Comparative study of the thermal performance of four different parallel flow shell and tube heat exchangers with different performance indicators

Greek symbols

ρ

density (kg m⁻³)

ε

turbulence kinetic energy dissipation rate

k

turbulence kinetic energy

μ

dynamic viscosity (kg/m s)

λ

thermal conductivity (W m⁻¹ K⁻¹)

η

entropy generation number ratio

γ

equivalent temperature difference (K)

λ*

dimensionless length of the duct

τ

dimensionless temperature difference

Subscripts

w

wall

in

inlet

out

outlet

i,j

tensor

eff

effective term

rod

round rod

t

tube side

s

shell side

diss

dissipation

Abbreviations

RRB

round rod baffle

PPB

plain plate baffle

WSB

wavy curve plate baffle

PSB

polygonal curve plate baffle

STHX

shell and tube heat exchanger

PEC

performance evaluation criterion (–)

3.1 Physical model

A periodic flow unit duct is taken as the simplified model of the shell side to perform numerical simulation by using Fluent. The following assumptions are made: (1) the working fluid is water with constant thermal properties (density ρ = 999.7 kg/m³, thermal conductivity λ = 0.574 W/(m K), specific heat C _p = 4,191 (J/kg K), and μ = 0.001306 (Pa s)) remaining constant during the calculation process; (2) the fluid flow and heat transfer processes are turbulent and in steady state; (3) the heat exchanger is well insulated; (4) the viscous heating is neglected; and (5) the effect of gravity is negligible.

The actual length of the computational domain is four times the baffle distance plus 200 mm. That is, the length of the entrance section and exit section is equal to 100 mm and 100 mm to ensure the inlet uniformity and avoid outlet flow recirculating [25]. The schematic diagram of computational domain of STHX-RRB, STHX-PPB, STHX-WSB, and STHX-PSB is shown in Figure 4. The mesh of partial zoom out view of the selected baffle zone is shown in Figure 5.

Figure 4

Schematic diagram of computational domain of STHX-RRB, STHX-PPB, STHX-WSB and STHX-PSB.

Figure 5

Partial zoom out view of selected baffle zone of different unit duct mesh.

3.2 Numerical methods

The conservation equations of continuity, momentum, and energy for the turbulent heat transfer of fluid flowing on the shell side are presented in the tensor form as follows [26]:

Continuity equation:

Momentum equation:

Energy equation:

where T and p stand for fluid temperature and pressure, respectively; u is the fluid velocity; μ eff stands for effective dynamic viscosity, which is equal to the sum of laminar and turbulent dynamic viscosities, i.e., μ eff = μ l + μ t . The effective thermal conductivity, λ eff , is calculated by λ eff = λ l + μ t c p / Pr t , where λ l and Pr t are the laminar conductivity and turbulent Prandtl number, respectively. Negligible viscous dissipation is adopted in modeling the flow in this article because the working fluid is assumed to be incompressible and physical parameters are constant.

For simple and fully developed turbulence cases, the standard k–ε turbulence model is applicable, and it is easier to converge than other turbulence models. Thus, standard k−ε turbulence model together with the standard wall function is adopted for the current computation. The conservation equations of turbulent kinetic energy and its dissipation rate are given below [27]:

For turbulence kinetic energy k:

For turbulence kinetic energy dissipation rate ε:

where μ t = ρ C μ k 2 / ε ; G k = 2 μ t E i j E i j ; E i j = 1 / 2 ( ( ∂ u i / ∂ x j ) + ( ∂ u i / ∂ x j ) ) .

The constants for the current turbulent model are set as below:

The 3-D, double-precision, pressure-based solver is used. The nonslip boundary condition is adopted on all solid surfaces. The surfaces of the baffles are set as adiabatic. The velocity-inlet boundary condition and the pressure-outlet boundary condition are applied for the inlet and outlet, respectively. The other surfaces of the unit model are set as symmetry boundary conditions. SIMPLE algorithm is used for pressure–velocity coupling. The second-order upwind difference scheme is applied for energy and momentum computation. The second-order difference scheme is used for the pressure. The inlet bulk temperature is set as 283.15 K. The temperature of tube wall is set as a constant, 350 K. All equations take the convergent criteria of relative residual of 10⁻⁴ except energy taking 5 × 10⁻⁷.

