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Development and validation of Nusselt number and friction factor correlations for laminar flow in semi-circular zigzag channel of printed circuit heat exchanger
Accepted Manuscript Research Paper Development and Validation of Nusselt Number and Friction Factor Correlations for Laminar flow in Semi-circular Zigzag Channel of Printed Circuit Heat Exchanger Su-Jong Yoon, James O'Brien, Minghui Chen, Piyush Sabharwall, Xiaodong Sun PII: DOI: Reference:
S1359-4311(16)34274-0 http://dx.doi.org/10.1016/j.applthermaleng.2017.05.135 ATE 10449
To appear in:
Applied Thermal Engineering
Received Date: Revised Date: Accepted Date:
21 December 2016 16 May 2017 20 May 2017
Please cite this article as: S-J. Yoon, J. O'Brien, M. Chen, P. Sabharwall, X. Sun, Development and Validation of Nusselt Number and Friction Factor Correlations for Laminar flow in Semi-circular Zigzag Channel of Printed Circuit Heat Exchanger, Applied Thermal Engineering (2017), doi: http://dx.doi.org/10.1016/j.applthermaleng. 2017.05.135
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Development and Validation of Nusselt Number and Friction Factor Correlations for Laminar flow in Semi-circular Zigzag Channel of Printed Circuit Heat Exchanger
Su-Jong Yoona*, James O’Briena, Minghui Chenb, Piyush Sabharwalla and Xiaodong Sunb a
Idaho National Laboratory, 2525 Fremont Ave., Idaho Falls, ID 83415
b
Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI 48109
Tel: 208-526-5407, Fax: 208-526-2930, *Email: [email protected]
Abstract Friction factor and heat transfer correlations have been developed for printed circuit heat exchangers (PCHEs) as a function of geometric parameters. These correlations summarize the thermal hydraulic performance of PCHEs and allow for accurate determination of their cost and effectiveness.. Although there have been previous studies of the thermal-hydraulic performance of zigzag channel PCHEs, explicit correlations for friction factor or Nusselt number as a function of geometric parameters have not yet been reported. In this study, computational fluid dynamic (CFD) predictions were used to aid in the development of friction factor and Nusselt number correlations for laminar flow in semicircular zigzagchannel PCHEs. Two CFD models were developed to investigate the thermal-hydraulic characteristics of fluid flow in zigzag channels. A single-channel isothermal CFD model was used to investigate friction factors in PCHE zigzag channels. A two-channel CFD model was used to investigate the Nusselt number in zigzag channels and the effect of temperature-dependent fluid properties on the pressure loss. The effects of geometric parameters such as relative length ratio, zigzag angle and radius of curvature of bend were investigated. From the extensive CFD analysis database, friction factor and Nusselt number correlations were developed by implementing a least squares method with a non-linear Generalized Reduced Gradient algorithm. The friction factor correlation is valid for 50 ≤ Re ≤ 2000, 5° ≤ α ≤ 45° and 4.09 ≤ lR/Dh ≤ 32.73. The Nusselt number correlations are valid for 200 ≤ Re ≤ 2000, Pr ≤ 1.0, 5° ≤ α ≤ 45° and 4.09 ≤ lR/Dh ≤ 12.27. The correlations were validated against experimental data from The Ohio State University (OSU) and Korea Advanced Institute of Science and Technology (KAIST). The friction factor for zigzag-channel PCHEs is mainly influenced by the zigzag channel geometry while the Nusselt number is influenced by the overall heat exchanger design including the plenum sections.
1. Introduction A Printed Circuit Heat Exchanger (PCHE) is a type of compact heat exchanger which has high heat transfer surface area density and efficiency that result in reduced size, weight and cost. As illustrated in Fig. 1, PCHEs consist of many straight or zigzag microchannels, which are fabricated by diffusionbonding the chemically etched plates [1]. The diffusion-bonding technique permits application of PCHEs over a wide range of temperatures and pressures [2], as shown in Fig. 2. Due to many advantages, PCHEs have been highlighted as a promising heat exchanger type for high temperature reactor systems (HTRs) and small modular reactors (SMRs). For instance, PCHEs are strong candidate heat exchangers for use in the intermediate heat transfer loop (IHTL) of HTRs for both power generation and non-electric applications. PCHEs with zigzag channels are used for industrial applications since the heat transfer performance of zigzag channels is superior to that of straight channels. Many researchers have made dedicated efforts to examine the thermal-hydraulic performance of PCHEs. Lee and Kim numerically investigated the effect of the channel angle and ellipse aspect ratio on the thermal-hydraulic performance of PCHEs [3] and optimized the heat exchanger design by using the hybrid multi-objective evolutionary algorithm (hybrid MOEA) coupled with the response surface approximation (RSA) model [4]. They also compared the thermal-hydraulic performance of various channel cross-sectional shapes and configurations [5]. Ngo et al. [6] developed empirical correlations of Nusselt numbers and pressure drop factors of S-shaped and zigzag-channel microchannel heat exchangers (MCHEs). Fatima et al. [7] conducted a numerical study on the thermal-hydraulic behavior of supercritical CO2 in PCHEs with semicircular zigzag channels and compared the results with experimental data obtained at University of Wisconsin [8]. Song et al. [9] also investigated the thermal-hydraulic characteristics of zigzag-channel PCHEs with CO2 and water experimentally. Berbish et al. [10] experimentally investigated forced convection heat transfer and pressure loss characteristics in a straight semicircular duct with Reynolds numbers ranging from 8,242 to 57,794. Kar [11] performed computational fluid dynamics (CFD) analyses on the thermal hydraulic characteristics in straight and sinusoidal ducts with a semi-elliptical crosssection. Kim et al. [12] conducted experiments and CFD simulations to investigate the thermal-hydraulic performance of the PCHE with semicircular zigzag channels within a Reynolds number range from 350 to 1,200. Kim and No [13] performed experiments and numerical simulations of PCHEs using a heliumwater test loop and proposed Fanning friction factor and Nusselt number correlations for a system analysis code. Kwon et al. [14] conducted experiments and CFD simulations to investigate the effect of zigzag angle on the heat transfer and pressure drop characteristics in a zigzag channel PCHE. Seo et al. [15] reviewed recent research on PCHEs and experimentally investigated the thermal hydraulic performance of straight-channel PCHEs by using the Wilson plot method. Ma et al. [16] investigated local thermal-hydraulic performance of zigzag-channel PCHEs with a commercial CFD code, FLUENT. In this study, the effect of zigzag angle on local Nusselt number and pressure drop was examined. Meshram et al. [17] evaluated the performance of PCHEs with straight and zigzag channels and developed Nusselt number and friction factor correlations as a function of geometric parameters such as zigzag pitch and angle. However, their correlations did not account for possible different flow regimes by not considering Reynolds number as one of the key variables, thus restricting the application of the correlation. Van Abel et al. [18] conducted a numerical study on supercritical CO2 in a zigzag-channel PCHE by comparing with experimental data and investigated the sensitivity on the chosen turbulence model and the radius of curvature at the bends.
To develop new PCHE designs and to evaluate their thermal-hydraulic performance, accurate information on friction factor and heat transfer coefficients are required as a function of geometric parameters. Although previous studies have investigated the thermal-hydraulic performance of PCHEs, the effects of the geometric parameters and their combinations were not investigated sufficiently. A few numerical studies [3, 13, 14 and 17] investigated the effects of design parameters and their combinations on the PCHE thermal-hydraulic performance but the relationship between the design parameters and heat exchanger thermal-hydraulic characteristics is stilled needed. Consequently, CFD analyses were conducted to investigate the effects of the geometric parameters on the thermal hydraulic characteristics in zigzag-channel PCHEs for Reynolds numbers ranging from 50 to 2,000. As a result, friction factor and Nusselt number correlations for a semi-circular zigzag channels were developed as a function of geometric parameters.
2. Flow Characteristics in Zigzag Channel The use of zigzag channels enhances heat transfer, but also results in higher pressure loss due to form losses. Several geometric parameters have a significant influence on both heat transfer and pressure loss. Basically, the zigzag channel can be considered as a series of multiple bends in the channel. The magnitude of bend loss coefficient is influenced by the degree of flow turbulence, inlet velocity profile shape, the Reynolds number, the channel length-to-diameter ratio, the bend angle, the channel aspect ratio, radius of curvature, etc. The pressure loss in the zigzag channel can be divided into frictional loss and the minor losses. Generally, the velocity profile in the zigzag channel is unstable and asymmetric so that the friction factor for a fully-developed flow would not be applicable. If the relative length, which is the length of straight channel between two bends, is not long enough to form a fully-developed velocity profile, flow maldistribution from the upstream bend to the downstream bend results in additional pressure loss. In very low Reynolds number ranges, where flow separation and re-attachment at turning points in the zigzag channel do not occur, the flow velocity and profile in the channel would be stable so that the frictional pressure drop is dominant. At high Reynolds numbers, greater than 2×105, the local loss coefficient of any elbows and bends can be considered constant and independent of the Reynolds number [19]. However, the characteristics of the bend loss coefficient in the laminar regime, where the Reynolds number is less than 2×103, is complex. In the case of a pair of 90o z-shaped channels, the increase of relative length ratio between the axes of two single bends first leads to a sharp increase of the total resistance coefficient and then, when a certain maximum is reached, it gradually decreases to a value roughly to twice the resistance coefficient of a single bend [19]. The angle of bend is the most influential geometric parameter. As the angle of the zigzag channel increases, the pressure loss of the channel increases significantly. However, if the bend of the channel is not sharp-edged, i.e. rounded corner, the pressure loss of the zigzag channel becomes lower than sharp-edged channel. Based on the review on flow characteristics in zigzag channel, four geometric parameters of hydraulic diameter, relative length, zigzag angle and radius of curvature were selected to develop the friction factor and Nusselt number correlations.
3. CFD Modeling of Zigzag Channel
To investigate the thermal-hydraulic characteristics of flow through the zigzag channel numerically, single- and two-channel CFD models have been developed. The set of computational domains for each model was generated by varying major geometric parameters, e.g. zigzag angle, relative length of straight channel between two bends, etc. Details of the geometric configuration and the numerical methods used for CFD simulation of zigzag channel are described in Section 3.1 and 3.2.
3.1 Geometric configuration of the zigzag-channel model Figure 3 shows the geometries of two CFD models. First, a single zigzag channel with a fully-developed periodic boundary condition was simulated under isothermal conditions to investigate the effects of the zigzag angle on pressure drop. The simulations were parameterized using (α), the ratio of the relative length of the straight starting section to the hydraulic diameter (lR/Dh) and relative radius of curvature, (R/Dh) of the bend. Three sensitivity studies for the single-channel model were conducted to evaluate the mesh dependency, the fully-developed periodic boundary condition, and flow model dependency. Second, a two-channel model was developed to investigate the heat transfer characteristics of zigzag channels. The diameter of the semicircular channel was 2.0 mm for both models. Constant fluid properties were employed for the single-channel model while temperature-dependent properties were used for the twochannel model. In this work, the zigzag angle ranged from 5° to 45°, the relative length ratio ranged from 4.09 to 32.7, and the Reynolds number ranged from 100 to 2,000. The radius of curvature ranged from 0.5 mm to 1.0 mm. As for the two-channel CFD model, the height (H), width (W) and thickness of the solid body between the two channels (δ) were 3.2 mm, 2.5 mm and 0.6 mm, respectively. The ranges of the geometric parameters are summarized in Table I.
