**3. Thermodynamic analysis of plate heat exchangers**

If the heat transfer in a HEX is assumed to be only between the fluids in it and the absence of a heat loss in the center, it can be written in the PHEX with the following relation [6–9]:

	- = Heat given by the hot fluid (W).
	- = Heat received by the cold fluid (W).
	- *K* = Total heat transfer coefficient (W/m2 oK)

$$
\dot{Q} = \mathbf{K} \, A \, \Delta T\_m \tag{1}
$$

The temperatures given and received during the cooling and heating of hot and cold fluids can be found by the mass fluxes of fluids and the enthalpy of entrances and exits and can be written as [10]:

$$
\dot{Q} = \dot{m} \left( h\_i - h\_o \right) \tag{2}
$$

**235**

**Figure 7.**

*Temperature distribution in a parallel flow HEX [1].*

*Plate Heat Exchangers: Artificial Neural Networks for Their Design*

layers are brought to an additional thermal resistance during heat transfer. This contamination resistance (or factor), as indicated by the Rf symbol, can be found as follows, in the sense that the thermal resistances of the heat transfer surfaces are

*f*

*R*

1 1

Due to the roughness of the metal surfaces, there is a contact resistance between

( *A B* )

*Q A* 

Consequently, the total heat transfer coefficient at the surface of the HEX can be

1 2 3 1 ,1 2 ,2 3 2 11 2 3 2 1 1 <sup>1</sup> *RR R R ft t f <sup>K</sup>*

 δ

 λ

In the construction of the heat calculations of the heat exchangers, the expression of the mean logarithmic temperature difference (ΔTm) is required if Eq. (1) is used. The mean logarithmic temperature difference value is determined by the flow rate in the heat exchanger. **Figure 7** and **Figure 8** show temperature distributions

The mean logarithmic temperature difference (ΔTm) can be expressed as:

*m*

*T*

∆

ln

*T T <sup>T</sup> <sup>T</sup>*

∆ −∆ ∆ = <sup>∆</sup>

*T T*

these two surfaces due to the poor contact between the two metals. The contact resistance on two surfaces causes a decrease in temperature on these surfaces. In order to take these situations into consideration, a resistance definition can be made

*t*

*R*

δ

along the length of the HEX when the flow is parallel and opposite.

αλ

*dirty clean*

*K K* = − (4)

<sup>−</sup> <sup>=</sup> (5)

δ

λα

(7)

= + ++ ++ ++ + − − (6)

*DOI: http://dx.doi.org/10.5772/intechopen.95376*

dirty and clean [1]:

as follows [1]:

found by the following Eq. [1]:

If the temperatures of the fluids change when the heat is taken and given, the amount of heat that passes is [10]:

$$
\dot{Q} = \dot{m}\_h \mathcal{c}\_{ph} \left( T\_{hi} - T\_{ho} \right) = \dot{m}\_c \mathcal{c}\_{pc} \left( T\_{co} - T\_{ci} \right) \tag{3}
$$

After a certain period of operation, particles, metal salts or various chemical elements may accumulate in the fluids on the heat exchanger surfaces. Occasionally, due to corrosive effects, an oxidation layer may form on these surfaces. All these

*Heat Transfer - Design, Experimentation and Applications*

**3. Thermodynamic analysis of plate heat exchangers**

following relation [6–9]:

**Figure 6.**

*Asymmetric best use at PHE.*

Etilen Propilen (Epdm) Florokarbon (Fpm)

PTFE Encapsulated NBR

Nitrile (Nbr) Hnbr

Viton Gf

**Table 2.**

*Q* = Heat in the HEX (W).

and exits and can be written as [10]:

amount of heat that passes is [10]:

= Heat given by the hot fluid (W). = Heat received by the cold fluid (W). *K* = Total heat transfer coefficient (W/m2 oK)

*Gasket materials commonly used in plate heat exchangers.*

If the heat transfer in a HEX is assumed to be only between the fluids in it and the absence of a heat loss in the center, it can be written in the PHEX with the

The temperatures given and received during the cooling and heating of hot and cold fluids can be found by the mass fluxes of fluids and the enthalpy of entrances

If the temperatures of the fluids change when the heat is taken and given, the

After a certain period of operation, particles, metal salts or various chemical elements may accumulate in the fluids on the heat exchanger surfaces. Occasionally, due to corrosive effects, an oxidation layer may form on these surfaces. All these

*Q KA T* = ∆ *<sup>m</sup>* (1)

*Q mh h* ( *i o* ) = − (2)

*Q mc T T m c T T h ph hi ho c pc co ci* ( ) ( ) = −= − (3)

