Parallel Heat Exchanger Research Articles

Computational fluid dynamic simulation of earth air heat exchanger: A thermal performance comparison between series and parallel arrangements

This study uses Ansys Fluent to compare the thermal performance of series and parallel earth air heat exchanger (EAHE) systems for cooling. The model was validated using experimental data from published literature and matched simulation results. The sensitivity study examined how length, diameter, and ground surface coverings affected EAHE performance. The relationship between effectiveness and the Number Transfer Unit (NTU) of EAHE was explored along with the soil thermal regime. The simulation results show that the series EAHE can achieve a lower temperature drop than parallel. With an input air temperature of 32 °C and EAHE lengths varying from 10 m to 50 m (in 10 m increments), the 50 m EAHE produced the lowest outlet air temperatures: 27.2 °C for the series configuration and 28.8 °C for the parallel arrangement. Changing the EAHE diameter (4–6 in) results in a ±0.2 °C outlet air temperature drop for both setups. EAHE performed best under short grass soil cover, yielding 1.4 °C and 0.5 °C lower outlet air temperatures, for series and parallel arrangements than the asphalt cover. The pressure drop increased proportionally with EAHE's length. Simulation results indicate a ±6 times larger pressure loss in the series design compared to the parallel setup. The effectiveness-NTU relationship shows that parallel EAHEs are 15 % more effective than series ones.

Experimental and numerical assessment of thermal characteristics of PCM in a U-shaped heat exchanger using porous metal foam and NanoPowder

The utilization of Latent Heat Thermal Energy Storage (LHTES) has gained significant attention to address the disparity between energy supply and demand. One of the key advantages lies in the use of phase change materials (PCM). The purpose of this research is to overcome this obstacle by focusing on enhancing the thermal efficiency of an advanced thermal energy storage system specifically designed for solar domestic and industrial application. to overcome this obstacle by focusing on enhancing the thermal efficiency of an advanced thermal energy storage system specifically designed for solar domestic and industrial application. Through computational and experimental studies, a novel and small LHTES system with parallel U-shaped heat exchanger (USHX) has been created and investigated. To improve performance, two approaches are employed: optimizing thermal efficiency by dispersing nano-sized graphite powders into the paraffin material, and/or incorporating metal foams. The PCM is RT35HC, and the hot/cold heat transfer fluid is H2O, which travels via the U-shaped tube. The model incorporates the enthalpy-porosity technique to account for phase change phenomena. After comparing the numerical outcomes with the experiments herein run, data are shown in terms of liquid fraction, temperature evolution, stored energy, and a dimensionless parameter that characterizes the phase change process. The findings suggest that the proposed methods for enhancing heat transfer can enhance the thermal efficiency of systems. The outcomes illustrate that by addition of all methods, reduces the melting time by 13.39 %, 60.77 %, and 71.93 %, when compared to system with pure PCM.

Experimental validation of a design model for latent thermal energy storage extended with sensible heat

Latent thermal energy storage can be a key technology for a green energy transition by matching fluctuating heat demand and supply. In order to implement a storage system, it needs to be designed which requires estimating the outlet temperature of a system for a given geometry and time history of the heat transfer fluid’s mass flow rate and inlet temperature. Currently, design methods are either overly simplistic, focusing solely on e.g. the phase change time or requiring the solution of partial differential equations which can be computationally expensive. The present paper proposes a novel approach where a latent thermal energy storage system is decomposed into a heat transfer fluid vessel, a sensible storage system and a storage system with only latent heat. Computationally inexpensive models are available for all three of these sub heat exchangers. A heat exchanger model is obtained by connecting the sub heat exchangers in parallel. This novel approach is used to model an industrial scale shell and tube latent thermal energy storage heat exchanger. The predicted outlet temperature is compared to the measured outlet temperature and the design model obtains good agreement.

