this mode. It is connected with the capillary wave that enforced with thermal expansion of fluid. Weakly nonlinear analysis shows that supercritical branching of standing rolls is possible.

Prediction of annular pressure caused by thermal expansion by considering the variability of fluid properties

To accurately predict annular pressure build-up, which is caused by the thermal expansion of fluid in deepwater wells, a prediction model should consider that the thermal expansion coefficient and compressibility of the fluid vary with temperature and pressure. By means of 2-D Lagrangian interpolation, polynomial expressions are proposed to obtain the expansion coefficient and compressibility of water. A series of laboratory experiments is conducted to validate these polynomials. The numerical and experimental errors and error propagation are analyzed, indicating that the relative errors between the theoretical and experimental data are acceptable in engineering. To apply these polynomials to the estimation of annular pressure, the prediction model is improved. A case study and some crucial factors are analyzed. The results show that compared with published models that did not properly consider the fluid properties, the improved model, which is based on the proposed interpolation polynomials, advances the prediction accuracy. It is important to adopt the improved model and interpolation polynomials to predict the risk in engineering because, in most cases, previously published models may lead to unacceptable errors and an underestimation of the pressure. These findings help engineers predict annular pressure more accurately and contribute to the safety design of deepwater wells.

Research and development of compressible foam for pressure management in casing annulus of deepwater wells

Casing annulus pressure build-up owing to fluid thermal expansion in sealed annuli severely threaten the casing safety and wellbore integrity of deepwater wells. To mitigate the annular pressure with stronger effect and lower cost, a new type of syntactic foam with high compression ratio, which consists of hollow glass microspheres and a resin matrix system, was developed in this study. It can provide extra space for the annular fluid to expand when the annular pressure exceeds the specific foam crush pressure, allowing the foam to collapse and consequently, mitigate the annular pressure. The foam compression behavior is divided into three stages according to a hydrostatic compression test, and the annular pressure is mainly influenced by the microspheres volume and temperature. To optimize the parameters of the compressible foam, a novel model considering fluid thermal expansion, casing elastic deformation, and foam compression behavior in multi-annuli is established. The results indicate that the compressible foam can reduce the trapped annular pressure effectively and permanently. In a case study well, the annular pressure no longer exceeds the maximum acceptable annular pressure in each annulus with optimized design of the foam wraps and a suitable number of installed foam casing joints.

Modeling Thermal Pressurization Around Shallow Dikes Using Temperature‐Dependent Hydraulic Properties: Implications for Deformation Around Intrusions

AbstractPressurization and flow of groundwater around igneous intrusions depend in part on the hydraulic diffusivity of the host rocks and processes that enhance diffusivity, such as fracturing, or decrease diffusivity, such as mineral precipitation during chemical alteration. Characterizing and quantifying the coupled effects of alteration, pore pressurization, and deformation have significant implications for deformation around intrusions, geothermal energy, contact metamorphism, and heat transfer at mid‐ocean ridges. Fractures around dikes at Ship Rock, New Mexico, indicate that pore pressures in the host rocks exceeded hydrostatic conditions by at least 15 MPa following dike emplacement. Hydraulic measurements and petrographic analysis indicate that mineral precipitation clogged the pores of the host rock, reducing porosity from 0.25 to <0.10 and reducing permeability by ~5 orders of magnitude. Field data from Ship Rock are used to motivate and constrain numerical models for thermal pore fluid pressurization adjacent to a meter‐scale dike, using temperature‐dependent hydraulic properties in the host rock as a proxy for porosity loss by mineral precipitation during chemical alteration. Reduction in permeability by chemical alteration has a negligible effect on pressurization. However, reduction in porosity by mineral precipitation increases fluid pressure by constricting pore volume and is identified as a potentially significant source of pressure. A scaling relationship is derived to determine when porosity loss becomes important; if permeability is low enough, pressurization by porosity loss outweighs pressurization by thermal expansion of fluids.

