**3. Mathematical modeling**

Due to the large number of plate types and pass arrangements, there are many possible configurations of a particular PHE design. As a result, a number of mathematical modeling approaches have been proposed for the calculation of performance. Two different modeling approaches are described below.

#### **3.1. Model 1**

**Figure 9.** Feed connection of a PHE.

172 Heat Transfer Studies and Applications

**3. MATHEMATICAL MODELING**

flow. The following assumptions are made:

**3.1 Model 1**

performance. Two different modeling approaches are described below.

**Figure 10.** (a). Vertical flow plate [9]. (b). Diagonal flow plate [9].

The PHE operates at steady state;

Perfect mixture in the end of each pass;

The velocity profile in the channels is flat (plug flow);

more likely to occur in the array of vertical flow [7].

distribution is more likely to occur in the array of vertical flow [7].

The plates of a PHE can provide vertical or diagonal flow, depending on the arrangement of the gaskets. For vertical flow, the inlet and outlet of a given stream are located on the same side of the heat exchanger, whereas for diagonal flow they are on opposite sides. Assembly of the plate pack involves alternating between the "A" and "B" plates for the respective flows. Mounting of the plate pack in vertical flow mode only requires an appropriate gasket config‐ uration, because the A and B arrangements are equivalent (they are rotated by 180°, as shown in Figure 10a). This is not possible in the case of diagonal flow, which requires both types of mounting plate (Figure 10b). To identify each type of flow, Gut (2003) considered the binary parameter *Yf* (*Yf* = 1 for diagonal flow and *Yf* = 0 for vertical flow). Poor flow distribution is

Figure 9. Feed connection of a PHE.

The plates of a PHE can provide vertical or diagonal flow, depending on the arrangement of the gaskets. For vertical flow, the inlet and outlet of a given stream are located on the same side of the heat exchanger, whereas for diagonal flow they are on opposite sides. Assembly of the plate pack involves alternating between the "A" and "B" plates for the respective flows. Mounting of the plate pack in vertical flow mode only requires an appropriate gasket configuration, because the A and B arrangements are equivalent (they are rotated by 180°, as shown in Figure 10a). This is not possible in the case of diagonal flow, which requires both types of mounting plate (Figure 10b). To identify each type of flow, Gut (2003) considered the binary parameterܻ (ܻ = 1 for diagonal flow and ܻ = 0 for vertical flow). Poor flow

(a)

(b)

Figure 10a. Vertical flow plate [9]. Figure 10b. Diagonal flow plate [9].

Due to the large number of plate types and pass arrangements, there are many possible configurations of a particular PHE design. As a result, a number of mathematical modeling approaches have been proposed for the calculation of

 A mathematical model was developed to simulate the general configuration of a PHE operating under steady state conditions, characterized using six different parameters [6]. In this model, the parameters considered are the number of channels, the number of passes for each side, the fluid locations, the feed connection locations, and the type of channel

The main flow is divided equally among the channels that make up each pass;

A mathematical model was developed to simulate the general configuration of a PHE oper‐ ating under steady state conditions, characterized using six different parameters [6]. In this model, the parameters considered are the number of channels, the number of passes for each side, the fluid locations, the feed connection locations, and the type of channel flow. The following assumptions are made:


The last assumption listed above implies an overall heat transfer coefficient *U* constant throughout the process, which is quite reasonable for compact heat exchangers operating without phase change [10]. In the absence of this consideration, the energy balance in the channels would result in a nonlinear system of ordinary first order differential equations, which would make the simulation much more complex. It has also been found that the results obtained assuming a constant overall heat transfer coefficient are very close to those found without such a restriction [6]. Thus, this assumption is not a limiting factor for the evaluation of a PHE.

Applying the energy conservation law to a given volume of control of a generic channel *i* with dimensions *WP*, *δx* and *b* (Figure 11) and neglecting variations of kinetic and potential energy, the enthalpy change of the fluid passing through the volume is equal to the net heat exchanged by the two adjacent channels. This can be described by a system of differential equations:

$$\frac{d\theta\_1}{d\eta} = \mathbf{s}\_1 \alpha^l \left(\theta\_2 - \theta\_1\right) \tag{1}$$

