# Unit Model Costing¶

The IDAES Process Modeling Framework includes support for incorporating costing of unit operations into a flowsheet to allow for calculation and optimization of process costs. Cost Correlations are implemented using unit costing sub-modules to allow users to easily develop and incorporate their own costing models.

## Introduction¶

Note

This is a work in progress, and costing is currently only implemented for pressure changers and heat exchanger.

All unit models within the core IDAES model library include a get_costing method which can be called to include cost correlations for an instance of that unit. The get_costing method for each unit takes a number of arguments used to specify the basis for costing each piece of equipment. Details are given for each unit model later in this documentation, however, all get_costing methods take the following two arguments:

• module - this argument specifies the costing module to use when constructing the constraints and associated variables. if not provided, this defaults to the standard IDAES costing module.
• year - this argument sets the year to which all costs should be normalized (CE index 2010 to 2019)

When get_costing is called on an instance of a unit model, a new sub-block is created on that unit named costing (i.e. flowsheet.unit.costing). All variables and constraints related to costing will be constructed within this new block (see detailed documentation for each unit for details on these variables and constraints).

In addition, the first time get_costing is called for a unit operation within a flowsheet, an additional costing block is created on the flowsheet object (i.e. flowsheet.unit.costing) in order to hold any global parameters relating to costing. The most common of these paramters is the cost normalization parameter based on the year selected by the user.

Note

The global paramters are created when the first instance of get_costing is called and use the values provided there for initialization. Subsequent get_costing calls use the existing paramters, and do not change the initialized values. i.e. any “year” argument provided to a get_costing call after the first will be ignored.

Table 1. Main Variables added to the unit block (“self.costing”).

Variable Symbol Units Notes
Purchase cost $$purchase\_cost$$ dollars Purchase cost
Base cost $$base\_cost$$ unitless Base cost

## Example¶

Below is a simple example of how to add cost correlations to a flowsheet including a heat exchanger using the default IDAES costing module.

from pyomo.environ import (ConcreteModel, SolverFactory)
from pyomo.util.calc_var_value import calculate_variable_from_constraint
from idaes.core import FlowsheetBlock
from idaes.generic_models.unit_models.heat_exchanger import \
(HeatExchanger, HeatExchangerFlowPattern)
from idaes.generic_models.properties import iapws95
from idaes.core.util.model_statistics import degrees_of_freedom

m = ConcreteModel()
m.fs = FlowsheetBlock(default={"dynamic": False})

m.fs.properties = iapws95.Iapws95ParameterBlock()

m.fs.unit = HeatExchanger(default={
"shell": {"property_package": m.fs.properties},
"tube": {"property_package": m.fs.properties},
"flow_pattern": HeatExchangerFlowPattern.countercurrent})
# set inputs
m.fs.unit.shell_inlet.flow_mol[0].fix(100)     # mol/s
m.fs.unit.shell_inlet.enth_mol[0].fix(3500)    # j/s
m.fs.unit.shell_inlet.pressure[0].fix(101325)  # Pa

m.fs.unit.tube_inlet.flow_mol[0].fix(100)
m.fs.unit.tube_inlet.enth_mol[0].fix(4000)
m.fs.unit.tube_inlet.pressure[0].fix(101325.0)

m.fs.unit.area.fix(1000)  # m2
m.fs.unit.overall_heat_transfer_coefficient.fix(100)  # W/m2K

m.fs.unit.initialize()
m.fs.unit.get_costing(module=costing, length_factor='12ft')
# initialize costing equations
calculate_variable_from_constraint(
m.fs.unit.costing.base_cost,
m.fs.unit.costing.base_cost_eq)

calculate_variable_from_constraint(
m.fs.unit.costing.purchase_cost,
m.fs.unit.costing.cp_cost_eq)

opt = SolverFactory('ipopt')
opt.options = {'tol': 1e-6, 'max_iter': 50}
results = opt.solve(m, tee=True)


## Units¶

It is important to highlight that the costing method interrogates the property package to determine the units of this model, if the user provided the correct units in the metadata dictionary (see property models for additional information), the model units will be converted to the right units. For example: in this example area is in m^2, while the cost correlations for heat exchangers require units to be in ft^2. Therefore, the costing method will convert the units to ft^2. The use of Pyomo-unit conversion tools is under development.

