Modeling of Supercritical CO2 Shell-and-Tube Heat Exchangers Under Extreme Conditions: Part II: Heat Exchanger Model

Krishna, Akshay Bharadwaj; Jin, Kaiyuan; Ayyaswamy, Portonovo S.; Catton, Ivan; Fisher, Timothy S.

doi:10.1115/1.4053511

Abstract

Heat exchangers play a critical role in supercritical CO₂ Brayton cycles by providing necessary waste heat recovery. Supercritical CO₂ thermal cycles potentially achieve higher energy density and thermal efficiency operating at elevated temperatures and pressures. Accurate and computationally efficient estimation of heat exchanger performance metrics at these conditions is important for the design and optimization of sCO₂ systems and thermal cycles. In this paper (Part II), a computationally efficient and accurate numerical model is developed to predict the performance of shell-and-tube heat exchangers (STHXs). Highly accurate correlations reported in Part I of this study are utilized to improve the accuracy of performance predictions, and the concept of volume averaging is used to abstract the geometry and reduce computation time. The numerical model is validated by comparison with computational fluid dynamics (CFD) simulations and provides high accuracy and significantly lower computation time compared to existing numerical models. A preliminary optimization study is conducted and the advantage of using supercritical CO₂ as a working fluid for energy systems is demonstrated.

Introduction

Heat exchangers are devices that facilitate the transfer of thermal energy between two fluids without having to be mixed and are critical components of a variety of energy systems for providing efficient energy management and significant heat recovery [1–3]. Heat exchangers provide efficient heating/cooling in thermal cycles and facilitate in the enhancement of thermal efficiency of a cycle. Using supercritical CO₂ (sCO₂) at extreme temperatures and pressures, thermal systems can achieve unprecedentedly high energy conversion efficiency, lower carbon footprint, and low capital cost [4–8]. Good chemical stability, high density at high temperatures, and a low critical point (7 MPa and 300 K) make sCO₂ a more attractive candidate fluid for operation in energy systems than other supercritical fluids [9–11]. By using sCO₂ as the working fluid, heat exchangers are highly compact and operate under high temperatures and pressures [12,13]. The design and fabrication process of sCO₂ heat exchangers is thus complex and requires further investigation for accurate prediction of performance [14–17].

Among the various heat exchanger configurations, shell-and-tube heat exchangers (STHXs) are exceedingly popular owing to their ability to operate at elevated temperatures and pressures, their structural simplicity, design adaptability, and ease of maintenance [18–21]. These advantages make STHXs a highly manufactured component, as over 35% of all heat exchangers use the shell-and-tube configuration [22]. A fluid flow (tube-side flow) runs inside the tubes and heats or cools the second flow (shell-side flow) that runs between tube outer surfaces and the shell. Heat transfer is enhanced by having multiple passes on the shell side. This enhancement is usually achieved by placing baffles in the fluid stream on the shell side to guide the flow.

Thermohydraulic performance predictions of the heat exchangers can be achieved by experimental testing. However, accurate measurement of thermohydraulic performance is extremely complicated, and experimental testing is neither cost nor time efficient [23]. Numerical modeling of the heat exchangers has become an accurate and time-efficient method to obtain detailed temperature, velocity, and pressure fields in a given design. Modeling the entire STHX by representing the tubes in detail using computational fluid dynamics (CFD) is computationally expensive. The concept of distributed resistance has been introduced, and early work was carried out, by Patankar and Spalding [24], Butterworth [25], Sha [26], and Sha et al. [27]. Using this approach, tube bundle details are eliminated, and the model is simplified based on volumetric porosity and surface permeability. Although this approach significantly reduces computation time and provides accurate solutions, hours are typically required to obtain the solution.

Correlation-based modeling is an approach used to design and predict thermohydraulic performance of heat exchangers. Correlation-based approaches are computationally efficient but require highly accurate correlations for thermal and hydraulic performance in order to accurately predict heat exchanger performance. The Kern method [28] and the Bell-Delaware method [29] are commonly used correlation-based approaches. The Kern method is only suitable for initial sizing of the STHX, as this method provides a conservative estimate of performance. The Bell-Delaware method provides an accurate estimation of the shell-side performance but does not have the capability to provide a detailed description of temperature and flow fields within an STHX.

Advances in the modeling of multiphysics transport through heterogeneous media with volume averaging theory (VAT) [30] have allowed engineers to simulate flow and heat transfer in thermal devices in mere seconds on a laptop, in comparison to the many hours (or longer) required for CFD. Volume averaging is a method of abstracting the geometry for computation of complex problems and follows a porous medium approach where the volume occupied by the fluids and the solid are represented in terms of their volume fractions. The flow and temperature fields are described nonlocally, and the topology of the problem can be embedded into the governing equations to allow for the complete treatment of conjugate effects [31–33]. Most solutions of the governing equations have been obtained through numerical methods such as finite element and finite difference schemes [34,35]. Closure must be obtained theoretically, numerically, or experimentally for the transport equations in order to account completely for geometric complexities [36,37]. In Part I of this study, closure was demonstrated, and correlations for friction and Colburn factor were developed. By closing the transport equations, simulations can be executed in express time, and this advancement enables accurate and efficient performance prediction of thermal devices [38], as well as optimization routines that span broad parameter spaces efficiently.

Previous studies provide modeling approaches that are either less accurate in performance predictions or resource-intensive and time-consuming. A simultaneously efficient and accurate methodology is essential to perform feasible multiparameter optimization studies. The objective of this study is to develop and demonstrate a computationally efficient and accurate numerical model for performance predictions of STHXs. We utilize highly accurate correlations for thermohydraulic performance estimation developed in Part I of this study [39] and abstract the geometry of the STHX using the concept of volume averaging. Our approach combines the advantages of correlation-based numerical modeling and the distributed resistance porous media approach to provide a highly accurate and time-efficient modeling methodology for STHXs.

Methodology

In this section, the methodology used to define the STHX geometry and governing equations for the model development are introduced. The model algorithm is reported in the latter half of this section.