**Figure 11.** Scheme of a fluid control volume

$$\frac{d\theta\_1}{d\eta} = \mathbf{s}\_i \alpha^{\text{side}(i)} \left(\theta\_{i-1} + 2\theta\_i + \theta\_{i+1}\right) \\ \text{side}(i) = \{I, II\} \tag{2}$$

$$\frac{d\theta\_1}{d\eta} = \mathbf{s}\_i \alpha^{\text{side}(N\_C)} \left(\theta\_{N\_c - 1} - \theta\_{N\_c}\right) \\ \text{side}\left(\mathbf{N}\_C\right) = \{I, II\} \tag{3}$$

where *si* is a constant that represents the flow direction in the channels (s = 1 for upward flow and s = - 1 for downward flow); *θ* is the adimensional temperature:

$$\Theta = \frac{T\_i - T\_{\text{cold},in}}{T\_{hot,in} - T\_{\text{cold},in}} \tag{4}$$

and

$$\alpha^{\prime} = \frac{A\_p \text{LIN}^{\prime}}{\dot{M}^{\prime} \boldsymbol{c}\_p^{\prime}}, \text{ } \alpha^{\prime\prime} = \frac{A\_p \text{LIN}^{\prime\prime}}{\dot{M}^{\prime\prime} \boldsymbol{c}\_p^{\prime\prime}}\tag{5}$$

*AP* is the plate area, *U* is the overall heat transfer coefficient, *N* is the number of channels per pass, *M*˙ is the mass flow and *cp* is the specific heat.

This system of linear differential equations can be written in the matrix form as:

#### Modeling and Design of Plate Heat Exchanger http://dx.doi.org/10.5772/60885 175

$$\frac{d\overline{\theta}}{d\eta} = \stackrel{\cdot}{M}\_{\cdot} \overline{\theta} \tag{6}$$

where

( ) () *side i*

 qq

*i NN C*

*s side i I II*

( ) ( ) *<sup>C</sup> c c*

where *si* is a constant that represents the flow direction in the channels (s = 1 for upward flow

*i cold in hot in cold in*

, , ,

*I II*

*I I II II p p*

*AP* is the plate area, *U* is the overall heat transfer coefficient, *N* is the number of channels

*A UN A UN Mc M c*

 a

*T T T T*

*I P P II*

This system of linear differential equations can be written in the matrix form as:

*s side N I II*

1 1 2 {, }

<sup>1</sup> {, }




= = , & & (5)

*i i ii*

 qq

 q

*d*

**Figure 11.** Scheme of a fluid control volume

174 Heat Transfer Studies and Applications

q

h

1 ( )

a

1 ( )

a

per pass, *M*˙ is the mass flow and *cp* is the specific heat.

a

*side N*

and s = - 1 for downward flow); *θ* is the adimensional temperature:

q

*d*

*d*

q

h

*d*

and

$$
\bar{M} = \begin{bmatrix}
+d\_2 & -2d\_2 & +d\_2 & & 0 & \cdots & 0 \\
0 & +d\_3 & -2d\_3 & & & +d\_3 & \vdots \\
& \vdots & & & & & & 0 \\
0 & \cdots & 0 & & +d\_{N\_c-1} & -2d\_{N\_c-1} & +d\_{N\_c-1} \\
0 & \cdots & 0 & & 0 & +d\_{N\_c} & -d\_{N\_c}
\end{bmatrix}
$$

$$d\_i = \begin{cases} s\_i \alpha^l & \text{if } i \text{ is odd} \\ s\_i \alpha^{ll} & \text{if } i \text{ is even} \end{cases} \\ i = 1, \dots, N\_{\odot}$$

The boundary conditions, which are dependent on the PHE configuration, can be divided into three different categories:

**1.** *Fluid inlet temperature*: In the channels of the first pass, the fluid inlet temperature is the fluid feed temperature.

$$
\theta\_i(\eta) = \theta\_{fluid, in} i \text{ } \epsilon \text{ first pass} \tag{7}
$$

**2.** *Change of pass temperature*: The temperature at the beginning of the channels of a particular pass is equal to the arithmetic average of the temperatures in the channels of the previous pass.

$$\theta\_i(\eta) = \frac{1}{N} \sum\_{\substack{j \ \epsilon \text{ previous} \\ \text{pass}}}^N \theta\_j(\eta) \text{ i } \epsilon \text{ new pass} \tag{8}$$

**3.** *Fluid outlet temperature*: The outlet temperature of the fluid is the arithmetic average of the outlet temperatures of the channels of the last pass.

