**2.1 Hydraulic equations**

an immiscible mixture using a countercurrent spiral heat exchanger for two-phase flow. The new correlation was based on the experimental data, and the results were agreement with the theoretical correlation. Khorshidi and Heidari [3] fabricated a spiral heat exchanger geometry to study the performance, the examination showed that spiral heat exchanger is an excellent option to transfer heat especially from fouling fluids and also a computational fluid dynamic was presented to determine a previous design. Ramachandran et al. [4] determined the heat transfer behaviour for a system of two fluids by implementing a countercurrent spiral plate heat exchanger; the data were obtained from varying mass fraction inlets and demonstrated efficient results between the experiment and the correlation. Maruyama et al. [5] measured the thermal effectiveness of a cross-flow spiral plate heat exchanger; the aim was to convert radiation energy from a combustion chamber. Wang [6] analysed the thermal performance of a spiral plate heat exchanger used as an adsorber in a refrigeration process, the flows were configured to follow a spiral trajectory and the spiral exchanger resulted appropriated for a refrigeration system. Bahiraei et al. [7, 8] presented a study to evaluate the thermal and hydraulic performance of a spiral plate heat exchanger under a turbulent flow of a nanofluid. The experimental procedure was to determine the effects due to the spiral geometry

**Figure 1.**

*Cross-flow and spiral flow arrangement.*

*Low-temperature Technologies*

varying the flow rate to define the optimal operational conditions.

outlet temperature, pressure drop and heat transfer area [11–15].

**2. Empirical thermal and hydraulic model**

**208**

The thermal and hydraulic concept is an innovative tool to achieve designs for heat exchangers and applies to all types of heat exchangers. Previously, researches have been using this procedure [9, 10]. Compact and conventional exchangers are employed to develop two duties heating and cooling. Nevertheless, they behave differently from each other due to their geometrical configuration, effectiveness,

The current study is organised to describe four main purposes: 1. To present two design methods by a thermal and hydraulic procedure. The first design is for a cooler using a cross-flow arrangement (liquid-gas) without phase change, to evaluate if a spiral plate heat exchanger can take part as a radiator of the cooling system car. The second approach is to size a vertical spiral heat exchanger condenser for a cryogenic operation. 2. To extend the operational activities of spiral plate heat exchangers. 3. To improve the spiral thermal and hydraulic performance by modifying the spacing plate. 4. A numeric analysis applying computational fluid dynamics to validate the method.

The method to design spiral plate heat exchangers includes two main equations, the film heat transfer coefficient and the pressure drop, which both are functions of the fluid properties, heat load, geometrical standard parameters, flow section area and

The hydraulic equations were presented by Minton [16]. These correlations are a function of a flow along the spiral channel which is separated by studs to give support to the plates. Factor 1.5 in Eq. 1 (**Table 1**) supposes 17 studs per square foot. Every stud has a diameter of 0.3125, and, then, in every 0.118 in<sup>2</sup> , a stud is installed [16]. Eq. 2 (**Table 1**) has the same approach; however, the pressure drop is negligible because the fluid flows across the plate width, and value close to zero represents minor influence even by installing the studs [16]. Calculating the Reynolds number and the critical Reynolds number values is possible to select the correct equation to describe the hydraulic operation of the spiral plate heat exchangers. The equations are developed for the three flow regimes: laminar, transition and turbulent. The hydraulic correlations used in this study are presented in **Table 1**.

#### **2.2 Thermal equations**

The thermal model equations were introduced by Minton, although Sander [17] proposed the heat transfer correlation earlier.

Eq. (3) describes the heat transfer coefficient for a liquid fluid flowing by the spiral side. Similarly, Minton presented 11 mechanisms to determine the heat transfer coefficient as a function of flow configuration, type of service (condensing and heating) and a vertical nucleate boiling. Eq. (4) is for gas fluid where the Reynolds number is higher than 10,000. Even when this number is above critical Reynolds number, gases have low heat capacity, and they have poor heat transfer coefficient values (**Table 2**).


#### **Table 1.**

*Correlations for pressure drop [16].*


#### **Table 2.**

*Correlations for heat transfer coefficient [16].*

The Reynolds number and critical Reynolds number are represented by Eqs. (5) and (6):

$$Re = 10,000 \left(\frac{F}{H\mu}\right) \tag{5}$$

$$Re\_{\varepsilon} = 20,000 \left(\frac{D\_{\varepsilon}}{D\_{H}}\right)^{0.32} \tag{6}$$

where F is the flow rate in lb./hr., H is the plate width in inches, μ is the viscosity in cp, De is the equivalent diameter in ft. and DH is the spiral diameter in ft.

