# Appendix III – Case Study Calculations #

The following lists the assumptions built into the Case Study modeled in an Excel spreadsheet (not included here), as well as the logic used in developing the model calculations.

Altitude – Sea Level

Nominal Height of Structure – 1000m

Height of HXT1 – 1000m (placed externally on structure)

Height of remaining HXT’s – 750m (placed internally in CUTs)

Hexagonal CUTs, Distance across Flats – 31.4m

Surface Air Condition – 25°C & 55% Relative Humidity & 101.325 kPa

## CALCULATION 1 – Air Temperatures and moisture content at top and bottom heat exchangers #

- Use dry lapse rate formula from Appendix II to calculate HXT1 temperature at 1000m
- Transfer this to HXB1 air temperature (assume 100% heat exchange efficiency)
- Use this air temperature to determine moisture content of air at bottom of CUT 1 from ASHRAE Chart #1
- Subtract this moisture content from ambient air moisture content to determine condensate production rate at HXB1
- Calculate air temperature just below HXT2 using formula for fully saturated lapse rate from Appendix II
- Use this air temperature to determine moisture content of air at HXT2 of Stage 1from ASHRAE Chart #5
- Use this moisture content to subtract from moisture content at bottom of Stage 1to determine condensate production from mist/cloud formation in Stage 1.
- Repeat these steps for Stage 2-5
- Totalize the moisture production rate per kg of air

## CALCULATION 2 & 3 – Matching heat absorption/rejection in adjacent CUT to balance air flow rate #

- Nominate 5 m/s air flow rate in Stage 5 and calculate heat absorption rate by HXB5 from ambient air
- Heat rejection from HXT5 must match that in 1 above. Calculate associated air flow rate using temperatures established in CALCULATION 1.
- Use air flow rate from 2 to calculate heat rejection from HXB4 then repeat steps 1 & 2 above for remaining Stages.
- Calculate total surface heat rejected by CHE by addition of individual stages – 28.9 GW.

## CALCULATION 4 – Determining height of neutral buoyancy stages. #

- Using temperatures from CALCULATION 1, determine volume and thus negative buoyancy of the space between HXB and HXT. This is the load which has to be lifted by the space above HXT.
- Using the load from 1, and the temperatures from CALCULATION 1 OR any geo-thermal temperature assumed to be available. The Case Study assumed geo-thermal temperature of 25K above surface ambient air temp. This serves to limit the total height of the structure required. Geo-thermal heat is deemed to be advantageous for this as well as for above freezing defrost heat in winter to de-ice HX’s as necessary.
- Repeat these steps for all stages.

## CALCULATION 5 – Determining height of stages to deliver air speeds identified in CALCULATION 2&3. #

- Use the Moody Chart and Darcy-Weisbach equation to determine the pressure drop through the height of the system due to friction at the specified stage air speed.
- Determine the additional height required to deliver the extra positive buoyancy to overcome this frictional pressure drop.
- An iterative process was used, since step 2 added to total height and thus pressure drop.
- Note that pressure drop across the HXs was ignored since the authors had no data for same. A proper study would be required to add additional height to overcome actual pressure drop across the HXs.

## CALCULATION 6 – Determining the Convection Rate in Stage 1 hydraulic circuit with various pipe sizes #

- Determine the density difference in the warm and cold “legs” of the hydraulic circuit. Determine the resulting pressure differential which acts to drive circulation.
- Use the Moody Chart and Darcy-Weisbach equation to determine the equilibrium flow velocity associated with the pressure drop in 1.
- Calculate the rate of Heat Convection from 2, knowing temperature differential.
- Note that pressure drop across the HXs was ignored since the authors had no data for same. A proper study would be required to add additional height to overcome actual pressure drop across the HXs.