**4. Design of a plate heat exchanger**

#### **4.1. Basic equations for the design of a plate heat exchanger**

The methodology employed for the design of a PHE is the same as for the design of a tubular heat exchanger. The equations given in the present chapter are appropriate for the chevron type plates that are used in most industrial applications.

#### *4.1.1. Parameters of a chevron plate*

The main dimensions of a *chevron* plate are shown in Figure 14. The corrugation angle, *β*, usually varies between extremes of 25° and 65° and is largely responsible for the pressure drop and heat transfer in the channels.

**Figure 14.** Parameters of a chevron plate.

The corrugations must be taken into account in calculating the total heat transfer area of a plate (effective heat transfer area):

$$A\_p = \spadesuit \mathcal{W}\_p \square\_p \tag{14}$$

where

AP = plate effective heat transfer area

Φ = plate area enlargement factor (range between 1.15 and 1.25)

WP = plate width

LP = plate length

The enlargement factor of the plate is the ratio between the plate effective heat transfer area, *AP* and the designed area (product of length and width *WP*.*L <sup>P</sup>*), and lies between 1.15 and 1.25. The plate length *L <sup>P</sup>* and the plate width *WP* can be estimated by the orifices distances. *L <sup>V</sup>* , *L <sup>H</sup>* , and the port diameter *Dp* are given by Eq. (15) and Eq. (16) [5].

$$L\_p \approx L\_V - D\_p \tag{15}$$

$$\mathcal{W}\_p \approx L\_H + D\_p \tag{16}$$

For the effective heat transfer area, the hydraulic diameter of the channel is given by the equivalent diameter, *De*, which is given by:

$$D\_e = \frac{2b}{\Phi} \tag{17}$$

where b is the channel average thickness.

#### *4.1.2. Heat transfer in the plates*

The heat transfer area is expressed as the global design equation:

$$Q = UA\Lambda T\_M \tag{18}$$

where *U* is the overall heat transfer coefficient, *A* is the total area of heat transfer and *ΔTM* is the effective mean temperature difference, which is a function of the inlet and outlet fluid temperatures, the specific heat, and the configuration of the exchanger. The total area of heat transfer can be given by:

$$A = \mathcal{N}\_p A\_p \tag{19}$$

where *NP* is the number of plates. The end plates, which do not exchange heat, are not taken into account in determining the area. The inner plates are usually called thermal plates in order to distinguish them from the adiabatic end plates. The overall heat transfer coefficient can be determined by:

$$LI = \frac{1}{\frac{1}{h\_{\text{hot}}} + \frac{t\_p}{k\_p} + \frac{1}{h\_{\text{cold}}} + R\_{f,\text{cold}} + R\_{f,\text{hot}}} \tag{20}$$

where

The corrugations must be taken into account in calculating the total heat transfer area of a plate

The enlargement factor of the plate is the ratio between the plate effective heat transfer area, *AP* and the designed area (product of length and width *WP*.*L <sup>P</sup>*), and lies between 1.15 and 1.25. The plate length *L <sup>P</sup>* and the plate width *WP* can be estimated by the orifices distances. *L <sup>V</sup>* ,

For the effective heat transfer area, the hydraulic diameter of the channel is given by the

where *U* is the overall heat transfer coefficient, *A* is the total area of heat transfer and *ΔTM* is the effective mean temperature difference, which is a function of the inlet and outlet fluid

*e <sup>b</sup> <sup>D</sup>* <sup>2</sup>

The heat transfer area is expressed as the global design equation:

*A WL P PP* = Φ. . (14)

*PV p LLD* » - (15)

*WLD PH p* » + (16)

<sup>Φ</sup> <sup>=</sup> (17)

*Q UA TM* = D (18)

(effective heat transfer area):

182 Heat Transfer Studies and Applications

AP = plate effective heat transfer area

equivalent diameter, *De*, which is given by:

where b is the channel average thickness.

*4.1.2. Heat transfer in the plates*

Φ = plate area enlargement factor (range between 1.15 and 1.25)

*L <sup>H</sup>* , and the port diameter *Dp* are given by Eq. (15) and Eq. (16) [5].

where

WP = plate width

LP = plate length

*hhot* = convective heat transfer coefficient of the hot fluid


The convective heat transfer coefficient, *h* , depends on the fluid properties, fluid velocity, and plate geometry.