3.3 Data reduction

Some formulas used in the postprocessing are defined as [28,29,30]:

3.4 Grid independence and code validation

In order to validate the solution independency of the grid, the grid dependency is checked for three different grids of STHX-RRB which have 2.2, 2.9, and 3.5 million cells, respectively. For the heat transfer coefficient, the error between the finest grid having 2.2 million cells and the grid having 3.5 million cells is less than 2% for the heat transfer coefficient and less than 3% for the friction coefficient. Therefore, the settings of the grid of the latter (3.5 million cells) are used for further investigations in the current study.

To validate the reliability of present numerical model, the results of nonstaggered tubes supported by RRB are compared with the data obtained by Dong et al. [17] the working fluid, the boundary conditions, and the baffle distance of them are all the same. The average Nu is plotted in Figure 6. For the unit model, it was found that the Nu was relatively smaller than the experimental data, especially in higher Reynolds number. This is mainly because the boundary is set as symmetry wall condition, thus there is no heat and mass transfer on that boundary in numerical calculation. While in fact there is plenty of turbulence and secondary flow leading to heat and mass transfer on the virtual boundary in the experiment. Figure 6 shows that the relative maximum deviation is within 16%. Such agreements show the reliability of the present physical model and numerical method.

Figure 6

The model validation of STHX-RRB.

3.5 Performance indicators

Based on the first law of thermodynamics, the transferred energy ratio under the same pump consumption can be evaluated using equation (13) [21].

Based on the second law of thermodynamics, the nondimensional entropy generation number ratio η can be used to weigh the irreversibility. For the problem presented in this article, the entropy generation of a duct with constant wall temperature is directly given below [22].

where τ = ( T w − T in ) / T w , λ ⁎ = L / D o , Ec = V in 2 / c p T w and St = h / ρ V in c p .

It is worth noting that if η < 1 and the smaller of the η, the better of the thermal–hydraulic performance and heat enhancement.

For the optimization of heat transfer process, a new physical quantity entransy dissipation extremum principle was proposed, which was initially referred to as the heat transfer potential capacity. For tube under uniform temperature condition, the entransy balance equation is directly given below [23].

It is deemed that the larger of γ, the more entransy dissipation, the more irreversibility of the heat transfer and the less heat can be transferred.

In this article, the abovementioned three performance indicators are used to compare the overall thermal–hydraulic performance of STHX-RRB, STHX-PPB, STHX-WSB, and STHX-PSB.

4.1 Inlet Re effect

For Re in the range of 10,849–32,547, the variation in different performance indicators of STHX-RRB, STHX-PPB, STHX-WSB, and STHX-PSB is shown in Figure 7. The baffle distance is 200 mm. Other geometrical parameters of these four STHXs are the same with those listed in Table 1.

Figure 7

The effect of inlet Re on different performance indicators. (a) Nu comparisons for different Re, (b) f-factor comparisons for different Re, (c) PEC comparisons for different Re, (d) N _s comparisons for different Re, (e) η comparisons for different Re, (f) γ comparisons for different Re.

Figure 7(a) shows that the Nu of STHX-RRB, STHX-PPB, STHX-WSB, and STHX-PSB increases with an increase in Re. The increasing rate is calculated based on the results of STHX-RRB. It can be found that the average increasing rate is 20.9% for STHX-PPB, 15.2% for STHX-WSB, and 23.9% for STHX-PSB. This indicates that the heat transfer enhancement ability order of STHX-RRB, STHX-PPB, STHX-WSB, and STHX-PSB is STHX-RRB < STHX-WSB < STHX-PPB < STHX-PSB.

Figure 7(b) shows that the f-factor of STHX-RRB, STHX-PPB, STHX-WSB, and STHX-PSB decreases with an increase in Re. Similarly, compared with STHX-RRB, it can be found that the average increasing rate is 142.0% for STHX-PPB, 154.6% for STHX-WSB, and 242.4% for STHX-PSB. This indicates that the power consumption order of STHX-RRB, STHX-PPB, STHX-WSB, and STHX-PSB is STHX-RRB < STHX-PPB < STHX-WSB < STHX-PSB.