The STHX modeled in this study is shown in Fig. 1.

Fig. 1

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Isometric view of the STHX

Definition of Geometric Parameters.

To perform temperature and flow field calculations using volume averaging, a unit cell is defined as shown in Fig. 2. The temperature and flow fields through the STHX can be obtained by serially repeating the unit cell over the volume of the STHX and solving for each unit cell individually.

Fig. 2

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Front view (left) and side view (right) of a unit cell of the STHX

A schematic of flow over the various tube banks and definitions of various geometric parameters associated with the tubes, fins, and the tube bundle are provided in Fig. 3.

Fig. 3

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Schematic of the staggered tube bank (left). Side view of the selected volume depicting the cross-sectional and heat transfer areas (top right). Geometric parameters (bottom right) associated with (a) bare, (b) disk-finned, and (c) cylindrical pin-finned tube banks.

Abstraction of the unit cell geometry is carried out by defining three key parameters – volume fraction,

ϕ

⁠, specific wetted surface area,

S_{w}

⁠, and VAT hydraulic diameter,

D_{VAT}

⁠. These parameters are given by

ϕ_{i} = \frac{Volume occupied by component i in the unit cell}{Total volume of the unit cell}

(1)

S_{w_{j}} = \frac{Tube area wetted by fluid j + Fin area wetted by fluid j}{Total volume of the unit cell}

(2)

D_{VA T_{j}} = \frac{4 ϕ_{j}}{S_{w_{j}}}

(3)

where component $i$ is the internal fluid stream, external fluid stream, or the solid material, and fluid $j$ is the internal or external fluid stream.

As explained in Part I of this study, Reynolds number is a measure of the ratio between inertial and viscous forces in the flow, as well as its turbulent character. In order to compute the thermal and hydraulic performance for flow over bundles, another form of hydraulic diameter is introduced as a length scale. This hydraulic diameter is a function of the minimum free-flow area and the total area of flow and is defined as

D_{h} = \frac{2 A_{c} P_{l}}{A_{H T}}

(4)

where

A_{c}

is the cross-sectional free flow area between two tubes,

P_{l}

is the longitudinal pitch, and

A_{H T}

is the heat transfer area between two tubes as shown in Fig. 3. Reynolds number is calculated as

R e_{D_{h}} = \frac{u_{\max} D_{h}}{ν}

(5)

R e_{VAT} = \frac{u_{avg} D_{VAT}}{ν}

(6)

where $u_{\max}$ is the maximum velocity of flow located at the minimum free flow area, $A_{\min}$ ⁠, and $u_{avg}$ is the average velocity located at the frontal area, $A_{f r}$ ⁠.

Modified specific wetted surface area,

S_{w m}

⁠, is defined using the concept of fin efficiency in order to account for the actual wetted area for heat transfer. The parameter is given by

S_{w m_{j}} = \frac{Tube area wetted by fluid j + (Fin area wetted by fluid j \times η_{f})}{Total volume of the unit cell}

(7)

where fluid

j

is the internal or external fluid stream and

η_{f}

is the fin efficiency defined below:

η_{f} = \frac{q_{f}}{q_{f, \max}}

(8)

Disk fins and cylindrical pin-fins are the fin geometries under consideration. The efficiencies of these types of fins are given by [40]

Disk fins

η_{D F} = C_{1} \frac{K_{1} (m r_{o t}) I_{1} (m r_{f c}) - I_{1} (m r_{o t}) K_{1} (m r_{f c})}{I_{0} (m r_{o t}) K_{1} (m r_{f c}) + K_{0} (m r_{o t}) I_{1} (m r_{f c})}

(9)

where $C_{1} = \frac{2 r_{o t}}{m (r_{f c}^{2} - r_{o t}^{2})}$ ⁠, $r_{f c} = (D_{f} / 2) + (δ_{f} / 2)$ ⁠, $r_{o t} = D_{o t} / 2$ ⁠, and $m = \sqrt{\frac{2 h_{1}}{κ_{s} δ_{f}}}$ .

Cylindrical pin-fins

η_{CPF} = \frac{\tanh (m L_{c})}{m L_{c}}

(10)

where $L_{c} = (\frac{D_{f} - D_{o t}}{2}) + (\frac{δ_{f}}{4})$ and $m = \sqrt{\frac{4 h_{1}}{κ_{s} δ_{f}}}$ . The number of tubes in the bundle is high, and they are packed closely together. The high density of solid phase in the bundle causes majority of the flow resistance and heat transfer to occur in the bundle of the tubes and not due to the presence of the baffles. Therefore, we neglect the effect of the baffles in the evaluation of the external flow characteristics. By making use of the unit cell concept, the volume fraction, specific wetted surface area, and the modified specific wetted surface area for the external flow stream of the various tube bundle configurations are defined as:

Bare tube bundle

ϕ_{1, B T} = 1 - \frac{N_{t} π D_{o t}^{2} B}{4 V_{T}}

(11)

{S_{w}}_{1, B T} = \frac{N_{t} π D_{o t} B}{V_{T}}

(12)

{S_{w m}}_{1, B T} = {S_{w}}_{1, B T}

(13)

Disk-finned tube bundle

ϕ_{1, D F} = 1 - \frac{N_{t}}{4 V_{T}} [π D_{o t}^{2} B + N_{f} π (D_{f}^{2} - D_{o t}^{2}) δ_{f}]

(14)

{S_{w}}_{1, D F} = \frac{N_{t} N_{f}}{2 V_{T}} [2 π D_{o t} (P_{f} - δ_{f}) + 2 π D_{f} δ_{f} + π (D_{f}^{2} - D_{o t}^{2})]

(15)

{S_{w m}}_{1, D F} = \frac{N_{t} N_{f}}{2 V_{T}} [2 π D_{o t} (P_{f} - δ_{f}) + 2 π D_{f} δ_{f} η_{f} + π (D_{f}^{2} - D_{o t}^{2}) η_{f}]

(16)