Eq. (7) describes the equivalent diameter:

$$\mathbf{D}\_{\mathbf{c} = \begin{bmatrix} \frac{(d\_{\theta}H)^{0.625} \mathbf{1}\_{\mathcal{D}} \end{bmatrix}}{\mathbf{1}\_{\mathcal{D}} + H}} \tag{7}$$

where ds is the channel spacing between plates.

#### **2.3 Geometric additional equations**

Auxiliary equations are needed to complete the thermal and hydraulic spiral plate model, for instance, the spiral outside diameter is determined by Eq. (8):

$$D\_s = \left[\mathbf{15.36L}(d\_{\kappa} + d\_{sh} + \mathbf{2x})\right]^{0.5} \tag{8}$$

L is the plate length; dsc and dsh are the cold and cold spacing cannel, respectively; and x is the plate thickness.

The total heat transfer area is defined by the two spiral plates and is shown in Eq. (9):

$$A = H(\mathfrak{L}L) \tag{9}$$

**2.4 Thermal performance**

*Recommended spacing plate values for standard plate widths [16].*

flow rate.

0.0158 0.0190 0.0254

**Table 3.**

cold stream *CPc* ¼ *Fccpc*:

turns and the CP ratio:

**211**

The validation of the method was carried out by two forms: the calculation of the NTU method and a numerical simulation with the commercial software for computational fluid dynamics ANSYS Fluent. The number of transfer units was implemented to calculate the thermal effectiveness of the spiral plate exchanger, as shown in Eq. (12):

**Plate spacing (m) Plate width (m)** 0.00476 0.101

*Designing Spiral Plate Heat Exchangers to Extend Its Service and Enhance the Thermal…*

*DOI: http://dx.doi.org/10.5772/intechopen.85345*

0.00635 0.457

0.00793 1.524 0.00952 1.778

0.0127 For more than 1.778 m

*NTU* <sup>¼</sup> *UA*

where U is the overall heat transfer coefficient, A is the total heat transfer area and CPmin is the minimum CP for the stream with minimum heat capacity times the

The overall heat transfer coefficient was calculated by the following equation:

The CP ratio was set by Eq. (14) based on the hot stream *CPh* ¼ *Fhcph* and the

*<sup>R</sup>* <sup>¼</sup> *CPmin CPmax*

Bes and Roetzel [12, 19] reported an analytical equation to calculate thermal effectiveness. Correlation 15 contains the number of transfer units, the number of

*<sup>U</sup>* <sup>¼</sup> <sup>1</sup> 1 *hh* <sup>þ</sup> *<sup>x</sup> kA* <sup>þ</sup> <sup>1</sup> *hc*

*CPmin*

(13)

(12)

0.152 0.304 0.304

0.457 0.609 0.609 0.762 0.914 1.219

(14)

Dongwu presented an entire description to calculate the number of spiral turns. This equation is based on plate length L, spiral semicircles, plate spacing, core diameter d and plate thickness. These geometric values are presented in Eq. (10). Due to a constant spacing of the two rolled passages, the number of turn equation gives an effective accuracy [18]:

$$N = \frac{-\left(d - \frac{t}{2}\right) + \sqrt{\left(d - \frac{t}{2}\right)^2 + \frac{4tL}{\pi}}}{2t} \tag{10}$$

where d is the core diameter of the first turn at the centre of the spiral heat exchanger and t is composed by Eq. (11):

$$\mathbf{t} = d\_{\rm sh} + d\_{\rm sc} + 2\mathbf{x} \tag{11}$$

The improvement of the spiral heat exchangers depends on geometrical variables. For instance, the spacing plate can increase the pressure drop if the separation between the spiral plates decreases. That means the heat exchanger needs more pumping energy, besides this, improves the heat transfer coefficient and the thermal effectiveness. The variation of the spacing channel allows to enhance the heat transfer and optimise the thermal and hydraulic performance.