Figure 7(c) shows that STHX-RRB has the largest PEC. The average PEC of STHX-PPB, STHX-WSB, and STHX-PSB is 90.1%, 84.4%, and 82.3% that of STHX-RRB. This indicates that the overall thermal–hydraulic performance order of STHX-RRB, STHX-PPB, STHX-WSB, and STHX-PSB is STHX-PSB < STHX-WSB < STHX-PPB < STHX-RRB. The reason why STHX-PPB has better overall thermal–hydraulic performance than STHX-WSB and STHX-PSB is that the pressure drop of STHX-PSB increases fast while the Nu of STHX-WSB increases slow.

Figure 7(d) and (e) shows the variations in N _s and η with different Re. It can be found that the N _s increases with an increase in Re. The N _s and η of STHX-RRB, STHX-PPB, STHX-WSP, and STHX-PSP follow the order: STHX-PSB < STHX-PPB < STHX-WSP < STHX-RRB. This indicates that STHX-PSB has a better ability to reduce irreversibility compared to the other three. Therefore, STHX-PSB is recommended if minimum entropy generation is expected.

Figure 7(f) shows the variation in γ with different Re. It can be found that the γ increases with an increase in Re. The relative relation of γ between four different baffles is the same with those of N _s and η. This implies that STHX-PSB can reduce irreversibility more effectively than others. This further proves the accuracy of entransy dissipation theory.

4.2 Baffle distance effect

For baffle distance is in the range of 110–350 mm; the variation in different performance indicators of STHX-RRB, STHX-PPB, STHX-WSB and STHX-PSB is shown in Figure 8. The inlet Re is 10,849 and other geometrical parameters of these four STHXs are the same as those listed in Table 1.

Figure 8

The effect of baffle distance on different performance indicators. (a) Nu comparisons for different baffle distance, (b) f-factor comparisons for different baffle distance, (c) N _s comparisons for different baffle distance, (d) N _s comparisons for different baffle distance, (e) η comparisons for different baffle distance, (f) γ comparisons for different baffle distance.

Figure 8(a) shows that the Nu decreases with an increase in baffle distance. The Nu of STHX-RRB, STHX-PPB, STHX-WSB, and STHX-PSB with a baffle distance of 110 mm is, respectively, 133%, 148.8%, 157%, and 157.4% times of those whose baffle distance is 350 mm. This indicates that the increase in baffle distance can significantly decrease the heat transfer rate. The reason is that an increased baffle distance would decrease the turbulence intensity of the jet flow along the flow passage.

Figure 8(b) shows that the f-factor decreases with an increase in baffle distance. The f-factor of STHX-RRB, STHX-PPB, STHX-WSB, and STHX-PSB with a baffle distance of 110 mm is, respectively, 243.2%, 312.7%, 320.7%, and 315.6% times of those whose baffle distance is 350 mm. This indicates that an increase in baffle distance can decrease the f-factor. This means that when RRB is replaced by PPB/WSB/PSB, the baffle distance should be enlarged to avoid excess pressure drop.

Figure 8(c) depicts the change in PEC versus baffle distance. It can be found that the PEC of STHX-PPB, STHX-WSB, and STHX-PSB follows the order of STHX-PSB < STHX-WSB < STHX-PPB at every baffle distance. Quantitatively, the PEC of STHX-PPB, STHX-WSB, and STHX-PSB is on average 87.7%, 83.7%, and 80.6% that of STHX-RRB over the investigated baffle distance. This means that the STHX-PPB has the best overall thermal–hydraulic performance among the above three.

Figure 8(d) and (e) shows the variations in N _s and η in different baffle distance. It can be found that the N _s decreases with an increase in baffle distance. The reason is that an increase in heat transfer area results in the reduction of irreversibility. In addition, it can be seen that the N _s and η of STHX-RRB, STHX-PPB, STHX-WSP, and STHX- PSB follow the order: STHX-PSP < STHX-PPB < STHX-WSP < STHX-RRB. This further proves that STHX-PSB has a better ability to reduce irreversibility than the other three. Therefore, a relative larger baffle distance is recommended if a minimum entropy generation is expected.

