INGB 472 semester assignment [100]

General

This is not a GA assignment and no GA is tested in this module. Therefore, there will not be a second submission opportunity and the mark received is final. The mark received will be a combination of mathematical modelling, programming and report writing. This mark will then contribute 50% towards your final module mark.

Submission November 1st at 08h00 (8 AM). A submission window will be allowed for those who struggle to submit. This window will close on November 1st at 23h00 (11 PM).

25% will be deducted for late submissions within the first 24 hours, 50% within the following 24 hours and no submission will be marked thereafter. No exceptions will be made, and it is your responsibility to ensure the work is uploaded before the deadline(s).

This is an individual assignment and no “teamwork” is allowed. If significant similarities between work exists, it will be forwarded to the NWU’s disciplinary office for a formal investigation.

A marking rubric will not be provided beforehand and only after the assignments have been graded.

Background

Underground mining requires air cooling to provide a save working environment for the workforce. As a result, central air cooling is typically available. However, due to the expansion of mining tunnels, over time, the outskirts of the mine are typically above safe working temperatures.

Additional or larger air chiller plants can be installed, but at great costs. As a result, the mining industry requires other means of providing cooled air in the warmer parts where the central cooling does not have the required effect. If cooled air cannot be provided, production time (and revenue) will be lost.

One solution is mobile air-cooling units. These units comprise refrigeration cycle, exactly as a typical home fridge. To cool the surrounding air, cooling water is required for these units. The cooling water is used to transport the heat extracted from the air away from the immediate environment. If the heat cannot be extracted from the environment, then no air cooling is possible.

Water flows are, however, not constant. This can result in sub-optimal decision making when choosing which air-cooling units to invest in.

What further complicates the decision is that these units do not always operate at the same efficiency, to be discussed in the next section.

Thermodynamic working principles

Before any modelling can commence, a basic working principle of these cooling systems must be grasped, which is the coefficient of performance or COP. The COP provides a ration of air cooling in kilo Watts [kW] over electricity costs (compressor work) in kilo Watts [kW] that are required to operate the system. Therefore:

?????????

COP=

???????????????????????

????????? is then the air-cooling rate at any time and ??????????????????????? the electrical input (compressor work) required to deliver the air-cooling.

Air-cooling units under investigation

There are four air cooling units to choose from and the required information for both units are provided in the following tables. The first table provides the minimum and maximum compressor work required by each system, together with the water flow that will result in the compressor work. It can be assumed that there exists a linear correlation between the compressor work [kW] and water flows [kg/s].

Units Minimum compressor work [kW] Water flow required for minimum air-cooling rate [kg/s] Maximum compressor work [kW] Water flow required for maximum air-cooling

rate [kg/s]

Unit I 32 2.0 50 6.0

Unit II 40 2.2 55 7.0

Unit III 26 2.0 64 7.0

Unit IV 33 2.5 69 8.0

The following tables provide the COP intervals in terms of water flow conditions for each unit.

Unit 1: water flows kg[s] 2.0 – 3.0 3.0 – 4.0 4.0 – 6.0

COP 2.5 3.0 4.0

Unit 2: water flows kg[s] 2.2 – 3.5 3.5 – 5.5 5.5 – 7.0

COP 2.5 3.0 4.0

Unit 3: water flows kg[s] 2.0 – 4.0 4.0 – 6.0 6.0 – 7.0

COP 2.3 2.5 3.3

Unit 4: water flows kg[s] 2.5 – 4.0 4.0 – 6.0 6.0 – 8.0

COP 2.4 3.0 3.5

Very important to note. No unit can be operational if water flows are below its minimum level. Take Unit 4; if the water flow rate is 1.95 [kg/s], then the unit will shut down and produce zero cooling for that interval. Further importance, the unit cannot utilise more than its maximum flow rate rating. If 10.0 [kg/s] water is available for Unit 4, it can only draw 69 [kW] electrical power for the compressor and provide 69x3.5 = 241.5 [kW] of air cooling.

What is provided

You will receive an Excel file with two six-hourly available water flow data over a time period. The flows within the six-hour intervals may be assumed to be constant. These profiles are a typical representation of waterflows that can be expected during any time of the year when production occurs at two different locations. The data sets are labelled Data 1 and Data 2.