3.2 Numerical method Steady-state numerical solutions for laminar flow simulation of the single-channel model and conjugate heat and mass transfer simulation of the two-channel model were obtained using the commercial CFD code, STARCCM+ Ver.10.060.01.R8. The advection terms were discretized by a second-order upwind convection scheme. Both models used a polyhedral mesh with boundary layer elements. In the singlechannel model, the fully-developed periodic boundary condition was specified on inlet and outlet boundaries to minimize the computational cost and time. In the two-channel CFD model, periodic boundary conditions were specified on the top and bottom walls, and on the left and right side walls. Mass flow inlet and pressure outlet boundary conditions were specified for the two-channel model. The front and back walls were specified as adiabatic walls. Constant fluid properties were used in the singlechannel model while temperature-dependent fluid properties were used in the two-channel model. The working fluid is a helium gas. The constant and temperature-dependent Helium properties were obtained from the National Institute of Standards Technology (NIST) Chemistry WebBook database [20]. The solid material of the two-channel model was Alloy 617. A constant density of 8,360 kg/m3 was specified for Alloy 617. The temperature–dependent thermal conductivity and specific heat of Alloy 617 are tabulated in Table II [21]. The apparent friction factor of the zigzag channel is determined as follows:
(1) where Ax-s is the cross-sectional area of the channel, Dh is the hydraulic diameter, L is the total travel length, ṁ is the mass flow rate, ∆P is the pressure drop between the inlet and outlet of the channel and ρ is the fluid density. The heat transfer rate of the heat exchanger is obtained from the total enthalpy change of the fluid through the heat exchanger as follows: (2) where cp is the specific heat of fluid, Ti and To are the fluid temperatures at the inlet and outlet, respectively. For a given boundary condition, the heat transfer rate of the heat exchanger can be determined by Eqn. (2). The size of heat exchanger can be determined by the Log Mean Temperature Difference (LMTD) method as follows: (3) where A is an overall heat transfer surface area, U is the overall heat transfer coefficient and ∆TLM is the log mean temperature difference. In the LMTD method, the overall heat transfer coefficient can be obtained from the convective heat transfer coefficients of hot and cold fluids and the thermal conductivity of solid body as follows: (4) where hh and hc are the averaged heat transfer coefficients of the cold and hot fluid sides, respectively, ks is the thermal conductivity of the solid heat exchanger body and δ is the plate thickness between the cold and hot channels. In the heat exchanger design process, the Nusselt number correlation is employed to determine the local heat transfer coefficient in Eqn. (5). The Nusselt number is defined as follows: (5) where kf is the thermal conductivity of fluid, Re is the Reynolds number, Pr is the Prandtl number and Ω represents the geometric parameters. In Eqns. (2)-(5), the mass flow rate and the inlet and outlet temperatures of the heat exchanger are input variables for the heat exchanger design. However, the convective heat transfer coefficients in Eqns. (4) and (5) are unknown variables which are dependent on heat exchanger design and thermal hydraulic conditions. To develop the Nusselt number correlation, the heat transfer coefficient in Eqn. (5) needs to be determined from the CFD analysis result. In the CFD analysis, the heat transfer coefficient can be determined by Eqn. (6) as follows:
(6) where is the averaged heat transfer coefficient, A are the heat transfer surface area, Tm is the fluid bulk temperature and Tw is the wall temperature. The subscripts h and c mean the hot fluid side and cold fluid side, respectively. In Eqn. (6), the heat transfer surface area of each fluid is determined based on the design of heat exchanger. The fluid bulk temperature and wall temperature can be determined as follows: (7) (8) where u is the fluid velocity, ρ is the fluid density and cp is the specific heat of fluid. The heat transfer rate in Eqns. (2) and (6), the fluid bulk in Eqn. (7) and wall temperature in Eqn. (8) of the given heat exchanger geometry are obtained from the CFD analysis result and the heat transfer coefficient in Eqn. (6) can be determined. Consequently, the Nusselt number can be obtained by Eqn. (5) from the CFD analysis results. The relative deviation of one variable with respect to another one (reference) is determined as follows formula. (9)
4. Sensitivity Tests for CFD Models In order to assess the reliability and accuracy of the CFD models, sensitivity tests for mesh dependency, geometry scale-up and flow models have been performed. In following sections, details of sensitivity tests were described. 4.1 Mesh dependency test A mesh sensitivity analysis was conducted by the Grid Convergence Index (GCI) method [22]. This test was carried out using the single-channel model with a zigzag angle of 45° and a relative length ratio of 8.18. A mass flow rate corresponding to a Reynolds number of 1000 was specified for the inlet boundary. Three polyhedral mesh structures were constructed to evaluate the GCI of the fine mesh. The values of fapp·Re obtained by the coarse, medium and fine mesh structures were denoted as ϕ3, ϕ2 and ϕ1, respectively. The apparent order p is determined as follows: (10) (11)
(12) where ε21=ϕ2 − ϕ1, ε32=ϕ3 − ϕ2, r21=h2/h1 and r32=h3/h2. h denotes the mesh size. The extrapolated solution is determined by Eqn. (13). (13) The approximate relative error is determined by Eqn. (14). (14) The grid convergence index is determined by Eqn. (15). (15) Figure 4 shows the results of the mesh dependency test. The number of computational cells for the coarse, medium and fine mesh for the single-channel model were 125,870, 596,511 and 3,365,149, respectively. The values of GCI for medium and fine meshes were 3.33% and 0.49%, respectively. The fine mesh structure resulted in very similar apparent friction factors to the extrapolated solution. The relative error of the fine mesh to the extrapolated solution was 0.39%. It is concluded that the fine mesh used in this study is reliable to analyze the pressure drop through the zigzag channel. To minimize the numerical error due to the mesh, the parameters of the fine mesh were applied to both the single-channel and two-channel models. The mesh size of the two-channel model varies from approximately 20 million cells to 53 million cells according to the relative length of the channel.
4.2 Geometry scale-up tests The fully-developed periodic boundary condition was employed in the single-channel model to reduce the size of the computational domain. A geometry scale-up study was performed to evaluate the numerical uncertainty that might be caused by this modeling with a periodic boundary condition. To evaluate the effect of the number of bends in the computational domain, the number of bends was varied from 2 to 8 while the fully-developed periodic boundary condition was implemented to inlet and outlet boundaries. In this test, the relative length ratio (lR/Dh) and the zigzag angle were 8.183 and 45°. Figure 5 shows the values of fapp·Re as a function of Reynolds number. The result did not vary with the number of bends in the computational domain. The maximum relative deviation of the fapp·Re value among three cases was less than 0.7%. Therefore, the required computational time and cost of the simulation could be reduced by the two-bend CFD model with the fully-developed periodic boundary condition. The geometry scale-up test for the two-channel CFD model was implemented by varying the number of bends in the computational domain. Since the inlet boundary condition in the two-channel CFD model was uniform distribution of mass flow rate, the velocity profile at the entrance region would be different from the downstream of the channel. To evaluate the effect of the flow developing at the entrance region, the apparent fanning friction factors of two-channel models with 10 bends, 20 bends and 40 bends were
investigated. The zigzag angle and relative length ratio for this test were 15° and 4.09, respectively. To determine the local pressure drop, the zigzag channel was divided into sub-channels along the flow direction. The local friction factor was evaluated from the local pressure drop of each sub-channel. Figure 6 shows the local friction factor of the hot fluid side as a function of location from the inlet. The local friction factors for increasing number of bends lie on the same line. The apparent friction factor as a function of number of bends is depicted in Fig. 7. Based on the result of a 40-bend model, the relative deviations of apparent friction factor of 10 bends and 20 bends cases were evaluated to be 11.2 and 3.6 %, respectively. The expected apparent friction factor for 100 bends was 0.0644. Therefore, the two-channel model with 20 bends was employed to obtain the results for thermal-hydraulic performance of zigzag channel.
4.3 Flow model sensitivity test The flow regime is considered to be laminar flow because the Reynolds number studied in this work was less than 2,000 for all cases. However, the velocity profile in the semi-circular zigzag channel would be far from the fully-developed laminar flow due to its wavy geometry. The momentum change of the fluid flow due to flow separation and re-attachment at the bends is the main reason for increased the pressure drop in the zigzag channel. Therefore, accurate prediction of the separated flow is of importance for the analysis in a zigzag channel. To evaluate the flow model sensitivity for the separated flow and its influence on the pressure loss, the simulations were conducted using both the laminar solver and a RANSbased turbulent solver. As for the turbulence model, the Abe-Kondho-Nagano (AKN) k-ε model, standard low Reynolds number k-ε model, realizable k-ε two-layer model, and Shear Stress Transport (SST) k-ω model were employed. To maximize the magnitude of the flow separation, the zigzag channel with an angle of 45° was selected. Figure 8 shows the fanning friction factor – Reynolds number product of the 45° channel as a function of Reynolds number. There is no significant difference in pressure loss for the different turbulence models, but the pressure drop predicted by the turbulence models slightly decreased, compared to the laminar solution, as the Reynolds number increased. Consequently, a turbulence model sensitivity analysis would be necessary to evaluate the model uncertainty for turbulent flow in zigzag channels where the Reynolds number is greater than 2,000.
5.
Development of Friction Factor Correlation
The pressure drop of the fluid flow in the zigzag channel is the sum of the frictional loss and minor losses, including form loss due to the changes in the flow direction and/or flow cross-sectional area, etc. In this study, the minor losses at the entrance or exit of the channel are neglected because the fully-developed periodic boundary condition was implemented at the interface between the inlet and outlet of the zigzag channel. Also, the minor losses due to flow acceleration or deceleration associated with heating or cooling are neglected because it is an isothermal flow. In Kim and NO’s study [13], a linear correlation of the apparent fanning friction factor of zigzag channel was analytically proposed as a function of Reynolds number as follows:
(16) where C1 corresponds to the frictional loss and C2 corresponds to the form loss. Although this correlation seems to have the correct format, the relationship between the model coefficients and geometric parameters were not examined in their work. The correlation of friction factor as a function of geometric parameters is essential to the design of new heat exchangers and evaluation of the heat exchanger pressure drop of. In this section, therefore, the relationship between the friction factor and geometric parameters were examined by the CFD analysis for both sharp-edged and round-edged zigzag channels.
5.1 Friction factor of sharp-edged channel As the Reynolds number approaches 0, the model coefficient C1 will approach 15.78 which is the theoretical fanning friction factor –Reynolds number product for fully-developed flow in a semi-circular straight channel. The coefficient C1 is assumed to be a constant value of 15.78. On the other hand, the model coefficient C2 should be a function of geometric parameters. To develop the correlation for constant C2, the effects of zigzag angle, relative length ratio and radius of curvature were examined. The apparent friction factors were obtained in the Reynolds number range from 50 to 2,000. Figure 9 shows the values of C2·Re as a function of Reynolds number over a range of zigzag angles and relative length ratios. The value of C2·Re generally showed a linear relation with the Reynolds number. The value of C2 could be obtained from of the slope of a linear fit for each case. The velocity profile for fluid flow downstream of the bend in a zigzag channel is asymmetric due to the centrifugal force caused by flow direction change at the bend. This asymmetric velocity profile reverts to a symmetric shape as the fluid flow passes through the straight channel. Figure 10 shows the value of C2 as a function of relative length ratio (lR/Dh). It is found that C2 decreased exponentially as the relative length ratio increased. A correlation for C2 is given by Eqn. (17) as a function of relative length ratio. (17) Since the model coefficient C2 strongly depends on the zigzag angle, the model coefficients C3, C4 and C5 should also be a function of zigzag angle. The values of model coefficients were determined through the least squares method with a non-linear Generalized Reduced Gradient (GRG) algorithm. Figure 11 show the values of C3, C4 and C5 as a function of zigzag angle, respectively. The correlations for the model coefficients obtained from the GRG algorithm as a function of zigzag angle are given by Eqns. (18-20). (18) (19) (20)
Substituting Eqns. (17-20) into Eqn. (16) yields the apparent fanning friction factor correlation for semicircular zigzag channels as follows: (21) where α is the zigzag angle (rad) and lR/Dh is the relative length ratio. The apparent friction factor correlation for sharp-edged zigzag channels is valid for 50 ≤ Re ≤ 2,000. Figure 12 shows a comparison of values of fapp·Re for sharp-edged zigzag channels between the CFD results and the correlation. In general, the correlation shows good agreement with the CFD results within a relative deviation of ± 15%. A few relative deviations over 15% were observed at a Reynolds number less than 50.
5.2 Friction factor of round-edged zigzag channel To investigate the effect of bend radius of curvature, the apparent friction factor of zigzag channels was analyzed for a radius of curvature ranging from 0.5 mm to 1.0 mm. For this series of simulations, the zigzag angle and relative length ratio were fixed at 25° and 8.183, respectively. Figure 13 shows fapp·Re for zigzag channels as a function of bend radius of curvature. In these plots, the radius of curvature of 0.0 mm represents a sharp-edged zigzag channel. The value of fapp·Re decreased as the radius of curvature increased. The effect of the roundness of the bend became more significant at higher Reynolds numbers. The value of fapp·Re with a radius of curvature of 1.0 mm (R/Dh=0.818) was reduced by 15.86% compared to the sharp-edged zigzag channel at a Reynolds number of 2,000. The influence of edge radius of curvature also became more significant as the zigzag angle increased and the relative length ratio (lR/Dh) decreased. The value of fapp·Re of round-edged zigzag channel with zigzag angle of 45° and relative length ratio of 4.09 was reduced by 54% compared to the value for the sharp-edged channel. Figure 14 shows the apparent friction factor of a round-edged zigzag channel with a radius of curvature of 1.0 mm as a function of Reynolds number. Since the difference of fapp·Re between round-edged and sharpedged for zigzag angle of 5° was negligible, the result of zigzag angle of 5° was used to develop the correlation for round-edged zigzag channel. The apparent friction factor for a round-edged zigzag channel was developed as a function of Reynolds number, zigzag angle and relative length ratio in the same manner which was used to develop the correlation for a sharp-edged zigzag channel. The apparent friction factor for a round-edged zigzag channel with a radius of curvature less than 1.0 mm could be obtained by interpolation of two correlations of sharp and round-edged zigzag channels. The apparent friction factor of the round-edged zigzag channel is given by Eqn. (22). (22) The correlation for apparent friction factor for round-edged zigzag channels is valid for 50 ≤ Re ≤ 2,000. Figure 15 shows a comparison of values of fapp·Re for round-edged zigzag channels between the CFD results and the developed correlation. Since the value of fapp·Re of round-edged zigzag channels was not
quite linear compared to that of sharp-edged zigzag channel, the maximum deviation between the correlation and the CFD results was slightly increased up to ± 20%.
