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.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

This chapter discusses the basic design methods for two fluid heat exchangers. We discuss the log-mean temperature difference (LMTD) method, the effectiveness ε � NTU method, dimensionless mean temperature difference (Ψ � P) and (P<sup>1</sup> – 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 (ε � NTUo) and reduced length and reduced period (Λ � π) methods for regenerators.

#### 2. Governing equations

The energy rate balance is

$$\frac{d\mathcal{E}\_{cv}}{dt} = \dot{\mathcal{Q}} - \dot{\mathcal{W}} + \sum\_{i} \dot{m}\_{i} \left( h\_{i} + \frac{V\epsilon^{2}}{2} + g\mathbf{z}\_{i} \right) - \sum\_{\epsilon} \dot{m}\_{\epsilon} \left( h\_{\epsilon} + \frac{V\epsilon^{2}}{2} + g\mathbf{z}\_{\epsilon} \right) \tag{1}$$

For a control volume at steady state, dEcv dt ¼ 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 <sup>W</sup>\_ <sup>¼</sup> 0 and single-stream (only one inlet and one exit) and from the steady-state form the heat transfer rate becomes simply [1–3]

$$
\dot{Q} = \dot{m}(h\_2 - h\_1) \tag{2}
$$

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

$$
\dot{Q} = \dot{m}(h\_{h1} - h\_{h2}) \tag{3}
$$

For cold fluids,

$$
\dot{Q} = \dot{m}(h\_{c2} - h\_{c1})\tag{4}
$$

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

$$
\dot{Q} = \mathcal{U} A \Delta T\_{lm} \tag{5}
$$

where U is the overall heat transfer coefficient, whose unit is W/m2 oC and ΔTlm 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.

Basic Design Methods of Heat Exchanger Basic Design Methods of Heat Exchanger 3 http://dx.doi.org/10.5772/67888 11

$$
\mathcal{U}IA = \mathcal{U}\_o A\_o = \mathcal{U}\_i A\_i = \frac{1}{\mathcal{R}\_t} \tag{6}
$$

where Ai ¼ πDiL and Ao ¼ πDoL and U is the overall heat transfer coefficient based on that area. Rt is the total thermal resistance and can be expressed as [1]

$$R\_t = \frac{1}{UA} = \frac{1}{h\_i A\_i} + R\_w + \frac{R\_{fi}}{A\_i} + \frac{R\_{fo}}{A\_o} + \frac{1}{h\_o A\_o} \tag{7}$$

where Rf is fouling resistance (factor) and Rw is wall resistance and is obtained from the following equations.

For a bare plane wall

This chapter discusses the basic design methods for two fluid heat exchangers. We discuss the log-mean temperature difference (LMTD) method, the effectiveness ε � NTU method, dimensionless mean temperature difference (Ψ � P) and (P<sup>1</sup> – 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

(ε � NTUo) and reduced length and reduced period (Λ � π) methods for regenerators.

Vi<sup>2</sup> <sup>2</sup> <sup>þ</sup> gzi � �

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 <sup>W</sup>\_ <sup>¼</sup> 0 and single-stream (only one inlet and one

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

where U is the overall heat transfer coefficient, whose unit is W/m2 oC and ΔTlm is log-mean

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

�<sup>X</sup> e

m\_ <sup>e</sup> he þ

dt ¼ 0. Changes in the kinetic and potential energies of

<sup>Q</sup>\_ <sup>¼</sup> <sup>m</sup>\_ <sup>ð</sup>h<sup>2</sup> � <sup>h</sup>1<sup>Þ</sup> (2)

<sup>Q</sup>\_ <sup>¼</sup> <sup>m</sup>\_ <sup>ð</sup>hh<sup>1</sup> � hh2<sup>Þ</sup> (3)

<sup>Q</sup>\_ <sup>¼</sup> <sup>m</sup>\_ <sup>ð</sup>hc<sup>2</sup> � hc1<sup>Þ</sup> (4)

<sup>Q</sup>\_ <sup>¼</sup> UAΔTlm (5)

Ve<sup>2</sup> <sup>2</sup> <sup>þ</sup> gze � �

(1)

2. Governing equations

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

dEcv

For a control volume at steady state, dEcv

dt <sup>¼</sup> <sup>Q</sup>\_ � <sup>W</sup>\_ <sup>þ</sup><sup>X</sup>

i

m\_ <sup>i</sup> hi þ

exit) and from the steady-state form the heat transfer rate becomes simply [1–3]

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

The energy rate balance is

For hot fluids,

For cold fluids,

temperature difference.

to the cold fluid by convection.

