Time-varying Boundary Conditions Research Articles

A new front surface heat flux calibration method for a 1-D nonlinear thermal system with a time-varying back boundary condition

A novel calibration methodology is presented for resolving the front surface heat flux in a one-dimensional nonlinear sample with an unknown time-varying back boundary condition. This method combines the attributes of the recently reported linear two-probe calibration equation and the nonlinear one-probe calibration equation. The proposed calibration formulation is expressed in terms of a Volterra integral equation of the first type. This functional equation relates the rescaled unknown front surface heat flux to two rescaled calibration front surface heat fluxes and corresponding rescaled temperature data at the two in-depth probes when the sample is subjected to the unknown boundary conditions. A localized Tikhonov regularization scheme is introduced for generating the family of predictions based on the Tikhonov parameter spectrum. The L-curve strategy is used to extract the optimal prediction. This paper studies the effectiveness of this new calibration equation for two engineering materials, stainless steel 304 and a carbon composite. Results show that properly regularized predictions are stable and accurate in the presence of significant noise. Moreover, it is demonstrated that a preferred back boundary condition strategy for calibration tests reduces the ill-conditioning effects of the kernel. This new calibration method does not require the specification of the probe locations, although knowledge of the thermophysical properties is necessary.

The study of thermal field of an object represented in the basis of green functions

Existing means of thermal field modeling for the construction of the real time spatial distribution of the field were analyzed. The expediency of using the method of integral transformations for solving heat conduction problems was shown. Based on the comparison of methods of thermal fields construction method of integral transformations in the basis of the Green's functions was chosen. Construction of Green's functions for the use in the experimental facility is performed by the method of Laplace integral transforms. Software, generated based on the use of Matlab environment, allows to adjust the visualized field pattern by actual measured temperature values. Temperature point values of the experimental facility field were measured by NAT thermocouples, that are connected to a PC through OWEN 150UM controller interface, which makes it possible not only to form an array of measured values of the temperature, but also to adjust the time-varying boundary conditions of the calculated heat equation. The main purpose of the research is to assess the performance of the developed algorithms to reduce the field frame modeling time, which would allow to use the resulting pattern of the thermal field distribution in the industrial thermal facilities control process. The advantage of the thermal field constructing algorithm in the basis of the Green's function is the ability of the independent calculation of all points of the investigated area, allowing in the future to speed up the computation by applying high-performance computing.

A practical model for sunlight disinfection of a subtropical maturation pond

Maturation ponds are a type of waste stabilisation pond (WSP) designed to reduce carbon, nutrients and pathogens in the final stages of a WSP wastewater treatment system. In this study, a one-dimensional plug-flow pond model is proposed to predict temperature and E. coli concentration distributions and overall pond disinfection performance. The model accounts for the effects of vertical mixing and ultraviolet light-dependent die-off rate kinetics. Measurements of radiation, wind-speed, humidity and air temperature are recorded for model inputs and good agreement with measured vertical temperature distributions and outlet E. coli concentrations is found in an operational, subtropical maturation pond. Measurements and the model both show a diurnal pattern of stratification during daylight hours and natural convective mixing at night on days corresponding to low wind speeds, strong heat input from solar radiation and clear night skies. In the evenings, the thermal stratification is shown to collapse due to surface energy loss via longwave radiation which triggers top-down natural convective mixing. The disinfection model is found to be sensitive to the choice of die-off kinetics. The diurnal mixing pattern is found to play a vital role in the disinfection process by ensuring that pathogens are regularly transported to the near-surface layer where ultraviolet light penetration is effective. The model proposed in this paper offers clear advantages to pond designers by including geographical specific, time-varying boundary conditions and accounting for the important physical aspects of vertical mixing and sunlight inactivation processes, yet is computationally straightforward.