5.3 Effect of temperature-dependent fluid properties on friction factor The friction factor for zigzag channels was developed from isothermal simulations in which the fluid properties were constant. For determining heat transfer rate, the fluid properties vary because of the temperature change in each fluid side. The friction factors obtained from the two-channel CFD model with heat transfer were compared to the friction factor correlation. When the fluid density is changed by heating or cooling, the fluid flow is accelerated or decelerated. The minor losses due to this flow acceleration or deceleration were considered for the comparison of friction factor. The minor loss due to flow acceleration/deceleration can be evaluated as given by Eqn. (23). (23) The apparent friction factors accounting for the flow acceleration/deceleration effect were compared to those calculated by the correlation developed in Eqn. (21), as shown in Fig. 16. The apparent friction factors from the two-channel model simulations agreed with the friction factor correlation within the relative deviation of ± 15%. The ratio of minor loss to total pressure loss ranged from 0.8% up to 13.5% with decreasing the zigzag angle and increasing Reynolds number. The evaluated minor loss given by Eqn. (23) is an approximate value because the inlet and outlet densities were used for calculation, but not local densities throughout the channel. Hence, the predictive capability of the correlation developed for the apparent friction factor of the zigzag channel could be improved by developing a methodology to evaluate minor losses due to flow acceleration/deceleration more accurately.
6.
Development of Heat Transfer Correlation
The Nusselt number for fully-develop laminar flow in a straight pipe with a uniform boundary condition is a constant, independent of Reynolds number. However, fluid flow in the zigzag channel does not achieve a fully-developed condition. It is reported in several instances in the literature that the Nusselt number for zigzag channels does not achieve a constant value, even for laminar periodic fully-developed flow conditions. Nusselt numbers for zigzag channel geometries increase with increasing Reynolds number, even in the laminar regime. To develop an explicit correlation for Nusselt number, the basic power-law form was used as follows: (24) where C6, C7 and C8 are the model coefficients for the Nusselt number correlation.
6.1 Determination of the model constant for Prandtl number The Prandtl number is defined as the ratio of momentum diffusivity to thermal diffusivity. It is a function of specific heat, dynamic viscosity and thermal conductivity of the fluid. It is assumed that the value for the exponent on Prandtl number in Eqn. (24) is independent of the geometric parameters. To determine the model coefficient C8 for the Nusselt number correlation, a sensitivity test was conducted by varying the Prandtl number from 0.1 to 50 with three zigzag angles of 15°, 30° and 45°. The mass flow rates of hot and cold fluid sides were specified at 5×10-5 kg/s. Figure 17 shows the resulting Nusselt numbers as a function of Prandtl number. The exponent on Prandtl number, C8, for low Prandtl number was higher than that for high Prandtl number. The Prandtl number exponent was independent of zigzag angle, but varied with the value of the Prandtl number and the heat transfer mode. The exponent for Prandtl number for Pr ≤ 1.0 was higher than that for Pr ≥ 5.0. For the hot fluid side where the fluid is being cooled, the Prandtl number exponent was slightly lower than that for the cold fluid side. For low Prandtl number, less than 1.0, the exponents of Prandtl number, C8, for hot and cold fluid sides were 0.56 and 0.58, respectively. For higher Prandtl numbers greater than 5.0, the values of C8 for hot and cold fluid sides were 0.31 and 0.35, respectively. In this study, the exponents of Prandtl number for low Prandtl number were used to develop the Nusselt number correlation since the Prandtl number of helium, 0.66, corresponds to the low Prandtl number region.
6.2 Effect of radius of curvature on Nusselt number The effect of radius of curvature on Nusselt number was minor compared to the effect on the friction factor as shown in Fig. 18. The variation in Nusselt number was less than approximately 4.8 % while it was as high as 41% for friction factor. The change in heat transfer surface area associated with the radius of curvature was very small so that its impact on the overall heat transfer was negligible whereas the radius of curvature did have an effect on the local velocity field in zigzag channels. The major advantage of round-edged zigzag channel is that the pressure loss through the heat exchanger could be significantly reduced with a much lower reduction in Nusselt number. Based on this analysis, the radius of curvature was excluded from the geometric parameter for Nusselt number correlation.
6.3 Development of Nusselt number correlation for zigzag channel To develop a Nusselt number correlation for zigzag channels, the characteristic behavior of the flow needs to be examined first. As shown in Fig. 19, a periodic flow pattern transitions to a chaotic pattern at higher Reynolds numbers. The flow pattern at Reynolds numbers less than 470 was stable and periodic. Irregular and rotational secondary vector fields on the cross-section of the channel increased in strength with increasing Reynolds number. A region of reversed flow downstream of the bends was observed for Reynolds numbers greater than 470, as shown in Fig. 20. The occurrence of the secondary flows downstream of the bends affected the heat transfer characteristics of the zigzag channel. Nusselt number was linearly proportional to the Reynolds number where the zigzag angle and Reynolds number were low so that there is no reversed flow downstream of the bend. Nusselt numbers for higher Reynolds numbers and zigzag angles were proportional to Ren. Based on these observations, separate Nusselt number
correlations were developed for two regions as depicted in Fig. 21. In this map, the critical value, which results in re-circulating secondary flows, of zigzag angle (αc) was 15° and the critical Reynolds number (Rec) was 550, regardless of the relative length ratio. Nusselt number was depicted as a function of Reynolds number for respective relative length ratios and zigzag angles in Fig. 22 and Fig. 23. As expected, the Nusselt number increased with increasing Reynolds number. It also increased with zigzag angle, but decreased with increasing relative length ratio. The dependency of Nusselt number on the relative length ratio was deemed to be diminished as the zigzag angle decreased. To determine the model coefficients C6 and C7 in Eqn. (24), a non-linear least squares regression analysis was performed. The procedure to determine the correlation coefficients is depicted in Fig. 24. First, the matrices of Nusselt number obtained from the CFD simulations and Eqn. (24) were constructed and then the correlation coefficients which minimize the sum of squared residuals were determined using a nonlinear GRG algorithm. As a result, the matrices of coefficients as a function of relative length ratio were constructed for each zigzag angle. The functional form of the coefficient models could be determined from the matrices of coefficients re-constructed as a function of relative length ratio and zigzag angle. To optimize the coefficient models, the squared residuals of Nusselt number between the CFD analysis result and correlations were evaluated. Finally, the constants for the coefficient models which minimized the sum of squared residuals were determined. In Region-A, where the reversed secondary flow did not occur, the Nusselt numbers were linearly proportional to the Reynolds number. Therefore, a linear equation of Nusselt number as a function of Reynolds number was employed for region A instead of Eqn. (24). The Nusselt number correlation for Region-A was developed and is given by Eqn. (25). (25) Equation (25) can be used to evaluate the Nusselt number for either the hot fluid side or the cold fluid side in Region-A. In Region-B, Nusselt number correlations were developed based on Eqn. (24) as follows: (26) (27) where the unit of zigzag angle (α) in the correlations is in radians. The Nusselt number correlations are valid for 100 ≤ Re ≤ 2000 and Pr ≤ 1.0. Figures 25 shows the Nusselt number comparison between CFD results and correlation. For both the hot and cold fluid sides, the correlation agreed with the CFD simulation results within a relative deviation of ±10%.
6.4 Validation of friction factor and Nusselt number correlations In order to gain confidence in the correlations obtained both for Nusselt number and fanning friction factor for the zig-zag channel geometries, comparisons were made with experimental data obtained at The Ohio State University (OSU) [24] and at the Korean Advanced Institute of Science and Technology
(KAIST) [12, 13], qualified with uncertainty analysis. The root mean square method was used to analyze and quantify the contribution made by the uncertainty in one variable to the overall uncertainty in the result. The root mean square method is illustrated by showing the relative uncertainty evaluation for both Reynolds number and Nusselt number, respectively. (28)
(29) Further details on uncertainty analysis for each measured variable can be found in the literature [23, 24]. Figure 25 compares the experimentally determined fanning friction factor values with the correlation obtained from CFD analysis. Note that the OSU PCHE is a round-edged zigzag channel PCHE whereas the KAIST PCHE has a sharp-edged zigzag channel. The CFD-based friction factor correlation for the round-edge case shows good agreement with the OSU PCHE experimental data. The average relative deviation was 9.95% and the minimum and maximum relative deviation was 0.7% and 24%, respectively. The CFD-based sharp-edged friction factor correlation agreed very well with Kim’s correlation [12] with a maximum deviation less than 1.0%. The CFD-based friction factor correlation is valid for Reynolds numbers up to 2000 and Kim’s empirical correlation is valid for Reynolds numbers up to 1200. Thus, the CFD-based correlations and Kim’s correlation are extrapolated for this comparison. The comparison results showed that the extrapolated correlations can accurately predict the friction factor of zigzag channel for both sharp-edged and round-edged zigzag channel PCHEs. Figure 26 shows the comparison of experimental and CFD-based Nusselt numbers for both the hot side and cold side, showing a relatively large deviation ranging between 28% to 49% for the hot side and 23% to 43% for the cold side, respectively. The CFD-based Nusselt number correlations overestimated the experimental data. Differences between the experimental configuration and the CFD model are responsible for this deviation. In the CFD simulation, a uniform flow distribution is assumed among all the channels while in the experiment, some flow misdistribution is probable due to manifold effects mainly in the inlet plenums, leading to different mass flow rates in some channels. The CFD model only simulated the core region while the experimental data included the plenum effects. In the experiment, heat loss was not accounted for which could provide incorrect values for heat load and heat transfer effectiveness, respectively. In addition, the temperature measurement carried out in the experiment was upstream and downstream of the inlet and outlet flanges, adding more uncertainty to the exact temperature difference across the heat exchanger. The friction factor was mainly influenced by the zigzag channel geometry while the Nusselt number was influenced by overall heat exchanger design including the inlet plenum part. CFD analysis with twochannel model or single channel model would be sufficient to investigate the friction factor of zigzag channel. However, since there are many variations of the inlet plenum design, it is not possible to obtain the correlation which can predict the Nusselt number precisely independent of inlet plenum designs. CFD analysis with the two-channel model is good for predicting the core heat transfer while it could result in incorrect overall heat transfer of PCHE. Thus, for better determination of heat transfer characteristic modeling of the plenum section is very essential, but it would be computationally intensive.
7.
Conclusion
Friction factor and Nusselt number correlations developed as a function of geometric parameters are required to design compact heat exchangers, such as PCHEs and to evaluate their thermal-hydraulic performance. Furthermore, these correlations are necessary for cost analysis and the overall effectiveness of the system. In this study, an extensive CFD analysis was conducted to develop friction factor and Nusselt number correlations for laminar flow in a semi-circular zigzag channel. Hydraulic diameter, relative length ratio, zigzag angle and bend radius of curvature were selected as geometric parameters. A least squares method using a non-linear Generalized Reduced Gradient algorithm was used to determine the model coefficients of the correlations. The following points summarize the results:
Friction factor correlations for sharp-edged and round-edged zigzag channels for PCHE have been developed. The friction factor correlations were valid for 50 ≤ Re ≤ 2000, 5° ≤ α ≤ 45° and 4.09 ≤ lR/Dh ≤ 32.73. The correlation exponent on the Prandtl number for low-Prandtl numbers (< 1.0) is 0.56 for the hot fluid side and 0.58 for the cold fluid side, respectively. As for high-Prandtl numbers (>5.0), the exponent is 0.31 for hot fluid side and 0.35 for cold fluid side. CFD simulations with a two-channel model have been conducted to develop Nusselt number correlations for zigzag channels which are valid for 200 ≤ Re ≤ 2000, Pr ≤ 1.0, 5° ≤ α ≤ 45° and 4.09 ≤ lR/Dh ≤ 12.27. Note that the friction factor of the zigzag channel PCHE is mainly influenced by the zigzag channel geometry while the Nusselt number is influenced by overall heat exchanger design including the plenum sections. CFD analysis with a two-channel model is good for predicting friction factor, but could overestimate or underestimate the thermal performance of zigzag channel PCHEs according to the inlet and outlet plenum designs.