Cylindrical pin-finned tube bundle

ϕ_{1, CPF} = 1 - \frac{N_{t}}{8 V_{T}} [2 π D_{o t}^{2} B + N_{f} N_{a f} π δ_{f}^{2} (D_{f} - D_{o t})]

(17)

{S_{w}}_{1, CPF} = \frac{N_{t} N_{f}}{2 V_{T}} [2 π D_{o t} P_{f} + N_{a f} π δ_{f} (D_{f} - D_{o t})]

(18)

{S_{w m}}_{1, CPF} = \frac{N_{t} N_{f}}{2 V_{T}} [2 π D_{o t} P_{f} + N_{a f} π δ_{f} (D_{f} - D_{o t}) η_{f}]

(19)

All geometric parameters in the equations above are shown in Figs. 2 and 3. Because no internal inserts/augmentations exist, the internal flow stream parameters are the same for all tube bundle configurations under consideration. These parameters are given by

ϕ_{2} = \frac{N_{t} π D_{i t}^{2} B}{4 V_{T}}

(20)

{S_{w}}_{2} = \frac{N_{t} π D_{i t} B}{V_{T}}

(21)

{S_{w m}}_{2} = {S_{w}}_{2}

(22)

Total volume of the unit cell

V_{T} = \frac{{tan}^{- 1} (\frac{L_{y}}{2 L_{h}}) π D_{i s}^{2} B}{720} + \frac{L_{h} L_{y} B}{2} - t_{s} L_{y} B

(23)

The volume fraction of the solid is defined as a function of the volume fraction of the fluid streams.

ϕ_{s} = 1 - (ϕ_{1} + ϕ_{2})

(24)

Governing Equations.

The following assumptions are made for the derivation of the governing equations:

The working fluid is steady, incompressible, and Newtonian.
Conduction in the fluid is negligible since the flow has high Peclet number.
Solid is discontinuous in the y-direction, and the thermal conductivity of the solid material (Haynes282) is approximately 200 $\times$ higher than that of the working fluid (CO₂). Therefore, solid conduction is neglected in the y-direction.

The resulting equations for continuity and momentum for both fluid streams are given below.

Continuity equation

ϕ_{j} (\frac{\partial \bar{u_{j}}}{\partial x} + \frac{\partial \bar{v_{j}}}{\partial y} + \frac{\partial \bar{w_{j}}}{\partial z}) = 0

(25)

where $j = 1$ for external flow and $j = 2$ for internal flow.

External flow momentum equation

ϕ_{1} \frac{\partial \bar{P}}{\partial y} = \frac{1}{2} ρ_{1} f_{1} {S_{w}}_{1} \bar{v_{1}^{2}}

(26)

Internal flow momentum equation

ϕ_{2} \frac{\partial \bar{P}}{\partial x} = \frac{1}{2} ρ_{2} f_{2} {S_{w}}_{2} \bar{u_{2}^{2}}

(27)

where $f_{1}$ and $f_{2}$ and the friction factors for the external and internal fluid streams, respectively, and $v$ and $u$ are the velocities in the y- and x-directions as shown in Fig. 2.

The energy equations for external and internal fluid streams, and the solid phase are obtained by performing an energy balance on the control volume in Fig. 4 and are given by

Fig. 4

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Control volume used to derive the energy equations

Fluid-phase energy equation (external flow)

{\bar{(ρ C_{p})}}_{1} ϕ_{1} \bar{v_{1}} \frac{\partial T_{1}}{\partial y} = h_{1} S_{w m 1} ({\bar{T}}_{s o} - {\bar{T}}_{1})

(28)

Fluid-phase energy equation (internal flow)

{\bar{(ρ C_{p})}}_{2} ϕ_{2} \bar{u_{2}} \frac{\partial T_{2}}{\partial x} = h_{2} {S_{w m}}_{2} ({\bar{T}}_{s i} - {\bar{T}}_{2})

(29)

Solid-phase energy equation

ϕ_{s} \bar{κ_{s}} \frac{\partial^{2} \bar{T_{s}}}{\partial x^{2}} = h_{1} {S_{w m}}_{1} ({\bar{T}}_{s o} - {\bar{T}}_{1}) + h_{2} {S_{w m}}_{2} ({\bar{T}}_{s i} - {\bar{T}}_{2})

(30)

where

h_{1}

and

h_{2}

are the heat transfer coefficients of the external and internal fluid streams, respectively. Bulk solid temperature,

\bar{T_{s}}

⁠, is defined by means of a resistance network as shown in Fig. 5 and is given by

{\bar{T}}_{s} = \frac{\ln (r / r_{i}) {\bar{T}}_{s o} + \ln (r_{o} / r) {\bar{T}}_{s i}}{\ln (r_{o} r_{i} / r^{2})}

(31)

where $r = \frac{r_{i} + r_{o}}{2}$ ⁠.

Fig. 5

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Cross-section of a half-tube (left) and resistance network to define the tube wall temperatures (right)

Flow Correlations and Correction Factors.

Closure is required to solve for the governing equations. Correlations for the friction factor and heat transfer coefficient need to be obtained for both the external and internal flow streams.

External Flow.

Part I of this study reports the development of Colburn and friction factor correlations for the external flow with various tube bundle configurations. The unified form of the correlations applicable to bare, disk-finned, and cylindrical pin-finned tube banks is given below.