## IDAES Costing Module¶

A default costing module has been developed primarily based on purchase cost correlations from the following reference with some exceptions (noted in the documentation as appropiate).

Process and Product Design Principles: Synthesis, Analysis, and Evaluation. Seider, Seader, Lewin, Windagdo, 3rd Ed. John Wiley and Sons. Chapter 22. Cost Accounting and Capital Cost Estimation

Users should refer to the reference above for details of the costing correlations, however, a summary of this methods is provided below.

Table 2. Cost basis for each unit model.

Unit Model Basis Units
heat exchanger $$area$$ ft^2
pump $$fluid_{work}$$ ft^3/s
compressor $$mechanical_{work}$$ hp
turbine $$mechanical_{work}$$ hp

### Heat Exchanger Cost¶

The purchse cost is computed based on the base unit cost and three correction factors (Eq. 22.43 in Seider et al.). The base cost is computed depending on the heat exchanger type selected by the user:

$self.costing.purchase\_cost = pressure\_factor*material\_factor*L\_factor*self.costing.base\_cost*(CE_{index}/500)$
$self.costing.base\_cost = \exp{(\alpha_{1} - \alpha_{2}*\log{area*hx\_os} + \alpha_{3}*(\log{area*hx\_os})^{2})}$

where:

• pressure_factor - is the pressure design correction factor
• material_factor - is the construction material correction factor
• length_factor - is the tube length correction factor
• CE_index - is a global parameter for Chemical Enginering cost index for years 2010-2019
• hx_os - heat exchanger oversize factor (default = 1)

The heat exchanger costing method has three arguments, hx_type = heat exchanger type, FM_Mat = construction material factor, and FL = tube lenght factor.

• material factor (Mat_factor): ‘stain_steel’*, ‘carb_steel’
• tube length (length_factor): ‘8ft’, ‘12ft’*, ‘16ft’, ‘20ft’

where ‘*’ corresponds to the default options, FL and FM_MAT are pyomo-mutable parameters fixed based on user selection.

Table 3. Base cost factors for heat exchanger type.

Tube Length (ft) $$\alpha_{1}$$ $$\alpha_{2}$$ $$\alpha_{3}$$
U-tube 11.3852 0.9186 0.09790
Kettle_vap 12.2052 0.8709 0.09005

Table 4. Tube-Length correction factor.

Tube Length (ft) FL
8 1.25
12 1.12
16 1.05
20 1.00

Construction material correction factor (FM_Mat) can be computed with Eq. 22.44 (Seider et al.)

$material\_factor = a + (\frac{area}{100})^{b}$

Table 5. Materials of construction factors.

Materials of Construction
Shell / Tube a b
carbon steel/carbon steel 0.00 0.00
carbon steel/brass 1.08 0.05
carbon steel/stainless steel 1.75 0.13
carbon steel/monel 2.1 0.13
carbon steel/titanium 5.2 0.16
carbon steel/Cr-Mo steel 1.55 0.05
Cr-Mo steel/Cr-Mo steel 1.7 0.07
stainless steel/stainless steel 2.7 0.07
monel/monel 3.3 0.08
titanium/titanium 9.6 0.06

Note that Mat_factor argument should be provided a string, for example: Mat_factor:’carbon steel/carbon steel’.

### Pressure Changer Cost¶

The costing of a pressure changer unit model is more complicated, because the pressure changer model can be imported into the flowsheet object representing a pump, turbine, compressor, or a simply pressure changer (fan, blower, etc.). The get_costing method currently supports costing of pumps, turbines, and compressors. The method authomatically interrogates the flowsheet object to determine if the unit is being used as a pump, turbine, or compressor.

The get_costing method authomatically determines if the unit model is being used as a pump, turbine, or compressor based on the compressor and thermodynamic_assumption configuration arguments provided by the user where creating the unit model. A summary of the decision logic is shown below.

Unit Type compressor thermodynamic_assumption
Turbine False Any
Pump True pump
Mover True not pump

Additionally, some unit types have different sub-types which can be costed appropiately. In these cases, an additional argument is provided to get_costing to identify the sub-type to use which is detailed below.