Acknowledgment This work was supported by the United States Department of Energy Office of Nuclear Energy, Very High Temperature Reactor Technology Development Office. Idaho National Laboratory (INL) is operated for the United States Department of Energy Office of Nuclear Energy by Battelle Energy Alliance, LLC, under contract No. DE-AC07-05ID14517. The authors also gratefully acknowledge helpful discussion with Junsoo Yoo at INL.
NOMENCLATURE A C
Area or Overall heat transfer surface area, m2 Model coefficient
cp D Dh f GCI H h k L lR Nu P p Pr R Re T U u W α Δ δ μ ρ
Specific heat, J/(kg·K) Diameter, m Hydraulic diameter, m Friction factor Grid convergence index Height, m Heat transfer coefficient, W/(m2·K) Thermal conductivity, W/(m·K) Length, m Relative length, m Mass flow rate, kg/s Nusselt number Pressure, Pa Order of scheme Prandtl number Heat transfer rate, W Radius of curvature, m Reynolds number Temperature, K Overall heat transfer coefficient, W/(m2·K) Fluid velocity, m/s Width, m Zigzag angle, rad Differential Plate thickness, m Fluid dynamic viscosity, Pa·sec Fluid density, kg/m3
Subscripts accel Acceleration app Apparent c Cold fluid side f Fluid h Hot fluid side HX Heat exchanger i Inlet LM Log mean temperature difference m Fluid bulk o Outlet ref Reference s Solid w Wall
References 1. D. Southall, R.L. Pierres and S.J. Dewson, Design Considerations for Compact Heat Exchangers, Proceedings of ICAPP’08, June 8-12, Anaheim, CA, USA, , 2008. 2. D. Southall and S.J. Dewson, Innovative Compact Heat Exchangers, Proceedings of ICAPP’10, June 13-17, San Diego, CA, USA, 2010. 3. Sang-Moon Lee and Kwang-Yong Kim, A Parametric Study of the Thermal-Hydraulic Performance of a Zigzag Printed Circuit Heat Exchanger, Proceedings of the ASME/JSME 8 th Thermal Engineering Joint Conference, March 13-17, Honolulu, Hawaii, USA, 2011. 4. Sang-Moon Lee and Kwang-Yong Kim, Optimization of Zigzag Flow Channels of a Printed Circuit Heat Exchanger for Nuclear Power Plant Application, Journal of Nuclear Science and Technology, Vol. 49 (3), pp.343-351, 2012. 5. Sang-Moon Lee and Kwang-Yong Kim, Comparative Study on Performance of a Zigzag Printed Circuit Heat Exchanger with Various Channel Shapes and Configurations, Heat and Mass Transfer, Vol. 49 (7), pp.1021-1028, 2013. 6. T.L. Ngo, Y. Kato, K. Nikitin and T. Ishizuka, Heat Transfer and Pressure Drop Correlations of Microchannel Heat Exchangers with S-Shaped and Zigzag Fins for Carbon Dioxide Cycles, Experimental Thermal and Fluid Science, Vol. 32, pp.560-570, 2007. 7. Roma Fatima, Alan Kruizenga, Mark Anderson and Devesh Ranjan, Numerical Investigation of Thermal Hydraulic Behavior of Supercritical Carbon Dioxide in Compact Heat Exchangers, 2014 Supercritical CO2 Power Cycle Symposium, Sept. 9-10, Pittsburgh, PA, USA, 2014. 8. A. Kurizenga, Mark Anderson, Roma Fatima, Machael Corradini, Aaron Towne and Devesh Ranjan, Heat Transfer of Supercritical Carbon Dioxide in Printed Circuit Heat Exchanger Geometries, Journal of Thermal Science and Engineering Applications, Vol. 3(3), p. 031002, 2011. 9. H. Song, J. Van Meter, S. Lomperski, D. Cho, H.Y. Kim and A.Tokuhiro, Experimental Investigations of a printed circuit heat exchanger for supercritical CO2 and water heat exchange, NTHAS5-N002, Nov. 26-29, Jeju, South Korea, 2006. 10. N.S. Berbish, M. Moawed, M. Ammar and R.I. Afifi, Heat Transfer and Friction Factor of Turbulent Flow through a Horizontal Semi-circular duct, Heat and Mass Transfer, Vol. 47, pp.377-384, 2011. 11. S.P. Kar, CFD Analysis of Printed Circuit Heat Exchanger, M.S Thesis, National Institute of Technology Rourkela, 2007. 12. I.H. Kim, Experimental and Numerical Investigations of Thermal-Hydraulic Characteristics for the Design of a Printed Circuit Heat Exchanger (PCHE) in HTGRs, Ph.D. Dissertation, Korea Advanced Institute of Science Technology (KAIST), 2012. 13. I.H. Kim and H.C. No, Thermal Hydraulic Performance Analysis of a Printed Circuit Heat Exchanger using a Helium-Water Test Loop and Numerical Simulations, Applied Thermal Engineering, Vol. 31, pp.4064-4073, 2011. 14. O.K. Kwon, M.J. Choi and Y.J. Choi, Heat Transfer and Pressure Drop Characteristics in Zigzag Channel Angles of Printed Circuit Heat Exchangers, Korean Journal of Air-Conditioning and Refrigeration Engineering, Vol. 21 (9), pp.475-482, 2009. 15. J.W. Seo, Y.H. Kim, D.S. Kim, Y.D. Choi and K.J. Lee, Heat Transfer and Pressure Drop Characteristics in Straight Microchannel of Printed Circuit Heat Exchangers, Entropy, Vol. 17, pp.3438-3457, 2015.
16. T. Ma, L. Li, X.Y. Xu, Y.T. Chen and Q.W. Wang, Study on Local Thermal-Hydraulic Performance and Optimization of Zigzag-type Printed Circuit Heat Exchanger at High Temperature, Energy Conversion and Management, Vol. 104, pp. 55-66, 2015. 17. A. Meshram, A.K. Jaiswal, S.D. Khivsara, J.D. Ortega, C. Ho, R. Bapat and P. Dutta, Modeling and Analysis of a Printed Circuit Heat Exchanger for Supercritical CO2 Power Cycle Applications, Applied Thermal Engineering, In Press, Corrected Proof, 2016. 18. E.N. Van Abel, M.H. Anderson and M.L. Corradini, Numerical Investigation of Pressure Drop and Local Heat Transfer of Supercritical CO2 in Printed Circuit Heat Exchanger, Supercritical CO2 Power Cycle Symposium, May 24-25, Boulder, CO, 2011. 19. I.E. Idelchik, Handbook of Hydraulic Resistance 3rd Ed., Begell House Inc., New York, 1996. 20. National Institute of Standards and Technology, NIST Chemistry WebBook: NIST Standard Reference Database Number 69, http://webbook.nist.gov/chemistry/, accessed on July 20, 2016. 21. INCONEL 617 Technical Bulletin, Publication Number SMC-029, Special Metals Corporation, Huntington, WV, 2005. 22. P.J. Roache, Quantification of uncertainty in computational fluid dynamics, Annual Review of Fluid Mechanics, Vol. 29, pp. 123–160, 1997. 23. S.K. Mylavarapu, Design, Fabrication, Performance Testing, and Modeling of Diffusion Bonded Compact Heat Exchangers in a High-Temperature Helium Test Facility, Doctoral Dissertation, The Ohio State University, 2011. 24. M. Chen, X. Sun, R.N. Christensen, I. Skavdahl, V. Utgikar and P. Sabharwall, Pressure Drop and Heat Transfer Characteristics of a High-Temperature Printed Circuit Heat Exchanger, Applied Thermal Engineering, 108, 2016.
(a) PCHE section (crossflow configuration) (b) Etched plates Fig. 1 Printed circuit heat exchanger [1]
Fig. 2 Temperature and pressure capabilities of heat exchangers [2]
Hot fluid side
D
W D
R
H
δ
Cold fluid side
Solid body
α lR/2
(a) Single-channel CFD model geometry.
(b) Two-channel CFD model geometry.
(c) Mesh structure on frontal surface of the two-channel model.
(d) Eagle-eye view of mesh structure of the two-channel model.
Fig. 3 Schematic diagram of geometry and mesh structure of CFD models.
Fig. 4 Result of mesh dependency test.
Fig. 5 The effect of number of bends in the computational domain with the fully-developed periodic boundary condition
Fig. 6 Local friction factor of hot channel as a function of distance from the inlet according to the number of bends (α=15°, lR/Dh=4.09)
Fig. 7. Apparent friction factor of hot channel as a function of number of bends (α=15°, lR/Dh=4.09)
Fig. 8 Result of flow model dependency test
(a) α = 45°
(b) α = 40°
(c) α = 35°
(d) α = 30°
(e) α = 25°
(f) α = 20°
(g) α = 15°
(h) α = 10°
(i) α = 5° Fig. 9 The plots of fapp·Re for various relative lengths and zigzag angles of sharp-edged zigzag channel
Fig. 10 Model coefficient (C2) as a function of relative length ratio (lR/Dh)
Fig. 11 Comparison of fapp·Re of sharp-edged zigzag channel (CFD vs. Correlation)
Fig. 12 fapp·Re as a function of the relative radius of curvature (R/Dh) (α = 25°, lR/Dh = 8.183)
(a) α = 45°
(b) α = 40°
(c) α = 35°
(d) α = 30°
(e) α = 25°
(f) α = 20°
(g) α = 15° (h) α = 10° Fig. 13 The plots of fapp·Re for various relative lengths and zigzag angles of round-edged zigzag channel
Fig. 14 Comparison of fapp·Re of round-edged zigzag channel (CFD vs. Correlation)
Fig. 15 Comparison of fapp·Re (Isothermal correlation vs. Two-channel model)
(a) Hot fluid side (b) Cold fluid side Fig. 16 Plots of Nusselt number as a function of Prandtl number
(a) Hot fluid side (b) Cold fluid side Fig. 17 Effect of radius of curvature on Nusselt number and apparent friction factor (α=45°, lR/Dh =8.183)
(a) Reynolds number = 285
(b) Reynolds number = 470
(c) Reynolds number = 565
(d) Reynolds number = 657
(e) Reynolds number = 842 (f) Reynolds number = 1845 Fig. 18 Streamwise velocity (x-velocity) magnitudes and tangential velocity vector fields on the crosssection of 15th bend from the inlet of hot fluid side (α=15°, lR/Dh=8.183)
Fig. 19 Normalized flow x-velocity profiles along to line A-A’ (α=15°, lR/Dh=8.183)
Fig. 20 Region map for Nusselt number correlation (αc=15°, Rec=550)
(a) α = 45°
(b) α = 40°
(c) α = 35°
(d) α = 30°
(e) α = 25°
(f) α = 20°
(g) α = 15°
(h) α = 10°
(i) α = 5° Fig. 21 Nusselt number as a function of Reynolds number (Hot fluid side)
(a) α = 45°
(b) α = 40°
(c) α = 35°
(d) α = 30°
(e) α = 25°
(f) α = 20°
(g) α = 15°
(h) α = 10°
(i) α = 5° Fig. 22 Nusselt number as a function of Reynolds number (Cold fluid side)
Fig. 23 Schematic diagram of procedure to determine the functional form of coefficients for Nusselt number correlation
(a) Hot fluid side (b) Cold fluid side Fig. 24 Comparison of Nusselt number (Correlation vs. CFD result)
Fig. 25 Comparisons of friction factor (Experimental data vs. correlations)
(a) Hot fluid side (b) Cold fluid side Fig. 26 Comparisons of Nusselt Number (Experimental data vs. correlations)
TABLE I. Geometric Parameters Variable Channel diameter, D, (mm) Hydraulic diameter, Dh, (mm) Height of solid body, H, (mm) Width of solid body, W, (mm) Thickness between two channels, δ, (mm) Zigzag angle, α, (°) Sub-channel length, lR, (mm) Radius of curvature, R, (mm)
Values 2.0 1.222 3.2 2.5 0.6 5, 10, 15, 20, 25, 30, 35, 40, 45 5, 6, 8, 10, 15, 20, 40 0.5, 0.6, 0.75, 0.9, 1.0
TABLE II. Thermo-physical Properties of Alloy 617 Temperature, K 293.15 373.15 473.15 573.15 673.15 773.15 873.15 973.15 1073.15 1173.15 1273.15
Thermal conductivity, W/(m·K) 13.4 14.7 16.3 17.7 19.3 20.9 22.5 23.9 25.5 27.1 28.7
Specific Heat, J/(kg·K) 419 440 465 490 515 536 561 586 611 636 662
S1359-4311(16)34274-0 http://dx.doi.org/10.1016/j.applthermaleng.2017.05.135 ATE 10449
To appear in:
Applied Thermal Engineering
Received Date: Revised Date: Accepted Date:
21 December 2016 16 May 2017 20 May 2017
Please cite this article as: S-J. Yoon, J. O'Brien, M. Chen, P. Sabharwall, X. Sun, Development and Validation of Nusselt Number and Friction Factor Correlations for Laminar flow in Semi-circular Zigzag Channel of Printed Circuit Heat Exchanger, Applied Thermal Engineering (2017), doi: http://dx.doi.org/10.1016/j.applthermaleng. 2017.05.135
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Development and Validation of Nusselt Number and Friction Factor Correlations for Laminar flow in Semi-circular Zigzag Channel of Printed Circuit Heat Exchanger
Su-Jong Yoona*, James O’Briena, Minghui Chenb, Piyush Sabharwalla and Xiaodong Sunb a
Idaho National Laboratory, 2525 Fremont Ave., Idaho Falls, ID 83415
b
Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI 48109
Tel: 208-526-5407, Fax: 208-526-2930, *Email: [email protected]
Abstract Friction factor and heat transfer correlations have been developed for printed circuit heat exchangers (PCHEs) as a function of geometric parameters. These correlations summarize the thermal hydraulic performance of PCHEs and allow for accurate determination of their cost and effectiveness.. Although there have been previous studies of the thermal-hydraulic performance of zigzag channel PCHEs, explicit correlations for friction factor or Nusselt number as a function of geometric parameters have not yet been reported. In this study, computational fluid dynamic (CFD) predictions were used to aid in the development of friction factor and Nusselt number correlations for laminar flow in semicircular zigzagchannel PCHEs. Two CFD models were developed to investigate the thermal-hydraulic characteristics of fluid flow in zigzag channels. A single-channel isothermal CFD model was used to investigate friction factors in PCHE zigzag channels. A two-channel CFD model was used to investigate the Nusselt number in zigzag channels and the effect of temperature-dependent fluid properties on the pressure loss. The effects of geometric parameters such as relative length ratio, zigzag angle and radius of curvature of bend were investigated. From the extensive CFD analysis database, friction factor and Nusselt number correlations were developed by implementing a least squares method with a non-linear Generalized Reduced Gradient algorithm. The friction factor correlation is valid for 50 ≤ Re ≤ 2000, 5° ≤ α ≤ 45° and 4.09 ≤ lR/Dh ≤ 32.73. The Nusselt number correlations are valid for 200 ≤ Re ≤ 2000, Pr ≤ 1.0, 5° ≤ α ≤ 45° and 4.09 ≤ lR/Dh ≤ 12.27. The correlations were validated against experimental data from The Ohio State University (OSU) and Korea Advanced Institute of Science and Technology (KAIST). The friction factor for zigzag-channel PCHEs is mainly influenced by the zigzag channel geometry while the Nusselt number is influenced by the overall heat exchanger design including the plenum sections.
