Sinusoidal Wavy Channel Research Articles

On the effect of wave amplitude and corrugation profile of certain types of corrugated channels on thermal performance and entropy generation

This study aims to examine experimentally and numerically the thermal-hydraulic performance and entropy generation of wavy channels: sinusoidal and triangular, and compare them with respect to the straight channel. Various wavy amplitudes and corrugation profiles are considered in a sinusoidal wavy channel and a triangular wavy channel and assessed with the straight channel. Experimental test rig is built and the experiments are carried out on laminar air flow and used to verify the simulation results. The main findings show that the HT rate, fluid flow, and entropy generation are affected by the corrugation profile and wave amplitude of the wavy channel. The maximum Nusselt number of 11.17 occurred in a sinusoidal wavy channel with a wave amplitude equal to 1.75, and high friction factor also happens in this model. It was found that the sinusoidal and triangular wavy channels with A = 0.75 and a flow rate range of 1–1.75 have the highest overall thermal performance factor. Also, the total entropy generation of the straight channel was lower than all corrugated channel models. In conclusion, it has been found that the best compromise between enhanced HT and the accompanying increase in both pressure drop and entropy generation is occurred in case of A = 0.75 at flow rate ranged from 1 to 1.67 LPM.

Ternary nanofluid cooling of an elastic plate by using double sinusoidal wavy channels under different magnetic fields

Ternary nanofluid cooling of an elastic plate by using double sinusoidal wavy channels under different magnetic fields

Effect of slip boundary conditions on unsteady pulsatile nanofluid flow through a sinusoidal channel: an analytical study

A novel analysis of the pulsatile nano-blood flow through a sinusoidal wavy channel, emphasizing the significance of diverse influences in the modelling, is investigated in this paper. This study examines the collective effects of slip boundary conditions, magnetic field, porosity, channel waviness, nanoparticle concentration, and heat source on nano-blood flow in a two-dimensional wavy channel. In contrast to prior research that assumed a constant pulsatile pressure gradient during channel waviness, this innovative study introduces a variable pressure gradient that significantly influences several associated parameters. The mathematical model characterising nano-blood flow in a horizontally wavy channel is solved using the perturbation technique. Analytical solutions for fundamental variables such as stream function, velocity, wall shear stress, pressure gradient, and temperature are visually depicted across different physical parameter values. The findings obtained for various parameter values in the given problem demonstrate a significant influence of the amplitude ratio parameter of channel waviness, Hartmann number of the magnetic field, permeability parameter of the porous medium, Knudsen number due to the slip boundary, volume fraction of nanoparticles, radiation parameter, Prandtl number, and heat source parameters on the flow dynamics. The simulations provide valuable insights into the decrease in velocity with increasing magnetic field and its increase with increasing permeability and slip parameters. Additionally, the temperature increases with increasing nanoparticle volume fraction and radiation parameter, while it decreases with increasing Prandtl number.

Numerical analysis of the effect of wave amplitude on thermohydraulic performance in a heat exchanger with sinusoidal wavy channels

With the increasing application of heat exchangers, the research has become increasingly comprehensive. Previous studies mainly focused on comparing the amplitudes to investigate the performance of heat exchangers, while the study of variable amplitude channel structures was limited. This study investigated the thermal-hydraulic performance of seven different sinusoidal channels with inclination angles of 20° and 30°, and their combinations. Using the CFD solver, simulations were conducted to assess the heat transfer capabilities of sinusoidal channels under various Reynolds numbers and different amplitude conditions, compared to straight channels and single amplitude sinusoidal channels. The results indicated that within one period, both the pressure drop and temperature difference exhibited a ripple trend of gradually decreasing amplitude. When evaluating the comprehensive performance of the channels, three trends emerged across the seven models, with the performance of the cold fluid and the hot fluid within the same group showing opposite trends. This suggested that more variations could enhance the overall performance of the hot fluid while reducing the performance of the cold fluid. For structures with the same amplitude, in terms of amplitude variation frequency, models with six periods per group exhibited higher overall heat transfer performance than those with three periods per group, with an average increase of 36.37%. For the arrangement of amplitudes (taking the hot fluid inlet as an example), configurations with a smaller amplitude at the inlet outperformed those with a larger amplitude, indicating that in combined amplitude models, a structure with a larger group and a smaller amplitude at the inlet (taking the hot fluid as an example) should be designed.

