Abstract

Concentrating solar power (CSP) development has focused on increasing the energy conversion efficiency and lowering the capital cost. To improve performance, CSP research is moving to high-temperature and high-efficiency designs. One technology approach is to use inexpensive, high-temperature heat transfer fluids and storage, integrated with a high-efficiency power cycle such as the supercritical carbon dioxide (sCO2) Brayton power cycle. The sCO2 Brayton power cycle has strong potential to achieve performance targets of 50% thermal-to-electric efficiency and dry cooling at an ambient temperature of up to 40 °C and to reduce the cost of power generation. Solid particles have been proposed as a possible high-temperature heat transfer or storage medium that is inexpensive and stable at high temperatures above 1000 °C. The particle/sCO2 heat exchanger (HX) provides a connection between the particles and sCO2 fluid in emerging sCO2 power cycles. This article presents heat transfer modeling to analyze the particle/sCO2 HX design and assess design tradeoffs including the HX cost. The heat transfer process was modeled based on a particle/sCO2 counterflow configuration, and empirical heat transfer correlations for the fluidized bed and sCO2 were used to calculate heat transfer area and estimate the HX cost. A computational fluid dynamics simulation was applied to characterize particle distribution and fluidization. This article shows a path to achieve the cost and performance objectives for a particle/sCO2 HX design by using fluidized-bed technology.

1 Introduction

Concentrating solar power (CSP) collects solar heat and uses it to drive a thermal power cycle for electricity generation. CSP systems are typically deployed as large, centralized power plants to take advantage of the economies of scale. A key advantage of certain CSP systems—in particular, parabolic troughs and power towers—is the ability to incorporate thermal energy storage (TES) for flexible power generation. TES is less expensive and provides more capacity than electric storage using batteries, and it allows CSP plants to increase the capacity factor and dispatch power as needed—for example, to cover evening demand peaks. To improve solar thermal-to-electricity conversion efficiency (and subsequently, the economic competitiveness for power generation), next-generation CSP systems are moving toward operating at a higher temperature for high ideal Carnot efficiency. The power tower technology surpasses parabolic trough and linear Fresnel systems and can deliver the high-temperature conditions necessary to drive high-efficiency power cycles, which is the focus of future development by the US Department of Energy [1].

Current CSP plants use oil, nitrite salt, or steam as a heat transfer fluid (HTF) to carry solar thermal energy to the working fluid of a power block. These fluids have properties that limit plant performance; for example, synthetic oil has an upper limit on temperature of 400 °C, nitrite salt is limited to 560 °C, and direct-steam generation requires complex controls and has limited storage capacity. Next-generation CSP technologies, namely, Gen3 CSP, are pursuing higher operating temperatures for higher power cycle efficiency and lower cost of TES [2]. For higher temperature conditions, solid particles are an attractive storage option because the stable, low-cost particles can work at temperatures above 1000 °C and can store the energy in hot particles at relatively low storage containment cost [2]. According to Black & Veatch reports to the DOE, a particle-based storage system can enable high-temperature and high power cycle efficiency (>50%) at a lower cost than the liquid HTFs with readily available materials [3,4]. The allowable temperature range for particles matches well with the supercritical carbon dioxide (sCO2) Brayton power cycle suitable for CSP applications. The sCO2 working fluid provides higher temperature operability than steam, and it can achieve better efficiency than conventional power cycles. Supercritical CO2 operated in a closed-loop Brayton cycle offers the potential of equivalent or higher cycle efficiency versus supercritical steam cycles [5]. A sCO2 Brayton cycle system has a smaller weight and volume, lower thermal mass, and less complex power blocks compared with steam Rankine cycles because of the higher density of the fluid and a simpler cycle design. The power block compactness and high efficiency make the sCO2 Brayton cycle attractive for the scales and temperature ranges associated with the CSP integration. The combination of the particle thermal system with the sCO2 Brayton cycle may address the temperature, efficiency, and cost barriers associated with current molten nitrate salt CSP systems.

Figure 1 shows one example of a sCO2 power cycle integrated with a particle-based CSP system. The illustrated system uses a particle heating receiver for solar energy collection. Hot particles from a solar receiver fall into the hot particle containment silo for storage. When discharging the thermal energy, hot particles then flow through the particle/sCO2 heat exchanger (HX). The cold particles return to the cold silo and circulate back to the receiver when solar flux is available. Alternatively, the particles could be indirectly heated by other HTFs (such as compressed gas from a gas-phase receiver) and provide only the storage capability [1]. In addition, heating particles via an electric heater or pumped thermal system is recently being considered in newly inceptive research work for grid-scale electricity storage [6]. The inexpensive, stable solid particles have high potential for storing large amounts of energy to promote grid resilience and reliability in future renewable-dominated power generation.

Fig. 1
Particle/sCO2 heat exchanger integration in a CSP system
Fig. 1
Particle/sCO2 heat exchanger integration in a CSP system
Close modal

The sCO2 power cycle shown in Fig. 1 is a high-efficiency recompression cycle. The link between the particle thermal system and the sCO2 power cycle in a CSP plant is a particle-to-sCO2 HX. The sCO2 power cycle is often optimized for a high turbine inlet temperature and pressure to achieve high cycle efficiency [5]. The operation targets for a particle/sCO2 HX are to heat sCO2 to ≥720 °C [1]. The operating parameters of sCO2 flow impose a high-stress requirement on the heat transfer tubes, which often need large quantities of expensive high-temperature superalloys.

The challenge of a particle/sCO2 HX design is to achieve both high thermal performance and low cost. A key performance metric required to limit the HX cost is a high particle/sCO2 heat transfer rate to reduce the heat transfer surface area and the superalloy usage. Thus, a critical challenge is to achieve efficient heat transfer into and out of the particles in the particle HX [1]. Fluidized-bed and moving-bed particle heat exchangers are often used in industry for power generation or chemical processes. Performance and cost characterization for a moving packed-bed particle/sCO2 HX has recently been reported [7]. A particle/sCO2 fluidized-bed HX has the potential to offer efficient heat transfer but may be more complicated than a moving-bed design. Thus, we used modeling methods to analyze a fluidized bed (FB) HX design including empirical and computational fluid dynamic (CFD) method to show the potential of a fluidized-bed design with efficient heat transfer to justify its complexity. In selecting analysis methods, few published research studies examine the real total heat transfer coefficient at high temperatures over broad operating conditions. In addition to the difficulty and cost of a hot test, accurate heat transfer measurements must separate radiation and convection at high temperature. Multiphase discrete element method or Eulerian simulations can offer insight into fundamental behavior, but are computationally expensive and scale limited with many physical and mathematical assumptions. Improved certainty in the heat transfer coefficient is a major need in FB HX design and testing. This article presents a high-level theoretical study of parametric sensitivity to show the potential of the fluidized-bed design using modeling techniques including empirical and CFD methods. Our analysis development can guide a full-scale HX design for minimizing experimental testing and design iterations. To explore the design factors on selecting a HX configuration, the following section compares the heat transfer processes of a moving packed bed, granular flow, and fluidized bed. This article evaluates the heat transfer capabilities of these particle flow regimes and presents analysis results for a fluidized-bed particle/sCO2 HX.