**234**

layers are brought to an additional thermal resistance during heat transfer. This contamination resistance (or factor), as indicated by the Rf symbol, can be found as follows, in the sense that the thermal resistances of the heat transfer surfaces are dirty and clean [1]:

$$R\_f = \frac{1}{K\_{\text{dity}}} - \frac{1}{K\_{\text{clean}}} \tag{4}$$

Due to the roughness of the metal surfaces, there is a contact resistance between these two surfaces due to the poor contact between the two metals. The contact resistance on two surfaces causes a decrease in temperature on these surfaces. In order to take these situations into consideration, a resistance definition can be made as follows [1]:

$$\mathcal{R}\_t = \frac{\left(T\_A - T\_B\right)}{\frac{\dot{Q}}{A}}\tag{5}$$

Consequently, the total heat transfer coefficient at the surface of the HEX can be found by the following Eq. [1]:

$$\frac{\mathbf{1}}{K} = \frac{\mathbf{1}}{\alpha\_1} + R\_{f1} + \frac{\delta\_1}{\lambda\_1} + R\_{t,1-2} + \frac{\delta\_2}{\lambda\_2} + R\_{t,2-3} + \frac{\delta\_3}{\lambda\_3} + R\_{f2} + \frac{\mathbf{1}}{\alpha\_2} \tag{6}$$

In the construction of the heat calculations of the heat exchangers, the expression of the mean logarithmic temperature difference (ΔTm) is required if Eq. (1) is used. The mean logarithmic temperature difference value is determined by the flow rate in the heat exchanger. **Figure 7** and **Figure 8** show temperature distributions along the length of the HEX when the flow is parallel and opposite.

The mean logarithmic temperature difference (ΔTm) can be expressed as:

$$
\Delta T\_m = \frac{\Delta T\_1 - \Delta T\_2}{\ln \frac{\Delta T\_1}{\Delta T\_2}} \tag{7}
$$

**Figure 7.** *Temperature distribution in a parallel flow HEX [1].*

**Figure 8.** *Temperature distribution in the reverse flow HEX [1].*

If the last equation is moved to Eq. (1):

$$\dot{Q} = \frac{\mathbf{K} \mathbf{A} (\mathbf{A} T\_1 - \mathbf{A} T\_2)}{\ln \frac{\mathbf{A} T\_1}{\mathbf{A} T\_2}} \tag{8}$$

expression is obtained.

The efficiency of HEXs can be calculated with the help of the following Equation [11]:

$$\mathfrak{s} = \frac{\dot{\mathbf{Q}}}{\dot{\mathbf{Q}}\_{\text{max}}} \tag{9}$$

*C mc h h ph* = and *C mc c c pc* = are the thermal capacity values of hot and cold fluids, the actual heat transfer in the HEX can be written as [12, 13]:

$$\dot{\mathbf{Q}} = \mathbf{C}\_h \left( T\_{hi} - T\_{ho} \right) = \mathbf{C}\_c \left( T\_{co} - T\_{ci} \right) \tag{10}$$

If the Qmax value is defined as the maximum possible heat transition, it can be written as follows, provided that it is smaller than the Ch or Cc thermal capacity outputs [14, 15]:

$$\dot{\mathbf{Q}}\_{\text{max}} = \mathbf{C}\_{\text{min}} \left( T\_{hl} - T\_{cl} \right) \tag{11}$$

**237**

**Figure 9.**

*Plate Heat Exchangers: Artificial Neural Networks for Their Design*

calculated as the weighted sum of the input signals and transformed by the transfer function. The artificial neurons are provided with the capability of learning by adjusting the weights in accordance with the preferred learning algorithm.

The data obtained from the experimental work [16] were used to obtain some relativity in order to use it in PHEXs and ANN method was used for this purpose. ANN method is used in many engineering applications. The most important advantages of this method are rapid formation, simple formation and high learning capacity. Two methods are used in thermal calculations in PHEXs. These methods are mean logarithmic temperature difference and effectiveness-NTU methods. In this study, the data obtained from the experiments [16] (hot water inlet–outlet temperature, cold water inlet–outlet temperature, flow rate and plate surface angle) are quite suitable for using the mean logarithmic temperature difference method. These experimental data are the input variables of the network created for the ANN method. The effectiveness and the heat transfer rate of the PHEX are output variables. All data and calculated results obtained from experimental data can be modeled with ANN

method and can be used to obtain results depending on different variables.

MATLAB Toolbox was used for ANN methodology. In the training of the data, the number of neurons in the hidden layer was changed between 3 and 12. 80% of the 227 data obtained from the experiments were randomly selected for training and 20% for testing. The generated network has 4 input variables as hot water inlet temperature (Thi), cold water inlet temperature (Tci), fluid flow (m) and plate surface angle (β). Effectiveness (ε) is output variable. The generated artificial neural network is shown schematically in **Figure 9**. In the created network, LOGSIG is selected as the transfer function, Forward Back Prop for the network type, and TRAINLM and TRAINSCG as the training function. The number of epochs used was 1000 values. Each network is run 10 times in order to get the best value. To obtain the best result from ANN method, different algorithms and hidden neurons are used in different numbers. The values of root mean square error

obtained for heat transfer rate and effectiveness value are given in **Tables 3** and **4**. The performance evaluation criteria obtained from the ANN model are given

In determining the heat transfer rate and effectiveness of the PHEX, the equations obtained from the ANN method were used. In the above equations, Ei is the neuron summation function and Fi is the neuron activation function. In represents the input variables and bn represents the bias value. The coefficients used in the formulas represent the weight values of each neuron's summation function of the hidden layer of the training network. In the above equations, hot water inlet temperature (Thi), cold water inlet temperature (Tc), fluid flow (m) and plate surface angle (β) are used in ANN as 4 input variables. The weight coefficients and bias values used for the determination of the heat transfer rate and effectiveness are

the TRAINLM-5 training function. For the best effectiveness value, R<sup>2</sup>

value for heat transfer rate was obtained as 0.999636 for

) and coefficient of variation (cov)

value was

*DOI: http://dx.doi.org/10.5772/intechopen.95376*

(RMSE), the coefficient of determination (R2

0.999565 for the TRAINLM-12 training function.

given in **Tables 6** and **7** respectively.

