Basic Design Methods of Heat Exchanger

Abstract

Heat exchangers are devices that transfer energy between fluids at different temperatures by heat transfer. These devices can be used widely both in daily life and industrial applications such as steam generators in thermal power plants, distillers in chemical industry, evaporators and condensers in HVAC applications and refrigeration process, heat sinks, automobile radiators and regenerators in gas turbine engines. This chapter discusses the basic design methods for two fluid heat exchangers.

1. Introduction

Heat exchangers (HE) are devices that transfer energy between fluids at different temperatures by heat transfer. Heat exchangers may be classified according to different criteria. The classification separates heat exchangers (HE) in recuperators and regenerators, according to construction is being used. In recuperators, heat is transferred directly (immediately) between the two fluids and by opposition, in the regenerators there is no immediate heat exchange between the fluids. Rather this is done through an intermediate step involving thermal energy storage. Recuperators can be classified according to transfer process in direct contact and indirect contact types. In indirect contact HE, there is a wall (physical separation) between the fluids. The recuperators are referred to as a direct transfer type. In contrast, the regenerators are devices in which there is intermittent heat exchange between the hot and cold fluids through thermal energy storage and release through the heat exchanger surface or matrix. Regenerators are basically classified into rotary and fixed matrix models. The regenerators are referred to as an indirect transfer type.

This chapter discusses the basic design methods for two fluid heat exchangers. We discuss the log-mean temperature difference (LMTD) method, the effectiveness $\epsilon -\text{NTU}$ method, dimensionless mean temperature difference ($\Psi -P$) and (P₁ – P₂) to analyse recuperators. The LMTD method can be used if inlet temperatures, one of the fluid outlet temperatures, and mass flow rates are known. The ε – NTU method can be used when the outlet temperatures of the fluids are not known. Also, it is discussed effectiveness-modified number of transfer units ($\epsilon -{\text{NTU}}_{\text{o}}$) and reduced length and reduced period ($\Lambda -\pi$) methods for regenerators.

2. Governing equations

The energy rate balance is

$\frac{d{E}_{cv}}{dt}={\displaystyle \dot{Q}}-{\displaystyle \dot{W}}+{\displaystyle {\displaystyle \sum }_i}{{\displaystyle \dot{m}}}_i\left(h_i+\frac{V{i}^2}{2}+g{z}_i\right)-{\displaystyle {\displaystyle \sum }_e}{{\displaystyle \dot{m}}}_e\left(h_e+\frac{V{e}^2}{2}+g{z}_e\right)$E1

For a control volume at steady state, $\frac{d{E}_{cv}}{dt}=0$. Changes in the kinetic and potential energies of the flowing streams from inlet to exit can be ignored. The only work of a control volume enclosing a heat exchanger is flow work, so ${\displaystyle \dot{W}}=0$ and single-stream (only one inlet and one exit) and from the steady-state form the heat transfer rate becomes simply [1–3]

${\displaystyle \dot{Q}}={\displaystyle \dot{m}}(h_2-h_1)$E2

For single stream, we denote the inlet state by subscript 1 and the exit state by subscript 2.

For hot fluids,

${\displaystyle \dot{Q}}={\displaystyle \dot{m}}\left(h_{h1}-h_{h2}\right)$E3

For cold fluids,

${\displaystyle \dot{Q}}={\displaystyle \dot{m}}\left(h_{c2}-h_{c1}\right)$E4

The total heat transfer rate between the fluids can be determined from

${\displaystyle \dot{Q}}=UA\text{Δ}{T}_{lm}$E5

where U is the overall heat transfer coefficient, whose unit is W/m^{2 o}C and $\text{Δ}{T}_{lm}$ is log-mean temperature difference.

3. Overall heat transfer coefficient

A heat exchanger involves two flowing fluids separated by a solid wall. Heat is transferred from the hot fluid to the wall by convection, through the wall by conduction and from the wall to the cold fluid by convection.