#### *4.1.3. Design methods*

There are two main approaches used in the design of PHEs, namely the log-mean temperature difference and the thermal effectiveness methods. For the first method, the rate of heat transfer is given by:

$$Q = \mathsf{LLA}(F\Delta T\_{lu}) \tag{21}$$

where *ΔTlm* is the log-mean temperature difference, given by Eq. (22) and *F* is the log-mean temperature difference correction factor.

$$
\Delta T\_{lu} = \frac{\Delta T\_1 - \Delta T\_2}{\ln \left( \Delta T\_1 / \Delta T\_2 \right)} \tag{22}
$$

Where

$$\begin{aligned} \Delta T\_1 &= \begin{cases} T\_{\text{hot},in} - T\_{\text{cold},out} & \text{if } \text{countercurrent} \\ T\_{\text{hot},in} - T\_{\text{cold},in} & \text{if } \text{countercurrent} \end{cases} \\ \Delta T\_2 &= \begin{cases} T\_{\text{hot},out} - T\_{\text{cold},in} & \text{if } \text{countercurrent} \\ T\_{\text{hot},out} - T\_{\text{cold},out} & \text{if } \text{countercurrent} \end{cases} \end{aligned}$$

The correction factor is a function of the heat exchanger configuration and the dimensionless parameters *R* and *PC*. For purely countercurrent or concurrent (single-pass) arrangements, the correction factor is equal to one, while for multi-pass arrangements, it is always less than one. However, because the end channels of the PHE only exchange heat with one adjacent channel, different to the inner channels that exchange heat with two adjacent channels, purely coun‐ tercurrent or concurrent flow is only achieved in two extreme situations. These are:


The adimensional parameters *R* e *PC* are defined as:

$$\mathbf{R} = \frac{T\_{\text{hot,in}} - T\_{\text{hot,out}}}{T\_{\text{cold,out}} - T\_{\text{cold,in}}} = \frac{\{\dot{M}\mathcal{L}\_p\}\_{\text{cold}}}{\{\dot{M}\mathcal{L}\_p\}\_{\text{hot}}} \tag{23}$$

$$P\_C = \frac{T\_{\text{cold,out}} - T\_{\text{cold,in}}}{T\_{\text{hot,in}} - T\_{\text{cold,in}}} = \frac{\Delta T\_{\text{cold}}}{\Delta T\_{\text{max}}} \tag{24}$$

The second method provides a definition of heat exchanger effectiveness in terms of the ratio between the actual heat transfer and the maximum possible heat transfer, as shown in Eq. (25):

$$E = \frac{Q}{Q\_{\text{max}}} \tag{25}$$

The actual heat transfer can be achieved by an energy balance:

$$\mathcal{Q} = \left(\dot{M}c\_p\right)\_{\text{hot}} \left(T\_{\text{hot},in} - T\_{\text{hot},out}\right) \tag{26}$$

Modeling and Design of Plate Heat Exchanger http://dx.doi.org/10.5772/60885 185

$$\mathcal{Q} = \left(\dot{M}c\_p\right)\_{\text{cold}} \left(T\_{\text{cold,out}} - T\_{\text{cold,in}}\right) \tag{27}$$

Thermodynamically, *Qmax* represents the heat transfer that would be obtained in a pure countercurrent heat exchanger with infinite area. This can be expressed by:

$$\mathcal{Q}\_{\text{max}} = \left(\dot{\mathcal{M}}\mathcal{c}\_p\right)\_{\text{min}} \Delta T\_{\text{max}} \tag{28}$$

Using Eqs. (26), (27) and (28), the PHE effectiveness can be calculated as the ratio of tempera‐ tures:

$$E = \begin{cases} \frac{\Delta T\_{hot}}{\Delta T\_{max}} & \text{if } R > 1\\ \frac{\Delta T\_{cold}}{\Delta T\_{max}} & \text{if } R < 1 \end{cases} \tag{29}$$

The effectiveness depends on the PHE configuration, the heat capacity rate ratio (*R*), and the number of transfer units (NTU). The NTU is a dimensionless parameter that can be considered as a factor for the size of the heat exchanger, defined as:

$$NTUI = \frac{ULA}{(\dot{M}c\_p)\_{mlu}}\tag{30}$$

#### *4.1.4. Pressure drop in a plate heat exchanger*

Where

184 Heat Transfer Studies and Applications

neglected.