3. Overall heat transfer coefficient

$$R\_w = \frac{t}{kA} \tag{8}$$

where t is the thickness of the wall

For a cylindrical wall

$$R\_w = \frac{\ln(\frac{r\_o}{r\_i})}{2\pi Lk} \tag{9}$$

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

$$\mathcal{U}\_o = \frac{1}{\frac{r\_o}{r\_i}\frac{1}{h\_i} + \frac{r\_o}{r\_i}\ R\_{fi} + \frac{r\_o}{k}\ln\left(\frac{r\_o}{r\_i}\right) + \mathcal{R}\_{fo} + \frac{1}{h\_o}}\tag{10}$$

where Rfi and Rfo are fouling resistance of the inside and outside surfaces, respectively. or

$$\mathcal{U}\_o = \frac{1}{\frac{r\_o}{r\_i}\frac{1}{h\_i} + R\_{f\ell} + \frac{r\_o}{k}\ln\left(\frac{r\_o}{r\_i}\right) + \frac{1}{h\_o}}\tag{11}$$

where Rft is the total fouling resistance, given as

$$R\_{fi} = \frac{A\_o}{A\_i} R\_{fi} + R\_{fo} \tag{12}$$

For finned surfaces,

$$
\dot{Q} = \eta h A \Delta T \tag{13}
$$

where η is the overall surface efficiency and

$$\eta = 1 - \frac{A\_f}{A} (1 - \eta\_f) \tag{14}$$

where Af is fin surface area and η<sup>f</sup> is fin efficiency and is defined as

$$
\eta\_f = \frac{\dot{Q}\_f}{\dot{Q}\_{f,\text{max}}} \tag{15}
$$

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

$$
\eta\_{f, \text{long}} = \frac{1}{mL} \tag{16}
$$

$$\eta\_{f, \text{insulated}} = \frac{\tanh(mL)}{mL} \tag{17}$$

where L is the fin length.

For straight triangular fins,

$$
\eta\_{f, \text{triangle}} = \frac{1}{mL} \frac{I\_1(2mL)}{I\_0(2mL)} \tag{18}
$$

For straight parabolic fins,

$$\eta\_{f, \text{parabolic}} = \frac{2}{1 + \sqrt{\left(2mL\right)^2 + 1}}\tag{19}$$

For circular fins of rectangular profile,

$$\eta\_{f, \text{rectangular}} = \mathcal{C} \frac{K\_1(mr\_1)I\_1(mr\_{2c}) - I\_1(mr\_1)K\_1(mr\_{2c})}{I\_0(mr\_1)K\_1(mr\_{2c}) - K\_0(mr\_1)I\_1(mr\_{2c})} \tag{20}$$

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

$$m = \sqrt{2h/kt} \tag{21}$$

where t is the fin thickness.

and

$$C = \frac{2r\_1/m}{r\_{2c}^2 - r\_1^2} \tag{22}$$

where

Basic Design Methods of Heat Exchanger Basic Design Methods of Heat Exchanger 5 http://dx.doi.org/10.5772/67888 13

$$r\_{2c} = r\_2 + t/2\tag{23}$$

For pin fins of rectangular profile,

$$
\eta\_{f, \text{pin, rectangular}} = \frac{\tanh mL\_c}{mL\_c} \tag{24}
$$

where

<sup>η</sup> <sup>¼</sup> <sup>1</sup> � Af

<sup>η</sup><sup>f</sup> <sup>¼</sup> <sup>Q</sup>\_ <sup>f</sup> <sup>Q</sup>\_ <sup>f</sup> ,max