$UA=U_o{A}_o=U_i{A}_i=\frac{1}{R_t}$E6

where $A_i=\pi D_{i}L$ and $A_o=\pi D_{o}L$ and U is the overall heat transfer coefficient based on that area. R_t is the total thermal resistance and can be expressed as [1]

$R_t=\frac{1}{UA}=\frac{1}{h_i{A}_i}+R_w+\frac{R_{fi}}{A_i}+\frac{R_{fo}}{A_o}+\frac{1}{h_{o }{A}_o}$E7

where R_f is fouling resistance (factor) and R_w is wall resistance and is obtained from the following equations.

For a bare plane wall

$R_w=\frac{t}{kA}$E8

where t is the thickness of the wall

For a cylindrical wall

$R_w=\frac{\ln (\frac{r_o}{r_i})}{2\pi Lk}$E9

The overall heat transfer coefficient based on the outside surface area of the wall for the unfinned tubular heat exchangers,

$U_o=\frac{1}{\frac{r_o}{r_i}\frac{1}{h_i}+\frac{r_o}{r_i} R_{fi}+\frac{r_o}{k}\ln \left(\frac{r_o}{r_i}\right)+R_{fo}+\frac{1}{h_o}}$E10

where R_fi and R_fo are fouling resistance of the inside and outside surfaces, respectively.

or

$U_o=\frac{1}{\frac{r_o}{r_i}\frac{1}{h_i}+R_{ft}+\frac{r_o}{k}\ln \left(\frac{r_o}{r_i}\right)+\frac{1}{h_o}}$E11

where R_ft is the total fouling resistance, given as

$R_{ft}=\frac{A_o}{A_i}{R}_{fi}+R_{fo}$E12

For finned surfaces,

${\displaystyle \dot{Q}}=\eta hA\text{Δ}T$E13

where η is the overall surface efficiency and

$\eta =1-\frac{A_f}{A}(1-{\eta }_f)$E14

where A_f is fin surface area and η_f is fin efficiency and is defined as

${\eta }_f=\frac{{{\displaystyle \dot{Q}}}_f}{{{\displaystyle \dot{Q}}}_{f,\text{max}}}$E15

Constant cross-section of very long fins and fins with insulated tips, the fin efficiency can be expressed as

${\eta }_{f,\text{long}}=\frac{1}{mL}$E16

${\eta }_{f,\text{insulated}}=\frac{\text{tanh}(mL)}{mL}$E17

where L is the fin length.

For straight triangular fins,

${\eta }_{f,\text{triangular}}=\frac{1}{mL}\frac{I_1(2mL)}{I_0(2mL)}$E18

For straight parabolic fins,

${\eta }_{f,\text{parabolic}}=\frac{2}{1+\sqrt{{\left(2mL\right)}^2+1}}$E19

For circular fins of rectangular profile,

${\eta }_{f,\text{rectangular}}=C\frac{K_1\left(m{r}_1\right)I_1\left(m{r}_{2c}\right)-I_1\left(m{r}_1\right)K_1\left(m{r}_{2c}\right)}{I_0\left(m{r}_1\right)K_1\left(m{r}_{2c}\right)-K_0\left(m{r}_1\right)I_1\left(m{r}_{2c}\right)}$E20

where the mathematical functions I and K are the modified Bessel functions and

$m=\sqrt{2h/kt}$E21

where t is the fin thickness.

and

$C=\frac{2r_1/m}{r_{2c}^2-r_1^2}$E22

where

$r_{2c}=r_2+t/2$E23

For pin fins of rectangular profile,

${\eta }_{f,\text{pin},\text{rectangular}}=\frac{\text{tan}hm{L}_c}{m{L}_c}$E24

where

$m=\sqrt{4h/kD}$E25

and corrected fin length, L_c, defined as

$L_c=L+D/4$E26

where L is the fin length and D is the diameter of the cylindrical fins. The corrected fin length is an approximate, yet practical and accurate way of accounting for the loss from the fin tip is to replace the fin length L in the relation for the insulated tip case.

A is the total surface area on one side

$A=A_u+A_f$E27

The overall heat transfer coefficient is based on the outside surface area of the wall for the finned tubular heat exchangers,

$U_o=\frac{1}{\frac{A_o}{A_i}\frac{1}{{\eta }_i{h}_i}+\frac{A_o}{A_i}\frac{R_{fi}}{{\eta }_i}+A_o{R}_w+\frac{R_{fo}}{{\eta }_o}+\frac{1}{{\eta }_o{h}_o}}$E28

where A_o and A_i represent the total surface area of the outer and inner surfaces, respectively.

