# Review of Design and Modeling of Regenerative Heat Exchangers

## Abstract

**:**

## 1. Introduction

## 2. Description of the Regenerator

#### 2.1. Classification of Regenerative Heat Exchangers

#### 2.2. Basic Geometric Characteristics of Packed Bed

**Voidage, -**

_{b}is the total volume of the packed bed (m

^{3}), V

_{p}is the volume of the packed bed material (m

^{3}), and V

_{m}is the free volume of the packed bed (m

^{3}).

**Particle diameter, m**

_{p}(m), according to the specific surface given by Ergun [2], which has the same ratio of the surface to the volume as the given particle and is given by

_{p}is the particle surface area (m

^{2}).

**Sphericity, -**

_{s}is the surface area of a sphere that has the same volume as the particle (m

^{2}), A

_{p}is the particle surface area (m

^{2}), and V

_{p}is the volume of the particle (m

^{3}).

**Hydraulic diameter of packed bed, m**

_{h}is the hydraulic radius (m), a is the absolute specific surface (m

^{2}), and a

_{r}is the relative specific surface (m

^{−1}).

**Absolute specific surface, (m**

^{−1})**Relative specific surface, (m**

^{−1})## 3. Mathematical Model of the Regenerative Heat Exchanger

#### 3.1. Energy Balance of the Regenerator

_{p}, then the mean temperature change of packed bed T

_{b}at time t can be expressed in the form

_{b}is the mass of packed bed (kg), T

_{b}is the mean temperature of packed bed (°C), T

_{g}is the temperature of the gas flowing through the bed (°C), h

_{t}is the total heat transfer coefficient (W∙m

^{−2}∙K

^{−1}) between the flowing gas and the bed material, C

_{p}

_{,b}is the heat capacity of packed bed (J∙kg

^{−1∙}K

^{−1}), and A is the total heat transfer area (m

^{2}). In a cooling period, where the gas temperature, T

_{g}, is lower than the bed temperature, T

_{b}, the gas temperature increases over time while the bed temperature decreases $\frac{d{T}_{b}}{dt}<0$. During the heating period (T

_{g}> T

_{b}), the gas outlet temperature decreases with time, while the bed temperature increases $\frac{d{T}_{b}}{dt}>0$.

_{g}is the mass flow rate of gas (kg∙s

^{−1}), C

_{p}

_{,g}is the heat capacity of gas (J∙kg

^{−1∙}K

^{−1}), M

_{g}is mass of gas resident in the regenerator (kg), and L is the height of regenerator (m).

_{t}is the surface heat transfer coefficient, usually a convective coefficient to which may be added a radiative component.

_{b}is the thermal conductivity of packing material of regenerator (W∙m

^{−1}∙K

^{−1}), h

_{lum}is the lumped heat transfer coefficient (W∙m

^{−2}∙K

^{−1}), h

_{c}is the convective heat transfer coefficient (W∙m

^{−2}∙K

^{−1}), and h

_{r}is the radiative heat transfer coefficient (W∙m

^{−2}∙K

^{−1}). The lumped heat transfer coefficient incorporates the surface convective heat transfer coefficient, h

_{c}, and the resistance to heat transfer within the regenerator packing, as represented by the $\frac{d}{2\left(n+2\right)k}{\varphi}_{H}$ therm. The total heat transfer coefficient can be used in the conventional model of the thermal performance of the regenerator, set out in the differential equations.

^{2}∙s

^{−1}), P is the length of period for heating and cooling process (s), ε = 2.7 for plates, ε = 9.9 for cylinders, and ε = 27.0 for spheres, Ω′ is the reduced time for hot period, and Ω″ is the reduced time for cold period.

#### 3.2. Differential Equations

_{REG}= η’

_{REG}= η″

_{REG}), an estimate of the thermal ratio is given by

_{0}describing the effect of nonlinear variations of temperature using a factor is given in Reference [4].

_{0}results in a greater effect of both the nonlinear variations of temperature and the corresponding truncation error.