Heating and defrosting performance assessment of dual-source heat pump operational modes under various ambient conditions

Multifunctional dual-source heat pump systems, featuring two sets of heat exchangers (air-to-refrigerant and water-to-refrigerant) in series or parallel, aim to boost efficiency. However, these systems often require extensive piping, leading to power losses, higher energy consumption, and increased installation space requirements. Furthermore, few comparative studies have assessed the effectiveness of dual-source versus single-source heat pump systems under various ambient conditions. This study introduces a novel air-water dual-source evaporator design that enables efficient heat exchange using air, water, or both. With this design, the heat pump system can be tuned according to three distinct modes: pure water-source mode (WSHP), pure air-source mode (ASHP), and dual air-water-source mode (AWSHP). The heat pump's efficiency across these modes was evaluated under various weather conditions, adhering to the CNS15466 and GB/T21362 standards. The findings reveal a strong correlation between the operational modes and ambient temperatures, indicating that maximum efficiency is achieved with the WSHP mode below 7 °C, AWSHP mode between 10 °C and 35 °C, and ASHP mode above 35 °C. Moreover, three defrosting methods were evaluated: hot gas bypass, hot water bypass, and a combination of both. The hot water bypass, in conjunction with the proposed dual-source evaporator design, efficiently performed full defrosting, boosting system efficiency. The ability of the heat pump to switch between operating modes enhances the overall energy efficiency during space heating and defrosting, reduces installation space requirements and compressor oil loss, and facilitates hot water production. The proposed system is ideal for various industrial applications requiring efficient operation under diverse ambient conditions.

A study on how efficient measures for secondary district heating system performance can be encouraged by motivational tariffs

BackgroundDistrict Heating (DH) is a technology that provides heating and domestic hot water to buildings and is an important technology for supporting the European energy transition. As such the heating systems increasingly resort to renewable heat sources and waste heat, it is even more important that they operate in the most efficient way possible. DH companies have access to the primary network of which they can impact system performance. To maximize the efficiency of the system, however, it is important that the system at the building level, known as the secondary system, is also efficient; otherwise, overall system efficiency is reduced. To increase system efficiency, return temperatures from the secondary system into the primary system have been targeted through motivational tariffs. There is limited information on how to establish a motivational tariff that motivates the customer to improve both the primary and secondary systems, which is a gap that this paper aims to fill.ResultsIn this paper, the impacts of retrofit actions in secondary systems are assessed through simulations. The identified relevant refurbishment measures to lower the return temperature to the primary system are variable flow pumps, low-temperature radiators, parallel heat exchangers, and a pass-through DHW system. Apart from simulated refurbishments, we also identify that the secondary system sometimes generates excess heat, which is valuable to recover, especially during peak load periods for the primary system. Hence, motivational tariffs targeting secondary system efficiency should also encompass an incentive for the customer to make use of waste heat in the secondary circuit to lower peak demand for the DH system.ConclusionsTo date, the most commonly used parameters introduced to customers are linked to the flow of water through the customer’s asset and the bonus malus principle. The results from simulations show that DH companies can introduce additional parameters to support customers in guiding their secondary system to perform more efficiently. Increased overall system efficiency has a positive impact on both costs and emissions.

Coupld thermo-hydraulic modelling of heat storage in an embankment

Coupld thermo-hydraulic modelling of heat storage in an embankment

CFD-Based Approach to Propose a Zigzag-Shaped Tube Heat Exchanger without Fins

This study explores the application of zigzag-shaped, finless tubes in enhancing heat transfer performance within heat exchangers. Using three-dimensional numerical simulations, we examined the heat transfer per unit area and the volume of the pressure drop, comparing these findings with a traditional parallel tube heat exchanger. This innovative design strategy involved arranging zigzag-shaped tubes at varying distances, and the thermal transfer and frictional characteristics were tested at different air speeds. This research suggests that the introduction of zigzag heat exchangers, as opposed to traditional fin-and-tube designs, led to a significant improvement in heat transfer. This enhancement is attributed to the swirling flow created around the zigzag tubes, which increased the total heat transfer area. Furthermore, we found that the heat transfer area increased by 14.2%, 32.1%, and 63.9% for tube zigzag angles of 30°, 45°, and 60°, respectively, when compared to a parallel tube heat exchanger. Consequently, the zigzag-shaped tube heat exchanger demonstrated not only superior heat transfer, but also a reduction in frictional pressure loss.