4. Thermal design for recuperators

Four methods are used for the recuperator thermal performance analysis: log-mean temperature difference (LMTD), effectiveness-number of transfer units ($\epsilon -\text{NTU}$), dimensionless mean temperature difference ($\Psi -P$) and (P₁ – P₂) methods.

4.1. The log-mean temperature difference (LMTD) method

The use of the method is clearly facilitated by knowledge of the hot and cold fluid inlet and outlet temperatures. Such applications may be classified as heat exchanger design problems; that is, problems in which the temperatures and capacity rates are known, and it is desired to size the exchanger.

4.1.1. Parallel and counter flow heat exchanger

Two types of flow arrangement are possible in a double-pipe heat exchanger: parallel flow and counter flow. In parallel flow, both the hot and cold fluids enter the heat exchanger at the same end and move in the same direction, as shown in Figure 1. In counter flow, the hot and cold fluids enter the heat exchanger at opposite end and flow in opposite direction, as shown in Figure 2.

Figure 1.

Parallel flow in a double-pipe heat exchanger.

Figure 2.

Counter flow in a double-pipe heat exchanger.

The heat transfer rate is

${\displaystyle \dot{Q}}=UA\text{Δ}{T}_{lm}$E29

where $\text{Δ}{T}_{lm}$ is log-mean temperature difference and is

$\text{Δ}{T}_{lm}=\frac{\text{Δ}{T}_1-\text{Δ}{T}_2}{\text{ln}(\frac{\text{Δ}{T}_1}{\text{Δ}{T}_2})}$E30

Then,

${\displaystyle \dot{Q}}=UA\frac{\text{Δ}{T}_1-\text{Δ}{T}_2}{\text{ln}(\frac{\text{Δ}{T}_1}{\text{Δ}{T}_2})}$E31

where the endpoint temperatures, $\text{Δ}{T}_1$ and $\text{Δ}{T}_2$, for the parallel flow exchanger are

$\text{Δ}{T}_1=T_{hi}-T_{ci}$E32

$\text{Δ}{T}_2=T_{ho}-T_{co}$E33

where T_hi is the hot fluid inlet temperature, T_ci is the cold fluid inlet temperature, T_ho is the hot fluid outlet temperature and T_co is the cold fluid outlet temperature.

The endpoint temperatures, $\text{Δ}{T}_1$ and $\text{Δ}{T}_2$, for the counter flow exchanger are

$\text{Δ}{T}_1=T_{hi}-T_{co}$E34

$\text{Δ}{T}_2=T_{ho}-T_{ci}$E35

4.1.2. Multipass and cross-flow heat exchanger

In compact heat exchangers, the two fluids usually move perpendicular to each other, and such flow configuration is called cross-flow. The cross-flow is further classified as unmixed and mixed flow, depending on the flow configuration, as shown in Figures 3 and 4.

Figure 3.

Both fluids unmixed.

Figure 4.

One fluid mixed and one fluid unmixed.

Multipass flow arrangements are frequently used in shell-and-tube heat exchangers with baffles (Figure 5).

Figure 5.

One shell pass and two tube passes.

Log-mean temperature difference $\text{Δ}{T}_{lm}$ is computed under assumption of counter flow conditions. Heat transfer rate is

${\displaystyle \dot{Q}}=UAF\text{Δ}{T}_{lm,cf}$E36

where F is a correction factor and non-dimensional and depends on temperature effectiveness P, the heat capacity rate ratio R and the flow arrangement.

$P=\frac{T_{c2}-T_{c1}}{T_{h1}-T_{c1}}$E37

$R=\frac{T_{h1}-T_{h2}}{T_{c2}-T_{c1}}$E38

The value of P ranges from 0 to 1. The value of R ranges from 0 to infinity. If the temperature change of one fluid is negligible, either P or R is zero and F is 1. Hence, the exchanger behaviour is independent of the specific configuration. Such would be the case if one of the fluids underwent a phase change.

Correction factor F charts for common shell-and-tube and cross-flow heat exchangers are shown in Figures 6–10.

Figure 6.

One shell pass and any multiple of two tube passes.