Perturbations to the Spatial and Temporal Characteristics of the Diurnally-Varying Atmospheric Boundary Layer Due to an Extensive Wind Farm

The effect of extensive terrestrial wind farms on the spatio-temporal structure of the diurnally-evolving atmospheric boundary layer is explored. High-resolution large-eddy simulations of a realistic diurnal cycle with an embedded wind farm are performed. Simulations are forced by a constant geostrophic velocity with time-varying surface boundary conditions derived from a selected period of the CASES-99 field campaign. Through analysis of the bulk statistics of the flow as a function of height and time, it is shown that extensive wind farms shift the inertial oscillations and the associated nocturnal low-level jet vertically upwards by approximately 200 m; cause a three times stronger stratification between the surface and the rotor-disk region, and as a consequence, delay the formation and growth of the convective boundary layer (CBL) by approximately 2 h. These perturbations are shown to have a direct impact on the potential power output of an extensive wind farm with the displacement of the low-level jet causing lower power output during the night as compared to the day. The low-power regime at night is shown to persist for almost 2 h beyond the morning transition due to the reduced growth of the CBL. It is shown that the wind farm induces a deeper entrainment region with greater entrainment fluxes. Finally, it is found that the diurnally-averaged effective roughness length for wind farms is much lower than the reference value computed theoretically for neutral conditions.

Vacuum excitation by sudden appearance and disappearance of a Dirichlet wall in a cavity

Vacuum excitation by time-varying boundary conditions is not only of fundamental importance but also has recently been confirmed in a laboratory experiment. In this paper, we study the vacuum excitation of a scalar field by the instantaneous appearance and disappearance of a both-sided Dirichlet wall in the middle of a 1D cavity, as toy models of bifurcating and merging spacetimes, respectively. It is shown that the energy flux emitted positively diverges on the null lines emanating from the appearance and disappearance events, which is analogous to the result of Anderson and DeWitt. This result suggests that the semiclassical effect prevents the spacetime both from bifurcating and merging. In addition, we argue that the diverging flux in the disappearance case plays an interesting role to compensate for the lowness of ambient energy density after the disappearance, which is lower than the zero-point level.

Estimation of hydrodynamic pattern change of Ichamati River using HEC RAS model, West Bengal, India

The physically-based hydrodynamic model can simulate the water flow dynamics of a stream network against time varying boundary conditions that can be implemented for the case of Ichamati River. Such river models are often an important component of flood forecasting, tidal fluctuation system that forecasts river levels in flood prone regions of the middle stream of Ichamati river. The measurements of river bed depth/slope, water quality (river cross section), floodplain mapping and boundary condition flow are essential for the set up of a river model. But one can use proxy approaches relying mostly on remote sensing data from space platforms for the purpose. In this study, we set up the one dimensional river analysis system (RAS) model of the hydrologic engineering center (HEC) over the stream network of the Ichamati river basins. Good quality in situ measurements of river hydraulics (cross section, slope, flow) were available only for the upstream and middle stream flood prone region of the basin. The de-siltation of the stretch of Ichamati river under survey i.e. middle part showed impacts which were positive in ways more than one. The river velocity in the top strata is 0.3 m/s from model and the river regained its ecological flow. Also, the increased salinity of the river helped in cultivation of different species of fishes, which are socio economically relevant in the adjoining areas. Due to the increased capacity of the channel in the middle part possibility of flood has also decreased. The total volume of silt accumulated has been estimated 277,589.92 m3. The rate of sedimentation was then calculated and found to be 16,328.82 m3/month silt deposition during monsoon and 11,109 m3 silt deposition during non monsoon period. It may be concluded that out of 6 % annual silt deposition, 5 % silt deposition took place during the month between June and October and 1 % silt deposition during the month between November and May. Silt excavation along 20 km stretch of Ichamati river showed distinct rejuvenation of the river stretch. The program of silt excavation along other stretches of river Ichamati study area would certainly bring positive impact on the river ecology and environment. Ecological flow needs to be maintained in rivers in the upstream due to chocking the silt deposition. Ecological flow in Ichamati river is absent in many stretches of the river due to heavy silt deposition.

Explicit Solution for Cylindrical Heat Conduction

The heat equation is a partial differential equation that elegantly describes heat conduction or other diffusive processes. Primary methods for solving this equation require time-independent boundary conditions. In reality this assumption rarely has any validity. Therefore it is necessary to construct an analytical method by which to handle the heat equation with time-variant boundary conditions. This paper analyzes a physical system in which a solid brass cylinder experiences heat flow from the central axis to a heat sink along its outer rim. In particular, the partial differential equation is transformed such that its boundary conditions are zero which creates a forcing function in the transform PDE. This transformation constructs a Green’s function, which admits the use of variation of parameters to find the explicit solution. Experimental results verify the success of this analytical method. KEYWORDS: Heat Equation; Bessel-Fourier Decomposition; Cylindrical; Time-dependent Boundary Conditions; Orthogonality; Partial Differential Equation; Separation of Variables; Green’s Functions