2 Comparison of Particle Flow Regimes in a Particle/sCO2 Heat Exchanger

Particle-phase heat transfer depends strongly on specific details of the system configuration including particle/wall contact, particle mixing, flow regime and flow condition, and particle properties. As such, a high degree of variability exists in publicly available measurements or theoretical predictions of heat transfer coefficients in solids-phase systems. Often, the heat transfer studies were conducted with different particle types and flow configurations. Thus, direct comparison of measured or simulated values is challenging given dissimilarity in particle properties, particle-bed designs, and flow conditions.

Improving the particle heat transfer in a particle heat exchanger is beneficial because the particle-side heat transfer is lower than that of the sCO2 fluid, and thus, particle-side heat transfer controls heat exchanger size and layout. Several characteristic flow regimes exist, and Fig. 2 shows three flow patterns proposed for CSP [710]. A moving packed bed relies on particle conduction heat transfer from particles to parallel panels [7]. Particles move downward slowly by gravity as plug flow with nearly no lateral movement in the normal direction to the panels. The gap distance between two parallel panels and particle thermal conductivity determines the particle heat transfer rate. Solid particles often have low thermal conductivity and limit the effectiveness of heat transfer. Measures to enhance particle heat transfer depending on the gap between the parallel panels could be limited by particle flowability. A granular flow moving bed improves particle mixing; however, granular flow over round tubes may introduce a top stagnant region and a bottom void around the round tube as shown in Fig. 2(b) [8], which hinders the particle/tube heat transfer. In contrast, a fluidized bed has a potential for high heat transfer rates due to improved mixing, particle/heat transfer surface contact, and particle refresh rates at the heat transfer surfaces. Figure 2(c) shows a schematic of a fluidized bed with fluidization gas distributed from the bottom of the bed, while particles enter from the bed top and flow out to the bottom.

Fig. 2
Flow pattern and flow regime effects on particle-to-wall heat transfer
Fig. 2
Flow pattern and flow regime effects on particle-to-wall heat transfer
Close modal

Table 1 summarizes the comparison among HX operating regimes. Fluidized-bed heat transfer rates have been extensively studied and reported in the literature, and they can exceed 500 W/m2/K under the relevant particle size and temperature conditions [10,14,15]. But the parasitic loads, heat losses, components, and cost of a fluidization system are concerns for the cost and efficiency of a small-scale particle heat exchanger in CSP applications [16]. Moving packed-bed heat transfer is comparatively simple, does not require additional fluidization gas supply, and does not incur the parasitic loads and heat losses of a FB. However, particle/wall heat transfer coefficients can be lower than those achieved with fluidization; thus, a moving-bed HX may require a larger heat transfer surface area and increased use of expensive alloys for high-temperature conditions. Granular flow conditions with enhanced particle mixing and particle/wall contact fall between the two extremes of moving packed bed and fluidized bed and are comparatively less studied in the literature. Theoretical studies show that granular flow can not only improve upon moving-bed heat transfer rates but also exhibit heat transfer characteristics that are a strong function of particle/wall contact conditions achieved through surface configurations [17,18].

Table 1

Comparison of particle flow regimes in a particle bed for a sCO2 HX

Particle flow patternsRange of heat transfer coefficientsStrengthWeakness
Moving packed-bed heat exchanger100–250 W/m2/K [11]
  • No need for gas fluidization

  • Simple system and operation

  • Limited heat transfer rate

  • Slow particle flow to maintain particle wall contact

  • Critical to avoid particle bridging in the flow path

Granular flow moving-bed heat exchanger200–300 W/m2/K [8,12]
  • Low parasitic power

  • Heat transfer may be improved over a moving packed bed

  • Particle flow stagnation can incur unreliable particle-to-bed wall contact and limit heat transfer

Fluidized-bed heat exchanger>300 W/m2/K and up to 1000 W/m2/K [13]
  • Potentially high heat transfer coefficient

  • High gas/solid mixing and contact with heat transfer surfaces

  • Fluidizing gas parasitic power and heat loss

  • Complex system and operation

Particle flow patternsRange of heat transfer coefficientsStrengthWeakness
Moving packed-bed heat exchanger100–250 W/m2/K [11]
  • No need for gas fluidization

  • Simple system and operation

  • Limited heat transfer rate

  • Slow particle flow to maintain particle wall contact

  • Critical to avoid particle bridging in the flow path

Granular flow moving-bed heat exchanger200–300 W/m2/K [8,12]
  • Low parasitic power

  • Heat transfer may be improved over a moving packed bed

  • Particle flow stagnation can incur unreliable particle-to-bed wall contact and limit heat transfer

Fluidized-bed heat exchanger>300 W/m2/K and up to 1000 W/m2/K [13]
  • Potentially high heat transfer coefficient

  • High gas/solid mixing and contact with heat transfer surfaces

  • Fluidizing gas parasitic power and heat loss

  • Complex system and operation

A particle/sCO2 FB HX could be complicated due to the fluidizing gas loop and complexity in operation at small scale (on the order of 100s kWth). But the reduced use of expensive alloys through enhanced FB heat transfer may result in lower costs and may justify a FB HX design at a commercial scale (on the order of 100 MWth). This article shows the fluidization characterization and heat transfer performance in an FB HX. The modeling results assess the heat transfer effect on the specific cost of the heat exchanger. The objective of this study focuses on FB design for a particle/sCO2 HX to investigate its potential to obtain a high heat transfer coefficient and to reduce the use of expensive alloy by minimizing the HX heat transfer area and size.