Corrugation characteristics effect of channel on heat transfer and pressure Drop: Experimental study

Four sinusoidal wavy channels were examined experimentally under turbulent flow conditions of air to clarify the potentials of such configuration in terms of heat transfer and hydrodynamic performance. The subjected heat flux on these wavy walls (sinusoidal wavy plates) were (500, 750, and 1000) W /m2. The ratio of corrugation wavelength (λ) to corrugation amplitude (a) had a range of (5.7–12.6), and the four wavy channels are scrutinized at zero angles (0°) of phase shift. According to the applied range of Reynolds number which is (1.932 × 103–3.286 × 103) and duct entrance dimension, the supplied inlet velocities were (0.3––0.67) m/s. The collected outcomes of the tested wavy sections showed a gained in heat transfer coefficient of (11–39 %) compared to the conventional duct, this increase is originally came out due to the occurred gained of the enhancement factors which is (1.1–1.4). regarding the pressure drop, the acquired decay of the wavy channels is (17–45 %) compared to the conventional channels. Also the results showed the corrugation angle 54.5° is the best in terms of thermal performance factor TPF at Re ≈ 3300 with appreciable high pressure drop, while the corrugation angle of 32.5° has a poor TPF at Re ≈ 1950.

Numerical study and machine learning on local flow and heat transfer characteristics of supercritical carbon dioxide mixtures in a sinusoidal wavy channel PCHE

The ambient temperature has a great influence on the practical application of the supercritical carbon dioxide(S-CO2) Brayton cycle. The introduction of mixtures is an effective way to change the critical characteristics of the working fluid, which allows the system to better match the ambient temperature. The mixtures not only affect the overall performance of the power system, but also affect the flow and heat transfer processes in various heat exchange equipment. In this investigation, the physical model of a sinusoidal wavy channel printed circuit heat exchangers (PCHE) is built, and numerical simulations are conducted firstly on the flow and heat transfer characteristics of three S-CO2 mixtures (CO2-He, CO2-Ke and CO2-Xe). Effects of the mole fraction and type of the additive gasses on the thermal-hydraulic performance are discussed, as well as the variations of the local heat transfer coefficients and pressure drop along the flow direction. It's found that transfer coefficient and pressure drop of CO2-He increase with the increase of the additive gas mole fraction n, with maximum heat increases of 396 % and 1147 % against pure S-CO2, respectively. With the increase of n, heat transfer coefficient and pressure drop of CO2-Kr and CO2-Xe decrease. Maximum decreases of 71 % and 38 % can be seen for the heat transfer coefficient and pressure drop against pure CO2 for CO2-Kr, and 80 % and 64 % against pure CO2 for CO2-Xe, respectively. CO2-Xe yields the best thermal-hydraulic performance of the PCHE among the three mixtures. Secondly, machine learning is adopted to predict the local flow and heat transfer characteristics of S-CO2 mixtures in response to the significant variation of S-CO2 mixtures along the flow direction within the PCHE. Four machine learning models including Support vector machine (SVR), Artificial neural network (ANN), Random forest (RF) and Extreme Boosting Tree (XGBoost) are used and the corresponding prediction performance are analysed. It is found that machine learning is efficient and accurate in the prediction. Among the four machine learning models, XGBoost has a strong fitting ability to the local Nu and f, with an R2 of 0.9992 and 0.9718 on the test set, respectively. The prediction results of the XGBoost model can reflect the variations of local Nusselt number and friction factor along the flow direction well. However, ANN has stronger generalization ability in the prediction under new working conditions. The use of machine learning can greatly help the design and optimization of PCHEs with S-CO2 mixtures as working fluids.

Forced convection of non-Newtonian nanofluid in a sinusoidal wavy channel with response surface analysis and sensitivity test

A numerical study on the forced convection of water-Cu power-law non-Newtonian nanofluid in a wavy channel has been done. The temperature of the horizontal walls is larger than that of the input fluid. The governing equations are solved by the SIMPLE algorithm-based finite volume method. The effects of pertinent parameters power-law index, Reynolds number, and nanoparticle volume fraction have been varied. The obtained results are presented in terms of the local and average Nusselt number (Nu) for the rate of heat transfer, streamlines, and isotherms. The correlation for the average Nusselt number is obtained using the Response Surface Analysis (RSM) for the flow parameters. The relationship between input parameters and output was observed by plotting the response surface and contours that were obtained by RSM. The sensitivity of the output response to all the input parameters is also observed. This study concludes that heat transfer enhanced in wavy channels by adding nanoparticles and increasing the Reynolds number. It is also observed that shear-thinning non-Newtonian nanofluids can be more effective in heat transfer enhancement than shear-thickening fluids.