*Schematic representation of artificial neural network [17].*

in **Table 5**. The best R2

## **3.1 Determination of heat transfer rate and effectiveness in plate heat exchangers using alternative an approach**

Artificial Neural Network (ANN) was designed for the generalizations of biological nervous systems' mathematical models. When the simplified neurons were introduced, the first steps were taken towards the neural networks, which are also known as connectionist models or parallel distributed processing. The artificial neurons are the main component of the process, and they are also known as simply neurons and nodes. In order to represent the effects of synapses in a simplified mathematical neuron model, the connection weights modulating the effects of associated input signals are utilized, while the nonlinear characteristics of neurons are represented by using the transfer function. Then, the impulse of neuron is

#### *Plate Heat Exchangers: Artificial Neural Networks for Their Design DOI: http://dx.doi.org/10.5772/intechopen.95376*

*Heat Transfer - Design, Experimentation and Applications*

If the last equation is moved to Eq. (1):

*Temperature distribution in the reverse flow HEX [1].*

expression is obtained.

Equation [11]:

**Figure 8.**

outputs [14, 15]:

( 1 2 ) 1 2

(8)

∆ ∆ ∆ ∆

*T*

max

(9)

*Q CT T CT T h hi ho c co ci* ( ) ( ) = −= − (10)

*Q CTT* max min ( *hi ci* ) = − (11)

<sup>=</sup> *Q Q* ε

*C mc h h ph* = and *C mc c c pc* = are the thermal capacity values of hot and cold

If the Qmax value is defined as the maximum possible heat transition, it can be written as follows, provided that it is smaller than the Ch or Cc thermal capacity

fluids, the actual heat transfer in the HEX can be written as [12, 13]:

**3.1 Determination of heat transfer rate and effectiveness in plate heat** 

Artificial Neural Network (ANN) was designed for the generalizations of biological nervous systems' mathematical models. When the simplified neurons were introduced, the first steps were taken towards the neural networks, which are also known as connectionist models or parallel distributed processing. The artificial neurons are the main component of the process, and they are also known as simply neurons and nodes. In order to represent the effects of synapses in a simplified mathematical neuron model, the connection weights modulating the effects of associated input signals are utilized, while the nonlinear characteristics of neurons are represented by using the transfer function. Then, the impulse of neuron is

**exchangers using alternative an approach**

ln <sup>−</sup> <sup>=</sup> *KA T T <sup>Q</sup> <sup>T</sup>*

The efficiency of HEXs can be calculated with the help of the following

**236**

calculated as the weighted sum of the input signals and transformed by the transfer function. The artificial neurons are provided with the capability of learning by adjusting the weights in accordance with the preferred learning algorithm.

The data obtained from the experimental work [16] were used to obtain some relativity in order to use it in PHEXs and ANN method was used for this purpose. ANN method is used in many engineering applications. The most important advantages of this method are rapid formation, simple formation and high learning capacity. Two methods are used in thermal calculations in PHEXs. These methods are mean logarithmic temperature difference and effectiveness-NTU methods. In this study, the data obtained from the experiments [16] (hot water inlet–outlet temperature, cold water inlet–outlet temperature, flow rate and plate surface angle) are quite suitable for using the mean logarithmic temperature difference method. These experimental data are the input variables of the network created for the ANN method. The effectiveness and the heat transfer rate of the PHEX are output variables. All data and calculated results obtained from experimental data can be modeled with ANN method and can be used to obtain results depending on different variables.

MATLAB Toolbox was used for ANN methodology. In the training of the data, the number of neurons in the hidden layer was changed between 3 and 12. 80% of the 227 data obtained from the experiments were randomly selected for training and 20% for testing. The generated network has 4 input variables as hot water inlet temperature (Thi), cold water inlet temperature (Tci), fluid flow (m) and plate surface angle (β). Effectiveness (ε) is output variable. The generated artificial neural network is shown schematically in **Figure 9**. In the created network, LOGSIG is selected as the transfer function, Forward Back Prop for the network type, and TRAINLM and TRAINSCG as the training function. The number of epochs used was 1000 values. Each network is run 10 times in order to get the best value.