Figure 7.

Two shell passes and four-tube passes.

Figure 8.

Single pass cross flow with one fluid mixed and the other unmixed.

Figure 9.

Single pass cross flow with both fluids unmixed.

Figure 10.

Two shell passes and any multiple of four tube passes.

4.1.3. The procedure to be followed with the LMTD method

1. Select the type of heat exchanger.

2. Calculate any unknown inlet or outlet temperatures and the heat transfer rate.

3. Calculate the log-mean temperature difference and the correction factor, if necessary.

4. Calculate the overall heat transfer coefficient.

5. Calculate the heat transfer surface area.

6. Calculate the length of the tube or heat exchanger

4.2. The ε – NTU method

If the exchanger type and size are known and the fluid outlet temperatures need to be determined, the application is referred to as a performance calculation problem. Such problems are best analysed by the NTU-effectiveness method [4, 5].

Capacity rate ratio is

$C^{*}=\frac{C_{\text{min}}}{C_{\text{max}}}$E39

where C_min and C_max are the smaller and larger of the two magnitudes of C_h and C_c, respectively, and C_h and C_c are the hot and cold fluid heat capacity rates, respectively.

Heat exchanger effectiveness εis defined as

$\epsilon =\frac{{\displaystyle \dot{Q}}}{{{\displaystyle \dot{Q}}}_{\text{max}}}=\frac{\text{Actual heat transfer rate}}{\text{Maximum possible heat transfer rate}}$E40

where

${{\displaystyle \dot{Q}}}_{\text{max}}={\left({\displaystyle \dot{m}}{c}_p\right)}_c(T_{h1}-T_{c1})\text{if}{C}_c<C_h$E41

or

${{\displaystyle \dot{Q}}}_{\text{max}}={\left({\displaystyle \dot{m}}{c}_p\right)}_h(T_{h1}-T_{c1})\text{if}{C}_h<C_c$E42

where $C_c={\displaystyle \dot{m_c}}{c}_{pc}$ and $C_h={\displaystyle \dot{m_h}}{c}_{ph}$ are the heat capacity rates of the cold and the hot fluids, respectively, and ${\displaystyle \dot{m}}$ is the rate of mass flow and c_p is specific heat at constant pressure.

Heat exchanger effectiveness is therefore written as

$\epsilon =\frac{C_h(T_{h1}-T_{h2})}{C_{\text{min}}(T_{h1}-T_{c1})}=\frac{C_c(T_{c2}-T_{c1})}{C_{\text{min}}(T_{h1}-T_{c1})}$E43

The number of transfer unit (NTU) is defined as a ratio of the overall thermal conductance to the smaller heat capacity rate. NTU designates the non-dimensional heat transfer size or thermal size of the exchanger [4, 5].

$\text{NTU}=\frac{UA}{C_{\text{min}}}=\frac{1}{C_{\text{min}}}{{\displaystyle \int }}^{\text{}}UdA$E44

In evaporator and condenser for parallel flow and counter flow,

$C^{*}=\frac{C_{\text{min}}}{C_{\text{max}}}=0$E45

and

$\epsilon =1-e^{-\text{NTU}}$E46

The effectivenesses of some common types of heat exchangers are also plotted in Figures 11–16.

Figure 11.

Effectiveness of parallel flow.

Figure 12.

Effectiveness of counter flow.

Figure 13.

Effectiveness of one shell pass and 2, 4, 6,… tube passes.

Figure 14.

Effectiveness of two shell passes and 4, 8, 12,… tube passes.

Figure 15.

Effectiveness of cross flow with both fluids unmixed.

Figure 16.

Effectiveness of cross flow with one fluid mixed and the other unmixed.