The adimensional parameters *R* e *PC* are defined as:

*R*

*C*

The actual heat transfer can be achieved by an energy balance:

*hot in cold out hot in cold in hot out cold in hot out cold out*

, ,

if countercurrent f concurrent

f countercurrent if concurrent

*T Ti T Ti*

, , , ,

, ,

tercurrent or concurrent flow is only achieved in two extreme situations. These are:

The correction factor is a function of the heat exchanger configuration and the dimensionless parameters *R* and *PC*. For purely countercurrent or concurrent (single-pass) arrangements, the correction factor is equal to one, while for multi-pass arrangements, it is always less than one. However, because the end channels of the PHE only exchange heat with one adjacent channel, different to the inner channels that exchange heat with two adjacent channels, purely coun‐

**i.** when the PHE has only one thermal plate, so that only two channels are formed by

**ii.** when the number of thermal plates is sufficiently large that the edge effect can be

*hot in hot out p cold cold out cold in p hot*

*cold out cold in cold*

*hot in cold in max*

The second method provides a definition of heat exchanger effectiveness in terms of the ratio between the actual heat transfer and the maximum possible heat transfer, as shown in Eq. (25):

*max*

( *p hot in hot out* ) ( ) *hot*

*<sup>Q</sup> <sup>E</sup>*

*T T Mc*

*T T Mc* , , , ,

*T T <sup>T</sup> <sup>P</sup> TT T* , , , ,


the end plates and the thermal plate, with each stream flowing through one channel;

( ) ( )

& (23)


*<sup>Q</sup>*<sup>=</sup> (25)

*Q Mc T T* , , <sup>=</sup> - & (26)

&

*T T*

*T T*

*T*

1

ìï - D = <sup>í</sup> ï - î

ìï - D = <sup>í</sup> ï - î

*T*

2

The pressure drop is an important parameter that needs to be considered in the design and optimization of a plate heat exchanger. In any process, it should be kept as close as possible to the design value, with a tolerance range established according to the available pumping power. In a PHE, the pressure drop is the sum of three contributions:


The pressure drop in the manifolds and ports should be kept as low as possible, because it is a waste of energy, has no influence on the heat transfer process, and can decrease the uni‐ formity of the flow distribution in the channels. It is recommended to keep this loss lower than 10% of the available pressure drop, although in some cases it can exceed 30% [3].

$$
\Delta P = \frac{2f \text{L}\_{\text{V}} \text{P} \text{G}\_{\text{C}}^{\text{-2}}}{\rho \text{D}\_{\text{e}}} + 1,4 \frac{\text{G}\_{\text{p}}}{2 \,\rho} + \rho g \text{L}\_{\text{V}} \tag{31}
$$

where *f* is the Fanning fator, given by Eq. (33), *P* is the number of passes and *GP* is the fluid mass velocity in the port, given by the ratio of the mass flow, *M*˙ , and the flow cross-sectional area, *πDP*<sup>2</sup> / 4.

$$\mathcal{G}\_p = \frac{4\dot{M}}{\pi \mathcal{D}\_p^{\,^2}}\tag{32}$$

$$f = \frac{K\_p}{Re^{\text{""}}} \tag{33}$$

The values for *Kp* and *m* are presented in Table 3 as function of the Reynolds number for some *β* values.