To obtain the best result from ANN method, different algorithms and hidden neurons are used in different numbers. The values of root mean square error (RMSE), the coefficient of determination (R2 ) and coefficient of variation (cov) obtained for heat transfer rate and effectiveness value are given in **Tables 3** and **4**. The performance evaluation criteria obtained from the ANN model are given in **Table 5**. The best R2 value for heat transfer rate was obtained as 0.999636 for the TRAINLM-5 training function. For the best effectiveness value, R<sup>2</sup> value was 0.999565 for the TRAINLM-12 training function.

In determining the heat transfer rate and effectiveness of the PHEX, the equations obtained from the ANN method were used. In the above equations, Ei is the neuron summation function and Fi is the neuron activation function. In represents the input variables and bn represents the bias value. The coefficients used in the formulas represent the weight values of each neuron's summation function of the hidden layer of the training network. In the above equations, hot water inlet temperature (Thi), cold water inlet temperature (Tc), fluid flow (m) and plate surface angle (β) are used in ANN as 4 input variables. The weight coefficients and bias values used for the determination of the heat transfer rate and effectiveness are given in **Tables 6** and **7** respectively.

**Figure 9.** *Schematic representation of artificial neural network [17].*


*The bold values are the best values of root mean square error (RMSE), the coefficient of determination (R2) and coefficient of variation (cov) obtained from the ANN model for to estimate of heat transfer rate and effectiveness.*

#### **Table 3.**

*The statistical values of the network for predicting the heat transfer rate.*

Heat transfer rate in the plate heat exchanger can be calculated by the following equations depending on the hot water inlet temperature (Thi), cold water inlet temperature (Tc), fluid flow (m) and plate surface angle (β) [17].

$$\begin{array}{l} \mathrm{E}\_{6} = -112.6039 \mathrm{F}\_{1} + 83.284 \mathrm{F}\_{2} - 89.934 \mathrm{F}\_{3} \\ -70.4233 \mathrm{F}\_{4} + 229.471 \mathrm{1F}\_{5} - 107.9219 \end{array} \tag{12}$$

$$
\dot{\mathbf{Q}} = \left(\frac{\mathbf{1}}{\mathbf{1} + \mathbf{e}^{-E\_k}}\right) \mathbf{3} \mathbf{9} \mathbf{2} \mathbf{0} \tag{13}
$$

Similarly, the effectiveness in the PHEX can be calculated from the following equations depending on the hot water inlet temperature (Thi), cold water inlet temperature (Tc), fluid flow (m) and plate surface angle (β) [17].

$$\begin{aligned} \mathbf{E}\_{13} &= 21.3112 \mathbf{F}\_1 - 2.7185 \mathbf{F}\_2 + 21.2966 \mathbf{F}\_3 + 7.2653 \mathbf{F}\_4 - 0.056092 \mathbf{F}\_5 \\ &- 26.4603 \mathbf{F}\_6 - 19.2568 \mathbf{F}\_7 - 12.4354 \mathbf{F}\_8 + 3.0543 \mathbf{F}\_9 - 46.6099 \mathbf{F}\_{10} \\ &+ 58.215 \mathbf{F}\_{11} + 0.28895 \mathbf{F}\_{12} - 1.5743 \end{aligned} \tag{14}$$

$$\mathfrak{g} = \frac{\mathbf{1}}{\mathbf{1} + \mathbf{e}^{-E\_{1j}}} \tag{15}$$

**239**

**Table 6.**

**Table 5.**

**Neuron position** (**wni**)

**Table 4.**

*Plate Heat Exchangers: Artificial Neural Networks for Their Design*

**Algorithm-Neuron RMSE cov R2** Lm-3 0.0102575 0.022758 0.999475 Lm-4 0.0096802 0.021477 0.999532 Lm-5 0.0134191 0.029773 0.999101 Lm-6 0.0097002 0.021522 0.999530 Lm-7 0.0099292 0.022030 0.999508 Lm-8 0.0097623 0.021660 0.999524 Lm-9 0.0100210 0.022234 0.999498 Lm-10 0.0096225 0.021350 0.999538 Lm-11 0.0109986 0.024403 0.999396 **Lm-12 0.0093323 0.020706 0.999565** SCG-3 0.0124508 0.027625 0.999226 SCG-4 0.0109349 0.024261 0.999403 SCG-5 0.0129422 0.028715 0.999164 SCG-6 0.0114423 0.025387 0.999346 SCG-7 0.0121871 0.027040 0.999258 SCG-8 0.0118729 0.026343 0.999296 SCG-9 0.0112524 0.024966 0.999368 SCG-10 0.0101688 0.022562 0.999484 SCG-11 0.0109066 0.024199 0.999406 SCG-12 0.0113416 0.025164 0.999358 *The bold values are the best values of root mean square error (RMSE), the coefficient of determination (R2) and coefficient of variation (cov) obtained from the ANN model for to estimate of heat transfer rate and effectiveness.*

**Thermodynamic Values Method Comparison Parameters**

*Performance evaluation criteria obtained from the ANN model.*

*The statistical values of the network for predicting the effectiveness.*

Heat transfer rate ANN 0.999636 52.2255 0.01954 Effectiveness ANN 0.999565 0.0093323 0.020706