#### 3.3. Calculation Methods

#### 3.4. Selected Mathematical Model

- The effect of the reversals can be neglected, that is, the rapid gas temperature transients which are associated with the residual gas in the regenerator being replaced by the gas flowing in the opposite direction at the reversal can be ignored.
- The entrance gas temperatures in both periods remain constant.
- The mass flow rates of the heating and cooling gases do not vary throughout each period.
- Heat transfer between gas and solid can be represented in terms of an overall heat transfer coefficient relating gas temperature to mean solid temperature. Furthermore, the rate of heat transfer in the packed bed at any height is represented by the time variation of the mean solid temperature.
- The heat capacity of the gas in the channels of the packed bed at any instant is small relative to the heat capacity of the solid and, therefore, can be neglected.
- The heat transfer coefficients and the thermal properties of the heat storing mass and the gas do not vary throughout a period and are identical at all parts of the regenerator in that period.
- Longitudinal thermal conductivity is neglected.

**Boundary conditions**

- The inlet temperatures for both hot and cold cycles are constant.
- The surface temperatures along the length of the regenerator at the end of the hot/cold period are the same as those at the beginning of the following cold/hot period. Since the gases flow in opposite directions in successive cycles, the boundary conditions are expressed by the equations

#### 3.4.1. Linear Model

**Case 1—taken from Reference [9]**

#### 3.4.2. Nonlinear Model

## 4. Pressure Drops

^{−1}), and λ

_{k}is the friction factor, most often given in the form

_{1}and k

_{2}are constants, and b is the exponent. Re

_{m}is the modified Reynolds number given by Ergun as follows:

_{m}> 700.

_{1}= 190, k

_{1}= 2.00, and k

_{2}= 0.77 can be used for the cylindrical shapes of the particles. For other particles, the constants K

_{1}= 155, k

_{1}= 1.42, and k

_{2}= 0.83 can be used.

_{m}< 400.

_{h}is the hydraulic diameter (m).

## 5. Voidage Calculation

_{p}. In the randomly packed bed, other than spherical particles, the influence of particle orientation affects the local distribution of the voidage, which affects the mean voidage. This means that different medium voidage can be achieved with each filling of the packing.

_{b}= 0.4, while Reference [35] suggested that ε

_{b}= 0.373, a value derived from their own data (subscript b means a bulk region).

_{p}≤ 50),

_{p}≤ 26.3),

_{p}≤ 14.5),

_{p}≤ 50; 0.42 < $\psi $ < 1.0),

_{p}ratio is shown in Figure 5. It can be seen that, up to value 10, the voidage strongly depends on this ratio. From this value, the voidage is approximately constant.

## 6. Heat Transfer Calculation

_{c}is the convective heat flux density (W∙m

^{−2}), q

_{r}is the radiative heat flux density (W∙m

^{−2}), and the total heat transfer coefficient is defined as

_{lum}.

#### 6.1. Convective Heat Transfer Coefficient

_{g}is the thermal conductivity (W∙m

^{−1}∙K

^{−1}), Re is the Reynolds number (-), and Pr is the Prandtl number (-).

_{1}, a

_{2}, and n are model constants. The values of these constants given by Reference [44] are a

_{1}= 2.0, a

_{2}= 1.1, and n = 0.6. Values of these constant for different packed cells were mentioned in Reference [45].

#### 6.2. Radiative Heat Transfer Coefficient

_{2}and H

_{2}O vapor. In these circumstances, the radiation heat transfer must be considered.

_{g}and α

_{g}are the emissivity and mean absorptivity of gases (-), respectively, ε

_{b}is the emissivity of bed material (-), and σ is Stefan Boltzmann’s constant, 5.67 × 10

^{−8}(W∙m

^{−2}∙K

^{−1}). The fraction of $\frac{{\epsilon}_{b}+1}{2}$ is sometimes called the emissivity correction factor.

_{r}using the equation mentioned in Reference [48].

_{b}, in a given space and on the partial pressures of the respective radiant gas components. Several methods can be found in the literature.

_{b}is determined from a known equation for the case when L

_{b}> 1 m,

_{b}< 1 m,

^{3}), and P is the mean inner surface of the channel in the packed bed (m

^{2}).

_{CO}

_{2}is the emissivity of carbon dioxide (-), ε

_{H}

_{2O}is the uncorrected emissivity of water vapor (-), C

_{H2O}is Beer’s law correction factor for water vapor (-), and C

_{SO}is the spectral overlap correction factor (-).