1. Introduction A Printed Circuit Heat Exchanger (PCHE) is a type of compact heat exchanger which has high heat transfer surface area density and efficiency that result in reduced size, weight and cost. As illustrated in Fig. 1, PCHEs consist of many straight or zigzag microchannels, which are fabricated by diffusionbonding the chemically etched plates [1]. The diffusion-bonding technique permits application of PCHEs over a wide range of temperatures and pressures [2], as shown in Fig. 2. Due to many advantages, PCHEs have been highlighted as a promising heat exchanger type for high temperature reactor systems (HTRs) and small modular reactors (SMRs). For instance, PCHEs are strong candidate heat exchangers for use in the intermediate heat transfer loop (IHTL) of HTRs for both power generation and non-electric applications. PCHEs with zigzag channels are used for industrial applications since the heat transfer performance of zigzag channels is superior to that of straight channels. Many researchers have made dedicated efforts to examine the thermal-hydraulic performance of PCHEs. Lee and Kim numerically investigated the effect of the channel angle and ellipse aspect ratio on the thermal-hydraulic performance of PCHEs [3] and optimized the heat exchanger design by using the hybrid multi-objective evolutionary algorithm (hybrid MOEA) coupled with the response surface approximation (RSA) model [4]. They also compared the thermal-hydraulic performance of various channel cross-sectional shapes and configurations [5]. Ngo et al. [6] developed empirical correlations of Nusselt numbers and pressure drop factors of S-shaped and zigzag-channel microchannel heat exchangers (MCHEs). Fatima et al. [7] conducted a numerical study on the thermal-hydraulic behavior of supercritical CO2 in PCHEs with semicircular zigzag channels and compared the results with experimental data obtained at University of Wisconsin [8]. Song et al. [9] also investigated the thermal-hydraulic characteristics of zigzag-channel PCHEs with CO2 and water experimentally. Berbish et al. [10] experimentally investigated forced convection heat transfer and pressure loss characteristics in a straight semicircular duct with Reynolds numbers ranging from 8,242 to 57,794. Kar [11] performed computational fluid dynamics (CFD) analyses on the thermal hydraulic characteristics in straight and sinusoidal ducts with a semi-elliptical crosssection. Kim et al. [12] conducted experiments and CFD simulations to investigate the thermal-hydraulic performance of the PCHE with semicircular zigzag channels within a Reynolds number range from 350 to 1,200. Kim and No [13] performed experiments and numerical simulations of PCHEs using a heliumwater test loop and proposed Fanning friction factor and Nusselt number correlations for a system analysis code. Kwon et al. [14] conducted experiments and CFD simulations to investigate the effect of zigzag angle on the heat transfer and pressure drop characteristics in a zigzag channel PCHE. Seo et al. [15] reviewed recent research on PCHEs and experimentally investigated the thermal hydraulic performance of straight-channel PCHEs by using the Wilson plot method. Ma et al. [16] investigated local thermal-hydraulic performance of zigzag-channel PCHEs with a commercial CFD code, FLUENT. In this study, the effect of zigzag angle on local Nusselt number and pressure drop was examined. Meshram et al. [17] evaluated the performance of PCHEs with straight and zigzag channels and developed Nusselt number and friction factor correlations as a function of geometric parameters such as zigzag pitch and angle. However, their correlations did not account for possible different flow regimes by not considering Reynolds number as one of the key variables, thus restricting the application of the correlation. Van Abel et al. [18] conducted a numerical study on supercritical CO2 in a zigzag-channel PCHE by comparing with experimental data and investigated the sensitivity on the chosen turbulence model and the radius of curvature at the bends.
To develop new PCHE designs and to evaluate their thermal-hydraulic performance, accurate information on friction factor and heat transfer coefficients are required as a function of geometric parameters. Although previous studies have investigated the thermal-hydraulic performance of PCHEs, the effects of the geometric parameters and their combinations were not investigated sufficiently. A few numerical studies [3, 13, 14 and 17] investigated the effects of design parameters and their combinations on the PCHE thermal-hydraulic performance but the relationship between the design parameters and heat exchanger thermal-hydraulic characteristics is stilled needed. Consequently, CFD analyses were conducted to investigate the effects of the geometric parameters on the thermal hydraulic characteristics in zigzag-channel PCHEs for Reynolds numbers ranging from 50 to 2,000. As a result, friction factor and Nusselt number correlations for a semi-circular zigzag channels were developed as a function of geometric parameters.
2. Flow Characteristics in Zigzag Channel The use of zigzag channels enhances heat transfer, but also results in higher pressure loss due to form losses. Several geometric parameters have a significant influence on both heat transfer and pressure loss. Basically, the zigzag channel can be considered as a series of multiple bends in the channel. The magnitude of bend loss coefficient is influenced by the degree of flow turbulence, inlet velocity profile shape, the Reynolds number, the channel length-to-diameter ratio, the bend angle, the channel aspect ratio, radius of curvature, etc. The pressure loss in the zigzag channel can be divided into frictional loss and the minor losses. Generally, the velocity profile in the zigzag channel is unstable and asymmetric so that the friction factor for a fully-developed flow would not be applicable. If the relative length, which is the length of straight channel between two bends, is not long enough to form a fully-developed velocity profile, flow maldistribution from the upstream bend to the downstream bend results in additional pressure loss. In very low Reynolds number ranges, where flow separation and re-attachment at turning points in the zigzag channel do not occur, the flow velocity and profile in the channel would be stable so that the frictional pressure drop is dominant. At high Reynolds numbers, greater than 2×105, the local loss coefficient of any elbows and bends can be considered constant and independent of the Reynolds number [19]. However, the characteristics of the bend loss coefficient in the laminar regime, where the Reynolds number is less than 2×103, is complex. In the case of a pair of 90o z-shaped channels, the increase of relative length ratio between the axes of two single bends first leads to a sharp increase of the total resistance coefficient and then, when a certain maximum is reached, it gradually decreases to a value roughly to twice the resistance coefficient of a single bend [19]. The angle of bend is the most influential geometric parameter. As the angle of the zigzag channel increases, the pressure loss of the channel increases significantly. However, if the bend of the channel is not sharp-edged, i.e. rounded corner, the pressure loss of the zigzag channel becomes lower than sharp-edged channel. Based on the review on flow characteristics in zigzag channel, four geometric parameters of hydraulic diameter, relative length, zigzag angle and radius of curvature were selected to develop the friction factor and Nusselt number correlations.
3. CFD Modeling of Zigzag Channel
To investigate the thermal-hydraulic characteristics of flow through the zigzag channel numerically, single- and two-channel CFD models have been developed. The set of computational domains for each model was generated by varying major geometric parameters, e.g. zigzag angle, relative length of straight channel between two bends, etc. Details of the geometric configuration and the numerical methods used for CFD simulation of zigzag channel are described in Section 3.1 and 3.2.
3.1 Geometric configuration of the zigzag-channel model Figure 3 shows the geometries of two CFD models. First, a single zigzag channel with a fully-developed periodic boundary condition was simulated under isothermal conditions to investigate the effects of the zigzag angle on pressure drop. The simulations were parameterized using (α), the ratio of the relative length of the straight starting section to the hydraulic diameter (lR/Dh) and relative radius of curvature, (R/Dh) of the bend. Three sensitivity studies for the single-channel model were conducted to evaluate the mesh dependency, the fully-developed periodic boundary condition, and flow model dependency. Second, a two-channel model was developed to investigate the heat transfer characteristics of zigzag channels. The diameter of the semicircular channel was 2.0 mm for both models. Constant fluid properties were employed for the single-channel model while temperature-dependent properties were used for the twochannel model. In this work, the zigzag angle ranged from 5° to 45°, the relative length ratio ranged from 4.09 to 32.7, and the Reynolds number ranged from 100 to 2,000. The radius of curvature ranged from 0.5 mm to 1.0 mm. As for the two-channel CFD model, the height (H), width (W) and thickness of the solid body between the two channels (δ) were 3.2 mm, 2.5 mm and 0.6 mm, respectively. The ranges of the geometric parameters are summarized in Table I.