3 Modeling of a Fluidized-Bed Heat Exchanger

The particle/sCO2 FB HX analysis uses known information from particle characterization and the gas/solid two-phase flow conditions and operating mechanisms. The basic characterization of FB HX was based on a conceptual design developed in a DOE SunShot National Laboratory Multiyear Partnership (SuNLaMP) project intended to demonstrate a 100-kWth particle/sCO2 HX [19]. The FB HX is assumed to be a basic counterflow configuration with particle-side flow against the direction of the sCO2 flow. Particle lateral flow in a fluidized bed has been demonstrated in a lab test setup [20]. The high-temperature/high-pressure sCO2 working fluid flows inside horizontal tubes and is heated by the hot particles. Mathematical modeling of the basic particle/sCO2 flow configuration provides a basis for the sensitivity analysis on the effects of FB HX design on performance and cost.

The base set of model input parameters for a 100-kWth HX is listed in Table 2 and is used with the design conditions specified in Table 3 and the physical properties of CARBO Accucast ID50 particles [19]. Parameters are held constant at these values in the following analysis unless otherwise specified. Tube wall thickness is adjusted for each tube size based on interpolation between the nearest schedule 40 tube sizes. The tube material is assumed to be a high nickel alloy with high strength at high temperature because the tubes are subjected to >700 °C fluid temperature with nearly 20 MPa internal pressure.

Table 2

Base-case model parameters

Tube parametersHeat transfer parameters
Tube OD0.0254 mNu for CO2-side h (hCO2)Eq. (7)
Tube ID0.0204 mSolids-side h (hp)Eqs. (12)(14)
Ntubes10Particle average diameter280 μm
Particle density3363 kg/m3
Tube thermal conductivity15 W/m/Kɛmf0.5
Tube wall emissivity0.9Particle sphericity (φ)0.9
Tube alloy cost75 $/kgFluidizing gas velocity1.5 umf at Tp,out
Tube wall density9000 kg/m3Bed emissivity0.8
Tube parametersHeat transfer parameters
Tube OD0.0254 mNu for CO2-side h (hCO2)Eq. (7)
Tube ID0.0204 mSolids-side h (hp)Eqs. (12)(14)
Ntubes10Particle average diameter280 μm
Particle density3363 kg/m3
Tube thermal conductivity15 W/m/Kɛmf0.5
Tube wall emissivity0.9Particle sphericity (φ)0.9
Tube alloy cost75 $/kgFluidizing gas velocity1.5 umf at Tp,out
Tube wall density9000 kg/m3Bed emissivity0.8
Table 3

Operating parameters for fluidized-bed heat exchanger

ConditionValueUnits
Inlet solids temperature775°C
Outlet solids temperature570°C
Average solids temperature672°C
Inlet CO2 temperature550°C
Outlet CO2 temperature700°C
Particle flow0.41kg/s
CO2 flow0.55kg/s
ΔP_CO20.01MPa
ConditionValueUnits
Inlet solids temperature775°C
Outlet solids temperature570°C
Average solids temperature672°C
Inlet CO2 temperature550°C
Outlet CO2 temperature700°C
Particle flow0.41kg/s
CO2 flow0.55kg/s
ΔP_CO20.01MPa

3.1 Model Configuration.

An ideal counterflow configuration between the fluidized particles and sCO2 flow inside a tube is illustrated schematically in Fig. 3 as the HX design and modeling basis. The choice of tube dimensions includes tube inner diameter and outer diameter (OD), length, and the number of tubes immersed in the particle flow. We developed both a one-dimensional (1D) distributed parameter model for the particle and sCO2 HX and a two-dimensional (2D) CFD model to simulate the particle flow and heat transfer inside the fluidized bed. Figure 3 illustrates the ideal counterflow configuration for the 1D flow and heat transfer simulations of the particle/sCO2 HX along the axial coordinate (z). The FB HX design is unique and different from a conventional fluidized bed shown in Fig. 2(c). In the simulated design, fluidized hot particles flow horizontally from the inlet to the exit (opposite the direction of sCO2 flow in the tubes) while being fluidized in a perpendicular direction by the fluidizing gas flowing upward. The hot particles are distributed from one side and then fill the fluidized bed containing sCO2 tubes for heat transfer to the sCO2 working fluid, and finally the cooled particles exit to the opposite side. The fluidizing gas enters the fluidized bed from a bottom gas distributor and exits through the bed top. Inlet fluidizing gas can be heated by the exiting hot fluidizing gas to recuperate heat. That fluidized particles move horizontally, which was demonstrated by experiment in a SuNLaMP project [20]. This flow configuration creates a near counterflow heat transfer path between particles and sCO2 flow stream as shown in Fig. 3, which can be approximated by the 1D model.

Fig. 3
Simulation schematic for a counterflow heat exchanger configuration
Fig. 3
Simulation schematic for a counterflow heat exchanger configuration
Close modal

Table 3 presents the operating conditions of the FB HX used as inputs to the models. These conditions correspond to operating parameters of a high-efficiency sCO2 power cycle and targeted particle CSP system operating conditions. The sCO2 and particle flow rates presented in Table 3 come from overall heat and mass balances for a 100-kW FB HX. The simulated particles inside the FB HX are based on the particle size and properties of CARBO Accucast ID50.

The 1D distributed model provides a parametric evaluation of heat transfer surface area, cost, and performance for the HX design according to the tube dimensions, number of tubes, and relation to the heat transfer coefficients on both particle and sCO2 sides. The energy balances for the sCO2 and particle flows are represented by Eqs. (1) and (2), and the initial and boundary conditions follow Eqs. (5) and (6). Inner and outer tube wall surface temperatures are related to particle and CO2 temperatures based on steady-state convective/conductive heat transfer resistances in Eqs. (3) and (4):
m˙CO2,tubeCpCO2dTCO2dz=hCO2(Tw,iTCO2)(πDi)
(1)
m˙pCppdTpdz=Ntubeshp(Tw,oTp)(πDo)
(2)
Tw,i=TCO2+2DihCO2(TpTCO2)[2DihCO2+1kwln(DoDi)+2Dohp]1
(3)
Tw,o=Tp2Dohp(TpTCO2)[2DihCO2+1kwln(DoDi)+2Dohp]1
(4)
z=0:TCO2=TCO2,inTp=Tp,out
(5)
z=Ltube:TCO2=TCO2,outTp=Tp,in
(6)

The 1D conservation equations were solved using finite-difference methods with the particle and sCO2 heat transfer coefficients evaluated at local conditions based on empirical heat transfer correlations for sCO2 flowing in a tube and particle wall heat transfer in a fluidized bed. The heat transfer coefficients on the sCO2 and particle sides close the energy balance equations (Eqs. (1)(4)) and are presented in the following sections. The model is solved iteratively for the tube length (Ltube) and, correspondingly, the total heat exchanger surface area and tube material volume required to achieve the specified outlet temperature given a specified number of tubes and tube sizing parameters. The HX design considers low fluidization velocity with small-size particles (<300 µm) for optimum performance. The effects of tube diameter, tube length, and tube number on HX design, material usage, and material cost were investigated for the 100-kWth prototype.