 1.8443 10,5531 2.1624 −1.2641 18.7258 56.9616 −52.142 27.1211 −0.0728 −20.0211 −0.38653 0.42041 −2.0398 −5.9629 4.8897 −2.6099 2.2846 3.6669 8.8609 −7.3451 −40.8109 37.1115 −105.8138 33.7376 27.9695

*The weight coefficients and bias values used for the determination of the heat transfer rate.*

**I T** <sup>1</sup> ( **hi**) **I T** <sup>2</sup> ( **ci**) **I**<sup>3</sup> (*m* ) **I**<sup>4</sup> (β) **bn**

**R2 RMSE cov**

*DOI: http://dx.doi.org/10.5772/intechopen.95376*


#### *Plate Heat Exchangers: Artificial Neural Networks for Their Design DOI: http://dx.doi.org/10.5772/intechopen.95376*

*The bold values are the best values of root mean square error (RMSE), the coefficient of determination (R2) and coefficient of variation (cov) obtained from the ANN model for to estimate of heat transfer rate and effectiveness.*

#### **Table 4.**

*Heat Transfer - Design, Experimentation and Applications*

**Algorithm-Neuron RMSE cov R2** Lm-3 53.5148 0.02003 0.999618 Lm-4 52.7873 0.01975 0.999628 **Lm-5 52.2255 0.01954 0.999636** Lm-6 58.6761 0.02196 0.999540 Lm-7 54.9293 0.02056 0.999597 Lm-8 63.3763 0.02372 0.999464 Lm-9 57.5448 0.02153 0.999558 Lm-10 60.1653 0.02251 0.999517 Lm-11 58.8812 0.02203 0.999537 Lm-12 129.645 0.04852 0.997758 SCG-3 95.9341 0.03590 0.998772 SCG-4 76.4545 0.02861 0.999220 SCG-5 75.8600 0.02839 0.999232 SCG-6 61.0010 0.02283 0.999503 SCG-7 71.3101 0.02669 0.999321 SCG-8 61.6170 0.02306 0.999493 SCG-9 63.3772 0.02372 0.999464 SCG-10 54.6721 0.02046 0.999601 SCG-11 62.3508 0.02333 0.999481 SCG-12 59.8349 0.02239 0.999522 *The bold values are the best values of root mean square error (RMSE), the coefficient of determination (R2) and coefficient of variation (cov) obtained from the ANN model for to estimate of heat transfer rate and effectiveness.*

Heat transfer rate in the plate heat exchanger can be calculated by the following equations depending on the hot water inlet temperature (Thi), cold water inlet

> 6 123 4 5

70.4233F 229.4711F 107.9219

**6 <sup>1</sup> <sup>3920</sup>**

Similarly, the effectiveness in the PHEX can be calculated from the following equations depending on the hot water inlet temperature (Thi), cold water inlet

13 1 2 3 4 5

<sup>−</sup> <sup>=</sup> <sup>+</sup> **<sup>13</sup> 1 1** *<sup>E</sup> e*

E 21.3112F 2.7185F 21.2966F 7.2653F 0.056092F 26.4603F 19.2568F 12.4354F 3.0543F 46.6099F

= −+ +− − − − +−

ε

6 7 8 9 10

(15)

−+ − (12)

(13)

(14)

E 112.6039F 83.284F 89.934F

=− + −

**<sup>1</sup>** *<sup>Q</sup> <sup>E</sup> <sup>e</sup>* <sup>−</sup> <sup>=</sup> +

temperature (Tc), fluid flow (m) and plate surface angle (β) [17].

*The statistical values of the network for predicting the heat transfer rate.*

temperature (Tc), fluid flow (m) and plate surface angle (β) [17].

11 12

++ −

58.215F 0.28895F 1.5743

**238**

**Table 3.**

*The statistical values of the network for predicting the effectiveness.*


#### **Table 5.**

*Performance evaluation criteria obtained from the ANN model.*


#### **Table 6.**

*The weight coefficients and bias values used for the determination of the heat transfer rate.*

#### *Heat Transfer - Design, Experimentation and Applications*


**Table 7.**

*Weight coefficients and bias values used to determine of the effectiveness.*


#### **Table 8.**

*Comparison of the actual values of the heat transfer rate with the estimated values obtained in the ANN model.*

**241**

**Table 9.**

*Plate Heat Exchangers: Artificial Neural Networks for Their Design*

determined that this value is an acceptable error value.