Internal Flow Heat Transfer Rate for Parallel Heat Exchanger Pipes One Pass of Rice Husk to Air

The research was conducted to analyze the heat transfer rate of heat exchangers from black steel pipes with rice husk as an energy source. This can be used as a recommendation to replace solar energy in the post-harvest drying process, especially for small farmers in rural areas. Research by converting rice husk biomass into heat energy using a heat exchanger of black steel pipes arranged parallel to one air fluid flow. Heat exchanger pipes are placed inside the furnace with dimensions of 50 cm x 50 cm x 80 cm. Forced convection is applied through a fan placed in the drying chamber with a constant speed of 2 m/s. It was found that the ambient temperature increased by an average of 11.70% without adding rice husk mass during the test. The average ambient temperature (Ti) is 305.10 K and ranges from 300.60 to 307.40 K which can be increased to (To) 340.80 K with a range of 317.25 to 366.30 K. This temperature is the out the heat exchanger pipes that can be used for the drying process. The value of the heat transfer rate reached an average of 39.19 W with a range of 18.62-63.19 W. The pattern of distribution of heat transfer rates follows the trend of temperature distribution. It was found that when rice husk turns into charcoal it causes a high heat transfer rate. High heat transfer rates are needed in the drying process of a product so that the required time is shorter and optimal. In addition, rice husk biomass can be used as alternative energy because it is easy to obtain, especially in agricultural areas that grow rice.

Evaluating the long-term sustainability of geothermal energy utilization from deep coal mines

Deep underground mines offer a great potential as low-enthalpy geothermal resource to be adopted in direct regional heating. In this paper, a parallel horizontal ground heat exchangers (pHGHE) system is introduced to utilize the geothermal resource in deep coal mines, however, the sustainability of the system in a long-run performance has not been well addressed in the previous study. This work presented a quantitative evaluation on the long-term sustainability of pHGHE system installed in backfilled stopes of deep coal mines. A novel numerical model is applied to characterize the heat transfer occurring between the horizontal pipes and backfill stopes. The proposed model was first verified to an analytical solution to determine the model reliability and accuracy. The validated numerical model was further applied to a typical underground backfilled stope of Anju coal mine in China. The results demonstrated that the maximum sustainable specific heat extraction rate must not exceed 40 W/m to maintain a 50-year exploitation period. A total thermal capacity of 17 GWh could be realized during each extraction cycle under the current conditions, which could provide heating to more than 170,000 m2 of residential building area. Accordingly, sensitivity analysis was performed to identify the key parameters influencing system sustainability. In addition, the effects of groundwater flow on the thermal interaction and long-term sustainability of the pHGHE systems were examined. Groundwater flow can be beneficial to the long-term sustainability of the system.This work confirms the feasibility and sustainability of geothermal resource utilization in backfill stopes of underground coal mines. The insights gained in this study can provide technical guidance for the design of geothermal systems in similar coal mines.

Flow distribution in the air cooler of HTGR passive cavity cooling system

Passive Cavity Cooling System (PCCS) is an important safety related system for High Temperature Gas-cooled Reactor (HTGR). Air cooler is one of the key components of this system. The non-uniform flow distribution is a common issue encountered in similar Parallel Tube Heat Exchanger (PTHX). Improving flow uniformity is favorable for both thermal hydraulic performance and structure safety. In this work, the flow distribution characteristics of a compact parallel tube air cooler with U-type header arrangement are theoretically and numerically investigated. First, a theoretical discrete model is proposed for flow distribution in PTHX. It reveals that the static pressure distributions in the headers are influenced by static pressure recovery, friction resistance and local pressure drop, which is directly related with flow uniformity. Then, the flow uniformity of Z-type and U-type header arrangements is numerically investigated and compared in detail. Moreover, the effects of thermal hydraulic parameter and geometrical parameter on flow uniformity are successively evaluated. Numerical results show that the flow uniformity of PCCS air cooler with U-type header arrangement is more excellent than that of Z-type header arrangement. Thermal hydraulic parameters, such as tube-side water temperature and mass flow rate, have little impact on flow uniformity. Nevertheless, the flow distribution characteristics are very sensitive to the geometrical parameters, including the insertion angle configuration, the flow area ratio of header to parallel heat transfer tubes (AR) and the cross-sectional area ratio of distribution header to collection header (DCR). Reasonable insertion angle configuration and decreasing AR can effectively enhance the flow uniformity of PCCS air cooler. For header match, it should be avoided that the distribution header diameter below the collection header diameter. As a result, the optimal structure design determined in this work is an air cooler with U-type header arrangement, where the inlet insertion angle is 0°, the outlet insertion angle is 180°, and the diameters of distribution header and collection header are both 120 mm. Its flow uniformity is improved by more than 50% compared with present design. This work reports a promising design modification, as the potential structure optimization for HTGR PCCS air cooler.