Homogenization of high-frequency wave propagation in linearly elastic layered media using a continuum Irving-Kirkwood theory

This article presents an application of a recently developed continuum homogenization theory, inspired by the classical work of Irving and Kirkwood, to the homogenization of plane waves in layered linearly elastic media. The theory explicitly accounts for the effects of microscale dynamics on the macroscopic definition of stress. It is shown that for problems involving high-frequency wave propagation, the macroscopic stress predicted by the theory differs significantly from classical homogenized stress definitions. The homogenization of plane waves is studied to illustrate key aspects and implications of the theory, including the characteristics of the homogenized macroscopic stress and the influence of frequency on the determination of an intermediate asymptotic length scale. In addition, a method is proposed for predicting the homogenized stress field in a one-dimensional bar subjected to a frequency-dependent forced vibration using only knowledge of the boundary conditions and the material’s dispersion solution. Furthermore, it is shown that due to the linearity of the material, the proposed method accurately predicts the homogenized stress for any time-varying displacement or stress boundary condition that can be expressed as a sum of time-periodic signals.

A new approach and results of wall and air temperature dynamic analysis in underground spaces

In this paper our primary aim is to define the changes of air and internal wall temperature in underground spaces in time domain. As an additional aim the change of heat flux through the wall in time domain has been calculated. Based on the heat balance, the dynamic basic equation of the space has been defined. The basic equation is a differential equation which contains the internal heat sources and the heat capacity of the space. For solving the basic equation, the initial condition, the time-varying boundary condition of the third kind and the Fourier's conductivity differential equation are necessary. The convolution integral of the solution function has been obtained by the use of the integral-differential equation acquired by substituting the temperatures and heat fluxes into the basic equation. The solution of the acquired equation can be obtained in a numerical way. Our new mathematical approach to the solution of the physical model makes it possible to investigate the air and wall temperatures, as well as the heat flow through the wall in underground spaces.

Atmospheric composition in the Eastern Mediterranean: Influence of biomass burning during summertime using the WRF-Chem model

The composition of the atmosphere over the Aegean Sea (AS) during an ‘Etesian’ outbreak under the influence of biomass burning (BB) activity is investigated. Simulations with the fully coupled WRF-Chem model during the Aegean-GAME campaign (29/8-9/9/2011) are used to examine the BB effect over the region. Two distinct Etesian flow patterns characterized by different transport conditions are analysed. The influence of the off-line calculated BB emissions on the atmospheric chemical composition over the AS under these conditions is estimated. In addition, sensitivity runs are used to examine the influence of the biogenic emissions calculated on-line and the realistic representation of the stratosphere-troposphere exchange processes are investigated through the time-varying chemical boundary conditions from the MOZART global chemical transport model. The horizontal and vertical distributions of gaseous and aerosol species are simulated under long-range transport conditions and interpreted in relation to the evolution of the Planetary Boundary Layer (PBL). In the case of a weaker synoptic system (medium-range transport conditions), even a small variability of meteorological parameters in limited areas become critical for the spatial distribution of gases and aerosols. The BB activity increases O3, PM2.5 and organic matter concentrations up to 5.5 ppb, 5.8 μg m−3 and 3.3 μg m−3, respectively. The spatial extent of the simulated BB plumes is further examined by comparison with airborne measurements of hydrogen cyanide (HCN). The estimated effect of biogenic emissions on O3 and PM2.5 concentrations is either positive or negative (±6 ppb for O3 and up to ± 1 μg m−3 for PM2.5) depending on the emission algorithm employed. The realistic representation of the chemical boundary conditions reproduces an observed layer rich in O3 above 4 km, but also increases O3 concentrations inside the PBL by up to 40%.

Nitsche’s method for parabolic partial differential equations with mixed time varying boundary conditions

We investigate a finite element approximation of an initial boundary value problem associated with parabolic Partial Differential Equations endowed with mixed time varying boundary conditions, switching from essential to natural and vice versa. The switching occurs both in time and in different portions of the boundary. For this problem, we apply and extend the Nitsche’s method presented in [Juntunen and Stenberg, Math. Comput. (2009)] to the case of mixed time varying boundary conditions. After proving existence and numerical stability of the full discrete numerical solution obtained by using the θ-method for time discretization, we present and discuss a numerical test that compares our method to a standard approach based on remeshing and projection procedures.

