An iterative regularization method in estimating the transient heat-transfer rate on the surface of the insulation layer of a double circular pipe

In this study, a conjugate gradient method based inverse algorithm is applied to estimate the unknown space- and time-dependent heat-transfer rate on the surface of the insulation layer of a double circular pipe heat exchanger using temperature measurements. It is assumed that no prior information is available on the functional form of the unknown heat-transfer rate; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The accuracy of the inverse analysis is examined by using simulated exact and inexact temperature measurements. Results show that an excellent estimation on the space- and time-dependent heat-transfer rate can be obtained for the test case considered in this study.     

An iterative regularization method in estimating the base temperature for non-Fourier fins

An inverse non-Fourier fin problem is examined in the present study by an iterative regularization method, i.e., conjugate gradient method (CGM), in estimating the unknown base temperature of non-Fourier fin based on the boundary temperature measurements. Results obtained in this inverse problem will be justified based on the numerical experiments where three different temperature distributions are to be determined. Results show that the inverse solutions can always be obtained with any arbitrary initial guesses of the base temperature. Moreover, the drawbacks of previous study for this identical inverse problem, such as (1) the inverse solutions become poor when the frequency of base temperature is increased, (2) the estimations depend strongly on the size of grids, (3) the estimations are sensitive to the measurement errors and (4) the uncertainty of using the concept of future time step, can all be avoided by applying this algorithm. Finally, it is concluded that accurate base temperatures can be estimated in the present study.     

An inverse problem in estimating the reaction functions and solute concentration simultaneously in a reversible process

An inverse algorithm basing on the Iterative Regularization Method (IRM) is applied in this study in determining the unknown time-dependent reaction functions and solute concentration in the solution, i.e. three unknown time-dependent functions, simultaneously in a reversible process by using measurements of concentration components. It is assumed that no prior information is available on the functional form of the unknown functions in the present study, it can thus be classified as function estimation for the inverse calculations. The accuracy of this inverse problem is examined by using the simulated exact and inexact concentration measurements in the numerical experiments. Results show that the estimation of the time-dependent reaction functions and solute concentration in the solution can be obtained in a very short CPU time on a HP d2000 2.66 GHz personal computer. Moreover, the sensors should be placed as close to the boundary as possible to obtain better estimations.     

Inverse analysis for estimating the unsteady inlet temperature distribution for two-phase laminar flow in a channel

An inverse thermal problem was considered for two-phase laminar flow in a parallel plate duct. The inlet temperature, which varies temporally as well as spatially, was estimated when measured temperatures were available at downstream of the duct. In the present study, the problem is solved through a minimization of an objective function by using two regularization methods, i.e., the iterative conjugate gradient method (CGM) and the Tikhonov regularization method (TRM). The effects of the functional form of inlet temperature profile, the number of the measurement points and the measurement errors are investigated and discussed. The computational accuracy and efficiency of these two regularization method are compared and discussed.     

Transient heat transfer in simultaneously developing channel flow with step change in inlet temperature

The transient heat transfer in forced convection for simultaneously developing laminar flow inside a parallel-plate channel is studied by solving the steady momentum equation with the generalized integral transform technique and the transient energy equation through a hybrid approach that combines the integral transform method with a second-order accurate finite-differences scheme. Semi-analytical results are then presented for the fluid bulk temperature and local Nusselt number along the channel as a function of position and time.     

The effect of inlet velocity distribution on the temperature field of a rotating circular pipe

The temperature field in the developing region of a circular rotating pipe was analysed by assuming three different velocity distributions at the entrance. It was noted that the shape of the inlet velocity distribution has a weak influence when the pipe is stationary. However, when the pipe is subjected to large rotating velocity significant difference was realized. This is due to the size of the separation bubble which depends strongly upon the shape of inlet velocity profile.     

On the iterative regularization of inverse heat conduction problems by conjugate gradient method

The convergence and regularization properties of the conjugate gradient algorithm applied to the inverse heat conduction problem are considered for a time-dependent boundary heat flux. An analysis based on both numerical and analytical results clearly shows that the convergence process of the algorithm is strongly frequency-dependent and provides in this way a very efficient regularization mechanism against the destabilizing effect of random errors in the input data.     

Analytical solution of Graetz problem in pipe flow with periodic inlet temperature variation

The paper reports an analytical solution for the temperature field in a fully developed pipe flow subject to periodic (of any shape) inlet temperature variation. The solution is given in term of a series of Kummer functions for the cases of uniform and constant wall temperature and wall heat flux, thus comprising also the adiabatic wall case. A “fully developed” region for the fluctuating component of the fluid temperature is also evidenced and closed-form solutions are given. An interpretation of the temperature field as superposition of travelling thermal waves is presented and discussed.     

