Optimal operation of simple refrigeration cycles

Abstract

The paper focuses on the operation of simple refrigeration cycles. With equipment given, there are, from a control and operational point of view, five steady-state degrees of freedom; the compressor power, the heat transfer in the condenser, the heat transfer in the evaporator, the choke valve opening and the active charge in the cycle. Different designs for affecting the active charge, including the location of the liquid receiver, are discussed. With a given load (e.g. given cooling duty) the compressor power is set. Furthermore, it is usually optimal to maximize the heat transfer. The two remaining degrees of freedom (choke valve and active charge) may be used to set the degree of super-heating and sub-cooling. It is found that super-heating should be minimized whereas some sub-cooling is optimal. For a simple ammonia cycle, sub-cooling gives savings in compressor power of about 2%. In this paper, refrigeration (cooling) cycles are considered, but the same principles apply to heat pumps.

Introduction

Cyclic processes for heating and cooling are widely used and their power ranges from less than 1 kW to above 100 MW. In both cases vapour compression cycle is used to “pump” energy from a low to a high temperature level.

The first application, in 1834, was to produce ice for storage of food, which led to the refrigerator found in most homes (Nagengast, 1976). Another well-known system is the air-conditioner (A/C). In colder regions a cycle operating in the opposite direction, the “heat pump”, has recently become popular. These two applications have also merged together to give a system able to operate in both heating and cooling mode.

In Fig. 1 a schematic drawing of a simple cycle is shown together with a typical pressure–enthalpy diagram for a sub-critical cycle. The cycle works as follows.

The low pressure vapour (4) is compressed by supplying work W_s to give a high pressure vapour with high temperature (1). The vapour is cooled to its saturation temperature in the first part of the condenser, condensed in the middle part and possibly sub-cooled in the last part to give the liquid (2). In the choke valve, the pressure is lowered to its original value, resulting in a two-phase mixture (3). This mixture is vaporized and possibly super-heated in the evaporator (4) closing the cycle.

The choke valve may be replaced by an expander for improved efficiency, but this is not considered here. The coefficient of performance for a refrigeration cycle (refrigerator, A/C) is defined as

$$\text{COP}=\frac{Q_{\text{c}}}{W_{\text{s}}}=\frac{\dot{m}(h_4-h_3)}{\dot{m}(h_1-h_4)}$$

The COP is typically around 3 which indicates that 33% of the heat duty is added as work (e.g. electric power).

In this paper, the objective is to optimize the operation of a given cycle (Fig. 1) in terms of maximize the COP, or specifically to minimize the compressor power W_s for a given cooling load Q_C. We consider only steady-state operation. The model equations are summarized in Table 1. Note that pressure losses in piping and equipment are neglected. We also assume that the temperature of the hot (T_H) and cold (T_C) source are constant throughout the heat exchanger. This assumption holds for a cross flow heat exchanger. In practice, there may be some operational constraints, for example, maximum and minimum pressure constraints, which are not considered here.

In industrial processes, especially in cryogenic processes such as air separation and liquefaction of natural gas (LNG process), more complex refrigeration cycles are used in order to improve the thermodynamic efficiencies. These modifications lower the temperature differences in the heat exchangers and include cycles with mixed refrigerants, several pressure levels and cascaded cycles. Our long term objective is to study the operation of such processes. However, as a start we need to understand the simple cycle in Fig. 1.

An important result from this study is the degree of freedom analysis given in Section 2. We find that the “active” charge plays an important role in operation of cyclic processes. This is also directly applicable to more complex designs. Unlike an open process, a closed cyclic process does not have boundary conditions on pressures imposed by the flows in and out of the system. Instead the pressure level is indirectly given by the external temperatures, heat exchanger sizes, load and the active charge. The active charge is defined as the total mass accumulated in the process equipment in the cycle, mainly in the condenser and evaporator, but excluding any adjustable mass in liquid receivers (tanks).

The effect of a change in active charge on operation depends on the specific design. Intuitively, it seems that an increase in active charge must increase the pressure, and indeed this is true in most cases. For example, this is the case for the models used in this paper with plug-flow in the heat exchangers. Then more liquid in the condenser gives more sub-cooling which, effectively reduces cooling and pressure increases. Similarly more liquid in the evaporator gives less super-heating effectively increasing heat transfer and pressure increases. However, there may be designs where the effect of charge on pressure is opposite. For example, consider a well-mixed flooded condenser where the heat transfer coefficient U to liquid is larger than to vapour. An increase in charge (liquid) may then improve cooling and pressure decreases. In any case, the main point is that the “active” charge is a degree of freedom that affects the operation of the system, and this paper focuses on how to use it effectively.

Although there is a vast literature on the thermodynamic analysis of refrigeration cycles, there are very few authors who discuss their operation and control. Some discussions are found in text books such as Stoecker (1998), Langley (2002) and Dossat (2002), but these mainly deal with more practical aspects. Svensson (1994) and Larsen, Thybo, Stoustrup, and Rasmussen (2003) discuss operational aspects. A more comprehensive recent study is that of Kim, Pettersen, and Bullard (2004) who consider the operation of trans-critical CO₂ cycles. They discuss the effect of “active charge” and consider alternatives for placing the receiver.

The paper also discuss super-heating and sub-cooling. In the literature, it is generally taken for granted that there for a given cycle should be no sub-cooling and super-heating (ΔT_sub = 0 °C and ΔT_sup = 0 °C) in optimal operation. For example, Stoecker (1998, p. 57) states that

The refrigerant leaving industrial refrigeration condensers may be slightly sub-cooled, but sub-cooling is not normally desired since it indicates that some of the heat transfer surface that should be used for condensation is used for sub-cooling. At the outlet of the evaporator it is crucial for protection of the compressor that there be no liquid, so to be safe it is preferable for the vapor to be slightly super-heated.

In this study, we confirm that super-heating is not optimal. The issue of sub-cooling is less clear. Of course, sub-cooling in itself is always optimal, as less refrigerant needs to be circulated. The issue is whether sub-cooling is optimal for a given cold source temperature and a given condenser area, because sub-cooling will reduce the temperature driving forces which must be compensated by increasing the pressure. We find, contrary to popular belief, that with given equipment, sub-cooling in the condenser may give savings in energy usage (compressor power) in the order of 2%. An ammonia case study is presented to obtain numerical results.