Viscous fluid damping in a laterally oscillating finger of a comb-drive micro-resonator based on micro-polar fluid theory

Viscous damping is a dominant source of energy dissipation in laterally oscillating micro-structures. In micro-resonators in which the characteristic dimensions are comparable to the dimensions of the fluid molecules, the assumption of the continuum fluid theory is no longer justified and the use of micro-polar fluid theory is indispensable. In this paper a mathematical model was presented in order to predict the viscous fluid damping in a laterally oscillating finger of a micro-resonator considering micro-polar fluid theory. The coupled governing partial differential equations of motion for the vibration of the finger and the micro-polar fluid field have been derived. Considering spin and no-spin boundary conditions, the related shape functions for the fluid field were presented. The obtained governing differential equations with time varying boundary conditions have been transformed to an enhanced form with homogenous boundary conditions and have been discretized using a Galerkin-based reduced order model. The effects of physical properties of the micro-polar fluid and geometrical parameters of the oscillating structure on the damping ratio of the system have been investigated.

Clinical assessment of intraventricular blood transport in patients undergoing cardiac resynchronization therapy.

In the healthy heart, left ventricular (LV) filling generates different flow patterns which have been proposed to optimize blood transport by coupling diastole and systole. This work presents a novel image-based method to assess how different flow patterns influence LV blood transport in patients undergoing cardiac resynchronization therapy (CRT). Our approach is based on solving the advection equation for a passive scalar field from time-resolved blood velocity fields. Imposing time-varying inflow boundary conditions for the scalar field provides a straightforward method to distinctly track the transport of blood entering the LV in the different filling waves of a given cardiac cycle, as well as the transport barriers which couple filling and ejection. We applied this method to analyze flow transport in a group of patients with implanted CRT devices and a group of healthy volunteers. Velocity fields were obtained using echocardiographic color Doppler velocimetry, which provides two-dimensional time-resolved flow maps in the apical long axis three-chamber view of the LV. In the patients under CRT, the device programming was varied to analyze flow transport under different values of the atrioventricular conduction delay, and to model tachycardia (100 bpm). Using this method, we show how CRT influences the transit of blood inside the left ventricle, contributes to conserving kinetic energy, and favors the generation of hemodynamic forces that accelerate blood in the direction of the LV outflow tract. These novel aspects of ventricular function are clinically accessible by quantitative analysis of color-Doppler echocardiograms.

Reduced order model analysis method via proper orthogonal decomposition for variable coefficient of transient heat conduction based on boundary element method

Boundary element method (BEM) is widely used in engineering analysis, especially in solving the transient heat conduction problem because of the advantage that only boundary of the problem needs to be discretized into elements. The general procedure of solving the variable-coefficient transient heat conduction problem by using the BEM is as follows. First, the governing differential equations are transformed into the boundary-domain integral equations by adopting the basic solution of the linear and homogeneous heat conduction problemGreen function. Second, domain integrals in the integral equation are converted into boundary integrals by the radial integral method or the dual reciprocity method. Finally, the time difference propulsion technology is used to solve the discrete time differential equations. A large number of practical examples verify the correctness and validity of the BEM in solving the variable coefficient of transient heat conduction problem. However, two deficiencies are encountered when the system of time differential equations is solved with the time difference method, i.e., one is the stability of the algorithm, which is closely related to the time step size, and the other is time-consuming when the freedom degree of the problem is large and all specified time steps are considered, because a system of linear equations needs to be solved in each time step. Therefore, in this paper we present a reduced order model analysis method of solving the variable-coefficient transient heat conduction problem based on BEM by using the model reduction method of proper orthogonal decomposition (POD). For variable-coefficient transient heat conduction problems, the discrete integral equations which are suitable for order reduction operation are deduced by using the BEM, the reduced order model is established by using the model reduction method of POD, and a lowdimensional approximate description of the transient heat conduction problem under time-varying boundary condition is obtained by projection of the initial discrete integral equations on some few dominant POD modes obtained from the problem under constant boundary conditions. First, for a variable coefficient transient heat conduction problem, boundary-domain integral equations are established and the domain integrals are transformed into boundary integrals by using the radial integration method. Second, the time differential equations with discrete format which is suitable for order reduction operation are obtained by reorganizing the integral equations. Third, the POD modes are developed by calculating the eigenvectors of an autocorrelation matrix composed of snapshots which are clustered by the given results obtained from experiments, BEM or other numerical methods for transient heat transfer problem with constant boundary conditions. Finally, the reduced order model is established and solved by projecting the time differential equations on reduced POD modes. Examples show that the method developed in this paper is correct and effective. It is shown that 1) the low order POD modes determined under constant boundary conditions can be used to accurately analyze the temperature field of transient heat conduction problems with the same geometric domain but a variety of smooth and time-varying boundary conditions; 2) the establishment of low order model solves the problem of heavy workload encountered in BEM where a set of large linear equations will be formed and solved in each time step when using the time difference method to solve the large time differential equations.