#### Turbine Cost Model¶

The turbine cost is based on the mechanical work of unit (work_mechanical), this correlation has been obtained using the NETL Report (DOE/NETL 2015).

$self.costing.purchase\_cost = 580*(mechanical_{work})^{0.81}$

DOE/NETL, 2015, report. Cost and performance Baseline for Fossil Energy Plants. Volume 1a: Bituminous Coal (PC) and Natural Gas to Electricity. Revision 3

#### Pump Cost Model¶

Three subtypes are supported for costing of pumps, which can be set using the “pump_type” argument.

1. Centrifugal pumps (pump_type=’centrifugal’)
2. External gear pumps (pump_type=’external’)
3. Reciprocating Plunger pumps (pump_type=’reciprocating’)
##### Centrifugal Pump¶

The centrifugal pump cost has two main components, the cost of the pump and the cost of the motor. The pump cost is based on the fluid work (work_fluid), pump head, and size factor. Additional arguments are required:

• pump_type_factor = ‘1.4’ (see Table 6)
• pump_motor_type_factor = ‘open’, ‘enclosed’, ‘explosion_proof’

Based on user’s inputs the get_costing method builds base_cost and purchase_cost for both the pump and the motor. The unit purchase cost is obtained by adding the motor and pump costs.

$self.costing.purchase\_cost = self.costing.pump\_purchase\_cost + self.costing.motor\_purchase\_cost$

To compute the purchase cost of the centrifugal pump, first we obtain the pump size factor (S) with Eq. 22.13, then we obtain the base cost with Eq. 22.14. Finally, the purchase cost of the pump is obtained in Eq. 22.15. (Seider et al.)

$S = QH^{0.5}$
$self.costing.pump\_base\_cost = \exp{(9.7171 - 0.6019*\log{S} + 0.0519*(\log{S})^{2})}$
$self.costing.pump\_purchase\_cost = F_{T}*material_\factor*self.costing.pump\_base\_cost*(CE_{index}/500)$

where:

• S is the pump size factor (self.costing.size_factor)
• Q is the volumetric flowrate in gpm (depending on the model this variable can be found as self.unit.properties_in.flow_vol)
• H is the head of the pump in ft (self.pump_head; which is defined as $$H = \Delta P/\rho_{liq}$$)
• FT is a parameter fixed based on the pump_type_factor argument (users must wisely select this factor based on the pump size factor, pump head range, and maximum motor hp)
• material_factor is the material factor for the pump

Table 6. Pump Type factor (Table 22.20 in Seider et al.).

Case FT factor # stages Shaft rpm Case-split Pump Head range (ft) Maximum Motor Hp
‘1.1’ 1.00 1 3600 VSC 50 - 900 75
‘1.2’ 1.50 1 1800 VSC 50 - 3500 200
‘1.3’ 1.70 1 3600 HSC 100 - 1500 150
‘1.4’ 2.00 1 1800 HSC 250 - 5000 250
‘2.1’ 2.70 2 3600 HSC 50 - 1100 250
‘2.2’ 8.90 2+ 3600 HSC 100 - 1500 1450

For more details on how to select the FT factor, please see Seider et al.

Table 7. Materials of construction factors for centrifugal pumps and external gear pumps.

Material Factor FM_MAT
cast iron 1.00
ductile iron 1.15
cast steel 1.35
bronze 1.90
stainless steel 2.00
hastelloy C 2.95
monel 3.30
nickel 3.50
titanium 9.70

Electric Motor:

A centrifugal pump is usually driven by an electric motor, the self.costing.motor_purchase_cost is calculated based on the power consumption.