$$\Theta\_{\text{fluid,out}}\left(\eta\right) = \frac{1}{N} \sum\_{j \text{ } \epsilon \text{ last} }^{N} \theta\_{j}\left(\eta\right) \tag{9}$$

The analytical solution is given by Eq. (10), where *λ<sup>i</sup>* and *z*¯*<sup>i</sup>* are, respectively, the eigenvalues and eigenvectors of matrix *M* = :

$$\overline{\partial}\left(\eta\right) = \sum\_{i=1}^{N\_c} c\_i \overline{z}\_i e^{\lambda\_i \eta} \tag{10}$$

Application of Eq. (10) in the boundary condition equations for the fluid inlet and change of pass enables the creation of a linear system of *Nc* equations for *ci* variables. After solving the linear system, the outlet temperatures can be determined by the use of the outlet boundary conditions, hence enabling the thermal effectiveness to be determined.

**Example:** Creation of the linear system of *Nc* equations:

In order to illustrate the generation of the linear system, a PHE containing 7 thermal plates (or 8 channels), with the cold fluid making two passes and the hot fluid making one pass, is shown in Figure 12.

**Figure 12.** PHE streams.

Applying Eq. (10), the following analytical solution can be achieved:

$$\overline{\boldsymbol{\Theta}}\left(\eta\right) = \boldsymbol{c}\_{1} \begin{bmatrix} \boldsymbol{z}\_{1,1} \\ \boldsymbol{z}\_{2,1} \\ \boldsymbol{z}\_{2,1} \\ \vdots \\ \boldsymbol{z}\_{8,1} \end{bmatrix} \boldsymbol{e}^{\boldsymbol{\lambda}\_{1}\eta} + \boldsymbol{c}\_{2} \begin{bmatrix} \boldsymbol{z}\_{1,2} \\ \boldsymbol{z}\_{2,2} \\ \boldsymbol{z}\_{2,2} \\ \vdots \\ \boldsymbol{z}\_{8,2} \end{bmatrix} \boldsymbol{e}^{\boldsymbol{\lambda}\_{2}\eta} + \dots + \boldsymbol{c}\_{8} \begin{bmatrix} \boldsymbol{z}\_{1,8} \\ \boldsymbol{z}\_{2,8} \\ \boldsymbol{z}\_{2,8} \\ \vdots \\ \boldsymbol{z}\_{8,8} \end{bmatrix} \boldsymbol{e}^{\boldsymbol{\lambda}\_{8}\eta}$$

Using the boundary condition equations (7) and (8) for all the channels of the PHE under investigation, the equations presented in Table 1 are generated.


**Table 1.** Boundary condition equations.

The analytical solution is given by Eq. (10), where *λ<sup>i</sup>*

= :

**Example:** Creation of the linear system of *Nc* equations:

( ) *<sup>c</sup>*

q h

conditions, hence enabling the thermal effectiveness to be determined.

Applying Eq. (10), the following analytical solution can be achieved:

l h

= + ++

*zz z cz e cz e cz e*

1,1 1,2 1,8 1 2,1 2 2,2 8 2,8

éù éù éù êú êú êú

> l h

1 2 8

...

l h

*zz z*

8,1 8,2 8,8

ëû ëû ëû MM M

( )

q h *N*

=

*i*

l h

*i i i cze* 1

Application of Eq. (10) in the boundary condition equations for the fluid inlet and change of pass enables the creation of a linear system of *Nc* equations for *ci* variables. After solving the linear system, the outlet temperatures can be determined by the use of the outlet boundary

In order to illustrate the generation of the linear system, a PHE containing 7 thermal plates (or 8 channels), with the cold fluid making two passes and the hot fluid making one pass, is shown

and eigenvectors of matrix *M*

176 Heat Transfer Studies and Applications

in Figure 12.

**Figure 12.** PHE streams.

and *z*¯*<sup>i</sup>*

are, respectively, the eigenvalues

<sup>=</sup> å (10)

These equations can be written in the following way:

$$\begin{aligned} \theta\_1 \left( \eta = 0 \right) - \left[ \theta\_5 \left( \eta = 0 \right) + \theta\_7 \left( \eta = 0 \right) \right] / \, \Big/ \, \, \mathbf{2} = 0\\ \theta\_2 \left( \eta = 1 \right) &= 1 \\ \theta\_3 \left( \eta = 0 \right) - \left[ \theta\_5 \left( \eta = 0 \right) + \theta\_7 \left( \eta = 0 \right) \right] / \, \, \mathbf{2} = 0\\ \theta\_4 \left( \eta = 1 \right) &= 1 \\ \theta\_5 \left( \eta = 1 \right) &= 0 \\ \theta\_5 \left( \eta = 1 \right) &= 1 \\ \theta\_7 \left( \eta = 1 \right) &= 0 \\ \theta\_8 \left( \eta = 1 \right) &= 1 \end{aligned}$$