_{CO}

_{2}) is a function of temperature and the product (p

_{CO}

_{2}L), where p

_{CO}

_{2}is the partial pressure of carbon dioxide in the gases (Pa) and L

_{b}is the mean beam length (m). Thus,

_{H}

_{2O}) is a function of temperature and (p

_{H}

_{2O}L

_{b}), where p

_{H}

_{2O}is the partial pressure of water vapor (Pa) and L

_{b}is the mean beam length (m). Thus,

_{b}< 500 °C, n = 0.4 for T

_{b}> 900 °C, and n = 0.45 for 500 °C < T

_{b}< 900 °C.

#### 6.3. Heat Losses

_{w}is the wall area (m

^{2}), T

_{w}is the wall temperature (°C), T

_{inf}is the ambient temperature (°C), and h

_{o}is the outside convective heat transfer coefficient (W∙m

^{−2}∙K

^{−1}).

_{o}is calculated from a correlation of Churchill and Chu [54] for plane surfaces as

^{−2}), $\beta =\frac{1}{{T}_{f}}=\frac{2}{{T}_{w}+{T}_{inf}}$ is the thermal expansion coefficient (K

^{−1}), α is the thermal diffusivity (m

^{2}∙s

^{−1}), and ν is the kinematic viscosity of gas (m

^{2}∙s

^{−1}) as defined in the temperature of Tf. Validity of the correlation is in the range 10

^{−1}< Ra < 10

^{12}. This correlation can be used provided the curvature effect is not too significant. This represents the limit where boundary layer thickness is small relative to cylinder diameter D. The correlations for vertical plane walls can be used when D/L ≥ 35/Gr

^{0.25}where Gr is the Grashof number.

## 7. Software Implementation of the Model

#### Case Study

^{−6}. According to the calculation results, the equilibrium was reached after 18 cycles, and the calculation time was 48 milliseconds. It is obvious that the calculation is fast. The main results are shown in Table 6, and graphical dependencies of temperatures and pressures are shown in the figures below. These are only basic graphical outputs provided by the software. Similarly, the output text protocol contains much more data than shown in Table 6.

## 8. Summary

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Greek symbols | |

a | absolute specific surface (m^{−1}) |

a_{r} | relative specific surface (m^{−1}) |

A_{p} | particle surface area or heat transfer surface area (m^{2}) |

A_{s} | surface area of a sphere that has the same volume as the particle (m^{2}) |

A_{w} | wall area (m^{2}) |

b | exponent in the Equation (41) (-) |

$\overline{c}$ | velocity of gas based on the empty cross-section of the bed (m∙s^{−1}) |

C_{p} | heat capacity (J∙kg^{−1∙}K^{−1}) |

C_{H2O} | Beer’s law correction factor for water vapor (-) |

C_{SO} | spectral overlap correction factor (-) |

D | diameter of packed bed or regenerator (m) |

d_{h} | hydraulic diameter of packed bed (m) |

d_{V} | diameter of a sphere that has the same volume as the particle (m) |

d_{p} | particle diameter which has the same surface-to-volume ratio as the given particle (m) |

e | effective roughness height (m) |

g | gravity acceleration (m∙s^{−2}) |

h_{c} | convective heat transfer coefficient (W∙m^{−2}∙K^{−1}) |

h_{lum} | lumped heat transfer coefficient (W∙m^{−2}∙K^{−1}) |

h_{o} | outside convective heat transfer coefficient (W∙m^{−2}∙K^{−1}) |

h_{r} | radiative heat transfer coefficient (W∙m^{−2}∙K^{−1}) |

h_{t} | total heat transfer coefficient (W∙m^{−2}∙K^{−1}) |

k_{1}k_{2} | constants in the Equation (41) (-) |

K/K_{0} | ratio specified in the Equation (30) (-) |

L | height of regenerator (m) |

L_{b} | mean beam length (m) |

m | number of section (-) |

M_{b} | mass of packed bed (kg) |

m_{g} | mass flow rate of gas (kg∙s^{−1}) |

M_{g} | mass of gas resident in the regenerator (kg) |

n | number of cycles (-) |

Nu | Nusselt number (-) |

P | length of period (s) |

P | mean inner surface of the channel in the packed bed in Equations (70) and (71) (m^{2}) |