3.2 Numerical method Steady-state numerical solutions for laminar flow simulation of the single-channel model and conjugate heat and mass transfer simulation of the two-channel model were obtained using the commercial CFD code, STARCCM+ Ver.10.060.01.R8. The advection terms were discretized by a second-order upwind convection scheme. Both models used a polyhedral mesh with boundary layer elements. In the singlechannel model, the fully-developed periodic boundary condition was specified on inlet and outlet boundaries to minimize the computational cost and time. In the two-channel CFD model, periodic boundary conditions were specified on the top and bottom walls, and on the left and right side walls. Mass flow inlet and pressure outlet boundary conditions were specified for the two-channel model. The front and back walls were specified as adiabatic walls. Constant fluid properties were used in the singlechannel model while temperature-dependent fluid properties were used in the two-channel model. The working fluid is a helium gas. The constant and temperature-dependent Helium properties were obtained from the National Institute of Standards Technology (NIST) Chemistry WebBook database [20]. The solid material of the two-channel model was Alloy 617. A constant density of 8,360 kg/m3 was specified for Alloy 617. The temperature–dependent thermal conductivity and specific heat of Alloy 617 are tabulated in Table II [21]. The apparent friction factor of the zigzag channel is determined as follows:
(1) where Ax-s is the cross-sectional area of the channel, Dh is the hydraulic diameter, L is the total travel length, ṁ is the mass flow rate, ∆P is the pressure drop between the inlet and outlet of the channel and ρ is the fluid density. The heat transfer rate of the heat exchanger is obtained from the total enthalpy change of the fluid through the heat exchanger as follows: (2) where cp is the specific heat of fluid, Ti and To are the fluid temperatures at the inlet and outlet, respectively. For a given boundary condition, the heat transfer rate of the heat exchanger can be determined by Eqn. (2). The size of heat exchanger can be determined by the Log Mean Temperature Difference (LMTD) method as follows: (3) where A is an overall heat transfer surface area, U is the overall heat transfer coefficient and ∆TLM is the log mean temperature difference. In the LMTD method, the overall heat transfer coefficient can be obtained from the convective heat transfer coefficients of hot and cold fluids and the thermal conductivity of solid body as follows: (4) where hh and hc are the averaged heat transfer coefficients of the cold and hot fluid sides, respectively, ks is the thermal conductivity of the solid heat exchanger body and δ is the plate thickness between the cold and hot channels. In the heat exchanger design process, the Nusselt number correlation is employed to determine the local heat transfer coefficient in Eqn. (5). The Nusselt number is defined as follows: (5) where kf is the thermal conductivity of fluid, Re is the Reynolds number, Pr is the Prandtl number and Ω represents the geometric parameters. In Eqns. (2)-(5), the mass flow rate and the inlet and outlet temperatures of the heat exchanger are input variables for the heat exchanger design. However, the convective heat transfer coefficients in Eqns. (4) and (5) are unknown variables which are dependent on heat exchanger design and thermal hydraulic conditions. To develop the Nusselt number correlation, the heat transfer coefficient in Eqn. (5) needs to be determined from the CFD analysis result. In the CFD analysis, the heat transfer coefficient can be determined by Eqn. (6) as follows:
(6) where is the averaged heat transfer coefficient, A are the heat transfer surface area, Tm is the fluid bulk temperature and Tw is the wall temperature. The subscripts h and c mean the hot fluid side and cold fluid side, respectively. In Eqn. (6), the heat transfer surface area of each fluid is determined based on the design of heat exchanger. The fluid bulk temperature and wall temperature can be determined as follows: (7) (8) where u is the fluid velocity, ρ is the fluid density and cp is the specific heat of fluid. The heat transfer rate in Eqns. (2) and (6), the fluid bulk in Eqn. (7) and wall temperature in Eqn. (8) of the given heat exchanger geometry are obtained from the CFD analysis result and the heat transfer coefficient in Eqn. (6) can be determined. Consequently, the Nusselt number can be obtained by Eqn. (5) from the CFD analysis results. The relative deviation of one variable with respect to another one (reference) is determined as follows formula. (9)
4. Sensitivity Tests for CFD Models In order to assess the reliability and accuracy of the CFD models, sensitivity tests for mesh dependency, geometry scale-up and flow models have been performed. In following sections, details of sensitivity tests were described. 4.1 Mesh dependency test A mesh sensitivity analysis was conducted by the Grid Convergence Index (GCI) method [22]. This test was carried out using the single-channel model with a zigzag angle of 45° and a relative length ratio of 8.18. A mass flow rate corresponding to a Reynolds number of 1000 was specified for the inlet boundary. Three polyhedral mesh structures were constructed to evaluate the GCI of the fine mesh. The values of fapp·Re obtained by the coarse, medium and fine mesh structures were denoted as ϕ3, ϕ2 and ϕ1, respectively. The apparent order p is determined as follows: (10) (11)
(12) where ε21=ϕ2 − ϕ1, ε32=ϕ3 − ϕ2, r21=h2/h1 and r32=h3/h2. h denotes the mesh size. The extrapolated solution is determined by Eqn. (13). (13) The approximate relative error is determined by Eqn. (14). (14) The grid convergence index is determined by Eqn. (15). (15) Figure 4 shows the results of the mesh dependency test. The number of computational cells for the coarse, medium and fine mesh for the single-channel model were 125,870, 596,511 and 3,365,149, respectively. The values of GCI for medium and fine meshes were 3.33% and 0.49%, respectively. The fine mesh structure resulted in very similar apparent friction factors to the extrapolated solution. The relative error of the fine mesh to the extrapolated solution was 0.39%. It is concluded that the fine mesh used in this study is reliable to analyze the pressure drop through the zigzag channel. To minimize the numerical error due to the mesh, the parameters of the fine mesh were applied to both the single-channel and two-channel models. The mesh size of the two-channel model varies from approximately 20 million cells to 53 million cells according to the relative length of the channel.
4.2 Geometry scale-up tests The fully-developed periodic boundary condition was employed in the single-channel model to reduce the size of the computational domain. A geometry scale-up study was performed to evaluate the numerical uncertainty that might be caused by this modeling with a periodic boundary condition. To evaluate the effect of the number of bends in the computational domain, the number of bends was varied from 2 to 8 while the fully-developed periodic boundary condition was implemented to inlet and outlet boundaries. In this test, the relative length ratio (lR/Dh) and the zigzag angle were 8.183 and 45°. Figure 5 shows the values of fapp·Re as a function of Reynolds number. The result did not vary with the number of bends in the computational domain. The maximum relative deviation of the fapp·Re value among three cases was less than 0.7%. Therefore, the required computational time and cost of the simulation could be reduced by the two-bend CFD model with the fully-developed periodic boundary condition. The geometry scale-up test for the two-channel CFD model was implemented by varying the number of bends in the computational domain. Since the inlet boundary condition in the two-channel CFD model was uniform distribution of mass flow rate, the velocity profile at the entrance region would be different from the downstream of the channel. To evaluate the effect of the flow developing at the entrance region, the apparent fanning friction factors of two-channel models with 10 bends, 20 bends and 40 bends were
investigated. The zigzag angle and relative length ratio for this test were 15° and 4.09, respectively. To determine the local pressure drop, the zigzag channel was divided into sub-channels along the flow direction. The local friction factor was evaluated from the local pressure drop of each sub-channel. Figure 6 shows the local friction factor of the hot fluid side as a function of location from the inlet. The local friction factors for increasing number of bends lie on the same line. The apparent friction factor as a function of number of bends is depicted in Fig. 7. Based on the result of a 40-bend model, the relative deviations of apparent friction factor of 10 bends and 20 bends cases were evaluated to be 11.2 and 3.6 %, respectively. The expected apparent friction factor for 100 bends was 0.0644. Therefore, the two-channel model with 20 bends was employed to obtain the results for thermal-hydraulic performance of zigzag channel.
4.3 Flow model sensitivity test The flow regime is considered to be laminar flow because the Reynolds number studied in this work was less than 2,000 for all cases. However, the velocity profile in the semi-circular zigzag channel would be far from the fully-developed laminar flow due to its wavy geometry. The momentum change of the fluid flow due to flow separation and re-attachment at the bends is the main reason for increased the pressure drop in the zigzag channel. Therefore, accurate prediction of the separated flow is of importance for the analysis in a zigzag channel. To evaluate the flow model sensitivity for the separated flow and its influence on the pressure loss, the simulations were conducted using both the laminar solver and a RANSbased turbulent solver. As for the turbulence model, the Abe-Kondho-Nagano (AKN) k-ε model, standard low Reynolds number k-ε model, realizable k-ε two-layer model, and Shear Stress Transport (SST) k-ω model were employed. To maximize the magnitude of the flow separation, the zigzag channel with an angle of 45° was selected. Figure 8 shows the fanning friction factor – Reynolds number product of the 45° channel as a function of Reynolds number. There is no significant difference in pressure loss for the different turbulence models, but the pressure drop predicted by the turbulence models slightly decreased, compared to the laminar solution, as the Reynolds number increased. Consequently, a turbulence model sensitivity analysis would be necessary to evaluate the model uncertainty for turbulent flow in zigzag channels where the Reynolds number is greater than 2,000.
5.
Development of Friction Factor Correlation
The pressure drop of the fluid flow in the zigzag channel is the sum of the frictional loss and minor losses, including form loss due to the changes in the flow direction and/or flow cross-sectional area, etc. In this study, the minor losses at the entrance or exit of the channel are neglected because the fully-developed periodic boundary condition was implemented at the interface between the inlet and outlet of the zigzag channel. Also, the minor losses due to flow acceleration or deceleration associated with heating or cooling are neglected because it is an isothermal flow. In Kim and NO’s study [13], a linear correlation of the apparent fanning friction factor of zigzag channel was analytically proposed as a function of Reynolds number as follows:
(16) where C1 corresponds to the frictional loss and C2 corresponds to the form loss. Although this correlation seems to have the correct format, the relationship between the model coefficients and geometric parameters were not examined in their work. The correlation of friction factor as a function of geometric parameters is essential to the design of new heat exchangers and evaluation of the heat exchanger pressure drop of. In this section, therefore, the relationship between the friction factor and geometric parameters were examined by the CFD analysis for both sharp-edged and round-edged zigzag channels.
5.1 Friction factor of sharp-edged channel As the Reynolds number approaches 0, the model coefficient C1 will approach 15.78 which is the theoretical fanning friction factor –Reynolds number product for fully-developed flow in a semi-circular straight channel. The coefficient C1 is assumed to be a constant value of 15.78. On the other hand, the model coefficient C2 should be a function of geometric parameters. To develop the correlation for constant C2, the effects of zigzag angle, relative length ratio and radius of curvature were examined. The apparent friction factors were obtained in the Reynolds number range from 50 to 2,000. Figure 9 shows the values of C2·Re as a function of Reynolds number over a range of zigzag angles and relative length ratios. The value of C2·Re generally showed a linear relation with the Reynolds number. The value of C2 could be obtained from of the slope of a linear fit for each case. The velocity profile for fluid flow downstream of the bend in a zigzag channel is asymmetric due to the centrifugal force caused by flow direction change at the bend. This asymmetric velocity profile reverts to a symmetric shape as the fluid flow passes through the straight channel. Figure 10 shows the value of C2 as a function of relative length ratio (lR/Dh). It is found that C2 decreased exponentially as the relative length ratio increased. A correlation for C2 is given by Eqn. (17) as a function of relative length ratio. (17) Since the model coefficient C2 strongly depends on the zigzag angle, the model coefficients C3, C4 and C5 should also be a function of zigzag angle. The values of model coefficients were determined through the least squares method with a non-linear Generalized Reduced Gradient (GRG) algorithm. Figure 11 show the values of C3, C4 and C5 as a function of zigzag angle, respectively. The correlations for the model coefficients obtained from the GRG algorithm as a function of zigzag angle are given by Eqns. (18-20). (18) (19) (20)
Substituting Eqns. (17-20) into Eqn. (16) yields the apparent fanning friction factor correlation for semicircular zigzag channels as follows: (21) where α is the zigzag angle (rad) and lR/Dh is the relative length ratio. The apparent friction factor correlation for sharp-edged zigzag channels is valid for 50 ≤ Re ≤ 2,000. Figure 12 shows a comparison of values of fapp·Re for sharp-edged zigzag channels between the CFD results and the correlation. In general, the correlation shows good agreement with the CFD results within a relative deviation of ± 15%. A few relative deviations over 15% were observed at a Reynolds number less than 50.
5.2 Friction factor of round-edged zigzag channel To investigate the effect of bend radius of curvature, the apparent friction factor of zigzag channels was analyzed for a radius of curvature ranging from 0.5 mm to 1.0 mm. For this series of simulations, the zigzag angle and relative length ratio were fixed at 25° and 8.183, respectively. Figure 13 shows fapp·Re for zigzag channels as a function of bend radius of curvature. In these plots, the radius of curvature of 0.0 mm represents a sharp-edged zigzag channel. The value of fapp·Re decreased as the radius of curvature increased. The effect of the roundness of the bend became more significant at higher Reynolds numbers. The value of fapp·Re with a radius of curvature of 1.0 mm (R/Dh=0.818) was reduced by 15.86% compared to the sharp-edged zigzag channel at a Reynolds number of 2,000. The influence of edge radius of curvature also became more significant as the zigzag angle increased and the relative length ratio (lR/Dh) decreased. The value of fapp·Re of round-edged zigzag channel with zigzag angle of 45° and relative length ratio of 4.09 was reduced by 54% compared to the value for the sharp-edged channel. Figure 14 shows the apparent friction factor of a round-edged zigzag channel with a radius of curvature of 1.0 mm as a function of Reynolds number. Since the difference of fapp·Re between round-edged and sharpedged for zigzag angle of 5° was negligible, the result of zigzag angle of 5° was used to develop the correlation for round-edged zigzag channel. The apparent friction factor for a round-edged zigzag channel was developed as a function of Reynolds number, zigzag angle and relative length ratio in the same manner which was used to develop the correlation for a sharp-edged zigzag channel. The apparent friction factor for a round-edged zigzag channel with a radius of curvature less than 1.0 mm could be obtained by interpolation of two correlations of sharp and round-edged zigzag channels. The apparent friction factor of the round-edged zigzag channel is given by Eqn. (22). (22) The correlation for apparent friction factor for round-edged zigzag channels is valid for 50 ≤ Re ≤ 2,000. Figure 15 shows a comparison of values of fapp·Re for round-edged zigzag channels between the CFD results and the developed correlation. Since the value of fapp·Re of round-edged zigzag channels was not
quite linear compared to that of sharp-edged zigzag channel, the maximum deviation between the correlation and the CFD results was slightly increased up to ± 20%.