**Table 3** shows some recommended spacing plate values, and the values depend on the calculated plate width.


*Designing Spiral Plate Heat Exchangers to Extend Its Service and Enhance the Thermal… DOI: http://dx.doi.org/10.5772/intechopen.85345*

#### **Table 3.**

The Reynolds number and critical Reynolds number are represented by Eqs. (5)

*F Hμ* � �

*DH* � �0*:*<sup>32</sup>

h i (7)

*A* ¼ *H*ð Þ 2*L* (9)

<sup>0</sup>*:*<sup>5</sup> (8)

where F is the flow rate in lb./hr., H is the plate width in inches, μ is the viscosity

*<sup>e</sup>*<sup>¼</sup> ð Þ *dsH* <sup>0</sup>*:*6251*:*<sup>3</sup> ð Þ *ds*þ*<sup>H</sup>* <sup>0</sup>*:*<sup>25</sup>

Auxiliary equations are needed to complete the thermal and hydraulic spiral plate model, for instance, the spiral outside diameter is determined by Eq. (8):

*Ds* ¼ ½ � 15*:*36*L d*ð Þ *sc* þ *dsh* þ 2*x*

L is the plate length; dsc and dsh are the cold and cold spacing cannel, respec-

The total heat transfer area is defined by the two spiral plates and is shown in

Dongwu presented an entire description to calculate the number of spiral turns.

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>d</sup>* � *<sup>t</sup>* 2 � �<sup>2</sup> <sup>þ</sup> <sup>4</sup>*tL*

*π*

*t* ¼ *dsh* þ *dsc* þ 2*x* (11)

<sup>2</sup>*<sup>t</sup>* (10)

This equation is based on plate length L, spiral semicircles, plate spacing, core diameter d and plate thickness. These geometric values are presented in Eq. (10). Due to a constant spacing of the two rolled passages, the number of turn equation

> 2 � � <sup>þ</sup>

where d is the core diameter of the first turn at the centre of the spiral heat

The improvement of the spiral heat exchangers depends on geometrical variables. For instance, the spacing plate can increase the pressure drop if the separation between the spiral plates decreases. That means the heat exchanger needs more pumping energy, besides this, improves the heat transfer coefficient and the thermal effectiveness. The variation of the spacing channel allows to enhance the heat

**Table 3** shows some recommended spacing plate values, and the values depend

*<sup>N</sup>* <sup>¼</sup> � *<sup>d</sup>* � *<sup>t</sup>*

transfer and optimise the thermal and hydraulic performance.

(5)

(6)

*Re* ¼ 10*;* 000

*Rec* <sup>¼</sup> <sup>20</sup>*;* <sup>000</sup> *De*

in cp, De is the equivalent diameter in ft. and DH is the spiral diameter in ft.

*D*

Eq. (7) describes the equivalent diameter:

**2.3 Geometric additional equations**

tively; and x is the plate thickness.

gives an effective accuracy [18]:

on the calculated plate width.

**210**

exchanger and t is composed by Eq. (11):

Eq. (9):

where ds is the channel spacing between plates.

and (6):

*Low-temperature Technologies*

*Recommended spacing plate values for standard plate widths [16].*

#### **2.4 Thermal performance**

The validation of the method was carried out by two forms: the calculation of the NTU method and a numerical simulation with the commercial software for computational fluid dynamics ANSYS Fluent. The number of transfer units was implemented to calculate the thermal effectiveness of the spiral plate exchanger, as shown in Eq. (12):

$$NTU = \frac{UA}{CP\_{min}}\tag{12}$$

where U is the overall heat transfer coefficient, A is the total heat transfer area and CPmin is the minimum CP for the stream with minimum heat capacity times the flow rate.

The overall heat transfer coefficient was calculated by the following equation:

$$U = \frac{1}{\left(\frac{1}{h\_h} + \frac{\chi}{kA} + \frac{1}{h\_\epsilon}\right)}\tag{13}$$

The CP ratio was set by Eq. (14) based on the hot stream *CPh* ¼ *Fhcph* and the cold stream *CPc* ¼ *Fccpc*:

$$R = \frac{C P\_{\min}}{C P\_{\max}} \tag{14}$$

Bes and Roetzel [12, 19] reported an analytical equation to calculate thermal effectiveness. Correlation 15 contains the number of transfer units, the number of turns and the CP ratio:

$$\varepsilon = \frac{\mathbf{1} - e^{(R-1)NTU}}{\mathbf{1} - Re^{(R-1)NTU}} \tag{15}$$

The next expressions were used to calculate the hot outlet temperature and the cold outlet temperature. These equations depend on the CP ratio, effectiveness and maximum temperature difference. The procedure to calculate the hot temperature at the exit of the spiral flow is as follows:

If the *CPh* >*CPc*, then the correct form of the equation is

$$T\_{hout} = T\_{hin} - \varepsilon (T\_{hin} - t\_{cin}) \tag{16}$$

If the *CPh* <*CPc*, then the equation takes the next form:

$$T\_{hout} = T\_{hin} - \epsilon R (T\_{hin} - t\_{cin}) \tag{17}$$

Next, the procedure to calculate the cold temperature at the exit of the axial flow is shown in Eqs. (18) and (19).