#### *4.1.5. Experimental heat transfer and friction correlations for the chevron plate PHE*

Due to the wide range of plate designs, there are various parameters and correlations available for calculations of heat transfer and pressure drop. Despite extensive research, there is still no generalized model. There are only certain specific correlations for features such as flow patterns, parameters of the plates, and fluid viscosity, with each correlation being limited to its application range. In this chapter, the correlation described in [14] was used.

$$N\mu = \mathcal{C}\_h (Re)^u (Pr)^{1/3} \left(\frac{\mu}{\mu\_w}\right)^{0.17} \tag{34}$$

where *μw* is the viscosity evaluated at the wall temperature and the dimensionless parameters Nusselt number (*Nu*), Reynolds number (*Re*) and Prandtl number (*Pr*) can be defined as:

$$Nu = \frac{hD\_c}{k} \text{ , } Re = \frac{G\_\text{C} D\_c}{\mu} \text{ , } Pr = \frac{c\_p \mu}{k} \text{ } \tag{35}$$

In Reynolds number equation, *GC* is the mass flow per channel and may be defined as the ratio between the mass velocity per channel *m*˙ and the cross sectional area of the flow channel (*bWP*):

$$G\_{\mathbb{C}} = \frac{\dot{m}}{bW\_p} \tag{36}$$

The constants *Ch* and *n*, which depend on the flow characteristics and the chevron angle, are given in Table 3.

#### **4.2. Optimization**

(32)

(34)

where *f* is the Fanning fator, given by Eq. (33), *P* is the number of passes and *GP* is the fluid mass velocity in the port, given by the ratio of the mass flow, *M*˙ , and the flow cross-sectional

*P*

*p m K*

The values for *Kp* and *m* are presented in Table 3 as function of the Reynolds number for some

Due to the wide range of plate designs, there are various parameters and correlations available for calculations of heat transfer and pressure drop. Despite extensive research, there is still no generalized model. There are only certain specific correlations for features such as flow patterns, parameters of the plates, and fluid viscosity, with each correlation being limited to

*w*

m

m

ç ÷ è ø

0,17

m

== = (35)

*bW* <sup>=</sup> & (36)

*<sup>f</sup> Re* <sup>=</sup> (33)

*P*

*4.1.5. Experimental heat transfer and friction correlations for the chevron plate PHE*

its application range. In this chapter, the correlation described in [14] was used.

*h*

*Nu C Re Pr*

*n*

1/3 ( )( )

æ ö <sup>=</sup> ç ÷

where *μw* is the viscosity evaluated at the wall temperature and the dimensionless parameters Nusselt number (*Nu*), Reynolds number (*Re*) and Prandtl number (*Pr*) can be defined as:

*<sup>p</sup> <sup>e</sup> C e hD G D c*

*k k* , ,

m

In Reynolds number equation, *GC* is the mass flow per channel and may be defined as the ratio between the mass velocity per channel *m*˙ and the cross sectional area of the flow channel

*P*

The constants *Ch* and *n*, which depend on the flow characteristics and the chevron angle, are

*Nu Re Pr*

*C*

*<sup>m</sup> <sup>G</sup>*

*<sup>M</sup> <sup>G</sup> D* <sup>2</sup> 4 p= &

area, *πDP*<sup>2</sup> / 4.

186 Heat Transfer Studies and Applications

*β* values.

(*bWP*):

given in Table 3.

Any industrial process, whether at the project level or at the operational level, has aspects that can be enhanced. In general, the optimization of an industrial process aims to increase profits and/or minimize costs. Heat exchangers are designed for different applications, so there can be multiple optimization criteria, such as minimum initial and operational costs, minimum volume or area of heat transfer, and minimum weight (important for space applications).


**Table 3.** Constants for the heat transfer and pressure drop calculation in a PHE with chevron plates [14]

The optimization problem is formulated in such a way that the best combination of the parameters of a given PHE minimizes the number of plates. The optimization method used is based on screening [15], where for a given type of plate, the number of thermal plates is the objective function that has to be minimized. In order to avoid impossible or non-optimal solutions, certain inequality constraints are employed. An algorithm has been proposed in a screening method that uses MATLAB for optimization of a PHE, considering the plate type as the optimization variable [16]. For each type of plate, local optimal configurations are found (if they exist) that employ the fewest plates. Comparison of all the local optima then gives a global optimum.