The actual values of the heat transfer rate and the predicted values and error values obtained in the ANN model are given in **Table 8**. Eq. (16) is used in the

is the data obtained from experiments and Ap

obtained from ANN. As seen in **Table 8**, the highest error value is 1.37. It has been

Similarly, the actual values of the effectiveness and the predicted values and error values obtained in the ANN model are given in **Table 9**. As seen in **Table 9**, the highest error value is 1.23. This value is also determined to be an acceptable

In all HEXs there is a close physical and economic relationship between heat transfer and pressure drop. In a heat exchanger intended to be designed for a constant heat capacity, increasing fluid velocities increases the heat transfer coefficient and allows smaller heat exchangers (compact). In this way, a smaller size or more

> **Flow rate** *m* **(kg/s)**

21.2 15.4 0.167 30 0.40 0,396 0.95 44.6 37.1 0.167 30 0.41 0,409 0.22 51.1 43.6 0.167 30 0.41 0,407 0.73 31.8 26.6 0.239 30 0.42 0,423 0,71 39.3 34 0.239 30 0.43 0,435 1.23 22.4 18.9 0.321 30 0.43 0,426 0.91 29.1 25.5 0.321 30 0.42 0,423 0.69 34.8 28.4 0.263 60 0.44 0,445 1.14 41.4 34.9 0.263 60 0.45 0,453 0.62 53.3 46.8 0.263 60 0.48 0,477 0.63 26.4 21.9 0.39 60 0.44 0,439 0.20 45.2 40.8 0.39 60 0.50 0,501 0.22 21.8 19.1 0.517 60 0.44 0,443 0.77 32.1 29 0.517 60 0.45 0,452 0.40 38.8 35.7 0.517 60 0.48 0,481 0.13 45.7 42.5 0.517 60 0.50 0,505 0.90 54.3 51 0.517 60 0.52 0,521 0,21

*Comparison of the actual values of the effectiveness with the estimated values obtained in the ANN model.*

*e p e A A Error x A*

**100**

**Plate surface angle β( o )**

<sup>−</sup> <sup>=</sup> (16)

is the estimates

**Effectiveness Error (%) Actual** 

**ANN**

**values**

*DOI: http://dx.doi.org/10.5772/intechopen.95376*

calculation of error values [18].

**3.2 Pressure drop in heat exchangers**

**Temperature Tci(°C)**

Where, Ae

error value.

**Temperature Thi(°C)**

*Plate Heat Exchangers: Artificial Neural Networks for Their Design DOI: http://dx.doi.org/10.5772/intechopen.95376*

*Heat Transfer - Design, Experimentation and Applications*

*Weight coefficients and bias values used to determine of the effectiveness.*

**Flow rate** *m* **(kg/s)**

*Comparison of the actual values of the heat transfer rate with the estimated values obtained in the* 

32.8 25 0.167 30 2167 2191 1.09 44.6 37.1 0.167 30 2167 2144 1.08 51,1 43.6 0.167 30 2167 2166 0.07 58.1 50.1 0.167 30 2377 2354 0.97 31.8 26.6 0.239 30 2195 2204 0.40 39.3 34 0.239 30 2295 2284 0.48 44.7 39.4 0.239 30 2295 2319 1.05 37.9 34.3 0.321 30 2144 2115 1.37 45.5 41.9 0.321 30 2144 2159 0.68 34.8 28.4 0.263 60 3077 3103 0.85 53.3 46.8 0.263 60 3407 3393 0.40 19.2 15.7 0.39 60 2608 2609 0.06 26.4 21.9 0.39 60 3259 3254 0.17 37.5 33.1 0.39 60 3422 3421 0.04 45.2 40.8 0.39 60 3585 3572 0.37 32.1 29 0.517 60 3023 3055 1.06 45.7 42.5 0.517 60 3455 3419 1.04

**Plate surface angle β( o )**

**Heat transfer rate (W)**

**ANN**

**Actual values**

**Error (%)**

**Temperature Tci(°C)**

**I T** <sup>1</sup> ( **hi**) **I T** <sup>2</sup> ( **ci**) **I m**<sup>3</sup> ( ) **I**<sup>4</sup> (β) **bn**

 11.1556 11.7598 39.362 9.2238 47.3869 −37.921 46.173 −7.0158 −4.478 −2.4622 16.3938 −38.169 −35.378 −4.640 39.5329 13.9183 −1.3878 −10.187 13.3089 3.5955 92.1152 8.6141 28.7158 −49.57 −16.817 31.4252 −30.344 3.3642 25.2838 −23.704 −14.906 11.1061 −9.057 −9.2724 10.1669 23.2713 −4.3873 −49.1203 20.2264 4.1424 68.8739 −72.556 34.7909 −44.281 13.6961 −0.13074 3.2502 1.8366 11.4214 −8.3915 −10.7511 12.7919 −3.6727 13.7598 −7.913 35.966 58.1377 91.3142 3.7576 −46.231

**Neuron position** (**wni**)

**Table 7.**

**Temperature Thi(°C)**

**240**

**Table 8.**

*ANN model.*

The actual values of the heat transfer rate and the predicted values and error values obtained in the ANN model are given in **Table 8**. Eq. (16) is used in the calculation of error values [18].

$$Error = \frac{\left| A^{\epsilon} - A^{p} \right|}{A^{\epsilon}} \text{x100} \tag{16}$$

Where, Ae is the data obtained from experiments and Ap is the estimates obtained from ANN. As seen in **Table 8**, the highest error value is 1.37. It has been determined that this value is an acceptable error value.

Similarly, the actual values of the effectiveness and the predicted values and error values obtained in the ANN model are given in **Table 9**. As seen in **Table 9**, the highest error value is 1.23. This value is also determined to be an acceptable error value.

