Optimizing chevron plate heat exchangers based on the second law of thermodynamics and genetic algorithm

Abstract

The aim of the current work is to determine optimal geometries and flow conditions of the chevron plate heat exchangers based on entropy generation minimization approach (a combination of the second law of the thermodynamics and heat transfer and fluid-flow equations). The optimization process is carried out by considering the entropy generation as target function. The all effective parameters are taken into account including chevron angle (30° ≤ β ≤ 60°), surface enlargement factor (1.1 ≤ ϕ ≤ 1.4), dimensionless plate width (19 ≤ ${\mathcal{W}}$ ≤ 79), Prandtl number (2.6 ≤ Pr ≤ 6.4) and Reynolds number (1000 ≤ Re ≤ 8000). The results indicate that for each surface enlargement factor, there is an optimum chevron angle. Also, by increasing chevron angle, the optimum values of dimensionless plate width, working fluid Prandtl number and Reynolds number decrease. After presenting a comprehensive sensitivity analysis, the genetic algorithm is utilized to find optimum conditions at (a) designing and (b) operating situations. In the first situation, the optimization process reveals optimum chevron angle, surface enlargement factor, dimensionless plate width, Prandtl number and Reynolds number. For the second situation, a useful and practical correlation is developed for obtaining optimum Reynolds number as a function of the geometrical parameters.

Graphic abstract

Appendices

Appendix 1

In this section, formulation for the rate of entropy generation, $\dot{S}'_{\mathrm{gen}}$ (W mK⁻¹), is derived for internal flow in a heat plate exchanger. Consider the flow passage of cross-section of a PHE (Fig. 11). The bulk properties of the stream $\dot{m}$ are $T, P, h, \rho , s$. When heat is transferred to stream at a rate of $q^{\prime}$, temperature difference is ΔT. Focusing on slice of thickness dx as a system, the rate of entropy generation is given by the second law of thermodynamics as:

Fig. 11

Heat transfer in a PHE cross-section

$${{d}}\dot{S}'_{\mathrm{gen}} = \dot{m}{{d}}s - \frac{{q^{\prime}{{d}}x}}{T + \Delta T}$$

(13)

The first law of thermodynamics is applied to same system as:

$$\dot{m}{{d}}h = q^{\prime}{{d}}x$$

(14)

In addition for any pure substance:

$$\frac{dh}{dx} = T\left( {\frac{ds}{sx}} \right) + \frac{1}{\rho }\left( {\frac{dp}{dx}} \right)$$

(15)

Substituting $d\dot{S}'_{\rm gen}$ given by Eq. (13) and dh given by Eq. (14) into Eq. (15) yields the entropy generation rate per unit length:

$$\dot{S}'_{\mathrm{gen}} = \frac{{q^{\prime}\Delta T}}{{T^{2} \left( {1 + \tau } \right)}} + \frac{{\dot{m}}}{\rho T}\left( { - \frac{dp}{dx}} \right)$$

(16)

Dimensionless temperature difference τ is negligible as compared to unity, as a result:

$$\dot{S}'_{\mathrm{gen}} = \frac{{q^{\prime}\Delta T}}{{T^{2} }} + \frac{{\dot{m}}}{\rho T}\left( { - \frac{dp}{dx}} \right)$$

(17)

The relationship between heat transfer rate q^′ and wall-bulk fluid temperature is expressed in the form of Stanton number:

$$St = \frac{{\frac{{q^{\prime}}}{2w\Delta T}}}{{c_{\mathrm{p}} \left( {\dot{m}/A_{\mathrm{c}} } \right)}}$$

(18)

The friction characteristics of the fluid inside a duct are usually reported by correlation of the friction factor:

$$f = \frac{{\rho d_{e} }}{{2\left( {\dot{m}/A_{\mathrm{c}} } \right)^{2} }}\left( { - \frac{dP}{dx}} \right)$$

(19)

Substituting ΔT given by Eq. (18) and $\frac{{{\text{d}}p}}{{{\text{d}}x}}$ given by Eq. (19) into Eq. (17), the entropy generation rate per unit length for an internal fluid flow could be written as Eq. (8):

$$\dot{S}'_{\mathrm{gen}} = \frac{{q{\prime}^{2} d_{\mathrm{e}} }}{{4T^{2} \dot{m}C_{\mathrm{p}} St}} + \frac{{2\dot{m}^{3} f}}{{\rho^{2} Td_{\mathrm{e}} A_{\mathrm{c}}^{2} }}$$

(20)

Appendix 2

The following equation can be used for determining the goodness evaluation parameter of R²:

$$R^{2} = 1 - \frac{{{\text{SS}}_{\mathrm{residuals}} }}{{{\text{SS}}_{\mathrm{total}} }}$$

(21)

where the sum-of-squares of the residuals (SS_residuals) from the regression line (fitted curve) has n - K degrees of freedom, where n is the number of data points and K is the number of parameters fit by the regression. The total sum-of-squares (SS_total) is the sum of the squares of the distances from a horizontal line through the mean of all Y values.