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

<sup>η</sup><sup>f</sup> ,long <sup>¼</sup> <sup>1</sup>

<sup>η</sup><sup>f</sup> ,insulated <sup>¼</sup> tanhðmL<sup>Þ</sup>

mL

I1ð2mLÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2mLÞ

<sup>2</sup> <sup>þ</sup> <sup>1</sup>

<sup>η</sup><sup>f</sup> ,triangular <sup>¼</sup> <sup>1</sup>

<sup>η</sup><sup>f</sup> ,parabolic <sup>¼</sup> <sup>2</sup>

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

1 þ

<sup>η</sup><sup>f</sup> ,rectangular <sup>¼</sup> <sup>C</sup> <sup>K</sup>1ðmr1ÞI1ðmr2cÞ � <sup>I</sup>1ðmr1ÞK1ðmr2c<sup>Þ</sup>

<sup>m</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffi

<sup>C</sup> <sup>¼</sup> <sup>2</sup>r1=<sup>m</sup> r2 <sup>2</sup><sup>c</sup> � <sup>r</sup><sup>2</sup> 1

where Af is fin surface area and η<sup>f</sup> is fin efficiency and is defined as

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

expressed as

where L is the fin length. For straight triangular fins,

For straight parabolic fins,

where t is the fin thickness.

and

where

For circular fins of rectangular profile,

<sup>A</sup> <sup>ð</sup><sup>1</sup> � <sup>η</sup>f<sup>Þ</sup> (14)

mL (16)

mL (17)

<sup>I</sup>0ð2mL<sup>Þ</sup> (18)

<sup>q</sup> (19)

<sup>I</sup>0ðmr1ÞK1ðmr2cÞ � <sup>K</sup>0ðmr1ÞI1ðmr2c<sup>Þ</sup> (20)

2h=kt p (21)

(15)

(22)

$$m = \sqrt{4h/kD} \tag{25}$$

and corrected fin length, Lc, defined as

$$L\_c = L + D/4\tag{26}$$

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

$$A = A\_{\mathfrak{u}} + A\_{\mathfrak{f}} \tag{27}$$

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

$$\mathcal{U}\_o = \frac{1}{\frac{A\_o}{A\_i}\frac{1}{\eta\_i h\_i} + \frac{A\_o}{A\_i}\frac{R\_{\hat{\imath}i}}{\eta\_i} + A\_o R\_w + \frac{R\_{\hat{\imath}o}}{\eta\_o} + \frac{1}{\eta\_o h\_o}}\tag{28}$$

where Ao and Ai 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 (ε � NTU), dimensionless mean temperature difference (Ψ � P) and (P<sup>1</sup> – 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.

The heat transfer rate is

4.1.1. Parallel and counter flow heat exchanger

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

Figure 1. Parallel flow in a double-pipe heat exchanger.

Figure 2.

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

$$
\dot{Q} = \mathcal{U} A \Delta T\_{lm} \tag{29}
$$

where ΔTlm is log-mean temperature difference and is

$$
\Delta T\_{lm} = \frac{\Delta T\_1 - \Delta T\_2}{\ln(\frac{\Delta T\_1}{\Delta T\_2})} \tag{30}
$$

Then,

$$\dot{Q} = UA \frac{\Delta T\_1 - \Delta T\_2}{\ln(\frac{\Delta T\_1}{\Delta T\_2})} \tag{31}$$

where the endpoint temperatures, ΔT<sup>1</sup> and ΔT2, for the parallel flow exchanger are

$$
\Delta T\_1 = T\_{hi} - T\_{ci} \tag{32}
$$

$$
\Delta T\_2 = T\_{h\nu} - T\_{\alpha\nu} \tag{33}
$$

where Thi is the hot fluid inlet temperature, Tci is the cold fluid inlet temperature, Tho is the hot fluid outlet temperature and Tco is the cold fluid outlet temperature.