4.2.1. The procedure to be followed with the ε – NTU method

1. For the rating analysis:

   1. Calculate the capacity rate ratio

   2. Calculate NTU.

   3. Determine the effectiveness.

   4. Calculate the total heat transfer rate.

   5. Calculate the outlet temperatures.

2. For the sizing problem:

   1. Calculate the effectiveness.

   2. Calculate the capacity rate ratio.

   3. Calculate the overall heat transfer coefficient.

   4. Determine NTU.

   5. Calculate the heat transfer surface area.

   6. Calculate the length of the tube or heat exchanger

4.3. The Ψ – P method

The dimensionless mean temperature difference is [4]

$\psi =\frac{\text{Δ}{T}_m}{T_{hi}-T_{ci}}=\frac{\text{Δ}{T}_m}{\text{Δ}{T}_{\text{max}}}$E47

$\psi =\frac{\epsilon }{\text{NTU}}=\frac{P_1}{{\text{NTU}}_1}=\frac{P_2}{{\text{NTU}}_2}$E48

where P is the temperature effectiveness and the temperature effectivenesses of fluids 1 and 2 are defined as, respectively

$P_1=\frac{T_{1,o}-T_1}{T_{2,i}-T_{1,i}}$E49

$P_2=\frac{T_{2,i}-T_{2,o}}{T_{2,i}-T_{1,i}}$E50

$\psi =\{\begin{array}{c}\frac{F{P}_1\left(1-R_1\right)}{ln\left[\frac{\left(1-R_1{P}_1\right)}{\left(1-P_1\right)}\right]} \text{for} R_1\ne 1\\ F\left(1-P_1\right) \text{for} R_1=1\end{array}$E51

where 1 and 2 are fluid stream 1 and fluid stream 2, respectively, and R is the heat capacity ratio and defined as

$R_1=\frac{C_1}{C_2}=\frac{T_{2,i}-T_{2,o}}{T_{1,o}-T_{1,i}}$E52

$R_2=\frac{C_2}{C_1}=\frac{T_{1,o}-T_{1,i}}{T_{2,i}-T_{2,o}}$E53

$R_1=\frac{1}{R_2}$E54

Non-dimensional mean temperature difference as a function for P₁ and R₁ with the lines for constant values of NTU₁ and the factor is shown in Figure 17.

Figure 17.

Non-dimensional mean temperature difference as a function for P₁ and R_1.

The heat transfer rate is given by

$q=UA\mathrm{\Psi }\left(T_{hi}-T_{ci}\right)$E55

4.3.1. The procedure to be followed with the Ψ – P method

1. Calculate NTU₁.

2. Calculate F factor.

3. Calculate R₁ with the lines for constant values of NTU₁ and the F factor superimposed in Figure 17.

4. Plot the dimensionless mean temperature Ψ as a function of P₁ and R₁ in Figure 17.

5. Calculate the heat transfer rate.

4.4. The P_l – P₂ method

The dimensionless mean temperature difference is [4]

$\psi =\frac{\epsilon }{\text{NTU}}=\frac{P_1}{{\text{NTU}}_1}=\frac{P_2}{{\text{NTU}}_2}$E56

P₁ –P₂ chart for 1–2 shell and tube heat exchanger [2] with shell fluid mixed is shown in Figure 18.

Figure 18.

P₁ – P₂ chart for 1–2 shell and tube heat exchanger with shell fluid mixed.

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

4.4.1. The procedure to be followed with the P₁ – P₂ method

1. Calculate NTU₁ or NTU₂.

2. Calculate R₁ or R₂.

3. Plot P₁ as a function of R₁ with NTU₁ or P₂ as a function of R₂ with NTU₂ in Figure 18.

4. Calculate the dimensionless mean temperature Ψ.

5. Calculate the heat transfer rate.

5. Thermal design for regenerators

Two methods are used for the regenerator thermal performance analysis: $\epsilon -{\text{NTU}}_{\text{o}}$ and $\Lambda -\pi$ methods, respectively, for rotary and fixed matrix regenerators.

5.1. The ε – NTU_o method

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

${\text{NTU}}_{\text{o}}=\frac{1}{C_{\text{min}}}\left[\frac{1}{\frac{1}{{(hA)}_h}+\frac{1}{{(hA)}_c}}\right]$E57

$C^{*}=\frac{C_{\text{min}}}{C_{\text{max}}}$E58

$C_r^{*}=\frac{C_r}{C_{\text{min}}}$E59

$C_r=M_w{c}_{w}N$E60

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

$M_w=A_{rc}{H}_r{\rho }_m{S}_m$E61

where A_rc is the rotor cross-sectional area, H_r is the rotor height, ρ_m is the matrix material density and S_m is the matrix solidity.

