Problem $3$ Statement:

A small commercial building has a heating load of $\backslash (250,000\backslash$mathrm$\{$Btu$\}/\backslash$mathrm$\{$hr$\}\backslash )$ sensible and $\backslash (30,000\backslash$mathrm$\{$Btu$\}/\backslash$mathrm$\{$hr$\}\backslash )$ latent. The building is divided into three zones, each with different room conditions, occupancy levels, and outdoor air requirements. The heating load varies throughout the day and year, and the building uses a mix of radiant and forced$-$air heating systems. Additionally, energy efficiency measures are to be considered. Assuming a $\backslash (45^\{\backslash$circ$\}\backslash$mathrm$\{$F$\}\backslash )$ temperature rise for the heating unit, and that the total heating load is distributed among the zones as follows: Zone $1$ with $100,000$ Btu$/$hr$,$ Zone $2$ with $\backslash (100,000\backslash$mathrm$\{$Btu$\}/\backslash$mathrm$\{$hr$\}\backslash ),$ and Zone $3$ with $\backslash (50,000\backslash$mathrm$\{$Btu$\}/\backslash$mathrm$\{$hr$\}\backslash ).$ Assume further that the heating load values for different times of the day and seasons are estimated based on typical usage patterns and occupancy levels.

Additionally, the building has the following constraints and considerations:

$-$ Energy Source: The building uses a combination of natural gas and electric heating, with a maximum allowable natural gas consumption of $\backslash (60\backslash \%\backslash )$ of the total heating load.

$-$ Heat Recovery: The building is equipped with a heat recovery system that can recover $\backslash (15\backslash \%\backslash )$ of the total heating load.

$-$ Zoning Control: Each zone must be independently controllable to maintain specified conditions.

$-$ Energy Efficiency: Improved insulation reduces the heating load by $10\backslash \%,$ and high$-$efficiency heating systems reduce the heating load by $\backslash (15\backslash \%\backslash ).$

$-$ Environmental Impact: The heating system must comply with local environmental regulations, which limit $\backslash (\backslash$mathrm$\{$CO$\}\_\{2\}\backslash )$ emissions to a maximum of $20$ lbs per $1,000$ Btu of heating load.

$-$ System Limitations: The forced$-$air system cannot exceed a maximum air supply rate of $6,000$ cfm$,$ and the radiant system cannot exceed $\backslash (60\backslash \%\backslash )$ of the total heating load.

$-$ Optimization Criteria: Minimize energy costs while meeting all constraints and heating requirements. The cost of natural gas is $\backslash (\backslash$$ $0\mathrm{.}50\backslash )$ per therm $\backslash ((100,000\backslash$mathrm$\{$Btu$\})\backslash )$ and the cost of electricity is $\backslash (\backslash$$ $0\mathrm{.}10\backslash )$ per kWh$.$

$-\backslash (\backslash$quad $\backslash )$ Energy Storage: Incorporate an energy storage system capable of storing up to $10,000$ Btu$/$hr for balancing peak loads and optimizing energy use.

$-$ Demand Response: Implement a demand response strategy to reduce energy consumption during peak hours by $\backslash (5\backslash \%\backslash )$ without compromising comfort.

Determine the air quantity to be supplied by the unit using the following methods:

Method $($a$)$: Use a psychrometric chart with the following conditions:

$-$ Zone $1$: $\backslash (70^\{\backslash$circ$\}\backslash$mathrm$\{$F$\}\backslash )$ and $\backslash (30\backslash \%\backslash )$ relative humidity, $10$ occupants

$-$ Zone $2$: $\backslash (75^\{\backslash$circ$\}\backslash$mathrm$\{$F$\}\backslash )$ and $\backslash (40\backslash \%\backslash )$ relative humidity, $20$ occupants

$-$ Zone $3$: $\backslash (68^\{\backslash$circ$\}\backslash$mathrm$\{$F$\}\backslash )$ and $\backslash (35\backslash \%\backslash )$ relative humidity, $15$ occupants

$-\backslash (\backslash$quad $\backslash )$ Outdoor air: $\backslash (35^\{\backslash$circ$\}\backslash$mathrm$\{$F$\}\backslash )$ and $\backslash (50\backslash \%\backslash )$ relative humidity

Method $($b$)$: Calculate the air quantity based on the sensible heat transfer for each zone and the outdoor air $(2$ Points$).5$

Method $($c$)$: Adjust calculations for varying heating loads throughout the day and year, assuming the following: $(2$ Points$).$

$-$ Morning $(6$ AM $-12$ PM$)$: $200,000$ Btu$/$hr $($sensible$),20,000$ Btu$/$hr $($latent$)$

$-\backslash (\backslash$quad $\backslash )$ Afternoon $(12$ PM $-6$ PM$)$: $\backslash (250,000\backslash$mathrm$\{$Btu$\}/\backslash$mathrm$\{$hr$\}\backslash )($sensible$),30,000$ Btu$/$hr $($latent$)$

$-$ Evening $(6$ PM $-12$ AM $)$: $150,000$ Btu$/$hr $($sensible$),\backslash (15,000\backslash$mathrm$\{$Btu$\}/\backslash$mathrm$\{$hr$\}\backslash )($latent$)$

$-$ Winter: $\backslash (300,000\backslash$mathrm$\{$Btu$\}/\backslash$mathrm$\{$hr$\}\backslash )($sensible$),\backslash (40,000\backslash$mathrm$\{$Btu$\}/\backslash$mathrm$\{$hr$\}\backslash )($latent$)$

$-$ Summer: $\backslash (200,000\backslash$mathrm$\{$Btu$\}/\backslash$mathrm$\{$hr$\}\backslash )($sensible$),\backslash (20,000\backslash$mathrm$\{$Btu$\}/\backslash$mathrm$\{$hr$\}\backslash )($latent$)$

Method $($d$)$: Incorporate energy efficiency measures, different types of heating systems, comply with local environmental regulations, and optimize for minimum energy costs, while including energy storage and demand response strategies. $(2$ Points$).$