The endpoint temperatures, ΔT<sup>1</sup> and ΔT2, for the counter flow exchanger are

$$
\Delta T\_1 = T\_{hi} - T\_{\alpha \nu} \tag{34}
$$

$$
\Delta T\_2 = T\_{ho} - T\_{ci} \tag{35}
$$

#### 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.

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

Figure 3. Both fluids unmixed.

Figure 4. One fluid mixed and one fluid unmixed.

Then,

<sup>Q</sup>\_ <sup>¼</sup> UA <sup>Δ</sup>T<sup>1</sup> � <sup>Δ</sup>T<sup>2</sup> lnð ΔT<sup>1</sup> ΔT<sup>2</sup>

where Thi is the hot fluid inlet temperature, Tci is the cold fluid inlet temperature, Tho is the hot

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

Multipass flow arrangements are frequently used in shell-and-tube heat exchangers with

and mixed flow, depending on the flow configuration, as shown in Figures 3 and 4.

where the endpoint temperatures, ΔT<sup>1</sup> and ΔT2, for the parallel flow exchanger are

fluid outlet temperature and Tco is the cold fluid outlet temperature.

4.1.2. Multipass and cross-flow heat exchanger

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

baffles (Figure 5).

Figure 3. Both fluids unmixed.

The endpoint temperatures, ΔT<sup>1</sup> and ΔT2, for the counter flow exchanger are

<sup>Þ</sup> (31)

ΔT<sup>1</sup> ¼ Thi � Tci (32)

ΔT<sup>2</sup> ¼ Tho � Tco (33)

ΔT<sup>1</sup> ¼ Thi � Tco (34)

ΔT<sup>2</sup> ¼ Tho � Tci (35)

Figure 5. One shell pass and two tube passes.

Log-mean temperature difference ΔTlm is computed under assumption of counter flow conditions. Heat transfer rate is

$$
\dot{Q} = \mathcal{U}AF\Delta T\_{lm,cf} \tag{36}
$$

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.

$$P = \frac{T\_{c2} - T\_{c1}}{T\_{h1} - T\_{c1}} \tag{37}$$

$$R = \frac{T\_{h1} - T\_{h2}}{T\_{c2} - T\_{c1}} \tag{38}$$

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.

behaviour is independent of the specific configuration. Such would be the case if one of the

Correction factor F charts for common shell-and-tube and cross-flow heat exchangers are

fluids underwent a phase change.

1810 Heat Exchangers Heat Exchangers– Design, Experiment and Simulation – Design, Experiment and Simulation

Figure 7. Two shell passes and four-tube passes.

Figure 8. Single pass cross flow with one fluid mixed and the other unmixed.

Figure 6. One shell pass and any multiple of two tube passes.

shown in Figures 6–10.

Figure 10. Two shell passes and any multiple of four tube passes.

#### 4.1.3. The procedure to be followed with the LMTD method


#### 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

$$\mathbf{C}^\* = \frac{\mathbf{C}\_{\text{min}}}{\mathbf{C}\_{\text{max}}} \tag{39}$$

where Cmin and Cmax are the smaller and larger of the two magnitudes of Ch and Cc, respectively, and Ch and Cc are the hot and cold fluid heat capacity rates, respectively.

Heat exchanger effectiveness εis defined as

$$\varepsilon = \frac{\dot{Q}}{\dot{Q}\_{\text{max}}} = \frac{\text{Actual heat transfer rate}}{\text{Maximum possible heat transfer rate}} \tag{40}$$

where

$$\dot{Q}\_{\text{max}} = (\dot{m}c\_p)\_c (T\_{h1} - T\_{c1}) \text{ if } \mathbb{C}\_c < \mathbb{C}\_h \tag{41}$$

or

$$\dot{Q}\_{\text{max}} = (\dot{m}c\_p)\_h (T\_{h1} - T\_{c1}) \text{ if } \mathbb{C}\_h < \mathbb{C}\_c \tag{42}$$

where Cc ¼ m\_<sup>c</sup> cpc and Ch ¼ m\_<sup>h</sup> cph are the heat capacity rates of the cold and the hot fluids, respectively, and m\_ is the rate of mass flow and cp is specific heat at constant pressure.