The convection conductance ratio is

${\left(hA\right)}^{*}=\frac{\left(hA\right){ }_{C_{\text{min}}}}{{\left(hA\right)}_{C_{\text{max}}}}$E62

Most regenerators operate in the range $\text{of }0.25\le {\left(hA\right)}^{\text{*}}<4$. The effect of ${\left(hA\right)}^{\text{*}}$on the regenerator effectiveness can usually be ignored.

A is the total matrix surface area and given as

$A=A_{rc}{H}_r\beta F_{rfa}$E63

where A_rc is the rotor cross-sectional area, H_r is the rotor height, β is the matrix packing density and F_rfa 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.

$A_h=\left(\frac{{\alpha }_h}{{360}^{°}}\right)A$E64

$A_c=\left(\frac{{\alpha }_c}{{360}^{°}}\right)A$E65

where ${\alpha }_h$ and ${\alpha }_c$ are disk sector angles of hot flow and cold flow in degree, respectively.

The regenerator effectiveness is

$\epsilon =\frac{q}{q_{\text{max}}}$E66

$q_{\text{max}}=C_{\text{min}}\left(T_{hi}-T_{ci}\right)$E67

5.1.1. The counter flow regenerator

The regenerator effectiveness for $\epsilon \le 0.9$ is

$\epsilon ={\epsilon }_{cf}\left(1-\frac{1}{9C_r^{*1.93}}\right)$E68

where ${\epsilon }_{cf}$ is the counter flow recuperator effectiveness and is determined as

${\epsilon }_{cf}=\frac{1-\text{exp}\left[-{\text{NTU}}_o\left(1-C^{*}\right)\right]}{1-C^{*}\text{exp}\left[-{\text{NTU}}_o\left(1-C^{*}\right)\right]}$E69

The counter flow regenerator effectiveness as a function of NTU_o and for C* = 1 is presented in Figure 19. The regenerator effectiveness increases with $C_r^{*}$ for given values of NTU_o and $C^{*}$. The range of the optimum value of $C_r^{*}$ is between 2 and 4 for optimum regenerator effectiveness.

Figure 19.

The counter flow regenerator effectiveness as a function of NTU_o and for C* = 1.

5.1.2. The parallel flow regenerator

The parallel flow regenerator effectiveness as a function of NTU_o and for C* = 1 and (hA)* = 1 is presented in Figure 20.

Figure 20.

The parallel flow regenerator effectiveness as a function of NTU_o and for C* = 1 and (hA)* = 1.

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

1. Calculate the capacity rate ratio.

2. Calculate (hA)*.

3. Calculate (C_r)*.

4. Calculate NTU_o.

5. Determine the effectiveness.

6. Calculate the total heat transfer rate.

7. Calculate the outlet temperatures.

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]

$\Lambda =bL$E70

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

${\text{Λ}}_{\text{h}}={\left(\frac{hA}{C}\right)}_h=nt{u}_h$E71

${\text{Λ}}_{\text{c}}={\left(\frac{hA}{C}\right)}_c=nt{u}_c$E72

The reduced period is

$\pi =c{P}_h \text{or} c{P}_c$E73

where b and c are constants.

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

${\pi }_h={\left(\frac{hA}{C_r}\right)}_h$E74

${\pi }_c={\left(\frac{hA}{C_r}\right)}_c$E75

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:

Regenerator
Balanced	$\frac{{\Lambda }_h}{{\Pi }_h}=\frac{{\Lambda }_c}{{\Pi }_c}\text{or}\gamma =1$
Unbalanced	$\frac{{\Lambda }_h}{{\Pi }_h}\ne \frac{{\Lambda }_c}{{\Pi }_c}$
Symmetric	${\pi }_h={\pi }_c$
Unsymmetric	${\pi }_h\ne {\pi }_c$
Symmetric and balanced	${\Lambda }_h={\Lambda }_c, {\pi }_h={\pi }_c$
Unsymmetric and balanced	$\frac{{\Lambda }_h}{{\Pi }_h}=\frac{{\Lambda }_c}{{\Pi }_c}$
Long	$\Lambda /\Pi >5$

Table 1.