If the *CPh* <*CPc*, then the equation takes the next form:

$$t\_{\rm out} = t\_{\rm cin} + \varepsilon (T\_{\rm fin} - t\_{\rm cin}) \tag{18}$$

**Figure 2** shows the sequential steps to design the spiral plate heat exchanger. A

) 11.57

The numerical simulation was performed by the software ANSYS Fluent. The mathematical model κ-ε solved the three balances: mass, energy and momentum.

where k is the thermal conductivity, ρ is the density, μ is the dynamic viscosity, cp is the specific heat, v is the velocity, T is the temperature and P is the pressure

∇ � ð Þ¼ *ρv* 0∇ (20)

**Water Air**

∇ � ð Þ¼� *ρvv* ∇*P* þ ∇ � ð Þ *μ*∇*v* (21)

∇ � ð Þ¼ *ρvcpT* ∇ � ð Þ *k*∇*T* (22)

visual basic programming code was used to achieve these calculations.

Thermal effectiveness 0.84

*Designing Spiral Plate Heat Exchangers to Extend Its Service and Enhance the Thermal…*

Height (m) 0.41 Length (m) 0.54 Width (m) 0.028

Outlet temperature (°C) 80.95 86.75 Mass flow (kg/h) 4200 5200 Inlet temperatures (°C) 98 20

**2.5 Computational fluid dynamics (CFD)**

Conservation of mass:

*Car radiator design (car radiator) [20].*

Total heat transfer area (m2

*DOI: http://dx.doi.org/10.5772/intechopen.85345*

Conservation of momentum:

Conservation of energy:

(**Table 4**).

**Figure 3.**

**213**

*Mesh of the spiral plate heat exchanger.*

**Table 4.**

If the *CPh* >*CPc*, then the correct equation is

$$\mathbf{t}\_{\rm out} = \mathbf{t}\_{\rm cin} + \varepsilon \mathbf{R} (T\_{\rm hin} - \mathbf{t}\_{\rm cin}) \tag{19}$$

**Figure 2.** *Flow chart to design the cross-flow spiral heat exchanger.*

*Designing Spiral Plate Heat Exchangers to Extend Its Service and Enhance the Thermal… DOI: http://dx.doi.org/10.5772/intechopen.85345*


#### **Table 4.**

*<sup>ε</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>e</sup>*ð Þ *<sup>R</sup>*�<sup>1</sup> *NTU*

at the exit of the spiral flow is as follows:

*Low-temperature Technologies*

is shown in Eqs. (18) and (19).

**Figure 2.**

**212**

*Flow chart to design the cross-flow spiral heat exchanger.*

If the *CPh* >*CPc*, then the correct form of the equation is

If the *CPh* <*CPc*, then the equation takes the next form:

If the *CPh* <*CPc*, then the equation takes the next form:

If the *CPh* >*CPc*, then the correct equation is

The next expressions were used to calculate the hot outlet temperature and the cold outlet temperature. These equations depend on the CP ratio, effectiveness and maximum temperature difference. The procedure to calculate the hot temperature

Next, the procedure to calculate the cold temperature at the exit of the axial flow

<sup>1</sup> � *Re*ð Þ *<sup>R</sup>*�<sup>1</sup> *NTU* (15)

*Thout* ¼ *Thin* � *ε*ð Þ *Thin* � *tcin* (16)

*Thout* ¼ *Thin* � *εR T*ð Þ *hin* � *tcin* (17)

*tcout* ¼ *tcin* þ *ε*ð Þ *Thin* � *tcin* (18)

*tcout* ¼ *tcin* þ *εR T*ð Þ *hin* � *tcin* (19)

*Car radiator design (car radiator) [20].*

**Figure 2** shows the sequential steps to design the spiral plate heat exchanger. A visual basic programming code was used to achieve these calculations.