3.2 s-CO2 Heat Transfer Coefficient.

The model computes the sCO2 heat transfer coefficient at local conditions from Gnielinski’s correlation for turbulent flow in tubes [21]:
NuCO2=(f/8)(Re1000)Pr1+12.7(f/8)1/2(Pr2/31)
(7)
f4{1.737ln[0.269εwDi2.185Reln(0.269εwDi+14.5Re)]}2
(8)

The Reynolds and Prandtl numbers characterizing the sCO2 flow conditions were verified to conform to the Reynolds and Prandtl number limits for correlation validity.

3.3 Characterization of Particle Flow and Heat Transfer in a Fluidized Bed.

Fluidized-bed design and characterization have been studied extensively for chemical reactors, coal-fired boilers, and particulate products and processes [10]. However, FB design is still specific to the particle media and application and relies on both theoretical and experimental works to optimize the design and proper operations. Here, a high-level description of characterization methods provides fundamental performance parameters for evaluating the FB HX performance and cost.

3.3.1 Hydrodynamic Characterization of a Fluidized Bed.

The fluidization condition is a result of the fluidizing gas flow velocity exceeding the minimum fluidization velocity and interacting with the particles. Minimum fluidization velocity Umf is the superficial gas velocity at which the bed just starts to fluidize. The fluidized bed behaves as a liquid, and the total pressure drop through the bed is equal to the total hydrostatic pressure drop of the bed. It represents the minimum superficial gas velocity at which a fluidized bed can be operated. In most cases, one must operate at U > Umf to avoid channeling, defluidized zones, and grid leakage and to maximize the throughput. Many correlations to predict minimum fluidization velocity were proposed for various particles and operation conditions. In this case of small particles, a simple equation, Umf=(dp2(ρsρg)g/159μ)(εmf3ϕs2/1εmf), was used [9], when Rep,mf < 20, where ɛmf is the void fraction at minimum fluidization and assumed a value of 0.5 [9], and the subscript mf stands for minimum fluidization. ϕs is the particle sphericity and has a value of 0.9 for the type of particles selected.

The FB pressure drop and the flow rate affect the fluidizing gas compression power and heat loss. The computed fluidization conditions and overall auxiliary equipment selection criteria can be used for preliminary estimates of the FB performance in a tradeoff study of the HX size, layout, parasitic power, and heat loss. Bed pressure drop over a unit height and minimum fluidization velocity can be estimated from Eqs. (9) and (10), where Eq. (10) is a derivation of Eq. (9) at minimum fluidization condition in terms of dimensionless parameters [9]:
ΔpH=(1εmf)(ρpρg)g=150(1εmf)2εmf3μumf(ϕsdp)2+1.751εmfεmf3ρfumf2ϕsdp
(9)
Ar=150(1εmf)εmf3ϕs2(umfdpρgμ)+1.75εmf3ϕs(umfdpρgμ)2
(10)
where Ar is the Archimedes number expressed as Ar = g(φdp)3ρg(ρpρg)/μ2, and ɛmf is the void fraction at minimum fluidization conditions that is determined by the bed expansion at incipient fluidization.
FB power consumption is a concern for operating efficiency, and it depends on the bed pressure drop and the fluidizing gas flow rate. To calculate the compression power, we assumed that the fluidization gas is an ideal gas, such that the ideal compressor power can be calculated from the isentropic process in Eq. (11):
w˙s,ideal=γ1γpivi[(popi)(γ1)/γ1]
(11)
where γ is the ideal gas index (1.4 for a diatomic gas), pi is the gas inlet pressure, and vi is the gas volumetric flow rate. The actual compressor power is w˙s,comp=w˙s,ideal/ηcomp. In this study, the compressor power only considers the bed pressure drop, without adding the needed distributor, air ducts, and bag-filter pressure drop. The compressor efficiency is assumed to be 75%.

3.3.2 Particle-Side Heat Transfer Coefficient in a Fluidized Bed.

The particle-side heat transfer can be a limiting factor for the performance of a particle/sCO2 HX; thus, it is the focus for developing a particle-bed HX. Particle heat transfer coefficients are affected by many factors including particle physical properties, size, and operating conditions. The selection of the fluidization velocity is critical to maximize the particle heat transfer to the working fluid while minimizing the parasitic power consumption of the gas compressor and reducing the heat carried out by the exit gas. FB heat transfer experiments reported in the literature imply that the highest heat transfer coefficient occurs at typically two to three times the minimum fluidization velocity [10]. For small particle sizes, the minimum fluidization velocity is often low, in the range of 0.05–0.3 m/s depending on temperature.

Numerous correlations for heat transfer coefficients between fluidized particles and immersed tube or wall surfaces exist in the literature, and there is a large degree of variability between correlations [10]. Two such correlations are provided in Eqs. (12) and (13) for submerged horizontal or vertical tubes from the correlations of Vreedenberg [10] (Eq. 12) and Borodulya [14] (Eq. 13), respectively:
Nut=hcDtkg=420(ρpρgRet)0.3(μg2gρp2dp3)0.3(Prg)0.3
(12)
Nup=hcdpkg=0.74Ar0.1(ρpρg)0.14(CppCpg)0.14(1ε)2/3+0.46RePr(1ε)2/3ε
(13)

The Reynolds number in Eq. (12) is based on the tube diameter, whereas that in Eq. (13) is based on the particle diameter and particle gas relative velocity. As such, Eq. (13) does not depend on the chamber geometry or the tube size.

Radiative heat transfer between the particles and walls can play a significant role in particle heat transfer at high temperatures. The radiative heat transfer coefficient in the FB HX can be expressed as [10] follows:
hr=qrTwTp=σ(Tw2+Tp2)(Tw+Tp)(1/ew)+(1/ep)1
(14)
where Tw and Tp are the heat transfer surface and particle temperatures, respectively; ew is the heat transfer surface emissivity, usually in an oxidized condition under high temperatures with a value near 0.9; and ep is the equivalent emissivity for a particle cluster, which can be more absorptive than a single particle. For FB particle clusters, ep is in the range of 0.7–1.0 according to Ozkaynak and Chen [22].