#### **5. Formulation of the optimization problem**

Minimize:

$$N\_p = f\left(N\_{\mathbb{C}'}P^l, P^{ll}, \phi, Y\_{h'} \text{plate type}\right) \tag{37}$$

Subject to:

$$\mathcal{N}\_{\mathcal{C}}{}^{\min} \le \mathcal{N}\_{\mathcal{C}} \le \mathcal{N}\_{\mathcal{C}}{}^{\max} \tag{38}$$

$$
\Delta P\_{hot} \le \Delta P\_{hot}^{\quad max} \tag{39}
$$

$$
\Delta \mathbf{P}\_{cold} \le \Delta \mathbf{P}\_{cold}^{\text{max}} \tag{40}
$$

$$
\boldsymbol{\upsilon}\_{hvt} \ge \boldsymbol{\upsilon}\_{hvt}^{\quad \text{min}} \tag{41}
$$

$$
\boldsymbol{\upsilon}\_{cold} \ge \boldsymbol{\upsilon}\_{cold}^{\min} \tag{42}
$$

$$E^{\text{univ}} \le E \le E^{\text{max}} \tag{43}$$

If closed-form model is considered, as the closed-form equations are limited for some number of passes, there are two more constraints:

$$P^l \le P^{l, \text{max}} \text{ if using closed -- form model} \tag{44}$$

$$P^{\text{ill}} \le P^{\text{ill,max}} \text{ if using closed } -\text{form model} \tag{45}$$

Depending on the equipment model, the number of plates can vary between 3 and 700. The first constraint (38) is imposed according to the PHE capacity. Constraints (39) and (40) can also be imposed, depending on the available pumping power. The velocity constraints are usually imposed in order to avoid dead spaces or air bubbles inside the set of plates. In practice, velocities less than 0.1 m/s are not used [5].

The optimization problem is solved by successively evaluating the constraints, reducing the number of configurations until the optimal set (OS) is found (if it exists). The screening process begins with the identification of an initial set (IS) of possible configurations, considering the channel limits. A reduced set (RS) is generated by considering the velocity and pressure drop constraints. The constraint of thermal effectiveness is then applied to the RS, in increasing order of the number of channels. Configurations with the smallest number of channels form the local optima set. The global optimum can therefore be found by comparing all the local optima. It is important to point out that the global optimum configuration may have a larger total heat transfer area. However, it is usually more economical to use a smaller number of large plates than a greater number of small plates [17]. The optimization algorithm is described in Table 4.

**5. Formulation of the optimization problem**

of passes, there are two more constraints:

velocities less than 0.1 m/s are not used [5].

( ) *I II N f N P P Y plate type PC h* = , , ,, , f

*max*

*max*

*min*

*min*

If closed-form model is considered, as the closed-form equations are limited for some number

Depending on the equipment model, the number of plates can vary between 3 and 700. The first constraint (38) is imposed according to the PHE capacity. Constraints (39) and (40) can also be imposed, depending on the available pumping power. The velocity constraints are usually imposed in order to avoid dead spaces or air bubbles inside the set of plates. In practice,

(37)

*min max N NN C CC* £ £ (38)

*hot hot* D £D *P P* (39)

*cold cold* D £D *P P* (40)

*hot hot v v* ³ (41)

*cold cold v v* ³ (42)

*min max E EE* £ £ (43)

*I I max P P* , £ - if using closed form model (44)

*II II max P P* , £ - if using closed form model (45)

Minimize:

188 Heat Transfer Studies and Applications

Subject to:

In Step 5, both methods can be used. The model using algebraic equations has the limitation of only being applicable to PHEs that are sufficiently large not to be affected by end channels and channels between adjacent passes. Industrial PHEs generally possess more than 40 thermal plates, although the limitation of the number of passes can still be a drawback. The major advantage of the model using differential equations is its general applicability to any config‐ uration, without having to derive a specific closed-form equation for each configuration.



**Table 4.** Optimization algorithm.

However, a drawback is the highly complex implementation of the simulation algorithm (see Table 5), in contrast to the second model, which is very simple.


