5. Thermal design for regenerators

Two methods are used for the regenerator thermal performance analysis: ε � NTUo and Λ � π methods, respectively, for rotary and fixed matrix regenerators.

#### 5.1. The ε – NTUo method

4. Plot the dimensionless mean temperature Ψ as a function of P<sup>1</sup> and R<sup>1</sup> in Figure 17.

NTU <sup>¼</sup> <sup>P</sup><sup>1</sup>

NTU1

P<sup>1</sup> –P<sup>2</sup> chart for 1–2 shell and tube heat exchanger [2] with shell fluid mixed is shown in

3. Plot P<sup>1</sup> as a function of R<sup>1</sup> with NTU1 or P<sup>2</sup> as a function of R<sup>2</sup> with NTU2 in Figure 18.

<sup>¼</sup> <sup>P</sup><sup>2</sup> NTU2

(56)

5. Calculate the heat transfer rate.

2618 Heat Exchangers Heat Exchangers– Design, Experiment and Simulation – Design, Experiment and Simulation

The dimensionless mean temperature difference is [4]

4.4.1. The procedure to be followed with the P<sup>1</sup> – P<sup>2</sup> method

Figure 18. P<sup>1</sup> – P<sup>2</sup> chart for 1–2 shell and tube heat exchanger with shell fluid mixed.

4. Calculate the dimensionless mean temperature Ψ.

1. Calculate NTU1 or NTU2.

5. Calculate the heat transfer rate.

2. Calculate R<sup>1</sup> or R2.

<sup>ψ</sup> <sup>¼</sup> <sup>ε</sup>

where 1 and 2 are one shell pass and two tube passes, respectively.

4.4. The P<sup>l</sup> – P<sup>2</sup> method

Figure 18.

The ε – NTUo method was developed by Coppage and London in 1953. The modified number of transfer units is [4]

$$\text{NTU}\_0 = \frac{1}{\mathbb{C}\_{\text{min}}} \left[ \frac{1}{\frac{1}{(hA)\_h} + \frac{1}{(hA)\_c}} \right] \tag{57}$$

$$\mathbf{C}^\* = \frac{\mathbf{C}\_{\text{min}}}{\mathbf{C}\_{\text{max}}} \tag{58}$$

$$\mathbf{C}\_r^\* = \frac{\mathbf{C}\_r}{\mathbf{C}\_{\min}} \tag{59}$$

$$\mathbf{C}\_r = M\_w \mathbf{c}\_w \mathbf{N} \tag{60}$$

where cw is the specific heat of wall material, N is the rotational speed for a rotary regenerator and Mw is matrix mass and determined as

$$M\_w = A\_{rl} H\_r \rho\_m S\_m \tag{61}$$

where Arc is the rotor cross-sectional area, Hr is the rotor height, ρ<sup>m</sup> is the matrix material density and Sm is the matrix solidity.

The convection conductance ratio is

$$(hA)^{\*} = \frac{(hA)\_{\mathbb{C}\_{\text{min}}}}{(hA)\_{\mathbb{C}\_{\text{max}}}} \tag{62}$$

Most regenerators operate in the range of 0:25 ≤ ðhAÞ � < 4. The effect of ðhAÞ � on the regenerator effectiveness can usually be ignored.

A is the total matrix surface area and given as

$$A = A\_{rc} H\_r \beta F\_{rfa} \tag{63}$$

where Arc is the rotor cross-sectional area, Hr is the rotor height, β is the matrix packing density and Frfa is the fraction of rotor face area not covered by radial seals.

The hot and cold gas side surface areas are proportional to the respective sector angles.

$$A\_h = \left(\frac{\alpha\_h}{360^\circ}\right)A\tag{64}$$

$$A\_c = \left(\frac{\alpha\_c}{360^\circ}\right)A\tag{65}$$

where α<sup>h</sup> and α<sup>c</sup> are disk sector angles of hot flow and cold flow in degree, respectively.

The regenerator effectiveness is

$$
\varepsilon = \frac{q}{q\_{\text{max}}} \tag{66}
$$

$$\eta\_{\text{max}} = \mathbb{C}\_{\text{min}}(T\_{hi} - T\_{ci}) \tag{67}$$

#### 5.1.1. The counter flow regenerator

The regenerator effectiveness for ε ≤ 0:9 is

$$
\varepsilon = \varepsilon\_{\ell\prime} \left( 1 - \frac{1}{9\mathcal{C}\_r^{\*1.93}} \right) \tag{68}
$$

where εcf is the counter flow recuperator effectiveness and is determined as

$$\varepsilon\_{\mathcal{f}} = \frac{1 - \exp[-\text{NTU}\_o(1 - \mathbb{C}^\*)]}{1 - \mathbb{C}^\* \exp[-\text{NTU}\_o(1 - \mathbb{C}^\*)]} \tag{69}$$

Figure 19. The counter flow regenerator effectiveness as a function of NTU<sup>o</sup> and for C\* = 1.