3.3.3 Computational Fluid Dynamics Modeling of Fluidized-Bed Heat Transfer.

Empirical correlations provide guidance in selecting design and operating conditions, but existing literature correlations contain a high degree of uncertainty, may not be valid over the desired operating range, and may not address design-specific concerns including tube sizing, tube spacing, and effects of gas maldistribution or dead zones. A 2D Eulerian-Eulerian hydrodynamic and heat transfer CFD model using the software ansys/fluent was developed to further investigate these factors. A detailed description of the conservation equations and selected constitutive models is described elsewhere [23]. The model includes standard 2D continuity, momentum, and energy equations for each phase. Interphase momentum and energy exchange are described by the models of Gidaspow [24] and Gunn [25], respectively. Kinetic energy associated with random particle motion is described via a partial differential equation for granular temperature [26] with the diffusion coefficient suggested by Syamlal et al. [27]. Kinetic, collisional, and frictional components of the solid-phase viscosity are assessed from the expressions of Gidaspow [28], Syamlal et al. [27], and Schaeffer [29], respectively.

In this analysis, the CFD model is intended to verify the relationship between convective/conductive heat transfer coefficients and fluidizing gas flow velocity and considers only convective/conductive phenomena. Radiative transfer was omitted to reduce the computational burden and to avoid introducing additional uncertainty associated with modeling radiative transfer with particulate media. However, the radiation effect is included in heat exchanger sizing via Eq. (14) in the energy balance. Constitutive models for effective phase thermal conductivity are critical for capturing convective/conductive heat transfer rates at wall boundaries. Here, we consider three different models. The model of Kuipers et al. [30,31] is based on the effective packed-bed thermal conductivity derived by Zehner and Schlünder [32], which is commonly employed in the literature but was originally derived for the core region of the packed bed and does not account for the disruptive presence of solid walls on local bed-void fraction distributions. The local porosity distribution in the vicinity of the solid walls can differ substantially from that in the bed core [31,33], and overestimation of near-wall effective solids thermal conductivity when these wall effects are neglected can at least partially explain the tendency of Eulerian-based models to overpredict wall-to-bed heat transfer rates in the literature [34,35]. The model of Patil et al. [31] is based on the Kuipers et al.’s model for the core region of the fluidized bed, but modifies the voidage distribution in the near-wall region to reduce the solids fraction in the region less than half of a particle diameter away from the wall. Finally, the model suggested by Legawiec and Ziólkowski establishes a separate constitutive model for effective thermal conductivity in the near-wall region (less than half of a particle diameter away from the wall) based explicitly on near-wall particle contacts [34,36,37].

The calculation domain consisted of a simplified 2D cross section including an array of eight 0.0127-m OD tubes. A primary goal of this study is to evaluate the sensitivity of FB heat transfer performance to gas velocity, including gas velocity approaching minimum fluidization. This case may not be well represented by existing empirical correlations, and thus, the motivation for the CFD study is to improve confidence in expected performance under the desired low-velocity fluidization conditions. The computational mesh consisted of 2D quadrilateral elements with a maximum size of 1 mm and was refined in the vicinity of the tube walls with a minimum wall distance of 5 μm. Minimum wall distances between 2 and 60 μm were evaluated for a single set of conditions. Heat transfer rates decreased with the increasing boundary layer resolution, and mesh-independent solutions were achieved with a 5-μm first layer element thickness. Time-averaged heat transfer coefficients with a 5-μm first layer element thickness differed from those with a 60-μm first layer element thickness by approximately 13%. The solution was initialized with stagnant fluid/solid phases with an initial packed bed extending slightly above the uppermost tube and a packed-bed solids fraction of 0.6. A uniform gas inflow velocity was applied to the bottom boundary starting at time t = 0, and the transient simulation was conducted with a fixed time step of 2.5 × 10−4 s. All simulations maintain a fixed tube wall temperature, with the initial bed temperature and gas inflow temperature set 50 K above the tube wall temperature. All calculations were continued until t = 7–10 s to achieve approximately steady time-averaged heat transfer coefficients.

4 Results and Discussion

Simulation of the exemplary FB HX shown in Fig. 3 includes particle and sCO2 heat transfer through a counterflow scheme. Sensitivity of operational characteristics and equipment costs to design and operating parameters was evaluated for a range of parameters.

4.1 Characterizing the Performance of a Fluidized Bed Heat Exchanger.

The particle-side performance is critical to designing and sizing the FB HX to minimize parasitic power and reduce capital and operating cost. Proper design of the fluidized bed (including low fluidizing gas velocity, flow rate, and optimized bed height) can limit power consumption. Here, we target parasitic power consumption less than 1% of the HX rated power, which is the ratio of the compressor power to the HX thermal load.

There is no precise theoretical determination of operating targets for FB design. Values in the literature were used in the characterization calculations reported here, but more rigorous investigation and experimental testing are needed to verify and improve accuracy of predictive calculations for the fluidized bed, particularly under high-temperature operation. Figures 4 and 5 show the calculation for umf, bed pressure drop, and compressor power based on an assumed ɛmf value of 0.5. This value of ɛmf is approximate, and experimental measurements of this relation as a function of particle size and temperature would improve certainty of detailed FB characterization.

Fig. 4
Effect of particle size on minimum fluidization velocity at different temperatures. The calculations were based on base-case model parameters listed in Table 2, where ɛmf was assumed a value of 0.5, particle diameter is 280 µm, the particle sphericity ϕs has a value of 0.9, and particle density is 3.363 kg/m3.
Fig. 4
Effect of particle size on minimum fluidization velocity at different temperatures. The calculations were based on base-case model parameters listed in Table 2, where ɛmf was assumed a value of 0.5, particle diameter is 280 µm, the particle sphericity ϕs has a value of 0.9, and particle density is 3.363 kg/m3.
Close modal
Fig. 5
The compressor power of various fluidizing air flow rates for the 100 kW HX and FB pressure drop versus different bed heights. The calculation used similar particle properties as shown in Fig. 4. Ideal gas index of 1.4 was used in computing the air compression power.
Fig. 5
The compressor power of various fluidizing air flow rates for the 100 kW HX and FB pressure drop versus different bed heights. The calculation used similar particle properties as shown in Fig. 4. Ideal gas index of 1.4 was used in computing the air compression power.
Close modal

If the bed voidage at minimum fluidization does not vary with temperature and particle size, then Eq. (9) indicates that the bed pressure drop is nearly constant across both particle size and fluidization velocity and that it varies little with the bed temperature. The particle-bed height inside a fluidized bed affects the gas compressor power consumption and the size of a gas blower or compressor.