Analysis of heat transfer in simultaneously developing pulsating laminar flow in a pipe with constant wall temperature

Numerical studies of flow and heat transfer in a circular tube under pulsating flow condition were carried out in the laminar regime. The flow at the inlet consists of a fixed part and a pulsating component, which varies sinusoidally in time. The flow was both thermally and hydrodynamically developing while the tube wall was kept at a uniform temperature. The solution of two-dimensional Navier–Stokes equation was performed using the SIMPLE algorithm with the momentum interpolation technique of Rhie and Chow. By analysing the data generated from the simulation, it is observed that in the range of present study (frequency: 0–20 Hz; amplitude: < 1.0), pulsation has no effect on time-averaged heat transfer, although the Nu distribution varies in time in the near-entry region of the pipe.     

Inverse method in simultaneously estimate internal heat generation and root temperature of the T-shaped fin

This study employed the finite element method and rearranged the matrix to establish an inverse operation without iteration, in order to simultaneously estimate the internal heat generation and root temperature of the T-shaped fin. The results of numerical validation showed that the optimized the T-shaped fin has a good effect because of better efficiency of heat transfer. Regardless of the configurations, the estimation of internal heat generation is obviously affected by the measurement error. Nevertheless, the measurement of temperature in future time can reduce sensitivity of internal heat generation and root temperature to the measurement error.     

Use of the reference stress method in estimating the life of pipe bends under creep conditions

The effect that the change of ovality of pipe bends, due to creep under internal pressure, has on the stationary-state stress state has been investigated using a reference stress approach. Attention has been focused on practical pipe bend geometries and loading conditions in power plant applications.Prediction of failure times based on stationary-state stresses, for cases in which ovality changes are neglected and in which ovality is included, indicate that the change of ovality can be very significant in some practical situations. Failure times obtained using a creep continuum damage mechanics approach have also been compared with those obtained using the stationary-state predictions with ovality changes included. These results indicate that even sophisticated damage mechanics analyses are inadequate unless the effects of the changing ovality which occurs are taken into account.     

Mass transfer at a pipe inlet zone in relation to impingement corrosion

Rates of mass transfer required to predict the rate of diffusion controlled corrosion of the inlet zone of a tube fitted with a perpendicular inlet nozzle were studied using the diffusion controlled dissolution of copper in acidified FECl₃. Variables studied were inlet nozzle diameter, solution velocty and physical properties of the solution. The rate of corrosion of the tube inlet zone was found to increase with decreasing inlet nozzle diameter and increasing solution velocity. The mass transfer coefficient of the diffusion controlled corrosion of the tube inlet zone was related to different variables by dimensionless equations. Implications of the present results to the design and protection of pipelines against corrosion in industry is highlighted.     

Recovering both the space-dependent heat source and the initial temperature by using a fast convergent iterative method

In this paper, we solve two types of inverse heat source problems: one recovers an unknown space-dependent heat source without using initial value, and another recovers both the unknown space-dependent heat source and the initial value. Upon inserting the adjoint Trefftz test functions into Green’s second identity, we can retrieve the unknown space-dependent heat source by an expansion method whose expansion coefficients are derived in closed form. We assess the stability of the closed-form expansion coefficients method by using the condition numbers of coefficients matrices. Then, numerical examples are performed, which demonstrates that the closed-form expansion coefficient method is effective and stable even when it imposes a large noise on the final time data. Next, we develop a coupled iterative scheme to recover the unknown heat source and initial value simultaneously, under two over specified temperature data at two different times. A simple regularization technique is derived to overcome the highly ill-posed behavior of the second inverse problem, of which the convergence rate and stability are examined. This results in quite accurate numerical results against large noise.     

Simultaneous estimation of heat-transfer rate and coolant fluid velocity in a transpiration cooling process

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent heat-transfer rate on the heated surface and coolant fluid velocity in a transpiration cooling process using temperature measurements of the porous medium and coolant fluid. It is assumed that no prior information is available on the functional forms of the unknown heat-transfer rate and coolant fluid velocity; hence the procedure is classified as the function estimation inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The accuracy of the inverse analysis is examined by using simulated exact and inexact temperature measurements. Results show that an excellent estimation on the time-dependent heat-transfer rate and coolant fluid velocity can be obtained for the test case considered in this study.     

The BFGS method for estimating the interface temperature and convection coefficient in ultrasonic welding

An inverse heat transfer problem is investigated in the present study by the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method to predict the unknown time-dependent heat generation at the weld interface and convection heat transfer coefficient during an ultrasonic metal welding process based on the knowledge of temperature measurements taken on the horn. With known temperature data at some locations on the horn, the inverse solution was rapidly obtained by solving nonlinear direct problem, Central Finite Difference and Simple Step Method. The proposed method which did not need solving adjoint and sensitivity problem revealed the characteristics of high efficiency, lower iterations for a computational algorithm and high accuracy for estimating values even when measurement error was considered. Besides, a comparison of the BFGS method with some previous methods (i.e. CGM, SCGM) was established. These results show that an excellent estimation on interfacial heat generation (or temperature), as well as a convection heat transfer coefficient, can be simultaneously obtained in this study. The current methodology will provide a useful tool to optimize welding conditions in ultrasonic welding.     