5.3 Effect of temperature-dependent fluid properties on friction factor The friction factor for zigzag channels was developed from isothermal simulations in which the fluid properties were constant. For determining heat transfer rate, the fluid properties vary because of the temperature change in each fluid side. The friction factors obtained from the two-channel CFD model with heat transfer were compared to the friction factor correlation. When the fluid density is changed by heating or cooling, the fluid flow is accelerated or decelerated. The minor losses due to this flow acceleration or deceleration were considered for the comparison of friction factor. The minor loss due to flow acceleration/deceleration can be evaluated as given by Eqn. (23). (23) The apparent friction factors accounting for the flow acceleration/deceleration effect were compared to those calculated by the correlation developed in Eqn. (21), as shown in Fig. 16. The apparent friction factors from the two-channel model simulations agreed with the friction factor correlation within the relative deviation of ± 15%. The ratio of minor loss to total pressure loss ranged from 0.8% up to 13.5% with decreasing the zigzag angle and increasing Reynolds number. The evaluated minor loss given by Eqn. (23) is an approximate value because the inlet and outlet densities were used for calculation, but not local densities throughout the channel. Hence, the predictive capability of the correlation developed for the apparent friction factor of the zigzag channel could be improved by developing a methodology to evaluate minor losses due to flow acceleration/deceleration more accurately.
6.
Development of Heat Transfer Correlation
The Nusselt number for fully-develop laminar flow in a straight pipe with a uniform boundary condition is a constant, independent of Reynolds number. However, fluid flow in the zigzag channel does not achieve a fully-developed condition. It is reported in several instances in the literature that the Nusselt number for zigzag channels does not achieve a constant value, even for laminar periodic fully-developed flow conditions. Nusselt numbers for zigzag channel geometries increase with increasing Reynolds number, even in the laminar regime. To develop an explicit correlation for Nusselt number, the basic power-law form was used as follows: (24) where C6, C7 and C8 are the model coefficients for the Nusselt number correlation.
6.1 Determination of the model constant for Prandtl number The Prandtl number is defined as the ratio of momentum diffusivity to thermal diffusivity. It is a function of specific heat, dynamic viscosity and thermal conductivity of the fluid. It is assumed that the value for the exponent on Prandtl number in Eqn. (24) is independent of the geometric parameters. To determine the model coefficient C8 for the Nusselt number correlation, a sensitivity test was conducted by varying the Prandtl number from 0.1 to 50 with three zigzag angles of 15°, 30° and 45°. The mass flow rates of hot and cold fluid sides were specified at 5×10-5 kg/s. Figure 17 shows the resulting Nusselt numbers as a function of Prandtl number. The exponent on Prandtl number, C8, for low Prandtl number was higher than that for high Prandtl number. The Prandtl number exponent was independent of zigzag angle, but varied with the value of the Prandtl number and the heat transfer mode. The exponent for Prandtl number for Pr ≤ 1.0 was higher than that for Pr ≥ 5.0. For the hot fluid side where the fluid is being cooled, the Prandtl number exponent was slightly lower than that for the cold fluid side. For low Prandtl number, less than 1.0, the exponents of Prandtl number, C8, for hot and cold fluid sides were 0.56 and 0.58, respectively. For higher Prandtl numbers greater than 5.0, the values of C8 for hot and cold fluid sides were 0.31 and 0.35, respectively. In this study, the exponents of Prandtl number for low Prandtl number were used to develop the Nusselt number correlation since the Prandtl number of helium, 0.66, corresponds to the low Prandtl number region.
6.2 Effect of radius of curvature on Nusselt number The effect of radius of curvature on Nusselt number was minor compared to the effect on the friction factor as shown in Fig. 18. The variation in Nusselt number was less than approximately 4.8 % while it was as high as 41% for friction factor. The change in heat transfer surface area associated with the radius of curvature was very small so that its impact on the overall heat transfer was negligible whereas the radius of curvature did have an effect on the local velocity field in zigzag channels. The major advantage of round-edged zigzag channel is that the pressure loss through the heat exchanger could be significantly reduced with a much lower reduction in Nusselt number. Based on this analysis, the radius of curvature was excluded from the geometric parameter for Nusselt number correlation.
6.3 Development of Nusselt number correlation for zigzag channel To develop a Nusselt number correlation for zigzag channels, the characteristic behavior of the flow needs to be examined first. As shown in Fig. 19, a periodic flow pattern transitions to a chaotic pattern at higher Reynolds numbers. The flow pattern at Reynolds numbers less than 470 was stable and periodic. Irregular and rotational secondary vector fields on the cross-section of the channel increased in strength with increasing Reynolds number. A region of reversed flow downstream of the bends was observed for Reynolds numbers greater than 470, as shown in Fig. 20. The occurrence of the secondary flows downstream of the bends affected the heat transfer characteristics of the zigzag channel. Nusselt number was linearly proportional to the Reynolds number where the zigzag angle and Reynolds number were low so that there is no reversed flow downstream of the bend. Nusselt numbers for higher Reynolds numbers and zigzag angles were proportional to Ren. Based on these observations, separate Nusselt number
correlations were developed for two regions as depicted in Fig. 21. In this map, the critical value, which results in re-circulating secondary flows, of zigzag angle (αc) was 15° and the critical Reynolds number (Rec) was 550, regardless of the relative length ratio. Nusselt number was depicted as a function of Reynolds number for respective relative length ratios and zigzag angles in Fig. 22 and Fig. 23. As expected, the Nusselt number increased with increasing Reynolds number. It also increased with zigzag angle, but decreased with increasing relative length ratio. The dependency of Nusselt number on the relative length ratio was deemed to be diminished as the zigzag angle decreased. To determine the model coefficients C6 and C7 in Eqn. (24), a non-linear least squares regression analysis was performed. The procedure to determine the correlation coefficients is depicted in Fig. 24. First, the matrices of Nusselt number obtained from the CFD simulations and Eqn. (24) were constructed and then the correlation coefficients which minimize the sum of squared residuals were determined using a nonlinear GRG algorithm. As a result, the matrices of coefficients as a function of relative length ratio were constructed for each zigzag angle. The functional form of the coefficient models could be determined from the matrices of coefficients re-constructed as a function of relative length ratio and zigzag angle. To optimize the coefficient models, the squared residuals of Nusselt number between the CFD analysis result and correlations were evaluated. Finally, the constants for the coefficient models which minimized the sum of squared residuals were determined. In Region-A, where the reversed secondary flow did not occur, the Nusselt numbers were linearly proportional to the Reynolds number. Therefore, a linear equation of Nusselt number as a function of Reynolds number was employed for region A instead of Eqn. (24). The Nusselt number correlation for Region-A was developed and is given by Eqn. (25). (25) Equation (25) can be used to evaluate the Nusselt number for either the hot fluid side or the cold fluid side in Region-A. In Region-B, Nusselt number correlations were developed based on Eqn. (24) as follows: (26) (27) where the unit of zigzag angle (α) in the correlations is in radians. The Nusselt number correlations are valid for 100 ≤ Re ≤ 2000 and Pr ≤ 1.0. Figures 25 shows the Nusselt number comparison between CFD results and correlation. For both the hot and cold fluid sides, the correlation agreed with the CFD simulation results within a relative deviation of ±10%.
6.4 Validation of friction factor and Nusselt number correlations In order to gain confidence in the correlations obtained both for Nusselt number and fanning friction factor for the zig-zag channel geometries, comparisons were made with experimental data obtained at The Ohio State University (OSU) [24] and at the Korean Advanced Institute of Science and Technology
(KAIST) [12, 13], qualified with uncertainty analysis. The root mean square method was used to analyze and quantify the contribution made by the uncertainty in one variable to the overall uncertainty in the result. The root mean square method is illustrated by showing the relative uncertainty evaluation for both Reynolds number and Nusselt number, respectively. (28)
(29) Further details on uncertainty analysis for each measured variable can be found in the literature [23, 24]. Figure 25 compares the experimentally determined fanning friction factor values with the correlation obtained from CFD analysis. Note that the OSU PCHE is a round-edged zigzag channel PCHE whereas the KAIST PCHE has a sharp-edged zigzag channel. The CFD-based friction factor correlation for the round-edge case shows good agreement with the OSU PCHE experimental data. The average relative deviation was 9.95% and the minimum and maximum relative deviation was 0.7% and 24%, respectively. The CFD-based sharp-edged friction factor correlation agreed very well with Kim’s correlation [12] with a maximum deviation less than 1.0%. The CFD-based friction factor correlation is valid for Reynolds numbers up to 2000 and Kim’s empirical correlation is valid for Reynolds numbers up to 1200. Thus, the CFD-based correlations and Kim’s correlation are extrapolated for this comparison. The comparison results showed that the extrapolated correlations can accurately predict the friction factor of zigzag channel for both sharp-edged and round-edged zigzag channel PCHEs. Figure 26 shows the comparison of experimental and CFD-based Nusselt numbers for both the hot side and cold side, showing a relatively large deviation ranging between 28% to 49% for the hot side and 23% to 43% for the cold side, respectively. The CFD-based Nusselt number correlations overestimated the experimental data. Differences between the experimental configuration and the CFD model are responsible for this deviation. In the CFD simulation, a uniform flow distribution is assumed among all the channels while in the experiment, some flow misdistribution is probable due to manifold effects mainly in the inlet plenums, leading to different mass flow rates in some channels. The CFD model only simulated the core region while the experimental data included the plenum effects. In the experiment, heat loss was not accounted for which could provide incorrect values for heat load and heat transfer effectiveness, respectively. In addition, the temperature measurement carried out in the experiment was upstream and downstream of the inlet and outlet flanges, adding more uncertainty to the exact temperature difference across the heat exchanger. The friction factor was mainly influenced by the zigzag channel geometry while the Nusselt number was influenced by overall heat exchanger design including the inlet plenum part. CFD analysis with twochannel model or single channel model would be sufficient to investigate the friction factor of zigzag channel. However, since there are many variations of the inlet plenum design, it is not possible to obtain the correlation which can predict the Nusselt number precisely independent of inlet plenum designs. CFD analysis with the two-channel model is good for predicting the core heat transfer while it could result in incorrect overall heat transfer of PCHE. Thus, for better determination of heat transfer characteristic modeling of the plenum section is very essential, but it would be computationally intensive.
7.
Conclusion
Friction factor and Nusselt number correlations developed as a function of geometric parameters are required to design compact heat exchangers, such as PCHEs and to evaluate their thermal-hydraulic performance. Furthermore, these correlations are necessary for cost analysis and the overall effectiveness of the system. In this study, an extensive CFD analysis was conducted to develop friction factor and Nusselt number correlations for laminar flow in a semi-circular zigzag channel. Hydraulic diameter, relative length ratio, zigzag angle and bend radius of curvature were selected as geometric parameters. A least squares method using a non-linear Generalized Reduced Gradient algorithm was used to determine the model coefficients of the correlations. The following points summarize the results:
Friction factor correlations for sharp-edged and round-edged zigzag channels for PCHE have been developed. The friction factor correlations were valid for 50 ≤ Re ≤ 2000, 5° ≤ α ≤ 45° and 4.09 ≤ lR/Dh ≤ 32.73. The correlation exponent on the Prandtl number for low-Prandtl numbers (< 1.0) is 0.56 for the hot fluid side and 0.58 for the cold fluid side, respectively. As for high-Prandtl numbers (>5.0), the exponent is 0.31 for hot fluid side and 0.35 for cold fluid side. CFD simulations with a two-channel model have been conducted to develop Nusselt number correlations for zigzag channels which are valid for 200 ≤ Re ≤ 2000, Pr ≤ 1.0, 5° ≤ α ≤ 45° and 4.09 ≤ lR/Dh ≤ 12.27. Note that the friction factor of the zigzag channel PCHE is mainly influenced by the zigzag channel geometry while the Nusselt number is influenced by overall heat exchanger design including the plenum sections. CFD analysis with a two-channel model is good for predicting friction factor, but could overestimate or underestimate the thermal performance of zigzag channel PCHEs according to the inlet and outlet plenum designs.
