3.1.3. *UA*¯ Value Based Comparison

The plot depicted in Figure 9 displays the comparison of propane, i-butane, propane/i-butane (0.2/0.8), and propane/i-butane (0.8/0.2). Figure 9 is an alternative representation of the results presented in Figures 3 and 6. Essentially, the difference between Figures 3 and 9 is that the net power outputs are plotted as a function of *UA*¯ instead of minimum pinch point temperature differences. The left most values (lowest *UA*¯ and net power output) plotted in Figure 9 thereby correspond to the solutions with minimum pinch point temperature difference values of 10 ◦C, while the right most values (highest *UA*¯ and net power output) correspond to the solutions with minimum pinch point temperature values of 1 ◦C. Note that the solutions with 1 ◦C minimum pinch point temperature difference for the mixtures are outside the boundaries of the plot. The filled markers correspond to the solutions with minimum pinch point temperature differences of 5 ◦C.

In the comparison between propane and propane/i-butane (0.2/0.8), propane achieved the highest performance for *UA*¯ values below 4000 kW/◦C. The plots in Figure 9 suggest that the benefits of using mixtures were highest when the value of *UA*¯ for the condenser was high. Furthermore, by comparing the net power outputs of the fluids based on the same values of the condenser *UA*¯ , the increase in net power output was lower compared to the case when the net power outputs were compared based on the same minimum pinch point temperature differences. For example, the net power output of the mixture propane/i-butane (0.8/0.2) was 7.0% higher compared to propane for condenser minimum pinch point temperature differences of 3 ◦C. Simultaneously, the *UA*¯ value of

the condenser was 91% higher for the mixture compared to propane. When the net power outputs were compared based on condenser *UA*¯ values of 4300 kW/◦C (corresponding to Δ*Tpp*,*cond* = 3 ◦C for the mixture and Δ*Tpp*,*cond* = 1 ◦C for propane) the net power output of the mixture was 1.7% larger compared to that of propane. Comparing the fluids based on the same *UA*¯ values was similar to the comparison based on equal mean temperature differences, since the variation in heat transfer rate was insignificant. These results indicate that the comparison based on the same minimum pinch point temperature differences resulted in an optimistic estimation of the thermodynamic benefit of using zeotropic mixtures, while the comparisons based on the same *UA*¯ values and mean temperature difference resulted in conservative estimations of the benefits of using zeotropic mixtures.

#### 3.1.4. Heat Transfer Area Based Comparison

Figure 10 shows the results of the fluid comparison when the fluids are rated on equal values of heat transfer area for the condenser. Similar to Figure 9, the plots in Figure 10 were generated based on the optimized solutions from Figure 3. The heat transfer areas were calculated based on the outputs (pressures, temperatures and mass flow rates) from the thermodynamic ORC process model by using the shell-and-tube heat exchanger model with conditions listed in Table 3. For each fluid, the lowest condenser areas in Figure 10 correspond to minimum pinch point temperature difference values of 10 ◦C, while the highest condenser areas correspond to minimum pinch point temperature values of 1 ◦C. The filled markers denote the solutions with a minimum pinch point temperature difference of 5 ◦C.

**Figure 9.** Net power output as a function of condenser *UA*¯ .

**Figure 10.** Net power output as a function of condenser heat transfer area.

By comparing Figures 9 and 10, the influence of the variation in *U*¯ can be observed. The heat transfer performance of propane was higher than that of i-butane. Furthermore, the overall heat transfer coefficient of propane was also larger than those of the mixtures. In the comparisons depicted in Figure 10, the larger heat transfer coefficient of propane increased the feasibility of this fluid compared to the *UA*¯ value based method presented in Figure 9.

At condenser areas below 2000 m2, propane was the highest performing fluid. At around 2000 m2, there was a crossing of the curves for propane and propane/i-butane (0.8/0.2), and at higher condenser areas the (0.8/0.2) mixture achieved the highest net power outputs. The curve crossing occurred where the minimum pinch point temperature difference in the condensers was around 2 ◦C for propane and 5 ◦C for the mixture.

## *3.2. Fluid Performance Ranking*

In this section, it is demonstrated how the outcome of a fluid performance ranking can be affected by the employed modeling method. The demonstration is based on the fluid selection study presented by Andreasen et al. [10] for a 120 ◦C liquid water heat source, and the fluids considered for comparison are the 30 optimum fluids identified in that study. In the present study, the net power output of the ORC unit was maximized for each of the 30 fluids by employing the modeling conditions listed in Table 1. Supercritical pressures were allowed and the maximum pressure was limited to 100 bar. The high pressure in the system was limited to be below 80% of the critical pressure for subcritical systems and above 120% of the critical pressure for transcritical systems. For transcritical ORC solutions, the degree of superheating was replaced by the temperature difference between the expander inlet temperature and the critical temperature as decision variable which was varied in the range from 0 ◦C to 100 ◦C. For mixtures, the composition was optimized.