Topology optimization of unsteady flow problems using the lattice Boltzmann method

This article demonstrates and discusses topology optimization for unsteady incompressible fluid flows. The fluid flows are simulated using the lattice Boltzmann method, and a partial bounceback model is implemented to model the transition between fluid and solid phases in the optimization problems. The optimization problem is solved with a gradient based method, and the design sensitivities are computed by solving the discrete adjoint problem. For moderate Reynolds number flows, it is demonstrated that topology optimization can successfully account for unsteady effects such as vortex shedding and time-varying boundary conditions. Such effects are relevant in several engineering applications, i.e. fluid pumps and control valves.

Advancing river modelling in ungauged basins using satellite remote sensing: the case of the Ganges–Brahmaputra–Meghna basin

ABSTRACTRiver modelling is the process of simulating the water flow dynamics of a stream network against time-varying boundary conditions. Such river models are often an important component of any flood forecasting system that forecasts river levels in flood-prone regions. However, large river basins such as the Ganges, Brahmaputra, and Meghna (GBM), Indus, Irrawaddy, Salween, Mekong, and Niger in the developing world are mostly ungauged as they lack the necessary and routine in situ measurements of river bed depth/slope, bathymetry (river cross section), flood plain mapping, and boundary condition flows for setting up of a river model. For such basins, proxy approaches relying primarily on remote-sensing data from space platforms may be the only way to overcome the lack of in situ data. In this study, we share our experience in setting up the one-dimensional River Analysis System model of the Hydrologic Engineering Center over the stream network of the GBM basin. Good-quality in situ measurements of river hydraulics (cross section, slope, flow) were available only for the basin's downstream and flood-prone region, which comprises 7% of the total basin area. For the remaining 93% of the basin area, data from the following satellite sensors were used to build a functional river model: (a) Shuttle Radar Topography Mission to derive river network and adjust river bed profiles; (b) Landsat–MODIS for updating river network and flow direction generated by elevation data; (c) radar altimetry data to build the depth versus width relationship at river locations; and (d) satellite precipitation-based hydrologic modelling of lateral flows into major rivers. We measured the success of our approach by systematically testing how well the basin-wide river model could simulate river-level dynamics at two measured downstream low-lying locations. This paper summarizes the key hurdles faced and offers a step-by-step ‘rule book’ approach to setting up river models for large ungauged river basins around the world. By following these rules in a systematic way, the root mean squared error for river-level simulation was reduced from 3 to 1 m. Such a guide can be useful for setting up river hydraulic models for flood forecasting systems in ungauged basins such as the Niger, Mekong, Irrawaddy, and Indus.

A 1D/3D Methodology for the Prediction and Calibration of a High Performance Motorcycle SI Engine

Nowadays internal combustion engines development is widely supported by 1D and 3D codes. On the other hand, an extensive experimental activity at test bench involves increased development cost and time-to-market of a new engine. For this reason, a numerical methodology capable to provide a reliable estimation of engine performance starting from a reduced set of measured data represents a very promising approach.In this paper, a hierarchical 1D/3D numerical procedure is proposed with reference to a motorcycle naturally aspirated spark ignition engine to predict its performance even in absence of experimental data.To this aim, the real engine is geometrically characterized through a reverse engineering process. Different measurements including 3D cylinder geometry, main lengths and diameters of intake /exhaust systems and valve lift profiles, are carried out. Then, a 1D model of the whole engine is realized within the GT-Power™ code, while a 3D model of the sole cylinder is developed within ANSYS Fluent™ environment.The exchange of data between 1D and 3D models starts with preliminary 3D CFD analyses, performed to evaluate the discharge coefficients of intake and exhaust valves. The latter are passed to 1D model to compute the time-varying boundary conditions at the intake and exhaust head ducts, under motored operation. Multi-cycle 3D CFD analyses are hence carried out to describe the in-cylinder mean and turbulent flow fields in motored conditions. The mass-averaged 3D results are then used to tune the turbulence sub-model included in the 1D engine model.The last step of the procedure is the computation of the engine performance under fired conditions at full load by means of the 1D simulation. The numerical/experimental comparison of performance parameters demonstrates that the proposed methodology is capable to satisfactory describe the overall engine behavior even in absence of detailed experimental data.