Acknowledgment This work was supported by the United States Department of Energy Office of Nuclear Energy, Very High Temperature Reactor Technology Development Office. Idaho National Laboratory (INL) is operated for the United States Department of Energy Office of Nuclear Energy by Battelle Energy Alliance, LLC, under contract No. DE-AC07-05ID14517. The authors also gratefully acknowledge helpful discussion with Junsoo Yoo at INL.
NOMENCLATURE A C
Area or Overall heat transfer surface area, m2 Model coefficient
cp D Dh f GCI H h k L lR Nu P p Pr R Re T U u W α Δ δ μ ρ
Specific heat, J/(kg·K) Diameter, m Hydraulic diameter, m Friction factor Grid convergence index Height, m Heat transfer coefficient, W/(m2·K) Thermal conductivity, W/(m·K) Length, m Relative length, m Mass flow rate, kg/s Nusselt number Pressure, Pa Order of scheme Prandtl number Heat transfer rate, W Radius of curvature, m Reynolds number Temperature, K Overall heat transfer coefficient, W/(m2·K) Fluid velocity, m/s Width, m Zigzag angle, rad Differential Plate thickness, m Fluid dynamic viscosity, Pa·sec Fluid density, kg/m3
Subscripts accel Acceleration app Apparent c Cold fluid side f Fluid h Hot fluid side HX Heat exchanger i Inlet LM Log mean temperature difference m Fluid bulk o Outlet ref Reference s Solid w Wall
References 1. D. Southall, R.L. Pierres and S.J. Dewson, Design Considerations for Compact Heat Exchangers, Proceedings of ICAPP’08, June 8-12, Anaheim, CA, USA, , 2008. 2. D. Southall and S.J. Dewson, Innovative Compact Heat Exchangers, Proceedings of ICAPP’10, June 13-17, San Diego, CA, USA, 2010. 3. Sang-Moon Lee and Kwang-Yong Kim, A Parametric Study of the Thermal-Hydraulic Performance of a Zigzag Printed Circuit Heat Exchanger, Proceedings of the ASME/JSME 8 th Thermal Engineering Joint Conference, March 13-17, Honolulu, Hawaii, USA, 2011. 4. Sang-Moon Lee and Kwang-Yong Kim, Optimization of Zigzag Flow Channels of a Printed Circuit Heat Exchanger for Nuclear Power Plant Application, Journal of Nuclear Science and Technology, Vol. 49 (3), pp.343-351, 2012. 5. Sang-Moon Lee and Kwang-Yong Kim, Comparative Study on Performance of a Zigzag Printed Circuit Heat Exchanger with Various Channel Shapes and Configurations, Heat and Mass Transfer, Vol. 49 (7), pp.1021-1028, 2013. 6. T.L. Ngo, Y. Kato, K. Nikitin and T. Ishizuka, Heat Transfer and Pressure Drop Correlations of Microchannel Heat Exchangers with S-Shaped and Zigzag Fins for Carbon Dioxide Cycles, Experimental Thermal and Fluid Science, Vol. 32, pp.560-570, 2007. 7. Roma Fatima, Alan Kruizenga, Mark Anderson and Devesh Ranjan, Numerical Investigation of Thermal Hydraulic Behavior of Supercritical Carbon Dioxide in Compact Heat Exchangers, 2014 Supercritical CO2 Power Cycle Symposium, Sept. 9-10, Pittsburgh, PA, USA, 2014. 8. A. Kurizenga, Mark Anderson, Roma Fatima, Machael Corradini, Aaron Towne and Devesh Ranjan, Heat Transfer of Supercritical Carbon Dioxide in Printed Circuit Heat Exchanger Geometries, Journal of Thermal Science and Engineering Applications, Vol. 3(3), p. 031002, 2011. 9. H. Song, J. Van Meter, S. Lomperski, D. Cho, H.Y. Kim and A.Tokuhiro, Experimental Investigations of a printed circuit heat exchanger for supercritical CO2 and water heat exchange, NTHAS5-N002, Nov. 26-29, Jeju, South Korea, 2006. 10. N.S. Berbish, M. Moawed, M. Ammar and R.I. Afifi, Heat Transfer and Friction Factor of Turbulent Flow through a Horizontal Semi-circular duct, Heat and Mass Transfer, Vol. 47, pp.377-384, 2011. 11. S.P. Kar, CFD Analysis of Printed Circuit Heat Exchanger, M.S Thesis, National Institute of Technology Rourkela, 2007. 12. I.H. Kim, Experimental and Numerical Investigations of Thermal-Hydraulic Characteristics for the Design of a Printed Circuit Heat Exchanger (PCHE) in HTGRs, Ph.D. Dissertation, Korea Advanced Institute of Science Technology (KAIST), 2012. 13. I.H. Kim and H.C. No, Thermal Hydraulic Performance Analysis of a Printed Circuit Heat Exchanger using a Helium-Water Test Loop and Numerical Simulations, Applied Thermal Engineering, Vol. 31, pp.4064-4073, 2011. 14. O.K. Kwon, M.J. Choi and Y.J. Choi, Heat Transfer and Pressure Drop Characteristics in Zigzag Channel Angles of Printed Circuit Heat Exchangers, Korean Journal of Air-Conditioning and Refrigeration Engineering, Vol. 21 (9), pp.475-482, 2009. 15. J.W. Seo, Y.H. Kim, D.S. Kim, Y.D. Choi and K.J. Lee, Heat Transfer and Pressure Drop Characteristics in Straight Microchannel of Printed Circuit Heat Exchangers, Entropy, Vol. 17, pp.3438-3457, 2015.
16. T. Ma, L. Li, X.Y. Xu, Y.T. Chen and Q.W. Wang, Study on Local Thermal-Hydraulic Performance and Optimization of Zigzag-type Printed Circuit Heat Exchanger at High Temperature, Energy Conversion and Management, Vol. 104, pp. 55-66, 2015. 17. A. Meshram, A.K. Jaiswal, S.D. Khivsara, J.D. Ortega, C. Ho, R. Bapat and P. Dutta, Modeling and Analysis of a Printed Circuit Heat Exchanger for Supercritical CO2 Power Cycle Applications, Applied Thermal Engineering, In Press, Corrected Proof, 2016. 18. E.N. Van Abel, M.H. Anderson and M.L. Corradini, Numerical Investigation of Pressure Drop and Local Heat Transfer of Supercritical CO2 in Printed Circuit Heat Exchanger, Supercritical CO2 Power Cycle Symposium, May 24-25, Boulder, CO, 2011. 19. I.E. Idelchik, Handbook of Hydraulic Resistance 3rd Ed., Begell House Inc., New York, 1996. 20. National Institute of Standards and Technology, NIST Chemistry WebBook: NIST Standard Reference Database Number 69, http://webbook.nist.gov/chemistry/, accessed on July 20, 2016. 21. INCONEL 617 Technical Bulletin, Publication Number SMC-029, Special Metals Corporation, Huntington, WV, 2005. 22. P.J. Roache, Quantification of uncertainty in computational fluid dynamics, Annual Review of Fluid Mechanics, Vol. 29, pp. 123–160, 1997. 23. S.K. Mylavarapu, Design, Fabrication, Performance Testing, and Modeling of Diffusion Bonded Compact Heat Exchangers in a High-Temperature Helium Test Facility, Doctoral Dissertation, The Ohio State University, 2011. 24. M. Chen, X. Sun, R.N. Christensen, I. Skavdahl, V. Utgikar and P. Sabharwall, Pressure Drop and Heat Transfer Characteristics of a High-Temperature Printed Circuit Heat Exchanger, Applied Thermal Engineering, 108, 2016.
(a) PCHE section (crossflow configuration) (b) Etched plates Fig. 1 Printed circuit heat exchanger [1]
Fig. 2 Temperature and pressure capabilities of heat exchangers [2]
Hot fluid side
D
W D
R
H
δ
Cold fluid side
Solid body
α lR/2
(a) Single-channel CFD model geometry.
(b) Two-channel CFD model geometry.
(c) Mesh structure on frontal surface of the two-channel model.
(d) Eagle-eye view of mesh structure of the two-channel model.
Fig. 3 Schematic diagram of geometry and mesh structure of CFD models.
Fig. 4 Result of mesh dependency test.
Fig. 5 The effect of number of bends in the computational domain with the fully-developed periodic boundary condition
Fig. 6 Local friction factor of hot channel as a function of distance from the inlet according to the number of bends (α=15°, lR/Dh=4.09)
Fig. 7. Apparent friction factor of hot channel as a function of number of bends (α=15°, lR/Dh=4.09)
Fig. 8 Result of flow model dependency test
(a) α = 45°
(b) α = 40°
(c) α = 35°
(d) α = 30°
(e) α = 25°
(f) α = 20°
(g) α = 15°
(h) α = 10°
(i) α = 5° Fig. 9 The plots of fapp·Re for various relative lengths and zigzag angles of sharp-edged zigzag channel
Fig. 10 Model coefficient (C2) as a function of relative length ratio (lR/Dh)
Fig. 11 Comparison of fapp·Re of sharp-edged zigzag channel (CFD vs. Correlation)
Fig. 12 fapp·Re as a function of the relative radius of curvature (R/Dh) (α = 25°, lR/Dh = 8.183)
(a) α = 45°
(b) α = 40°
(c) α = 35°
(d) α = 30°
(e) α = 25°
(f) α = 20°
(g) α = 15° (h) α = 10° Fig. 13 The plots of fapp·Re for various relative lengths and zigzag angles of round-edged zigzag channel
Fig. 14 Comparison of fapp·Re of round-edged zigzag channel (CFD vs. Correlation)
Fig. 15 Comparison of fapp·Re (Isothermal correlation vs. Two-channel model)
(a) Hot fluid side (b) Cold fluid side Fig. 16 Plots of Nusselt number as a function of Prandtl number
(a) Hot fluid side (b) Cold fluid side Fig. 17 Effect of radius of curvature on Nusselt number and apparent friction factor (α=45°, lR/Dh =8.183)
(a) Reynolds number = 285
(b) Reynolds number = 470
(c) Reynolds number = 565
(d) Reynolds number = 657
(e) Reynolds number = 842 (f) Reynolds number = 1845 Fig. 18 Streamwise velocity (x-velocity) magnitudes and tangential velocity vector fields on the crosssection of 15th bend from the inlet of hot fluid side (α=15°, lR/Dh=8.183)
Fig. 19 Normalized flow x-velocity profiles along to line A-A’ (α=15°, lR/Dh=8.183)
Fig. 20 Region map for Nusselt number correlation (αc=15°, Rec=550)
(a) α = 45°
(b) α = 40°
(c) α = 35°
(d) α = 30°
(e) α = 25°
(f) α = 20°
(g) α = 15°
(h) α = 10°
(i) α = 5° Fig. 21 Nusselt number as a function of Reynolds number (Hot fluid side)
(a) α = 45°
(b) α = 40°
(c) α = 35°
(d) α = 30°
(e) α = 25°
(f) α = 20°
(g) α = 15°
(h) α = 10°
(i) α = 5° Fig. 22 Nusselt number as a function of Reynolds number (Cold fluid side)
Fig. 23 Schematic diagram of procedure to determine the functional form of coefficients for Nusselt number correlation
(a) Hot fluid side (b) Cold fluid side Fig. 24 Comparison of Nusselt number (Correlation vs. CFD result)
Fig. 25 Comparisons of friction factor (Experimental data vs. correlations)
(a) Hot fluid side (b) Cold fluid side Fig. 26 Comparisons of Nusselt Number (Experimental data vs. correlations)
TABLE I. Geometric Parameters Variable Channel diameter, D, (mm) Hydraulic diameter, Dh, (mm) Height of solid body, H, (mm) Width of solid body, W, (mm) Thickness between two channels, δ, (mm) Zigzag angle, α, (°) Sub-channel length, lR, (mm) Radius of curvature, R, (mm)
Values 2.0 1.222 3.2 2.5 0.6 5, 10, 15, 20, 25, 30, 35, 40, 45 5, 6, 8, 10, 15, 20, 40 0.5, 0.6, 0.75, 0.9, 1.0
TABLE II. Thermo-physical Properties of Alloy 617 Temperature, K 293.15 373.15 473.15 573.15 673.15 773.15 873.15 973.15 1073.15 1173.15 1273.15
Thermal conductivity, W/(m·K) 13.4 14.7 16.3 17.7 19.3 20.9 22.5 23.9 25.5 27.1 28.7
Specific Heat, J/(kg·K) 419 440 465 490 515 536 561 586 611 636 662