Analysis of the wellhead growth in HPHT gas wells considering the multiple annuli pressure during production

In high-temperature and high-pressure (HPHT) gas wells, the wellhead growth caused by temperature and pressure effects during production might damage the well integrity. A calculation model of the wellhead growth produced by temperature and pressure effects was built. For the case well, the maximum pressures of annulus A, B and C are 64 MPa, 48 MPa and 38 MPa, respectively. The maximum production and intervention time are 114.5 × 104 m3/d and 540 d, respectively. Based on the calculation process, the maximum wellhead growth is 412.7 mm. The axial load caused by the multiple annuli pressure is second only to that caused by the casing axial temperature difference. Wellhead growth increases with the annulus fluid thermal expansion coefficient and decreases with the annulus fluid isothermal compression coefficient. The increasing annulus temperature difference might aggravate the effect of annulus fluid thermal properties on the wellhead growth. Selecting the casing with greater wall thickness and lower thermal expansion coefficient can reduce the wellhead growth. The annulus width has little effect on the wellhead growth while the annulus length will significantly change the wellhead growth. The wellbore multiple annuli pressure can increase the wellhead growth prominently. The annulus pressure management shall be introduced into the production. Optimizing the well structure and production plan and installing the wellhead monitoring equipment contribute to mitigating the wellhead growth.

A coupled THMC modeling application of cemented coal gangue-fly ash backfill

Cemented coal gangue-fly ash backfill (CGFB) is prepared by mixing coal gangue, fly ash, cement and water. The CGFB mixtures are filled into mined-out areas of coal mines for ground control and waste disposal. Once placed underground, the performance of CGFB is subjected to the thermal (T), hydraulic (H), mechanical (M) and chemical (C) processes and THMC coupling effects. Understanding the coupled THMC influence on CGFB is crucial for predicting the behavior of CGFB. This paper develops a THMC coupling model to describe and analyze the coupled THMC processes that occur in CGFB. The developed model considers the courses of thermal conduction and convection, fluid flow, suction development, chemical shrinkage, thermal expansion, and binder hydration. The simulation and prediction results of the developed model are compared with the experimentally tested data from 3 case studies. Good agreement is found between the modeling results and experimental data, validating the capability of the developed model to well characterize the properties of CGFB under coupled THMC loadings. The modeling results can also contribute to a better design of stable, durable and environment-friendly CGFB structures.

Fracture development and fluid pathways in shales during granite intrusion

Shales generally act as barriers to subsurface fluid flow due to their low permeability. Magmatic intrusion can produce fractures in host rocks, but the relationship between fracture development and permeability changes in shales is still not clear. A 48m-wide granitic dike intruded into the Cambrian black shales at 131.8±1.5Ma in the Luocun section of the Lower Yangtze region (China). Shale samples from the Luocun section were collected to investigate the influence of granite intrusion on fracture parameters, density, porosity and permeability of shales. Based on the cross-cutting relationship, cements and characteristics of microfractures, three phases of fractures in the Luocun shales were recognized and related with the stress evolution. The widespread, narrow and straight shear fractures (Phase-I) reflect initial fracturing under compression around the intrusion. With continued magma intrusion, thermal reaction of organic matter produced CH4, CO2 and H2O, which joined with siliceous fluids from the granitic dike and increased pore fluid pressure in surrounding shales. High pore fluid pressure and thermal expansion counteracted the regional compression and caused wide, curving extension fractures (Phase-II) in shales near the dike. Meanwhile siliceous fluids transported carboniferous residues through the open pore space and fractures, sealed the fractures and silicified the shales. In the late cooling stage, pore fluid pressure and thermal contraction produced bedding-parallel, narrow extension fractures (Phase-III) in shales near the dike. The small shear displacement along Phase-II and Phase-III fractures implies that they originated as extension fractures and immediately reactivated as shear planes.The effective porosity of shale samples increases from ~0.5% at distance ≥12.95m to ~2% at 10.05m and ~4% at 0.5m from the granitic dike. Pressure dependence of plug permeability of shale samples follow an exponential equation. At 6.9MPa, plug permeability of the shales increases from 1.81×10−19m2 at 27.45m to 4.98×10−16m2 at 0.5m. Within ~37% of the dike width, the fracture density, effective porosity, plug permeability, as well as the equivalent vitrinite reflectance from our previous study, increase sharply with a decreasing distance from the dike, suggesting coincidence of a narrow contact aureole with the “permeability aureole” due to localized hydrofracturing and hydrothermal circulation. Because Phase-II and Phase-III fractures only occur within ~37% of the dike width and they were filled by microcrystalline quartz and carboniferous residues, they should be effective fluid pathways and allow fast escape of shale gas during granite intrusion.