Colburn factor

j_{H_{1, VAT}} = 0.47 {(\frac{D_{h}}{D_{E}})}^{0.53} {(\frac{P_{t}}{D_{o t}})}^{- 0.21} {(\frac{P_{l}}{D_{o t}})}^{- 0.19} {(\frac{D_{f}}{D_{o t}})}^{0.12} {(\frac{P_{f} - δ_{f}}{P_{f}})}^{- 0.38} R e_{D_{h_{1}}}^{- 0.40}

(32)

Friction factor

f_{1, VAT} = 4 \times 0.54 {(\frac{D_{h}}{D_{E}})}^{0.62} {(\frac{P_{t}}{D_{o t}})}^{0.40} {(\frac{P_{l}}{D_{o t}})}^{- 0.20} {(\frac{D_{f}}{D_{o t}})}^{- 0.45} {(\frac{P_{f} - δ_{f}}{P_{f}})}^{- 0.23} R e_{D_{h_{1}}}^{- 0.23} {(\frac{D_{VAT}}{{D_{h}}_{1}})}^{3}

(33)

Heat transfer coefficient is obtained using the following relation:

h_{1} = {j_{H}}_{1} R e_{D_{h_{1}}} \Pr^{1 / 3} \frac{κ_{1}}{D_{h}}

(34)

The total pressure drop in each unit cell for external flow is a summation of the pressure drop in the tube bundle and the pressure drop in the turning region

Δ P_{1} = Δ P_{1, tbc} + Δ P_{1, turn}

(35)

Pressure drop in the tube bundle is defined as

Δ P_{1, t b} = \frac{1}{2} ρ_{1} f_{1, VAT} \bar{v_{1}^{2}} \frac{L_{y}}{D_{VA T_{1}}}

(36)

Pressure drop in the turning region is calculated using the correlation developed by Bell [29]

Δ P_{1, turn} = \frac{{\dot{m}}_{1}^{2}}{ρ_{1} N_{t c} A_{c} A_{w}} ζ_{l}

(37)

where ${\dot{m}}_{1}$ is the external fluid mass flow rate, $ζ_{l}$ is the correction for baffle leakage effects, and $A_{w}$ is the cross section area of the window region as shown in Fig. 2.

The Colburn and friction factor correlations assume ideal perpendicular flow over the tube bundle. In an STHX, the external flow will experience bypass, leakage, and other secondary effects that influence the heat transfer coefficient and pressure drop. We employ correction factors developed by Bell [29] to account for these effects. The modified equations take the form:

h_{1 c} = h_{1} J_{c} J_{l} J_{b} J_{s}

(38)

where $J_{c}$ is the correction for baffle configuration, $J_{l}$ is the correction for baffle leakage effects, $J_{b}$ is the correction for bundle bypass effects, and $J_{s}$ is the correction for larger baffle spacing at the inlet and outlet unit cells.

The corrected external flow pressure drop is

{Δ P}_{1, tbc} = {Δ P}_{1, t b} ζ_{l} ζ_{b} ζ_{s}

(39)

where $ζ_{l}$ is the correction for baffle leakage effects, $ζ_{b}$ is the correction for bundle bypass effects, and $ζ_{s}$ is the correction for larger baffle spacing at the inlet and outlet unit cells.

Internal Flow.

In the present study, U-tubes are used in the STHX. Estimation of the heat transfer coefficient for internal flow uses the Gnielinski correlation [41] (valid for

3000 \leq R e_{D_{it}} \leq 5 \times 10^{6}

and

0.5 \leq \Pr \leq 2000

⁠).

h_{2} = \frac{(f_{2} / 8) ({Re}_{D_{i t}} - 1000) P r}{1 + 12.7 {(f_{2} / 8)}^{0.5} (\Pr^{0.67} - 1)} \frac{κ_{2}}{D_{i t}}

(40)

The internal flow friction factor is obtained using Petukhov formula [42] (valid for

3000 \leq R e_{D_{it}} \leq 5 \times 10^{6}

⁠) and is given by

f_{2} = {(0.790 {\ln Re}_{D_{i t}} - 1.64)}^{- 2}

(41)

For flow in the laminar regime, the Nusselt number is

{Nu}_{2} = 4.36

⁠, and the friction factor is

f_{2} = 64 / {Re}_{D_{h_{2}}}

⁠. The total pressure drop for internal flow in the STHX is the summation of friction losses in the straight region, contraction and expansion losses as the flow enters and leaves the tubes, and pressure losses in the u-bends of the tube.

Δ P_{2} = Δ P_{2, c e} + Δ P_{2, u b} + \sum_{1}^{2 N} Δ P_{2, s t}

(42)

where N is the total number of unit cells. Internal pressure drop in the straight region,

Δ P_{2, s t}

⁠, due to friction losses in one unit cell is

Δ P_{2, s t} = \frac{1}{2} ρ_{2} f_{2} \bar{u_{2}^{2}} \frac{B}{D_{i t}}

(43)

Contraction and expansion losses,

Δ P_{2, c e}

⁠, are computed using correlations developed by Kays and London [43]

Δ P_{2, c e} = \frac{1}{2} (K_{c} + K_{e}) \frac{G_{2}^{2}}{\bar{ρ_{2}}}

(44)

where

K_{c}

and

K_{e}

are contraction and expansion coefficients, and

G_{2}

is the mass velocity of internal flow. Pressure losses in the u-bends,

Δ P_{2, u b}

⁠, are calculated using the following correlation [44]:

Δ P_{2, u b} = \frac{1}{2} ρ_{2} u_{2}^{2} (K_{b} + f_{2} \frac{π R_{b}}{D_{i t}})

(45)

where $K_{b}$ is the bend loss coefficient and $R_{b}$ is the bend radius.

The performance of a heat exchanger can be quantified by its effectiveness,

ϵ

⁠. The effectiveness of a heat exchanger is defined as the ratio of actual rate of heat transfer to the maximum possible heat transfer rate

ϵ = \frac{Actual heat transfer}{Maximum possible heat transfer}

(46)

Shell-and-Tube Heat Exchanger Numerical Model.

The indexing of the STHX is shown in Fig. 6. External fluid enters the STHX in unit cell “2N” and flows through to unit cell “N + 1” in a serpentine pattern. The fluid then turns to the second half of the STHX and follows a similar flow pattern until it reaches unit cell “1,” where it exits the heat exchanger. The internal fluid enters the first unit cell and flows straight through to unit cell “N.” The flow then encounters the U-bend and enters unit cell “N + 1” and flows through to unit cell “2N,” where it exits the heat exchanger.