p_{CO2} | partial pressure of carbon dioxide in the gases (Pa) |

p_{H2O} | partial pressure of water vapor (Pa) |

Pr | Prandtl number (-) |

q | total heat flux density (W∙m^{−2}) |

q_{c} | convective heat flux density (W∙m^{−2}) |

q_{r} | radiative heat flux density (W∙m^{−2}) |

r | refers to the distance (m) |

r_{h} | hydraulic radius (m) |

Ra | Rayleigh number (-) |

Re Re_{m} Re_{l} | Reynolds number (-) |

S | refers to the time (s) |

t | time (s) |

T_{b} | mean temperature of packed bed (°C) |

T_{g} | temperature of the gas flowing through the bed (°C) |

T_{w} | wall temperature (°C) |

T_{inf} | ambient temperature (°C) |

V | mean channel volume in Equations (70) and (71) (m^{3}) |

V_{b} | total volume of the packed bed (m^{3}) |

V_{p} | volume of the material of the packed bed (m^{3}) |

V_{m} | free volume of the packed bed (m^{3}) |

Superscripts | |

′ | refers to heating period |

″ | refers to cooling period |

Substricpts | |

b | packed bed |

g | gas |

H | harmonic |

i | input |

m | mean value or modified (for Reynolds number) |

o | output |

r | refers to distance |

ref | reference value |

S | refers to time |

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**Figure 1.**Some types of geometry of storage materials [1].

**Figure 2.**Possibilities of connection of regenerative heat exchanger: (

**a**) connection of regenerative heat exchanger for heating or cooling media; (

**b**) connection for cleaning of flue gases or waste gasses —option 1; (

**c**) connection for cleaning of flue gases or waste gasses—option 2.

**Figure 4.**Influence of number sections on outlet temperature obtained by the basic calculation model: (

**a**) for the heating period; (

**b**) for the cooling period.

**Figure 9.**The graphical dependences of gas temperatures along the bed: (

**a**) for the cooling period; (

**b**) for the heating period.

Nonsymmetric | |||
---|---|---|---|

Symmetric | Balanced | Unbalanced | |

Parameters | Λ, Π | Λ′, Π′, Λ”, Π″ | Λ′, Π′, Λ”, Π″ |

Relationships | Λ = Λ′ = Λ″ Π = Π′ = Π″ | Π′/Π″ = Λ′/Λ″ = k ≠ 1 | Π′/Π″ ≠ Λ′/Λ″ |

Thermal ratios | η_{REG} = η′_{REG} = η″_{REG} | η_{REG} = η′_{REG} | η_{REG} ≠ η′_{REG} |

γ | 1 | 1 | ≠1 |

Hot | Cold | ||
---|---|---|---|

T_{g}_{,i} | 1000 | 0 | °C |

Λ | 6 | 3.5 | - |

Π | 6 | 3.5 | - |

P | 12 | 6 | s |

K/K_{0} | 0.73 | - |

Autor(s) | Equation | Range of Validity |
---|---|---|

Erdim [17] | ${\lambda}_{v}=160+2.81R{e}_{m}^{0.904}$ | $2<R{e}_{m}<3600$ |

Fahien and Schriver [18] | ${\lambda}_{k}=q\frac{{f}_{1L}}{R{e}_{m}}+\left(1-q\right)\left({f}_{2}+\frac{{f}_{1T}}{R{e}_{m}}\right)$ $q=exp\left(-\frac{{\epsilon}^{2}\left(1-\epsilon \right)}{12.6}R{e}_{m}\right)$ ${f}_{1L}=\frac{136}{{\left(1-\epsilon \right)}^{0.38}}$ ${f}_{1\mathrm{T}}=\frac{29}{{\left(1-\epsilon \right)}^{1.45}{\epsilon}^{2}}$${f}_{2}=\frac{1.87{\epsilon}^{0.75}}{{\left(1-\epsilon \right)}^{0.26}}$ | NA it can be consider $0.2<R{e}_{l}<700$ |

KTA [19] | ${\lambda}_{k}=\frac{160}{R{e}_{m}}+\frac{3}{R{e}_{Erg}^{0.1}}$ | $1<R{e}_{m}<\mathrm{100,000}$ |