Figure 5 shows the compressor power (relative to the HX capacity) and bed pressure drop as a function of the bed height and fluidization velocity (relative to the minimum fluidization velocity umf). The percentage of compressor power in Fig. 5 is the ratio to the heat capacity of the HX at 570 °C air inlet temperature.

The pressure drop across the HX is directly proportional to the bed height. The allowable particle-bed height can be determined from the gas compressor power consumption and the availability of the gas blower or compressor. To balance the fluidizing air pressure head and the flow rate, the bed height should be in the range of the allowable compressor or blower compression pressure, but with an adequate height to limit the cross-sectional area to reduce the air volumetric flow rate. From the compressor power level in Fig. 5, bed heights between 3 and 5 m are still feasible because conventional air blowers or compressors can support the requirements of pressure drops and compressor power.

The correlations in Eqs. (12) and (13) predict convective heat transfer coefficients in the range of 200–900 W/m2/K as shown in Fig. 6. Calculations in Fig. 6 assume a fixed dp = 280 μm, bed voidage at minimum fluidization ɛmf = 0.5, and tube size Dt = 2.5 cm. Bed voidage in Eq. (13) was related to the gas velocity from the correlation shown in the study by Gunn and Hilal [38].

Fig. 6
Convective particle heat transfer coefficient with dp = 280 µm using Borodulya equation for vertical tubes (v) (Eq. (13)) and Vreedenberg equation for horizontal tubes (h) (Eq. (12))
Fig. 6
Convective particle heat transfer coefficient with dp = 280 µm using Borodulya equation for vertical tubes (v) (Eq. (13)) and Vreedenberg equation for horizontal tubes (h) (Eq. (12))
Close modal

Various empirical correlations can predict a wide range of heat transfer coefficient values and trends, as commonly noted in the literature and illustrated in Fig. 6 [10,15]. The differences in trends shown in Fig. 6 arise partially from the discrepancy between horizontal and vertical tubes and characteristic differences in how bubble growth moving upward through the bed can be expected to impact wall-to-bed heat transfer rates. However, similar order-of-magnitude discrepancies are observed among horizontal tube correlations illustrated in Sec. 4.2 The empirical correlations may not be capable of completely capturing complex relationships among convective heat transfer coefficients, gas velocity, and bed voidage with generality over wide ranges of conditions. Thus, the predictions should be interpreted with caution and must be verified through HX design iterations and experimental testing. However, all correlations predict an increase in convective heat transfer coefficient with a temperature that can be traced back to an increase in the gas-phase thermal conductivity and viscosity with temperature.

Figure 7 shows the calculated radiative heat transfer coefficients for typical emissivity (ew = 0.9 and ep = 0.9) and temperature conditions. The radiative heat transfer coefficient increases with both the particle temperature and the heat transfer surface temperature. In the range of temperatures relevant to the particle/sCO2 HX, the radiative heat transfer coefficient is expected to be about 100–200 W/m2/K. Radiative heat transfer adds to the convective heat transfer and augments the overall particle heat transfer.

Fig. 7
Radiative heat transfer between the particles and the tube walls typical emissivity (ew = 0.9, ep = 0.9) and temperature conditions
Fig. 7
Radiative heat transfer between the particles and the tube walls typical emissivity (ew = 0.9, ep = 0.9) and temperature conditions
Close modal

4.2 Computational Fluid Dynamic Modeling of Fluidized-Bed Heat Transfer.

The fluidized-bed CFD models described in Sec. 3.3.3 are applied in many other fluidized-bed heat transfer studies in the literature, but are most commonly employed to simulate to low-temperature conditions (<100 °C). Numerous empirical correlations for heat transfer rates exist, but these correlations contain substantial uncertainty, particularly regarding the relationship between heat transfer rates and fluidization velocity. Here, we use the CFD-simulated heat transfer predictions to compare with empirical correlations at high-temperature conditions (>650 °C) for the purpose of (1) verifying the overall expected order of magnitude of the particle-side heat transfer coefficient before the parametric study of the FB HX design and (2) assessing the feasibility of using near-minimum fluidization velocity at the high-temperature conditions.

Figure 8 illustrates contours of solid volume fraction at a single snapshot in time for a gas inflow velocity of 0.06 m/s and 0.1 m/s at average heat exchanger conditions (675 °C). At these conditions, the minimum fluidization velocity is estimated to be between 0.045 and 0.070 m/s. As expected, the CFD simulations indicate that the injection of gas leads to the formation of bubbles, which coalesce into larger bubbles as they move upward through the bed. Bubble size, bed voidage, and bed expansion all increase with gas velocity, but the bed remains relatively dense with a model-predicted time-averaged solids fraction of 0.563, 0.509, and 0.466 for gas velocity of 0.06, 0.10, and 0.15 m/s, respectively. Model-predicted time-averaged pressure drop is 5.65, 5.44, and 5.43 kPa for the gas velocity of 0.06, 0.10, and 0.15 m/s, respectively. When normalized to the time-averaged expanded bed height (calculated from the initial bed height, initial solids fraction, and time-averaged solids fraction), the resulting pressure drop is 18.2, 15.8, and 14.4 kPa/m, respectively, which agree reasonably well with the 16.5 kPa/m pressure drop predicted by Eq. (9) and shown in Fig. 5 at an assumed bed voidage at minimum fluidization of ɛmf = 0.5. At the lowest fluid velocity, the comparatively smaller bubbles tend to move upward in an orderly and regular fashion. Increasing the gas inflow velocity produced progressively more chaotic and random flow patterns and increased lateral bubble movement between the tubes.

Fig. 8
Solids volume fraction contours at t = 1.5 s for (a) uf = 0.06 m/s and (b) uf = 0.10 m/s
Fig. 8
Solids volume fraction contours at t = 1.5 s for (a) uf = 0.06 m/s and (b) uf = 0.10 m/s
Close modal

Instantaneous local heat transfer rates are characterized by rapid variations attributed to the passage of bubbles, followed by subsequent restoration of contact between the tube wall and newly refreshed solids. Local minima arise when a bubble passes along the tube surface, and correspondingly, the solids fraction is exceedingly low in the immediate vicinity of the tube. Sharp peaks correspond to initial solid contact immediately following bubble passage and result from high conductive heat transfer rates upon initial contact between the refreshed hot solids and comparatively cooler tube walls. Figure 9 illustrates a representative example of the instantaneous convective heat transfer coefficient as a function of gas velocity for the left-side tube in the bottom row at an angle of 135 deg relative to the top of the tube. Computed heat transfer rates with a low, near-minimum fluidization velocity (0.06 m/s) exhibit a more regular, repeating time sequence of peak characteristic of the orderly vertical bubble movement around the tubes visible in Fig. 8. Comparatively higher velocity conditions induce larger bubble sizes and more random, chaotic hydrodynamic characteristics, resulting in both sharper peaks and longer duration minima in local heat transfer.