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

${\Lambda }_h={\Lambda }_c=\Lambda ={\Lambda }_m=\frac{hA}{{\displaystyle \dot{m}}{c}_p}=ntu$E76

${\pi }_h={\pi }_c=\pi ={\pi }_m=\frac{hAP}{M_w{c}_w}$E77

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

$Q=C_h{P}_h\left(T_{hi}-T_{ho}\right)=C_c{P}_c\left(T_{co}-T_{ci}\right)$E78

The maximum possible heat transfer is

$Q_{\text{max}}={\left(CP\right)}_{\text{min}}\left(T_{hi}-T_{ci}\right)$E79

The effectiveness for a fixed-matrix regenerator is

$\epsilon =\frac{Q}{Q_{\text{max}}}=\frac{{\left(CP\right)}_h\left(T_{hi}-T_{ho}\right)}{{\left(CP\right)}_{\text{min}}\left(T_{hi}-T_{ci}\right)}=\frac{{\left(CP\right)}_c\left(T_{co}-T_{ci}\right)}{{\left(CP\right)}_{\text{min}}\left(T_{hi}-T_{ci}\right)}$E80

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

Figure 21.

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

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

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

1. Calculate the reduced length.

2. Calculate the reduced period.

3. Calculate C*.

4. Calculate (C_r)*.

5. Calculate NTU_o.

6. Determine the effectiveness.

7. Calculate the total heat transfer rate.

8. Calculate the outlet temperatures.

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 ($\epsilon -\text{NTU}$), dimensionless mean temperature difference ($\Psi -P$) and (P₁ – 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 ($\epsilon -\text{NTU}$) method can be used. ($\Psi -P$) and (P₁ – P₂) methods are graphical methods. The (P₁ – 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 ($\epsilon -{\text{NTU}}_o$) and reduced length and reduced period ($\Lambda -\pi$) methods for the regenerators. ($\Lambda -\pi$) method is generally used for fixed matrix regenerators.

Nomenclature

A

Total heat transfer surface area of heat exchanger, total heat transfer surface area of all matrices of a regenerator, m²

A_f

Fin surface area, m²

A_rc

Rotor cross-sectional area, m²

C

Flow stream heat capacity rate, W/K

C_W

Matrix heat capacity rate, W/K

c_p

Specific heat at constant pressure, J/kgK

c_w

Specific heat of wall material, J/kgK

d, D

Diameter, m

E

Total energy, kJ

F_rfa

Fraction of rotor face area not covered by radial seals.

H_r

Rotor height

h

Specific enthalpy, kJ/kg

k

Thermal conductivity, W/mK

L

Length of heat exchanger, m

${\displaystyle \dot{m}}$

Mass flow rate, kg/s

M_w

The total mass of all matrices of a regenerator, kg

N

Rotational speed for a rotary regenerator, rev/s, rpm

NTU

Number of transfer units

ntu_c

Number of transfer units based on the cold fluid side

ntu_h

Number of transfer units based on the hot fluid side

P

Temperature effectiveness for one fluid stream

${\displaystyle \dot{Q}}$

Heat transfer rate, kW

r

Tube radius, m

R

Thermal resistance, m²K/W

R_f

Fouling resistance, fouling factor, m²K/W

S_m

Matrix solidity

T

Temperature, °C, K

T_c

Cold fluid temperature, °C, K

T_h

Hot fluid temperature, °C, K

t

Wall thickness, m

ΔT_lm

Log-mean temperature difference, °C, K

U

Overall heat transfer coefficient, W/m²K

V

Velocity, m/s

${\displaystyle \dot{W}}$

Power, kW

z

Elevation, m

Greek symbols

β

packing density for a regenerator, m²/m³

Δ

Difference

ε

Effectiveness

ρ_m

Matrix material density, kg/m³

η

Efficiency

Λ

Reduced length for a regenerator

$\pi$

Reduced period for a regenerator

Subscripts

c

Cold fluid

cf

Counter flow

cv

Control volume

e

Exit conditions

f

Fin, finned, friction

h

Hot

i

Inlet conditions, inner, inside

lm

logarithmic mean

max

Maximum

min

Minimum

o

Outer, outside, overall

u

Unfinned

1

Initial or inlet state, fluid 1

2

Final or exit state, fluid 2