Figure 8(f) shows that the variation in γ in different baffle distances. It can be found that the γ decreases with an increase in baffle distance, and the relative relation of the γ between four different baffles is similar to those of N _s and η. This demonstrates that STHX-PSB is a more effective parallel flow baffle to reduce irreversibility compared to others.

4.3 Baffle width effect

Figure 3 shows that for STHX-PPB, STHX-WSB, and STHX-PSB, the contact area between baffle and tube increases with an increase in baffle width b. As a result, the contact stress between the baffle and tube decreases. In other words, the increase in baffle width is beneficial to antivibration ability.

For baffle width in the range of 10–30 mm, the variation in different performance indicators of STHX-PPB, STHX-WSB, and STHX-PSB is shown in Figure 9. The inlet Re is 10,849 and all other geometrical parameters of these three STHXs are the same as those listed in Table 1.

Figure 9

The effect of baffle width on different performance indicators. (a) Nu comparisons for different baffle width, (b) f-factor comparisons for different baffle width, (c) PEC comparisons for different baffle width, (d) N _s comparisons for different baffle width, (e) η comparisons for different baffle width, (f) γ comparisons for different baffle width.

Figure 9(a) shows that the Nu increases with an increase in baffle width. The reason for this phenomenon is that with an increase in baffle width, the scope of zone with the smallest flow area also extended, which enhances the heat transfer and increases pressure drop. Besides, it can be seen that the Nu of STHX-PPB, STHX-WSB, and STHX-PSB with a baffle width of 30 mm is, respectively, on average 4.23%, 2.05%, and 3.97% higher than that with the baffle width of 10 mm. This means that the baffle width does not have a significant effect on the heat transfer characteristic.

Figure 9(b) shows that the f-factor increases with an increase in baffle width. It can be found that the f-factor of STHX-PPB, STHX-WSB, and STHX-PSB with a baffle width of 30 mm is, respectively, on average 8.22%, 13.01%, and 9.1% higher than that with baffle width of 10 mm. This means that baffle width has a significant effect on the pressure drop characteristic.

Figure 9(c) shows the changes in PEC under different baffle width. It can be found that the PEC does not monotonically increase or decrease with an increase in baffle width for every Re. The PEC of STHX-PPB and STHX-PSB with a baffle width of 30 mm is, respectively, on average 1.53% and 0.99% higher than that with baffle width of 10 mm. While the PEC of STHX-WSB with baffle width of 30 mm is −2.03% lower than that with baffle width of 10 mm. This means that an increase in the baffle width is beneficial in increasing the overall thermal–hydraulic performance of STHX-PPB and STHX-PSB while decreasing the baffle width is beneficial in increasing the overall thermal–hydraulic performance of STHX-WSB.

Figure 9(d) and (e) shows the changing tendency of N _s and η in different baffle width. It can be found that the N _s decreases with an increase in baffle width for a given Re. The reason is that increasing the baffle width results in the increase of the transferred heat. This indicates that increasing the baffle width can reduce irreversibility. Therefore, a relative large baffle width is recommended if a minimum entropy generation is expected.

Figure 9(f) shows that the variation in γ in different baffle width. It can be found that the γ increases with an increase in baffle width for a given Re. This result is consistent with the relationship between the entropy generation number and the baffle width.

4.4 Correlations for Nusselt number and friction factor

The Nu and f-factor correlations are fitted for STHX-RRB, STHX-PPB, STHX-WSB, and STHX-PSB. These correlations are shown in equations (18)–(25).

The adjusted residual square of the above fitted equations is 0.9963, 0.9995, 0.99936, 0.99909, 0.99863, 0.99893, 0.99904 and 0.99856. The above correlations are valid for Re in the range of 10,849–32,547, for baffle distance L _b in the range of 110–350 mm and baffle width b in the range of 10–30 mm.

4.5 Cost comparison

In this section, cost comparison is carried out for an STHX with different parallel flow baffles. The input design parameters of the STHX are listed in Table 3. The basic geometric parameters of the STHX with different parallel flow baffles are listed in Table 4. The cost of STHX is calculated with Hall’s method [23,24]. Corresponding equations are listed in equations (26)–(36). Here we assume that the capital investment cost per area of STHX-RRB, STHX-PPB, STHX-WSB, and STHX-PSB is the same. The thermal resistance of tube wall and fouling thermal resistance are neglected. The value of ζ is 0.6. The cost comparison results are listed in Table 5.

where C _tot is the total cost, C _i is the capital investment cost, C _op is the operating cost, A is the total tube outside the heat transfer area, elife is the equipment design life, δ is the rate of annual discount, k _el is the price of electricity, H is the hours of operation per year, P is the pumping power, ζ is the pump efficiency, Δp _t is the tube-side pressure drop, Δp _s is the shell-side pressure drop, V _t is the volume flow rate of tube side, and V _s is the volume flow rate of shell side.