What is required

You are required to solve for the optimal decision for four different scenarios. You will therefore have four optimisation models; all four must be kept separate. You will also simulate three different scenarios with three different simulation models. The four optimisation models are as follows:

1. Model I: you must solve for the single best (optimal) unit to be used under water profile Data 1. The objective must be to maximise accumulated air cooling over time (equivalent to the highest average). Times when sufficient water is available the unit may provide air cooling and for the times when sufficient water is not available, the unit must be switched off and provide zero air cooling. Each time step’s ai- cooling rates must directly be written to an Excel column. 2. Model II: you must solve for the single best (optimal) unit to be used under water profile Data 1. The objective must be to maximise accumulated air cooling over time (equivalent to the highest average). Times when sufficient water is available the unit may provide air cooling and for the times when sufficient water is not available, the unit must be switched off and provide zero air cooling. When the unit is switched off, it must remain switched off for at least three time-steps (i.e. 18 hours: time period unit switches off and at least the two following time periods.). Each time step’s air-cooling rates must directly be written to an Excel column.

3. Model III: you must solve for the single best (optimal) unit to be used under water profile Data 1. The objective must be to maximise accumulated air cooling over time (equivalent to the highest average). Times when sufficient water is available the unit may provide air cooling and for the times when sufficient water is not available, the unit must be switched off and provide zero air cooling. When the unit is switched off, it must remain switched off for at least three time-steps (i.e. 18 hours: time period unit switches off and at least the two following time periods.). Furthermore, the unit may only be restarted if the water flow from the last time period before it is restarted, is above the minimum operational value. This is to ensure that a unit is not restarted directly after a time period where there was not enough water flow to potentially have supported air cooling. Each time step’s air-cooling rates must directly be written to an Excel column.

4. Model IV: you must solve for the two best (optimal) units to be used in combination under water profile Data 2. The objective must be to maximise accumulated air cooling between the two units over time (equivalent to the highest average). The water flow rate must be split between the two units and your model must compute how this is done for each time-step. Times when sufficient water is available the unit may provide air cooling and for the times when sufficient water is not available, the unit must be switched off and provide zero air cooling. When the unit is switched off, it must remain switched off for at least three timesteps (i.e. 18 hours: time period unit switches off and at least the two following time periods.). *This is then an expansion of your Model II. Each time step’s air-cooling rates must directly be written to an Excel column (two different columns).

All optimisation modelling must be done in Excel’s SolverStudio GMPL language. You were introduced to SolverStudio during your third year and should have sufficient background knowledge of the software. If not, it is up to yourself to acquire the necessary skills. There will not be a lecture on how to use SolverStudio. Furthermore, each optimisation model must be run at a different Excel tab.

For the simulation part, you must use Excel, either logic programmed within Cells or Visual Basic. Three separate simulation models are required.

1. A single unit’s operational parameters, as provided in the tables, must be specified by the user. Your model must then simulate how that unit will operate under the water profile of Data 1. This means your model must calculate the air-cooling rate at each time step of the unit. All the rules of Model 1 apply, i.e. the unit may provide air cooling when sufficient water is available and zero otherwise. Your simulation model must then solve for maximum average air cooling over time. Each time step’s air-cooling rates must directly be written to a column.

2. A single unit’s operational parameters, as provided in the tables, must be specified by the user. Your model must then simulate how that unit will operate under the water profile of Data 1. This means your model must calculate the air-cooling rate at each time step of the unit. All the rules of Model 2 apply. Each time step’s air-cooling rates must directly be written to a column.

3. A single unit’s operational parameters, as provided in the tables, must be specified by the user. Your model must then simulate how that unit will operate under the water profile of Data 1. This means your model must calculate the air-cooling rate at each time step of the unit. All the rules of Model 3 apply. Each time step’s air-cooling rates must directly be written to a column.

You will write up a concise, but sufficient report that contains all your model formulations, both optimisation and simulation models. Explain what each constraint means, or each line of your simulation logic. This is to demonstrate that you understand what you have formulated and coded.

Your report must, furthermore, contain result discussions on all your models.

To be handed in:

You will have to submit your Excel file. Allow for instructions in your file on how external users must operate the model(s).

Your report must also be submitted. Your report must contain a declaration that the work provided is your own.