Fig. 6

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Indexing of the unit cells of the STHX: (a) front section view and (b) top section view

The transport equations given above form the basis for the computation of the temperature and flow field in the STHX. The computation is carried out with the help of a numerical model developed using Julia programming language [45]. The computer code solves for the temperature fields, velocity fields, and pressure drop of both streams by using the Gauss-Seidel iterative method. The discretized forms of the governing energy equations are given below.

External fluid energy equation

\frac{{\bar{(ρ C_{p})}}_{1} ϕ_{1} \bar{u_{1}} ({\bar{T}}_{y + 1} - {\bar{T}}_{y})}{Δ y} = h_{1} {S_{w}}_{1} ({\bar{T}}_{s o} - {\bar{T}}_{1})

(47)

Internal fluid energy equation

\frac{{\bar{(ρ C_{p})}}_{2} ϕ_{2} \bar{u_{2}} ({\bar{T}}_{x + 1} - {\bar{T}}_{x})}{Δ x} = h_{2} {S_{w}}_{2} ({\bar{T}}_{s i} - {\bar{T}}_{2})

(48)

Solid-phase energy equation

\frac{\bar{κ_{s}} ({\bar{T}}_{x - 1} - 2 {\bar{T}}_{x} + {\bar{T}}_{x + 1})}{Δ x^{2}} ϕ_{s} = h_{1} {S_{w}}_{1} ({\bar{T}}_{s o} - {\bar{T}}_{1}) + h_{2} {S_{w}}_{2} ({\bar{T}}_{s i} - {\bar{T}}_{2})

(49)

The temperature and flow fields through the STHX are obtained by combining the unit cells serially. The temperature and velocity through each unit cell are solved completely before moving on to the next unit cell. The temperatures at the outlet on one unit cell are taken to be the inlet temperatures of the subsequent unit cell.

The algorithm for performance computation of the STHX is outlined in Fig. 7. With inlet temperatures of both streams as inputs, an initial effectiveness is assumed for the STHX, and the corresponding outlet temperature for the external flow stream is calculated. The temperature and pressure fields are then iteratively solved in each unit cell, and the predicted external inlet temperature is recorded. If the predicted external inlet temperature is lower than the user-inputted temperature, an updated lower value of effectiveness is assumed (and vice versa), and the process is repeated until a converged solution is obtained. The thermophysical properties of the fluid for both internal and external flow streams are updated during every iteration using CoolProp [46] based on the temperature and pressure at the location of the STHX. The thermal properties of the solid are obtained from the Haynes International website and calculated in the STHX code using a piecewise cubic Hermite interpolator [47]. The updating of effectiveness is carried out using a binary search algorithm to increase the efficiency of computation. The STHX model is very versatile and has the capability to handle effects of varying baffle cut, variable baffle spacing, and effect of sealing strips among other geometric and thermodynamic state inputs. Any fluid that is available in CoolProp's database can be chosen as a working fluid for the STHX [45]. The STHX numerical model can be used to predict the performance of STHXs whose tube bundle configurations lie within the applicability ranges of the correlations developed in Part I of this study [39].

Fig. 7

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Algorithm of the computer code for the computation of temperature field, pressure field, flow field and effectiveness of the STHX

Results and Discussion

All STHX numerical model simulations in this study were performed on a computer with an Intel^® Core i7-9750H processor at a clock rate of 2.60 GHz. All CFD simulations in this study were performed on a computer with an Intel^® Xeon^® Gold 6230 processor at a clock rate of 2.10 GHz.

Model Validation.

Computational fluid dynamics simulations were performed using ansysfluent 2021 R2 to validate thermohydraulic performance predictions from the STHX numerical model. The domain used and the boundary conditions applied in the CFD simulations are shown in Fig. 8. The mass flow rate at the external flow outlet of the unit cell is varied to achieve a range of external flow Reynolds numbers. The tube material used is Haynes282, and the sCO₂ is the working fluid for both tube-side and shell-side flows. The interfaces between the tubes and the working fluids are coupled, and a U.S. National Institute of Standards and Technology real gas model is used for thermophysical properties of the working fluid. The working fluid is assumed to be three-dimensional, steady-state, and turbulent. The k– $ω$ shear-stress transport (k– $ω$ SST) model with automatic wall function treatment is used to predict heat transfer and turbulent flow along the tube bundle and inside tubes. A pressure-based coupled algorithm is used, and the RMS-type residual criteria for solution convergence are set to $10^{- 5}$ for energy and momentum balances.

Fig. 8

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(a) Schematic of the domain and boundary conditions in the CFD model for STHX numerical model validation, (b) temperature contours, and (c) velocity pathlines of the $R e_{D_{h}} = 6000$ case obtained from ansysfluent.

The comparison metrics are the external heat transfer rate (HTR), internal HTR, tube bundle pressure drop, and unit cell external pressure drop. External and internal HTRs are the amount of energy lost and gained by the external and internal fluid streams, respectively. Unit cell and tube bundle pressure drops of the external flow are obtained from ansysfluent by taking the difference between the pressures at the inlet and outlet of a unit cell and tube bundle, respectively. These metrics were similarly calculated from the STHX numerical model and are utilized to perform a comparative analysis. A mesh refinement and grid independence study was performed on a case discussed later with $R e_{D_{h}} = 6000$ ⁠. The external HTR and unit cell external pressure drop of the flow is plotted as a function of the element size in Fig. 9. Based on the trend observed in the plot, the mesh with an element size of 0.35 mm was employed for all CFD simulations below.