To achieve the matrix form is Eq. (10) is applied to the linear system:

$$
\vec{A}.\vec{C} = \vec{B}\tag{11}
$$

where

*A* = = eigenvalues and eigenvectors matrix

*<sup>C</sup>*¯ <sup>=</sup>*ci* ' s coefficients vector

*<sup>B</sup>*¯ = binary vector

where

$$\begin{aligned} \mathcal{B}\_i &= 0 \text{ if } \mathcal{B}\_i = \theta\_{\text{tot},in} \text{ } i \in \text{first pass} \\ \mathcal{B}\_i &= 1 \text{ if } \mathcal{B}\_i = \theta\_{\text{cold},in} \text{ } i \in \text{first pass} \\ \mathcal{B}\_i &= 0 \text{ if } \mathcal{B}\_i = \theta\_i(\eta) - \frac{1}{N} \sum\_{j \in \text{previous pass}}^N \theta\_j(\eta) \end{aligned}$$

#### **3.2. Model 2**

The assumption is made that any multi-pass PHE with a sufficiently large number of plates (so that end effects and inter-pass plates can be neglected) can be reduced to an arrangement consisting of assemblies of single-pass PHEs [11]. This enables the development of closed-form equations for effectiveness, as a function of the ratio between the heat capacities of the fluids and the number of transfer units, for the arrangements 1-1, 2-1, 2-2, 3-1, 3-2, 3-3, 4-1, 4-2, 4-3, and 4-4 (Table 2). In other words, most multi-pass plate heat exchangers can be represented by simple combinations of pure countercurrent and concurrent exchangers, so that a multipass PHE is therefore equivalent to combinations of smaller single-pass exchangers (Figure 13).

**Figure 13.** Equivalent configurations.

The assumptions considered are the same as in the first mathematical model. The derived formulas are only valid for PHEs with numbers of thermal plates sufficiently large that the end effects can be neglected. This condition can be satisfied, depending on the required degree of accuracy. For example, a minimum of 19 plates is recommended for an inaccuracy of up to 2.5% [12]. Elsewhere, a minimum of 40 thermal plates was used [11, 13]. In the formulas, *PCC* and *PP* are the thermal effectiveness for the countercurrent and concurrent flows, respectively, given by:

$$P\_{\rm CC}\left(NTU\_1, \mathbf{R\_1}\right) = \begin{cases} \frac{1 - e^{-NTU\_1\left(1 - R\_1\right)}}{1 - \mathbf{R\_1}e^{-NTU\_1\left(1 - R\_1\right)}} \text{ se } \mathbf{R\_1} \neq \mathbf{1} \\\\ \\\\ \frac{NTU\_1}{NTU\_1 + \mathbf{1}} \quad \text{se } \mathbf{R\_1} = \mathbf{1} \end{cases} \tag{12}$$

$$P\_p\left(NTL I\_{1}, R\_1\right) = \frac{1 - e^{-NTL\_1\left(1 + R\_1\right)}}{1 + R\_1} \tag{13}$$

**3.2. Model 2**

178 Heat Transfer Studies and Applications

**Figure 13.** Equivalent configurations.

given by:

The assumption is made that any multi-pass PHE with a sufficiently large number of plates (so that end effects and inter-pass plates can be neglected) can be reduced to an arrangement consisting of assemblies of single-pass PHEs [11]. This enables the development of closed-form equations for effectiveness, as a function of the ratio between the heat capacities of the fluids and the number of transfer units, for the arrangements 1-1, 2-1, 2-2, 3-1, 3-2, 3-3, 4-1, 4-2, 4-3, and 4-4 (Table 2). In other words, most multi-pass plate heat exchangers can be represented by simple combinations of pure countercurrent and concurrent exchangers, so that a multipass PHE is therefore equivalent to combinations of smaller single-pass exchangers (Figure 13).