Harrison, Brunner and Hecker [20] | ${\lambda}_{k}=\frac{119.8A}{R{e}_{m}}+\frac{4.63B}{R{e}_{Erg}^{\frac{1}{6}}}$ $A={\left(1+\pi \frac{{d}_{p}}{6\left(1-\epsilon \right)D}\right)}^{2}$$B=1-\text{}\frac{{\pi}^{2}{d}_{p}}{24D}\text{}$ | $0.32<Re<7700$ |

Carman [21] | ${\lambda}_{k}=\frac{180}{R{e}_{m}}+\frac{2.871}{R{e}_{m}^{0.1}}$ | $0.01<R{e}_{l}<\mathrm{10,000}$ |

Brauer [22] | ${\lambda}_{k}=\frac{160}{R{e}_{m}}+\frac{3.1}{R{e}_{m}^{-0.1}}$ | $0.01<R{e}_{Erg}<\mathrm{20,000}$ |

Eisfeld and Schnitlein [23] | ${\lambda}_{k}=\frac{{K}_{1}{M}^{2}}{R{e}_{m}}+\frac{M}{{B}_{W}}$ $M=1+\frac{2{d}_{p}}{3\left(1-\epsilon \right)D}$${B}_{W}=\left[{k}_{1}{\left(\frac{{d}_{p}}{D}\right)}^{2}+{k}_{2}\right]$ K _{1} = 154 k_{1} = 1.15 k_{2} = 0.87 | $0.01<Re<\mathrm{17,635}$ |

Ergun [2] | ${\lambda}_{k}=\frac{150}{R{e}_{m}}+1.75$ | $0.2<R{e}_{l}<700$ |

Hicks [24] | ${\lambda}_{k}=\frac{6.8}{R{e}_{m}^{0.2}}$ | $300<R{e}_{m}<\mathrm{60,000}$ |

Hot Gas | Cold Gas | ||
---|---|---|---|

Mass flowrate | 79.2 | 79.2 | kg∙h^{−1} |

Input temperature | 727 | 27 | °C |

Period | 600 | 600 | s |

Density | 0.51 | 0.51 | kg∙m^{−3} |

Dynamic viscosity | 364 × 10^{−7} | 364 × 10^{−7} | Pa∙s |

Heat capacity | 1060 | 1060 | J∙kg^{−1}∙K^{−1} |

Thermal conductivity | 0.046 | 0.046 | W∙m^{−1}∙K^{−1} |

Bed Diameter | 0.2 | m |
---|---|---|

Bed height | 1 | m |

Number of sections | 100 | - |

Type of packed bed | Ceramic balls | - |

Ball diameter | 0.03 | m |

Voidage | 0.38 | - |

Density | 3970 | kg∙m^{−3} |

Heat capacity | 765 | J∙kg^{−1}∙K^{−1} |

Therma conductivity | 15.8 | W∙m^{−1}∙K^{−1} |

Thermal diffusivity | 0.463 × 10^{−6} | m^{2}∙s^{−1} |

Cold Gas | Hot Gas | ||
---|---|---|---|

Input temperature | 27.0 | 727.0 | °C |

Out. temp. at the start of the period | 702.7 | 51.4 | °C |

Out. temp. at the end of the period | 576.2 | 178.2 | °C |

Heat transfer coefficient | 92.7 | 92.7 | W∙m^{−2∙}K^{−1} |

Velocity of gas | 3.6 | 3.6 | m∙s^{−1} |

Efficiency of regenerator | 87.8 | % | |

Heat transfer area | 3.9 | m^{2} | |

Mass of packed bed | 77.3 | kg | |

Mena pressure drop | 3141 | 3573 | Pa |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kilkovský, B.
Review of Design and Modeling of Regenerative Heat Exchangers. *Energies* **2020**, *13*, 759.
https://doi.org/10.3390/en13030759

**AMA Style**

Kilkovský B.
Review of Design and Modeling of Regenerative Heat Exchangers. *Energies*. 2020; 13(3):759.
https://doi.org/10.3390/en13030759

**Chicago/Turabian Style**

Kilkovský, Bohuslav.
2020. "Review of Design and Modeling of Regenerative Heat Exchangers" *Energies* 13, no. 3: 759.
https://doi.org/10.3390/en13030759