3.1. Calculation procedure for the operational optimization of multi-level refrigeration cycles

This section describes the proposed method for the operational optimization of multi-level refrigeration cycles. In this methodology, it is assumed that the process is designed for maximum heat recovery, the refrigeration rejects the absorbed heat to external cooling utility, e.g., cooling water, the number of cooling levels is given, and the compression shaft work dominates the process economics; thus, minimization shaft power consumption in the refrigeration compressors is set as the objective for optimization.

Figure 5. Matching two refrigeration levels against GCC.

The calculation procedure starts with constructing GCC, which is used to identify the total cooling duty Q<sup>e</sup> and the lowest refrigeration temperature level. To determine the temperature level (independent variable) and the cooling duty of each level (dependent variables), the part of GCC below the pinch, which is the point where the lowest driving forces between hot and cold streams are located, is modeled as a set of linear functions,

$$H = f(T) \tag{7}$$

$$T = T\_{\text{evap}} + \frac{\Delta T\_{\text{min}}}{2} \tag{8}$$

$$Q\_{\text{evap}\_i} = H(T\_{\text{evap}\_i}) - H(T\_{\text{evap}\_{i+1}}) \tag{9}$$

$$Q\_{\text{evap}\_l} = Q\_\epsilon - \sum\_{i=1}^{l-1} Q\_{\text{evap}\_i} \tag{10}$$

The second step is to decompose the complex refrigeration cycle into simple cycles. This allows the shortcut model to be used for estimating the shaft work requirement of each level. Finally, a nonlinear model is applied to find the optimal cooling temperature levels and duties of each level. The objective function is to:

$$\begin{aligned} \min \left( W \right) &= \sum\_{i=1}^{l-1} \frac{Q\_{\text{evap}\_i}}{\text{COP}\_i} \\\\ T\_{\text{evap}\_{i+1}} - T\_{\text{evap}\_i} &\ge \Delta T\_{\text{min}}, \ i = 1: I - 1 \\ T\_{\text{cond}} - T\_{\text{evap}\_l} &\ge \Delta T\_{\text{min}} \\ T\_{\text{evap}}^{\text{lb}} &\le T\_{\text{evap}\_i} \le T\_{\text{evap}}^{\text{ub}} \end{aligned} \tag{11}$$

Subject to:

optimizing the design conditions of a complex refrigeration cycle and/or the associated

Table 5. Predicted shaft work requirement for multi-level cycle – two heat sinks and a single heat source using HYSYS

Modelling approach Shaft work (kW) %Error

Multi-level cycle (HYSYS) 1615 + 276 = 1891 Two simple cycles (shortcut model) 704 + 1112 = 1816 4 Two simple cycles (HYSYS) 727 + 1166 = 1893 0.11

Heat source 1 40 3000 Heat sink 1 17 2227 Heat sink 2 37 –

Temperature (C) Duty (kW)

For systems working at sub-ambient temperatures, the power demand can be very high; the lower the source temperature, the more complicated the refrigeration system design, the larger the amount of energy consumed. However, there is a great chance to reduce of the compression energy consumption if heat integration technology is applied, such that most appropriate refrigeration levels and their duties are determined to match them against grand composite curve (GCC), as shown in Figure 5. The GCC provides the overall source and sink temperature profiles of a process and allows the minimum hot and cold utility requirements to be identified. Also, another key feature of the GCC is that it considers the integration among the process, heat exchanger network, and refrigeration system simultaneously [8]. Therefore, in this work, the GCC has been used in the optimization approach presented in Section 3.1 to find

the optimal operating conditions that minimize the compressor energy consumption.

3.1. Calculation procedure for the operational optimization of multi-level refrigeration

This section describes the proposed method for the operational optimization of multi-level refrigeration cycles. In this methodology, it is assumed that the process is designed for maximum heat recovery, the refrigeration rejects the absorbed heat to external cooling utility, e.g., cooling water, the number of cooling levels is given, and the compression shaft work dominates the process economics; thus, minimization shaft power consumption in the refrigeration

processing conditions, as will be seen in Section 3.

Table 4. Case data – one heat source and two sink.

Relative to complex cycle simulation in HYSYS.

.

and new shortcut model<sup>1</sup>

cycles

1

10 Refrigeration

3. Heat-integrated process and refrigeration

compressors is set as the objective for optimization.

Where W is the net power demand of the refrigeration cycle, Qevap<sup>i</sup> is the cooling duty of ith cooling level, COP<sup>i</sup> is the coefficient of performance which is calculated from the developed shortcut model presented in Section 3, T is the shifted temperature, ΔTmin is the minimum approach temperature, Tevap<sup>i</sup> is the evaporation temperature of ith cooling level, Tcond is the condensing temperature at which the refrigerant being condensed, I is the number of cooling levels, and lb and ub represent the lower and upper bounds, respectively.

The main features of this calculation procedure are: (1) using GCC to determine the temperature level and the duty of each stage; (2) the shaft work required of each stage calculated directly without going through the detailed refrigeration calculations or rigorous simulation; and (3) the constrained optimization problem can be solved easily using a simple optimization algorithm, such that available in MATLAB and Excel (i.e., Excel's Solver). The limitations of this approach can be summarized as follows: (1) the advantage of using economizer in minimizing shaft work consumption cannot be explored because the effect of its use cannot be represented in GCC [7], (2) only pure refrigerants are considered, and (3) heat is rejected to external utility rather than process heat sink streams, so the opportunities of the matching refrigeration system with process sink streams—which can provide significant energy savings —are missing. The implementation of the proposed optimization approach for minimizing the overall shaft work requirement of a complex refrigeration cycle is illustrated in Section 4.