The assumptions considered are the same as in the first mathematical model. The derived formulas are only valid for PHEs with numbers of thermal plates sufficiently large that the end effects can be neglected. This condition can be satisfied, depending on the required degree of accuracy. For example, a minimum of 19 plates is recommended for an inaccuracy of up to 2.5% [12]. Elsewhere, a minimum of 40 thermal plates was used [11, 13]. In the formulas, *PCC* and *PP* are the thermal effectiveness for the countercurrent and concurrent flows, respectively,

> ( ) ( )

*<sup>e</sup> se R*

1 1

1 1

1

(12)

(13)

1

*NTU se R*

1

*NTU R* ( )

1 1 1

*R*

1

1

<sup>ï</sup> <sup>=</sup> <sup>ï</sup> <sup>+</sup> <sup>î</sup>

1


*NTU R NTU R*


*R e*

1

ï - ï ï = í ï ï

1

*NTU*

<sup>1</sup> , <sup>1</sup>

( )

*P*

1 1

( )

*<sup>e</sup> P NTU R*

1 1

,

*P NTU R*

*CC*

**Table 2.** Closed formulas for multi-pass arrangement [11]

**Formulas Arrangements**

**Arrangement 331**

180 Heat Transfer Studies and Applications

*PCC* =*PCC*(*NT U*1, *R*1)

**Arrangement 411**

<sup>2</sup> (*PCC* <sup>+</sup> *PP* <sup>−</sup>

<sup>2</sup> (*PCC* <sup>+</sup> *PP* <sup>−</sup>

Arrangement 431 *<sup>P</sup>*<sup>1</sup> <sup>=</sup> *<sup>A</sup>* <sup>+</sup> *BD*(1 <sup>−</sup> *<sup>G</sup>*) <sup>+</sup> *BQE* (1 − *G*)(1 − *E*) − *QS*

*PCC* =*PCC*(*NT U*<sup>1</sup> / 2, *R*<sup>1</sup> / 2) *PP* =*PP*(*NT U*<sup>1</sup> / 2, *R*<sup>1</sup> / 2)

<sup>4</sup> (3*PCC* + *PP* −*rPCCPP*)

<sup>6</sup> (1−*rPCC*)(3*PCC* + 3*PP* −2*rPCCPP* −*rPP*2)

<sup>6</sup> (*PCC*<sup>2</sup> + 3*PP*2) + *PCCPP*(3−*rPCC* −2*rPP*)

<sup>6</sup> 3*PCC*<sup>2</sup> + *PP*<sup>2</sup> + *PCCPP*(3−2*rPCC* −*rPP*)

<sup>12</sup> (1−*rPCC*)(3−2*rPCC* −*rPP*)(*PCC* + 3*PP* −*rPCCPP* −2*rPP*2)

<sup>12</sup> (*PCC* + 3*PP* −*rPP*(*PCC* + 2*PP*))

*PCC* =*PCC*(*NT U*1, *R*<sup>1</sup> / 4) *PP* =*PP*(*NT U*1, *R*<sup>1</sup> / 4) **Arrangement 421**

> *PI* (1 − *PI* )(1 − *PI R*1) 1 − *PI* <sup>2</sup>*R*<sup>1</sup>

*R*1*PCC PP* <sup>4</sup> )

*PCC PPR*<sup>1</sup> <sup>2</sup> )

*P*<sup>1</sup> =*PCC* where:

*P*<sup>1</sup> =*PI* − *PI* <sup>2</sup>*R*<sup>1</sup> 4

where: *PI* <sup>=</sup> <sup>1</sup>

*P*<sup>1</sup> =*PI* −

where: *PI* <sup>=</sup> <sup>1</sup>

where: *<sup>A</sup>*<sup>=</sup> <sup>1</sup>

*B* =1− *A*

*<sup>D</sup>* <sup>=</sup> <sup>1</sup>

*<sup>E</sup>* <sup>=</sup> <sup>1</sup>

*<sup>G</sup>* <sup>=</sup> *<sup>r</sup>*

*<sup>H</sup>* <sup>=</sup> *<sup>r</sup>*

*Q* =1−

*r* = 3*R*<sup>1</sup> 4

*<sup>S</sup>* <sup>=</sup> *Er PCC* <sup>1</sup> <sup>−</sup> *r PCC* <sup>+</sup> *r PP*

(*PCC* + *PP*)

*PCC* =*PCC*(*NT U*<sup>1</sup> / 3, *r*) *PP* =*PP*(*NT U*<sup>1</sup> / 3, *r*)

<sup>2</sup> <sup>+</sup> *r PCC PP* 3