The counter flow regenerator effectiveness as a function of NTU<sup>o</sup> and for C\* = 1 is presented in Figure 19. The regenerator effectiveness increases with C <sup>r</sup> for given values of NTU<sup>o</sup> and C . The range of the optimum value of C <sup>r</sup> is between 2 and 4 for optimum regenerator effectiveness.

#### 5.1.2. The parallel flow regenerator

Ac <sup>¼</sup> <sup>α</sup><sup>c</sup> 360� � �

where α<sup>h</sup> and α<sup>c</sup> are disk sector angles of hot flow and cold flow in degree, respectively.

<sup>ε</sup> <sup>¼</sup> <sup>q</sup> qmax

<sup>ε</sup> <sup>¼</sup> <sup>ε</sup>cf <sup>1</sup> � <sup>1</sup>

<sup>ε</sup>cf <sup>¼</sup> <sup>1</sup> � exp½�NTUoð<sup>1</sup> � <sup>C</sup>�

where εcf is the counter flow recuperator effectiveness and is determined as

1 � C�

Figure 19. The counter flow regenerator effectiveness as a function of NTU<sup>o</sup> and for C\* = 1.

9C�1:<sup>93</sup> r

exp½�NTUoð1 � C�

!

The regenerator effectiveness is

2820 Heat Exchangers Heat Exchangers– Design, Experiment and Simulation – Design, Experiment and Simulation

5.1.1. The counter flow regenerator

The regenerator effectiveness for ε ≤ 0:9 is

A (65)

qmax ¼ CminðThi � TciÞ (67)

Þ�

Þ� (69)

(66)

(68)

The parallel flow regenerator effectiveness as a function of NTU<sup>o</sup> and for C\* = 1 and (hA)\* = 1 is presented in Figure 20.

Figure 20. The parallel flow regenerator effectiveness as a function of NTU<sup>o</sup> and for C\* = 1 and (hA)\* = 1.

5.1.3. The procedure to be followed with the ε – NTUo method


#### 5.2. The Λ – π method

This method is generally used for fixed matrix regenerators. The reduced length designates the dimensionless heat transfer or thermal size of the regenerator. The reduced length is [4]

$$
\Lambda = bL\tag{70}
$$

The reduced lengths for hot and cold sides, respectively, are

$$
\Lambda\_{\mathbf{h}} = \left(\frac{hA}{\mathbf{C}}\right)\_{\mathbf{h}} = ntu\_{\mathbf{h}} \tag{71}
$$

$$
\Lambda\_{\mathfrak{C}} = \left(\frac{hA}{\mathbb{C}}\right)\_{\mathfrak{C}} = ntu\_{\mathfrak{C}} \tag{72}
$$

The reduced period is

$$
\pi = cP\_h \text{ or } cP\_c \tag{73}
$$

where b and c are constants.

The reduced periods for hot and cold sides, respectively, are

$$
\pi\_h = \left(\frac{hA}{\mathbb{C}\_r}\right)\_h \tag{74}
$$

$$
\pi\_c = \left(\frac{hA}{\mathcal{C}\_r}\right)\_c \tag{75}
$$

Designations of various types of regenerators are given in Table 1. For a symmetric and balanced regenerator, the reduced length and the reduced period are equal on the hot and cold sides:

$$
\Lambda\_h = \Lambda\_c = \Lambda = \Lambda\_m = \frac{hA}{\dot{m}c\_p} = ntu
\tag{76}
$$

$$
\pi\_h = \pi\_c = \pi = \pi\_m = \frac{hAP}{M\_w c\_w} \tag{77}
$$

The actual heat transfer during one hot or cold gas flow period is

$$Q = \mathbb{C}\_h P\_h (T\_{hi} - T\_{ho}) = \mathbb{C}\_c P\_c (T\_{co} - T\_{ci}) \tag{78}$$

The maximum possible heat transfer is

$$Q\_{\text{max}} = (\text{CP})\_{\text{min}} (T\_{hi} - T\_{ci}) \tag{79}$$

The effectiveness for a fixed-matrix regenerator is

$$\varepsilon = \frac{\mathcal{Q}}{\mathcal{Q}\_{\text{max}}} = \frac{(\mathcal{C}P)\_{h}(T\_{hi} - T\_{ho})}{(\mathcal{C}P)\_{\text{min}}(T\_{hi} - T\_{ci})} = \frac{(\mathcal{C}P)\_{c}(T\_{co} - T\_{ci})}{(\mathcal{C}P)\_{\text{min}}(T\_{hi} - T\_{ci})} \tag{80}$$

The effectiveness chart for a balanced and symmetric counter flow regenerator is given in Figure 21.