Fig. 9

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Grid size refinement study: comparison of the (a) external heat transfer rate and (b) unit cell external pressure drop as a function of the mesh element size

To validate the STHX numerical model and demonstrate its applicability to sCO₂ processes at extreme temperatures and pressures, CFD simulations were carried out for a geometry with 39 tubes, $D_{o t} = 3.18 mm$ ⁠, $P_{t} = 8.41 mm$ ⁠, $P_{l} = 4.76 mm$ ⁠, and $B = 45.72 mm$ ⁠. By using sCO₂ as the working fluid for both streams, six simulations were performed by varying the external outlet mass flow rate, corresponding to a Reynolds number range from 2,000 to 12,000 for the external flow stream. Figure 10 provides a comparison of the external HTR, internal HTR, unit cell external pressure drop, and tube bundle pressure drop obtained from CFD and the STHX model; Table 1 summarizes the results. The average deviation between the STHX numerical model and CFD results for HTR and unit cell external pressure drop is less than 10% and the tube bundle pressure drop deviation is 6%, which demonstrates the accuracy of the developed STHX numerical model. Validation for flow over finned geometries would require a more advanced CFD study; therefore, the validation pertains to bare tube banks. The experimental data utilized for bare tube banks involve air flows at conditions similar to the flows over the finned tube banks. All data and correlations appear to translate well to finned tubes and therefore, the correlations and model are expected to exhibit high accuracy even for flows over finned tube banks.

Fig. 10

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Comparison of (a) external heat transfer rate, (b) internal heat transfer rate, (c) unit cell external pressure drop, and (d) tube bundle pressure drop between CFD simulations and the STHX numerical model

Table 1

Comparison of performance metrics obtained from the STHX numerical model and CFD simulations

	STHX numerical model						CFD
	$Δ P_{1}$ (Pa)		External		Internal		$Δ P_{1}$ (Pa)		External		Internal
$R e_{D_{h}}$	Unit cell	Tube bundle	HTR (kW)	$Δ T$ (⁠ $° C)$	HTR (kW)	$Δ T (° C)$	Unit cell	Tube bundle	HTR (kW)	$Δ T (° C)$	HTR (kW)	$Δ T (° C)$
2000	34.6	27.4	2.56	196.0	2.54	25.1	34.9	24.7	2.38	182.5	2.36	23.3
4000	116.6	89.5	3.62	144.8	3.60	35.6	128.3	83.2	3.19	127.5	3.16	31.3
6000	266.0	199.9	4.44	114.1	4.42	43.7	289.1	194.7	4.19	107.5	4.15	41.4
8000	470.6	348.2	5.03	95.3	5.01	49.6	534.2	360.3	4.59	85.9	4.53	44.3
10,000	715.5	523.0	5.47	82.9	5.45	54.0	845.1	550.9	5.04	76.2	5.01	50.1
12,000	1008.2	729.6	5.85	73.7	5.83	57.7	1178.0	785.6	5.36	67.3	5.29	52.1

	STHX numerical model						CFD
	$Δ P_{1}$ (Pa)		External		Internal		$Δ P_{1}$ (Pa)		External		Internal
$R e_{D_{h}}$	Unit cell	Tube bundle	HTR (kW)	$Δ T$ (⁠ $° C)$	HTR (kW)	$Δ T (° C)$	Unit cell	Tube bundle	HTR (kW)	$Δ T (° C)$	HTR (kW)	$Δ T (° C)$
2000	34.6	27.4	2.56	196.0	2.54	25.1	34.9	24.7	2.38	182.5	2.36	23.3
4000	116.6	89.5	3.62	144.8	3.60	35.6	128.3	83.2	3.19	127.5	3.16	31.3
6000	266.0	199.9	4.44	114.1	4.42	43.7	289.1	194.7	4.19	107.5	4.15	41.4
8000	470.6	348.2	5.03	95.3	5.01	49.6	534.2	360.3	4.59	85.9	4.53	44.3
10,000	715.5	523.0	5.47	82.9	5.45	54.0	845.1	550.9	5.04	76.2	5.01	50.1
12,000	1008.2	729.6	5.85	73.7	5.83	57.7	1178.0	785.6	5.36	67.3	5.29	52.1

Test Case.

A sample case was run to assess the outputs of the STHX model while evaluating computational efficiency. The inlet conditions for the external and internal flow streams and the STHX geometric parameters of the sample case are provided in Table 2.

Table 2

Inlet conditions and details of parameters used in the sample case

	External fluid	Internal fluid
Working fluid	sCO₂	sCO₂
Inlet temperature (°C)	800	320
Inlet pressure (bar)	80	250
Mass flow rate (kg/s)	0.1	0.1
Tube material	Haynes282
Tube outer diameter (OD)	2 mm
Transverse pitch	3 mm
Longitudinal pitch	5.8 mm
Number of U-tubes	100
Number of shell passes	12
Heat exchanger (HX) Length	500 mm

	External fluid	Internal fluid
Working fluid	sCO₂	sCO₂
Inlet temperature (°C)	800	320
Inlet pressure (bar)	80	250
Mass flow rate (kg/s)	0.1	0.1
Tube material	Haynes282
Tube outer diameter (OD)	2 mm
Transverse pitch	3 mm
Longitudinal pitch	5.8 mm
Number of U-tubes	100
Number of shell passes	12
Heat exchanger (HX) Length	500 mm

A grid refinement study was conducted by comparing the prediction accuracy against the computation time of the STHX numerical model for the sample case. Several grid sizes were investigated by varying the grid sizing from $10 \times 10$ to $100 \times 100$ ⁠, and the results are summarized in Table 3. By refining the grid size, the accuracy of performance predictions increases, but the time taken to compute the performance also increases. The difference between the predicted flow stream pressure drops for the $60 \times 60$ and $100 \times 100$ case is less than 1.4%, but the $60 \times 60$ case is 2.4 times more computationally efficient than the $100 \times 100$ case. The performance prediction is effectively grid independent after the $60 \times 60$ case and executes in about 7 s.