The ranking of the 30 working fluids based on fixed minimum pinch point temperature differences of Δ*Tpp*,*PrHE* = 10 ◦C and Δ*Tpp*,*cond* = 5 ◦C is listed in Table 4. The ranking was essentially the same as that presented by Andreasen et al. [10] except from the definition of the expander efficiency, which Andreasen et al. [10] based on a polytropic efficiency of 0.8, while an isentropic efficiency of 0.8 was employed in the present study. Moreover, the pressure constraints for subcritical and transcritical ORC solutions were not employed by Andreasen et al. [10]. This resulted in small variations in the fluid ranking compared to the previous results.

Table 5 lists the ranking of the 30 fluids when based on a fixed value of *UA*¯ *tot* = 3500 kW/◦C. This value is the sum of the primary heat exchanger and condenser *UA*¯ values for R218 in Table 4. The distribution of the total *UA*¯ value between the primary heat exchanger and condenser was included as a decision variable. The optimization of the *UA*¯ distribution enabled the power output of the ORC unit with R218 to increase from 1444 kW to 1460 kW. Despite this increase, the fluid was not ranked as the best performing fluid when the same value of *UA*¯ *tot* was assumed for all fluids. Instead, the results suggest that R143a was the highest performing fluid due to an increase in net power output from 1361 kW to 1521 kW. This large increase was due to the relatively low value of *UA*¯ *tot* resulting from the minimum pinch point temperature difference based optimization (see Table 4). When comparing the net power output and *UA*¯ value of R218 and R143a in Table 4, R218 reached 6% higher net power output with 22% higher *UA*¯ *tot*. The minimum pinch point temperature difference based method resulted in large differences in the requirement for the heat exchangers. When rated based on the *UA*¯ value method, the differences due to variations in the *UA*¯ values were eliminated, which enabled a fairer comparison of the fluids. The fluids R227ea, R1234yf and propylene also experienced a large increase in the ranking when rated on the *UA*¯ value based method rather than the minimum pinch point temperature difference based method.

Generally, the results indicate that, when the fluid ranking was based on equal values of *UA*¯ *tot*, the pure fluids performed better relative to the zeotropic mixtures than when the performance comparison was based on equal values of the minimum pinch point temperature difference. In the minimum pinch point temperature difference based comparison, 19 out of the 30 fluids were zeotropic

mixtures (including the predefined mixture R422A). In this case, the zeotropic mixtures enabled up to 13.6% higher performance compared to the best pure fluid in each mixture. In the *UA*¯ *tot* based comparison, only 12 fluids were zeotropic mixtures, since the composition of seven mixtures converged to 0% or 100% mole fraction. In this case, the performance increase of the zeotropic mixtures compared to the best pure fluid in each mixture was below 2.6%. High performance was reached for the transcritical mixture ethane/propane in both fluid rankings, but the composition changed from 88.3% (in Table 4) to 3.73% (in Table 5). The mixtures propane/butane, propane/i-butane, propane/pentane, propane/i-pentane and propane/hexane did not reach higher net power output than pure propane. It is also worth noting that, although ethane as a pure fluid reached the lowest net power output of the 30 fluids, it was possible to reach higher net power outputs than propane, butane, i-butane, pentane, and i-pentane by mixing small amounts of ethane in these fluids. The propane/i-butane mixture, which was analyzed thoroughly (see Section 3.1), did not reach higher performance than pure propane.


**Table 4.** Fluid selection and optimization based on fixed values of the minimum pinch point temperature difference Δ*Tpp*,*PrHE* = 10 ◦C and Δ*Tpp*,*cond* = 5 ◦C.

(‡) Δ*T* represents the degree of superheating for subcritical ORC units (sc) and the temperature difference between the turbine inlet and the critical temperature for transcritical ORC units (tc). (‡‡) Δ*Tg* represents the temperature glide of condensation. (‡‡‡)] <sup>Δ</sup>*W*˙ *<sup>r</sup>* represents the relative difference in net power output between the mixture (mix) and the best pure component in the mixture (pure), <sup>Δ</sup>*W*˙ *<sup>r</sup>* = (*W*˙ *net*,*mix* <sup>−</sup> *<sup>W</sup>*˙ *net*,*pure*)/*W*˙ *net*,*pure*.

Moreover, when the fluid ranking was based on equal values of the minimum pinch point temperature difference, 17 out of the 30 fluids were transcritical, but, in the fluid ranking based on equal values of *UA*¯ *tot*, only seven fluids were transcritical. The highest net power output was still achieved with transcritical ORC units, however the relative difference in net power output of the highest performing subcritical compared to the highest performing transcritical unit was reduced from 18.8% (R227ea compared to R218 in Table 4) to 1.1% (R227ea compared to R143a in Table 5). This indicates that the method assuming the same minimum pinch point temperature differences for all fluids resulted in optimistic performance estimations of the thermodynamic benefits of using transcritical ORC units and/or zeotropic mixtures, while the method assuming the same total *UA*¯ value resulted in a conservative estimations.

**Table 5.** Fluid selection and optimization based on fixed values of *UA*¯ *tot* = *UA*¯ *PrHE* + *UA*¯ *cond* = 3500 kW/◦C.