Heat exchanger effectiveness is therefore written as

$$\varepsilon = \frac{\mathbb{C}\_h (T\_{h1} - T\_{h2})}{\mathbb{C}\_{\text{min}} (T\_{h1} - T\_{c1})} = \frac{\mathbb{C}\_c (T\_{c2} - T\_{c1})}{\mathbb{C}\_{\text{min}} (T\_{h1} - T\_{c1})} \tag{43}$$

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].

$$\text{NTU} = \frac{\text{ULA}}{\text{C}\_{\text{min}}} = \frac{1}{\text{C}\_{\text{min}}} \int \text{LIdA} \tag{44}$$

In evaporator and condenser for parallel flow and counter flow,

$$\mathbf{C}^\* = \frac{\mathbf{C}\_{\text{min}}}{\mathbf{C}\_{\text{max}}} = \mathbf{0} \tag{45}$$

and

$$
\varepsilon = 1 - e^{-\text{NTU}} \tag{46}
$$

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

Figure 11. Effectiveness of parallel flow.

<sup>C</sup>� <sup>¼</sup> <sup>C</sup>min Cmax

where Cmin and Cmax are the smaller and larger of the two magnitudes of Ch and Cc, respec-

<sup>¼</sup> Actual heat transfer rate

where Cc ¼ m\_<sup>c</sup> cpc and Ch ¼ m\_<sup>h</sup> cph are the heat capacity rates of the cold and the hot fluids,

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

> <sup>¼</sup> <sup>1</sup> Cmin ð

<sup>C</sup>� <sup>¼</sup> <sup>C</sup>min Cmax

ε ¼ 1 � e

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

respectively, and m\_ is the rate of mass flow and cp is specific heat at constant pressure.

<sup>ε</sup> <sup>¼</sup> ChðTh<sup>1</sup> � Th2<sup>Þ</sup> CminðTh<sup>1</sup> � Tc1Þ

> NTU <sup>¼</sup> UA Cmin

In evaporator and condenser for parallel flow and counter flow,

Maximum possible heat transfer rate (40)

max ¼ ðmc\_ <sup>p</sup>ÞcðTh<sup>1</sup> � Tc1Þ if Cc < Ch (41)

<sup>Q</sup>\_ max ¼ ðmc\_ <sup>p</sup>ÞhðTh<sup>1</sup> � Tc1<sup>Þ</sup> if Ch <sup>&</sup>lt; Cc (42)

<sup>C</sup>minðTh<sup>1</sup> � Tc1<sup>Þ</sup> (43)

UdA (44)

¼ 0 (45)

�NTU (46)

<sup>¼</sup> CcðTc<sup>2</sup> � Tc1<sup>Þ</sup>

tively, and Ch and Cc are the hot and cold fluid heat capacity rates, respectively.

Heat exchanger effectiveness εis defined as

2012 Heat Exchangers Heat Exchangers– Design, Experiment and Simulation – Design, Experiment and Simulation

where

or

and

<sup>ε</sup> <sup>¼</sup> <sup>Q</sup>\_ Q\_ max

Heat exchanger effectiveness is therefore written as

thermal size of the exchanger [4, 5].

Q\_

(39)

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

2214 Heat Exchangers Heat Exchangers– Design, Experiment and Simulation – Design, Experiment and Simulation

Figure 14. Effectiveness of two shell passes and 4, 8, 12,… tube passes.