### **3.2 Pressure drop in heat exchangers**

In all HEXs there is a close physical and economic relationship between heat transfer and pressure drop. In a heat exchanger intended to be designed for a constant heat capacity, increasing fluid velocities increases the heat transfer coefficient and allows smaller heat exchangers (compact). In this way, a smaller size or more


#### **Table 9.**

*Comparison of the actual values of the effectiveness with the estimated values obtained in the ANN model.*

compact heat exchanger design can be achieved at the same capacity with a lower investment cost. On the other hand, increasing the velocities of the fluids causes the pressure drop in the heat exchanger to increase. This increases the investment cost of the pump due to the operating costs of the system and the growth of the pump or fan, as it increases the power of the pump or fan. Therefore, in the design of a HEX, the heat transfer and pressure drop must be considered together and the most appropriate solution for the system should be sought.

Since the flow model is very complex even in the simplest heat exchangers, approximate solutions and experimental findings are utilized in the determination of pressure drop as well as theoretical analyzes. The total pressure drop in a HEX is considered in two ways, the pressure drop in the straight pipe and the local pressure drop. The pressure drop in the straight pipe indicates the pressure drop from the rubbing in the flowing fluid in the fixed section piping or conduits. The local pressure drop is the loss of flow and direction changes in the flow. Unlike fluids at the same temperature, the natural convection caused by the temperature distribution in the heat exchangers may cause an additional pressure loss (or sometimes gambling).

The total pressure loss in a heat exchanger is summed separately from the pressure losses in each step of the exchanger.

#### **3.3 Local losses**

Cross-sectioning, rotation, separation, or coupling of fluid as it flows through a channel also causes pressure losses. These are generally called local losses. Changes in the velocity and direction of the fluid create Eddy movements (eddies) that cause energy loss. Although local losses occur at very short distances, they remain effective over a long period of time throughout the flow. These losses are generally [1];

$$
\Delta P\_y = \zeta \frac{\rho v^2}{2} \tag{17}
$$

form. Where ζ is called the local loss coefficient and can be found from the relevant sources for various local loss factors either using formulas or diagrams.

#### **3.4 Pressure loss caused by acceleration of the fluid and lifting force**

The pressure loss during the acceleration of the fluid, in fixed cross-sections,

$$
\Delta P\_{\rm iv} = \rho\_o \upsilon\_o^2 - \rho\_i \upsilon\_i^2 \tag{18}
$$

Where *v*g*, v*ç are the fluid velocity at inlet and outlet of the flow channel; *ρ*g*, ρ*ç, again indicate the density of the fluid at the inlet and outlet of the channel. The fluid is assumed to be incompressible, and in fluid fluids this value is the order of magnitude that other pressure losses can be neglected.

#### **3.5 Pressure loss in sealed plate heat exchangers**

The pressure loss in this type of heat exchanger is [19];

$$
\Delta p\_{gas\text{electrode}} = \mathcal{A}\_{gas\text{electrolyte}} \frac{L\_p \rho \nu^2}{d\_h \mathcal{Q}} \tag{19}
$$

**243**

*Plate Heat Exchangers: Artificial Neural Networks for Their Design*

**3.6 The power required to maintain fluid motion**

λ

, , Re<sup>−</sup> = **0 252**

Due to the protrusions on the plates, turbulence can pass through the Reynolds numbers, which are smaller than the values given in the flow flat exit channels. Therefore, in the case of sealed PHEXs, the flow at values such as Re > 100–400 is

First, the pressure losses in the HEX and the pressure losses in the piping up to the heat exchanger are calculated. Later, the fan or pump power required to move the fluid in this system by calculating the sum of the pressure losses in the heat

> ∑( ) <sup>=</sup> *m P <sup>t</sup> <sup>N</sup>* ∆ ρη

PHEXs are the most frequently used heat transfer equipments in energy applications and can be under various names such as evaporators and condensers in almost every stage of chemistry, petrochemical industry, power plants, cooling, heating and air conditioning process in various types and capacities. From the point of view of machine and chemical engineering education, plate heat exchangers are a very good application for this branch of science which contains all of the basic subjects of these engineering branches: materials, strength, thermodynamics and heat transfer science. As can be seen, PHEXs are a commonly used construction in our daily lives. For this reason, its design should be done in detail, analysis results should be obtained with analysis programs and studies should be done to improve the designs. Decreased the amount of heat transferred in the PHEX causes the performance of the HEX to decrease. This means loss of capacity in energy system with plate heat exchanger. The regulation of heat transfer permits the system dimensions to be kept at the proper values, thus decreasing system cost and operating costs. In this study, some equations were obtained to use in the plate heat exchangers by using the data obtained from the experimental work. ANN methodology was used for this purpose. As a result of the equations obtained for heat transfer and efficiency values in ANN application, approximate results were obtained at the value of 1.37 which is the highest error value for the real value heat transfer value and 1.23 for the efficiency value. When we look at the literature it is seen that these

*gasketplate* **1 22** (20)

(21)

*DOI: http://dx.doi.org/10.5772/intechopen.95376*

Turbulent flow;

assumed to be turbulent.

exchanger up to the HEX [20],

can be found in the equation.

values are acceptable error values.