Numerical simulation of wind loads on a parabolic trough solar collector using lattice Boltzmann and finite element methods

In this study, we evaluate lattice Boltzmann and finite element methods for wind load estimation of parabolic trough solar collectors. Mean, root-mean-square (RMS) and peak values of aerodynamic load coefficients of an isolated collector are estimated using large-eddy simulation and compared with experimental results obtained in a boundary layer wind tunnel. Despite their fundamental differences, the two numerical approaches yield similar values for the drag, lift and pitching moment indicating that the results are essentially independent of the numerical schemes. Time-varying inlet boundary conditions are obtained using an efficient synthetic generation technique. The statistics of the numerical boundary layer are investigated by comparing mean and turbulence intensity profiles at varying distances from the inlet as well as the spectra and autocorrelation at the position of the structure. Through appropriate modelling of the boundary layer, the numerical models are shown to reproduce the mean, RMS and peak load behaviour measured in the wind tunnel.

On the modeling of a piezoellectrically actuated micro-sensor for measurement of microscale fluid physical properties

This paper deals with the analysis of a novel micro-electromechanical sensor for measurement of microscale fluid physical properties. The proposed sensor is made up of a micro-beam with one end fixed and a micro-plate as a sensing element at its free end, which is immersed in a microscale fluid media. As fluids show different behavior in microscale than in macroscale, the microscale fluid media have been modeled based on micro-polar theory. So non-classical properties of fluid that are absent in macroscale flows need to be measured. In order to actuate the sensor longitudinally, an AC voltage is applied to the piezoelectric layers on the upper and lower surfaces of the micro-beam. Coupled governing partial differential equations of motion of the fluid field and longitudinal vibration of the micro-beam have been derived based on micro-polar theory. The obtained governing differential equations with time-varying boundary conditions have been simplified and transformed to an enhanced form with homogenous boundary conditions. Then, they have been discretized over the beam and fluid domain using Galerkin-based reduced-order model. The dynamic response of the sensing element for different piezoelectric actuation voltages and different exciting frequencies has been studied. It has been shown that by investigating damping and inertial effect fluid loading on response of the micro-beam, properties of a microscale fluid can be measured. At the end, effects of geometrical parameters of the sensor on the response of sensing element have been studied.

Simulation of dynamic pressure response of finite gas reservoirs experiencing time varying flux in the external boundary

Pressure transient response (PTR) of a hydrocarbon reservoir to the alteration of production or injection rate can be computed through solution of its associated diffusivity equation. The PTR is likely the most important data for characterizing a hydrocarbon reservoir, forecasting its future productive performance, designing the appropriate stimulation scenario, and optimizing its management activity.The aim of this study is to simulate the effect of a time varying boundary flux on the PTR of a bounded gas reservoir using a simple and straightforward approximate scheme. The physical model can be viewed as a finite homogeneous reservoir experiencing a constant production rate at its inner boundary and an exponential flux in its outer boundary. Since direct handling of the time varying boundary condition is often hard, it makes a traditional solution applying Laplace transform or/and its inverse very tedious and time consuming. Therefore in this study the orthogonal collocation (OC) method which is elegant in its simplicity and efficient in its application is employed for simulating the PTR of the considered model.Reliability of the OC methodology is verified by comparing its result with an exact analytical solution for the closed reservoir. The proposed OC method has predicted the exact analytical solution with the absolute average relative deviation (AARD %) of 0.36%. Thereafter a general approximate solution is proposed to describe the PTR of gas reservoir with the time varying flux in the external boundary. The effect of boundary type, boundary flux, and boundary radius on the characteristic shape of the derivative graph has been investigated. The numerical results indicate that the PTR of a gas reservoir that experiencing time varying flux in its outer border can be successfully simulated using this approximate approach.