Performance of parallel plate-fin heat exchanger for piston aero-engines with front-placed guide plate at high altitude

• Heat exchangers’ performance in aircraft is enhanced by installing guide plates. • New indicators relating thermal performance to structural parameters are proposed. • The effect of inclination angle on thermodynamic performance is discussed. • The shielding proportion would significantly affect the heat transfer capacity. • Correlations are obtained to predict high-altitude characteristics. The reduction in atmospheric density would hinder the normal operation of the cooling system of piston aero-engines, thus influencing the aircraft’s performance in all aspects. This paper first introduced a composite structure consisting of a plate-fin heat exchanger and a front-placed guide plate to strengthen the heat transfer capacity. The improvement was evaluated and predicted by CFD (computational fluid dynamics) simulations at different altitudes. The optimal layout parameters of the guide plate, such as the inclination angle and the shielding proportion, were also explored. To scientifically elucidate the relationship between heat dissipation, flow performance, and structure parameters, the conception of effective mass rate was proposed for the first time. The results showed that the enhancement of the composite structure in heat transfer rate achieved a maximum improvement rate of 34.9% at 20 km altitude. Also, different arrangements of the guide plate directly affected the thermal and flow performance. Below 11 km, the best inclination angle of the plate was 30°, while above 11 km, it became 45°. The optimal gap in partial shielding cases decreased within 7∼9 mm as the altitude increased. Finally, relevant correlations for predicting the flow and heat transfer characteristics of the heat exchanger at different altitudes were developed in this study.

An investigation on dehumidification and heat transfer characteristics of a minichannel heat exchanger for heat pump system of electric vehicles

Heat pump systems for electric vehicles must assure the system's energy efficiency and the comfort and safety of the passenger cabin. This project aims to analyze the dehumidification and heat transfer characteristics of a heat exchanger through experiments and theoretical modeling. In this paper, a parallel flow minichannel heat exchanger was experimentally and theoretically studied under various operating conditions. A distributed parameter model considering heat and mass transfer was developed and utilized to predict the heat exchanger's dehumidification and heat transfer characteristics. The results reveal that, with a fixed inlet air relative humidity and a low refrigerant mass flow rate, the dehumidification rate (DHR) increases and subsequently declines with the increasing inlet airflow. When the inlet air relative humidity and inlet airflow are constant, the lower the refrigerant mass flow rate, the greater the enthalpy humidity ratio. As the inlet airflow increases, the refrigerant mass flow rate and inlet air relative humidity stay constant, and the enthalpy humidity ratio rises. The proposed model's predictions match the experimental data rather well. The predicted heat transfer rates (HTRs) and DHRs have absolute errors of less than 4% and 10%, respectively, compared with the experimental results. Under various airflows for the OM3 operating conditions, the distributions of HTR and DHR demonstrate that the HTR percentage is lowest in flow path IV, no dehumidification is conducted in flow path IV, and the DHR percentage is largest in flow path II. The proposed model may be employed to assess the dehumidification and heat transfer performance of heat exchangers and optimize the design of heat exchangers in similar heat pump systems. Highlights Construct a test bench for the dehumidification and heat transfer performance evaluation of a minichannel evaporator. Analyze and discuss the dehumidification and heat transfer characteristics of the evaporator under different operating conditions. Develop a distributed parameter model of dehumidification and heat transfer for the evaporator.

Dynamic optimization analyses and algorithm design for the parallel heat exchange system in spacecraft

Dynamic optimization of the fluid loop is critical of the active thermal control system (ATCS) for future spacecraft. In this paper, the transient heat current model of the parallel heat exchanger system is constructed by the thermoelectric analogy method to analyze the optimal control problem of dynamic flow allocation. The continuous free time optimal flow control problem is transformed into a fixed-time nonlinear programming problem which can be solved by the sequential quadratic programming (SQP) algorithm through the control variable parameterization (CVP) and time-scaling methods. The simulation showed that the optimized flow allocation significantly reduced the process time required for system heat dissipation. Compared with the non-gradient particle swarm optimization (PSO) algorithm, the SQP method constructed in this paper decreased the optimization index by about 4%∼6% and the calculation time by about one order of magnitude. To diminish the sensitivity of the SQP algorithm to the initial value, this paper simplifies the model according to the system characteristics. Through the Pontryagin extremum principle, it is found and proved that, under the simplified model, the optimal flow rates are a set of time-independent variables, which significantly reduces the difficulty of the optimization problem, according to which a hybrid algorithm is designed that uses the optimal result of the simplified model as the initial value predictor. Compared with the average of using the random initial value, using the estimated initial value decreased the optimization index by more than 2% and the computation time by about 60%, under the same SQP algorithm.