**Table 5.** Simulation algorithm.

**Steps Mathematical relations Comments**

*Step 3.1.2* Verification of the pressure

190 Heat Transfer Studies and Applications

*Step 3.1.3* Verification of the velocity constraint for the fluid in side II. *<sup>v</sup> II*

*Step 3.1.4* Verification of the pressure

*Stage 3.2* Take *Yh* =1 and do the same

*Step 4:* Calculate the effectiveness in

*Step 5:* Verification of the thermal effectiveness constraint. The local optimal set of configurations (OS) is

**Table 4.** Optimization algorithm.

pure countercurrent flow, *ECC*. *ECC* ={ <sup>1</sup> <sup>−</sup> *<sup>e</sup>*

drop constraint in side II.

*Stage 3.3:* Combination of the configuration parameters.

as Stage 3.1.

determined.

*hot* = *GC*,*hot II ρhot*

*Pcold II* = *P II*

*NCP <sup>I</sup> P II*

Table 5), in contrast to the second model, which is very simple.

 and *Phot <sup>I</sup>* =*P <sup>I</sup>*

<sup>−</sup>*NTU* (1−*Cr*)

*NTU NTU* <sup>+</sup> <sup>1</sup> *if Cr* =1

<sup>−</sup>*NTU* (1−*Cr*) *if Cr* <1

*Step 6:* Find global optimum. By comparing all local optima, the

However, a drawback is the highly complex implementation of the simulation algorithm (see

1 − *Cre*

drop constraint in side I

allowable value, it is not necessary to evaluate configurations with smaller

*cold*

it is not necessary to evaluate configurations with greater number of

Analogous to Step 3.1.1.

the PHE is combined.

be discarded.

*E min* ≤*E* ≤*E max* The selected configurations in Step 4

of the possible number of passes of a

achieves the maximum allowable value,

passes selected for the sides I and II of

If *ECC* <sup>&</sup>lt;*<sup>E</sup> min* , these configurations can

are simulated in a crescent order of the number of channels to find the possible local optimum set (OS). The remaining configurations do not need to be simulated. Both modeling can be used.

global optimum is found.

, in a crescent order

*cold*

*<sup>C</sup>* . If *Δ<sup>P</sup> <sup>I</sup>*

number of passes.

calculated, *ΔP <sup>I</sup>*

passes.

. Analogous to stage 3.1.

*Yh <sup>ϕ</sup>* For *Yh* = 0 and *Yh* =1 , the number of

given element of *<sup>N</sup>*¯

Eq. (31) The cold fluid pressure drop is

Eq. (31) Analogous to Step 3.1.2.

#### **5.1. Simulation algorithm for the model using differential equations**

For the development of this algorithm, the boundary conditions equations are used in the algorithm form described previously [6] (see Table 6). The simulation algorithm is applied separately for each value of *ϕ* separately. The algorithm is presented below.

### **6. Case study**

A case study was used to test the developed algorithm and compare the two mathematical models. Data were taken from examples presented in [18]. A cold water stream exchanges heat with a hot water stream of process. As the closed-form equations only consider configurations with a maximum of 4 passes for each fluid, a case was chosen in which the reduced set only had configurations with less than 4 passes for each stream. Table 8 presents the data used. Only one type of plate was considered.

The RS was obtained by applying the optimization algorithm up to Step 3. The optimal set was found by applying Step 5. As only one type of plate was considered, the local optimum was





■ Optimal configurations

**Table 9.** Thermal effectiveness of RS for both mathematical models

#### **7. Conclusions**

In this chapter it was presented the development of two models for the design and optimization of plate heat exchangers. Both mathematical models were used to accomplish the heat exchanger design simulations. These methods use differential equations and closed-form equations based on the notion that a multi-pass PHE can be reduced to an arrangement consisting of assemblies of single-pass PHEs.

As a case study, an example obtained from the literature was used. The optimal sets were the same for both approaches, and agreement was achieved between the effectiveness values. The model using algebraic equations has the limitation of only being applicable to PHEs sufficiently large not to be affected by end channels and channels between adjacent passes. However, industrial PHEs generally possess more than 40 thermal plates. The major advantage of using this model is its general applicability to any configuration, without having to derive a specific closed-form equation for each configuration. However, its drawback is the highly complex implementation of the simulation algorithm, unlike the second approach, which is very simple.