General

This is not a GA assignment and no GA is tested in this module. Therefore, there will not be a second submission opportunity and the mark received is final. The mark received will be a combination of mathematical modelling, programming and report writing. This mark will then contribute 50% towards your final module mark.

Submission November 1st at 08h00 (8 AM). A submission window will be allowed for those who struggle to submit. This window will close on November 1st at 23h00 (11 PM).

25% will be deducted for late submissions within the first 24 hours, 50% within the following 24 hours and no submission will be marked thereafter. No exceptions will be made, and it is your responsibility to ensure the work is uploaded before the deadline(s).

This is an individual assignment and no “teamwork” is allowed. If significant similarities between work exists, it will be forwarded to the NWU’s disciplinary office for a formal investigation.

A marking rubric will not be provided beforehand and only after the assignments have been graded.

Background

Underground mining requires air cooling to provide a save working environment for the workforce. As a result, central air cooling is typically available. However, due to the expansion of mining tunnels, over time, the outskirts of the mine are typically above safe working temperatures.

Additional or larger air chiller plants can be installed, but at great costs. As a result, the mining industry requires other means of providing cooled air in the warmer parts where the central cooling does not have the required effect. If cooled air cannot be provided, production time (and revenue) will be lost.

One solution is mobile air-cooling units. These units comprise refrigeration cycle, exactly as a typical home fridge. To cool the surrounding air, cooling water is required for these units. The cooling water is used to transport the heat extracted from the air away from the immediate environment. If the heat cannot be extracted from the environment, then no air cooling is possible.

Water flows are, however, not constant. This can result in sub-optimal decision making when choosing which air-cooling units to invest in.

What further complicates the decision is that these units do not always operate at the same efficiency, to be discussed in the next section.

Thermodynamic working principles

Before any modelling can commence, a basic working principle of these cooling systems must be grasped, which is the coefficient of performance or COP. The COP provides a ration of air cooling in kilo Watts [kW] over electricity costs (compressor work) in kilo Watts [kW] that are required to operate the system. Therefore:

?????????

COP=

???????????????????????

????????? is then the air-cooling rate at any time and ??????????????????????? the electrical input (compressor work) required to deliver the air-cooling.

Air-cooling units under investigation

There are four air cooling units to choose from and the required information for both units are provided in the following tables. The first table provides the minimum and maximum compressor work required by each system, together with the water flow that will result in the compressor work. It can be assumed that there exists a linear correlation between the compressor work [kW] and water flows [kg/s].

Units Minimum compressor work [kW] Water flow required for minimum air-cooling rate [kg/s] Maximum compressor work [kW] Water flow required for maximum air-cooling

rate [kg/s]

Unit I 32 2.0 50 6.0

Unit II 40 2.2 55 7.0

Unit III 26 2.0 64 7.0

Unit IV 33 2.5 69 8.0

The following tables provide the COP intervals in terms of water flow conditions for each unit.

Unit 1: water flows kg[s] 2.0 – 3.0 3.0 – 4.0 4.0 – 6.0

COP 2.5 3.0 4.0

Unit 2: water flows kg[s] 2.2 – 3.5 3.5 – 5.5 5.5 – 7.0

COP 2.5 3.0 4.0

Unit 3: water flows kg[s] 2.0 – 4.0 4.0 – 6.0 6.0 – 7.0

COP 2.3 2.5 3.3

Unit 4: water flows kg[s] 2.5 – 4.0 4.0 – 6.0 6.0 – 8.0

COP 2.4 3.0 3.5

Very important to note. No unit can be operational if water flows are below its minimum level. Take Unit 4; if the water flow rate is 1.95 [kg/s], then the unit will shut down and produce zero cooling for that interval. Further importance, the unit cannot utilise more than its maximum flow rate rating. If 10.0 [kg/s] water is available for Unit 4, it can only draw 69 [kW] electrical power for the compressor and provide 69x3.5 = 241.5 [kW] of air cooling.

What is provided

You will receive an Excel file with two six-hourly available water flow data over a time period. The flows within the six-hour intervals may be assumed to be constant. These profiles are a typical representation of waterflows that can be expected during any time of the year when production occurs at two different locations. The data sets are labelled Data 1 and Data 2.