(‡) Δ*T* represents the degree of superheating for subcritical ORC units (sc) and the temperature difference between the turbine inlet and the critical temperature for transcritical ORC units (tc). (‡‡) Δ*Tg* represents the temperature glide of condensation. (‡‡‡) <sup>Δ</sup>*W*˙ *<sup>r</sup>* represents the relative difference in net power output between the mixture (mix) and the best pure component in the mixture (pure), <sup>Δ</sup>*W*˙ *<sup>r</sup>* = (*W*˙ *net*,*mix* <sup>−</sup> *<sup>W</sup>*˙ *net*,*pure*)/*W*˙ *net*,*pure*.

#### *3.3. Assessment of Modeling Methods*

The use of zeotropic mixtures provides the flexibility of adjusting the temperature profile of evaporation and condensation to obtain optimal alignment with the temperature profiles of sensible heat sources and sinks. This has an effect on the temperature difference between the heat exchanging fluid streams throughout the primary heat exchanger and the condenser. By basing the comparison of pure fluids and mixtures on equal values of the minimum pinch point temperature difference, there is no distinguishing between solutions where the minimum temperature difference value is reached at a single location in the heat exchanger (for example in case of pure fluids) and solutions where the temperature difference is close to the minimum value throughout the heat exchanger (in case of temperature profile matching) (see Figure 7). However, as demonstrated in Section 3.1, these two situations can result in very different mean temperatures and *UA*¯ values. Ultimately, when comparing pure fluids and zeotropic mixtures based on equal values of minimum pinch point

temperature difference, the zeotropic mixtures benefit from reduced irreversibilities in the heat transfer processes due to temperature profile matching. However, the additional cost related to the larger heat transfer equipment required for transferring the heat across a lower mean temperature difference is unaccounted for. Thereby, the estimation of the benefit of temperature profile matching is deemed to be optimistic when the comparison is based on equal values of minimum pinch point temperature differences, since the ORC unit employing the zeotropic mixtures are provided with the additional benefit of having heat exchangers with larger thermal capacity (larger *UA*¯ and lower mean temperature difference). Using this thermodynamic modeling approach for comparing the performance of pure fluids and zeotropic mixtures there is a risk of overestimating the benefits of zeotropic mixtures, which might not be representative of the actual performance difference when the cost of heat exchangers are taken into account in subsequent thermoeconomic analyses.

By use of alternative modeling methods, it is possible, with limited additional modeling effort, to take differences in mean temperature differences or *UA*¯ values into account. The added modeling complexity solely consists of discretizing the heat exchangers in a suitable number of control volumes, which enables to calculate the *UA*¯ values and mean temperature differences, and defining limiting values for either of them. In many situations it is even required to discretize the heat transfer process for identifying the location of the minimum pinch point temperature difference, for example when the temperature profile of the mixture is highly curved, or when the high pressure is supercritical or slightly below the critical pressure. It is beneficial to specify limits for the mean temperature difference or the *UA*¯ value of the heat exchangers rather than for the minimum pinch point temperature difference, since the mean temperature difference and the *UA*¯ value take the complete heat transfer process into account and are therefore more indicative of the heat transfer surface area and thereby the cost of the heat exchanger. The use of the methods assuming either the same mean temperature difference or the same *UA*¯ values for pure fluids and zeotropic mixtures, is deemed to result in conservative estimations of the thermodynamic performance benefit of using zeotropic mixtures.

Defining a constraint based on the mean temperature difference is beneficial in the sense that it is conceptually similar to the minimum pinch point temperature difference and it is therefore easier to define suitable limiting values. Defining heat exchanger constraints based on *UA*¯ values provides an additional option of optimizing the distribution of *UA*¯ to each of the heat exchangers by defining a limit for the total ORC unit *UA*¯ (sum of *UA*¯ for all heat exchangers in the unit). The drawback of defining constraints based on *UA*¯ values is that the *UA*¯ value depends on the capacity of the system, i.e., the net power output and heat transferred in the process. One way of addressing this issue is to define a base case fluid for which the heat exchanger minimum pinch point temperature differences are fixed at reasonable values. The *UA*¯ values of the heat exchangers for the base case solution can then be used as limiting values for the remaining simulations. The use of methods fixing the mean temperature differences or *UA*¯ values of heat exchangers does not account for differences in heat transfer coefficients. Such differences can be accounted for by including heat exchanger models in the optimization framework for estimation of heat transfer areas. However, this requires the use of heat transfer correlations which adds to the model complexity and computational time required for the simulations. In relation to fluid screening studies, it is desirable to limit the computational time of the model when many fluids are evaluated. An overview of the method assessment is shown in Table 6.

The thermodynamic modeling methods discussed in this paper do not account for effects of fluid properties on the design, efficiency and cost of the ORC expander. Such aspects are essential when selecting a fluid for an ORC unit. Astolfi et al. [45], Martelli et al. [46], and Meroni et al. [47] provided methods for including axial turbine design aspects in the framework of thermodynamic and thermoeconomic ORC process optimizations. Such methods are relevant to combine with the methods discussed in the present paper for preliminary design and optimization of ORC systems.


#### **Table 6.** Method assessment overview.