The effectiveness chart for a balanced and symmetric parallel flow regenerator is given in Figure 22.


Table 1. Designation of various types of regenerators for Λ – Π method.

Λ ¼ bL (70)

¼ ntuh (71)

¼ ntuc (72)

¼ ntu (76)

<sup>ð</sup>CPÞminðThi � Tci<sup>Þ</sup> (80)

(74)

(75)

(77)

π ¼ cPh or cPc (73)

The reduced lengths for hot and cold sides, respectively, are

3022 Heat Exchangers Heat Exchangers– Design, Experiment and Simulation – Design, Experiment and Simulation

The reduced periods for hot and cold sides, respectively, are

The actual heat transfer during one hot or cold gas flow period is

The maximum possible heat transfer is

The effectiveness for a fixed-matrix regenerator is

<sup>ε</sup> <sup>¼</sup> <sup>Q</sup> Qmax

The reduced period is

where b and c are constants.

<sup>Λ</sup><sup>h</sup> <sup>¼</sup> hA C 

<sup>Λ</sup><sup>c</sup> <sup>¼</sup> hA C 

h

c

<sup>π</sup><sup>h</sup> <sup>¼</sup> hA Cr 

<sup>π</sup><sup>c</sup> <sup>¼</sup> hA Cr 

Designations of various types of regenerators are given in Table 1. For a symmetric and balanced regenerator, the reduced length and the reduced period are equal on the hot and cold sides:

<sup>π</sup><sup>h</sup> <sup>¼</sup> <sup>π</sup><sup>c</sup> <sup>¼</sup> <sup>π</sup> <sup>¼</sup> <sup>π</sup><sup>m</sup> <sup>¼</sup> hAP

<sup>Λ</sup><sup>h</sup> <sup>¼</sup> <sup>Λ</sup><sup>c</sup> <sup>¼</sup> <sup>Λ</sup> <sup>¼</sup> <sup>Λ</sup><sup>m</sup> <sup>¼</sup> hA

<sup>¼</sup> <sup>ð</sup>CPÞhðThi � Tho<sup>Þ</sup> ðCPÞminðThi � TciÞ h

c

mc\_ <sup>p</sup>

Mwcw

Q ¼ ChPhðThi � ThoÞ ¼ CcPcðTco � TciÞ (78)

Qmax ¼ ðCPÞminðThi � TciÞ (79)

<sup>¼</sup> <sup>ð</sup>CPÞcðTco � Tci<sup>Þ</sup>

Figure 21. The effectiveness chart for a balanced and symmetric counter flow regenerator.

Figure 22. The effectiveness chart for a balanced and symmetric parallel flow regenerator.

5.2.1. The procedure to be followed with the Λ – π method


#### 6. Conclusion

This chapter has discussed the basic design methods for two fluid heat exchangers. The design techniques of recuperators and regenerators, which are two main classes, were investigated.

The solution to recuperator problem is presented in terms of log-mean temperature difference (LMTD), effectiveness-number of transfer units (ε NTU), dimensionless mean temperature difference (Ψ P) and (P<sup>1</sup> – P2) methods. The exchanger rating or sizing problem can be solved by any of these methods and will yield the identical solution within the numerical error of computation. If inlet temperatures, one of the fluid outlet temperatures, and mass flow rates are known, the LMTD method can be used to solve sizing problem. If they are not known, the (ε NTU) method can be used. (Ψ P) and (P<sup>1</sup> – P2) methods are graphical methods. The (P<sup>1</sup> – P2) method includes all major dimensionless heat exchanger parameters. Hence, the solution to the rating and sizing problem is non-iterative straightforward.

Regenerators are basically classified into rotary and fixed matrix models and in the thermal design of these models two methods: effectiveness-modified number of transfer units (ε NTUo) and reduced length and reduced period (Λ π) methods for the regenerators. (Λ π) method is generally used for fixed matrix regenerators.