It can be found from Table 5 that the C _i sequence of these four parallel flow STHX is STHX-PSB < STHX-PPB < STHX-WSB < STHX-RRB. The reason is that the PSB can enhance heat transfer rate more effectively than the other three. The C _op sequence is STHX-RRB < STHX-PSB < STHX-PPB < STHX-WSB. The reason is that the RRB has the smallest effect on pressure drop. The C _tot sequence of these four parallel flow STHX is as follows: STHX-PSB < STHX-PPB < STHX-WSB < STHX-RRB. This means that STHX-PSB exhibits an obvious economic benefit compared with the other three. The reason is that STHX-PSB has a better effect on heat enhancement although the C _op is the largest. Thus, the required total heat transfer area is also less. On the contrary, although the PEC of STHX-RRB is the largest, the heat enhancement is weak. Thus, the required total heat transfer area is large. This means that the STHX-PSB is recommended if the lowest total cost is expected for the design parameters given in Tables 3 and 4.

It is worth noting that cost estimation may be a more appropriate performance indicator than PEC, N _s , and γ for designers. Because cost estimation can reflect the effect of design parameters of both shell side and tube side, PEC can only reflect the effect of parameters of one side (shell side in this article). The N _s and γ are not intuitionistic for designers.

4.6 Heat transfer mechanism analysis

To better understand the heat transfer mechanism of STHX-RRB, STHX-PPB, STHX-WSB, and STHX-PSB, the 3D streamlines of them for inlet Re at 10,849 and baffle distance of 200 mm are plotted in Figure 10. The geometrical parameters of these four STHXs are the same as those listed in Table 1. The temperature variation in three cross-sections of the unit duct at position Z = 0.2 m, Z = 0 m, and Z = −0.2 m is shown in Figure 11. The field distribution contours on Y–Z plane are shown in Figure 12.

Figure 10

3D streamlines comparison of STHX-RRB, STHX-PPB, STHX-WSB and STHX-PSB.

Figure 11

Temperature contours comparisons of STHX-RRB, STHX-PPB, STHX-WSB and STHX-PSB at selected cross-sections.

Figure 12

The velocity, pressure and temperature contours of Y–Z plane of flow unit of STHX-RRB, STHX-PPB, STHX-WSB and STHX-PSB. (a) Velocity contour (m/s), (b) pressure contour (Pa), (c) temperature contour (K).

Figure 10 shows that the streamlines change their direction at the baffle zone. Some vortexes are generated at the back of the baffles. From Table 2, it can be found that the blocked area ratio order is as follow: STHX-PSB > STHX-WSB > STHX-PPB. Figure 10 shows that the velocity increasing rate order of STHX-RRB, STHX-PPB, STHX-WSB, and STHX-PSB at the baffle zone is the same as the blocked area ratio order mentioned above.

Figure 11 shows that, at cross-section Z = 0 m, the temperature of the water near the heated tube rises rapidly. However, an inner core no-heated region can also be seen in STHX-RRB and STHX-WSB. Furthermore, we can find that the no-heated zone in STHX-RRB is quite larger than that in STHX-WSB. This means that the working fluid in STHX-RRB shows a slower temperature rise compared with that in STHX-PPB, STHX-WSB, and STHX-PSB. This illustrates the low heat transfer capacity of STHX-RRB as demonstrated in Figure 7(a).

From Figure 12, we can find that the fluid velocity, pressure drop, and the temperature rapidly increase when the fluid flow passes the baffle. Meanwhile, the thickness of the boundary layer decreases. The variation in degree of velocity, pressure, and temperature follows an order of STHX-RRB < STHX-WSB < STHX-PPB < STHX-PSB. This further verifies the observation in Figure 10.