Fig. 9
Instantaneous local convective heat transfer coefficients for the bottom tube at θ = 135 deg (relative to the top of the tube). Base-case parameters and particle in Table 2 were used in the computation.
Fig. 9
Instantaneous local convective heat transfer coefficients for the bottom tube at θ = 135 deg (relative to the top of the tube). Base-case parameters and particle in Table 2 were used in the computation.
Close modal

Figure 10 compares the overall time-average, tube-average model-predicted heat transfer coefficients from each of the constitutive models for effective thermal conductivity against predictions from empirical correlations [3943] and packet-renewal models [44,45] for each high-temperature and near-ambient temperature conditions. Model-predicted heat transfer coefficients display a slight local maximum in heat transfer rates at about twice the minimum fluidization velocity followed by a slow decrease with the gas velocity. This behavior is commonly encountered in experimental systems, but rarely replicated in empirical heat transfer correlations [10]. The slight local maximum arises from the competing effects of an increase in both solids renewal rates at the tube surface and FB bubble fraction with an increase in velocity, and it can be captured within mechanistically based packet-renewal models (correlations shown in the studies by Kim et al. and Masoumifard et al. in Fig. 10). Eulerian model-predicted heat transfer coefficients that employ the effective thermal conductivity constitutive model of Kuipers et al. [30] exceed those predicted by any of the correlations or mechanistic models by a substantial margin despite the wide range, variable trends, and general uncertainty in empirical correlations. In contrast, model-predicted heat transfer rates that employ the near-wall modifications suggested by either Patil et al. [31] or Legawiec and Ziólkowski [36] produce convective/conductive heat transfer coefficients that fall well within the range expected from the preponderance of the empirical expressions.

Fig. 10
Model-predicted, time-average, tube-average heat transfer coefficients for all tubes for either (a) Twall = 625 °C, Tinitial = 675 °C, and comparison with empirical correlations at T = 650 °C and (b) Twall = 25 °C, Tinitial = 75 °C, and comparison with empirical correlations at T = 50 °C
Fig. 10
Model-predicted, time-average, tube-average heat transfer coefficients for all tubes for either (a) Twall = 625 °C, Tinitial = 675 °C, and comparison with empirical correlations at T = 650 °C and (b) Twall = 25 °C, Tinitial = 75 °C, and comparison with empirical correlations at T = 50 °C
Close modal

The Eulerian model predictions suggest that using near-minimum fluidization conditions is tenable from a heat transfer standpoint; thus, they support the selection of low-velocity operating conditions, which can reduce both the parasitic power requirement and heat loss from the fluidizing gas. The CFD model predicts a significant increase in heat transfer rates under high-temperature conditions when compared with low-temperature conditions. This increase originates purely from nonradiative heat transfer mechanisms because the simulation explicitly excludes contributions from radiative heat transfer. The improved heat transfer performance at high-temperature conditions results from an increase in gas-phase thermal conductivity with temperature because the controlling heat transfer resistance is commonly conduction through a gas film surrounding the solids.

4.3 Parametric Study of Fluidized-Bed Heat Exchanger Design.

The 1D model simulates the overall HX and provides a tool for sensitivity analysis on tube size, number, and operating conditions to assess the effects of design parameters, heat transfer coefficients, and overall conductance (UA) on the specific cost. All model input parameters are held at the values presented in Table 2 unless otherwise specified.

Figure 11 shows the average sCO2-side heat transfer coefficient as a function of the tube diameter and the tube number for a sCO2 pressure of 20 MPa. The computed heat transfer coefficients for the sCO2 flow range from 1000 to 5000 W/m2/K depending on the CO2 velocity. A small number of tubes and a small diameter increase the flow velocity and can achieve a high Reynolds number, thus obtaining high heat transfer coefficients. The 0.0127-m OD tube increases the heat transfer coefficient by nearly a factor of three when compared to a 0.01905-m OD tube; however, this heat transfer comes at an expense of higher pressure drop. More moderate tube sizes (e.g., the base-case sizing of 0.0254 m) produce reasonably high heat transfer (1000 W/m2/K) compared to the particle side and low enough flow velocity for a low pressure drop in the 100 kWth HX design.

Fig. 11
The average heat transfer coefficient on the sCO2 side versus the number of tubes and tube outer diameter. The sCO2 flow is at a pressure of 20 MPa, temperature from 550 °C inlet to 700 °C outlet.
Fig. 11
The average heat transfer coefficient on the sCO2 side versus the number of tubes and tube outer diameter. The sCO2 flow is at a pressure of 20 MPa, temperature from 550 °C inlet to 700 °C outlet.
Close modal

4.4 Performance and Cost Analysis of the Particle/sCO2 Heat Exchanger Design Tradeoff.

The tube size, area, and number of tubes determine the amount of materials required to make HX heat transfer surfaces and directly correlate with the HX cost. High heat transfer coefficients on both the particle and sCO2 sides are key to minimize heat transfer surface area and reduce the HX cost. Tradeoffs between sizing and number of flow tubes dictate material use and can be converted into the specific HX cost.

Figure 12 shows interrelationships between the tube OD, tube number, and tube length required to achieve the target inlet/outlet temperatures. The working conditions are met with the specified flowrate and inlet/outlet temperatures; thus, an increase in the thermal capacity of HX can be done by simply multiplying the unit number relative to this 100-kWth simulated case. Figure 12 indicates that for less than ten tubes, three OD cases—Do = 0.0127 m, 0.01905 m, and 0.0254 m—have nearly similar tube length. This is primarily because of high heat transfer coefficients on both the solids and sCO2 side with small tube diameter. The particle-side heat transfer coefficient used in Fig. 12 was computed using the Vreedenberg correlation (Eq. 12) at 1.5 times the minimum fluidization velocity and the radiative contribution in Eq. 14 as specified in the base-case parameters in Table 2.