Entropy generation of Al2O3/water nanofluid in corrugated channels

The flow of nanofluids in a corrugated channel has been shown to have a significant impact on heat transfer performance, and has therefore become an important area of research. The ob- jective of this paper is to understand the thermal behavior of Al2O3/water nanofluid in a sinu-soidal and square channel and to identify ways to optimize heat transfer performance in such configurations. For this purpose, a numerical simulation was conducted using ANSYS-Fluent software 16.0 on entropy generation and thermo-hydraulic performance of a wavy channel with the two corrugation profiles (sinusoidal and square). The analyses were carried out under laminar forced convection flow conditions with constant heat flux boundary conditions on the walls. The influence of various parameters, such as particle concentration (0–5%), particle di-ameter (10nm , 40nm and 60nm), and Reynolds number (200 < Re < 800) on the heat transfer, thermal, and frictional entropy generation, and Bejan number was analyzed. Moreover, the distribution of streamlines and static temperature contours has been presented and discussed, and a correlation equation for the average Nusselt number based on the numerical results is presented. One of the most significant results obtained is that the inclusion of nanoparticles (5% volume fraction) in the base fluid yielded remarkable results, including up to 41.92% and 7.03% increase in average Nusselt number for sinusoidal and square channels, respectively. The sinusoidal channel exhibited the highest thermo-hydraulic performance at Re= 800 and φ= 5%, approximately THP= 1.6. In addition, the increase of nanoparticle concentration from 0% to 5% at Re= 800 and dnp= 10nm, diminishes the total entropy generation by 28.39 % and 22.12 % for sinusoidal and square channels, respectively, but when the nanoparticle diameter decreases from 60nm to 10nm at ϕ= 5% and Re= 800, the total entropy generation in the sinusoidal channel decreases by 34.85%, whereas in the square channel, it decreases by 20.05%. Therefore, rather than using a square channel, it is preferable and beneficial to use small values of nanoparticle diameter and large values for each of ϕ and Re in the sinusoidal wavy channel. Overall, the study of nanofluid flow in a wavy channel can provide valuable insights into the behavior of nanofluids and their potential applications in a variety of fields, including manufacturing, energy produc-tion, mining, agriculture, and environmental engineering.

Flow and heat transfer in a meandering channel

Fluid flows occur due to internal or external forces such as wind, gravity, pressure gradients, side-wall motion, MHD, and free convection. This study examines how meanders impact heat transfer by studying the behavior of viscous fluid flow with streamwise vortices in a sinusoidal wavy meandering channel of non-uniform radius. The study simplifies the motion and energy equations governing the fluid flow using novel transformations and a regular perturbation method. By plotting graphs for different parameter values, such as Pr, Re, and Ec, it reveals that decreasing the wavelength leads to flow separation near the channel surface. However, the stream moves forward with a sudden meander disturbance, causing the flow to become rectilinear and independent of vertex-generating centrifugal forces. The study identifies a stream function using standard and established relations. The fluid flow patterns and temperature distribution behavior are shown in various plots, highlighting the significant impact of meanders on fluid flow.

Effects of Asymmetric Channel on the Convective Heat Transfer: A Computational Study

In the fields of engineering, especially in the heat transfer area, the contribution of the fluid flows through different channels is remarkable. Among the different available shapes in the channel, the sinusoidal wavy channels are considered more efficient in the heat transfer areas such as cooling of electronic devices, boilers, biomedical applications, etc., due to their increased recirculation capability of mixing the flowing fluids - which increases the heat transfer performance, consequently. The optimization of channel design and modelling has been an engineering problem in recent years. In this research, flat, symmetric sinusoidal, and asymmetric sinusoidal channels were designed - using ANSYS 2020 - to investigate the effects of channel shape, amplitude, and aspect ratio of sinusoidal wavy walls on the overall heat transfer performance. Several designs have been considered in the present study at different Reynolds numbers and different aspect ratios, keeping the wall temperature as constant. The results of this study exhibit that at an aspect ratio of 0.4, the asymmetric channel shows the highest heat transfer effect due to increased recirculation tendency.