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. Calculate the capacity rate ratio
	- 2. Calculate NTU.
	- 3. Determine the effectiveness.
	- 4. Calculate the total heat transfer rate.
	- 5. Calculate the outlet temperatures.
	- 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]

$$
\psi = \frac{\Delta T\_m}{T\_{hi} - T\_{ci}} = \frac{\Delta T\_m}{\Delta T\_{\text{max}}} \tag{47}
$$

$$
\psi = \frac{\varepsilon}{\text{NTU}} = \frac{P\_1}{\text{NTU}\_1} = \frac{P\_2}{\text{NTU}\_2} \tag{48}
$$

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

$$P\_1 = \frac{T\_{1,o} - T\_{1,i}}{T\_{2,i} - T\_{1,i}} \tag{49}$$

$$P\_2 = \frac{T\_{2,i} - T\_{2,o}}{T\_{2,i} - T\_{1,i}} \tag{50}$$

$$\psi = \begin{cases} \frac{FP\_1(1-R\_1)}{\ln\left[\frac{(1-R\_1P\_1)}{(1-P\_1)}\right]} & \text{for } R\_1 \neq 1\\ \frac{\ln\left[\frac{(1-R\_1)}{(1-P\_1)}\right]}{\ln\left[1-P\_1\right]\ln R\_1 = 1 \end{cases} \tag{51}$$

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

$$R\_1 = \frac{\mathcal{C}\_1}{\mathcal{C}\_2} = \frac{T\_{2,i} - T\_{2,o}}{T\_{1,o} - T\_{1,i}} \tag{52}$$

$$R\_2 = \frac{\mathcal{C}\_2}{\mathcal{C}\_1} = \frac{T\_{1,o} - T\_{1,i}}{T\_{2,i} - T\_{2,o}} \tag{53}$$

$$R\_1 = \frac{1}{R\_2} \tag{54}$$

Non-dimensional mean temperature difference as a function for P<sup>1</sup> and R<sup>1</sup> with the lines for constant values of NTU1 and the factor is shown in Figure 17.

The heat transfer rate is given by

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

1. Calculate the capacity rate ratio

2416 Heat Exchangers Heat Exchangers– Design, Experiment and Simulation – Design, Experiment and Simulation

3. Determine the effectiveness.

1. Calculate the effectiveness.

2. Calculate the capacity rate ratio.

3. Calculate the overall heat transfer coefficient.

6. Calculate the length of the tube or heat exchanger

<sup>ψ</sup> <sup>¼</sup> <sup>Δ</sup>Tm Thi � Tci

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

NTU1

where P is the temperature effectiveness and the temperature effectivenesses of fluids 1 and 2

<sup>P</sup><sup>1</sup> <sup>¼</sup> <sup>T</sup>1, <sup>o</sup> � <sup>T</sup>1,<sup>i</sup> T2,<sup>i</sup> � T1,<sup>i</sup>

<sup>P</sup><sup>2</sup> <sup>¼</sup> <sup>T</sup>2,<sup>i</sup> � <sup>T</sup>2, <sup>o</sup> T2,<sup>i</sup> � T1,<sup>i</sup>

� � for <sup>R</sup><sup>1</sup> 6¼ <sup>1</sup>

Fð1 � P1Þ for R<sup>1</sup> ¼ 1

FP1ð1 � R1Þ ln <sup>ð</sup><sup>1</sup> � <sup>R</sup>1P1<sup>Þ</sup> ð1 � P1Þ

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

ψ ¼

8 >>><

>>>:

<sup>¼</sup> <sup>Δ</sup>Tm ΔTmax

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

(47)

(48)

(49)

(50)

(51)

5. Calculate the heat transfer surface area.

The dimensionless mean temperature difference is [4]

4. Calculate the total heat transfer rate.

5. Calculate the outlet temperatures.

a. For the rating analysis:

2. Calculate NTU.

b. For the sizing problem:

4. Determine NTU.

4.3. The Ψ – P method

are defined as, respectively

$$q = \mathcal{U}A\Psi(T\_{hi} - T\_{ci})\tag{55}$$

Figure 17. Non-dimensional mean temperature difference as a function for P<sup>1</sup> and R1.

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


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

The dimensionless mean temperature difference is [4]

$$
\psi = \frac{\varepsilon}{\text{NTU}} = \frac{P\_1}{\text{NTU}\_1} = \frac{P\_2}{\text{NTU}\_2} \tag{56}
$$

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 Figure 18.

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

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

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