Thermal Analysis of Shell and Tube Type Heat Exchanger using Hybrid Nanofluid

A numerical analysis has been performed on a parallel flow shell and tube heat exchanger with different hybrid nanofluids in the present study. The heat exchanger consists of baffles which may increase the heat transfer characteristics. Hybrid nanofluids are used as a cooling medium. The author studied the effect of dissolved nanoparticles on Prandtl number and thermal transfer characteristics. The results show a convincing enhancement in the heat transfer characteristics by introducing hybrid nanoparticles into the primary fluid (water). The total concentration of nanoparticles was 2 vol%. Different combinations of nanoparticles as a coolant studied were Al₂O₃+MWCNT, Al₂O₃+Beo, Al₂O₃+AlN, and Al₂O₃+TiO₂ (in equal volume ratios). The heat transfer rate was enhanced maximum by 16.5 % for Al₂O₃+MWCNT/ water hybrid nanofluid. Moreover, Prandtl number was observed reduced using nanoparticles with a maximum reduction of 10.5 % for Al₂O₃+TiO₂/ water hybrid nanofluid.
 HIGHLIGHTS
 
 A numerical analysis was performed in a parallel flow shell and tube heat exchanger
 Nanofluids such as Al₂O₃+MWCNT, Al₂O₃+Beo, Al₂O₃+AlN, and Al₂O₃+TiO₂ (in equal volume ratios) was used as a coolant
 The effect of dissolved nanoparticles on Prandtl number and thermal transfer characteristics were studied
 The heat transfer rate was enhanced maximum by 16.5 % for Al₂O₃+MWCNT/ water hybrid nanofluid
 Prandtl number was observed reduced using nanoparticles with a maximum reduction of 10.5 % for Al₂O₃+TiO₂/ water hybrid nanofluid
 
 GRAPHICAL ABSTRACT

Experimental and CFD Analysis of GW70 based Cu Nanofluids in a Parallel Flow Heat Exchanger

The Nusselt number, overall heat transfer, and convective heat transfer coefficients of glycerol-water-based Cu nanofluids flowing in a parallel flow double pipe heat exchanger are estimated using CFD analysis. Single-phase fluid approach technique is used in the analysis. Ansys 19.0 workbench was used to create the heat exchanger model. Heat transfer tests with nanofluids at three flow rates (680<Re<1900) are carried out in a laminar developing flow zone. For testing, a 500 mm long concentric double pipe heat exchanger with tube dimensions of ID=10.2 mm, OD= 12.7 mm, and annulus dimensions of ID=17.0 mm, OD= 19.5 mm is employed. Copper is utilized for the tube and annulus material. This study employed three-particle volume concentrations of 0.2 percent, 0.6 percent, and 1.0 percent. The mass flow rates of hot water in the tube are 0.2, 0.017, and 0.0085 kg/s, while the mass flow rates of nanofluids in the annulus are 0.03, 0.0255, and 0.017 kg/s. The average temperature of nanofluids is 36°C, whereas hot water is 58°C. In comparison to base liquid, the overall heat transfer coefficient and convective HTC of 1.0 percent copper nanofluids at 0.03 kg/s are raised by 26.2 and 46.2 percent, respectively. The experimental findings are compared to CFD values, and they are in close agreement.

Improved effectiveness of a cryogenic counter-current parallel flow - Three fluid heat exchanger with three thermal communication due to Joule Thomson pressure drop

The pressure drop due to Joule-Thomson effect shows an improved effectiveness. Due to miniaturization Joule-Thomson effect are more prominent, hence it's effect should be considered. The current study on a counter-current, parallel flow cryogenic heat exchanger presents improved effectiveness due to the Joule-Thomson pressure drop. The investigation considered seven dimensionless parameters for the analysis, from which one to account for the Joule-Thomson effect. The effects are represented in non-dimensional parameters, presenting appreciable data towards improved effectiveness of the heat exchanger.