Analysis of SWCNT-water nanofluid flow in wavy channel under turbulent pulsating conditions: Investigation of homogeneous and discrete phase models

Improving thermal performance in heated channels has received much attention from many researchers due to its presence in various thermal systems. The present research numerically illustrated the effect of turbulent pulsating flow and SWCNT-water nanofluids in sinusoidal wavy channel on hydrothermal characteristics and entropy production. This study aims to provide a new thermal management approach for compact heat exchangers. Two various models for describing the motion of nanoparticles in a base fluid are applied. These models are Homogeneous single-phase, SPM and Lagrangian-Eulerian model (or discrete phase model), DPM. Governing equations are computationally solved utilizing the finite volume technique. This simulation covers Reynolds number, pulsation frequency, phase shift and nanoparticles concentration in the range of (4000 ≤ Re ≤ 7000), (1 ≤ f ≤ 5), (0° ≤ θ ≤ 180°) and (0% ≤ φ ≤ 3%), respectively. CFD simulations show that the pulsation frequency, f and phase shift, θ have a significant role in improving the heat transfer ratio and controlling the entropy generation through the wavy channel. Finally, it is recommended to use SWCNT/water nanofluid with a high-volume fraction (φ = 3%) as a working fluid in the considered channel due to its high heat transfer ratio and low thermal irreversibility.

Analytical and numerical study of capillary rise in sinusoidal wavy channel: Unveiling the role of interfacial wobbling

Capillary rise is ubiquitous in engineering applications and natural phenomena. In straight channels, the dynamics of capillary rise have been thoroughly investigated and are well understood. However, for nonuniform channels of varying radius, the dynamics remain largely unclear. In this study, the capillary rise in a sinusoidal wavy channel is investigated both analytically and numerically. Specifically, the capillary rate-of-rise of water in sinusoidal channels with different contraction frequencies and amplitudes is derived based on the principle of energy conservation. The change in capillary velocity and height over time is further validated by two-phase flow simulations based on the conservative level-set method. The results reveal a strong viscous dissipation in the interfacial region resulting from the wave-like wobbling motion of the liquid–air interface, constituting more than 50% of the total viscous dissipation when the channel profile changes rapidly. Failing to account for this interfacial effect will result in significant overestimations of the capillary velocity and erroneous predictions of the capillary rise curve, typically more than 4 times difference in the capillary velocity and more than 2.5 times difference in the time taken to arrive at the maximum height.

Thermal and hydrodynamic analysis of a conducting nanofluid flow through a sinusoidal wavy channel

The main objective of current study is investigation on thermo-hydraulic behavior of nanofluids in a sinusoidal wavy channel as a novel geometry under influences of thermophoresis and Brownian motion. Differential quadrature method and optimal homotopy asymptotic scheme have been employed to solve the momentum, continuity, energy and nanoparticle fraction equations. The validity of the proposed approaches is evaluated by comparing with previously published results. The impacts of some other physical parameters on temperature, nanoparticle concentration and velocity profiles will be analyzed in details. For instance, the results indicate that increment of the thermophoresis leads to enhance in values of temperature distribution. Also, the volume fraction of nanoparticle enhances by decreasing the parameter of thermophoresis.

Analytical investigation on heat transfer performance of H2O filled with SiO2 and TiO2 in a sinusoidal wavy channel

The peristaltic transport of hybrid nanofluid (SiO2–TiO2/H2O) flow in a sinusoidal wavy channel under the heat generation effect is reported in present analysis. We have simplified two dimensional equations of hybrid nanofluid for peristaltic transport by using long wavelength (δ≪1) and small Reynolds (Re) number assumptions. We also modeled temperature equation for hybrid nanofluid flow with the effect of heat generation. The two dimensional peristaltic flow equations of (SiO2–TiO2/H2O) are analytically solved by using perturbation method. The numerical integration and solutions of governing two dimensional equations are obtained in Mathematica software. The graphically behavior of solutions for temperature, pressure rise, streamlines and pressure gradient are elucidated with Matlab software. It is noticed that the pressure rise for hybrid nanofluid (SiO2–TiO2/H2O) increases for both amplitudes (a and b), solid fractional volume (ξ1) and heat generation (β). We observe that higher pressure gradient requires to retain same flux in narrow part of sinusoidal channel, also due to increase of amplitudes (a and b) pressure gradient increases. Moreover size of trapping bolus decreases by increase of channel width d and increases due to increase of heat generation parameter β.