$self.motor_purchase_cost = FT * self.costing.motor\_base\_cost * (CE_{index}/500) (Eq. 22.20)$
$self.costing.motor\_base\_cost = \exp{(5.8259 + 0.13141\log{PC} + 0.053255(\log{PC})^{2} + 0.028628(\log{PC})^{3} - 0.0035549(\log{PC})^{4})} (Eq. 22.19)$
$PC = \frac{P_{T}}{\eta_{P}\eta_{M}} = \frac{P_{B}}{\eta_{M}} = \frac{Q H \rho}{33000\eta_{P}\eta_{M}} (Eq. 22.16)$
$\eta_{P} = -0.316 + 0.24015*\log{Q} - 0.01199(\log{Q})^{2} (Eq. 22.17)$
$\eta_{M} = 0.80 + 0.0319\log{PB} - 0.00182(\log{PB})^{2} (Eq. 22.18)$

Efficiencies are valid for PB in the range of 1 to 1500Hp and Q in the range of 50 to 5000 gpm

where:

• motor_FT is the motor type correction factor
• PC is the power consumption in hp (self.power_consumption_hp; coded as a pyomo expression)
• Q is the volumetric flowrate in gpm (self.Q_gpm)
• PB is the pump brake hp (self.work)
• nP is the fractional efficiency of the pump
• nM is the fractional efficiency of the motor
• $$\rho$$ is the liquid density in lb/gal

Table 8. FT Factors in Eq.(22.20) and Ranges for electric motors.

Type Motor Enclosure 3600rpm 1800rpm
Open, drip-proof enclosure, 1 to 700Hp 1.0 0.90
Totally enclosed, fan-cooled, 1 to 250Hp 1.4 1.3
Explosion-proof enclosure, 1 to 25Hp 1.8 1.7
##### External Gear Pumps¶

External gear pumps are not as common as the contrifugal pump, and various methods can be used to correlate base cost. Eq. 22.21 in Seider et al. Here the purchase cost is computed as a function of the volumetric flowrate (Q) in gpm Eq. 22.22 in Seider et al.

$self.costing.pump\_base\_cost = \exp{(7.6964 + 0.1986\log{Q} + 0.0291(\log{Q})^{2})}$
$self.costing.pump\_purchase\_cost = material\_factor * self.costing.pump\_base\_cost * (CE_{index}/500)$
##### Reciprocating Plunger Pumps¶

The cost correlation method used here is based on the brake horsepower (PB).

$self.costing.pump\_base\_cost = \exp{(7.8103 + 0.26986\log{PB} + 0.06718(\log{PB})^{2})} (Eq. 22.23)$
$self.costing.pump\_purchase\_cost = material\_factor * self.costing.pump\_base\_cost * (CE_{index}/500) (Eq. 22.22)$

Table 9. Materials of construction factors for reciprocating plunger pumps.

Material Mat_factor
ductile iron 1.00
Ni-Al-Bronze 1.15
carbon steel 1.50
stainless steel 2.20

#### Mover¶

If the unit represents a “Mover”, the user can select to cost it as a compressor, fan, or blower. Therefore, the user must set the “mover_type” argument.

• mover_type= ‘compressor’ or ‘fan’ or ‘blower’ (uper/lower case sensitive)
##### Compressor Cost¶

The compressor cost is based on the mechanical work of the unit. Additional arguments are required to estimate the cost such as compressor type, driver mover type, and material factor (FM_MAT).

• compressor_type = ‘centrifugal’, ‘reciprocating’, ‘screw’
• driver_mover_type = ‘electrical_motor’, ‘steam_turbine’, ‘gas_turbine’
• Mat_factor = ‘carbon_steel’, ‘stain_steel’, ‘nickel_alloy’
$self.costing.purchase\_cost = F_{D} material\_factor self.costing.base\_cost$
$self.costing.base\_cost = \exp{(\alpha_{1} + \alpha_{2}*\log{mechanical_{work}})}$

where:

• FD is the driver mover type factor and FM is the construction material factor.

Table 10. Compressor type factors.

Compressor type $$\alpha_{1}$$ $$\alpha_{2}$$
Centrifugal 7.5800 0.80
Reciprocating 7.9661 0.80
Screw Compressor 8.1238 0.7243

Table 11. Driver mover type (for compressors only).

Mover type FD (mover_type)
Electric Mover 1.00
Steam Turbine 1.15
Gas Turbine 1.25

Table 12. Material of construction factor (for compressors only).

Material Mat_factor
Cast iron 1.00
Stainless steel 1.15
Nickel alloy 1.25