Analysis of entransy dissipation in heat exchangers

Abstract

The optimization of heat exchangers is an important topic. Entropy generation is used to describe the irreversibility of heat transfer processes and the principle of minimum entropy generation is sometimes used to optimize heat exchanger designs. This paper defines the heat exchanger thermal resistance based on its entransy dissipation and analyses various heat exchangers. Entropy generation analyses are also presented for comparison. The results indicate that the minimum entransy-dissipation-based thermal resistance always corresponds to the highest heat transfer rate, while the design with the minimum entropy generation is not always related to the design with the highest heat transfer rate.

Introduction

Energy consumption is continuously increasing and energy issues are becoming increasingly prominent. We need to improve the efficiency of energy utilization to reduce energy consumption and emissions of greenhouse gases from the fossil fuels. Heat exchangers are used to transfer thermal energy from one fluid to another, they are widely used in industry and daily life, and about 80% energy utilization are involved in heat transfer processes. Therefore, there is great interest in improving heat exchangers performance by optimizing the heat transfer processes in heat exchangers for high-efficiency energy utilization.

The heat transfer in heat exchangers is an irreversible process from the point of view of non-equilibrium thermodynamics. Onsager [1], [2] set up the fundamental equations for non-equilibrium thermodynamic processes and derived the principle of least energy dissipation using the variational principle. Prigogine [3] developed the principle of minimum entropy generation based on the idea that the entropy generation in a thermal system at steady-state should be minimum. However, both of these principles do not deal with heat transfer optimization. Many researchers have tried to establish a link between heat exchanger performance and its heat transfer irreversibility. McClintock [4] carried out a pioneering work in this area by applying the irreversibility concept to heat exchanger design. Bejan [5], [6], [7] explained two contributions to the exergy loss as heat transfer across a finite temperature difference and through the fluid friction in channels. He introduced an entropy generation number, N_s, to quantitatively estimate the entropy generation. The analysis of a counterflow heat exchanger showed that the entropy generation number, N_s, approaches zero in two limits: when the number of transfer unit approaches infinity, NTU → ∞, or the effectiveness approaches unity, ɛ → 1, which represents the ideal limit of zero driving temperature difference, and when the number of transfer unit approaches zero, NTU → 0, or the effectiveness approaches zero, ɛ → 0, which represents the heat transfer surface approaching zero. As shown in Fig. 1, the entropy generation number reaches maximum at ɛ = 0.5 and the heat exchanger effectiveness increases with the increasing entropy generation number for ɛ∈(0,0.5). On the right side, smaller entropy generation numbers result in larger effectiveness. Bejan referred to this symmetry behavior as the entropy generation paradox. The entropy generation paradox is not consistent with the principle of minimum entropy generation. Some scholars have tried to explain the entropy generation paradox in terms of the entropy generation in heat exchangers. Sekulic [8], [9] explained that temperature cross between hot stream and cold stream caused this paradox and examined the influence of parameters like the inlet temperature ratio, fluid flow heat capacity rate ratio and the effectiveness on the quality of the energy transformation for different types of heat exchangers. Hesselgreaves [10] and Ogiso [11] eliminated the entropy generation paradox by changing the dimensionless method and using a modified entropy generation number. Shah and Skiepko [12] analyzed 18 kinds of heat exchangers using the ɛ-NTU method to show that the heat exchanger effectiveness can have a maximum, an intermediate value or a minimum at the maximum irreversibility operating point depending upon the flow arrangement. Similarly, the heat exchanger effectiveness can be a minimum or a maximum at the minimum irreversibility operating point. Shah and Skiepko [12] believed that this paradox for the standalone heat exchanger is an intrinsic behavior of the temperature difference irreversibility function and can never be removed without violating the Second Law.

However, none of these analyses adequately explain the entropy generation paradox. The entropy generation paradox shows that the minimum entropy generation principle may not be appropriate for heat exchanger optimization, on the other words, problem may be existed to use entropy generation to describe the irreversibility of the heat transfer process in heat exchangers.

Guo et al. [13] recently developed a new physical quantity by using an analogy between heat and electricity. Entransy (formerly referred to as the heat transfer potential capacity, which corresponds to the electric potential energy in electrical fields) is defined as,

$$G=Q_{h}T/2$$

where Q_h is the internal thermal energy stored in an object, and T is the object’s absolute temperature. The unit of entransy, G, is J·K. The entransy of an object describes its heat transfer ability, as the electrical energy in a capacitor describes its charge transfer ability. During an irreversible heat transfer process, the thermal energy is conserved, but the entransy will be partially dissipated. The entransy equilibrium equation is obtained by multiplying the heat conduction equation by T.

$$\rho c_{v}T\frac{\partial T}{\partial t}=-\nabla ·\left(qT\right)+q·\nabla T$$

The left side of Eq. (2) represents the entransy variation with time, the first term on the right side is the entransy transfer associated with the heat transfer, while the second term is the local rate of entransy dissipation due to heat conduction. The last term can be written as the entransy dissipation rate per unit volume (W·K·m⁻³),

$$g_{\varphi }=-q·\nabla T=k{\left(\nabla T\right)}^2$$

Entransy dissipation is a yardstick to measure the loss of heat transfer ability during heat transfer processes. To understand the entransy dissipation in an irreversible heat transfer process, Guo et al. [13] discussed a simple heat transfer process of one-dimensional steady-state heat conduction in a plate with thickness d, where the input heat flux is equal to the output heat flux, q1 = q2. However, the input entransy flux is not equal to the output entransy flux due to dissipation during the heat transport. The entransy dissipation equals to q(T1 − T2), and T1, T2 are the surface temperatures.

In this paper, the heat exchanger thermal resistance is defined based on its entransy dissipation to analyze heat exchanger performance. Twenty heat exchanger flow arrangements including parallelflow, counterflow and 18 other flow arrangements are analyzed. The entropy generation numbers for these heat exchangers are also calculated for comparison.