modelConstantEvaporation

Information

Model of an organic Rankine cycle (ORC) as a bottoming cycle.

The thermodynamic cycle is steady-state while the evaporator and the condenser can be configured to have first order dynamics. The fluid stream 1 (using Medium1, port_a1, etc.) is the evaporator hot fluid, e.g., waste heat, and the stream 2 is the condenser cold fluid. The working fluid of the cycle is not based on a typical Modelica medium model. See the Thermodynamic Properties section of this document for the rational.

Cycle Architecture and Governing Equations

The implemented ORC is modeled based on the simplified cycle shown in the figure below. The cycle has two variants depending on the shape of the saturation lines of the working fluid and η_exp. For any given working fluid, the cycle is fully determined by providing the working fluid evaporating temperature T_w,eva, the working fluid condensing temperature T_w,con, the expander efficiency η_exp, and the pump efficiency η_pum. The superheating temperature difference ΔT_sup is minimized, meaning it is zero whenever possible; otherwise it assumes the smallest value not to cause the expander outlet state to fall under the two-phase region, i.e. the "dome". Subcooling after the condenser is not considered. The Thermodynamic Properties section of this document details how these state points are found.

An important assumption is that all heat is dissipated, i.e., the cycle is not controlled by thermal load.

The cycle processes the heat at a fixed T_w,eva provided by the user. The evaporator heat exchange is governed by

Q̇_eva = ṁ_h c_p,h (T_h,out - T_h,in),
Q̇_eva = ṁ_w (h_pum,out - h_exp,in),

where the subscripts are eva for evaporator, exp for expander, h for hot fluid of the evaporator, i.e. the fluid carrying heat, pum for pump, and w for working fluid.

The cycle accommodates the variable flow rate and temperature of the waste heat stream by changing the working fluid mass flow rate ṁ_w to maintain a constant pinch point (PP) temperature difference at the evaporator ΔT_pin,eva. This difference is found from

(T_pin,eva - T_h,out) (h_exp,in - h_pum,out) = (T_h,in - T_h,out) (h_eva,pin - h_pum,out),
ΔT_pin,eva = T_pin,eva - T_w,eva.

The condenser side uses the same equations with the evaporator variables replaced by their condenser counterparts where appropriate. Hence,

Q̇_con = ṁ_c c_p,c (T_c,out - T_c,in),
Q̇_con = ṁ_w (h_pum,in - h_exp,out),
(T_c,pin - T_c,in) (h_exp,out - h_pum,in) = (T_c,out - T_c,in) (h_con,pin - h_pum,in),
ΔT_con,pin = T_w,con - T_c,pin,

where the subscripts are con for condenser, and c for cold fluid in the condenser.

The electric power output of the expander is

P_exp = ṁ_w (h_exp,in - h_exp,out).

The electric power consumption of the pump is

P_pum = ṁ_w (h_pum,out - h_pum,in).

The pump work is

P_pum = ṁ_w  (p_eva - p_con) / (ρ_pum,in η_pum).

This takes advantage of the negligible density change of the liquid to avoid a property search in the subcooled liquid region.

In summary, the model has the following information flow:

Constraints

The ORC system controls ṁ_w to maintain the prescribed evaporator PP temperature difference set point. Although the model does not implement this as a control loop, an upper limit and a lower limit are imposed on ṁ_w to reflect the capacity constraints of a sized cycle.

• The working fluid mass flow rate ṁ_w will not go higher than a prescribed upper limit. Rather, ṁ_w stays at the user-specified upper limit and ΔT_pin,eva increases beyond its set point. This may happen when the incoming hot fluid has a high flow rate or a high incoming temperature, i.e., it carries more energy than the cycle is sized to process.
• When the working fluid mass flow rate ṁ_w needs to go below a prescribed lower limit, ṁ_w is set to zero and the cycle is switched off. This may happen when the incoming waste heat fluid has a low flow rate or a low incoming temperature, i.e., it carries too little energy.

How these constraints affect the cycle's behavior reacting to a variable waste heat fluid stream is demonstrated in AixLib.Fluid.CHPs.OrganicRankine.Validation.VariableSource.

Thermodynamic Properties

The thermodynamic properties of the working fluid are not computed by a typical Modelica medium model, but by interpolating data records in AixLib.Fluid.CHPs.OrganicRankine.Data. Specific enthalpy and specific entropy values are provided as support points on the saturated liquid line, the saturated vapor line, and a superheated vapor line (called the reference line). The values of these support points were obtained using CoolProp (https://www.coolprop.org; Bell et al., 2014) through its Python wrapper and stored as Modelica records. An example Python file is provided in AixLib/Resources/Python-Sources/MakeORCFluidRecord.py, but note that this file is not maintained. The records included in this library have ten data points for each line. It is recommended to have at least four points to take full advantage of the cubit Hermite spline interpolation that is set up in this model.

Thermodynamic state points in the cycle are determined by various schemes of interpolation and extrapolation.

• On the saturation line, the specific enthalpy, specific entropy or density, here labeled as y_A, are obtained using cubic Hermite spline interpolation as

  y_A = s(u_A,d)

  where s(·,·) is a cubic Hermite spline, u_A is the input property, and d are the support points. For the saturation curves, the user can configure the model to use either the saturation pressure or the saturation temperature for u_A. For the reference line u_A is the pressure.
• If the fluid is wet, the isentropic expander outlet point is in between the saturation lines. In this case, its enthalpy h_B is obtained from

  (h_B - h₁) / (s_B - s₁) = (h₂ - h₁) / (s₂ - s₁)

  where s_B is known because it equals the expander inlet entropy, and all other points are on the saturation line and therefore can be found as point A.
• C is a point in the superheated vapor region. This is the case for the expander inlet or outlet depending on the shape of the cycle. The isobaric lines are not straight in this section, but they are assumed linear so that the method for B can be applied using the saturated vapor line and the reference line, albeit with less accuracy.

The cycle can be completely defined by providing the following quantities: evaporating temperature T_eva or pressure p_eva, condensing temperature T_con or pressure p_con, expander efficiency η_exp, and pump efficiency η_pum. Most of the important state points can be found via the interpolation schemes described above. The only exceptions are the expander inlet, expander outlet, and the pump outlet.

• A dry cycle is a cycle where the expansion starts from the saturated vapor line and ends in the superheated vapor region. For either a dry fluid (a) or a wet fluid (b) undergoing such a cycle,

  h_exp,out - h_exp,in = (h_exp,out,ise - h_exp,in) η_exp

  where h_exp,out is solved and h_exp,in is known.
• A wet cycle is a cycle where the expansion starts from the superheated vapor region and ends on the saturated vapor line. In this scenario,

  h_exp,out - h_exp,in = (h_exp,out - h_exp,inl,ise) η_exp

  where h_exp,out is known and h_exp,in is solved. For this fluid and this η_exp, if the expansion started from the saturated vapor line, the outlet point would end up under the dome.

Implementation

References

Bell IH, Wronski J, Quoilin S, Lemort V. Pure and pseudo-pure fluid thermophysical property evaluation and the open-source thermophysical property library CoolProp. Industrial & engineering chemistry research. 2014 Feb 12;53(6):2498-508. https://doi.org/10.1021/ie4033999

Parameters

Components

Revisions

• January 29, 2024, by Hongxiang Fu:
  First implementation. This is for #3433.