Fig. 12
Tube length versus tube number with different tube sizes to guide geometry tradeoffs of a 100 kW HX. The calculation used the Vreedenberg correlation (Eq. (12)) at 1.5 times the minimum fluidization velocity and the radiative contribution (Eq. (14)) as specified in the base-case parameters in Table 2.
Fig. 12
Tube length versus tube number with different tube sizes to guide geometry tradeoffs of a 100 kW HX. The calculation used the Vreedenberg correlation (Eq. (12)) at 1.5 times the minimum fluidization velocity and the radiative contribution (Eq. (14)) as specified in the base-case parameters in Table 2.
Close modal

Figure 13 shows the relation between the specific cost (cost/UA) with respect to the heat transfer coefficients for each sCO2 and particle sides of the HX for the counterflow configuration in Fig. 3 and the base-case conditions. The calculations shown in Fig. 13 apply constant, specified heat transfer coefficients to each side of the HX to demonstrate parametric sensitivities. This differs from the calculations shown in Fig. 12, which apply local heat transfer coefficients based on the empirical correlations shown in Eqs. (7) and (13)(14). Note that the cost considered here represents only material usage based on the calculated heat transfer surface area. From Fig. 13, a solid-side heat transfer coefficient of at least 400 W/m2/K is essential to achieve the 5 $/(W/K) cost target assuming a material cost of 75 $/kg [46]. For a fluidized bed with 1000–3000 W/m2/K heat transfer coefficient on the sCO2 side, the specific cost is well below 10 $/(W/K) for heat transfer material only.

Fig. 13
Sensitivity study of the heat exchanger specific cost versus heat transfer coefficients (based on heat transfer surface material cost only)
Fig. 13
Sensitivity study of the heat exchanger specific cost versus heat transfer coefficients (based on heat transfer surface material cost only)
Close modal

This HX cost analysis considers only the size and material usage for the heat transfer surfaces, which are sensitive to the heat transfer processes. The total HX cost must also include manufacturing, manifolds, connection ducts, and packaging. However, reducing material usage for the heat transfer surfaces is the foremost step in HX cost reduction and indicates initial favorability of a FB HX design.

5 Conclusions

This article describes an initial design analysis for a fluidized-bed heat exchanger with the goals of reducing material usage under high-temperature/high-pressure conditions and promoting efficient operation by minimizing parasitic power requirements and heat loss. The model provides tradeoffs among tube diameter, length, and number. Fluidized-bed operation was characterized to study the particle fluidization conditions for high heat transfer and low parasitic power (low gas flow). We conducted a sensitivity analysis for the effects of critical design parameters (such as the heat transfer coefficient) on heat exchanger cost and provide design directions to minimize the heat transfer area for lower material cost.

The development of a high-temperature particle/sCO2 heat exchanger for a high-efficiency CSP system focused on the cost and performance of a fluidized-bed design. The heat transfer resistance is primarily on the low heat transfer coefficient side between solid particles and embedded tubes. Fluidized beds usually can obtain a high overall heat transfer coefficient, mainly due to the particle mixing and periodic contacts with the heat transfer surfaces. We developed simulation models based on a counterflow heat exchanger configuration that suggest that achieving a solid-side heat transfer coefficient ≥500 W/m2/K is critical to meeting the heat exchanger cost target. Literature data, empirical correlations, and computational fluid dynamics simulations all suggest that the fluidized-bed heat exchanger can obtain a >500 W/m2/K overall heat transfer coefficient at high temperature, even at low fluidizing gas flow rates. Enhancing particle heat transfer can reduce the heat transfer surface area and cost directly, and it is important for economic heat exchanger design and performance.

Acknowledgment

This work was authored by the National Renewable Energy Laboratory (NREL), operated by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. The work was funded by the DOE SuNLaMP program and Sandia National Laboratories (SNL). This article was prepared with the funding support from the DOE Advanced Research Projects Agency–Energy (ARPA-E) DAYS program and Solar Energy Technology Office (SETO) Generation 3 CSP. For discussion and project guidance, we thank Dr. Cliff Ho of SNL, Dr. Rajgopal Vijaykumar of DOE, and Professor Sheldon Jeter of Georgia Institute of Technology. We thank Mr. Bartev Sakadjian and Mr. Thomas Flynn of Babcock & Wilcox Company for technical inputs. The authors thank Mr. Don Gwinner and Mr. Mark Mehos for reviewing the paper and Mr. Patrick Davenport for the graphic work.

The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government or any agency thereof. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights.

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The authors attest that all data for this study are included in the paper. Data provided by a third party listed in Acknowledgment.

Nomenclature

     
  • e =

    emissivity

  •  
  • f =

    friction factor

  •  
  • g =

    gravitational acceleration (m2/s)

  •  
  • h =

    heat transfer coefficient (W/m2/K)

  •  
  • k =

    thermal conductivity (W/m/K)

  •  
  • p =

    pressure (Pa)

  •  
  • q =

    heat flux (W/m2)

  •  
  • u =

    gas velocity (m/s)

  •  
  • v =

    gas volumetric flow rate (m3/s)

  •  
  • z =

    tube axial coordinate (m)

  •  
  • γ =

    polytropic exponent

  •  
  • D =

    tube diameter (m)

  •  
  • H =

    bed height (m)

  •  
  • T =

    temperature (°C or K)

  •  
  • dp =

    particle diameter (m)

  •  
  • m˙CO2,tube =

    CO2 mass flow per tube (kg/s)

  •  
  • m˙p =

    solids mass flow rate (kg/s)

  •  
  • umf =

    gas velocity at minimum fluidization (m/s)

  •  
  • ws,comp =

    compressor power (W)

  •  
  • ws,ideal =

    ideal (isentropic) compressor power (W)

  •  
  • Ntubes =

    total number of tubes

  •  
  • Ar =

    Archimedes number, Ar = g(φdp)3ρg(ρpρg)/μ2

  •  
  • Nu =

    Nusslet number

  •  
  • Pr =

    Prandtl number

  •  
  • Re =

    Reynolds number

  •  
  • Cp =

    heat capacity (J/kg/K)

  •  
  • Δp =

    pressure drop (Pa)

  •  
  • ɛ =

    bed voidage

  •  
  • ɛw =

    wall roughness (m)

  •  
  • ɛmf =

    bed voidage at minimum fluidization

  •  
  • ηcomp =

    compressor efficiency

  •  
  • μ =

    fluid viscosity (kg/m/s)

  •  
  • ρ =

    density (kg/m3)

  •  
  • φ =

    particle sphericity

Subscript and Superscripts

     
  • c =

    convective

  •  
  • g =

    gas phase

  •  
  • i =

    inner

  •  
  • in =

    inlet

  •  
  • o =

    outer

  •  
  • out =

    outlet

  •  
  • p =

    particles

  •  
  • r =

    radiative

  •  
  • t =

    tube

  •  
  • w =

    wall

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