What is required

You are required to solve for the optimal decision for four different scenarios. You will therefore have four optimisation models; all four must be kept separate. You will also simulate three different scenarios with three different simulation models. The four optimisation models are as follows:

1. Model I: you must solve for the single best (optimal) unit to be used under water profile Data 1. The objective must be to maximise accumulated air cooling over time (equivalent to the highest average). Times when sufficient water is available the unit may provide air cooling and for the times when sufficient water is not available, the unit must be switched off and provide zero air cooling. Each time step’s ai- cooling rates must directly be written to an Excel column. 2. Model II: you must solve for the single best (optimal) unit to be used under water profile Data 1. The objective must be to maximise accumulated air cooling over time (equivalent to the highest average). Times when sufficient water is available the unit may provide air cooling and for the times when sufficient water is not available, the unit must be switched off and provide zero air cooling. When the unit is switched off, it must remain switched off for at least three time-steps (i.e. 18 hours: time period unit switches off and at least the two following time periods.). Each time step’s air-cooling rates must directly be written to an Excel column.

3. Model III: you must solve for the single best (optimal) unit to be used under water profile Data 1. The objective must be to maximise accumulated air cooling over time (equivalent to the highest average). Times when sufficient water is available the unit may provide air cooling and for the times when sufficient water is not available, the unit must be switched off and provide zero air cooling. When the unit is switched off, it must remain switched off for at least three time-steps (i.e. 18 hours: time period unit switches off and at least the two following time periods.). Furthermore, the unit may only be restarted if the water flow from the last time period before it is restarted, is above the minimum operational value. This is to ensure that a unit is not restarted directly after a time period where there was not enough water flow to potentially have supported air cooling. Each time step’s air-cooling rates must directly be written to an Excel column.

4. Model IV: you must solve for the two best (optimal) units to be used in combination under water profile Data 2. The objective must be to maximise accumulated air cooling between the two units over time (equivalent to the highest average). The water flow rate must be split between the two units and your model must compute how this is done for each time-step. Times when sufficient water is available the unit may provide air cooling and for the times when sufficient water is not available, the unit must be switched off and provide zero air cooling. When the unit is switched off, it must remain switched off for at least three timesteps (i.e. 18 hours: time period unit switches off and at least the two following time periods.). *This is then an expansion of your Model II. Each time step’s air-cooling rates must directly be written to an Excel column (two different columns).

All optimisation modelling must be done in Excel’s SolverStudio GMPL language. You were introduced to SolverStudio during your third year and should have sufficient background knowledge of the software. If not, it is up to yourself to acquire the necessary skills. There will not be a lecture on how to use SolverStudio. Furthermore, each optimisation model must be run at a different Excel tab.

For the simulation part, you must use Excel, either logic programmed within Cells or Visual Basic. Three separate simulation models are required.

1. A single unit’s operational parameters, as provided in the tables, must be specified by the user. Your model must then simulate how that unit will operate under the water profile of Data 1. This means your model must calculate the air-cooling rate at each time step of the unit. All the rules of Model 1 apply, i.e. the unit may provide air cooling when sufficient water is available and zero otherwise. Your simulation model must then solve for maximum average air cooling over time. Each time step’s air-cooling rates must directly be written to a column.

2. A single unit’s operational parameters, as provided in the tables, must be specified by the user. Your model must then simulate how that unit will operate under the water profile of Data 1. This means your model must calculate the air-cooling rate at each time step of the unit. All the rules of Model 2 apply. Each time step’s air-cooling rates must directly be written to a column.

3. A single unit’s operational parameters, as provided in the tables, must be specified by the user. Your model must then simulate how that unit will operate under the water profile of Data 1. This means your model must calculate the air-cooling rate at each time step of the unit. All the rules of Model 3 apply. Each time step’s air-cooling rates must directly be written to a column.

You will write up a concise, but sufficient report that contains all your model formulations, both optimisation and simulation models. Explain what each constraint means, or each line of your simulation logic. This is to demonstrate that you understand what you have formulated and coded.

Your report must, furthermore, contain result discussions on all your models.

To be handed in:

You will have to submit your Excel file. Allow for instructions in your file on how external users must operate the model(s).

Your report must also be submitted. Your report must contain a declaration that the work provided is your own.

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