Table 3

Grid refinement study for comparison of the accuracy of performance prediction against computation time of the STHX numerical model

	Heat transfer rate (kW)		$Δ T (° C)$			Total external pressure drop, $Δ P_{1}$ (bar)	Total internal pressure drop, $Δ P_{2}$ (bar)	Computation time (s)
Grid	External	Internal	External	Internal
$10 \times 10$	52.4	50.0	432.2	397.4		0.143	0.623	1.28
$20 \times 20$	51.8	50.7	427.6	403.0		0.134	0.568	2.01
$30 \times 30$	51.8	51.0	427.0	405.6		0.131	0.552	2.87
$40 \times 40$	51.7	51.1	426.0	406.2		0.130	0.544	4.56
$50 \times 50$	51.6	51.2	425.5	406.6		0.129	0.539	5.16
$60 \times 60$	51.6	51.3	425.1	406.9		0.128	0.536	7.35
$70 \times 70$	51.6	51.3	425.3	407.5		0.128	0.534	8.65
$80 \times 80$	51.6	51.3	425.1	407.6		0.128	0.532	12.5
$90 \times 90$	51.6	51.3	425.0		407.8	0.127	0.530	13.9
$100 \times 100$	51.5	51.3	424.5		407.4	0.127	0.529	17.6

	Heat transfer rate (kW)		$Δ T (° C)$			Total external pressure drop, $Δ P_{1}$ (bar)	Total internal pressure drop, $Δ P_{2}$ (bar)	Computation time (s)
Grid	External	Internal	External	Internal
$10 \times 10$	52.4	50.0	432.2	397.4		0.143	0.623	1.28
$20 \times 20$	51.8	50.7	427.6	403.0		0.134	0.568	2.01
$30 \times 30$	51.8	51.0	427.0	405.6		0.131	0.552	2.87
$40 \times 40$	51.7	51.1	426.0	406.2		0.130	0.544	4.56
$50 \times 50$	51.6	51.2	425.5	406.6		0.129	0.539	5.16
$60 \times 60$	51.6	51.3	425.1	406.9		0.128	0.536	7.35
$70 \times 70$	51.6	51.3	425.3	407.5		0.128	0.534	8.65
$80 \times 80$	51.6	51.3	425.1	407.6		0.128	0.532	12.5
$90 \times 90$	51.6	51.3	425.0		407.8	0.127	0.530	13.9
$100 \times 100$	51.5	51.3	424.5		407.4	0.127	0.529	17.6

The temperature and pressure fields within the STHX and the thermohydraulic performance of the STHX are evaluated using the numerical code. Temperature contours within the STHX obtained from the numerical model are shown in Fig. 11. Pressure profiles of internal and external flow through the STHX are given in Fig. 12.

Fig. 11

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(a) Front view and (b) top view of the temperature contours within the STHX obtained from the STHX numerical model for the sample case

Fig. 12

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Pressure profiles for external and internal flow streams through the STHX for the sample case. The naming convention for the x-axis is: “unit cell number—inlet (i) or outlet (o)”.

Preliminary Optimization Study.

To establish a pathway for achieving an optimal HX design, a preliminary optimization study with minimizing the STHX weight as the objective function was performed. The objective of the study is to demonstrate the minimization of STHX weight by selecting five parameters (STHX length, number of tubes, longitudinal pitch, transverse pitch, and number of baffles) as the variable design parameters. A discretized parameter search space is used in this study as shown in Table 4. STHX weight is calculated as the summation of the header, shell, tube, and baffle masses. The tube outer diameter is set to 1 mm to achieve attractive power density, and the thicknesses of tube and shell walls are set to be 20% of the outer diameter. Tubular Exchanger Manufacturers Association standards were followed in defining the baffle cut, baffle spacing, and minimum tube pitch (among other parameters) while selecting the design space [48,49]. Tubular Exchanger Manufacturers Association is an industry standard based on existing technologies, while microtubes are becoming an increasingly popular option for STHXs. The boundary conditions are given in Table 2, and constraints on HX effectiveness (⁠ $ϵ = 0.8$ ⁠) and HX power (⁠ $= 50 kW$ ⁠) are applied. The length of the STHX is a floating geometric parameter that helps constrain the effectiveness and power to 0.8 and 50 kW, respectively.

Fig. 13

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Effect of individual geometric parameters on the weight of various STHX designs

Table 4

Discretized search space for the various geometric design parameters

	Search space
Number of U-tubes	130, 150, 170, 190
Number of baffles	6, 8, 10, 12
Transverse pitch to tube OD ratio, $P_{t} / D_{o t}$	1.5, 1.9, 2.3, 2.7
Longitudinal pitch to tube OD ratio, $P_{l} / D_{o t}$	1.5, 1.9, 2.3, 2.7

The results from this study elucidate the effects of the various geometric parameters on performance metrics, demonstrate the ability of the STHX numerical model [45] to survey the parameter space, and narrow the design space. Figure 13 illustrates the variation in STHX weight as a function of the individual parameters obtained for the simulations conducted on various STHX geometries that satisfy the performance metrics. The optimal sCO₂ STHX design has a longitudinal pitch of 1.5 mm, transverse pitch of 1.9 mm, STHX length of 171.9 mm, 12 baffles (14 external passes), 130 U-tubes, and has a weight of 1.46 kg. This work demonstrates the capability of optimizing the design of the STHX to meet various performance requirements. The developed STHX model provides the possibility to perform more rigorous optimization studies using optimization techniques such as Particle Swarm Optimization and Genetic Algorithm. In addition, minimization of STHX cost can be adopted as a more suitable objective function to find an optimal STHX design that meets aggressive techno-economic metrics.

Air is one of the most used working fluids in the heat exchanger industry. The geometric parameters from the optimal design were used to perform a comparative analysis between STHXs with air and sCO₂ as the working fluids. The STHX length was utilized as a floating parameter in order to maintain a power rating of 50 kW for both heat exchangers. The geometric and performance parameters obtained from the numerical model are summarized in Table 5.