Modified methodology for determining the temperature profiles of inverted u-tubes steam generators used in pwr power plants

The most common and reliable methodology for determining temperature profiles of Inverted U-tubes Steam Generators is using Computational Fluid Dynamics (CFD) programs. In this work, we developed a modified methodology, using the Wolfram Mathematica software, in order to determine, with good approximation, the temperature profiles of these kind of equipment. The first step was to determine expressions for the physical properties of the water in the operational conditions, like density, thermal conductivity, specific heat and dynamic viscosity. Geometrical parameters like tubes diameter and sub-channel flowing area, as well as the flow parameters like flow mass of primary and secondary fluid, were also considered for determining the numbers of Reynolds, Prandtl, Nusselt and, consequently, the variation of convective coefficients and the global heat transfer coefficient. With subroutines that use the method of the lines we were able to solve the partial differential equations applied to parallel and countercurrent heat exchangers with no phase change. The U-tubes SG were divided in two regions which the first one was calculated considering a parallel heat exchanger and the second one was calculated considering a countercurrent heat exchanger, depending on the flow direction of the primary and secondary circuit. During the phase change, a constant variation of the enthalpy was considered, making the primary fluid temperature decrease following a linear behavior. Using the developed methodology called “Enthalpy Ruler”, the encountered results were considered adequate, since the defined lengths are compatible with the constant variation of the enthalpy from the compressed liquid to saturated steam.

Parallel Heat-Exchangers Network Optimization Using Sliding Mode Extremum Seeking with integral Anti Wind Up Scheme

In this paper, we study the Sliding Mode Extremum Seeking (SMES) optimization method for a Heat Exchangers Network. The studied network is constructed from plate Heat Exchangers, in which several thermal sources are used to heat a common cold stream that is distributed between the several branches of the network. The considered optimization problem is the optimization of the thermal power gained from different hot sources. The control variables are the split ratios of the cold stream to the different heat exchangers. The dynamical model for general Heat Exchanger Network with (n) Heat Exchangers was driven and the special case of two Heat Exchangers in parallel was considered as a case study. The SMES algorithm was modified with an integral anti-windup scheme to constrain the search within the admissible region. The simulation results obtained by using Matlab program confirmed the effectiveness of the approach. Index Terms— Heat Exchangers Network, Sliding Mode Based Extremum Seeking, Real-time Optimization.

The effects of incident solar radiation on the collector efficiency using coolant hybrid nanofluid via simulation of solar tower system with the parallel heat exchangers

BackgroundGreen energy market will contribute to three fifth of the world's electric power generation and two fifths of the fuel market by the mid of the present century. The annual transient thermal performance simulation of a solar tower power plant with parallel heat exchanger is performed. MethodThe heat transfer process for the two tanks is performed in which a heat loss from the tanks is considered. Two-phase heat transfer analysis at the heat exchanger is studied. The hourly reflected radiation is determined from DNI data. Nanofluid coolants are used for the flat-plate solar collector. FindingsIt is found that the temperature profile of the melted salt at the exit of the receiver fluctuates with higher frequency. The appropriate performance of the solar collector using SiO2 dispersed nanoparticles in water, is also provided. GO-DW mono nanofluid shows a better performance; however, GO/SiO2-DW hybrid nanofluid has more cost-efficiency.

Numerical investigation on turbulent thermal performance of parallel flow heat exchanger with a novel polyhedral longitudinal vortex generator in shell side

This work presents a parallel flow shell and tube heat exchanger with a novel clamping plate baffle and polyhedral longitudinal vortex generator (PLVG) in shell side. The parametric effectiveness of geometric parameters (width, height, pitch and attack angle) of PLVG on turbulent thermal performance is numerically explored using the Taguchi method to carry out the parameter optimization. The flow structure and heat transfer mechanism of PLVG is presented and revealed. The results show that the PLVG has a higher PEC than the equilateral triangle cross sectioned wire coil (ETWC) and sinusoidal wavy tape (SWT). The sequence of the parametric effectiveness for the Nu and PEC is PLVG height > attack angle > width > pitch, for the f is PLVG height > width > attack angle > pitch.