Thermo-hydraulic and entropy generation analysis for non-Newtonian fluid flow through sinusoidal wavy wall channel

The present investigation deals with numerical simulation of thermo-hydraulic transport characteristics of non-Newtonian fluid flowing through a wavy channel subjected to isothermal condition. The power-law constitutive model is adopted to illustrate the rheological behavior of fluid. The influence of various parameters like index of power-law (n), Reynolds number (Re), amplitude of sinusoidal profile (A), etc. on flow, thermal field, and entropy generation characteristics are delineated in detail. Furthermore, the outcome of the sinusoidal channel is compared with that of conventional straight channel in order to check the performance of wavy-wall channels. Results divulge that reliance of the thermo-hydraulic characteristics on wall geometry is unequivocally affected due to the amplitude of sinusoidal profile. Augmentation in heat transfer rate for lower values of amplitude of sinusoidal profile over the conventional channel is inconsiderable although difference upraises considerably owing to two factors namely: higher amplitude of sinusoidal profile and Re. The heat transfer rate diminishes with increment in power-law index. Consequently, heat transfer rate is progressively conspicuous for shear thinning fluids compared to shear thickening fluids. For comparison of pressure drop in the wavy and equivalent straight channel taking into account heat transfer and corresponding enhancement of pressure drop, we have defined a performance parameter. The comparison of performance parameter illustrates that shear-thickening fluids are more appropriate for engineering applications. Two nondimensional numbers are defined to elucidate the behavior of entropy generation with various parameters like Reynolds number, amplitude of sinusoidal profile, and power-law index. The entropy generation increases as the amplitude of waviness increases. Furthermore, entropy generation also intensify with increment in index of power-law for a distinct Reynolds number and amplitude of waviness. Results obtained from present study can be used as a tool to select wall geometry for design of compact and economical energy exchange devices dealing with non-Newtonian power-law fluids.

Effect of Top Wall Configuration on Unsteady Flow and Heat Transfer Characteristics of Sinusoidal Wavy Channels

In the present work, three-dimensional numerical investigations are carried out to study the influence of top wall confinement on unsteady flow and heat transfer characteristics of sinusoidal wavy channels. Different channel confinements considered in the present study are channels having sinusoidal bottom wall and plane, in-phase, and out-of-phase configurations of the top wall. Computations are carried out by using open-source computational fluid dynamics package OpenFOAM. Numerical results are validated against the experimental results reported in the literature. The onset of unsteady flow and three-dimensionality in the channel has been identified by monitoring velocity at a point in the domain. Steady and unsteady flow characteristics have been presented with the help of temporal variation of velocity signals, phase space trajectories, power spectral density, instantaneous streamlines, and iso Q-surfaces. Heat transfer characteristics have been presented with the help of instantaneous Nusselt number contours and time-averaged values of Nusselt number.

Numerical investigation and design optimization of a novel polymer heat exchanger with ogive sinusoidal wavy tube

In this paper, a novel polymer bare-tube heat exchanger (BTHX) with an ogive tube shape and a sinusoidal wavy channel is proposed and investigated for the liquid-to-gas application. The thermal-hydraulic performance of the proposed polymer BTHX is calculated using a validated computational fluid dynamics (CFD) simulation model, and the performance is optimized by setting nine independent design variables. A selective-series CFD method with an approximation-assisted optimization technique is developed for the multi-objective optimization to reduce the computational time. As a result, the thermal-hydraulic performance of the optimized novel polymer BTHX shows 91.5% and 134.9% of that of the target aluminum louvered-fin micro-channel heat exchanger (MCHX), respectively. This is mainly due to the improved air-side heat transfer coefficient of the proposed polymer BTHX by 70.2% compared to the target aluminum MCHX. Compared to the recent teardrop-shaped polymer BTHX, the thermal-hydraulic performance of the proposed BTHX is significantly improved. Regarding the polymer thermal conductivity, it is found that the thermal performance degradation can be minimized with at least 8.0 W·m−1·K−1 of the thermal conductivity. The flow field of the proposed polymer BTHX is also discussed and shows that the unique structure of the sinusoidal wavy tube can reduce the air-side pressure drop and promote the mixing of the air flow, thereby improving the air-side convective heat transfer.

