Design Optimization

There are four cost elements. {`C_1`} is the oxygen plant annual cost {`C_1 = a_1 + a_2F`}, {`C_2`} is the capital cost of the storage vessel {`C_2 = s_1 \, s_2 \, V^{s_3}`}, {`C_3`} is the capital cost of the compressor {`C_3 = 0.2 \, b_1 \, b_2 \, H^{b_3}`}, and {`C_4`} is the compressor power cost {`C_4 = 0.001 \, b_4 \, t_1 \, H`}. In these equations {`a_1`}, {`a_2`}, {`s_1`}, {`s_2`}, {`s_3`}, {`b_1`}, {`b_2`}, {`b_3`}, {`b_4`} are cost parameters.

A Basic Oxygen Furnace (BOF) is a type of furnace used in steelmaking in which a stream of oxygen is blown into a bath of molten iron and scrap steel. This accelerates the oxidation process and removes impurities such as carbon, silicon, manganese, and phosphorus from the steel. The end result is a higher quality steel product.

The operation of a Basic Oxygen Furnace (BOF) is based on cycles. Each cycle consists of a low oxygen demand interval in which raw materials are introduced to the furnace, and a high oxygen demand interval in which the raw material is processed.  The low demand interval is 0.4 hours, and the high demand interval ({`t_1`}) is 0.6 hours, giving a total cycle time ({`t_2`}) of 1 hour.  Due to equipment limitations, the oxygen production rate ({`F`}) is held constant throughout the cycle.  Any excess oxygen produced during the low demand period is compressed and stored, to be withdrawn as needed during the high demand period.  The objective of this problem is to design an oxygen-supply and storage system that meets the oxygen demand of the furnace over the entire cycle while minimizing cost. The following figure shows the BOF system.

m.Obj(annual_cost)

The tank pressure is subject to the constraint:

'''Acknowledgements'''

The two design variables for this system are the maximum storage tank pressure ({`p`} in psia) and the oxygen production rate ({`F`} in tons/hour). The given constants are the low oxygen demand rate ({`D_0`}) at 2.4 tons/hour, the high oxygen demand rate ({`D_1`}) at 37 tons/hour, the gas constant ({`R`}) at 670.6 {`ft^{3}`} lb/{`in^2`}/(HP-hr), temperature ({`T`}) at 530 degrees Rankine, the compressibility factor 0.98 ({`z`}), unit conversion factor ({`k_1`}) at 14005.8, compressor efficiency ({`k_2`}) at 0.75, and the oxygen delivery pressure ({`p_0`}) at 200 psia.

!!!! Cost Functions

The two design variables for this system are the maximum storage tank pressure ({`p`} in psia) and the oxygen production rate ({`F`} in tons/hour). The given constants are the low oxygen demand rate ({`D_0`}) at 2.4 tons/hour, the high oxygen demand rate ({`D_1`}) at 37 tons/hour, the gas constant ({`R`}) at 670.6 {`({{ft}^{3} (lb/in^2)}/{HP - hr})`}, temperature ({`T`}) at 530 degrees Rankine, the compressibility factor 0.98 ({`z`}), unit conversion factor ({`k_1`}) at 14005.8, compressor efficiency ({`k_2`}) at 0.75, and the oxygen delivery pressure ({`p_0`}) at 200 psia.

The two design variables for this system are the maximum storage tank pressure ({`p`} in psia) and the oxygen production rate ({`F`} in tons/hour). The given constants are the low oxygen demand rate ({`D_0`}) at 2.4 tons/hour, the high oxygen demand rate ({`D_1`}) at 37 tons/hour, the gas constant ({`R`}) at 670.6 ({`{ft^3 psia}/{(HP - HR)})`}), temperature ({`T`}) at 530 degrees Rankine, the compressibility factor 0.98 ({`z`}), unit conversion factor ({`k_1`}) at 14005.8, compressor efficiency ({`k_2`}) at 0.75, and the oxygen delivery pressure ({`p_0`}) at 200 psia.

The annual cost ({`AC`}) of the BOF system is found using the following equation. Here {`d`} is an annual cost factor for the compressor and storage tank and {`N`} is the number of cycles in one year of operation.

There are four cost elements. $C_1$ is the oxygen plant annual cost {`C_1 = a_1 + a_2F`}, $C_2$ is the capital cost of the storage vessel {`C_2 = s_1 \, s_2 \, V^{s_3}`}, $C_3$ is the capital cost of the compressor {`C_3 = 0.2 \, b_1 \, b_2 \, H^{b_3}`}, and $C_4$ is the compressor power cost {`C_4 = 0.001 \, b_4 \, t_1 \, H`}. In these equations $a_1$, $a_2$, $s_1$, $s_2$, $s_3$, $b_1$, $b_2$, $b_3$, $b_4$ are cost parameters.

||{`s_2`}||660||$(ft^3)^{-b_3}$||

The maximum amount of oxygen stored ({`I_{max}`}) in the tank during the cycle can be calculated as:

{$H = \frac{I_{max}}{t_1} \frac{RT}{k_1k_2} \ln{ \frac{p}{p_0}}$}

The maximum tank pressure is subject to the constraint:

p = m.Var()            #Maximum tank pressure (psia)

%width=550px%Attach:oxygen_supply.png