Flow dynamics in transient heat transfer of n-decane at supercritical pressure

Turbulent heat transfer of hydrocarbon fuel at supercritical pressure plays a crucial role in regenerative cooling of aerospace propulsion systems. In this paper, flow dynamics in transient heat transfer of n-decane at a supercritical pressure of 5MPahas been numerically investigated, focusing on the effects of a number of key influential parameters, including the surface heat flux, surface heating rate, cooling tube length, and inlet flow velocity, on the transient responding behaviors. Results indicate that the transient responding process is dictated by two fundamental mechanisms: the initial thermoacoustic oscillation, which is caused by strong fluid thermal expansion, and the subsequent transient convection. The thermoacoustic oscillating magnitude increases as the surface heat flux, surface heating rate, and cooling tube length are increased, but it decreases as the inlet flow velocity is increased. The surface heating rate and cooling tube length also exert strong impacts on the oscillating frequency of the thermoacoustic wave. Moreover, the cooling tube length and inlet flow velocity significantly affect the second-stage transient convective process and thus the total transient responding time, which both increase as the cooling tube length is increased and/or the inlet flow velocity is decreased. Results obtained herein are helpful for fundamental understanding of the transient heat transfer mechanisms relevant to regenerative engine cooling processes.

The interaction of thermocapillary and vibrational instabilities in a two-layer system subjected to the tangential vibrations

The paper deals with the investigation of interaction of thermocapillary, parametric and frozen wave instability mechanisms in a two-layer system of immiscible incompressible viscous fluids of different densities (more dense fluid is located below). External boundaries are rigid. They are maintained at constant different temperatures. The system is subjected to the gravity force and horizontal vibrations of finite frequency and amplitude. The study is performed taking into account the interface deformations. The thermal expansion of fluids is neglected. It is found that vibrations make strong destabilizing effect on the longwave instability mode related to the interface deformations and stabilizing effect on Pearson instability mode.

Transient responses of turbulent heat transfer of cryogenic methane at supercritical pressures

A numerical study has been conducted to analyze transient responding behaviors of fluid flow and heat transfer of the cryogenic methane at supercritical pressures, the physical phenomena closely related to the regenerative rocket engine cooling application. A steady-state cold flow is instantly enforced with a constant surface heat flux to activate the transient heat transfer process. The effects of surface heat flux, inlet flow velocity, and pressure on transient responses are studied in detail to obtain fundamental understanding of the underlying physical mechanisms. Results indicate that the increased fluid temperature during the heat transfer process leads to the significantly decreased fluid density at a supercritical pressure and consequently causes strong fluid thermal expansion, which results in flow oscillations. The strong pressure effect on thermophysical property variations in the supercritical-pressure heat transfer process, particularly in the trans-critical region, can lead to further extension of the transient responding process at a low inlet flow velocity and/or under a high surface heat flux. Flow oscillations become stronger and last longer under a higher surface heat flux and/or at a lower inlet flow velocity. An increased operating pressure slightly decreases the transient responding time. Under the tested conditions in the present work, the maximum transient response time is around 20ms.

Non-similar solutions for natural convection from a moving vertical plate with a convective thermal boundary condition

In a recent paper by Makinde (Thermal Science, 2011, Vol. 15, Suppl. 1, pp. S137-S143.) the effect of thermal buoyancy along a moving vertical plate with internal heat generation was considered. The plate thermal boundary condition was a convective condition with a heat transfer coefficient proportional to x-1/2 . The fluid thermal expansion coefficient was proportional to 1-x and the internal heat generation was assumed to decay exponentially across the boundary layer and proportional to x-1 in order that the problem accepts a similarity solution. In the present work, the same problem without heat generation is considered, with constant heat transfer coefficient and constant thermal expansion coefficient which is more realistic and has much more practical applications. The present problem is non-similar and results are obtained with the direct numerical solution of the governing equations. The problem is governed by the Prandtl number, the non-dimensional distance along the plate and a convective Grashof number, which is introduced for the first time. It is found that the wall shear stress, the wall heat transfer and the wall temperature, all increase with increasing distance and the wall temperature tends to 1. The influence of the convective Grashof number is to increase the wall shear stress and the wall heat transfer and to reduce the wall temperature.