Peristaltically driven flow of hybrid nanofluid in a sinusoidal wavy channel with heat generation

The present analysis is reported to analyze the hybrid nanofluid (SiO 2 − CNT/H 2 O) flow as peristaltic transport under the influence of heat generation/absorption in a sinusoidal wavy channel. We have expressed the non-dimensional equations into two dimensional coordinate system by using vary long wavelength (δ < 1) and low Reynolds number assumptions. The non-dimensional hybrid nanofluid (SiO 2 − CNT/H 2 O) equations are solved approximately by using well known perturbation approach. The numerical solutions for temperature and two dimensional peristaltic flow are obtained by using Mathematica software. The obtaining solutions for temperature and peristaltic transport are expressed graphically with help of Matlab software. The graphical results show that pressure rise, temperature profile and pressure gradient of hybrid nanofluid (SiO 2 − CNT/H 2 O) flow increases due to increase of heat generation. The pressure rise and pressure gradient of hybrid nanofluid decreases due to increase of phase difference ϑ. Furthermore, pressure gradient and pressure rise of SiO 2 − CNT/H 2 O hybrid nanofluid also decreases by increase in channel width. The streamlines for peristaltic transport are also elucidated in present analysis. The streamlines show that bolus size increases due to increase of amplitude but decreases by increase of heat generation.

SADI approach programming on GPU: convective heat transfer of nanofluids flow inside a wavy channel

In this study, the numerical simulation of convective heat transfer of nanofluids (Al2O3/water and CuO/water) inside a sinusoidal wavy channel is performed using the Graphics Processing Units (GPU). The governing equations including stream-function, vorticity transport, and energy are discretized using the fourth-order Spline Alternating-Direction Implicit (SADI) approach in combination with the curvilinear coordinates mapping. The final tridiagonal-matrices are solved by Parallel-Thomas-Algorithm (PTA) on GPU. A homogenous one-phase model is also applied to consider the effective characteristics of the nanofluids flow. In the first part of the results section, the effects of nanoparticle volume fraction, Reynolds number, and amplitude of the wavy wall on average Nusselt number and the skin friction coefficient of the channel are investigated. In the second part of the results section, the ability of GPU to accelerate the computation (or runtimes) is compared to the classic Thomas algorithm that runs on a Central Processing Unit (CPU). The results demonstrate that the speedup of PTA against CPU runtime for the finest grid is around 18.32×.

Numerical investigation on thermal hydraulic performance of supercritical LNG in sinusoidal wavy channel based printed circuit vaporizer

As a new type of micro-channel heat exchange, printed circuit heat exchanger (PCHE) is a better choice as main cryogenic heat exchanger (MCHE) on LNG carriers and LNG floating terminals for high efficiency and compactness. In this work, a 3D-numerical model of PCHE employing SST k-ω turbulent model is built after validating against the classic experimental data of supercritical fluid. The cross section of S-LNG channel is a semi-circle with radius of 0.9 mm, size of solid region (stainless steel) is 2.15 mm × 1.6 mm, the overall length of the channel is 500 mm and the centerline formula of the channel is z = 0.5∙sin(0.5∙x). Based on this model, the thermal hydraulic performance of supercritical LNG (S-LNG) in sinusoidal wavy channel of printed circuit vaporizer (PCV) is investigated, and the effects of heat flux and operation pressure on flow and heat transfer are analyzed, which aim to improve the understanding of flow and heat transfer mechanism of S-LNG in sinusoidal wavy semi-circuit channel. Moreover, the simulation result indicates that the maximum gap of heat transfer coefficient between adjacent observation cross section can reach 1.28 kW·m−2·K−1, and the effect of buoyancy force is not domination on heat transfer; the reverse flow occurs nearly in every periodic segment, and the flowing “dead zone” and vortex appear close to the inner wall of cross section, especially near the region of included angle between the straight edge and the semi-circle curve; the lower intensity of both turbulent structure rebuilding and secondary flow appears under higher operating pressure, and after passing large specific heat region, the effect of operating pressure is inappreciable.