Simulation of dynamical response of a countercurrent heat exchanger to inlet temperature or mass flow rate change

This paper simulates the dynamical response of a countercurrent heat exchanger. The model, which is employed for this purpose for the first time, employs the numerical method based on the analytical solution of the energy equation. The details of this method are introduced in the paper. Furthermore, the transient behavior of the system is considered for two cases. In the first case, the system response to the inlet temperature change of both fluids is studied. In the second case, the inlet mass flow rate change and its effect on the system behavior are analyzed. The time constant variation is considered during the dynamical behavior of the system. It was shown that the time constant behaves differently for both of the fluids. Finally, a parametric study on the effect of the mass flow rate on the system response is carried out. Comparison of the present results with the experimental data and analytical results of the other researchers confirms that this model can be used for system response prediction with a high level of confidence. This method has great capability in reducing the mathematical calculation amount and CPU time.     

A three-dimensional inverse geometry problem in estimating simultaneously two interfacial configurations in a composite domain

A three-dimensional inverse geometry problem, i.e. shape identification problem, in estimating simultaneously two interfacial configurations in a composite domain with three regions is examined in the present work by using the conjugate gradient method (CGM) and commercial package CFD-ACE+. Two over-utilized conditions should be utilized in determining two gradient equations. Numerical experiments using different measurement errors and different interfacial configurations were performed to justify the validity of the conjugate gradient method in solving this inverse geometry problem. Finally it is concluded that accurate interfacial configurations can be estimated by utilizing the present inverse algorithm.     

An optical method of measuring the temperature in a fluidised bed combustor

The paper analyses the dynamic aspects of the temperature field in a fluidised bed of solids particles (e.g., sand) in which a gaseous fuel is being burned. Such a hot bed emits electromagnetic radiation within the visible range and this can be recorded using a digital video camera. This fact has been used to develop a method for measuring the bed’s temperature in the line of sight, through the quartz sides of the reactor. A solid probe is only used for calibration.Video recordings were obtained covering different regions of the bed over three wavelength bands, red, green and blue. In the course of an experiment, the mean temperature of the bed, measured with thermocouples, was raised from ambient to 1300 K, at a rate of ∼ 1 K/s. The data collected were used for calibration, with the brightness of individual pixels converted to a temperature scale. The calibration can then be used to investigate the dynamic temperature distribution within the field of view, in individual elements of the bed. This can also help the study of heat transfer in the bed, its distribution and dissipation.Using this method, it is possible to make direct observations of the intermittent combustion of gaseous fuels in a bubbling fluidised bed. The results provide direct proof that the temperature gradients observed within such beds are associated with exothermic processes within fast moving bubbles. The method could be adapted to studying, e.g., the combustion of solid fuels.     

A method of correlating forced convection heat-transfer data and a comparison with fluid friction

A general method for the correlation of forced convection heat-transfer data is proposed, which consists in plotting, against the Reynolds number, a dimensionless group representing the experimentally measured data from which film heat-transfer coefficients would be calculated, namely, , or its equivalent, h/cG, multiplied by the two-thirds power of the group, . Data are cited from the literature which show that the resulting plots of heat-transfer data for flow parallel to plane surfaces and for fully turbulent flow inside tubes, coincide (when the properties are taken at the “film” temperature) with the best data on fluid friction plotted in the customary manner, as the friction factor , against the Reynolds number. For flow at right angles to tubes, however, the friction and heat-transfer factors differ, the friction factors being higher.

The equations successfully employed for representing heat-transfer data in streamline flow inside tubes have been modified for plotting with the same coordinates as used for turbulent flow ; and a quantitative allowance is suggested for the effect of free convection at low velocities by including a function of the group, . There is seen to be no relation between heat transfer and friction in the viscous region.

The method of correlation here proposed is shown to be particularly valuable in the transition region between streamline and turbulent flow in tubes, since heat-transfer factors may show “dips” analogous to those for friction. The controlling variables in this region are fully discussed in the light of the available data.     

Wall boiling in a vertical annulus: Effect of inlet subcooling and mass flow rate

Numerical studies on low flow rate convection boiling in a vertical annulus has been carried out to predict effects of inlet subcooling and mass flow rate. The aspect ratio of vertical annulus is 352 while the annular gap is 3.5?mm. RPI wall boiling model is used for the development of present code and the results are verified with those available in literature. The results show that onset of significant void (OSV) can be delayed to achieve maximum heat transfer by increasing the liquid subcooling and liquid mass flow rate. The average Nusselt number increases almost linearly with increase in the mass flow rate as well as the inlet subcooling. At high heat flux, very high wall temperatures are observed with low subcooling and low mass flow rates. This should be avoided for enhanced safety.