"The authors declare no conflict of interest."

**Conflict of interest**

**4. Conclusions**

*Plate Heat Exchangers: Artificial Neural Networks for Their Design DOI: http://dx.doi.org/10.5772/intechopen.95376*

Turbulent flow;

*Heat Transfer - Design, Experimentation and Applications*

appropriate solution for the system should be sought.

sure losses in each step of the exchanger.

**3.3 Local losses**

compact heat exchanger design can be achieved at the same capacity with a lower investment cost. On the other hand, increasing the velocities of the fluids causes the pressure drop in the heat exchanger to increase. This increases the investment cost of the pump due to the operating costs of the system and the growth of the pump or fan, as it increases the power of the pump or fan. Therefore, in the design of a HEX, the heat transfer and pressure drop must be considered together and the most

Since the flow model is very complex even in the simplest heat exchangers, approximate solutions and experimental findings are utilized in the determination of pressure drop as well as theoretical analyzes. The total pressure drop in a HEX is considered in two ways, the pressure drop in the straight pipe and the local pressure drop. The pressure drop in the straight pipe indicates the pressure drop from the rubbing in the flowing fluid in the fixed section piping or conduits. The local pressure drop is the loss of flow and direction changes in the flow. Unlike fluids at the same temperature, the natural convection caused by the temperature distribution in the heat exchangers may cause an additional pressure loss (or sometimes gambling). The total pressure loss in a heat exchanger is summed separately from the pres-

Cross-sectioning, rotation, separation, or coupling of fluid as it flows through a channel also causes pressure losses. These are generally called local losses. Changes in the velocity and direction of the fluid create Eddy movements (eddies) that cause energy loss. Although local losses occur at very short distances, they remain effective over a long period of time throughout the flow. These losses are generally [1];

> ∆ =*<sup>y</sup> <sup>v</sup> <sup>P</sup>* ρ ζ

**3.4 Pressure loss caused by acceleration of the fluid and lifting force**

magnitude that other pressure losses can be neglected.

The pressure loss in this type of heat exchanger is [19];

**3.5 Pressure loss in sealed plate heat exchangers**

form. Where ζ is called the local loss coefficient and can be found from the relevant sources for various local loss factors either using formulas or diagrams.

The pressure loss during the acceleration of the fluid, in fixed cross-sections,

2 2 ∆= − *P vv iv o o i i* ρ

Where *v*g*, v*ç are the fluid velocity at inlet and outlet of the flow channel; *ρ*g*, ρ*ç, again indicate the density of the fluid at the inlet and outlet of the channel. The fluid is assumed to be incompressible, and in fluid fluids this value is the order of

> ∆ = *<sup>p</sup> gasketplate gasketplate*

*<sup>p</sup> <sup>d</sup>*

λ

 ρ

2

(19)

2

*h L v*

ρ

2 2

(17)

(18)

**242**

$$\mathcal{A}\_{\text{gasketplate}} = \mathbf{1}, \mathbf{22} \,\mathrm{Re}^{-0.252} \tag{20}$$

Due to the protrusions on the plates, turbulence can pass through the Reynolds numbers, which are smaller than the values given in the flow flat exit channels. Therefore, in the case of sealed PHEXs, the flow at values such as Re > 100–400 is assumed to be turbulent.

#### **3.6 The power required to maintain fluid motion**

First, the pressure losses in the HEX and the pressure losses in the piping up to the heat exchanger are calculated. Later, the fan or pump power required to move the fluid in this system by calculating the sum of the pressure losses in the heat exchanger up to the HEX [20],

$$N = \frac{\dot{m}\sum (\Delta P)\_t}{\rho \eta} \tag{21}$$

can be found in the equation.

#### **4. Conclusions**

PHEXs are the most frequently used heat transfer equipments in energy applications and can be under various names such as evaporators and condensers in almost every stage of chemistry, petrochemical industry, power plants, cooling, heating and air conditioning process in various types and capacities. From the point of view of machine and chemical engineering education, plate heat exchangers are a very good application for this branch of science which contains all of the basic subjects of these engineering branches: materials, strength, thermodynamics and heat transfer science. As can be seen, PHEXs are a commonly used construction in our daily lives. For this reason, its design should be done in detail, analysis results should be obtained with analysis programs and studies should be done to improve the designs. Decreased the amount of heat transferred in the PHEX causes the performance of the HEX to decrease. This means loss of capacity in energy system with plate heat exchanger. The regulation of heat transfer permits the system dimensions to be kept at the proper values, thus decreasing system cost and operating costs.

In this study, some equations were obtained to use in the plate heat exchangers by using the data obtained from the experimental work. ANN methodology was used for this purpose. As a result of the equations obtained for heat transfer and efficiency values in ANN application, approximate results were obtained at the value of 1.37 which is the highest error value for the real value heat transfer value and 1.23 for the efficiency value. When we look at the literature it is seen that these values are acceptable error values.

#### **Conflict of interest**

"The authors declare no conflict of interest."

*Heat Transfer - Design, Experimentation and Applications*