Table 5

Geometric and performance parameters of the optimal STHX design using sCO₂ and air as the working fluids

STHX Working fluid	sCO₂	Air
Number of U-tubes	130
Number of baffles	12
Transverse pitch (mm)	1.9
Longitudinal pitch (mm)	1.5
STHX length (mm)	171.9	182.8
STHX power (kW)	50
STHX weight (kg)	1.46	1.54
STHX volume (m³)	0.000274	0.000289

STHX Working fluid	sCO₂	Air
Number of U-tubes	130
Number of baffles	12
Transverse pitch (mm)	1.9
Longitudinal pitch (mm)	1.5
STHX length (mm)	171.9	182.8
STHX power (kW)	50
STHX weight (kg)	1.46	1.54
STHX volume (m³)	0.000274	0.000289

The sCO₂ STHX weighs 6.3% lesser and is 5.5% more compact than the STHX with air as the working fluid for the same power rating. This result indicates the benefit of using sCO₂ as the working fluid and implies that highly compact and efficient power cycles could be achieved by utilizing sCO₂ for future applications.

Conclusion

This article reports a computationally efficient and accurate numerical model developed in Julia programming to solve the performance of STHXs. The geometry of the heat exchanger is modeled using the concept of volume averaging. Highly accurate correlations for external flow (reported in Part I of this study) and existing correlations for internal flow are utilized to predict the thermohydraulic behavior of flow over and inside the tube banks. The developed model has the following highlights:

The model is highly accurate (<10% error for STHX heat transfer rate and pressure drop predictions) and computationally efficient (<7 s per case).
The model can predict the performance for bare, disk-finned, and pin-finned STHX and is capable of accounting for the effects of varying geometric parameters.
The model is validated against CFD data with sCO₂ as the working fluid for both internal and external flows. The validated operating temperature and pressure are as high as 800 °C and 250 bar. Theoretically, the model can predict the performance of STHXs using any working fluid available in the CoolProp database.

This study provides necessary basis for STHX design with various working fluids and operating conditions. Multiparameter optimization can be conducted for compact sCO₂ HXs and the optimal design with superior energy density or lowest capital cost can be efficiently obtained from a large design search space.

Acknowledgment

The authors would like to thank ARPA-E and UCLA for their funding, and Dr. Nasr Ghoniem, Dr. Xiaochun Li, Mr. Zachary Wong, and Honeywell Aerospace for their research collaboration and insightful feedback.

Funding Data

Advanced Research Projects Agency-Energy (ARPA-E), U.S. Dept. of Energy, in collaboration with Honeywell Aerospace (Award No. DE-AR0001131; Funder ID: 10.13039/100006133).

Nomenclature

A_C =: cross-sectional free flow area between two tubes, m²
A_fr =: frontal area, m²
A_HT =: heat transfer area between two tubes, m²
A_min =: minimum free-flow area, m²
A_w =: cross-sectional area of flow in the window region, m²
$B$ =: baffle spacing, m
C_p =: specific heat, J/kg-K
CPF =: cylindrical pin-fins
D_E =: effective diameter, m
D_f =: fin diameter, m
D_h =: hydraulic diameter, m
D_is =: shell inner diameter, m
D_it =: tube inner diameter, m
D_os =: shell outer diameter, m
D_ot =: tube outer diameter, m
D_VAT =: hydraulic diameter as defined by VAT, m
DF =: disk fins
f =: friction factor
$G$ =: mass velocity, kg/m²-s
h =: heat transfer coefficient, W/m²-K
I₀, I₁, K₀, K₁ =: Bessel functions
j_H =: Colburn factor
L_h =: width of tube bundle (including longitudinal baffle), m
L_y =: height of tube bundle, m
N_af =: number of fins in the annular direction
N_f =: number of fins on one tube in the axial direction
N_t =: number of tubes in one unit cell
Nu =: Nusselt number
P_d =: diagonal pitch, m
P_f =: fin pitch, m
P_l =: longitudinal pitch, m
P_t =: transverse pitch, m
Pr =: Prandtl number
q_f =: actual fin heat transfer rate
q_f,_max =: maximum possible fin heat transfer rate
Re =: Reynolds number
sCO₂ =: supercritical carbon dioxide
S_w =: specific wetted surface area, m⁻¹
S_wm =: modified specific wetted surface area, m⁻¹
$T$ =: temperature, ° C
t_s =: longitudinal baffle thickness, m
u, v, w =: velocities in the x, y, and z directions
u_avg =: average velocity of flow at the front plane, m/s
u_max =: velocity of flow at the minimum free-flow area, m/s
VAT =: volume averaging theory
V_T =: total volume of unit cell, m³
x, y, z =: Cartesian coordinates

c =: corrected
ce =: contraction and expansion
s =: tube wall
si =: tube inner wall
so =: tube outer wall
st =: straight flow region in the tube
tb =: tube bundle
turn =: external flow turning region
ub =: U-bend region
1 =: external fluid
2 =: internal fluid

$δ_{f}$ =: fin thickness, m
$η_{f}$ =: fin efficiency
$κ$ =: thermal conductivity, W/m-K
$ν$ =: dynamic viscosity, Pa·s
$ρ$ =: density, kg/m³
$ϕ$ =: volume fraction

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Modeling of Supercritical CO₂ Shell-and-Tube Heat Exchangers Under Extreme Conditions: Part II: Heat Exchanger Model

Abstract

Introduction

Methodology

Definition of Geometric Parameters.

Governing Equations.

Flow Correlations and Correction Factors.

External Flow.

Internal Flow.

Shell-and-Tube Heat Exchanger Numerical Model.

Results and Discussion

Model Validation.

Test Case.

Preliminary Optimization Study.

Conclusion

Acknowledgment

Funding Data

Nomenclature

References

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Modeling of Supercritical CO2 Shell-and-Tube Heat Exchangers Under Extreme Conditions: Part II: Heat Exchanger Model

Abstract

Introduction

Methodology

Definition of Geometric Parameters.

Governing Equations.

Flow Correlations and Correction Factors.

External Flow.

Internal Flow.

Shell-and-Tube Heat Exchanger Numerical Model.

Results and Discussion

Model Validation.

Test Case.

Preliminary Optimization Study.

Conclusion

Acknowledgment

Funding Data

Nomenclature

References

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Modeling of Supercritical CO₂ Shell-and-Tube Heat Exchangers Under Extreme Conditions: Part II: Heat Exchanger Model